Ganitasara Sangraha

Author(s): Mahaviracharya, M Rangacharya
Publisher: Government of Madras
Catalog link: https://jainqq.org/explore/020800/1

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THE GANITA-SARA-SANGRAHA OF MAHAVIRACARYA WITH [PHICE, 2 rupees & annas.] 10 733 ENGLISH TRANSLATION AND NOTES BY Acharya Shri Kailassagarsuri Gyanmandir M. RANGACARYA, M.A., Rao Bahadur, PROFESSOR OF SANSKRIT AND COMPARATIVE PHILOLOGY, PRESIDENCY COLLEGE, AND CURATOR, GOVERNMENT ORIENTAL MANUSCRIPTS LIBRARY, MADRAS. Published under the Orders of the Government of Madras. MADRAS: PRINTED BY THE SUPERINTENDENT, GOVERNMENT PRESS, 1912. For Private and Personal Use Only [3 shillings 8 me. I

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namra sUcana isa grantha ke abhyAsa kA kArya pUrNa hote hI niyata samayAvadhi meM zIghra vApasa karane kI kRpA kareM.. jisase anya vAcakagaNa isakA upayoga kara sakeM. For Private and Personal Use Only

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THE GANITA-SARA-SANGRAHA OF MAHAVIRACARYA WITH ENGLISH TRANSLATION AND NOTES BY M. RANGACARYA, M.A., Rao Bahadur, PROFESSOR OF SANSKRIT AND COMPARATIVE PHILOLOGY, PRESIDENCY COLLEGE, AND CURATOR, GOVERNMENT ORIENTAL MANUSCRIPTS LIBRARY, MADRAS. Published under the Orders of the Government of Madras. MADRAS: PRINTED BY THE SUPARINTENDENT, GOVERNMENT PRESS, 1912, For Private and Personal Use Only

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mahAvIrAcAryapraNItaH gaNita sArasa Gya ha ma. raGgAcAryeNa parizodhitaH AGgalabhASAnuvAdaTIkAbhyAM saha rAjakIyAjJAnusAreNa prakAzitazca / Acharya Shri Kailassagarsuri Gyanmandir cenna puSa rAjakIya mudrAkSarazAlAyAM (suparinTeNDeNTAkhyaina) tanirvAhakeNa mudritaH / 1912. For Private and Personal Use Only

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Shri Mahavir Jain Aradhana Kendra Gutturals TABLE OF TRANSLITERATION. Palatals Linguals Dentals Labials www.kobatirth.org .. Consonants k, kh, g. gb, n, h, b. ka, sva, ga, gha, Ga, cha, : c, ch, j, jb, n, y, a ca, cha, ja, jha, Ja, ya, za t, th, d, dh, p, r, s Ta, Tha, Da, Dha, Na, 2, - t. th, d, dh, n, 1, 8 ta, tha, da, dha, na, la, sa p, ph, b, bh, m, v pa, pha, ba, bha, ma, va For Private and Personal Use Only a 37 371 P Vowels. i, i i, I F, F *,* 1 la Acharya Shri Kailassagarsuri Gyanmandir u, u u, U Diphthongs. e (e) ai. e o (0) au. art 3ft.

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GENERAL CONTENTS. Page Preface ... .. *** Introduction by Dr. David Eugene Smith ... ... ... ... ... vii-xyii ... xix-xxiv Contents of the text in Sanskrit Text: ... ... ... ... ... ... ... ... ... ... 19-v 1-158 Contents of the translation in English Translation .. . .. . ... . ... ... .. 1-286 APPENDIX 1.- Sapskrit words denoting numbers with their ordinary and numerical signification ... ... 287-245 APPENDIX II.-Sanskrit words used in the translation and their explanation ... ... ... ... ... ... ... ... ... 296-304 APPENDIX III.--- Answers to problems ... 305-322 APPENDIX IV.--Tables of measures ... ... ... 323-325 For Private and Personal Use Only

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Shri Mahavir Jain Aradhana Kendra Page Line X (Preface) 28 XIV 3 XX 8 XXIII iv 7 12244 73 25 footnote 27 29 53 62 65 66 73 74 75 76 ADDENDA ET CORRIGENDA TO THE GANITASARASANGRAHA. do. 12 13 31 20 29 1 5 8 14 18 11 3 3 CHEROICON 6 11 2 25 9 1 26 27 27, 28 8,9 2 3 moksa omit www.kobatirth.org vAlakA saGalita 5: Sakalya-samihita Mahaviracary a Do. 'much of before Rastrakuta. Aryabhata rAzi tantu mAtRkam durga gaccha. paca For 1 141 Text. Translation. of associated with in of what remained of what remained thereafter. Su S add "namber of" after "any" (n1)2-n-2 pn do. of that whole collection of bees. from in order to in the end. 25 moksa. Sakalya-sa hita. Mahaviracarya. do. Aryabhata. trairAzi vallikA. saGkalita. E: kantu. mAtRkA. durgA. gaccha.. SaT. Acharya Shri Kailassagarsuri Gyanmandir Read 1 41 3 of 3 (n)-n1-2 pn. For Private and Personal Use Only associated with in of the last (fraction). of this last fraction. of this last. (of that whole collection of bees). from in order to in the end. 5

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Shri Mahavir Jain Aradhana Kendra Page 78 79 82 + 87 265 287 289 292 293 footnote:.. do. 90 105 108 footnote 111 do. 113 do. 294 *** 138 217 footnote 223 do. 224 do. 301 303 314 323 do. do. do. Line 5 4 5 4 13 16 24 2 WTUNNI 11 29 8 34 6 14 19 Suvarna 5 (c-d2) after a2-b2 7 add before 2 } 3 9 30 01040 00 00 15 5 www.kobatirth.org 18 23 For SS samula substituted for e and b S.samula after known quantity' (ii) 324 palas 10,000 kargus 5 puranus 2 angulas 35 20 881 example in stauza 101 (Na+62) 2 20 9 ambIdha wordly mani nali Paal cubic kSAyakasamya, ktva vyAma * stambrema choosen Sotkas For Private and Personal Use Only Read Se amula. substituted for cand a. Sesamula. Acharya Shri Kailassagarsuri Gyanmandir add which is subtracted from or added to this. specified fractional part of the unknown collective quantity.' 32 palas. 100,000 kar us. 5 puranas. 226 angulas. 36 GO $81 examples in stanzas 100} and 101. Suvarna (c2-d"). ( a 2 + b 2 ) 2 20 ambudhi worldly. muni. nIla. kSAyakasamyaktva. vyoma. stamberama. Pula. cube. chosen. Srokas.

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tlve E F A C E. Soon after I was appointed Professor of Sanskrit and Comparative Philology in the Presidency College at Madras, and in that capacity took charge of the office of the Curator of the Government Oriental Manuscripts Library, the late Mr. G. H. Stuart, who was then the Director of Public Instruction, asked me to find out if in the Manuscripts Library in my charge there was any work of value capable of throwing new light on the history of Hindu mathematics, and to publish it, if found, with an English translation and with such notes as were necessary for the elucidation of its contents. Accordingly the mathematical manuscripts in the Library were examined with this object in view; and the examination revealed the existence of three incomplete manuscripts of Mahaviracarya's Ganita-sara-sangraha. A cursory perusal of these manuscripts made the value of this work evident in relation to the history of Hindu Mathematics. The late Mr. G. H. Stuart's interest in working out this history was so great that, when the existence of the manuscripts and the historical value of the work were brought to his notice, he at once urged me to try to procure other manuscripts and to do all else that was necessary for its proper publication. He gave me much advice and encouragement in the early stages of my endeavour to publish it; and I can well guess how it would bave gladdened his heart to see the work published in the form he desired. It has been to me a source of For Private and Personal Use Only

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vu GANITASARASANGRAHA. very keen regret that it did not please Providence to allow him to live long enough to enable me to enhance the value of the publication by means of his continued guidance and advice; and my consolation now is that it is something to have been able to carry out what he with scholarly delight imposed upon me as a duty. Of the three manuscripts found in the library one is written on paper in Grantha characters, and contains the first five chapters of the work with a running commentary in Sanskrit; it has been denoted here by the letter P. The remaining two are palm-leaf manuscripts in Kanarese characters, one of them containing, like P, the first five chapters, and the other the seventh chapter dealing with the geometrical measurement of areas. In both these manuscripts there is to be found, in addition to the Sanskrit text of the original work, a brief statement in the Kanarese language of the figures relating to the various illustrative problems as also of the answers to those same problems. Owing to the common characteristics of those manuscripts and also owing to their not overlapping one another in respect of their contents, it has been thought advisable to look upon them as one manuscript and denote them by K. Another manuscript, denoted by M, belongs to the Government Oriental Library at Mysore, and was received on loan from Mr. A. Mahadeva Sastri, B.A., the Carator of that institution. This manuscript is a transcription on paper in Kanarese characters of an original palm-leaf manuscript belonging to a Jaina Pandit, and contains the whole of the work with a short commentary in the Kanarese language by one Vallabha, who claims to be the author of also a Telugu commentary on the same For Private and Personal Use Only

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PREFACE. work. Although incorrect in many places, it proved to be of great value on account of its being complete and containing the Kanarese commentary; and my thanks are specially due to Mr. A. Mahadeva Sastri for his leaving it sufficiently long at my disposal. A fifth manuscript, denoted by B, is a transcription on paper in Kanarese characters of a palm-leaf manuscript found in a Jaina monastery at Mudbidri in South Canara, and was obtained through the kind effort of Mr. R. Krishnamacharyar, M.A., the Sub-assistant Inspector of Sanskrit Schools in Madras, and Mr. U. B. Venkataramanaiya of Mudbidri. This manuscript also contains the whole work, and gives, like K, in Kanarese a brief statement of the problems and their answers. The endeavour to secure more manuscripts having proved fruitless, the work has had to be brought out with the aid of these five manuscripts ; and owing to the technical character of the work and its elliptical and often riddle-like language and the inaccuracy of the mauuscripts, the labour involved in bringing it out with the translation and the requisite notes has been heavy and trying. There is, however, the satisfaction that all this labour has been bestowed on a worthy work of considerable historical value. It is a fortunate circumstance about the Ganita-sarasangraha that the time when its author Mahaviracarya lived may be made out with fair accuracy. In the very first chapter of the work, we have, immediately after the two introductory stanzas of salutation to Jina Mahavira, six stanzas describing the greatness of a king, whose name is said to have been Cakrika-bhanjana, and who appears to have been commonly known by the title of Amoghavarsa Nrpatanga; and in the last of these For Private and Personal Use Only

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GANITASARASANGRAHA. six stanzas there is a benediction wishing progressive prosperity to the rule of this king. The results of modern Indian epigraphical research show that this king Amoghavarsa Nrpatunga reigned from A.D. 814 or 815 to A.D. $77 or 878.* Since it appears probable that the author of the Ganita-sara-sangraha was in some way attached to the court of this Rastrakuta king Amoghavarsa Nrpatunga, we inay consider the work to belong to the middle of the ninth century of the Christian era. It is now generally accepted that, among well-known early Indian mathematicians Aryabhata lived in the fifth, Varahamihira in the sixth, Brahmagupta in the seventh and Bhaskaracarya in the twelfth century of the Christian era; and chronologically, therefore, Mahaviracarya comes between Brahmagupta and Bhaskaracarya. This in itself is a point of historical noteworthiness; and the further fact that the author of the Ganita-sara-sangraha belonged to the Kanarese speaking portion of South India in his days and was a Jaina in religion is calculated to give an additional importance to the historical value of his work. Like the other mathematicians mentioned above, Maha. viracarya was not primarily an astronomer, although he knew well and has himself remarked about the usefulness of mathematics for the study of astronomy. The study of mathematics seems to have been popular among Jaina scholars; it forms, in fact, one of their four anuyogas or auxiliary sciences indirectly serviceable for the attainment of the salvation of soul-liberation known as moksa. * A comparison of the Ganita-sara-sangraha with the corresponding portions in the Brahmasphuta-siddhanta of * Vide Nilgund Inscription of the time of Amoghavarsa I, A.D. 866 ; edited by J. F. Fleet, Ph.D., 0.1.E., in Epigraphia Indica, vol. VI, pp. 98-108, For Private and Personal Use Only

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PREFACE. Brahmagupta is calculated to lead to the conclusion that, in all probability, Mahaviracarya was familiar with the work of Brahmagupta and endeavoured to improve upon it to the extent to which the scope of his Ganita-sarasangraha permitted such improvement. Mahaviracarya's classification of arithmetical operations is simpler, his rules are fuller and he gives a large number of examples for illustration and exercise. Prthudakasvamin, the wellknown commentator on the Brahmasphu ta-siddhanta, could not have been chronologically far removed from Mahaviracarya, and the similarity of some of the examples given by the former with some of those of the latter naturally arrests attention. In any case it cannot be wrong to believe, that, at the time, when Mahaviracarya wrote his Ganita-sara-sangraha, Brahmagupta must have been widely recognized as a writer of authority in the field of Hindu astronomy and mathematice. Whether Bhaskaracarya was at all acquainted with the Ganitasara-sangraha of Mahaviracarya, it is not quite easy to say. Since neither Bhaskaracarya nor any of his known commentators seem to quote from him or mention him by name, the natural conclusion appears to be that Bhaskaracarya's Siddhanta-siromani, including his Lilavati and Bijagaaita, was intended to be an improvement in the main upon the Brahmasphuta-siddhanta of Brahmagupta. The fact that Mahaviracarya was a Jaina might have prevented Bhaskaracarya from taking note of him; or it may be that the Jaina mathematician's fame had not spread far to the north in the twelfth century of the Christian era. His work, however, seems to have been widely known and appreciated in Southern India. So early as in the course of the eleventh century and perhaps For Private and Personal Use Only

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xii GANITASXBASANGRAHA. under the stimulating influence of the enlightened rule of Rajarajanarendra of Rajahmundry, it was translated into Telugu in verse by Pavulari Mallana ; and some manuscripts of this Telugu translation are now to be found in the Government Oriental Manuscripts Library here at Madras. It appeared to me that to draw suitable attention to the historical value of Mahaviracarya's Gamita-sara-sangraha, I could not do better than soek the help of Dr. David Eugene Smith of the Columbia University of New York, whose knowledge of the history of mathematics in the West and in the East is known to be wide and comprehensive, and who on the occasion when he met me in person at Madras showed great interest in the contemplated publication of the Ganila-sarasangraha and thereafter read a paper on that work at the Fourth International Congress of Mathematicians held at Rome in April 1908. Accordingly I requested him to write an introduction to this edition of the Ganita-sarasangraha, giving in brief outline what he considers to be its value in building up the history of Hindu mathematics. My thanks as well as the thanks of all those who may as scholars become interested in this publication are there. fore due to him for his kindness in having readily complied with my request; and I feel no doubt that his introduction will be read with great appreciation. Since the origin of the decimal system of notation and of the conception and symbolic representation of zero are considered to be important questions connected with the history of Hindu mathematics, it is well to point out here that in the Ganita-sara-sangraha twenty-four notational places are mentioned, commencing with the units place and ending with the place called mahaksobha, For Private and Personal Use Only

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PREFACE. and that the value of each succeeding place is taken to be ten times the value of the immediately preceding place. Although certain words forming the names of certain things are utilized in this work to represent various numerical figures, still in the numeration of numbers with the aid of such words the decimal system of notation is almost invariably followed. If we took the words moon, eye, fire, and sky to represent respectively 1, 2, 3 and 0, as their Sanskrit equivalants are understood in this work, then, for instance, fire-sky-m:sun-eye would denote the vumber 2103, and moon.eye-sky-fire would denote 3021, since these nominal numerals denoting numbers are generally repeateil in order from the units place upwards. This combination of nominal numerals and the decimal system of notation has been adopted obviously for the sake of securing metrical convenience and avoiding at the same time cumbrous ways of mentioning numerical expressions; and it may well be taken for granted that for the ase of such nominal numerals as well as the decimal system of notation Mahaviracarya was indebted to his predecessors. The decimal system of notation is distinctly described by Aryabhata, and there is evidence in his writings to show that he was familiar with nominal numerals, Even in his brief mnemonic method of reperesenting numbers by certain combinations of the consonants and vowels found in the Sanskrit language, the decimal system of notation is taken for granted; and ordinarily 19 notational places are provided for therein. Similarly in Brahmagupta's writings also there is evidence to show that he was acquainted with the use of nominal numerals and the decimal system of notation. Both Aryabhata and Brahmiagupta claim that their astronomical works For Private and Personal Use Only

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xiv GANITASARASANGRAHA. are related to the Brahma-siddhanta; and in a work of this name, which is said to form a part of what is called Sakalya-samihita and of which a manuscript copy is to be found in the Governmert Oriental Manuscripts Library here, numbers are expressed mainly by nominal numerals used in accordance with the decimal system of notation. It is not of course meant to convey that this work is necessarily the same as what was known to Arayabhata and Brahmagupta; and the fact of its using nominal numerals and the decimal system of notation is mentioned here for nothing more than what it may be worth. It is generally recognized that the origin of the conception of zero is primarily due to the invention and practical utilization of a system of notation wherein the several numerical figures used have place-values apart from what is called their intrinsic value. In writing out a number according to such a system of notation, any notational place may be left empty when no figure with an intrinsic value is wanted there. It is probable that owing to this very reason the Sanskrit word sunya, meaning.empty', came to denote the zero; and when it is borne in mind that the English word 'cipher' is derived from an Arabic word having the same meaning as the Sanskrit Sunya, we may safely arrive at the conclusion that in this country the conception of the zero came naturally in the wake of the decimal system of notation : and so early as in the fifth century of the Christian era, Aryabhata is known to have been fully aware of this valuable mathematical conception. And in regard to the question of a symbol to represent this conception, it is well worth bearing in mind that opera For Private and Personal Use Only

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PREFACE. tions with the zero cannot be carried on-not to say cannot be even thought of easily-without a symbol of some sort to represent it. Mahaviracarya gives, in the very first chapter of his Ganita-sara-sangraha, the results of the operations of addition, subtraction, multiplication and division carried on in relation to the zero quantity; and although he is wrong in saying that a quantity, when divided by zero, remains unaltered, and should have said, like Bhaskaracarya, that the quotient in such a case is infinity, still the very mention of operations in relation to zero is enough to show that Mahaviracarya must have been aware of some symbolic representation of the zero quantity. Since Brahmagupta, who must have lived at least 150 years before Mahaviracarya, mentions in his work the results of operations in relation to the zero quantity, it is not unreasonable to suppose that before his time the zero must have had a symbol to represent it in written calculations. That even Aryabhata knew such a symbol is not at all improbable. It is worthy of note in this connection that in enumerating the nominal numerals in the first chapter of his work, Mahaviracarya mentions the names denoting the nine figures from I to 9, and then gives in the end the names denoting zero, calling all the ten by the name of sankhya: and from this fact also, the inference may well be drawn that the zero had a symbol, and that it was well known that with the aid of the ten digits and the decimal system of notation numerical quantities of all values may be definitely and accurately expressed. What this known zero-symbol was, is, however, a different question. The labour and attention bestowed upon the study and translation and annotation of the Ganita-sara-sangraha For Private and Personal Use Only

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xvi GANITASARASANGRAHA. have made it clear to me that I was justified in think. ing that its publication might prove useful in elucidating the condition of mathematical studies as they Hourished in South India annong the Jainas in the ninth century of the Christian era, and it has been to me a source of no small satisfaction to feel that in bringing out this work in this form, I have not wasted my time and thought on an unprofitable undertaking. The value of the work is undoubtedly more historical than mathematical. But it cannot be denied that the step by step construction of the history of Hindu culture is a worthy endeavour, and that even the most insignificant labourer in the field of such an endeavour deserves to be looked upon as a useful worker. Although the editing of the Ganitra-sara-sangraha has been to me a labour of love and duty, it has often been felt to be heavy and taxing; and I, therefore, consider that I am specially bound to acknowledge with gratitude the help which I have received in relation to it. In the early stage, when conning and collating and interpreting the manuscripts was the chief work to be done, Mr. M. B. Varadaraja Aiyangar, B.A., B.L., who is an Advocate of the Chief Court at Bangalore, co-operated with me and gave me an amount of aid for which I now offer him my thanks. Mr. K. Krishnaswami Aiyangar, B.A., of the Madras Christian College, has also rendered considerable assistance in this manner; and to him also I offer my thanks. Latterly I have had to consult on a few occasions Mr. P. V. Seshu Aiyar, B.A., L.T., Professor of Mathematical Physics in the Presidency College here, in trying to explain the rationale of some of the rules given in the work ; and I am much obliged to him for his ready For Private and Personal Use Only

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Shri Mahavir Jain Aradhana Kendra February 1912, Madras. www.kobatirth.org xvii willingness in allowing me thus to take advantage of his expert knowledge of mathematics. My thanks are. I have to say in conclusion, very particularly due to Mr. P. Varadacharyar, B.A., Librarian of the Government Oriental Manuscripts Library at Madras, but for whose zealous and steady co-operation with me throughout and careful and continued attention to details, it would indeed have been much harder for me to bring out this edition of the Ganita-sara-sangraha. } PREFACE. Acharya Shri Kailassagarsuri Gyanmandir M. RANGACHARYA. For Private and Personal Use Only

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( xix ) INTRODUCTION BY DAVID EUGENE SMITH, PROFESSOR OF MATHEMATICS IN TEACHERS' COLLEGE, COLUMBIA UNIVERSITY, NEW YORK. We have so long been accustomed to think of Pataliputra on the Ganges and of Ujjain over towards the western coast of India as the ancient habitats of Hindu mathematics, that we experience a kind of surprise at the idea that other centres equally iniportant existed among the multitude of cities of that great empire. In the same way we have known for a century, chiefly through the labours of such scholars as Colebrooke and Taylor, the works of Aryabhata. Brahmagupta, and Bhaskara, and have come to feel that to these men alone are due the noteworthy contributions to be found in native Hindu mathematics. Of course a little reflection shows this conolusion to be an incorrect one. Other great schools, particularly of astronomy, did exist, and other scholars taught and wrote and added their quota, small or large, to make up the sum total. It has, however, been a little discouraging that native scholars under the English supremacy have done so little to bring to light the ancient mathematical material known to exist and to make it known to the Western world. This neglect has not certainly been owing to the absence of material, for Sanskrit mathematical manuscripts are known, as are also Persian, Arabic, Chinese, and Japanese ; and many of these are well worth translating from the historical standpoint. It has rather been owing to the fact that it is hard to find a man with the requisite scholarship, who can afford to give his time to what is necessarily a labour of love.. For Private and Personal Use Only

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XX GANITASARASANGRAHA. It is a pleasure to know that such a man has at last appeared and that, thanks to his profound scholarship and great perseverance, we are now receiving new light upon the subject of Oriental mathematics, as known in another part of India and at a time about midway between that of Aryabhata and Bhaskara, and two centuries later than Brahmagupta. The learned scholar, Professor M. Rangacarya of Madras, some years ago became interested in the work of Mahaviracarya, and has now completed its translation, thus making the mathematical world his perpetual debtor; and I esteem it a high honour to be requested to write an introduction to so noteworthy a work. Acharya Shri Kailassagarsuri Gyanmandir Mahaviracarya appears to have lived in the court of an old much of Rastrakuta monarch, who ruled probably over much of what is now the kingdom of Mysore and other Kanarese tracts, and whose name is given as Amoghavarsa Nrpatunga. He is known to have ascended the throne in the first half of the ninth century A.D., so that we may roughly fix the date of the treatise in question as about 850. The work itself consists, as will be seen, of nine chapters, like the Bija-ganita of Bhaskara; it has one more chapter than the Kuttaha of Brahma-gupta. There is, however, no significance in this number, for the chapters are not at all parallel, although certain of the topics of Brahmagupta's Ganita and Bhaskara's Lilavati are included in the Ganita-sara-sangraha. In considering the work, the reader naturally repeats to himself the great questions that are so often raised --How much of this Hindu treatment is original? What evidences are there here of Greek influence? What relation was there between the great mathematical centres of India? What is the distinctive feature, if any, of the Hindu algebraic theory? Such questions are not new. Davis and Strachey, Colebrooke and Taylor, all raised similar ones a century ago, and they are by no means satisfactorily answered even yet. Nevertheless, we are making good progress towards their satisfactory solution in the not too distant future. The past century has seen several For Private and Personal Use Only

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INTRODUCTION. xxi Chinese and Japanese mathematical works' made more or less familiar to the West ; and the more important Arab treatises are now quite satisfactorily known. Various editions of Bbaskara have appeared in India, and in general the great treatises of the Orient have begun to be subjected to critical study. It would be strange, therefore, if we were not in a position to weigh up, with more certainty than before, the claims of the Hindu algebra. Certainly the persevering work of Professor Rangacarya has made this more possible than ever before. As to the relation hetween the East and the West, we should now be in a position to say rather definitely that there is no evidence of any considerable influence of Greek algebra upon that of India. The two subjects were radically different. It is true that Diophantus lived about two centuries before the first Aryabhata, that the paths of trade were open from the West to the East, and that the itinerant scholar undoubtedly carried learning from place to place. But the spirit of Diophantus, showing itself in a dawning symbolism and in a peculiar type of eqnation, is not seen at all in the works of the East. None of his problems, not a trace of his symbolism, and not a bit of his phraseology appear in the works of any Indian writer on algebra. On the contrary, the Hindu works have a style and a range of topics peculiarly their own. Their problems lack the cold, clear, geometric precision of the West; they are clothed in that poetic language which distinguishes the East, and they relate to subjects that find no place in the scientific books of the Greeks. With perhaps the single exception of Metrodorus, it is only when we come to the puzzle problems doubtfully attributed to Alcuin that we find anything in the West which resembles, even in a slight degree, the work of Alcuin's Indian contemporary, the author of this treatise. It therefore seems only fair to say that, although some knowledge of the scientific work of any one nation would, even in those remote times, naturally have been carried to other peoples by some wandering savant, we have nothing in the writings of the Hindu algebraists to show any direct influence of the West upon their problems or their theories. For Private and Personal Use Only

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xxii GANITASARASANGRAHA. When we come to the question of the relation between the different sections of the East, however, we meet with more difficulty. What were the relations, for example, between the school of Pataliputra, where Aryabhata wrote, and that of Ujjain, where both Brahmagupta and Bhaskara lived and taught? And what was the relation of each of these to the school down in South India, which produved this notable treatise of Mahaviracarya ? And, a still more interesting question is, what can we say of the influence exerted on China by Hindu scholars, or vice versa? When we find one set of early inscriptions, those at Nana Ghat, using the first three Chinese numerals, and another of about the same period using the later forms of Mesopotamia, we feel that both China and the West may have influenced Hindu science. When, on the other hand, we consider the problems of the great trio of Chinese algebraists of the thirteenth century, Ch'in Chiushaug, Li Yeh, and Chu Shih-chieh, we feel that Hindu algebra must have had no small influence upon the North of Asia, although it must be said that in point of theory the Chinese of that period naturally surpassed the earlier writers of India. The answer to the questions as to the relation between the schools of India cannot yet be easily given. At first it would seem a simple matter to compare the teratises of the three or four great algebraists and to note the similarities and differences. When this is done, however, the result seems to be that the works of Brahmagupta, Mahaviracarya, and Bhaskara may be described as similar in spirit but entirely different in detail. For example, all of these writers treat of the areas of polygons, but Mahaviracarya is the only one to make any point of those that are re-entrant. All of them touch upon the area of a segment of a circle, but all give different rules. The so-called janya operation (page 209) is akin to work found in Brahinagupta, and yet none of the problems is the same. The shadow problems, primitive cases of trigonometry and gnomonius, suggest a similarity among those three great writers, and yet those of Mahaviracarya are much better than the one to be found in either Brahmagupta or Bhaskara, and no questions are duplicated. For Private and Personal Use Only

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INTRODUCTION. xxiii In the way of similarity, both Brahmagupta and Mahaviracarya give the formula for the area of a quadrilateral, N (-a) (8-6) (8-c) (8-d) but neither one observes that it holds only for a cyclic figure. A few problems also show some similarity such as that of the broken tree, the one about the anchorites, and the common one relating to the lotus in the pond, but these prove only that all writers recognized certain stock problems in the East, as we generally do to-day in the West. But as already stated, the similarity is in general that of spirit rather than of detail, and there is no evidence of any close following of one writer by another. When it comes to geometry there is naturally more evidence of Western influence India seems never to have independently developed anything that was specially worthy in this science. Brahmagupta and Mahaviracarya both use the same incorrect rules for the area of a triangle and quadrilateral that is found in the Egyptian treatise of Ahmes. So while they seem to have been influenced by Western learning, this learning as it reached India could have been only the simplest. These rules had long since been shown by Greek scholars to be incorrect, and it seems not unlikely that a primitive geometry of Mesopotamia reached out both to Egypt and to India with the result of perpetuating these errors. It has to be borne in mind, however, that Mahaviracarya gives correct rules also for the area of a triangle as well as of a quadrilateral without indicating that the quadrilateral has to be cyclic. As to the ratio of the circumference to the diameter, both Brahmagupta and Maba viracarya used the old Semitic value 3, both giving also v 10 as a closer approximation, and neither one was aware of the works of Archimedes or of Heron. That Aryabhata gave 3.1416 as the value of this ratio is well known, although it seems doubtful how far be used it himself. On the whole the geometry of India seems rather Babylonian than Greek. This, at any rate, is the inference that one would draw from the works of the writers thus far known. As to the relations between the Indian and the Chinese algebra, it is too early to speak with much certainty. In the matter of For Private and Personal Use Only

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xxiv GANITASARASANGRAHA. problems there is a similarity in spirit, but we have not yet enough translations from the Chinese to trace any close resemblance. In each case the questions proposed are radically different from those found commouly in the West, and we must conclude that the algebraic taste, the purpose, and the method were all distinct in the two great divisions of the world as then known. Rather than assert that the Oriental algebra was influenced by the Occidental, we should say that the reverse was the case. Bagdad, subjected to the influence of both the East and the West, transmitted more to Europe than it did to India. Leonardo Fibonacci, for example, shows much more of the Oriental influence than Bhaskara, who was practically his contemporary, shows of the Occidental. Professor Rangacarya bas, therefore, by his great contribution to the history of mathematics confirmed the view already taking rather concrete form, that India developed an algebra of her own; that this algebra was set forth by several writers all imbued with the same spirit, but all reasonably independent of one another ; that India influenced Europe in the matter of algebra, more than it was influenced in return; that there was no pative geometry really worthy of the name; that trigonometry was practically non-existent save as imported from the Greek astronomers ; and that whatever of geometry was developed came probably from Mesopotamia rather than from Greece. His labours have revealed to the world a writer almost unknown to European scholars, and a work that is in many respects the most scholarly of any to be found in Indian matbemetical literature. They have given us further evidence of the fact that Oriental mathematics lacks the cold logic, the consecutive arrangement, an 1 the abstract character of Greek mathematics, but that it possesses a richness of imagination, an interest in problem-setting, and poetry, all of which are lacking in the treatises of the West, although abounding in the works of China and Japan. If, now, his labours shall lead others to bring to light and set forth more and more of the classics of the East, and in particular those of early and mediaeval China, the world will be to a still larger extent his debtor. For Private and Personal Use Only

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gaNita sA ra sa ya ha :: For Private and Personal Use Only

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CONTENTS. saMjJAdhikAraH maGgalAcaraNam gaNitazAstraprazaMsA saMjJA kSetraparibhASA... kAlaparibhASA dhAnyaparibhASA suvarNaparibhASA rajataparibhASA lohaparibhASA parikarmanAmAni dhanarNazanyaviSayakasAmAnyaniyamAH saGkhyAsaMjJA ... sthAnanAmAni... gaNakaguNanirUpaNam prathamaH parikarmavyavahAraH -- pratyutpannaH bhAgahAra: vargaH vargamUlam ... ghana: ghanamalam saGkAletam ... vyutkalitam ... dvitIyaH kAlAsavarNavyavahAraH bhinnapratyutpannaH bhinnabhAgahAraH :: :: :: :: :: :: :: : :: :: :: :: :: :: : :: :: :: :: ::::::: :: :: For Private and Personal Use Only

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CONTENTS. bhinnavargavargamUlaghanaghanamUlAni bhinnasaGkalitam bhinnavyutkAlatam kalAsavarNaSaDjAti: bhAgajAtiH ... prabhAgabhAgabhAgajAtI bhAgAnubandhajAti: bhAgApavAhajAtiH bhAgamATajAtiH tRtIyaH prakIrNakavyavahAraH-- bhAgajAtizeSajAtI mUlajAti: ... zeSamalajAtiH dviragrazeSamUlajAti: aMzamUlajAtiH bhAgasaMvargajAti: unAdhikAMzavargajAtiH mUlamizrajAti: bhinnadRzyajAtiH caturthaH trairAzikavyavahAraH-- trairAzikaH ... gatinivRttiH paJcasaptanavarAzikAH paJcamaH mizrakavyavahAraH saGkramaNasUtram paJcagazikavidhiH vRddhividhAnam prakSepakuTTIkAra: vallikAkuTIkAra viSamakuhakiAraH For Private and Personal Use Only

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CONTENTS. Page 83 93 102 110 sakalakuTTIkAra suvarNakuTTIkAraH vicitrakuTTIkAraH zreDhIbaddhasaGAlatam SaSTaH kSetragaNitavyavahAraH-- vyAvahArikagANitam sUkSmagaNitam janyavyavahAraH paizAcikavyavahAraH saptamaH khAtavyavahAraH khAtagaNitam citigANatam krakacikAvyavahAraH aSTamaH chAyAvyavahAraH 116 122 126 143 148 160 162 For Private and Personal Use Only

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gaNita sA ra saGgraha : mhaaviiraacaaryprnniitH| sNjnyaadhikaarH| maGgalAcaraNam / malayaM trijagatsAraM yasyAnantacatuSTayam / namastasmai jinendrAya mahAvIrAya tAyine // 1 // saGkhyAjJAnapradIpena jainendreNa mahAtviSA / prakAzitaM jagatsarvaM yena taM praNamAmyaham // 2 // prINitaH prANisa'syogho nirItiniravagrahaH / zrImatAmoghavarSeNa yena kheSTahitaSiNA // 3 // pAparUpAH parA yasya cittavRttihavirbhuji / bhasmasA'dAvamIyuste'vandhyakopo'bhavattataH // 1 // vazIkurvan jagatsarvaM svayaM nAnuvazaH paraiH / nAbhibhUtaH prabhustasmAdapUrvamakaradhvajaH // 5 // yo vikramakramAkrAntacakri'cakatakriyaH / cakrikAbhaJjano nAmnA cakrikAbhaJjano'asA // 1 // yo vidyAnadyadhiSThAno maryAdAvajavadikaH / ranagaoM yathAkhyAtacAritrajaladhirmahAn // 7 // vidhvastaikAntapakSasya syAdvAdanyAyavAdinaH / devasya nRpatuGgasya vardhatAM tasya zAsanama // 8 // - IM and B maha. ' praNIta:. * M Land K sadbhA. 'K, P and B bhavet. * Mand B To. . sau. B yo'yaM. ' vedina :. For Private and Personal Use Only

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gaNitasArasaGgrahaH gnnitshaastrprshNsaa| laukike vaidike vApi tathA sAmAyike'pi yH| vyApArastatra sarvatra saGkhyAnamupayujyate / / 9 // kAmatantre'rthazAstre ca gAndhave nATake'pi vaa| sUpazAstre tathA vaidye vAstuvidyAdivastuSu // 10 // chando'laGkArakAvyeSu tarkavyAkaraNAdiSu / kalAguNeSu sarveSu prastutaM gaNitaM param // 11 // sUryAdigrahacAreSu grahaNe grahasaMyutau / triprazne candravRttau ca sarvatrAGgIkRtaM hi tat // 12 // dvIpasAgarazailAnAM sngkhyaavyaasprikssipH| bhavanavyantarajyotirlokakalpAdhivAsinAma // 13 // nArakANAM ca sarveSAM zreNIbandhendrakotkarAH / prakIrNakapramANAdyA budhyante gaNitena te // 14 // prANinAM tatra saMsthAnamAyuraSTaguNAdayaH / yAtrAdyAssaMhitAdyAzca sarve te gaNitAzrayAH // 15 // bahubhirvipralApaiH kiM trailokaye sacarAcare / yatkiJcidvastu tatsarvaM gaNitena vinA na hi // 16 // tIrthakRdyaH kRtArthebhyaH pUjyebhyo jagadIzvaraiH / teSAM ziSyapraziSyebhyaH prasiddhAdguruparvataH // 17 // jaladheriva ratnAni pASANAdiva kAJcanam / zuktermuktAphalAnIva saGkhyAjJAna mahodadhe : // 18 // 1. syAt ; B cApi. * M and B daNDA . 1K, M and B baddhe. B ca. 5M and B purA. M vasu. K and M mahA. GK and M degkSipA.. . 'K and PTC for $17. For Private and Personal Use Only

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saMjJAdhikAraH kiduhRtya tatsAraM vakSye'haM matizaktitaH / alpaM granthamanalpArthaM gaNitaM sArasaGgraham // 19 // saMjJAmbhobhirathoM pUrNe parikarmoru vedike / kalAsavarNasaMrUDha luThatpAThInasaGkule // 20 // prakIrNakamahAgrAhe trairAzikataraGgiNi / mizraka vyavahArodyatsUktiratnAMzapiare // 21 // kSetra vistIrNa pAtAle khAtAkhya' sikatAkule | karaNaskandhasambandhacchAyAvelA virAjite // 22 // gaNakairguNasampUrNastadarthamaNayo'malAH / gRhyante karaNopAyaissArasaGgrahavAridhau // 23 // atha saMjJA / na zakyate'rtho boddhuM yatsarvasmin saMjJayA vinA / AdAvato'sya zAstrasya paribhASAbhidhAsyate // 24 // tatra tAvat kSetraparibhASA / jalAnalAdibhirnAzaM yo na yAti sa pudgalaH / paramANuranantaistairaNusso'trAdirucyate // 25 // trasareNuratastasmAdrathareNuH ziroruhaH / paramadhyajaghanyAkhyA bhogabhUkarmabhUbhuvA // 26 // lIkSA tilassa eveha sarSapo'tha yavo'Ggalam / krameNASTaguNAnyetadvayavahArAGgulaM matam // 27 = K saMjJAtoyasamA . 1M and B alpa . ' Mddha (Probably a scribe's mistake for ttha). 6 Pva. 6 K and P. P and B-LT. Acharya Shri Kailassagarsuri Gyanmandir P.. For Private and Personal Use Only 4 M and B saGkaTe. 7M and B va', 3

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gaNitasArasaGgrahaH tatpaJcakazataM proktaM pramANa mAnavedibhiH / bartamAnanarANAmaGgulamAtmAGgulaM bhavet // 28 // vyavahArapramANe dve' rAdhAnte laukike viduH / mAtmAGgalamiti tredhA tiryakpAdaH SaDaGgulaiH // 29 // pAdadvayaM vitastissyAttato hasto dvisaGgaNaH / daNDo hastacatuSeNa krozastadvisahasrakam // 30 // yojanaM caturaH krozAnprAhuH kSetravicakSaNAH / vakSyate'taH paraM kAlaparibhASA yathAkramama // 31 // atha kaalpribhaassaa| aNuraNvantaraM kAle vyatikrAmati yAvati / sa kAlassamayo'saGkhyaissamayairAvalirbhavet // 32 // saGkhyAtAvalirucchAsaH stokstuucchaassptkH| stokArasapta lavasteSAM sArdhASTAtriMzatA ghaThI // 33 // ghaThIdvayaM muhUto'tra muhUrtestriMzatA dinam / pazvanaustradinaiH pakSaH pakSau dvau mAsa iSyate // 34 // phaturmAsadvayena syAtribhistairayanaM matam / taddayaM vatsaro vakSye dhAnyamAnamataH param // 35 // atha dhaanypribhaassaa| viddhi SoDazikAstatra catasraH kuDahoM bhavet / kuDahAM zcaturaH prasthazcatuH prasthAnathADhakam // 31 // caturmirADhakaidroNo mAnI droNaizcaturguNaiH / ravArI mAnIcatuSaNa khAryaH paJca pravartikA // 37 // Ixnye. * K and B gr. ..vAM. For Private and Personal Use Only

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ga saMjJAdhikAraH seyaM caturguNA vAhaH kumbhaH paJca pravartikAH / ita : paraM suvarNasya paribhASA vibhASyate ' // 38 // atha suvarNaparibhASA / caturbhirgaNDakairguJjA guJjAH paJca paNo'STa te / / dharaNaM dharaNe karSaH palaM karSacatuSTayam // 39 // _atha rjtpribhaassaa| dhAnyadvayena gujaikA guJjAyugmena mASakaH / mASaSoDazakenAtra dharaNaM paribhASyate // 40 // taddayaM sArdhakaM karSaH purANAMzcaturaH palam / rUpye mAgadhamAnena prAhussaGkhyAnakovidAH // 41 // atha lohpribhaassaa| kalA nAma catuSpAdAH sapAdASaTUlA yavaH / yavaizcaturbhiraMzassyAdAgo'zAnAM catuSTayam // 42 // drANo bhAgaSaTrena dInAro'smAdisaNaH / dvau dInArau sateraM syAtsAhulAhe'tra sUrayaH / / 43 // I For the whole of dhAnyaparibhASA, P and B add what is given below as a lother reading and M has it in the original with the variations which are enclosed in brackets. AdyA SoDazikA tatra kuDa(hu)baH prastha aaddhk:| droNo mAnI tataH khArI krameNa (maza:*) caturAhatAH // (sahastraizca tribhiSSaDizzataizca vIhibhissamam / yassampUrNo'bhavatso'yaM kuDabaH paribhASyate // ). pratikAtra tAH paJca vAhastasyAzcaturguNaH / .5 kumbhassapAdavAhassyAt (pazca pravartikA: kumbhaH) svarNasaMjJAtha varNyate // -:: saterAkhyam. . * In Baleo. For Private and Personal Use Only

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gaNitasArasaGgrahaH palaiAdazabhissAdhaiH prasthaH palazatadvayam / tulA daza tulA mAra': saGkhyAdakSAH pracakSate // 44 // vastrAbharaNavetrANAM yugaLAnyatra viMzatiH / koTakAnantaraM bhASye* parikarmANi nAmata : // 45 // ___ atha parikarmanAmAni / AdimaM guNakAro'tra pratyutpanno'pi tdbhvet| dvitIyaM bhAgahArAkhyaM tRtIyaM kRtirucyate // 46 // caturtha vargamUlaM hi bhApyate paJcamaM dhanaH / / ghanamUlaM tataSSaSThaM saptamaM ca citissmRtam // 47 // tatsaGkalitamapyuktaM vyutkalitamato'STamam / tacca zeSamiti proktaM bhinnAnyaSTAvamUnyapi // 48 // atha dhanarNazUnyaviSayakasAmAnyaniyamAH / tADitaH vena rAziH va so'vikArI hato yutaH / hIno'pi vavadhAdiH vaM yoge khaM yojyarUpakam // 49 // jhaNayordhanayorghAte bhajane ca phalaM dhnm|.. RNaM dhanarNayostu syAtvarNayorvivaraM yutau // 50 // RNayordhanayoogo yathAsaGkhyamRNaM dhanam / zodhyaM dhanamaNaM rAzeH RNaM zodhyaM dhanaM bhavet // 11 // dhanaM dhanarNayorvoM mUle svarNe tayoH kramAt / RNaM svarUpato'vargoM yatastasmAnna tatpadama // 52 // atha sngkhyaasNjnyaaH| 'zazI somazca candrendU prAleyAMza rajanIkaraH / zvetaM himagu rUpazca mRgAGkazca kalAdharaH / 53 // 1Mt. M Di. 5M vidyAkalAsavarNasya. * Stanzas 53 to 68 ooour only in M, and are ben here, though erroneous 636068 DoordindagivcAkalAsavaNasya. here and there, as found in the original. * Used here in the 4th conjugation, active voice, For Private and Personal Use Only

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saMjJAdhikAraH hi he dvAvubhau yugalayugmaM ca locanaM dvayam / / dRSTinetrAmbakaM dvandvamakSicakSurnayaM dRzau // 54 // haranetraM puraM lokaM trai(tri)ratnaM bhuvanatrayam / " guNo vahniH ziravI jvalanaH pAvakazca hutAzanaH // 55 // ambudhiviSadhivArSiH payodhissAgaro gatiH / naladhibandhazcaturvedaH kaSAyassalilAkaraH // 56 // iSurvANaM zaraM zastraM mUtamindriyasAyakam / paJca vratAni viSayaH karaNIyastantusAyakaH // 17 // RtujIvo raso lekhyA dravyaJca SaTukaM kharana / kumAravadanaM varNaM zilImukhapadAni ca // 18 // zailamadriyaM bhadhro nagAcalamunigiri : / azvAvipannagA dvIpa dhAturvyasanamAtRkam // 59 // aSTauM tanurgajaH karma vasu vAraNapuSaram / dviradaM dantI digduritaM nAgAnIkaM karI yathA // 10 // nava nandaM ca randhrazca padArtha labdhakezavau / nidhiratnaM grahANAM ca durganAma ca saGkhyayA // 11 // AkAzaM gaganaM zUnyamambara vaM nabho viyat / / anantamantarikSaM ca viSNupAdaM divi smaret // 62 // atha sthAnanAmAni / ekaM tu prathamasthAnaM dvitIyaM dazasaMjJikam / tRtIyaM zatamityAhuH caturthaM tu sahastrakam // 63 // paJcamaM dazasAhastraM SaSThaM syAllakSameva ca / saptamaM dazalakSaM tu aSTamaM koThirucyate // 64 // For Private and Personal Use Only

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gaNitasArasaGgrahaH navamaM dazakovyastu dazamaM zatakoThayaH / / arbuda rudrasaMyuktaM nyarbudaM dvAdazaM bhavet // 15 // ravarva trayodazasthAnaM mahAravarvaM caturdazam / padmaM paJcadazaM caiva mahApadmaM tu SoDazam // 16 // kSoNI saptadazaM caiva mahAkSoNI dazASTakam / zaGkha navadazaM sthAnaM mahAzaGkha tu viMzakam // 67 / / kSityaikaviMzatisthAnaM mahAkSityA dviviMzakam / triviMzakamatha kSobhaM mahAkSobhaM caturnayam / / 68 // atha gaNakaguNanirUpaNam / laghukaraNohApohAnAlasyagrahaNadhAraNopAyaiH / vyaktikarAGkaviziSTairgaNako'STAbhirguNai yaH // 69 // iti saMjJA samAsena bhASitA munipuGgavaiH / vistareNAgamAddedyaM vaktavyaM yaditaH param // 70 // iti sArasaGgahe gaNitazAstre mahAvIrAcAryasya kRtau saMjJAdhikArasamAptaH // For Private and Personal Use Only

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prathamaH parikarmavyavahAraH. itaH paraM parikarmAbhidhAnaM prathamavyavahAramudAhariSyAmaH / pratyutpannaH / tatra' prathame pratyutpannaparikarmaNi karaNasUtraM yathA-- guNayadguNena guNyaM kavAThasandhikrameNa saMsthApya / rAzyarghakhaNDatatsthairanulomavilomamArgAbhyAm // 1 // __ atroddezakaH / dattAnyekaikasmai jinabhavanA yAmbujAni tAnyaSTau / vasatInAM caturuttaracatvAriMzacchatA'ya kati // 2 // nava padmarAgamaNayassamarcitA ekajinagRhe dRSTAH / sASTAzItidvizatImitavasatiSu te kiyantassyuH // 3 // 'catvAriMzaccaikonazatAdhikapuSyarAgamaNayo'AH / ekasmin jinabhavane sanavazate brUhi 'kati maNayaH // 4 // padmAni saptaviMzatire"kasmin jinagRhe pradattAni / sASTAnavatisahastre "sanavazate tAni kati kathaya // 5 // "ekaikasyAM vasatAvaSTottarazatasuvarNapadmAni / ekASTacatussaptakanavaSaTuJcASTakAnAM kim // 6 // 1K tatra ca. * KAnd B vinyasyobhau rAzI. K and B saGgaNayet. * B sya hi. 5 B nasyA. * B zatasya kati bhavanAnAma. 7 M and B catvAriMzadvayakA zatAdhikA. 3M icchA:. * M te kiyantassyu:. 10 M ekaikajinAlayAya dattAni. 11M prayuktanavazatagRhANAM kim, 12 This stanza is found only in M and B, For Private and Personal Use Only

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to gaNitasArasaGgrahaH zazivasuravarajalanidhinavapadArthabhayanayasamUhamAsthApya / himakaraviSanidhigatibhirguNite kiM' rAziparimANam / / 7 / / himagupayonidhigatizazivahnivratanicayamatra saMsthApya / saikAzItyA tvaM me guNayitvAcakSva tatsaGkhyAm // 8 // agnivasuravarabhayendriyazazalAJchanarAzimatra saMsthApya' / randhairguNayitvA me kathaya marave rAziparimANam // 9 // 'nandAyUtuzaracatustridvandvaikaM sthApya matra navaguNitam / AcAryamahAvIraiH kathitaM narapAlakaNThikAbharaNam // 10 // SaTvikaM paJcaSaTUzca sapta cAdau pratiSThitam / trayastriMzatsaGgaNitaM kaNThAbharaNamAdizat // 11 // hutavahagatizazimunibhirvasunayagaticandramatra saMsthApya / zailena tu guNayitvA kathayedaM ratnakaNThikAbharaNam // 12 // analAbdhihimagumunizaraduritAkSipayodhisomamAsthApya / zailena tu guNayitvA kathaya tvaM rAjakaNThikAbharaNam / / 13 / / giriguNadivigiriguNadivigiriguNanikaraM tathaiva guNaguNitam / punarevaM guNaguNitam ekAdinavottaraM viddhi // 14 // sapta zUnyaM dvayaM inheM paJcaikazca pratiSThitam / trayaHsaptatisaGgaNyaM "kaNThAbharaNamAdizet // 15 // jalanidhipayodhizazadharanayanadravyAkSinikaramAsthApya / guNite tu catuSSaSTayA kA saGkhyA gaNitavihRhi // 16 // 1 M and B kintasya. * pyam. M aho. meM zIghram. * B vinyasya. stanzas from 10 to 15 are found only in M and B. 1All the MES. read sthApya tatra. B ze. B nayaM. 10 All the MSS. give the metrically erroneous reading TTTOT TEHT DIE 1 For Private and Personal Use Only

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11 parikarmavyavahAraH zazAlenduravaikenduzUnyaikarUpaM nidhAya krameNAtra rAzipramANam / himAMzvagrarandhaiH prasantADite'smin bhavetkaNThikA rAjaputrasya yogyA // 17 // iti parikarmavidhau prathamaH pratyutpannaH samAptaH / / bhaaghaarH| dvitIye bhAgahAraparikarmaNi karaNasUtraM yathA 'vinyasya bhAjyamAnaM tasyAdhasthena bhAgahAreNa / sahazApavartavidhinA bhAgaM kRtvA phalaM pravadet // 18 // atha vA pratilomapathena bhajedAjyamadhasthena bhAgahAreNa / / sadRzApavartanavidhiryadyasti vidhAya tamapi tayoH // 19 // atroddezakaH / dInArASTasahastraM dAnavatiyutaM zatena saMyuktam / caturuttaraSaSTinarairbhaktaM ko'zo nurekasya // 20 // rUpAgrasaptaviMzatizatAni kanakAni yatra bhAjyante / saptatriMzatpuruSairekasyAMzaM mamAcakSva // 21 // dInAradazasahasraM trizatayutaM saptavargasammizram / navasaptatyA puruSairbhaktaM kiM labdhamekasya // 22 // 'ayutaM catvAriMzaJcatussahajaikazatayutaM henAm / navasaptativasatInAM dattaM vittaM kimekasyAH / / 23 // 1 This stanxa is not found in P. + This stanza is not found in P. sa. 3M ko'zo narekA Band K hemam. For Private and Personal Use Only

Page #46
12 gaNitasArasaGgrahaH 'saptadazatrizatayutAnyekatriMzatsahasrajambUni / bhaktAni navatriMzannarairvadaikasya bhAgaM tvam // 24 // 'yadhikadazatrizatayutAnyekatriMzatsahasrajambUni / saikAzItizatena prahRtAni naraidekAMzam // 25 // tridazasahastrI saikA ssssttidvishtiishsrssttyutaa| ratnAnAM navapuMsAM dattaikanaro'tra kiM labhate // 26 / / 'ekAdiSaDantAni krameNa hInAni hATakAni sarave / vidhujaladhibandhasaGkhyairnarairhatAnyekabhAgaH kaH // 27 // vyazItimizrANi catuzzatAni catussahasraghnanagAnvitAni / ratnAni dattAni jinAlayAnAM trayodazAnAM kathayaikabhAgam // 28 // iti parikarmavidhau dvitIyo bhAgahAraH samAptaH / / vargaH / tRtIye vargaparikarmaNi karaNasUtraM yathA dvisamavadho ghAto vA kheSTonayutadvayasya seSTakatiH / ekAdidvicayecchAgacchayutirvA bhavedvargaH // 29 // M reads the problem contained in this stanza thus: trizatayutaikatriMzatsahasrayuktA dazAdhikA: sata / bhaktAzcatvAriMzatpuruSerekonaistatra dInAram // * This stanza is found only in M. .. ekadvitricatuHpaJcaSaTrInAH krameNa sambhaktAH / saikacatuHzatasaMyutacatvAriMzajjinAlayAnAM kim // For Private and Personal Use Only

Page #47
13 parikarmavyavahAraH dvisthAnaprabhRtInAM rAzInAM sarvavargasaMyogaH / teSAM kramaghAtena dviguNena vimizrito vargaH // 30 // kRtvAntyakRti hanyAccheSapadaidviguNamantyamutsArya / zeSAnutsAyaivaM karaNIyo vidhirayaM varge // 31 // atroddezakaH / ekAdinavAntAnAM paJcadazAnAM dvisaGgaNASTAnAm / vratayugayozca rasAgnyozzaranagayornargamAcakSva // 32 // sASTAviMzatrizatI catussahakaSaSTiSaTchatikA / dvizatI SaTpaJcAzanmizrA vargIkRtA kiM syAt / / 33 // lekhyAguNeSubANadravyANAM zaragatitrisUryANAm / / guNaratnAmipurANAM varga bhaNa gaNaka yadi vetsi // 34 // saptAzItitrizatasahitaM SaTsahasraM punazca paJcatriMzacchatasamadhikaM saptanighnaM sahasram / dvAviMzatyA yutadazazataM 'vargitaM tatrayANAM brUhi tvaM me gaNaka guNavansaGguNathya pramANam // 35 // iti parikamaividhau tRtIyo vargassamAptaH / / vargamUlam / catuthai vargamUlaparikarmaNi karaNasUtraM yathA-- antyaujAdapahRtakatimUlena dviguNitena yugmahatau / labdhakRtistyAjyauje dviguNadalaM vargamUlaphalam // 36 // IP, Kand B rAziretatkRtInAm. For Private and Personal Use Only

Page #48
14 gaNitasArasaGgrahaH atroddezakaH / ekAdinavAntAnAM vargagatAnAM vadAzu me mUlam / atuviSayalocanAnAM dravyamahIdhendriyANAJca / / 37 // ekAgraSaSTisamadhika paJcazatopetaSaTsahasrANAm / Sar3ageMpaJcapaJcakaSaNNAmapi mUlamAkalaya // 38 // dravyapadArthanayAcalalekhyAlabdhyabdhi nidhinayAbdhInAm / zazinetrendriyayuganayajIvAnAJcApi kiM mUlam // 39 // candrAbdhigatikaSAyadravyartahutAzanatarAzInAm / / vidhulekhyendriyahimakaramunigirizazinAM ca mUlaM kim // 40 // dvAdazazatasya mUlaM SaNNavatiyatasya kathaya sazcintya / zataSaTkasyApi sarave paJcakavargeNa yuktasya / / 41 // aGkebhakarmAmbarazaGkarANAM somAkSivaizvAnarabhAskarANAm / candrartubANAbdhigatidvipAnAmAcakSva mUlaM gaNakAgraNIstvam // 42 // iti parikarmavidhau caturthaM vargamUlaM samAptam / / ghana : / pazcame ghanaparikamaNi karaNasUtraM yathA --- trisamAhatirghanassyAdiSTonayutAnyarAzighAto vA / alpaguNitaSTakRtyA kalito bRndena ceSTasya / / 43 // iSTAdidviguNeSTapracayeSTapadAnvayo'tha veSTakRtiH / vyekeSTahataikAdidvicayeSTapadai yayuktA vA // 44 / / "ekAdicayeSTapade pUrva rAzi pareNa saGgaNayet / guNitasamAsastriguNazcarameNa yuto ghano bhavati // 45 // | P and M vargagatAnAM zIghraM rUpAdinavAvasAnarAzInAm / mUlaM kathaya sakhe tvaM. M nava. 3 'This stanza is not found in P. For Private and Personal Use Only

Page #49
16 parikarmavyavahAraH antyAnyasthAnakRtiH parasparasthAnasaGgaNA trihtaa| puna revaM tadyo gassarvapadaghanAnvito bRndam / / 46 // "antyasya ghanaH kRtirapi sA trihatotsArya zeSaguNitA vA / zeSakRtisvyantyahatA sthApyotsAcaivamatra vidhiH // 17 // atroddezakaH / ekAdinavAntAnAM paJcadazAnAM zarekSaNasyApi / rasava yogirinagayoH kathaya ghanaM dravyalayozca // 48 // himakaragaganendUnAM nayagirizazinAM varenduvANAnAm / vada municandrayasInAM vRndaM caturudaviguNazazinAm // 49 // rAzinIkRto'yaM zatadvaya mizrita trayodazabhiH / tAMdaguNo'smAtriguNazcaturguNaH paJcaguNitazca // 50 // zatamaSTaSaSTiyuktaM dRSTamabhISTe ghane viziSTatamaiH / ekAdibhiraSTAnyairgaNitaM vada taddhanaM zIghram // 11 // bandhAmbara gaganandriyakezavAnAM saGkhyAH krameNa vinidhAya dhanaM gRhItvA / AcakSva labdhamadhunA karaNAnayoga gambharitAratarasAgarapAradRzvan / 52 / / iti parikarmavidhau paJcamo ghanassamAptaH // 1Mrapi. ___ M go vA. 3 This stanza is omitted in M. Che following stanzi is found as a TOTEC in P, K and B ; though not quite explicit, it mentions two of the processes above described : trisamaguNo'ntyasya ghanastadvargastriguNito hatazzeSaiH / utsArya zeSakRtiratha niSThA triguNA ghanastathAgre vA // * Instead of stanzas 48 and 49, M reads ekAdinavAntAnAM rudrANAM himakarendUnAm / vada municandrayatInAM bRndaM cturuddhigunnshshinaam|| For Private and Personal Use Only

Page #50
16 gaNitasArasagrahaH ghanamUlam / SaSThe ghanamUlaparikarmaNi karaNasUtraM yathAantyavanAdapahRtaghanamalarutitrihatibhAjite bhAjye / prAktrihatAptasya kRtizzodhyA zodhye ghane'tha ghanam // 13 // 'ghanamekaM ve apane ghana padakRtyA ajetriguNayAghanataH / pUrvatriguNAptakRtistyAjyAptaghanazca pUrvavallabdhapadaiH // 54 // atroddezakaH / ekAdinavAntAnAM ghanAtmanAM ratnazazinavAbdhInAm / 'nagarasavasuravartugajakSapAkarANAJca mUlaM kim // 55 // gatinayamadazivizazinAM muniguNavatvIkSanava'kharAgnInAm / 'vasuravayugaravAdrigatikaricandratUMnAM gRhANa padam // 56 // catuHpayodhyagnizarAkSidRSTi hayebharavavyomabhayekSaNasya / vadASTakarmAbdhiravaghAtibhAva dvivahiranartunagasya mUlam / / 57 // dravyAzvazailaduritaravavahayadribhayasya vadata ghanamUlam / navacandrahimagumunizazilabdhyambaravarayugasyApi // 18 // 'gatigajaviSayeSuvidhuvarAdrikaragatiyugasya bhaNa mUlam / lekhyAzvanaganavAcalapuraravaranayajIvacandramasAm // 59 // gtirvrduritebhaambhodhitaaydhvjaakssvikRtinvpdaarthdrvyvhiinducndr-| jaladharapatharandhraSvaSTakAnAM ghanAnAM gaNaka gaNitadakSAcakSva mUlaM parIkSya || 60 // iti parikarmavidhau SaSThaM ghanamUlaM samAptam // 1 This stanza is not found in M. 'M giri. M rasA. + M vidhapurakharasvarartujvalanadharANAM". This stanza is not found in M For Private and Personal Use Only

Page #51
parikarmavyavahAraH saGkalitam / saptame saGkalita parikarmaNi karaNasUtraM yathA rUpeNono gaccho dalIkRtaH pracayatADito mizraH / prabhaveNa padAbhyastassaGkalitaM bhavati sarveSAm // 61 // --- prakArAntareNa dhanAnayanasUtramekavihIno gaccha: pracayaguNo dviguNitAdisaMyuktaH / gacchAmyasto dvihRtaH prabhavetsarvatra saGkalitam // 62 // Adyuttara sarvadhanAnayanasUtram - padahatamurakhamAdidhanaM vyekapadArtha pracayaguNo gacchaH / uttaradhanaM 'tayoryogo dhanamUnottaraM mukhe'ntyadhane // 63 // antyadhanamadhyadhanasarvadhanAnayanasUtram -- "cayaguNitaikonapadaM sAdyantyadhanaM tadAdiyogArdham / madhyadhanaM tatpadavadhamuddiSTaM sarvasaGkalitam // 64 // atroddezakaH / This stanza is omitted in M. 3 Acharya Shri Kailassagarsuri Gyanmandir ekAdidazAntAdyAstAvatpracayAssamarcayanti dhanam / vaNijo daza daza gacchAsteSAM saGkalitamAkalaya // 65 // dvimukhatricayairmaNibhiH prAnarca zrAvakottamaH kazcit / paJcavasatIramISAM kA saGkhyA bUhi gaNitajJa // 66 // Adistrayazvo'STau dvAdaza gacchastrayo'pi rUpeNa / A saptakAtpravRddhAssarveSAM gaNaka bhaNa gaNitam // 67 // dvikRtirmukhaM cayo'STau nagarasahastre samacitaM gaNitam / gaNitAbdhisamuttaraNe bAhubalin tvaM samAcakSva // 68 // M tadUnA saika (va ?) padAptA yuti: prabhavaH / M balI. For Private and Personal Use Only 17

Page #52
18 gaNitasArasaGgrahaH gacchAnayanasUtram aSTottaraguNarAzerdviguNAdyuttaravizeSakRtisahitAt / mUlaM cayayutamarpitamAyUnaM cayahRtaM gacchaH // 69 // prakArAntareNa gacchAnayanasUtramaSTottaraguNarAzerDiguNAdyuttaravizeSakatisAhitAt / mUlaM kSepapadonaM dalitaM cayabhAjitaM gacchaH // 70 // atroddezakaH // AdiauM pracayo'STau dvau rUpeNA trayAtkramAbaddhau / ravAGkau rasAdrinetraM rakhenduharA vittamatra ko gacchaH // 71 // AdiH paJca cayo'STau guNaratnAgnidhanamatra ko gacchaH / SaT prabhavazva cayo'STau vadvicaturasvaM padaM kiM syAt // 72 // uttarAdyAnayanasutram-- AdidhanonaM gaNitaM padonapadakRtidalena sambhajitam / pracayastaddhanahInaM gaNitaM padabhAjitaM prabhavaH // 73 // AyuttarAnayanasUtramprabhavo gacchAptadhanaM vigataikapadArdhaguNitacayahInam / padahatadhanamAyUnaM nirekapadadalahRtaM pracayaH // 74 // prakArAntareNottarAdyAnayanasUtradvayam -- dvihataM saGkalitadhanaM gacchahRtaM dviguNitAdinA rahitam / vigataikapadavibhaktaM pracayassyAditi vijAnIhi // 75 // dviguNitasaGkalitadhanaM gacchahRtaM rUparahitagacchena / tADitacayena rahitaM dvayena sambhAjitaM prabhavaH // 76 // For Private and Personal Use Only

Page #53
parikarmavyavahAraH atroddezakaH / nava vadanaM tavapadaM bhAvAdhikazatadhanaM kiyAnpracayaH / paJca cayo'STa padaM SaTpaJcAzacchatadhanaM mukhaM kathaya 77 // sveSTAdyuttaragacchAnayanasUtram -- saGkalite sveSTahRte hAro gaccho'tra labdha iSTone / UnitamAdizzeSe vyekapadArthoddhRte pracayaH // 78 // atroddezakaH / catvAriMzatsahitA paJcazatI gaNitamatra sandRSTam / gacchapracayaprabhavAn 'gaNitajJaziromaNe kathaya // 79 // AdyuttaragacchasarvamizradhanavizleSaNe sUtratrayam - uttaradhanena rahitaM gacchanaikena saMyutena hRtam / mizradhanaM prabhavassyAditi gaNakaziromaNe viddhi // 80 // AdidhanonaM mizraM 'rUponapadArdhaguNitagacchena / saikena hRta pracayo gacchavidhAnAtpadaM mukhe saike // 81 // mizrAdapanIteSTau mukhagacchau pracayamizravidhilabdhaH / yo rAzista cayassyAtkaraNamidaM sarvasaMyoge // 82 // atroddezakaH / dvitrikapazcadazAgrA catvAriMzanmukhAdimizradhanam / tatra prabhavaM pracayaM gacchaM sarvaM ca me brUhi // 83 // M vigaNayya sakhe mamAcakSva. 2 M Acharya Shri Kailassagarsuri Gyanmandir padonapada kRtidalena saikena / bhaktaM praccayo'tra padaM gacchavidhAnAnmukhe saike // For Private and Personal Use Only 19

Page #54
20 gaNitasArasaGgrahaH dRSTadhanAdyuttarato dviguNatriguNadvibhAgatribhAgAdISTadhanAdyuttarAnayanasUtram --- dRSTavibhakteSTadhanaM dviSThaM tatpacayatADitaM pracayaH / tatprabhavaguNaM prabhavo 'guNabhAgasyeSTavittasya // 4 // atrodezakaH / samagacchazcatvAraHSaSTirmukhamuttaraM tato dviguNam / taDyAdi hatavibhaktasveSTasyAdyuttare brUhi // 85 / / iSTagacchapoLastAdyuttarasamadhanadviguNatriguNadvibhAgatribhAgAdidhanAnayanasUtram vyekAtmahato gacchastveSTano dviguNitAnyapadahInaH / mukhamAtmonAnyakRtidikeSTapadaghAtavarjitA pracayaH // 86 // va atroddeshkH| paJcASTagaccha puMso yastaprabhavottare samAnadhanam / dvitriguNAdidhanaM vA brUhi tvaM gaNaka vigaNayya // 87 // dvAdazaSoDazapadayorvyastaprabhavottare samAnadhanam / . dhyAdiguNabhAgadhanamapi kathaya tvaM gaNitazAstrajJa // 8 // asamAnottarasamagacchasamadhanasyAdyuttarAnayanasUtram-- adhikacayasyaikAdizcAdhikacayazeSacayavizeSo guNitaH / vigataikapadArthena sarUpazca muravAni mitra zeSacayAnAm // 89 // atroddezakaH / ekAdiSaDantacayAnAmekatritayapaJcasaptacayAnAm / navanavagacchAnAM samavittAnAM cAzu vada mukhAni sakhe // 90 // 1 M guNabhAgAdyuttarecchAyA:. : M guNa. . 3M gaNakamukhatilaka / For Private and Personal Use Only

Page #55
parikarmavyavahAraH visadRzAdisadRzagacchasamaMdhanAnAmuttarAnayanasUtram-adhikamukhasyaikacayazcAdhikamukhazeSamukhavizeSo bhaktaHH / vigataikapadArthena sarUpazca cayA bhavanti zeSamukhAnAm // 11 // atroddezakaH / ekatripaJcasaptanavaikAdazavadanapazcapaJcapadAnAm / samavittAnAM kathayottarANi gaNitAbdhipAradRzvan gaNakaH // 92 // atha guNadhanaguNasaGkalitadhanayossUtram - - padamitaguNahatiguNitaprabhavasrayAdguNadhanaM tadAdyUnam / ekonaguNavibhaktaM guNasaGkalitaM vijAnIyAt // 93 // guNasaGkalite anyadapi sUtram - 'samadalaviSamakharUpo guNaguNito vargatADito gaccha / rUponaH prabhavanno vyekottarabhAjitassAram // 94 // guNasaGkalitAntyadhanAnayane tatsaGkalitadhanAnayane ca sUtram -- guNasaGkalitAntyavanaM vigataikapadasya guNadhanaM bhavati / tadguNaguNaM mukhonaM vyakottarabhAjitaM sAram // 99 // guNadhanasyodAharaNam / svayaM gRhItvA triguNadhanaM pratipuraM samArjati / yaH puruSo'STanagaryAM tasya kiyattimAcakSva // 96 // guNadhanasyAdyuttarAnayanasUtram guNamAdivibhaktaM yatpadamitavadhasamaM sa eva cayaH / gacchataM guNitaM bhavetprabhavaH // 97 // 1M. samayati. For Private and Personal Use Only 21

Page #56
22 gaNitasArasaGgrahaH guNadhanasya gacchAnayanasUtram - mukhabhakke guNavitte yathA niragraM tathA guNena hRte / yAvatyo'tra zalAkAstAvAn gaccho guNadhanasya // 98 // guNasaGkalitodAharaNam / dInArapazcakAdidviguNaM dhanamarjayannaraH kazcit / prAvikSadaSTanagarIH kati jAtAstasya dInArAH // 99 // saptamukhAtriguNacayatrivargagacchasya kiM dhanaM vaNijaH / trikapazcakapaJcadazaprabhavaguNottarapadasthApi // 10 // guNasaGkalitottarAdyAnayanasUtram - asakRyekaM mukhahatavittaM yenoddhRtaM bhavetsa cayaH / vyekaguNaguNitagaNitaM nirekapadamAtraguNavadhAptaM prabhavaH // 101 // atroddezakaH / trimuravartugacchabANAGkAmbarajalanidhidhane kiyAnpracayaH / SaDNacayapazcapadAmbarazazihimagutrivittamatra mukhaM kim // 102 / / guNasaGkalitagacchAnayanasUtram - ekonaguNAbhyasta prabhavahRtaM rUpasaMyutaM vittam / yAvatkRtvo bhaktaM guNena tadvArasammitirgacchaH // 103 // atroddezakaH / triprabhavaM SaTuguNaM sAraM saptatyupetasaptazatI / saptAmA brUhi sarave kiyatpadaM gaNaka guNanipuNa // 104 / / paJcAdihiguNottare zaragiriokapramANe dhane saptAdi triguNe nagebhaduritastamberamartuprame / IH.. For Private and Personal Use Only

Page #57
parikarmavyavahAraH tryAsye paJcaguNAdhike hutavahopendrAkSa vahnidvipazvetAMzuddiradebhakarmakaradyamAne'pi gacchaH kiyAn // 105 // iti parikarmavidhau saptamaM saGkalitaM samAptam // Acharya Shri Kailassagarsuri Gyanmandir vyutkalitam / aSTame vyutkalita parikarmaNi karaNasUtraM yathA sapadeSTaM veSTamapi vyekaM dalitaM cayAhataM samukham / zeSeSTagacchaguNitaM vyutkalitaM khaSTa vittaM ca // 106 // prakArAntaraiNa vyutkalitadhanasveSTadhanAnayanasUtram - gacchasahiteSTamiSTaM caikonaM cayahataM dvihAdiyutam / zeSeSTapadArdhaguNaM vyutkalitaM svaSTavittamapi // 107 // cayaguNabhavavyutkalitaca nAnayane vyutkalitadhanasya zeSeSTagacchAnayane va sUtram iSTadhanonaM gaNitaM vyavakalitaM cayabhavaM guNotthaM ca / sarveSTagacchazeSe zeSapadaM jAyate tasya // 108 // zeSagacchasthAdyAnayanasUtrampracayaguNiteSTagacchassAdiH prabhavaH padasya zeSasya / prAktana eva cayassyAdgacchasyeSTasya tAveva ! 109 !! guNavyutkalitazeSagacchasthAdyAnayanasUtram - guNaguNite'pi cayAdI tathaiva bhedo'yamatrazeSapade / iSTapadamitiguNAhatiguNitaprabhavo bhavedvakam // 111 // M gaNitaM. For Private and Personal Use Only 23

Page #58
gaNitasArasaGgrahaH atroddezakaH / dvimukhastricayo gacchazcaturdaza khepmitaM padaM sapta / aSTanavaSaTrapaJca ca kiMvyutkAlataM samAkalaya // 111 / SaDAdiraSTau pracayo'tra SaTutiH padaM daza dvAdaza SoDazepsitam / mukhAdiranyasya tu paJcapaJcakaM zatadvayaM brUhi zataM vyayaH kiyAn // 112 // SaDDanamAno gacchaH pracayo'STau dviguNasaptakaM vakram / saptatriMzatveSTaM padaM samAcakSva phalamubhayama // 113 // aSTakRtirAdiruttaramUnaM catvAri SoDazAtra padam / iSTAni tattvakezavarudrArkapadAni kiM zeSam // 114 // guNavyutkalitasyodAharaNamacaturAdidviguNAtmakottarayuto gacchazcaturNA kRtirdaza vAJchApadamaGkasindhuragiridravyandriyAmbhodhayaH / / kathaya vyutkalita phalaM sakalasadbhUjAgrimaM vyAptavAn karaNaskandhavanAntaraM gaNitavinmattebhavikrIDitam // 115 // iti parikarmavidhAvaSTamaM vyutkalitaM samAptam / / iti sArasaGgrahe gaNitazAstre mahAvIrAcAryasya kRtau parikarmanAmA prathamo vyavahAraH samAptaH // 1M prA. For Private and Personal Use Only

Page #59
atha dvitIyaH kalAsavarNavyavahAraH / 'trilokarAjendrakirITakoTiprabhAbhirAlIdapadAravindam / nirmUlamunmUlitakarmavRkSaM jinendracandraM praNamAmi bhaktyA // 1 // itaH paraM kalAsavarNa dvitIyavyavahAramudAhariSyAmaH // bhinnapratyutpannaH / tatra bhinnapratyutpanne karaNasUtraM yathAguNayedaMzAnaMzaiArAn hArairghaTeta yadi teSAm / vajApavartanavidhividhAya taM bhinnaguNakAre // 2 // atroddezakaH / zuNThyAH palena labhate caturnavAMzaM paNasya yaH puruSaH / kimasau brUhi sarave tvaM triguNena palASTabhAgena // 3 // maricasya palasyArghaH paNasya saptASTamAMzako yatra / tatra bhavatkiM "mUlyaM palaSaTpaJcAMzakasya vada // 4 // kazcitpaNena labhate tripaJcabhAgaM palasya pippalyAH / navabhiH pardibhaktaiH kiM gaNakAcakSva guNayitvA // 5 // krINAti paNena vaNigrajIrakapalanavadazAMzakaM yatra / tatra paNaiH pazcAdhaiH kathaya tvaM kiM samagramate // 6 // dhyAdayo dvitayavRddhayoM'zakAjyAdayo dvayacayA harAH punaH / te dvaye dazapadAH kiyatphalaM brUhi tatra guNane dvayoIyoH // 7 // . iti bhinngunnkaarH| 1 This stanza is omitted in P. M mau. For Private and Personal Use Only

Page #60
gaNitasArasaGgrahaH bhinnbhaaghaarH| bhinnabhAgahAre karaNasUtraM yathA - aMzIkRtyacchedaM pramANarAzestataH kriyA guNavat / pramitaphale'nyaharane vicchidi vA sakalavacca bhAgahatau // 8 // atroddeshkH| hiGgoH palArdhamaulyaM paNatripAdAMzako avedyatra / tatrArdhe vikrINan palamekaM kiM naro labhate / / 9 // agaroH palASTamena triguNena paNasya viMzativyaMzAn / upalabhate yatra pumAnekena palena kiM tatra // 10 // paNapaJcamaizcatubhirnaravasya palasaptamo vyazItiguNaH / saMprApyo yatra syAdekena paNena kiM tatra // 11 // dhyAdirUpapariraddhiyujo'zA yAvadaSTapadamekavihInAH / hArakAstata iha dvitayAdyaiH kiM phalaM vada pareSu hateSu // 12 // ___ iti bhinnabhAgahAraH / bhinnavargavargamUlaghanaghanamUlAni // 'bhinnavargavargamUlaghanaghanamUleSu karaNasUtraM yathA kRtvAcchedAMzakayoH katikatimUle ghanaM ca ghanamUlam / tacchedaireMzahatau vargAdiphalaM bhavedbhinne // 13 // 1M bhinnavargabhinavargamUlabhinnaghanatanmaleSu. For Private and Personal Use Only

Page #61
kalAsavarNavyavahAraH. atroddezakaH / paJcakasaptanavAnAM dalitAnAM kathaya gaNaka varga tvam / SoDazaviMzatizatakadvizatAnAM ca tribhaktAnAm // 14 // trikAdirUpavayavRddhayoM'zA dvikAdirUpottarakA harAzca / padaM mataM dvAdaza vargameSAM vadAzu me tvaM gaNakAgragaNya // 15 // pAdanavAMzakaSoDazabhAgAnAM paJcaviMzatitamasya / SaTtriMzadbhAgasya ca kRtimUlaM gaNaka bhaNa zIghram // 16 // bhinne varge rAzayo vargitA ye teSAM mUlaM saptazatyAzca kiM syAt / vyaSTonAyAH paJcavauddhatAyA brUhi tvaM me vargamUlaM pravINa // 17 // ardhatribhAgapAdAH paJcAMzakaSaSThasaptamASTAMzAH / dRSTA navamazcaiSAM pRthak pRthagabrUhi gaNaka ghanam // 18 // tritayAdicatuzcayakoM'zagaNo dvimukhahicayo'tra harapracayaH / dazakaM padamAzu tadIyaghanaM kathaya priya sUkSmamate gaNite // 19 // zatakasya paJcaviMzasyASTavibhaktasya kathaya ghanamUlam / navayutasaptazatAnAM vizAnAmaSTabhaktAnAm // 20 // IM saptazatasyApi sane vyakonatriMzakAekAptasya // For Private and Personal Use Only

Page #62
9.8 gaNitasArasagrahaH bhinnaghane paridRSTaghanAnAM mUlamudagramate vada mitra / byUnazatadvayayugdvisahasyAzrApi navaprahatatrihatAyAH // 21 // iti bhinnavargavargamUlaghanaghanamUlAni / / bhinnasaGkalitam / bhinnasaGkalite karaNasUtraM yathA -- padamiSTaM pracayahataM dviguNaprabhavAnvitaM cayenonam / gacchAdhenAbhyastaM bhavati phalaM bhinnasaGkalite // 22 // . atroddezakaH / dvivyaMzaSSaDbhAgastricaraNabhAgo muravaM cayo gacchaH / dvau paJcamau tripAdo dvitryaMzo'nyasya kathaya kiM vittam / / 23 / / AdiH pracayo gacchastripaJcamaH paJcamastripAdAMzaH / sarvAMzaharau vRddhau dvitribhirA saptakAcca kA citiH / / 24 / / iSTagacchasyAyuttaravargarUpaghanarUpadhanAnayanasUtram --- padamiSTamekamAdivyekeSTadaloddhRtaM muravonapadam / pracayo vitta teSAM vargoM gacchAhataM bRndam // 25 // atroddezakaH / padamiSTaM dvivyaMzo rUpeNAMzo harazca saMvRddhaH / yAvaddazapadameSAM vada muravacayavargabRndAni // 26 // . 1 This stanza is not found in M. For Private and Personal Use Only

Page #63
kalAsavaNavyavahAraH. 29 iSTaghanadhanAdyuttaragacchAnayanasUtram -- iSTacaturthaH prabhavaH prabhavAtpracayo bhavedisaGgaNitaH / pracayazcaturabhyasto gacchasteSAM yutibRndam // 27 // atroddezakaH / dvimuravaikacayA aMzAstriprabhavaikottarA harA ubhaye / paJcapadA vada teSAM ghanadhanamuravacayapadAni sarave // 28 // dRSTadhanAdyuttarato dviguNatriguNadvibhAgatribhAgAdISTadhanAdyuttarAnayana . sUtram dRSTavibhakteSTadhanaM dviSThaM tatpracayatADitaM pracayaH / tatprabhavaguNaM prabhavo 'guNabhAgasyeSTavittasya // 29 // . atroddezakaH / prabhavasyoM rUpaM pracayaH paJcASTamassamAnapadam / icchAdhanamApi tAvatkathaya sarave ko mukhapracayI // 30 // pracayAdAdidiguNastrayodazASTAdazaM padaM kheSTam / . vittaM tu saptaSaSTiH SaDcanabhaktA vadAdicayau // 31 // 'mukhamekaM dvivyaMzaH pracayo gcchssmshcturnvmH|| dhanamiSTaM dvAviMzatirakAzItyA vadAdicayau // 32 / / gacchAnayanasUtram - dviguNacayaguNitavittAduttaradalamukhavizeSakRtisahitAt / mUlaM pracayArdhayutaM prabhavonaM cayahRtaM gacchaH // 33 // 1M guNabhAgAdyuttarAnayanasUtram / AM pracayena. * guNabhAgAdyuttarecchAyAH. * This stanza takes the place of stanza No. 31 in M and is omitted in B. 5 Instead of the following two stanzas M reads aSTottaraguNarAzItyAdinA isa.. panagaccha AnetavyaH and repeats stanza No. 70 given under parikarmavyavahAra. For Private and Personal Use Only

Page #64
30 gaNitasArasaGgrahaH prakArAntareNa tadevAha - dviguNacayaguNitavittAduttaradalamukhavizeSakatisahitAt / mUlaM kSepapadonaM pracayena hRtaM ca gacchassyAt // 34 // atroddezakaH / dvipaJcAMzo vakra triguNacaraNasyAdiha cayaH paDaMzassaptannastrikRtivihato vittamuditam / cayaH pazcASTAMzaH punarapi mukhaM vyaSTamamiti tricatvAriMzAsvaM priya vada padaM zIghramanayoH // 35 // AdyuttarAnayanasUtram - 'gacchAptagaNitamAdivigataikapadArdhaguNitacayahInam / padahatadhanamAyUnaM nireka padadalahataM pracayaH // 36 // atroddezakaH / tricaturthacatuHpaJcamacayagacche kheSuzazihataikatriMzad- / vitte tryaMzacatuHpaJcamamukhagacche ca vada mukhaM pracayaM ca // 37 // iSTagacchayowstAdyuttarasamadhanadviguNatriguNadvibhAgatribhAgadhanAnayanasUtram -- vyekAtmahato gacchassveSTano dviguNitAnyapadahInaH / / mukhamAtmonAnyakatiDhikeSTapadaghAtavarjitA pracayaH // 38 // atroddezakaH / ekAdiguNavibhAgasvaM vyastAdyuttare hi vada mitra / dviyazonaikAdazapaJcAMzakamizranavapadayoH // 39 // ___ K and B prabhavo gacchAptaphnam. For Private and Personal Use Only

Page #65
31 kalAsavarNavyavahAraH. guNadhanaguNasaGkalitadhanayoH sUtrampadAmitaguNahatiguNitaprabhavaH syAdguNadhanaM tadAbUnam / ekonaguNavibhaktaM guNasaGkalita vijAnIyAt // 40 // guNasaGkalitAntyadhanAnayane tatsaGkalitAnayane ca sUtram--- guNasaGkalitAntyadhanaM vigataikapadasya guNadhanaM bhavati / tadguNaguNaM muravonaM vyakottarabhAjitaM sAram / / 4 1 / / atroddezakaH / prabhavo'STamazcaturthaH pracayaH paJca padamatra guNaguNitam / guNasaGkalitaM tasyAnyadhanaM cAcakSva me zIghram / / 42 / / 'guNadhanasaGkalitadhanayorAdyuttarapadAnyapi pUrvoktasUtrairAnayet / samAneSTottaragacchasaGkalitaguNasaGkalitasamadhanasyAdyAnayanasatramamukhamekaM cayagacchAviSTau mukhavittarahitaguNacityA / hRtacayadhanamAdiguNaM mukhaM bhavedicitidhanasAmye // 43 // atroddezakaH / bhAvavArdhibhuvanAni padAnyambhodhipaJcamunayastrihatAste / uttarANi vadanAni kati syu. yugmasaGkalitavittasameSu // 44 // iti bhinnasaGkalitaM samAptam / / bhinnavyutkalitam / bhinnavyutkaline karaNasUtraM yathA - gacchAdhikaSTamiSTaM cayahatamUnottaraM dvihAdiyutam / zeSeSTapadArdhaguNaM vyutkalitaM kheSTavittaM ca // 45 // 1 Found cnly in B. For Private and Personal Use Only

Page #66
32 gaNitasArasaGgrahaH zeSagacchasyAdyAnayanasUtram - pracayA|naH prabhavo yutazcayaghneSTapadacayAdhIbhyAm / ' zeSasya padasyAdizcayastu pUrvokta eva bhavet // 46 // guNaguNite'pi cayAdI tathaiva bhedo'yamatra zeSapade / iSTapadamitaguNAhatiguNitaprabhavo bhavedvakam // 47 // __ atroddezakaH / pAdottaraM dalAsyaM padaM tripAdAMzakassamuddiSTaH / kheSTaM caturthabhAgaH kiM vyutkalitaM samAkalaya // 18 // prabhavo'dha paJcAzaH pracayo dvitryaMzako bhavedgacchaH / paJcASTAMzasveSTaM 'padamRNamAcakSva gaNitajJa / / 49 // AdizcaturthabhAgaH pracayaH paJcAMzakAstripaJcAMzaH / gaccho vAJchAgaccho dazamo vyavakalitamAnaM kim // 50 // tribhAgau dvau vakaM paJcamAMzazcayassyAt padaM trinaH pAdaH paJcamaskheSTagacchaH / SaDaMzassaptAMzo vA vyayaH ko vada tvam kalAvAsa prajJAcandrikAbhAsvadindo // 51 // dvAdazapadaM caturthottaramonapaJcakaM vadanam / tricatuHpaJcASTeSTapadAni vyutkAlatamAkalaya / / 52 / / ! pracayaguNiteSTagacchassAdiH prabhavaH padasya zeSasya / vayassyAdiSTasya prAktanAdeva // * M ca caturbhAgaH. kiM vyutkalitaM samAkalaya. For Private and Personal Use Only

Page #67
38 kalAsavarNavyavahAraH. guNasaGkalitavyutkalitodAharaNam / dvitribhAgarahitASTamukhaM dvitryaMzako guNacayo'STa padaM bhoH / mitra ratnagatipaJcapadAnISTAni zeSamukhavittapadaM kim // 53 // iti bhinnavyutkalitaM samAptam / / ' kalAsavarNaSaDjAtiH / / itaH paraM kalAsavaNe Sar3ajAtimudAhariSyAmaH --- bhAgaprabhAgAvatha bhAgabhAgo bhAgAnubandhaH parikIrtito'taH / bhAgApavAhassaha bhAgamAtrA Sar3ajAtayo'mutra kalAsavaNe // 54 // bhAgajAtiH / tatra bhAgajAtau karaNasUtraM yathA--- sadRzahatacchedahatI mithoM'zahArau samacchidAvaMzau / luptaikaharau yojyau tyAjyau vA bhAgajAtividhau / / 55 // 'prakArAntareNa samAnacchedamudrAvayitumuttarasUtram chedApavartakAnAM labdhAnAM cAhatI niruddhaH syAt / harahRtanirudvaguNite hArAMzaguNe samo hAraH / / 56 // 1 K and M add after this iti sArasaGgrahe mahAvIrAcAryasya kRtau dvitIyacyA . hArassAmaptaH . This, however, seems to be a mistake. * This and the stanza following are not found in M. For Private and Personal Use Only

Page #68
gaNitasArasaGghahaH atroddezakaH / jambUjambIranAraGgacocamocAmradADimam / ajhaiSIddaLaSaDbhAgadvAdazAMzakaviMzakaiH // 57 / / hemnastriMzacaturvizenASTamena yathAkramam / zrAvako jinapUjAyai tadyoge kiM phalaM vada // 58 / / aSTapaJcadazaM vizaM saptaSaTtriMzadaMzakam / ekAdazatriSaSTayaMzamekaviMzaM ca saGkSipa // 59 // 'ekaThikatrikAyekottaranavadazakaSoDazAntyaharAH / nijanijamuravapramAMzAsvaparAbhyastAzca kiM phalaM teSAm / / 60 // ekadikatrikAdyAzcaturAdyAzcaikaraddhikA hArAH / nijanijamukhapramAMzAH svAsannaparAhatAH kramazaH // 61 // viMzatyantAH SaDguNasaptAntAH paJcavargapazcimakAH / SatriMzatpAzcAtyAH sajhepe kiM phalaM teSAm // 62 / / 'candanaghanasArAgarukuGkamamakeSTa jinamahAya nrH| caraNadaLaviMzapakSamabhAgaiH kanakasya kiM zeSam // 63 // pAdaM paJcAMzamadhaiM triguNitadazamaM saptaviMzAMzakaJca svarNadvandvaM pradAya smita sitakamalaM styAnadadhyAjyadugdham / / J Stanzas Nos. 57 and 58 are omitted in P. 2 This stanza is found in K and B. 3 Stanzas Nos. 63 and 64 are found in K and B. 4M muru. For Private and Personal Use Only

Page #69
kalAsavarNavyavahAraH. zrIravaNDaM tvaM gRhItvAnaya jinasadanaprArcanAyAbravInmAmityadya zrAvakAryoM bhaNa gaNaka kiyaccheSamaMzAnvizodhya // 64 // 'aSTapaJcamuravau hArAvubhaye'pyekaddhikAH / 'triMzadantAH parAbhyastAzcaturguNitapazcimAH // 65 // 'svasvavakrapramANAMzA rUpAtsaMzodhya tadvayam / zeSaM sarave samAcakSva prottIrNagaNitArNava / / 66 // ekonaviMzatiratha kramAta trayoviMzatirTiSaSTizca / rUpavihInA triMzattatastrayoviMzatizataM syAt // 67 // paJcatriMzattasmAdaSTAzItikazataM vinirdiSTam / saptatriMzadamuSmAdaSTAnavatitrikonapaJcAzat / / 68 // catvAriMzacchatikA saikA ca punazzataM saSoDazakam / ekatrizadatassyAdvAnavatiH saptapaJcAzat / / 69 / / jyadhikA saptatirasmAtsapaJcapaJcAzadapi ca sA dviguNA / saptakatiH sacatuSA saptatirekonaviMzatidvizatam // 70 // hArA nirUpitA aMzA ekAdyakottarA amUn / prakSipya phalamAcakSva bhAganAtyabdhipAraga // 71 // atrAMzotpattau sUtram --- ekaM parikalpyAMzaM tairiSTaissamaharAMzakAn hanyAt / yadguNitAMzasamAsaH phalasadRzo'zArata eveSTAH // 72 // 1 This stanza is omitted in M. 3 This stanza is not found in M. 5 B prottIrNagaNitArNava. * B viMzatya. 4K and B bhAgajAtyabdhipAraga. For Private and Personal Use Only

Page #70
gaNitasArasaGghahaH ekAMzadvAnAM rAzInAM yutAvaMzAhArasyAdhikye satyaMzotpAdaka sUtram-- samahArakAMzakayutihRtayutyaMzo'za ekaTaddhInAm / zeSamitarAMzayutihatamanyAMzo'styevamA caramAt / / 73 / / atroddezakaH / navakadazaikAdazahatarAzInAM navatinavazatIbhaktA / nyUnAzItyaSTazatI saMyogaH ke'zakAH kathaya / / 74 / / 'chedotpattau sUtram-- rUpAMzakarAzInAM rUpAdyAstriguNitA harAH kramazaH / dvidvivyaMzAbhyastAvAdimacaramau phale rUpe // 75 // atroddezakaH / paJcAnAM rAzInAM rUpAMzAnAM yutirbhavedrUpam / SaNNAM saptAnAM vA ke hArAH kathaya gaNitajJa // 76 // viSamasthAnAnAM chedotpattA sUtram-- ekAMzakarAzInAM vyAdyA rUpottarA bhavanti harAH / svAsannaparAbhyastAssarve dalitAH phale rUpe // 77 // ekAzAnAmanekAMzAnAM caikAMze phale chedotpattau sUtramalabdhaharaH prathamasyacchedaH sasvAMzako'yamaparasya / prAk svapareNa hato'ntyaH svAMzenaikAMzake yoge // 78 // ___ atroddezakaH / saptakanavakatritayatrayodazAMzaprayuktarAzInAm / rUpaM pAdaH SaSThaH saMyogAH ke harAH kathaya // 79 // 1 B sadRzavRddhyaMzarAzInAM aMzotpAdakasUtram / For Private and Personal Use Only

Page #71
kalAsavarNavyavahAraH. ekAMzakAnAmekAMze'nekAMze ca phale chedotpattau sUtram seSTo hAro bhaktaH svAMzena niragramAdimAMzaharaH / tadyutihArApteSTaH zeSo'smAditthamitareSAm / / 80 // atroddezakaH / trayANAM rUpakAMzAnAM rAzInAM ke harA vada / phalaM caturthabhAgassyAccaturNAM ca trisaptamam // 81 // aikAMzAnAmanekAMzAnAM cAnekAMze phale chedotpattA sUtramiSTahatA dRSTAMzAH phalAMzasahazo yathA hi tadyogaH / nijaguNahRtaphalahArastahAro bhavati nirdiSTaH / / 82 // atroddezakaH / 'ekakAMzena rAzInAM trayANAM ke harA vada / dvAdazAptA trayoviMzatyazaMkA ca yutirbhavet / / 83 / / . trisaptakanavAMzAnAM trayANAM ke harA vada / / yUnapaJcAzadAptA trisaptatyaMzA yutirbhavet / / 84 / / ekAMzakayo rAzyorekAMze phale chedotpattau sUtramvAJchAhatayutihArazchedaH sa vyekavAJchayApto'nyaH / phalahArahAralabdhe svayogaguNite harau vA staH // 85 !! atroddezakaH / rAzyorekAMzayozchedI kau bhavetAM tayoryutiH / SaDazo dazabhAgo vA brUhi tvaM gaNitArthavit // 16 // 1Stanzas 83 and 84 are omitted in B. For Private and Personal Use Only

Page #72
38 gaNitasArasaGgrahaH __ekAMzakayoranekAMzayozca ekAMze'nekAMze'pi phale chedotpattIprathama sUtrama'iSTaguNAMzo'nyAMzaprayutaH zuddha hRtaH phalAMzena / iSTAptayutiharano haraH parasya tu tadiSTahatiH // 17 // atroddezakaH / rUpAMzakayo rAzyoH ko syAtAM hArakau yutiH pAdaH / paJcAMzo vA dvihatassaptakanavakAMzayozca vada // 8 // dvitIyasUtram-- phalahAratADitAMzaH parAMzasahitaH phalAMzakena hRtaH / syAdekasyacchedaH phalaharaguNito'yamanyasya // 89 // __ atroddezakaH / rAzidvayasya ko hArAvekAMzasyAsya saMyutiH / dvisaptAMzo bhaveddAha SaDaSTAMzasya ca priya // 90 // ardhavyaMzadazAMzakapaJcadazAMzakayutirbhavedrUpam / tyakte paJcadazAMze rUpAMzAvatra kau yojyau // 91 // dalapAdapaJcamAMzakaviMzAnAM bhavati saMyutI rUpam / saptaikAdazakAMzI ko yojyAviha vinA viMzam // 92 // yugmAnyAzrityacchedotpattau sUtram-- yugmapramitAn bhAgAnekaikAMzAn prakalpya phalarAzeH / tebhyaH phalAtmakebhyo dvirAzividhinA harAssAdhyAH // 93 // 1 P and B add as another reading. zuddhaM phalAMzabhakta: svAnyAMzayuto nijeSTaguNitAMzaH / For Private and Personal Use Only

Page #73
39 kalAsavarNavyavahAraH. 39 atroddeshkH| trikapaJcakatrayodazasaptanavaikAdazAMzarAzInAm / ke hArAH phalamekaM paJcAMzo vA caturguNitaH / / 94 / / ekasUtrotpannarUpAMzahAraissUtrAntarotpannarUpAMzahAraizca phale rUpe chedotpattau naSTabhAgAnayane ca sUtram--- vAJchitasUtrajahArA harA bhavantyanyasUtrajaharaghnAH / dRSTAMzaikyonaM phalamabhISTanaSTAMzamAnaM syAt // 95 // atroddezakaH / parahatidalanavidhAnAtrayodaza svaparasaGgaNavidhAnAt / bhAgAcatvAro'taH kati bhAgAssyuH phale rUpe // 96 // . prAksvaparahatavidhAnAtsaptasvAsannaparaguNArdhavidhAnAt / bhAgAstritayazcAtaH kati bhAgAssyuH phale rUpe // 97 / / rUpAMzakA dviSaTkadvAdazaviMzatiharA vinaSTo'tra / paJcamarAzI rUpaM sarvasamAsassa rAziH kaH // 98 // iti bhAgajAtiH / prabhAgabhAgabhAgajAtyossUtramaMzAnAM saGgaNanaM hArANAM ca prabhAgajAtau syAt / guNakAroM'zakarAzehariharo bhAgabhAgajAtividhau // 99 // prabhAgajAtAvuddezakaH / rUpAdhaM tryaMzAdhaM vyaMzArdhA) dalArdhapaJcAMzam / paJcAMzArdhatryaMzaM tRtIyabhAgArdhasaptAMzam // 100 // For Private and Personal Use Only

Page #74
gaNitasArasaGgrahaH daladaladalasaptAMzaM vyaMzavyaMzakadalArdhadalabhAgam / ardhavyaMzavyaMzakapaJcAzaM paJcamAMzadalam / / 101 // krItaM paNasya dattvA kokanadaM kundaketakIkumudam / jinacaraNaM prArcayituM prakSipyaitAn phalaM brUhi // 102 // rUpAdhU tryaMzakArdhAdhaM pAdasaptanavAMzakam / dvitribhAgadvisaptAMzaM dvisaptAMzanavAMzakam // 103 // dattvA paNadvayaM kazcidAnaiSInnUtanaM ghRtam / jinAlayasya dIpArtha zeSaM kiM kathaya priya // 104 // vyaMzAdvipaJcamAMzastRtIyabhAgAt trayAdezaSaDazaH / paJcASTAdazabhAgAt trayodazAMzo'STamAnnavamaH // 105 // navamAccatustrayodazabhAgaH paJcAMzakAt tripAdArdham / saGkSipyAcakSvaitAna prabhAgajAtau zramo'sti yadi // 106 // atraikAvyaktAnayanasUtram-- rUpaM nyasyAvyakte prAgvidhinA yatphalaM bhavettena / bhaktaM paridRSTaphalaM prabhAgajAtau tadajJAtam // 107 / / atroddezakaH / rAzeH kutazcidaSTAMzasyazapAdo'rdhapaJcamaH / SaSThatripAdapaJcAzaH kimavyaktaM phalaM dalam // 108 // anekAvyaktAnayanasUtramkRtvAjJAtAniSTAn phalasadRzI tadyutiryathA bhavati / vibhajeta pRthagvyaktairaviditarAzipramANAni // 109 // For Private and Personal Use Only

Page #75
kalA savarNavyavahAraH. atroddezakaH / rAzeH kutazcidadhai kutazcidaSTAMzaka tripaJcAMzaH / kasmAddizyaMzArdhaM phalamarthaM ke syurajJAtAH // 110 // bhAgabhAga jAtAvuddezakaH / SaTsaptabhAgabhAgastryaSTAMzAMzazcaturnavAMzAMzaH / tricaturthabhAgabhAgaH kiM phalametadyutau brUhi // 111 // dvitryaMzAptaM rUpaM tripAdabhaktaM dvikaM dvayaM cApi / dvitryaMzoddhRtamekaM navakAtsaMzodhya vada zeSam // 112 // iti prabhAgabhAgabhAgaMjAtI / 2 bhAgAnubandhajAtau sUtramharahatarUpeSvaMzAn sakSipa bhAgAnubandhajAtividhau / 'guNayAgrAMzacchedAvaMzayutacchedahArAbhyAm // 113 // rUpabhAgAnuvandha uddezakaH / 'dvitriSaTTASTaniSkANi dvAdazASTaSaDaMzakaiH / paJcASTamaistametAni viMzatezzodhaya priya // 114 // sArdhenaikena paGke sASTAMzairdazabhirhimama / sArdhAbhyAM kuGkumaM dvAbhyAM krItaM yoge kiyadbhavet // 115 // Acharya Shri Kailassagarsuri Gyanmandir 'sASTamASTau SaDaMzAn SaDRdvAdazAMzayutaM dvayam / trayaM pazcASTamopetaM viMzatazzodhaya priya // 116 // Breads guNayedaprAMzaharau sahitAMzaccheda .. * This stanza is not found in P. For Private and Personal Use Only 41 JI dvadet. 4 This stanza is found only in P,

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Shri Mahavir Jain Aradhana Kendra 42 www.kobatirth.org gaNitasArasaGgrahaH. saptASTau navadazamASakAn sapAdAn davA nA jinanilaye cakAra pUjAm / unmIlatkuravakakundajAtimallImAlAbhirgaNaka vadAzu tAn samasya // 117 // bhAgabhAgAnubandha uddezakaH / stvatryaMzapAdasaMyuktaM dalaM pacazako'pi ca / tryaMzasvakIyapaSThArdhasahitasatI kiyat // 118 // tryaMzAdyaMzakasaptamAMzacaramai ssvairanvitAdardhataH puSpANyardhaturIyapaJcanavarmasvIyairyutAtsaptamAt / gandhaM paJcamabhAgato'rdhacaraNatryaMzAMza kairmizritAddhUpaM cArcayituM naro jinavarAnAneSTa kiM tadyutau // 119 // svadalasahitaM pAdaM svatryaMzakena samanvita - dviguNanavamaM svASTAMzatryaMzakArdhavimizritam / navamamapi ca svASTAMzAdyarvapazcimasaMyutaM nijadalayutaM tryaMzaM saMzodhaya tritayAtpriya // 120 // svadalasahita pAdaM sasvapAdaM dazAMza nijadalayutaSaSThaM sasvakatryaMzamardham / caraNamapi sameta svavibhAgaM samasya priya kathaya samagraprajJa bhAgAnubandhe // 121 // Acharya Shri Kailassagarsuri Gyanmandir atrAgrAvyaktAnayanasUtram - labdhAtkalpitabhAgA rUpAnItAnubandhaphalabhaktAH / kramazaH khaNDasamAnAste'jJAtAMzapramANAni // 122 // ' B svacaraNAdyardhAntimaiH. For Private and Personal Use Only

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48 kalAsavarNavyavahAraH. atroddezakaH / kAzritsvakairardhatRtIyapAdairaMzo'paraH paJcacaturnavAMzaiH / anyastripaJcAzanavAMzakArdheyuto yutI rUpamihAzakAH ke // 123 // ko'pyaMzasvArdhapaJcAMzatripAdanavamairyutaH / ardhaM prajAyate zIghraM vadAvyaktapramAM priya // 124 // zeSeSTasthAnAvyaktabhAgAnayanasUtram - labdhAtkalpitabhAgAsavarNitairvyaktarAzibhirbhaktAH / kramazo rUpavihInAraveSTapadeSvaviditAMzAssyuH / / 125 // iti bhAgAnubandhajAtiH / atha bhAgApavAhajAtau sUtram-- harahatarUpeSvaMzAnapanaya bhAgApavAhajAtividhau / 'guNayAgrAMzacchedAvaMzonacchedahArAbhyAm // 126 // rUpabhAgApavAha uddezakaH / dhyaSTacaturdazakarSAH pAdArdhadvAdazAMzaSaSThonAH / savanAya narairdattAtIrthakatAM tadyutau kiM syAt // 127 / / triguNapAdadalatrihatASTamaivirahitA nava sapta nava kramAt / . B guNa yedagrAMzaharI rahitAMzacchedahArAbhyAm / 6-A For Private and Personal Use Only

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44 gaNitasArasaGgrahaH. priya vizodhya caturguNaSaTutaH kathaya zeSadhanapramiti drutam // 128 // bhAgabhAgApavAha uddezakaH / dviguNitapaJcamanavamatryaMzASTAMzadvisaptamAn kramazaH / svaSaDaMzapAdacaraNavyaMzASTamavarjitAn samasya vada // 129 // SaTsaptAMzasvaSaSThASTamanavamadazAMzauviyuktaH paNasya : syAtpazcadvAdazAMzasvakacaraNatRtIyAMzapaJcAMzakonaH / svahinyaMzadvipaJcAMzakadalaviyutaH paJcaSaDbhAgarAzidivyaMzo'nyasvapaJcASTamaparirahitastatsamAse phalaM kim // 130 / / ardhaM vyaSTamabhAgapAdanavamaisvIyavihInaM punaH khairaSTAMzakasaptamAMzacaraNairUnaM tRtIyAMzakam / adhyardhAtparizodhya saptamamapi svASTAMzaSaSThonitaM zeSaM brUhi parizramo'sti yadi te bhAgApavAhe sakhe // 131 // atrAgrAvyaktabhAgAnayanasUtramlabdhAtkalpitabhAgA rUpAnItApavAhaphalabhaktAH / kramazaH khaNDasamAnAste'jJAtAMzapramANAni // 132 // atroddezakaH / kazcitsvakai zvaraNapaJcamabhAgaSaSThaiH ko'pyaMzako dalaSaDaMzakapaJcamAMzaiH / hIno'paro dviguNapazcamapAdaSaSThaiH tatsaMyutirdalamihAviditAMzakAH ke // 133 // For Private and Personal Use Only

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kalAsavarNavyavahAraH. ko'pyazasvArdhaSaDbhAgapaJcamASTamasaptamaiH / vihIno 'jAyate SaSThassa ko'zo gaNitArthavit // 134 // zeSeSTasthAnAvyaktabhAgAnayanasUtram... labdhAtkalpitabhAgAsavarNitaiyaktarAzibhirbhaktAH / rUpAtpTathagapanItAsveSTapadeSvaviditAMzAssyuH // 135 // iti bhaagaapvaahjaatiH| bhAgAnubandhabhAgApavAhanAtyossarvAvyaktabhAgAnayanasUtram-- tyaktvaikaM kheSTAMzAn prakalpayedaviditeSu sarveSu / aitaistaM punaraMzaM prAguktairAnayetsUtraiH / / 136 // atroddezakaH / kazcidazo'zakaiH kaizcitpaJcabhissvairyuto dalam / viyukto vA bhavetpAdastAnaMzAn kathaya priya // 137 / / bhAgamAtRjAtau sUtram-- bhAgAdimajAtInAM svasvavidhirbhAgamAtRjAtau syAt / sA SadizatibhedA rUpaM chedo'cchido rAzeH // 138 // atroddeshkH| dhyaMzaH pAdo'rdhArgha paJcamaSaSThastripAdaha tamekam / . pazcArdhahataM rUpaM saSaSThamekaM sapaJcamaM rUpam / / 139 // svIyatRtIyayugdalamato nijaSaSThayuto dvisaptamo hInanavAMzamekamapanItadazAMzakarUpamaSTamaH / 'P, K and B tadyuti: for jAyate. For Private and Personal Use Only

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46 gaNitasArasaGgrahaH. khena navAMzakena rahitazcaraNassvaka paJcamojjhito . brUhi samasya tAn priya kalAsamakotpalamAlikAvidhau // 140 / iti bhAgamAtRjAtiH / iti sArasaGghahe gaNitazAstre mahAvIrAcAryasya kRtau kalAsavoM nAma dvitIyavyavahArassamAptaH / / For Private and Personal Use Only

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tR tI yaH prakIrNakavyavahAraH. praNutAnantaguNaughaM praNipatya jinezvaraM mahAvIram / praNatajagatrayavaradaM prakIrNaka gaNitamabhidhAsye // 1 // 'vidhvastadurnayadhvAntaH siddhaH syAdvAdazAsanaH / vidyAnando jino jIyAdvAdIndro munipuGgavaH // 2 // itaH paraM prakIrNakaM tRtIyavyavahAramudAhariSyAmaH-- bhAgazzeSo mUlakaM zeSamUlaM syAtAM jAtI dve dviraprAMzamUle / bhAgAbhyAso'noM'zavargo'tha mUla mizraM tasmAdinnadRzyaM dazAmUH // 3 // tatra bhAgajAtizeSajAtyorasUtram --- bhAgonarUpabhaktaM dRzyaM phalamatra bhAgajAtividhau / aMzonitarUpAhatihatamagraM zeSajAtividhau / / 4 / / bhAgajAtAvuddezakaH / dRSTo'STamaM pRthivyAM sambhasya tryaMzako mayA toye / pAdAMzaH zaivAle kaH stambhaH sapta hastAH rakhe // 5 // SaDDAgaH pATalISu bhramaravaratatestatribhAgaH kadambe pAdazrUtadrumeSu pradalitakusume campake paJcamAMzaH / protphullAmbhojaSaNDe ravikaradalite triMzadazo'bhireme tatraiko mattabhRGgo bhramati nabhasi kA tasya bRndasya saGkhyA // 6 // Band M omit this stanza. For Private and Personal Use Only

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Shri Mahavir Jain Aradhana Kendra 48 www.kobatirth.org gaNitasArasaGgrahaH. AdAyAmbhoruhANi stutizatamukharaH zrAvakastIrtha kRdbhayaH pUjAM cakre caturbhyo vRSabhajinavarAta tryaMzameSAmamuSya / tryaMzaM turyaM SaMDezaM tadanu sumataye tannavadvAdazAMzau zeSebhyo dvidvipadmaM pramuditamanasAdatta kiM tatpramANam // 7 // svavazIkRtendriyANAM dUrIkRtaviSakaSAyadoSANAm / zIlaguNAbharaNAnAM dayAGganAliGgitAGgAnAm // 8 // sAdhUnAM sadvRndaM sandRSTaM dvAdazo'sya tarkajJaH / svatryaMzavarjito'yaM saiddhAntazchAndasastayozzeSaH // 9 SaDno'yaM dharmakathI sa eva naimittikaH svapAdonaH / vAdI tayorvizeSaH SaGguNito'yaM tapasvI syAt // 10 // girizikharataTe mayopadRSTA yatipatayo navasaGguNASTasaGkhayAH / Acharya Shri Kailassagarsuri Gyanmandir ravikara paritApitojjvalAGgAH kathaya munIndrasamUhamAzu me tvam // 11 // phalabhAranamrakamre zAlikSetre zukAssamupaviSTAH / sahasotthitA manuSyaiH sarve santrAsitAssantaH // 12 // teSAmarthaM prAcImAya prati jagAma SaDbhAgaH / pUrvAyazeSaH svadalonaH svArdhavarjito yAmIm // 13 // yAmyAyazeSaH sa nairRti svadvipaJcabhAgonaH / yAmInairRtyazakaparizeSo vAruNImAzAm // 14 // nairRtya paravizeSo vAyavyAM sasvakatritaptAMzaH / vAyavyaparavizeSo yutasvasaptASTamaH saumIm // 15 // For Private and Personal Use Only

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prakIrNakavyavahAraH. vAyavyuttarayoyutiraizAnI svatribhAgayugahInA / dazaguNitASTAviMzatiravaziSTA vyoni kati kIrAH // 16 // kAcidvasantamAse prasUnaphalagucchabhAranamrodyAne / kusumAsavarasaraJjitazukakokilamadhupamadhuranisvananicite // 17 // himakaradhavale pRthule saudhatale sAndrarundramadutalpe / phaNiphaNanitambabimbA kanadamalAbharaNazobhAGgI / / 18 // pAThInajaTharanayanA ktthinstnhaarnmrtnumdhyaa| saha nijapatinA yuvatI rAtrau prItyAnuramamANA // 19 // praNayakalahe samutthe muktAmayakaNThikA tadabalAyAH / chinnAvanau nipatitA tatrayazazceTikAM prApat // 20 // SaDbhAgaH zayyAyAmanantarAnantarArdhamitibhAgAH / / SaTsaGkhyAnAstasyAH sarve sarvatra sampatitAH // 21 // ekAgraSaSTizatayutasahasramuktAphalAni dRSTAni / tanmauktikapramANaM prakIrNakaM vetsi cet kathaya / / 22 / / 'sphuradindranIlavarNa SaTpadabRndaM praphullitodyAne / dRSTaM tasyASTAMzo'zoke kuTaje SaDaMzako lInaH // 23 // . kuTajAzokavizeSaH SaDguNito vipulapATalISaNDe / pATalyazokazeSaH svanavAMzono vizAlasAlavane // 24 // pATalyazokazeSo yutassvasaptAMzakena madhukavane / paJcAMzassandRSTo vakuleSUtphullamukuleSu // 25 // tilakeSu kuravakeSu ca saraleSAneSu padmaSaNDeSu / vanakarikapolamUlapi santasthe sa evAMzaH // 26 // 1 M reads sphu ratendra For Private and Personal Use Only

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gaNitasArasaGgrahaH. kiJjalakapuJjapiJjarakaJjavane madhukarAstrayastriMzat / dRSTA bhramarakulasya pramANamAcakSva gaNaka tvam // 27 // goyUthasya kSitibhRti dalaM taddalaM zailamUle SaT tasyAMzA vipulavipine pUrvapUrvArdhamAnAH / santiSThante nagaranikaTe dhenavo dRzyamAnA dvAtriMzat tvaM vada mama sarave gokulasya pramANam // 28 // iti bhAgajAtyuddezakaH // zeSajAtAvuddezakaH / SaDbhAgamAmrarAze rAjA zeSasya paJcamaM raajnyii| turyatryaMzadalAni trayo'grahISuH kumAravarAH // 29 // zeSANi trINi cUtAni kaniSTho dArako grahIt / tasya pramANamAcakSva prakIrNakavizArada // 30 // carati girau saptAMzaH kariNAM SaSThAdimArdhapAzcAtyAH / pratizeSAMzA vipine SaDdRSTAssarAsi kati te syuH // 31 // koSThasya lebhe navamAMzamekaH pare 'STa bhAgAdidalAntimAMzAna / zeSasya zeSasya punaH purANA dRSTA mayA dvAdaza tatpramA kA // 32 / / iti zeSajAtyuddezakaH // atha mUlajAtau sUtramamUlA(gre chindyAdaMzonaikena yuktamUlakateH / dRzyasya padaM sapadaM vargitamiha mUlajAtau svam // 33 // For Private and Personal Use Only

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prakIrNakavyavahAraH. 51 atroddezakaH / dRSTo'TavyAmuSTrayUthasya pAdo mUle ca dve zailasAnau niviSTe / uSTrAstrighnAH paJca nadyAstu tIre kiM tasya syAduSTrakasya pramANam // 34 // zrutvA varSAbhramAlApaThahapaTuravaM zailazRGgoruraGge nATyaM cakre pramodapramaditazikhinAM SoDazAMzo'STamazca / vyaMzaH zeSasya SaSTho varavakulavane paJca mUlAni tasthuH punnAge paJca dRSTA bhaNa gaNaka gaNaM barhiNAM saGgaNayya / / 35 // carati kamalapaNDe sArasAnAM catoM navamacaraNabhAgI sapta mUlAni cAdrI / vikacavakulamadhye saptaninnASTamAnAH kati kathaya sarave tvaM pakSiNo dakSa sAkSAt // 36 // na bhAgaH kapibRndasya trINi mUlAni parvate / catvAriMzadvane dRSTA vAnarAstadgaNaH kiyAn // 37 // kalakaNThAnAma) sahakArataroH praphullazAkhAyAm / tilake'STAdaza tasthunI mUlaM kathaya pikanikaram // 38 // haMsakulasya dalaM vakule'sthAt paJca padAni tamAlakujAgre / FB reads kiM syAtteSAM kuJjarANAM pramANam / 1 Breads hasti. * Brrad nAgA:. For Private and Personal Use Only

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Shri Mahavir Jain Aradhana Kendra 52 www.kobatirth.org gaNitasArasaGgrahaH. atra na kiJcidapi pratidRSTaM tatpramitiM kathaya priya zIghram / / 39 / / iti mUlajAtiH // Acharya Shri Kailassagarsuri Gyanmandir atha zeSamUlajAtI sUtram - padadalavargayutAgrAnmUlaM saprAkpadArdhamasya kRtiH / dRzye mUlaM prApte phalamiha bhAgaM tu bhAgajAtividhiH // 40 // atroddezakaH // gajayUthasya tryaMza'zzeSapadaM ca trisaGguNaM sAnau / sarasi trihastinIbhirnAgo dRSTaH katIha gajAH // 41 // nirjantukapradeze nAnAdrumaSaNDamaNDitodyAne / AsInAnAM yaminAM mUlaM tarumUlayogayutam // 42 // zeSasya dazamabhAgo mUlaM navamo'tha mUlamaSTAMzaH / mUlaM saptamamUlaM SaSTho mUlaM ca paJcamo mUlaM // 43 // ete bhAgAH kAvyapravacanadharmapramANanayavidyAH / vAdacchandojyautiSamantrAlaGkArazabdajJAH || 44 // dvAdazatapaHprabhAvA dvAdaza bhedAGgazAstra kuzaladhiyaH / dvAdaza munayo dRSTAH kiyatI municandra yatisamitiH // 45 // mUlAni pazca caraNena yutAni sAnau zeSasya paJcanavamaH kariNAM nagAgre / mUlAni paJca sarasIjavane ramante nadyAstaTe SaDiha te dviradAH kiyantaH // 46 // iti zeSamUlajAtiH // * B reads zeSasya pada triguNaM. For Private and Personal Use Only

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58 prakIrNakavyavahAraH. atha dviragrazeSamUlajAtau sUtrammUlaM dRzyaM ca bhajedaMzakaparihANarUpaghAtena / pUrvAgramagrarAzau kSipedatazzeSamUlavidhiH // 47 // atroddezakaH / madhukara eko dRSTaH rave pajhe zeSapaJcamacaturtho / zeSavyaMzo mUlaM dvA'vAne te kiyantaH syuH // 48 // siMhAzcatvAro'drau pratizeSaSaDaMzakAdimArdhAntAH / mUle catvAro'pi ca vipine dRSTAH kiyantaste // 49 // taruNahariNIyugmaM dRSTaM dvisaGgANita vane kudharanikaTe zeSAH paJcAMzakAdidalAntimAH / vipulakalamakSetre tAsAM padaM tribhirAhataM kamalasarasItIre tasthurdazaiva gaNaH kiyAn // 50 // iti dviragrazeSamUlajAtiH // athAMzamUlajAtau sUtrambhAgaguNe mUlAgre nyasya padaprAptadRzyakaraNena / yallabdhaM bhAgahRtaM dhanaM bhavedaMzamUlavidhau // 51 // anyadapi mUtramdRzyAdazakabhaktAccaturguNAnmUlakRtiyutAnmUlam / sapadaM dalitaM vargitamaMzAbhyastaM bhavet sAram / / 52 // 1 B reads dvau cAne, For Private and Personal Use Only

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54 gaNitasArasaGgrahaH. atroddezakaH / padmanAlatribhAgasya jale mUlASTakaM sthitam / SoDazAGgulamAkAze jalanAlodayaM vada // 53 // dvitribhAgasya yanmalaM navaghnaM hastinAM punaH / zeSatripaJcamAMzasya mUlaM panissamAhatam // 54 // vigaladdAna'dhArArdragaNDamaNDaladantinaH / caturvizatirAdRSTA mayATavyAM kati dvipAH // 55 // kroDaughArdhacatuHpadAni vipinaM zArdUlavikrIDitaM prApuzzeSadazAMzamUlayugalaM zailaM catustADitam / zaSArdhasya padaM trivargaguNitaM vapraM varAhA vane dRSTAssaptaguNASTakapramita yasteSAM pramANaM vada // 56 // ityaMzamUlajAtiH // atha bhAgasaMvargajAtau sUtram khAMzAptaharAdUnAccaturguNAgreNa tadvareNa hatAt / mUlaM yojyaM tyAjyaM tacchede taddalaM vittam // 57 / / atroddezakaH / aSTamaM SoDazAMzaghnaM zAlirAzeH kRSIvalaH / caturviMzativAhAMzca lebhe rAziH kiyAn vada / / 58 // 1 B reads vArA. 2 After this stanza all the MSS. havo the following stanza; but it is simply a paraphrase of stanza No. 57: anyacca caturhatadRSTenonAdbhAgAhatyaMzahRtahArAt / tacchedena hatAnmalaM yojyaM tyAjyaM tacchede tadardha vittam // For Private and Personal Use Only

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55 prakIrNakavyavahAraH. zirivanAM SoDazabhAgaH svaguNazrUte tamAlapaNDe 'sthAt / zeSanavAMzaH svahatazcaturagradazApi kati te syuH // 59 // jale triMzadaMzAhato dvAdazAMzaH sthitazzeSaviMzo hataH SoDazena / trinighnena pate karA viMzatiH rakhe sarave sambhadairghyasya mAnaM vada tvam // 6 // iti bhAgasaMvargajAtiH // athonAdhikAMzavargajAnau sUtram----- svAMzakabhaktaharArdhaM nyUnayugadhikonitaM ca tadvargAt / vyUnAdhikavargAgrAnmUlaM svarNaM phalaM padeM zahRtam // 61 // 'hInAlApa udAharaNam / mahiSINAmaSTAMzo vyako vargIkRto vane ramate / / paJcadazAdrau dRSTAstRNaM carantyaH kiyantyastAH // 62 / / anekapAnAM dazamo dvivarjitaH svasaGgaNaH krIDati sallakIvane / caranti SaDurgamitA gajA girI kiyanta ete'tra bhavanti dantinaH / / 63 / / ____ "adhikAlApa udAharaNam / jambUrakSe paJcadazAMzo dvikayuktaH khenAbhyastaH kekikalasya dvikRtimAH / IM omits hIna. M omits this as well as the following stanza. For Private and Personal Use Only

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56 gaNitasArasaGgrahaH. paJcApyanye mattamayUrAssahakAre raMramyante mitra vadaiSAM parimANam // 64 // ityUnAdhikAMzavargajAtiH / / atha mUlamizrajAtau sUtrammizrakRtirUnayuktA vyadhikA ca dviguNamizrasambhaktA / vargIkRtA phalaM syAtkaraNamidaM mUlAma zravidhau // 65 // hInAlApa uddezakaH / mUlaM kapotabandasya dvAdazonasya cApi yat / tayoryoga' kapotASSaD dRSTAstannikaraH kiyAn // 66 // pArAvatIyasaGgha caturghanone'pi tatra yanmUlam / taddyayogaH SoDaza tadvande kati vihaGgAH syuH / / 67 // ___ adhikAlApa uddezakaH / rAjahaMsanikarasya yatpadaM. sASTaSaSTisahitasya caitayoH / saMyutirdikavihInaSaTkRtistadgaNe kati marAlakA vada // 68 // iti mUlamizrajAtiH // atha bhinnadRzyajAtau sUtram-- dRzyAMzone rUpe bhAgAbhyAsena bhAjite tatra / yallabdhaM tatsAraM prajAyate bhinnadRzyavidhau // 69 // - | B reads yoga: For Private and Personal Use Only

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prakIrNakavyavahAraH atroddezakaH / sikatAyAmaSTAMzassandRSTo'STAdazAMzasaGgaNitaH / stambhasyA 'dRSTaM stambhAyAmaH kiyAn kathaya // 70 // dvibhaktanavamAMzakamahatasaptaviMzAMzakaH pramodamavatiSThate karikulasya pRthvItale / vinIlajaladArUtirviharati tribhAgI nage vada tvamadhunA sarave karikulapramANaM mama / / 71 // sAdhUtkRtornavasati SoDazAMzakastribhAjitaH khakaguNito vanAntare / pAdo girau mama kathayAzu tanmiAMta prottIrNavAn jaladhisamaM prakIrNakam / / 02 // iti bhinnadRzyajAtiH // iti sArasabahe gaNitazAstre mahAvIrAcAryasya kRtI prakIrNako nAma tRtIyavyavahAraH samAptaH // B, M and K read gagane. For Private and Personal Use Only

Page #92
caturthaH trairAzikavyavahAraH / trilokabandhave tasmai kevalajJAnabhAnave / namaH zrIvardhamAnAya nirdhUtArivalakarmaNe // 1 // itaH paraM vairAzikaM caturthavyavahAramadAhariSyAmaH / tatra karaNasUtraM yathA trairAzike'tra sAraM phalamicchAsaGgaNaM pramANAptam / icchApramayossAmye viparIteyaM kriyA vyaste // 2 // pUrvArdhoddezakaH / divasaistribhissapAdairyojanaSaTraM caturthabhAgonama / gacchati yaH puruSo'sau dinayutavarSeNa kiM kathaya // 3 // vyardhASTAmirahobhiH krozASTAMzaM 'svapaJcamaM yAti / paGgassapaJcabhAgairvastribhiratra kiM brUhi // 4 // aGgalacaturthabhAgaM prayAti kITo dinASTabhAgena / meromUlAcchikharaM katibhirahobhissamAmoti / / 5 / / kArSApaNaM sapAda nirvizati tribhirahobhirardhayutaiH / yo nA purANazatakaM sapaNaM kAlena kenAsau // 6 // 'kRSNAgarusatvaNDaM dvAdazahastAyataM trivistAram / kSayametyaGgalamahnaH kSayakAlaH ko'sya vRttasya / / 7 / / svarNairdazamissA(droNADhakakuDaba mizritaH krItaH / vararAjamASavAhaH kiM hemazatena sAdhaiMna / / < || IP, K and M read sa for sva. * B reads satkRSNAgarukhaNvaM. For Private and Personal Use Only

Page #93
59 trairAzikavyavahAraH sAstribhiH purANaiH kuGkumapalamaSTabhAgasaMyuktam / saMprApyaM yatra syAt purANazatakena kiM tatra // 9 // sArdhAkasaptapalaizcaturdazAonitAH paNA 'labdhAH / dvAtriMzadAkapalaissapaJcamaiH kiM sarave brUhi // 10 // kArSApaNaizcaturbhiH paJcAMzayutaiH palAni rajatasya / SoDaza sArdhAni naro labhate kiM karSaniyutena // 11 // karpUrasyASTapalaivyaMzonai tra paJca dInArAn / bhAgAMzakalAyuktAn labhate kiM palasahasreNa // 12 // sArdhestribhiH paNairiha ghRtasya palapaJcakaM sapaJcAMzam / krINAti yo naro'yaM kiM sASTamakarSazatakena // 13 / / sArdheH paJcapurANaiH SoDaza sadalAni vastrayugalAni / labdhAni saikaSaSTayA karSANAM kiM sarave kathaya // 14 // vApI samacaturazrA slilviyuktaasstthstghnmaanaa| zailastasyAstIre samutthitAzzakharatastasya / / 15 / / vRttAGgulaviSambhA jaladhArA sphaTikanirmalA patitA / vApyantarajalapUrNA nagocchRitiH kA ca jalasaGkhyA // 16 // 'muddroNayugaM navAjyakuDavAn SaT taNDuladroNakAnaSTau vastrayugAni vatsasahitA gASaT suvarNatrayam / 1M and B read labhyAH , ___ * B reads samutthitA zi. * Band K read the following for this stanza : dugdhadroNayugaM navAjyakuDabAn SaT zarkarAdroNakAnaSToM cocaphalAni sAndradadhikhAryaSSaT purANatrayam / zrIkhaNDaM dadatA nRpeNa savanArtha SaDjinAgArake SaTtriMzacizateSu minna vada me taddattadugdhAdikam // 7-A For Private and Personal Use Only

Page #94
60 gaNitasArasaGgrahaH . sakrAntau dadatA narAdhipatinA SaDbhyo dvijebhyassakhe SaTtriMzatrizatebhya Azu vada kiM taddattamudgAdikam // 17 // iti trairAzikaH // vyastatrairAzike turIyapAdasyoddezakaH / kalyANakanakanavateH kiyanti navavarNakAni kanakAni / sASTAMzakadazavarNakasaguJjahemnAM zatasyApi // 18 // vyAsena daidhyeNa ca SaTkarANAM cInAmbarANAM trizatAni tAni / tripaJcahastAni kiyanti santi vyastAnupAtakramAvidvada tvam // 19 // iti vyastatrairAzikaH // vyastapaJcarAzika uddezakaH / paJcanavahastavistRtadairdhyAyAM cInavastrasaptatyAm / dvitrikaravyAsAyati tacchRtavastrANi kati kathaya // 20 // vyastasaptarAzika uddezakaH / vyAsAyAmodayato bahumANikye caturnavASTakare / dviSaDekahastamitayaH pratimAH kati kathaya tIrthakRtAm / / 21 // vyastanavarAzika uddeshkH| vistAradaiyodayataH karasya SaTtriMzadaSTapramitA navArdhA / zilA tayA tu dviSaDekamAnAstAH paJcakAryAH kati caityayogyAH / / 22 / iti vyastapazcasaptanavarAzikAH / / For Private and Personal Use Only

Page #95
trairAzikavyavahAraH gatinivRttau sUtram-- nijanijakAloddhRtayorgamananivRttyorvizeSaNAjAtAm / dinazuddhagatiM nyasya trairAzikavidhimataH kuryAt // 23 // atroddezakaH / krozasya paJcabhAgaM mauryAti dina trisaptabhAgena / 'vArthI vAtAvidyA pratyeti krozanavamAMzam // 24 // kAlena kena gacchet tripaJcabhAgonayojanazataM saa| saGkhyAbdhisamuttaraNe bAhubalistvaM samAcakSva // 25 // sapAdahema tridinaissapaJcamainaro'rjayan vyeti suvarNaturyakam / nijASTamaM paJcadinaidalonitaiH sa kena kAlena labheta saptatim // 26 // gandhebho madalubdhaSaTpadapadaprodbhinnagaNDasthalaH sArdhaM yojanapaJcamaM vanati yaSabhirdalonaidinaiH / pratyAyAti dinaistribhizca sadalaiH krozadvipaJcAMzakaM brUhi krozadalonayojanazataM kAlena kenAmuyAt // 27 // vApI payaHprapUrNA dazadaNDasamucchritAbjamiha jAtam / agalayugalaM sadalaM pravardhate sArdhadivasena // 28 // nissarati yantrato'mbhaH sArdhenAhAGgale saviMze de| zuSyati dinena salilaM sapaJcamAGgulakaminakiraNaiH // 29 // kUrmo nAlamadhastAt sapAdapaJcAGgalAni cAkRSati / sArdhestridinaiH padmaM toyasamaM kena kAlena // 30 // 'Band K read tasminkAle vAdhauM. For Private and Personal Use Only

Page #96
gaNitasArasaGgahaH dvAtriMzaddhastadIrghaH pravizati vivare paJcabhissaptamArdheH kRSNAhIndro dinasyAsuravapurajitaH sArdhasaptAGgulAni / pAdenAho'Ggale dve tricaraNasahite vardhate tasya pucchaM randhra kAlena kena pravizati gaNakottaMsa me brUhi so'yam // 31 // iti gatinivRttiH // paJcasaptanavarAzikeSu karaNasUtram'lAbhaM nItvAnyonyaM vibhajet pRthupatimalpayA pkyaa| guNayitvA jIvAnAM krayavikrayayostu tAneva // 32 // atroddezakaH / dvitricatuzzatayoge pazcAzatSaSTisaptatipurANAH / lAbhArthinA prayuktA dazamAseSvasya kA vRddhiH // 33 // hemnAM sArdhAzIte savyaMzena raddhiradhyardhA / satricaturthanavatyAH kiyatI pAdonaSaNmAsaiH // 34 // SoDazavarNakakAJcanazatena yo ralaviMzatiM labhate / dazavarNasuvarNAnAmaSTAzItidvizatyA kim // 35 // 1 reuds as variations the following: . prakArAntareNa sUtram saGkramya phalaM chindyAllaghupaGktyAnekarAzikA paktim / svaguNAmazvAdInAM krayavikrayayostu tAneva // anyadapi sUtramsaGakramya phalaM chindyAt pRthupaGktyabhyAsamarUpayA paGkatyA / azvAdInAM krayavikrayayorazvAdikAMzca saGkramya // B gives only the latter of these stan zag with the following variation in the second uarter: pRthupaGktyabhyAsamalpapatyAhatyA. For Private and Personal Use Only

Page #97
trairAzikavyavahAraH godhUmAnAM mAnIva nayatA yojanatrayaM lbdhaaH| SaSTiH paNAH saghAhaM kumbhaM dazayojanAni kati // 36 // bhANDapratibhANDasyoddezakaH / kastUrIkarSatrayamupalabhate dazabhiraSTabhiH kanakaiH / karSadvayakarpUraM mRganAbhitrizatakarSakaiH kati nA // 37 // panasAni SaSTimaSTabhirupalabhate'zItimAtuluGgAni / dazabhirmASairnavazatapanasaiH kati mAtuluGgAni // 38 // jIvakrayavikrayayoruddezakaH / SoDazavarSAsturagA viMzatirarhanti niyutakanakAni / dazavarSasaptisaptatiriha kati gaNakAgraNIH kathaya // 39 // svarNatrizatI mUlyaM dazavarSANAM navAGganAnAM syAt / SaTtriMzannArINAM SoDazasaMvatsarANAM kim // 40 // SaTkazatayuktanavaterdazamAsairvRddhiratra kA tasyAH / kaH kAlaH kiM vittaM viditAbhyAM bhaNa gaNakamukhamukura / / 41 // saptarAzika uddeshkH| tricaturvyAsAyAmau zrIkhaNDAvahato'STahemAni / SaNNavavistRtidairdhyA hastena caturdazAtra kati / / 42 // iti saptarAzikaH // BaddhnA at the end K, M and B read hemakarSAH for nA. For Private and Personal Use Only

Page #98
Shri Mahavir Jain Aradhana Kendra 64. www.kobatirth.org gaNitasArasaGgrahaH navarAzika uddezakaH / paJcASTatrivyAsa dairyodayAmbho dhatte vApI zAlinI vAhaSaTkam / Acharya Shri Kailassagarsuri Gyanmandir saptavyAsA hastataH SaSTidairvyAH pAtsadhoH kiM navAcakSva vidvan // 43 // iti sArasaGghahe gaNitazAstre mahAvIrAcAryasya kRtau trairAziko nAma caturthavyavahAraH. 1 The following stanza is found in K and B in addition to stanza No. 48. iSTAzItivyA sadaiyannatAmbho dhatte vApI zAlinI sArdhavAhau / hastAdaSTAyAmakAH SoDazAMcchrAH SaTkavyAsAH kiM catasrA vada tvam // For Private and Personal Use Only

Page #99
pazcamaH mizrakavyavahAraH. prAptAnantacatuSTayAn bhagavatastIrthasya kartRn jinAn siddhAn zuddhaguNAMstrilokamahitAnAcAryavaryAnapi / siddhAntArNavapAragAn bhavabhRtAM netRnupAdhyAyakAn sAdhUna sarvaguNAkarAn hitakarAn vandAmahe zreyase // 1 // itaH paraM mizragaNitaM nAma paJcamavyavahAramudAhariSyAmaH / tadyathAsakramaNasaMjJAyA viSamasaGkramaNasaMjJAyAzca sUtram yutiviyutidalanakaraNaM saGkramaNaM chedalabdhayo rAzyoH / sakramaNaM viSamamidaM prAhugaNitArNavAntagatAH // 2 // atroddezakaH / dvAdazasaGkhyArAzeAbhyAM saGkramaNamatra kiM bhavati / tasmAdrAzerbhaktaM viSamaM vA kiM tu saGkramaNam // 3 // -- pazcarAzikavidhiH // pazcarAzikavarUpavRddhyAnayanasUtramicchArAziH khasya hi kAlena guNaH pramANaphalaguNitaH / kAlapramANabhakto bhavati tadicchAphalaM gaNite // 4 // atroddeshkH| trikapazcakaSaTUzate paJcAzatSaSTisaptatipurANAH / lAmArthataH prayuktAH kA vRddhirmAsaSaTUsya // 5 // For Private and Personal Use Only

Page #100
gaNitasArasaGghahaH vyardhASTakazatayuktAstriMzatkArSApaNAH paNAzcASTau // mAsASTakena jAtA dalahInenaiva kA vRddhiH // 6 // SaSTyA sadvidRSTA paJca purANAH paNatrayavimizrAH / mAsadvayena labdhA zataddhiH kA tu varSasya // 7 // sArdhazatakaprayoge sArdhakamAsena paJcadaza lAbhaH / mAsadazakena labdhA zatatrayasyAtra kA vRddhiH // 8 // sASTazatakASTayoge triSaSTikArSApaNA vizA dattAH / saptAnAM mAsAnAM paJcamabhAgAnvitAnAM kim // 9 // mUlAnayanasUtram-- mUlaM svakAlaguNitaM svaphalena vibhAjitaM tdicchaayaaH| kAlena. majellabdhaM phalena guNitaM tAdicchA syAt // 10 // . atroddezakaH / paJcAdhakazatayoge paJca purANAndalonamAsI hau| vRddhiM labhate kazcit kiM mUlaM tasya me kathaya // 11 // saptatyAH sArdhamAsena phalaM paJcAmeva ca / vyardhASTamAse mUlaM kiM phalayossArdhayoIyoH // 12 // trikapaJcakaSaTUzate yathA navASTAdazAtha paJcakRtiH / paJcAMzakena mizrA SaTsu hi mAseSu kAni mUlAni / / 13 // kAlAnayanasUtramkAlaguNitapramANaM svaphalecchAbhyAM hRtaM tataH kRtvA / tadihecchAphalaguNitaM labdhaM kAlaM budhAH prAhuH // 14 // For Private and Personal Use Only

Page #101
mizrakavyavahAraH atroddezakaH / saptArdhazatakayoge vRddhistvaSTAyaviMzatirazItyA / kAlena kena labdhA kAlaM vigaNayya kathaya sarave // 15 // viMzatiSaTzatakasya prayogataH saptaguNaSaSTiH / vRddhirapi caturazItiH kathaya sarave kAlamAzu tvam // 16 // SaTUzatena hi yuktAH SaNNavatirdRddhiratra sandRSTA / saptottarapaJcAzat tripaJcamAgazca kaH kAlaH // 17 // bhANDapratibhANDasUtrambhANDasvamUlyabhaktaM pratibhANDaM bhANDamUlyasaGgaNitam / khecchAbhANDAbhyastaM bhANDapratibhANDamUlyaphalametat // 18 // . atroddezakaH / kItAnyaSTau zuNThyAH palAni SaDbhiH paNaiH sapAdAMzaiH / pippalyAH palapaJcakamatha pAdonaiH paNairnavabhiH // 19 // zuNThyAH palaizca 'kenacidazItibhiH kati palAni pipplyaaH| krItAni vicintya tvaM gaNitavidAcakSva me zIghram // 20 // .' iti mizrakavyavahAre paJcarAzikavidhiH samAptaH // vRddhividhAnam // itaH paraM mizrakavyavahAre vRddhividhAnaM vyAkhyAsyAmaH / mUladdhimizravibhAgAnayanasUtram-- rUpeNa kAlavRddhyA yutena mizrasya bhAgahAravidhim / kRtvA labdhaM mUlyaM vRddhirmUlonamizradhanam // 21 // 1 Both M and B have the arroneous reading kazcit tvazItibhiH saca palAni pippasyAH. For Private and Personal Use Only

Page #102
68 gaNitasArasAhaH bhtroddeshkH| pacakazataprayoge dvAdazamAsairdhanaM prayukre cet / sASTA catvAriMzanmizraM tanmUladdhI ke // 22 // punarapi mUlasaddhimizrAvibhAgasUtramicchAkAlaphalapnaM vakAlamUlena bhAjitaM saikam / sammizrasya vimaktaM labdhaM mUlaM vijAnIyAt // 23 // atroddezakaH / sArthadvizatakayoge mAsacatuSeNa kimapi dhanamekaH / datvA mizraM labhate kiM mUlyaM syAt trayastriMzat / / 24 // kAlavRddhimizravibhAgAnayanasUtrammUlaM svakAlaguNitaM vaphalecchAbhyAM hRtaM tataH kRtvaa| saikaM tenAptasya ca mizrasya phalaM hi vRddhiH syAt // 15 // atroddezakaH / paJcakazataprayoge phalArthinA yojitaiva dhanaSaSTiH / kAlaH khavRddhisahito viMzatiratrApi kaH kAlaH // 26 // ardhatrikasaptatyAH sArdhAyA yogayojitaM mUlam / pazcottarasaptazataM mizramazItiH svakAlavRddhyohi // 27 // vyardhacatuSAzItyA yuktA mAsadvayena sAdhaina / mUlaM catuzzataM SaTatriMzanmizraM hi kAlavRddhyorhi // 28 / / mUlakAlamiavibhAgAnayanasUtram vaphaloddhRtapramANaM kAlacaturvRddhitADitaM zodhyam / mizrakRtastanmUlaM mizre kriyate tu saGkramaNam // 29 // For Private and Personal Use Only

Page #103
mizrakavyavahAraH 69 atroddezakaH / saptatyA vRddhiriyaM catuHpurANAH phalaM ca paJcakRtiH / mizra nava pazcaguNAH pAdena yutAstu kiM mUlam // 30 // trikaSaSTyA datvaikaH kiM mUlaM kena kAlena / prApto'STAdazavRddhi SaTSaSTiH kAlamUlamizraM hi // 31 // adhyardhamAsikaphalaM SaSyAH pazcArdhameva sandRSTam / vRddhistu caturviMzatiratha SaSTirmUlayuktakAlazca // 32 // pramANaphalecchAkAlamizravibhAgAnayanasUtram mUlaM svakAlAdidvikRtiguNaM chinnamitaramUlena / mizrakRtizeSamUlaM mizre kriyate tu saGkramaNam // 33 // atroddezakaH / adhyardhamAsakasya ca zatasya phalakAlayozca mizradhanam / dvAdaza dalasaMmizraM mUlaM triMzatphalaM paJca / / 3,4 // mUlakAlavRddhimizravibhAgAnayanasUtrammizrAdUnitarAziH kAlastasyaiva rUpalAbhena / saikena bhajenmUlaM khakAlamUlonitaM phalaM mizram / / 35 // atroddeshkH| pazcakazataprayoge na jJAtaH kAlamUlaphalarAziH / tanmizra 'hAzItimUlaM kiM kAlahI ke // 36 // This wrong form is the reading found in th. MSS.; and the correct form Raif door Rot satisfy the exigencies of the metre, For Private and Personal Use Only

Page #104
gaNitasArasaGganchaH bahumUlakAlAdimizrAvibhAzanayanasUtramvibhajetsvakAlatADitamUlasamAsena phalasamAsahatam / kAlAbhyastaM mUlaM pRthak pRthak cAdizedRddhim // 37 / / atroddezakaH / catvAriMzatriMzaviMzatipaJcAzadatra mUlAni / mAsAH paJcacatustrikaSaT phalapiNDazcatustriMzat // 38 // bahumUlamizravibhAgAnayanasUtram-- khaphalaissvakAlabhaktaistadyutyA mUlamizradhanarAzim / / 'chindyAdazaM guNayet samAgamo bhavati mUlAnAm / / 39 / / atroddezakaH / dazatripaJcadazakA raddhaya iSavazcatustriSaNmAsAH / mUlasamAso dRSTazcatvAriMzacchatena saMmizrA // 40 // pazcArdhaSaDdazApi ca sArdhAH SoDaza phalAni ca triMzat / mAsAstu paJca SaT ravalu saptASTa dazApyazItiratha piNDaH // 41 bahukAlAmizravibhAgAnayanasUtram khaphalaiH svamUlabhaktaistadyutyA kAlamizradhanarAzim / 'chindyAdazaM guNayet samAgamo bhavati kAlAnAm // 42 // atroddezakaH / catvAriMzattriMzadizatipaJcAzadatra mUlAni / dazaSatripaJcadaza phalamaSTAdaza kAlamizradhanarAziH // 43 // pramANarAzau phalena tulyamicchArAzimUlaM ca tadicchArAzau vRddhi + The MSS. read fraTCUTT which does not seem to be correct. For Private and Personal Use Only

Page #105
pikavyavahAraH ca saMpIya tanmizrarAzau pramANarAzeH vRddhivibhAgAnayanasUtram kAlaguNitapramANaM parakAlahRtaM tadekaguNamizradhanAt / . itarArdhakRtiyutAt padamitarA|naM pramANaphalam // 44 // atroddeshkH| mAsacatuSkazatasya pranaSTaddhiH prayogamUlaM tat / vaphalena yutaM dvAdaza paJcakRtistasya kAlo'pi // 45 // mAsatritayAzItyAH pranaSTa vRddhiH svamUlaphalarAzeH / paJcamabhAgenonAthASTau varSeNa mUlasaDI ke // 46 // samAnamUladdhimizravibhAgasUtramanyonyakAlavinihatamizravizeSasya tasya bhAgAkhyam / kAlavizeSeNa hRte teSAM mUlaM vijAnIyAt // 47 // bhatroddezakaH / pazcAzadaSTapaJcAzanmizraM SaTSaSTireva ca / pazca saptaiva nava hi mAsAH kiM phalamAnaya // 48 // . triMzaccaikatriMzadivyaMzAH syuH punastrayastriMzat / satryaMzA mizradhanaM paJcatriMzaca gaNakAdAt // 49 // kazcinnara zcaturNA tribhizcaturbhizca paJcabhiH SabhiH / mAsairlabdhaM kisyAnmUlaM zIghraM mamAcakSva // 50 // samAnamUlakAlamizravibhAgasUtramanyonyavRddhisaGgaNamizravizeSasya tasya bhAgAkhyam / baDivizeSeNa hRte labdhaM mUlaM budhAH prAhuH // 51 // For Private and Personal Use Only

Page #106
72 gaNitasArasaGgahaH matroddezakaH / ekatripazcamizritaviMzatiriha kAlamUlayonizram / SaD daza caturdaza syu bhAH kiM mUlamatra sAmyaM syAt // 52 // pazcatriMzanmizraM saptatriMzaca navayutatriMzat / viMzatiraSTAviMzatiratha SaTtriMzacca vRddhidhanam // 53 // ubhayaprayogamUlAnayanasUtram-- rUpasyecchAkAlAdubhayaphale ye tayovizeSeNa / labdhaM vibhajenmUlaM svapUrvasaGkalpitaM bhavati // 54 / / atroddezakaH / udRttyA SaTUzate prayojito'sau punazca navakazate / mAsaistribhizca labhate saikAzIti krameNa mUlaM kim // 55 // triddhacaiva zate mAse prayuktazrASTabhizzate / lAbho'zItiH kiyanmUlaM bhavettanmAsayoIyoH // 56 // sRddhimUlavimocanakAlAnayanasUtrammUlaM vakAlaguNitaM phalaguNitaM tatpramANakAlAbhyAm / bhaktaM skandhasya phalaM mUlaM kAlaM phalAtprAgvat // 57 // atroddezakaH / mAse hi paJcaiva ca saptatInAM mAsadvaye'STAdazakaM pradeyam / ? This same rule is somewhat defectively stated again with . modifestion in rendingahus: punarapyubhayaprayogamUlAnayanasUtram - icchAkAlAdubhayaprayogavRddhiM samAnIya / tadaSantarabhaktaM labdhaM mUlaM vijAnIyAt // For Private and Personal Use Only

Page #107
mizrakavyavahAraH 78 skandhaM caturbhissahitA tvazItiH mUlaM bhavetko nu vimuktikAlaH // 58 // SaSTyA mAsikadRddhiH paJcaiva hi mUlamapi ca paJcatriMzat / mAsatritaye skandhaM tripaJcakaM tasya kaH kAlaH // 59 / / samAnavRddhimUlamizravibhAgasUtram-- mUlaiH svakAlaguNitaiIDivibhaktaistamAsakarvibhajet / mizraM vakAlanighnaM vRddhirmUlAni ca prAgvat // 60 // atroddezakaH / dvikaSaTcatuzzatake catussahasraM catuzzataM mizram / mAsadvayena vRddhayA samAni kAnyatra mUlAni // 61 // trikazatapaJcakasaptatipAdonacatuSkaSaSTiyogeSu / navazatasahasrasaGkhyA mAsatritaye samA yuktA // 62 // vimuktakAlasya mUlAnayanasUtram - skandhaM svakAlabhaktaM vimuktakAlena tADitaM vibhajet / nirmuktakAlatyA rUpasya hi saikayA mUlam // 63 / / atroddezakaH / / paJcakazataprayoge mAsau dvau skandhamaSTakaM datvA / mAsaiHSaSTibhiriha vai nirmuktaH kiM bhavenmUlam // 64 // dvau satripaJcabhAgau skandhaM dvAdazadinairdadAtyekaH / For Private and Personal Use Only

Page #108
gaNitasArasaGgahaH trikazatayoge dazabhirmAsairmuktaM hi mUlaM kim // 15 // vRddhiyuktahInasamAnamUlamizravibhAgasUtram kAlasvaphalonAdhikarUpoddhRtarUpayogahatamizre / prakSepo guNakAraH svaphalonAdhikasamAnamUlAni // 66 // bhatroddezakaH / trikapazcakASTakazataiH prayogato'STAsahasrapaJcazatam / viMzatisahitaM vRddhibhiruddhRtya samAni paJcabhirmAsaiH // 67 // trikaSaTrASTakaSaSTyA mAsadvitaye catussahasrANi / paJcAzadvizatayutAnyato'STamAsakaphalAhate sadRzAni // 18 // dvikapaJcakanavakazate mAsacatuSka trayodazasahasram / saptazatena ca mizrA catvAriMzatsaddhisamamUlAni // 69 // saikArdhakapaJcArdhakaSaDardhakAzItiyogayuktAstu / mAsASTake SaDadhikA catvAriMzacca SaTutizatAni // 70 // saGkalitaskandhamUlasya mUlavRddhi vimuktikAlAnayanasUtramskandhAptamUlacitiguNitaskandhecchAgraghAtiyutamUlaM syAt / skandhe kAlena phalaM skandhoddhRtakAlamUlahatakAlaH // 1 // atroddeshkH| kenApi saMprayuktA SaSTiH paJcakazataprayogeNa / / mAsatripaJcabhAgAt saptottaratazca saptAdiH // 72 // tatSaSTisaptamAMzakapadamitisaGkalitadhanameva / datvA tatsaptAMzakavRddhi prAdAcca citimUlam // FEX: is the reading found in the MSS.; fe is adopted as being more matinfactory reumationlly. For Private and Personal Use Only

Page #109
mizrakavyavahAraH 16 kiM taddhiH kA syAt kAlastahaNasya maukSiko bhavati / 73, / kenApi saMprayuktAzItiH paJcakazataprayogeNa // aSTAdyaSTottaratastadazItyaSTAMzagacchena / mUladhanaM datvASTAdyaSTottarato dhanasya mAsArdhAt / / 75 // vRddhi prAdAnmUlaM vRddhizca vimuktikAlazca / eSAM parimANaM kiM vigaNayya sakhe mamAcakSva // 76 // ekIkaraNasUtram-- vRddhisamAsa vibhajenmAsaphalaikyena labdhamiSTaH kAlaH / kAlapramANaguNitastAdiSTakAlena sambhaktaH // vRddhisamAsena hato mUlasamAsena bhAjito vRddhiH / 774 / atroddeshkH| yuktA catuzzatIha hikatrika paJcakacatuSazatena / mAsAH paJca caturditrayaH prayogaikakAlaH kaH // 7 // iti mizrakavyavahAre vRddhividhAnaM samAptam / / prakSepakakuTTIkAraH // itaH paraM mizrakavyavahAre prakSepakakuTTIkAragaNitaM vyAkhyA syAmaH / prakSepakakaraNamidaM savargavicchedanAMzayutihRtamizraH / prakSepakaguNakAraH kuTTIkAro budhaissamuddiSTam / / 79 / / atroddezakaH / dvitricatuSpar3abhAgaivibhAjyate dviguNaSaSTiriha hemrAma / bhUtyebhyo hi catubhyoM gaNakAcakSvAzu me bhAgAn // 8 // 8-A For Private and Personal Use Only

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76 gaNitasArasaGgrahaH prathamasyAMzatritayaM triguNottaratazca paJcabhirbhaktam / dInArANAM trizataM triSaSTisahitaM ka ekAMzaH // 813 // AdAya cAmbujAni pravizya sacchrAvako'tha jinanilayam / pUjAM cakAra bhaktyA pUjAhebhyo jinendrebhyaH // 823 // vRSabhAya caturthAMzaM SaSThAMzaM ziSTapArthAya / dvAdazamatha jinapataye tryaMzaM munisuvratAya dadau / / 83 // naSTASTakarmaNe jagadiSTAyAriSTanemaye'STAMzama / SaSThagnacaturbhAgaM bhaktyA jinazAntaye pradadau // 84 // kamalAnyazItimizrANyAyAtAnyatha zatAni catvAri / kusumAnAM bhAgAkhyaM kathaya prakSepakAkhyakaraNena // 853 // catvAri zatAni sarave yutAnyazItyA naraivibhaktAni / paJcabhirAcakSva tvaM dvitricatuHpaJcaSaNitaiH // 86 / / iSTaguNaphalAnayanasUtram-- bhaktaM zeSairmUlaM guNaguNitaM tena yojitaM prakSepam / tadrvyaM mUlyanaM kSepavibhaktaM hi mUlyaM syAt // 873 // asminnarthe punarapi sUtramphalaguNakArairhatvA paNAn phalaireva bhAgamAdAya / prakSepake guNAssyustrairAzikataH phalaM vadenmatimAn / / 88 // asminnarthe punarapi sUtramsvaphalahatAH svaguNanAH paNAstu tairbhavati pUrvavaccheSaH / iSTaphalaM nirdiSTaM trairAzikasAdhitaM samyak // 89 // For Private and Personal Use Only

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mizrakavyavahAraH __77 atroddezakaH / dvAbhyAM trINi tribhiH paJca paJcabhissapta mAnakaiH / dADimAmrakapitthAnAM phalAni gaNitArthavit // 903 // kapitthAt triguNaM hyAnaM dADimaM SaDguNaM bhavet / krItvAnaya sarave zIghraM tvaM SaTsaptatibhiH paNaiH // 913 // dadhyAjyakSIragharjinabimbasyAbhiSecanaM kRtavAn / jinapuruSo dvAsaptatipalaistrayaH pUritAH kalazAH / / 923 // dvAtriMzatprathamaghaTe punazcaturviMzatiditIyaghaTe / SoDaza tRtIyakalaze pRthak pRthak kathaya me kRtvA // 933 / / teSAM dadhighRtapayasAM tatazcaturviMzatighRtasya palAni / SoDaza payaHpalAni dvAtriMzad dadhipalAnIha // 943 // uttistrayaH purANAH puMsazvArohakasya tatrApi / sarve'pi paJcaSaSTiH kecidbhaganA dhanaM teSAm // 953 // sannihitAnAM dattaM labdhaM puMsA dazaiva caikasya / ke sannihitA bhanAH ke mama saJcintya kathaya tvama // 96 // iSTarUpAdhikahInaprakSepakakaraNasUtram--- piNDo'dhikarUpono hInottararUpasaMyutaH zeSAt / prakSepakakaraNamataH kartavyaM tairyutA hInAH // 97 // atroddezakaH / prathamasyaikAMzo'to dviguNadviguNottarAdbhajanti nraaH| catvAro'zaH kassyAdekasya hi saptaSaSTiriha // 98 // For Private and Personal Use Only

Page #112
gaNitasArasaGgahaH prathamAdadhyardhaguNAt triguNAdrUpottarAdvibhAjyante / sASTA saptatirobhizcaturbhirAptAMzakAn brUhi // 996 // prathamAdadhyardhaguNAH pazcArdhaguNottarANi rUpANi / paJcAnAM paJcAzatsaikA caraNatrayAbhyadhikA // 10 // prathamAtpazcArdhaguNAzcaturguNottaravihInabhAgena / bhaktaM naraizcaturbhiH paJcadazonaM zatacatuSkam / / 101 / / samadhanAonayanatajjyeSThadhanasaGkhyAnayanasUtram--- jyeSThadhanaM saikaM syAt svavikraye'ntyArpaguNamapaikaM tat / krayaNe jyeSThAnayanaM samAnayet karaNaviparItAt // 102 / / ___ atroddezakaH / hAvaSTau SaTrizanmUlaM nRNAM SaDeva caramArghaH / ekAdhaiNa krItvA vikrIya ca samadhanA jAtAH // 1033 // sAdhaiMkamardhamardhadvayaM ca sagRhya te trayaH puruSAH / krayavikrayau ca kRtvA SaDiH pazcArdhAtsamadhanA jAtAH // 10 // catvAriMzat saikA samadhanasaGkhyA SaDeva caramArthaH / AcakSva gaNaka zIghra jyeSThadhanaM kiM ca kAni mUlAni // 105 // samadhanasaGkhyA paJcatriMzadbhavanti yatra dInArAH / catvArazcaramA? jyeSThadhanaM kiM ca gaNaka kathaya tvm||10 // caramArthabhinnajAtI samadhanA_nayanasUtramtukhyApacchedadhanAntyArghAbhyAM vikrayakrayA! prAgvat / chedacchedakRtinAvanupAtAt samadhanAni bhinne'ntyApeM // 10 // For Private and Personal Use Only

Page #113
mizrakavyavahAraH ardhatripAdabhAgA dhanAni SaTpaJcamAMzakAzcaramArghaH / ekArpaNa krItvA vikrIya ca samadhanA jAtAH // 10 // punarapi antyAdhai bhinne sati samadhanAnayanasUtramjyeSThAMzadviharahatiH sAntyaharA vikrayo'ntyamUlyanaH / naiko yarivalaharaghnaH syAtkrayasaGkhacAnupAto'tha // 1093 // ___ atroddezakaH / ardha dvau vyaMzau ca trIn pAdAMzAMzca saGgRhya / vikrIya krItvAnte paJcabhiraGyaMzakaissamAnadhanAH // 110 // iSTaguNeSTasaGkhyAyAmiSTasaGkhyAsamarpaNAnayanasUtram-- antyapade svaguNahate kSipedupAntyaM ca tasyAntam / tenopAntyena bhajedyallabdhaM tadbhavenmUlam // 111 // atroddeshkH| kazcicchrAvakapuruSazcaturmuravaM jinagRhaM samAsAdya / pUjAM cakAra bhaktyA surabhINyAdAya kusumAni // 112 // dviguNamabhUdAdyamurave triguNaM ca caturguNaM ca paJcaguNan / sarvatra paJca paJca ca tatsaGkhyAmbhoruhANi kAni syuH|| 1136 // dvitricaturbhAgaguNAH paJcArdhaguNAstripaJcasaptASTau / bhaktarmaktyAhebhyo dattAnyAdAya kusumAni // 1146 / / iti mizrakavyavahAre prakSepakakuTIkAraH samAptaH // The following stanze is added in M after stanza No. 1101, but it is not tound B: bhatripAdabhAgA dhanAni ghaTpazcamAMzakAntyArghaH / ekArpaNa krItvA vikrIya ca samadhanA jAtA: // For Private and Personal Use Only

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asara , gaNitasArasaGgrahaH vallikAkuTThIkAraH // itaH paraM vallikAkuTTIkAragaNitaM vyAkhyAsyAmaH / kuTThIkAre vallikAgaNitanyAyasUtram chitvA chedena rAzi prathamaphalamapohyAptamanyonyabhaktaM sthApyordhvAdharyato'dho matiguNamayujAlpe'vaziSTe dhanarNam / chitvAdhaH svoparinopariyutaharabhAgo'dhikAgrasya hAraM chitvA chedena sAgrAntaraphalamadhikAyAnvitaM hAraghAtam // 1153 // atroddezakaH / jambUjambIrarambhAkramukapanasaravarjUrahintAlatAlIpunnAgAmrAdyanekadrumakusumaphalairnamrazAravAdhirUDham / bhrAmyagRGgAbjavApIzukapikakulanAnAdhvanivyAptadikkaM pAnthAH zrAntA vanAntaM zramanudamamalaM te praviSTAH prahRSTAH / / 116 // rAzitriSaSTiH kadalIphalAnAM sampIDya saMkSipya ca saptabhistaiH / pAnthestrayoviMzatibhirvizuddhA rAzestvamekasya vada pramANam // 1173 // rAzIn punarvAdaza dADimAnAM samasya saMkSipya ca paJcabhistaiH / pAnthairnarairviMzatibhirnirakabhaktAMstathaikasya vada pramANam // 118 / / dRSTAmrarAzIn pathiko yathaikatriMzatsamUhaM kurute vihInam / For Private and Personal Use Only

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mizrakavyavahAraH zeSe hRte saptatibhistrimizranarervizuddhaM kathayaikasaGkhyAm // 1193 / / Acharya Shri Kailassagarsuri Gyanmandir 81 dRSTAssaptatriMzatkapitthaphalarAzayo vane pathikaiH / saptadazApohya hRte vyekAzItyAMzakapramANaM kim || 1203 // For Private and Personal Use Only dRSTvA rAzimapahAya ca sapta pazcAkte'STabhiH punarapi pravihAya tasmAt / trINi trayodazabhirudda lite vizuddhaH pAnthairvane gaNaka me kathayaikarAzim // 1213 // tribhizcaturbhiH paJcabhirekaH kapitthaphalarAziH / bhakto rUpAgrastatpramANamAcakSva gaNitajJa || 1223 // dvAbhyAmekastribhirdhau ca caturbhirbhAjite trayaH / catvAri paJcabhizzeSaH ko rAzirvada me priya / / 1233 // dvAbhyAmekastribhizuddhazvaturbhirbhAjite trayaH / catvAri pazcabhizzeSaH ko rAzirvada me priya // 124 // dvAbhyAM nira ekAgrastribhirnAgro vibhAjitaH / caturbhiH paJcabhirbhako rUpAmro rAzireSa kaH // 129 // dvAbhyAmekastribhizzuddha caturbhirbhAjite trayaH / niragraH paJcabhirbhaktaH ko rAziH kathayAdhunA // 126 // dRSTA jambUphalAnAM pathi pathikajanai rAzayastatra rAzI sts tau navAnAM traya iti punarekAdazAnAM vibhaktAH / paJcAgrAste yatInAM caturadhikatarAH paJca te saptakAnAM kuTTIkArArthavinme kathaya gaNaka saJcintya rAzipramANam / / 1273 //

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gaNitasArasaGgahaH vanAntare dADimarAzayaste pAnthestrayassaptabhireka zeSAH / sapta trizeSA navabhirvibhaktAH paJcASTabhiH ke gaNaka hiragrAH / / 128 // . bhaktA dviyuktA navabhistu paJca yuktAzcaturbhizca SaDaSTabhistaiH / pAnthairjanaissaptabhirekayuktA catvAra ete kathaya pramANam // 129 // agrazeSavibhAgamUlAnayanasUtram - zeSAMzAgravadho yuk svAgreNAnyastadaMzakena guNaH / / pAvadAgAstAvadvicchedAH syustadagraguNAH // 1303 // atroddezakaH / mAnItavatyAmraphalAni puMsi prAgekamAdAya punastadardham / gate'graputre ca tathA jaghanyastatrAvazeSArdhamatho tamanyaH // 131 / / pravizya jainaM bhavanaM tripUruSaM prAgekamabhyarcya jinasya paade'| zeSatribhAgaM prathame'numAne tathA dvitIye ca tRtIyake tathA // 132 // zeSatribhAgaDha yatazca zeSatryaMzadvayaM cApi tatastribhAgAn / kRtvA caturviMzatitIrthanAthAn samarcayitvA gatavAn vizuddhaH // 133 // iti mizrakavyavahAre sAdhAraNakuTTIkAraH samAptaH // which does not seem to be oorreot here. B roade var 1 The M68. give for pAde. For Private and Personal Use Only

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mizrakavyavahAraH viSamakuTTIkAraH // itaH paraM viSamakuTTIkAraM vyAkhyAsyAmaH / viSamakuTTIkArasya sUtram-- matisaGgaNitau chedI yojyonatyAjyasaMyutau rAzihatau / bhinne kuTTIkAre guNakAro'yaM samuddiSTaH // 1353 // atroddezakaH / rAziH SaTrena hato dazAnvito navahato niravazeSaH / / dazabhirhAnazca tathA tadguNakau' ko mamAzu saGkathaya // 1353 // sakalakuTTIkAraH // sakalakuTTIkArasya sUtrambhAjyacchedAgrazeSaiH prathamahatiphalaM tyAjyamanyonyabhaktaM nyasyAnte sAgramUrvairupariguNayutaM taissamAnAsamAne / svarNanaM vyAptahArau guNadhanamRNayozcAdhikAgrasya hAra itvA hatvA tu sAmAntaradhanamAdhikAyAnvitaM hAraghAtam / / 1314 // atroddezakaH / saptottarasaptatyA yutaM zataM yojyamAnamaSTatriMzat / / saikazatadvayabhaktaM ko guNakAro bhavedatra // 137 // pazcatriMzat vyuttaraSoDazapadAnyeva hArAzca / dvAtriMzayadhikaikA vyuttarato'grANi ke dhanarNaguNAH // 13 // adhikAlparAzyormUlamizravibhAgasUtrama-- jyeSThanamahArAzerjaghanyaphalatADitonamapanIya / phalavargazeSabhAgo jyeSThA?'nyo guNasya viparItam // 1393 // B guNakArI. For Private and Personal Use Only

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84 gaNitasArasaGgrahaH atroddezakaH / navAnAM mAtuluGgAnAM kapitthAnAM sugandhinAm / saptAnAM mUlyasammizraM saptottarazataM punaH // 1403 // saptAnAM mAtuluGgAnAM kapitthAnAM sugandhinAm / navAnAM mUlyasammizramekottarazataM punaH // 14 13 // mUlye te vada me zIghraM mAtuluGgakapitthayoH / anayorgaNaka tvaM me kRtvA samyak pRthak pRthak // 142 // bahurAzimizratanmUlyamizravibhAgasUtramiSTapnaphalairUnitalAbhAdiSTAptaphalamasakRt / tairUnitaphalapiNDasacchedA guNayutAstadarghAssyuH // 143 // ___ atroddezakaH / atha mAtuluGgakadalIkapitthadADimaphalAni mizrANi / prathamasya saikaviMzatiratha dviramA dvitIyasya // 144 // viMzatiratha surabhINi ca punastrayoviMzatistRtIyasya / teSAM mUlyasamAsastrisaptatiH kiM phalaM ko'rghaH // 145 // jaghanyonamilitarAzyAnayanasUtrama-- paNyahRtAlpaphalonaizchindyAdalpanamUlyahIneSTam / kRtvA tAvatkhaNDaM tadUnamUlyaM jaghanyapaNyaM syAt // 146 // atrodezakaH / dvAbhyAM trayo mayUrAstribhizca pArAvatAzca catvAraH / haMsAH paJca caturbhiH paJcabhiratha sArasASaT ca // 147 // For Private and Personal Use Only

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mizrakavyavahAraH yatrArghastatra sarave SaTpaJcAzatpaNaiH khagAn krItvA / dvAsaptatimAnayatAmityuktvA mUlamevAdAt / katibhiH paNaistu vihagAH kati vigaNayyAzu jaaniiyaaH|| 149 // tribhiH paNaiH zuNThipalAni paJca caturbhirekAdaza pippalAnAm / aSTAbhirekaM maricasya mUlyaM SaTyAnayASTottaraSaSTimAza // 150 // iSTA(riSTamUlyauraSTavastupramANAnayanasUtram mUlyanaphalecchAguNapaNAntareSTanayutiviparyAsaH / . dviSThaH khadhaneSTaguNaH prakSepakakaraNamavaziSTam / / 151 / / atroddezakaH / tribhiH pArAvatAH paJca paJcabhissapta sArasAH / saptabhinava haMsAca navamizzivinastrayaH // 152 / / krIDArtha nRpaputrasya zatena zatamAnaya / ityuktaH prahitaH kazcit tena kiM kasya dIyate // 153 // vyastArghapaNyapramANAnayanasUtram'paNyaikyena paNaikyamantaramataH paNyeSTapaNyAntaraizchindyAtsaGkamaNe kRte tadubhayora? bhavetAM punaH / paNye te khalu paNyayogavivare vyastaM tayorarghayoH praznAnAM viduSAM prasAdanamidaM sUtraM jinendroditam / / 154 / / atroddezakaH / AdyamUlyaM yadekasya candanasyAgarostathA / palAni viMzatirmizraM caturagrazataM paNAH // 155 // INot found in any of the MSS. ooneuited. For Private and Personal Use Only

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86 gaNitasArasagrahaH kAlena vyatyayArghassyAtsaSoDazazataM paNAH / tayorarghaphale brUhi tvaM SaDaSTa pRthak pRthak // 15 // sUryarathAzveSTayogayojanAnayanasUtram arivalAptArivalayAjanasaGkhyAparyAyayojanAni syuH / tAnISTayogasaGkhyAninAnyekaikagamanamAnAni // 15 // atroddeshkH| ravirathaturagAssapta hi catvAro'zvA vahanti dhUryuktAH / yojanasaptatigatayaH ke vyUDhAH ke caturyogAH // 158 // sarvadhaneSTahInazeSapiNDAt svasvahastagatadhanAnayanasUtramrUponanaraivibhajet piNDIkRtabhANDasAramupalabdham / sarvadhanaM syAttasmAduktavihInaM tu hastagatam // 159 // atroddezakaH / vaNijaste catvAraH pRthak pRthak zaulkikena paripRSTAH / kiM bhANDasAramiti khalu tatrAhaiko vaNikacchreSThaH // 160 // AtmadhanaM vinigRhya dvAviMzatiriti tataH paro'vocat / tribhiruttarA tu viMzatiratha caturadhikaiva viMzatisturyaH !! 161 // saptottaraviMzatiriti samAnasArA nigRhya sarve'pi / UcuH kiM brUhi sarave pRthak pRthagbhANDasAraM me // 162 / / anyo'nyamiSTaratnasaGkhayAM dattvA samadhanAnayanasUtram--- puruSasamAsena guNaM dAtavyaM tadvizodhya paNyabhyaH / zeSaparasparaguNitaM khaM khaM hitvA maNermUlyam // 163 // For Private and Personal Use Only

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mizrakavyavahAraH 87 atrauddeshkH| prathamasya zakranIlAH SaT sapta ca marakatA dvitiiysy| vajrANyaparasyASTAvekaikArgha pradAya samAH // 164 // prathamasya zakranIlAH SoDaza daza marakatA dvitIyasya / vajrAstRtIyapuruSasyASTau dvau tatra datvaiva / / 165 // teSekaiko'nyAbhyAM samadhanatAM yAnti te trayaH puruSAH / tacchakanIlamarakatavajrANAM kiMvidhA arghAH // 166 // krayavikrayalAbhaiH mUlAnayanasUtramanyo'nyamUlyaguNite vikrayabhakte krayaM yadapalabdham / tenaikonena hRto lAbhaH pUrvAddhRtaM mUlyam // 167 / / atroddezakaH / tribhiH krINAti saptaiva vikrINAti ca paJcabhiH / nava prasthAn vaNik kiM syAllAbho dvAsaptatirdhanam // 168 // iti mizrakavyavahAre sakalakuTIkAraH samAptaH // suvarNakuTIkAraH // itaH paraM suvarNagaNitarUpakuTTIkAraM vyAkhyAsyAmaH / samasteSTavagairekIkaraNena saGkaravarNAnayanasUtram kanakakSayasaMvargoM mizrasvarNAhataH kSayo jJeyaH / paravarNapravimattaM suvarNaguNitaM phalaM henaH // 169 // For Private and Personal Use Only

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Shri Mahavir Jain Aradhana Kendra 88 gaNitasArasaGgrahaH atroddezakaH / ekakSayamekaM ca dvikSayamekaM trivarNamekaM ca / varNacatuSke ca dve paJcakSayikA catvAraH // 170 // sapta caturdazavarNAstriguNitapazcakSayAcASTau / etAnekIkRtya jvalane kSiptvaiva mizravarNaM kim / etamizrasuvarNa pUrvairbhaktaM ca kiM kimekasya / / 1713 // iSTavarNAnAmiSTasvavarNAnayanasUtram - svairasvairvarNa hatairmizraM svarNamizreNa bhAjitam / labdhaM varNaM vijAnIyAttadiSTAptaM pRthak pRthak / / 1723 // atroddezakaH / viMzatipaNAstu SoDaza varNA dazavarNaparimANaiH / parivartitA vada tvaM kati hi purANA bhavantyadhunA // 1733 // aSTottarazatakanakaM varNASTAMzatrayeNa saMyuktam / ekAdazavarNaM caturuttaradazavarNakaiH kRtaM ca kiM hema // 974 // ajJAtavarNAnavanasUtram kanakakSayasaMvargaM mizraM svarNaghnamizrataH zocyam / svarNena hRtaM varNaM varNavizeSeNa kanakaM syAt // 175 // ajJAtavarNasya punarapi sUtram svasvarNavarNavinihatayogaM svarNekyadRDhahatAcchodhyam / ajJAtavarNahemnA bhaktaM varNaM budhAH prAhuH || 1763 // atrAddezakaH / 'SaDjaladhivahnikanakaistrayodazASTartuvarNakaiH kramazaH / 1 Here vahni is substituted for ranala, and pratuvarNakaiH for STAvRtukSayaH, as thereby the reading will be better grammatically. For Private and Personal Use Only

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89 mizrakavyavahAraH ajJAtavarNahena : paJca vimizrakSayaM ca saikadaza / ajJAtavarNasaGkhayAM brUhi sarave gaNitatacvajJa // 178 / / caturdazaiva varNAni sapta svarNAni ttkssye'| caturavaNe dazotpannamajJAtakSayakaM vada // 179 / / ajJAtavAnayanasUtramvasvarNavarNavinihatayoga svarNekyaguNitadRDhavarNAt / tyaktvAjJAtasvarNakSayadRDhavarNAntarAhRtaM kanakam / / 180 // atroddeshkH| dvitricatuHkSayamAnAstristriH knkaastryodshkssyikH| varNayutirdaza jAtA bAhi sarave kanakaparimANam / / 181 / / sugmavarNamizrasuvarNAnayanasUtram jyeSThAlpakSayazodhita pakvavizeSAptarUpakaiH prAgvat / prakSepamataH kuryAdevaM bahuzo'pi vA sAdhyam // 182 // punarapi yugmavarNamizrasvarNAnayanasUtram iSTAdhikAntaraM caiva hIneSTAntarameva ca / ubhe te sthApayedhyastaM svarNaM prakSeSataH phalam // 183 // atroddeshkH| dazavarNasuvarNaM yat SoDazavarNena saMyutaM pakvam / dvAdaza cetkanakazataM dvibhedakanake pRthak pRthagbrUhi / / 184 // bahusuvarNAnayanasUtram vyekapadAnAM kramazaH svarNAnISTAni kalpayeccheSam / avyaktakanakavidhinA prasAdhayet prAktanAyeva / / 185 // I The reading in the Mss. is tatkSaya, which is obviously erroneous. For Private and Personal Use Only

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Shri Mahavir Jain Aradhana Kendra 90 www.kobatirth.org gaNitasArasaGgrahaH atroddezakaH / varNAzzarartunagavasumRDavizve nava ca pakkavarNaM hi / kanakAnAM SaSTizcet pRthak pRthak kanakamA kiM syAt // 186 // iyanaSTavarNAnayanasUtram -- Acharya Shri Kailassagarsuri Gyanmandir svarNAbhyAM hRtarUpe suvarNavarNAhate dviSThe / svasvatakena ca hInayute vyastato hi varNaphalam // 187 // atroddezakaH / SoDazadazakanakAbhyAM varNaM na jJAyate' pakvam / varNaM caikAdaza cedvarNau tatkanakayorbhavetAM kau // 188 // punarapi naSTavarNAnayanasUtram - ekasya kSayamiSTaM prakalpya zeSaM prasAdhayet prAgvat / bahukanakAnAmiSTaM vyekapadAnAM tataH prAgvat // 189 // atroddezakaH / dvAdazacaturdazAnAM svarNAnA samarasIkRte jAtam / varNAnAM dazakaM syAt tadvarNo brUhi sazcintya // 190 // aparArdhasyodAharaNam / saptanavazikhidazAnAM kanakAnAM saMyute pakvam / dvAdazavarNaM jAtaM kiM brUhi pRthak pRthagvarNam // 199 // parIkSaNazalAkAnayanasUtram paramakSayAptavarNAH sarvazalAkAH pRthak pRthagyojyAH / svarNaphalaM tacchodhyaM zalAkapiNDAt prapUNikA // 192 // Badds here yate / For Private and Personal Use Only

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mizrakavyavahAraH 91 atroddeshkH| vaizyAH svarNazalAkAzrikIrSavaH svrnnvrnnjnyaaH| cakruH svarNazalAkA dvAdazavarNaM tadAdyasya // 193 // caturuttaradazavarNaM SoDazavarNaM tRtIyasya / kanakaM cAsti prathamasyaikonaM ca dvitIyasya / / 194 // ardhArdhanyUnamatha tRtIyapuruSasya pAdonam / paravarNAdArabhya prathamasyaikAntyameva ca dyantyam // 195 // dhyantyaM tRtIyavaNijaH sarvazalAkAstu mASamitAH / zuddhaM kanakaM kiM syAt prapUraNI kA pRthak pRthak tvaM me / AcakSva gaNaka zIghraM suvarNagaNitaM hi yadi vetsi / / 1966 // vinimayavarNasuvarNAnayanasUtram --- krayaguNasuvarNavinimayavarNeSTanAntaraM punaH sthApyam / vyastaM bhavati hi vinimayavarNAntarahRtphalaM knk-| 1976 // atroddeshkH| SoDazavarNaM kanakaM saptazataM vinimayaM kRtaM lbhte| dvAdazadazavarNAbhyAM sASTasahastraM tu kanakaM kim // 1986 / / bahupadavinimayasuvarNakaraNasUtram varNanakanakamiSTasvarNenAptaM dRDhakSayo bhvti| prAgvatpraptAdhya labdhaM vinimayabahupadasuvarNAnAm // 1996 / / atroddeshkH| varNacaturdazakanakaM zatatrayaM vinimayaM prakurvantaH / varNedazadazavasunagaizca zatapaJcakaM svarNam / . 9-A For Private and Personal Use Only

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92 gaNitasArasaGgrahaH eteSAM varNAnAM pRthak pRthaka varNamAnaM kim // 201 // vinimayaguNavarNakanakalAbhAnayanasUtram varNannavarNayutihRtaguNayutimUlakSayanarUponena / AptaM labdhaM zodhyaM mUladhanAccheSavittaM syAt // 202 // tallabdhamUlayogAdvinimayaguNayogabhAjitaM lbdhm| prakSepakeNa guNitaM vinimayaguNavarNakanakaM syAt / / 203 / / atroddeshkH| kazcidvaNik phalArthI SoDazavarNaM zatadvayaM kanakam / yatkiJcidvinimayakRtamekAdyaM dviguNitaM yathA krmshH|| 204 / / dvAdazavasunavadazakakSayakaM lAbho dviragrazatam / zeSaM kiM syAdvinimayakAMsteSAM cApi me kathaya / / 205 / dRzyasuvarNavinimayasuvarNairmUlAnayanasUtram vinimayavaNenAptaM svAMzaM sveSTakSayanasaMmizrAt / aMzaikyonenAptaM dRzyaM phalamatra bhavati mUladhanam // 206 // atroddezakaH / vaNijaH kaMcit SoDazavarNakasauvarNagulakamAhRtya / tricatuHpaJcamabhAgAn krameNa tasyaiva vinimayaM kRtvA // 207 / / dvAdazadazanavavarNaiH saMyujya ca pUrvazeSaNa / mUlena vinA dRSTaM svarNasahasraM tu kiM mUlam // 208 // iSTAMzadAnena iSTavarNAnayanasya tadiSTAMzakayoH suvarNAnayanasya ca sUtram aMzAptakaM vyastaM kSiptveSTaghnaM bhavet suvrnnmyii| sA gulikA tasyA Api parasparAMzAptakanakasya // 209 // For Private and Personal Use Only

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mizrakavyavahAraH svadRDhakSayeNa varNI prakalpayetprAgvadeva ythaa| evaM taddvayayorapyumayaM sAmyaM phalaM bhavedyadi cet // 210 // prAkkalpaneSTavarNo gulikAbhyAM nizcayau bhavataH / no cetprathamasya tadA kiJcinnyUnAdhiko kSayau kRtvA // 211 // tatkSayapUrvakSayayorantarite zeSamatra sNsthaapy| trairAzikavidhilabdhaM varNoM tenonitAdhiko spssttau|| 212 // atroddezakaH / svarNaparIkSakavaNijau parasparaM yAcitau tataH prathamaH / / ardhaM prAdAt tAmapi gulikAM svasuvarNa Ayojya // 213 // varNadazakaM karomItyaparo'vAdIt tribhaagmaatrtyaa| labdhe tathaiva pUrNa dvAdazavarNaM karomi gulikaabhyaam|| 214 / / ubhayoH suvarNamAne varNI saJcintya gaNitatatvajJa / sauvarNagaNitakuzalaM yadi te'sti nigadyatAmAzu // 215 / / iti mizrakavyavahAre suvarNakuTTIkAraH samAptaH / / - vicitrakuTTIkAraH / itaH paraM mizrakavyavahAra vicitrakuTTIkAraM vyAkhyAsyAmaH / satyAnRtasUtram puruSAH saikeSTaguNA dviguNeSTonA bhavantyasatyAni / puruSakRtistairUnA satyAni bhavanti vacanAni // 216 // atroddeshkH| kAmukapuruSAH paJca hi vezyAyAzca priyAstrayastatra / pratyekaM sA brUte tvamiSTa iti kAni satyAni // 217 // For Private and Personal Use Only

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94 gaNitasArasaGgrahaH prastArayogabhedasya sUtrama-- ekAdyakottarataH padamUrdhvAdharyataH krmotkrmshH| sthApya pratilomannaM pratilomannena bhAjitaM sAram / / 218 // atroddezakaH / varNAzcApi rasAnAM kaSAyatiktAmlakaTukalavaNAnAm / madhurarasena yutAnAM bhedAn kathayAdhunA gaNaka / / 219 // vajendranIlamarakatavidrumamuktAphalaistu racitamAlAyAH / kati bhedA yutibhedAt kathaya sarave samyagAzu tvam // 220 // ketakyazokacampakanIlotpalakusumaracitamAlAyAH / kati bhedA yatibhedAtkathaya sakhe gaNitatatvajJa / / 221 // jJAtAjJAtalA bhairmUlAnayanasUtram--- ... lAbhonAmazrarAzeH prakSepakataH phalAni saMsAdhya / tena hRtaM tallabdhaM mUlyaM tvajJAtapuruSasya / / 222 // atroddezakaH / samaye kecidvaNijastrayaH kraya vikrayaM ca kurvIran / prathamasya SaT purANA aSTau mUlyaM dvitIyasya // 223 // na jJAyate tRtIyasya vyAptistairnaraistu SaNNavatiH / ajJAtasyaiva phalaM catvAriMzaddhi tenAptam / / 224 // kastasya prakSepo vaNijorubhayorbhavecca ko lAbhaH / pragaNayyAcakSva sarave prakSepaM yadi vijAnAsi // 225 // bhATakAnayanasUtram bharabhRtigatagamyahAta tyaktvA yojanadalanabhArakRteH / tanmUlonaM gampacchinnaM gantavyabhAjitaM sAram // 226 / / IM and B add ta here ; metrically it is faulty. For Private and Personal Use Only

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mizrakavyavahAraH 96 atroddezakaH / panasAni dvAtriMzannItvA yojanamasau dalonASTau / gRhNAtyanta TakamadhaiM bhagno'sya kiM deyam // 227 // dvitIyatRtIyayojanAnayanasya sUtram-. bharabhATakasaMvargo'dvitIyabhUtikRtivivarjitazchedaH / taddhRtyantarabharagatihatergatiH syAd dvitIyasya / / 228 // atroddezakaH / panasAni caturviMzatimA nItvA paJcayojanAni naraH / labhate tadbhRtimiha nava SaDbhRtiviyute dvitIyanRgatiH kaa|| 229 // bahupada'bhATakAnayanasya sUtram sannihitanarahateSu prAguttaramizriteSu maargessu|| vyArattanaraguNeSu prakSepakasAdhitaM mUlyam // 230 // atroddeshkH| zibikAM nayAnta puruSA viMzatiratha yojanadvayaM teSAm / vRtti-nArANAM viMzatyadhikaM ca saptazatam // 231 // krozadvaye niratto hAvubhayoH krozayostrayazcAnye / paJca naraH zeSArdhAbyAvRttAH kA bhRtisteSAm // 232 // iSTaguNitapoTalakAnayanasUtram----- saikaguNA svasveSTaM hitvAnyonyavazeSamitiH / apavartya yojya mUlaM(viSNoH) kRtvA vyekena mUlena // 233 // pUrvApavartarAzIn hatvA pUrvApavartarAziyuteH / / pRthageva pRthak tyaktvA hastagatAH svadhanasaGkhyAH syuH / / 234 / / tAH svasvaM hitvaiva tvazeSayogaM pRthak pRthak sthApya / svaguNanAH svakaragatairUnAH pohalakasaGkhyAH syuH // 23 // 1B o uits pada here. For Private and Personal Use Only

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gaNitasArasagrahaH atroddezakaH / mArge tribhirvaNigbhiH pohalakaM dRSTamAha tatraikaH / polakamidaM prApya dviguNadhano'haM bhaviSyAmi // 236 // hastagatAbhyAM yuvayAstriguNadhano'haM dvitIya Ahota / paJcaguNo'haM tvaparaH poTTalahastasthamAnaM kim // 237 // sarvatulyaguNakapolakAnayanahastagatAnayanasUtram-- vyekapadannavyekaguNeSTAMzavadhonitAMzayutiguNaghAtaH ! hastagatAH syurbhavati hi pUrvavadiSTAMzabhAjitaM poTTalakam // 238 // atroddezakaH / vaizyaiH paJcabhirekaM poTTalakaM dRSTamAha caikaikaH / poTTalakaSaSThasaptamanavamASTamadazamabhAgamAptvaiva // 239 // svasvakarasthena saha triguNaM triguNaM ca zeSANAm / gaNaka tvaM me zIghraM vada hastagataM ca poTTalakam / / 240 // iSTAMzeSTaguNapoTTalakAnayanasUtrama iSTaguNannAnyAMzAH seSTAMzAH saikanijaguNahatA yuktAH / DyUnapada STAMzanyUnAH saikeSTaguNahatA hastagatAH // 241 // atroddezakaH / dvAbhyAM pathi pathikAbhyAM poTTalakaM dRSTamAha tatraikaH / asyArdhaM samprApya higuNadhano'haM bhaviSyAmi // 242 // aparastryaMzadvitayaM triguNadhanastvatkarasthadhanAt / matkaradhanena sahitaM hastagataM kiM ca poTTalakam // 243 // dRSTaM pathi pathikAbhyAM poTTalakaM tadgRhItvA ca / dviguNamabhUdAdyastu svakarasthadhanena cAnyasya // For Private and Personal Use Only

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mizrakavyavahAraH hastasthadhanAdanyastriguNaM kiM karagataM ca poTalakam / 2443 // mArge naraizcaturbhiH poTTalakaM dRSTamAha tatrAdyaH / / poTTalakAmadaM labdhvA hyaSTaguNo'haM bhaviSyAmi // 245 // svakarasthadhanenAnyo navasaGgaNitaM ca zeSadhanAt / dazaguNadhanavAnaparastvakAdazaguNitadhanavAn syAt / poTTalakaM kiM karagatadhanaM kiyadbAhi gaNakAzu // 247 / / mArge naraiH poTTalakaM catubhirdaSTaM hi tasyaiva tadA babhUvuH / paJcAMzapAdArdhatRtIyabhAgAstavitripaJcanacaturguNA'tha // 248 // mArge tribhirvaNigbhiH poTTalakaM dRSTamAha tatrAdyaH / yadyasya caturbhAgaM labhe'hamityAha sa yuvayorDiguNaH // 249 // Aha tribhAgamaparaH svahastadhanasahitameva ca triguNaH / asyArdhaM prApyAhaM tRtIyapuruSazcaturnadhanavAn syAm / AcakSva gaNaka zIghraM kiM hastagataM ca poTTalakam // 250 // yAcitarUpairiSTaguNakahastagatAnayanasya sUtramyAcitarUpaikyAni svasaikaguNavardhitAni taiH prAgvat / hastagatAnA nItvA ceSTaguNaghneti sUtreNa // 2516 // sadRzacchedaM kRtvA saikeSTaguNAhRteSTaguNayutyA / rUponitayA bhaktAn tAneva karasthitAn vijAnIyAt // 2523 / / atroddezakaH / vaizyastribhiH parasparahastagataM yAcitaM dhanaM prathamaH / catvAryatha dvitIyaM paJca tRtIyaM naraM prArthya // 253 // IM and B read syuH ; and it is obviously inappropriate. For Private and Personal Use Only

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98 gaNitasArasaGgrahaH dviguNo'bhavavitIyaH prathamaM catvAri SaT tRtIyamagAt / triguNaM tRtIthapuruSaH prathamaM paJca dvitIyaM ca // 2543 / / SaT prArthyAbhUtpaJcakaguNaH svahastasthitAni kAni syuH / kathayAzu citrakuTTImizraM jAnAsi yadi gaNaka // 2553 / / puruSAstrayo'tikuzalAzcAnyonyaM yAcitaM dhanaM prathamaH / sa dvAdaza dvitIyaM trayodaza prArthya tatriguNaH // 256 / / prathamaM daza trayodaza tRtIyamabhyarthya ca dvitIyo'bhUt / paJcaguNito dvitIyaM dvAdaza daza yAcayitvAdyam // 257 // saptaguNitastRtIyo'bhavannaro vAJchitAni labdhAni / kathaya sakhe vigaNayya ca teSAM hastasthitAni kAni syuH // 25 // antyasyopAntyatulyadhanaM datvA samadhanAnayanasUtramvAJchAbhaktaM rUpaM sa upAntyaguNaH sarUpasaMyuktaH / zeSANAM guNakAraH saiko'ntyaH karaNametatsyAt / / 259 // atroddezakaH / vaizyAtmajAstrayaste mArgagatA jyeSThamadhyamakaniSThAH / khadhane jyeSTho madhyamadhanamAtra madhyamAya dadau // 2603 / / sa tu madhyamo jaghanyajadhanamAtraM yacchati smAsya / samadhanikAH syusteSAM hastagataM brUhi gaNaka saMcintya / / 2613 // vaizyAtmajAzca paJca jyeSThAdanujaH svakIyadhanamAtram / lebhe sarve'pyevaM samavittAH kiM tu hastagatam // 262 // For Private and Personal Use Only

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mizrakavyavahAraH vaNijaH paJca svasvAdadhu pUrvasya dattvA tu / samavittAH saJcintya ca kiM teSAM brUhi hastagatam // 2633 // vaNijaSSaT svadhanAvitribhAgamAtraM krameNa tajjyeSThAH / . vakhAnujAya dattvA samavittAH kiM ca hastagatam // 264 // parasparahastagatadhanasaGkhyAmAtradhanaM dattvA samadhanAnayanasUtramvAJchAbhaktaM rUpaM padayutamAdAvuparyuparyetat / saMsthApya saikavAJchAguNitaM rUponamitareSAm // 265 // atroddezakaH / vaNijastrayaH parasparakarasthadhanamekato'nyonyam / dattvA samavittAH syuH kiM syAddhastasthitaM dravyam // 2663 // vaNijazcatvAraste'pyanyonyadhanArdhamAtramanyasmAt / khalitya parasparataH samavittAH syuH kiyatkarasthadhanam / / 267. jayApajayayolImAnayanasUtramsvasvacchedAMzayutI sthApyordhvAdharyataH kramotkramazaH / anyonyacchedAMzakaguNitau vajApavartanakramazaH // 2686 // 'chedAMzakramavasthitatadantarAbhyAM krameNa sambhaktau / khAMzaharaghnAnyaharau vAJchAnau vyastataH karasthamitiH // 2693 // atroddeshkH| dRSTA kukuTayuddhaM pratyekaM tau ca kukkuTikau / uktau rahasyavAkyairmantrauSadhazaktimanmahApuruSeNa // 2703 // For Private and Personal Use Only

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Shri Mahavir Jain Aradhana Kendra 100 gaNitasArasaGgrahaH jayati hi pakSI te me dehi svarNaM hyavijayo'si dadyAM te / taddvitryaMzakamadyetyaparaM ca punaH sa saMsRtya // 2713 // tricaturthaM prativAJchatyubhayasmAdvAdazaiva lAbhaH syAt / tatkukuTikakarasthaM brUhi tvaM gaNakamukhatilaka // 272 // rAzilabdhaccheda mizravibhAgasUtram- mizrAdUnitasaGkhyA cheda: saikena tena zeSasya / bhAgaM hatvA labdhaM lAbhonitazeSa eva rAziH syAt // 2733 / / atroddezakaH / kenApi kimapi bhaktaM sacchedo rAzimizrito lAbhaH / paJcAzatribhiradhikA tacchedaH kiM bhavellabdham // 274 // iSTasaGkhyAyojyatyAjyavargamUlarAzyAnayanasUtram- yojyatyAjyayutiH sarUpaviSamAgrannArdhitA varNitA vyagrA bandhahRtA ca rUpasahitA tyAjyaikyazeSAgrayoH / zeSaikyArdhayutonitA phalamidaM rAzibhavedvAnchayostyAjyAtyAjyamahattvayoratha kRtermUlaM dadAtyeva saH // 275 // atroddezakaH / rAziH kazciddazabhiH saMyuktaH saptadazabhirapi hInaH / mUlaM dadAti zuddhaM taM rAzi syAnmamAzu vada gaNaka || 2763 / / rAziratAbharUno yaH so'STAdazabhiranvitaH kazcit / mUlaM yacchati zuddhaM vigaNayyAcakSva taM gaNaka || 2773 // For Private and Personal Use Only

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mizrakavyavahAraH rAzidvitryaMzonastri saptabhAgAnvitassa eva punaH / mUlaM yacchati ko'sau kathaya vicintyAzu taM gaNaka // / 278 iSTasaGkhyAhInayuktavargamUlAnayanasUtram - uddiSTo yo rAzistvarthIkRtavargito'tha rUpayutaH / yacchati mUlaM sveSTAtsaMyukte cApanIte ca // 2793 // atroddezakaH / || dazabhistammizro'yaM dazabhistairvarjitastu saMzuddham / yacchati mUlaM gaNaka prakathaya saJcintya rAzi me // 280 // iSTavargIkRta rAzidvayAdiSTaghnAdantaramUlAdiSTAnayana sUtram-saiSTavyeSTA varSIkRtyAtha vargitau rAzI / etAviSTaghnAvatha tadvizleSasya mUlamiSTaM syAt // 2813 // atroddezakaH / 101 yutahIna prakSepakaguNakArAnayanasUtram- saMvargiteSTazeSaM dviSThaM rUpeSTayutaguNAbhyAM tat / viparItAbhyAM vibhajetprakSepau tatra hInau vA // 285 // atroddezakaH / trika pazcakasaMvargaH paJcadazASTAdazaiva ceSTamapi / iSTaM caturdazAtra prakSepaH ko'tra hAnirvA // 285 // For Private and Personal Use Only yokIcirgIkRtazI guNitI tu saikasaptatyA / saddizleSapadaM syAdekottarasaptatizva rAzI kau || vigaNayya citrakuTTikagaNitaM yadi vetsi gaNaka me brUhi // 283 // "

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102 gaNitasArasaGgrahaH viparItakaraNAnayanasUtram-- pratyutpanne bhAgo bhAge guNito'dhike punazzodhyaH / varge mUlaM mUle vargoM viparItakaraNamidam // 286 // atrodezakaH / saptahRte ko rAzistriguNo vargIkRtaH zarairyuktaH / triguNitapaJcAMzahatastvardhitamUlaM ca paJcarUpANi // 287 // sAdhAraNazaraparidhyAnayanasUtram-- zaraparidhitrikamilanaM vargitametatpunastribhissahitam / dvAdazahate'pi labdhaM zarasaGkhyA syAtkalApakAviSTA // 288 // atroddezakaH / paridhizarA aSTAdaza tUNIrasthAH zarAH ke syuH / gaNitajJa yadi vicitre kuTThIkAre zramo'sti te kathaya // 289 / / iti mizrakavyavahAre vicitrakuTTIkAraH samAptaH / zreDhIbaddhasaGkalitam / itaH paraM mizrakagaNite zreDhIbaddhasaGkalitaM vyAkhyAsyAmaH / hInAdhikacayasaGkalitadhanAnayanasUtram vyekArdha padonAdhikacayaghAtonAnvitaH punaH prabhavaH / gacchAbhyasto hInAdhikacayasamudAyasaGkalitam // 290 // atroddezakaH / caturuttaradaza cAdirhAnacayastrINi paJca gacchaH kim / dvAvAdiIddhicayaH SaT padamaSTau dhanaM bhavedatra // 291 // For Private and Personal Use Only

Page #137
103 mizrakavyavahAraH. bhadhikahInottarasaGkalitadhane AdhuttarAnayanasUtramgacchavibhakte gaNite rUponapadArdhaguNitacayahIne / AdiH padahatavittaM cAcUnaM vyeka padadalahataH pracayaH // 292 // atroddezakaH / catvAriMzadgaNitaM gacchaH paJca trayaH pracayaH / na jJAyate'dhunAdiH prabhavo diH pracayamAcakSva / / 293 // zreDhIsaGkalitagacchAnayanasUtram --- AdivihIno lAbhaH pracayArdhahatassa eva rUpayutaH / gaccho lAbhena guNo gacchassaGkalitadhanaM ca sambhavati // 294 / / atroddezakaH / trINyuttaramAdideM vanitAbhizcotpalAni bhaktAni / ekasyA bhAgo'STau kati vanitAH kati ca kusumAni // 295 // vargasaGkalitAnayanasUtram saikeSTakatirdinA saikeSToneSTadalaguNitA / kRtighanacitisaGghAtastrikabhakto vargasaGkalitam // 296 // atrodeshkH| aSTASTAdazaviMzatiSaSTayekAzItiSaTkRtInAM ca / kRtighanacitisaGkalitaM vargacitiM cAzu me kathaya // 297 // iSTAdyuttarapadavargasaGkalitadhanAnayanasUtramdviguNaikonapadottarakRtihatiSaSThAMzamukhacayahatayutiH / vyekapadannA muravakRtisahitA pdtaadditessttkRticitikaa|| 298 // For Private and Personal Use Only

Page #138
104 gaNitasArasaGgrahaH punarapi iSTAdyuttarapadavargasaGkalitAnayanasUtram--- dviguNaikonapadottarakRtihatirekonapadahatAGgahatA / vyekapadAdicayAhatimukhakRtiyuktA padAhatA sAram // 299 // __ atroddezakaH / trINyAdiH paJca cayo gaccha: paJcAsya kathaya kRticitikAm / pazcAdistrINi cayo gaccha: saptAsya kA ca kRticitikA // 300 // ghanasaGkalitAnayanasUtram ---- gacchArdhavargarAzI rUpAdhikagacchavargasaGgaNitaH / ghanasaGkalitaM proktaM gaNite'smin gaNitatattvajJaiH // 30 1 // atroddeshkH| SaNNAmaSTAnAmapi saptAnAM paJcaviMzatInAM ca / SaTpaJcAzanmizritazatadvayasyApi kathaya ghanapiNDam / / 302 / / iSTAdyuttaragacchadhanasaGkalitAnayanasUtram ---- cityAdihatirmuravacayazeSanA pracayanighnacitivarge / Adau pracayAdUne viyutA yuktAdhike tu ghanacitikA // 303 / / atroddezakaH / Adistrayazcayo dvau gacchaH paJcAsya ghanacitikA / paJcAdissaptacayo gacchappaTa kA bhavecca ghanacitikA // 304 / / saGkalitasaGkalitAnayanasUtramdviguNaikonapadottarakRtihatiraGgAhRtA cayArdhayutA / AdicayAhatiyuktA vyeka padaghnAdiguNitena // saikaprabhavena yutA padadalaguNitaiva citicitikA / / 305: / / For Private and Personal Use Only

Page #139
mizrakavyavahAraH aMtroddezakaH / AdiSSaT paca cayaH padamapyaSTAdazAtha sandRSTam / ekAdyekottaracitisaGkalitaM kiM padASTadazakasya // 306 // catussaGkalitAnayanasUtram - saika padArdhapadAhatirazvairnihatA padonitA vyAptA / saikapadanA citiciticitikRtighana saMyutirbhavati // 307 // atroddezakaH / saptASTanavadazAnAM SoDazapaJcAzadekaSaSTInAm / brUhi catuHsaGkalitaM sUtrANi pRthak pRthak kRtvA // 108 // saGghAtasaGkalitAnayanasUtram - gacchastrirUpasahito gacchacaturbhAgatADitastaikaH / sapadapadakRtivinighno bhavati hi saGghAtasaGkalitam // 109 // atroddezakaH / saptakRteH SaTSaSTyAstrayodazAnAM caturdazAnAM ca / pazcAgraviMzatInAM kiM syAt saGghAtasaGkalitam // 110 // bhinnaguNasaGkalitAnayanasUtram - samadalaviSamakharUpaM guNaguNitaM vargatADitaM dviSTham / aMzAptaM vyekaM phalamAdyanpannaM guNonarUpahRtam // 3113 // atroddezakaH / dInArArthaM pazcasu nagareSu, cayastribhAgo'bhUt / AdizaH pAdo guNottaraM sapta bhinnaguNacitikA / 10 105 For Private and Personal Use Only

Page #140
106 gaNitasArasaGgrahaH kA bhavati kathaya zIghraM yadi te'sti parizramo gaNite // 313 // adhikahInaguNasaGkalitAnayanasUtram --- guNacitiranyAdihatA vipadAdhikahInasaGgaNA bhaktA / vyekaguNenAnyA phalarahitA hIne'dhike tu phalayuktA // 314 // _ atroddezakaH / pazca guNottaramAdiauM trINyadhikaM padaM hi catvAraH / adhikaguNottaracitikA kathaya vicintyiAzu gaNitatattvajJa // 315 // AdistrINi guNottaramaSTau hInaM dvayaM ca daza gacchaH / hInaguNottaracitikA kA bhavati vicintya kathaya gaNakAzu // 31 // AdyuttaragacchadhanamizrAdyuttaragacchAnayanasUtrammizrAduddhRtya padaM rUponecchAdhanena saikena / labdhaM pracayaH zeSaH sarUpapadabhAjitaH prabhavaH // 317 // atroddezakaH / AyuttarapadamizraM paJcAzaddhanamihaiva sandRSTam / gaNitajJAcakSva tvaM prabhavottarapadadhanAnyAzu // 318 / saGkalitagatidhruvagatibhyAM samAnakAlAnayanasUtram dhruvagatirAdivihAnazcayadalabhaktassarUpakaH kAlaH / dviguNo mArgastadgatiyogahRto yogakAlassyAt // 319 // atroddezakaH / kazcinnaraH prayAti tribhirAdA uttaraistathASTAbhiH / niyatagatirekaviMzatiranayoH kaH prAptakAlaH syAt // 320 // For Private and Personal Use Only

Page #141
mizraka vyavahAraH aparAdhadAharaNam / Sar3a yojanAni kazcitpuruSastvaparaH prayAti ca trINi / ubhayorabhimukhagatyoraSTottarazatakayojanaM gamyam / pratyekaM ca tayoH syAtkAlaH kiM gaNaka kathaya me zIghram // 329 // saGkalitasamAgamakAlayojanAnayanasUtram - ubhayorAdyozzeSazcayazeSahRto dvisaGguNaH saikaH / yugapatprayANayossyAnmArge tu samAgamaH kAlaH || 3223 // atroddezakaH / catvAryAdyaSTottarameko gacchatyatho dvitIyo nA / dvau pracayazca dazAdiH samAgame kastayoH kAlaH || 3233 // vRddhyuttarahInottarayossamAgamakAlAnayanasUtram zeSazcAdyorubhayozcayayutadala bhaktarUpayutaH / yugapatprayANakRtayormArge saMyogakAlaH syAt // 324 // 'atroddezakaH / pazvAdyaSTottarataH prathamo nAtha dvitIyanaraH / 107 For Private and Personal Use Only AdiH pazJcannanava pracayo hIno'STa yogakAlaH kaH || 3253 // zaghrigatimandagatyossamAgamakAlAnayanasUtram mandagatizIghragatyorekAzAgamanamatra gamyaM yat / tadgatyantarabhaktaM labdhadinaistaiH prayAti zIghro'lpam // 326 // atroddezakaH / navayojanAni kazcitprayAti yojanazataM gataM tena / pratidUto vrajati punastrayodazAmoti kairdivasaiH // 327 // 10-A

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108 gaNitasArasaGgrahaH viSamabANaissUNIrabANaparidhikaraNasUtram pariNAhastribhiradhiko dalito vargIkRtastribhirbhaktaH / saikazzarAstu parigherAnayane tatra viparItam // 3286 // atroddezakaH / nava paridhistu zarANAM saGkhyA na jJAyate punasteSAm / vyuttaradazabANAstatpariNAhazarAMzca kathaya me gaNaka // 329 / / zreDhIbaddhe iSTakAnayanasUtram -- taravargoM rUponastribhivibhaktastareNa saGgaNitaH / tarasaGkalite kheSTapratADite mizrataH sAram // 3303 // atroddezakaH / paJcataraikenAgraM vyavaghaTitA gaNitavinmizre / samacaturazrazreDhI katISTakAssyurmamAcakSva // 3313 // nandyAvartAkAraM catustarAH SaSTisamaghaTitAH / sarveSTakAH kati syuH zreDhIbaI mamAcakSva / / 3326 / / chandazzAstroktaSaTpratyayAnAM sUtrANi --- samadalaviSamaravarUpaM dviguNaM vargIkRtaM ca padamaGkhyA / saGkhyA viSamA saikA dalato gurureva samadalataH // 3333 // syAlaghurevaM kramazaH prastAro'yaM vinirdiSTaH / naSTAGkAdhai laghuratha tatkadale guruH punaH punaH sthAnam // 334 // rUpAdiguNottaratastaddiSTe lAGkasaMyutiH saikA / ekAdyakottarataH padamUrdhvAdharyataH kramotkramazaH // 335 // For Private and Personal Use Only

Page #143
109 mizrakavyavahAraH sthApya pratilomannaM pratilomannena bhAjitaM sAram / syAllaghugurukriyeyaM saGkhyA dviguNaikavarjitA sAdhvA / / 336 // atroddezakaH / saGkhyAM prastAravidhiM naSToddiSTe lgkriyaadhvaanau| SaTpratyayAMzca zIghraM vyakSaravRttasya me kathaya // 337 // iti mizrakavyavahAre zreDhIbaddhasaGkalitaM samAptam // iti sArasaGghahe gaNitazAstre mahAvIrAcAryasya kRtau mizrakagaNitaM nAma paJcamavyavahAraH samAptaH // For Private and Personal Use Only

Page #144
SaSThaH kSetragaNitavyavahAraH. siddhebhyo niSThitArthebhyo variSThebhyaH kRtAdaraH / abhipretArthasiddhayartha namaskurve punaH punaH // 1 // itaH paraM kSetragaNitaM nAma SaSThagaNitamudAhariSyAmaH / tadyathA kSetraM jinapraNItaM phalAzrayAyAvahArika sUkSmamiti / bhedAd dvidhA vicintya vyavahAraM spaSTametadabhidhAsye // 2 // tribhujacaturbhujavRttakSetrANi svasvabhedabhinnAni / gaNitArNavapAragatairAcAryaissamyaguktAni // 3 // tribhuja tridhA vibhinnaM caturbhujaM paJcadhASTadhA vRttam / avazeSakSetrANi hyeteSAM bhedabhinnAni // 4 // tribhujaM tu samaM dvisamaM viSamaM caturazramApa samaM bhavati / hidvisamaM dvisamaM syAtrisamaM viSamaM budhAH prAhuH // 5 // samavRttamardhavRttaM cAyatavRttaM ca kambukAvRttam / nimnonnataM ca vRttaM bahirantazcakravAlavRttaM ca // 1 // ___ vyAvahArikagaNitam / tribhujacaturbhujakSetraphalAnayanasUtramtribhujacaturbhujabAhupratibAhusamAsadalahataM gaNitam / neme jayutyadhaM vyAsaguNaM tatphalArdhamiha bAlendoH // 7 // atroddezakaH / tribhujakSetrasyASTau bAhupratibAhubhUmayo daNDAH / tadvyAvahArikaphalaM gaNayitvAcakSva me zIghram // 8 // For Private and Personal Use Only

Page #145
kSetragaNitavyavahAraH. 111 dvisamatribhujakSetrasyAyAmaH saptasaptatirdaNDAH / vistAro dvAviMzatiratha hastAbhyAM ca sammizrAH // 9 // tribhunakSetrasya bhujastrayodaza pratibhujasya paJcadaza / bhUmizcaturdazAsya hi daNDA viSamasya kiM gaNitam // 10 // gajadantakSetrasya ca pRSThe'STAzItiratra sandRSTAH / dvAsaptatirudare tanmUle'pi triMzadiha' daNDAH // 11 // kSetrasya daNDaSaSTirbAhupratibAhukasya gaNayitvA / samacaturazrasya tvaM kathaya sarave gaNitaphalamAzu // 12 // Ayatacatura zrasya vyAyAmaH saikaSaSTiriha daNDAH / vistAro dvAtriMzayavahAraM gaNitamAcakSva // 13 // daNDAstu saptaSaSTiDhisamacaturbAhukasya cAyAmaH / vyAsazcASTatriMzat kSetrasyAsya trayastriMzat // 14 // kSetrasyASTottarazatadaNDA bAhutraye murave cASTau / hastaistribhiryutAstatrisamacaturbAhukasya vada gaNaka // 15 // viSamakSetrasyASTatriMzaddaNDAH kSitirmurave dvAtriMzat / / paJcAzatprati bAhuH SaSTistvanyaH kimasya caturazre // 16 // parighodarastu daNDAstriMzatTaSThaM zatatrayaM dRSTam / navapaJcaguNo vyAso nemikSetrasya kiM gaNitam // 17 // hastau dvau viSkambhaH pRSThe'STASaSTiriha ca sandRSTAH / udare tu dvAtriMzahAlendoH kiM phalaM kathaya // 18 // The reading in both B and M is f ra: ; but as this is erroneous it is gorrooted into FTIGE So as to meet the requirements of the metre also. + B reads deka for prati. For Private and Personal Use Only

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112 gaNitasArasabAhaH uttakSetraphalAnayanasUtramtriguNIkRtaviSkambhaH paridhiAsArdhavargarAzirayam / triguNaH phalaM same'rthe vRtte'dha prAhurAcAryAH // 19 // atroddezakaH / vyAso'STAdaza vRttasya paridhiH kaH phalaM ca kim / vyAso'STAdaza vRttAdhaM gaNitaM kiM vadAzu me // 20 // AyatavRttakSetraphalAnayanasUtramvyAsArdhayuto dviguNita AyatavRttasya paridhirAyAmaH / viSkambhacaturbhAgaH pariveSahato bhavetsAram // 21 // atroddezakaH / kSetrasyAyatavRttasya viSkambho dvAdazaiva tu / AyAmastatra SaTtriMzat paridhiH kaH phalaM ca kim // 22 // zaGkAkAravRttasya phalAnayanasUtram --- vadanA|no vyAsastriguNaH paridhistu kambukAvRtte / valayArdhakRtivyaMzo muravArdhavargatripAdayutaH / 23 // atroddezakaH / vyAso'STAdaza hastA mukhavistAro'yamapi ca catvAraH / kaH paridhiH kiM gaNitaM kathaya tvaM kambukAratte // 24 // nimnonnatavRttayoH phalAnayanasUtram parigheza caturmAgo viSkambhaguNaH sa viddhi gaNitaphalam / catvAle kUrmanime kSetre nimnonnate tasmAt // 25 // For Private and Personal Use Only

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kSetragaNitavyavahAraH 118 atroddezakaH / catvAlakSetrasya vyAsastu bhasaGkhyakaH paridhiH / SaTpaJcAzadRSTaM gaNitaM tasyaiva kiM bhavati // 26 // kUrmanibhasyonnatavRttasyodAharaNamviSkambhaH paJcadaza dRSTaH paridhizca SaTtriMzat / kUrmanibhe kSetre kiM tasmin vyavahArajaM gaNitam // 27 // antazcakravAlavRttakSetrasya bahizcakravAlaTattakSetrasya va vyavahAraphalAnayanasUtram nirgamasahito vyAsastriguNo nirgamaguNoM bahirgaNitam / rahitAdhigamavyAsAdabhyantaracakravAlavRttasya // 28 // atroddezakaH / vyAso'STAdaza hastAH punarbahinirgatAstrayastatra / vyAso'STAdaza hastAzcAntaH punaradhigatAstrayaH kiM syAt // 29 // samavRttakSetrasya vyAvahArikaphalaM ca paridhipramANaM ca vyAsapramANaM ca saMyojya etatsaMyogasakyAmeva svIkRtya tatsaMyogapramANarAzeH sakAzAt pRthak paridhivyAsaphalAnAM saGkhyAnayanasUtram - gaNite dvAdazaguNite mizraprakSepakaM catuHSaSTiH / tasya ca mUlaM kRtvA paridhiH prakSepakapadonaH // 30 // atroddezakaH / paridhivyAsaphalAnAM mizraM SoDazazataM sahasrayutam / kaH paridhiH kiM gaNitaM vyAsaH ko vA mamAcakSva // 31 // For Private and Personal Use Only

Page #148
114 gaNitasArasaGgrahaH vAkAramardalAkAra paNavAkAravajjAkArANAM kSetrANAM vyAvahArika phalAnayanasUtram- Acharya Shri Kailassagarsuri Gyanmandir yavamuraja paNavazakrAyudhasaMsthAna pratiSThitAnAM tu / mukhamadhyasamAsArdhaM tvAyAmaguNaM phalaM bhavati // 32 // atroddezakaH / yavasaMsthAnakSetrasyAyAmo'zItirasya viSkambhaH / madhyazcatvAriMzatphalaM bhavetkiM mamAcakSva // 33 // AyAmo'zItirayaM daNDA mukhamasya viMzatirmadhye / catvAriMzatkSetre mRdaGgasaMsthAna ke brUhi // 34 // paNavAkArakSetrasyAyAmaH saptasaptatirdaNDAH / muvayorvistA'STau madhye daNDAstu catvAraH // 35 // vajrAkRtestathAsya kSetrasya SaDagranavatirAyAmaH / madhye sUcirmukhayostrayodaza tryaMzasaMyutA daNDAH || 31 // ubhayaniSedhAdikSetraphalAnayanasUtram - vyAsAttvAyAmaguNAdviSkambhArdhapradIrghamutsRjya / tvaM vada niSedhamubhayostadardhaparihINamekasya // 37 // atroddezakaH / AyAmaH SaTtriMzadvistAro'STAdazaiva daNDAstu / ubhayaniSedhe kiM phalamekaniSedhe ca kiM gaNitam // 38 // bahuvidhavajjAkArANAM kSetrANAM vyAvahArika phalAnayanasUtram --' rajjvardha kRtitryaMzo bAhavibhakto nirekabAhuguNaH / sarveSAmazravata phalaM hi bimbAntare caturthAMzaH // 39 // For Private and Personal Use Only

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116 116 kSetragaNitavyavahAraH atroddezakaH / SaDbAhukasya bAhoviSkambhaH paJca cAnyasya / vyAsastrayo bhujasya tvaM SoDazabAhukasya vada // 40 // tribhujakSetrasya bhujaH paJca pratibAharapi ca sapta dharA SaT / anyasya SaDaasya hyekAdiSaDantavistAraH // 41 // maNDalacatuSTayasya hi navaviSkambhasya madhyaphalam / SaTpazcacatuLasA uttatritayasya madhyaphalam // 42 // dhanurAkArakSetrasya vyAvahArikaphalAnayanasUtramkRtveSuguNasamAsaM bANArdhaguNaM zarAsane gaNitam / zaravargAtpaJcaguNAjjyAvargayutAtpadaM kASTham // 43 // atroddezakaH / jyA pAvaMzatireSA trayodazeSuzca kArmukaM dRSTam / kiM gaNitamasya kASThaM kiM vAcakSvAzu me gaNaka // 44 // bANaguNapramANAnayanasUtramguNacApakRtivizeSAt paJcahRtAtpadamiSuH samuddiSTaH / zaravargAtpazcaguNAdUnA dhanuSaH kRtiH padaM nIvA // 45 // atroddezakaH / asya dhanuHkSetrasya zaro'tra na jJAyate parasyApi / na jJAyate ca maurvI tadyamAcakSva gaNitajJa // 16 // bahirantazcaturazrakavRttasya vyAvahArikaphalAnayanasUtram bAho vRttasyedaM kSetrasya phalaM trisaMguNaM dalitam / abhyantare tadardha viparIte tatra caturazre // 47 // For Private and Personal Use Only

Page #150
116 gaNitasArasaGgrahaH atroddezakaH / pazcadazabAhukasya kSetrasyAbhyantaraM bahirgaNitam / caturazrasya ca vRttavyavahAraphalaM mamAcakSva // 48 // iti vyAvahArikagaNitaM samAptam / atha sUkSmagaNitam. itaH paraM kSetragaNite sUkSmagaNitavyavahAramudAhariSyAmaH / tadyathA -- AbAdhAvalambakAnayanasUtram --- mujakRtyantarabhUhRta bhUsaGkamaNaM tribAhukAbAdhe / tadbhujavargAntarapadamavalambakamAhurAcAryAH / / 49 // sUkSmagaNitAnayanasUtrammujayutyardhacatuSkADhujahInAdvAtitAtpadaM sUkSmam / athavA muravatalayutidalamavalambaguNaM na viSamacaturazre // 50 // atroddezakaH / tribhujakSetrasyASTau daNDA pUrvAhako samasya tvam / sUkSmaM vada gaNitaM me gaNitavidavalambakAbAdhe // 11 // dvisamatribhujakSetre trayodaza syurbhujadvaye daNDAH / daza bhUrasyAbAdhe athAvalamba ca sUkSmaphalam // 52 // viSamatribhujasya bhujA trayodaza pratibhujA tu pazcadaza / bhUmizcaturdazAsya hi kiM gaNitaM cAvalambakAbAdhe // 13 // 1 After this M adds the following :--tribhujakSetrasya apassthitabhAmisaMspRSTarekhAyA nAma avalamvakaH syAt / bhujadvayasaMyogasthAnamArabhya For Private and Personal Use Only

Page #151
117 kSetragaNitamyavahAraH itaH paraM pazcaprakArANAM caturazrakSetrANAM karNAnayanasUtram-- kSitihataviparItabhujau muravaguNamujamizritau guNacchedau / chedaguNI pratimujayoH saMvargayuteH padaM karNau // 14 // atroddezakaH / samacaturazrasya tvaM samantataH paJcabAhukasyAzu / karNa ca sUkSmaphalamapi kathaya sarave gaNitatattvajJa // 55 // Ayatacaturazrasya dvAdaza bAhuzca koTirapi pazca / karNaH kaH sUkSmaM kiM gaNitaM cAcakSva me zIghram // 16 // dvisamacaturazrabhUmiH SaTtriMzadvAhurekaSaSTizca / so'nyazcaturdazAsyaM karNaH kaH sUkSmagaNitaM kim // 57 / / vargastrayodazAnAM trisamacaturbAhu ke punarbhUmiH / sapta catuzzatayuktaM karNAvAdhAvalambagaNitaM kim // 58 / / viSamacaturazrabAhU trayodazAbhyastapaJcadazaviMzatikau / pazcaghano vadanamadhastrizataM kAnyatra karNamukhaphalAni || 59 // itaH paraM vRttakSetrANAM sUkSmaphalAnayanasUtrANi / tatra samavRttakSetrasya sUkSmaphalAnayanasUtram vRttakSetravyAso dazapadaguNito bhavetparikSepaH / vyAsacaturbhAgaguNaH paridhiH phalamardhamadhaM tat // 10 // ___ atroddezakaH / samavRttavyAso'STAdaza viSkambhazca SaSTiranyasya / dvAviMzatiraparasya kSetrasya hi ke ca paridhiphale // 61 // dvAdazaviSkambhasya kSetrasya hi cArdhavRttasya / SatriMzayAsasya kaH paridhiH kiM phalaM bhavati // // 12 // For Private and Personal Use Only

Page #152
Shri Mahavir Jain Aradhana Kendra 118 www.kobatirth.org gaNitasArasa jvandaH AyatavRttakSetrasya sUkSmaphalAnayanasUtram vyAsakRtiSSaGgaNitA dvisaGgaNAyAma kRtiyutA (padaM) paridhiH / vyAsacaturbhAgaguNazcAyatavRttasya sUkSmaphalam // 13 // atroddezakaH / AyatavRttAyAmaH SaTtriMzaddvAdazAsya viSkambhaH / kaH paridhiH kiM gaNitaM sUkSmaM vigaNayya me kathaya // 64 // zaGkhakArakSetrasya sUkSmaphalAnayanasUtram vadanAno vyAso dazapadaguNito bhavetparikSepaH / mukhadalarahitavyAsArdhavargamukhacaraNakRtiyogaH // 65 // dazapadaguNitaH kSetre kambunibhe sUkSmaphalametat / / 653 // atroddezakaH / nasUtram - Acharya Shri Kailassagarsuri Gyanmandir vyAso'STAdaza daNDA mukhavistAro'yamapi ca catvAraH / kaH paridhiH kiM gaNitaM sUkSmaM tatkambukAvRtte // 663 // bahizcakravAlavRttakSetrasya cAntazcakravAlavRttakSetrasya ca sUkSmaphalAnaya nirgamasahito vyAso dazapadanirgamaguNo bahirgaNitam / rahito'dhigamenAsAvabhyantaracakravAlavRttasya // 67 // atroddezakaH / vyAso'STAdaza daNDAH punarbahirnirgatAstrayo daNDAH / sUkSmagaNitaM vada tvaM bahirantazcakravAlavRttasya // 68 // vyAso'STAdaza daNDA antaH punaradhigatAzca catvAraH / sUkSmagaNitaM vada tvaM cAbhyantaracakravAlavRttasya // 69 // For Private and Personal Use Only

Page #153
119 kSetragaNitavyavahAraH yavAkArakSetrasya ca dhanurAkArakSetrasya ca sUkSmaphalAnayanasUtramiSupAdaguNazca guNo dazapadaguNitazca bhavati gaNitaphalam / yavasaMsthAnakSetre dhanurAkAre ca vijJeyam / / 706 // atroddeshkH| hAdazadaNDAyAmo muravadvayaM sUcirapi ca vistAraH / catvAro madhye'pi ca yavasaMsthAnasya kiM tu phalam / / 713 / / dhanurAkArasaMsthAne jyA caturviMzatiH punaH / catvAro'syeSuruddiSTassUkSmaM kiM tu phalaM bhavet // 723 // dhanurAkArakSetrasya dhanuHkASThavANapramANAnayanasUtram zaravargaH SaDguNito jyAvargasamanvitastu yastasya / mUlaM dhanurguNeSuprasAdhane tatra viparItam // 736 // viparItakriyAyAM sUtram-- guNacApakRtivizeSAttarkahatAtpadamiSuH samuddiSTaH / zaravargAt SaDDuNitAdUnaM' dhanuSaH kRteH padaM jIvA // 746 / / atroddeshkH| dhanurAkArakSetre jyA dvAdaza SaT zaraH kASTham / na jJAyate sarave tvaM kA jIvA kazzarastasya // 753 // mRdaGganibhakSetrasya ca paNavAkArakSetrasya ca vajAkArakSetrasya ca sUkSmaphalAnayanasUtram muravaguNitAyAmaphalaM khadhanuHphalasaMyutaM mRdaGganibhe / tatpaNavavajanibhayordhanuHphalonaM tayorubhayoH // 763 / / naSkrataH The reading in both Band M is 45 given above; but padaM javiA gives the required meaning. For Private and Personal Use Only

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120 gaNitasArasabAhaH atroddezakaH / caturvizatirAyAmo vistAro'STI muravadvaye / / kSetre mRdaGgasaMsthAne madhye SoDaza kiM phalam // 776 // caturviMzatirAyAmastathASTau muravayoIyoH / catvAro madhyaviSkambhaH kiM phalaM paNavAkRtau // 786 // caturviMzatirAyAmastathASTau muravayoIyoH / madhye sUcistathAcakSva vajAkArasya kiM phalam // 79 // nemikSetrasya ca bAlendrAkArakSetrasya ca ibhadantAkArakSetrasya ca sUkSmaphalAnayanasUtram - pRSThodarasaMkSepaH SaDbhakto vyAsarUpasaGgaNitaH / dazamUlaguNo nemerbAlendvibhadantayozca tasyArdham // 803 // atroddezakaH / pRSThaM caturdazodaramaSTau nemyAkRtau bhUmau / madhye catvAri ca tadbAlendoH kimibhadantasya // 813 // caturmaNDalamadhyasthitakSetrasya sUkSmaphalAnayanasUtramviSkambhavargarAzervRttasyaikasya sUkSmaphalam / tyaktvA samavRttAnAmantarajaphalaM caturNA syAt // 23 // atroddezakaH / golakacatuSTayasya hi parasparasparzakasya madhyasya / sUkSma gaNitaM kiM syAccatuSkavi kambhayuktasya // 83 // For Private and Personal Use Only

Page #155
121 kSetragaNitavyavahAraH. vRttakSetratrayasthAnyo'nyasparzanAjAtasyAntarAsthitakSetrasya sUkSmaphalAnapanasUtrama viSkambhamAnasamakatribhujakSetrasya sUkSmaphalam / / vRttaphalArdhavihInaM phalamannarajaM trayANAM syAt // 843 // . atroddezakaH / viSkambhacatuSkANAM vRttakSetratrayANAM ca / anyo'nyasTaSTAnAmantarajakSetrasUkSmagaNitaM kim // 853 // parazrakSetrasya karNAvalambakasUkSmaphalAnathanasUtrammujabhujakAtakRtivargA dvitritriguNA yathAkrameNaiva / zrutyavalambakaLatidhanakatayazca SaDake kSetre // 86 // atroddezakaH / mujaSaTUkSetre dvau dvau daNDau pratibhujaM syAtAm / asmin zrutyavalambakasUkSmaphalAnAM ca vargAH ke // 87 // vargasvarUpakaraNirAzInAM yutiptaGayAnayanasya ca teSAM vargasvarUpakaraNirAzInAM yathAkrameNa parasparaviyutitaH zeSasaGkhyAnayanasya ca sUtram kenApyapavartitaphalapadayogaviyogakRtihatAcchedAt / mUlaM padayutiviyutI rAzInAM viddhi karaNigaNitamidam // 8 // atrodeshkH| SoDazaSaTtriMzacchatakaraNInAM vargamUlapiNDaM me| maga caitatpadazeSaM kathaya sarave gaNitatacvajJa // 893 // hAta sUkSmagaNitaM samAptam // For Private and Personal Use Only

Page #156
122 gaNitasArasagrahaH. janyavyavahAraH. itaH paraM kSetragaNite janyavyavahAramudAhariSyAmaH / iSTasaGkhyAbIjAbhyAmAyatacaturazrakSetrAnayanasUtram vargavizeSaH koTissaMvargoM dviguNito bhavedvAhuH / dhargasamAsaH karNazcAyatacaturazrajanyasya // 90 // atroddezakaH / ekahike tu bIne kSetre janye tu saMsthApya / kathaya vigaNayya zIghraM koTibhujAkarNamAnAni // 91 / / bIje he trINi sarakhe kSetre janye tu saMsthApya / kathaya vigaNayya zIghraM koTibhujAkarNamAnAni // 92 / / punarapi bIjasaMjJAbhyAmAyatacaturazrakSetrakalpanAyAH sUtrambIjayuniviyutighAtaH koThistadvargayozca saGkramaNe / bAhuzrutI bhavetA janyavidhau karaNametadapi // 93 / / atroddezakaH / trikapacakabIjAbhyAM janyakSetraM sarave samutthApya / koTibhujAzrutisaGkhyAH kathaya vicintyAzu gaNitatacvajJa / / 94 / / iSTajanyakSetrAdvIjamaMjJasaGkhyayorAnayanasUtram - koThicchedAvAptyossaGkramaNe bAhudalaphalacchedau / bIne zrutISTakRtyoryogaviyogArdhamUle te // 955 // atroddezakaH / kasyApi kSetrasya ca SoDaza koTina bIje ke| triMzadayavAnyabAhubarbIje ke te zrutizcaturiMzat // 91 // For Private and Personal Use Only

Page #157
123 kSetragaNitavyavahAraH. koTisakhyAM jJAtvA bhujAkarNasaGkhyAnayanasya ca bhujasaGkhya jJAtvA koTikarNasaGkhyAnadhanasya ca karNasaGkhyAM jJAtvA koTibhujAsalyAnayanasya ca sUtram -- koTikRtezchedAptyossaGkamaNe zrutibhujau bhujakRtervA / athavA zrutISTakRtyorantarapadamiSTamapi ca koTibhuje // 97 // va atroddeshkH| kasyApi koTirekAdaza bAhuSSaSTiranyasya / zrutirekaSaSTiranyasyAnuktAnyatra me kathaya / / 986 // dvisamacaturazrakSetrasyAnayanaprakArasya sUtram--- janyakSetrabhujAhAraphalajaprAgjanyakovyoryutibhUrAsyaM viyutirbhujA zrutirathAlpAlpA hi koTirbhavet / AbAdhA mahatI zrutiH zrutirabhUjyeSThaM phalaM syAtphalaM bAhussyAdavalambako dvisamakakSetre caturbAhuke // 99 // atroddezakaH / caturazrakSetrasya dvisamasya ca paJcaSaTUbIjasya / / muravabhUmujAvalambakakarNAbAdhAdhanAni vada // 100 / / trisamacaturazrakSetrasya muravabhUbhunAvalambakakarNAbAdhAdhanAnayanasU bhujapadahatabIjAntarahatajanyadhanAptabhAgahArAbhyAm / tadbhunakoTibhyA ca dvisama iva trisamacaturazre // 101 // . atrodeshkH| caturaprakSetrasya trisamasyAsya dvikatrikakhabIjasya / mukhamabhujAvalambakakarNAbAdhAdhanAni vada // 1021 // 11-A For Private and Personal Use Only

Page #158
Shri Mahavir Jain Aradhana Kendra 124 tram gaNitasArasaGgrahaH. viSamacaturazrakSetrasya mukhabhUbhujAvalambaka karNAbAdhAdhanAnayanas jyeSThAspAnyonyahIna zrutihatabhujakoTI muje bhUmukhe te kothyoranyonyadobhyAM hatayutiratha dorghAtayukoTighAtaH / karNAvarapazrutinAvanadhikabhujakoTyAhatau lambakau tAvAbAce koTidorbhAvavanivivarake karNaghAtArthamarthaH // 103 // atrodezakaH / ekadvikadvika trikajanye cotthApya viSamacatura / mukha bhUbhujAbalambakakarNAbAdhAdhanAni vada / / 104 // punarapi viSamacaturazrAnayanasUtram- sva zrutitiguNito jyeSThabhujaH koTirapi tharA vadanam / karNAbhyAM saGguNitAvubhayabhujAvalpabhujakoTI // 1093 // jyeSTha koTiviyutirdvidhAlpabhujakoTitADitA yuktA / dvasvabhuja koTiyutiguNapRthukovyAlpazrutinako karNau // 106 // alpazrutihRtakarNAlpakoTibhujasaMhatI pRthaglambau / tadbhujayutiviyutiguNAtpadamAvAdhe phalaM zrutiguNArdham // 107 // ekasmAjjanyAgatacaturazrAddisamatribhujAnayanasUtram karNe mujadvayaM syAdvAhurdviguNIkRto bhavedbhUmiH / koTiravalambako'yaM dvisamatribhuje dhanaM gaNitam || 108 // atroddezakaH / trikapacakabIjotthahisamatribhujasya gaNaka jAhU hau / bhUmimavalambakaM ca pragaNayyAcakSva me zIghram // 109 // For Private and Personal Use Only

Page #159
125 kSetragaNitavyavahAraH. 125 viSamatribhujakSetrasya kalpanAprakArasya sUtram janyabhujArdha chitvA kenApicchedalabdhajaM cAbhyAm / koThiyutirbhUH karNau bhujau bhujA lambakA viSame // 110 // ___ atroddezakaH / he dvitribIjakasya kSetrabhujArthenaM cAnyamutthApya / tasmAdviSamatribhuje bhujabhUmyavalambakaM brUhi / / 111 // iti janyavyavahAraH samAptaH // paizAcikavyavahAraH. itaH paraM paizAcikavyavahAramudAhariSyAmaH / samacaturazrakSetre vA Ayatacatura zrakSetre vA kSetraphale rajjusaGkhyayA same sati, kSetraphale bAhusaGkhyayA same sati, kSetraphale karNasaGkhyayA same sati, kSetraphale rajvardhasaGkhyayA same sati, kSetraphale bAhostRtIyAMzasamayayA same sati, kSetraphale karNasaGkhyAyAzcaturthAMzasaGkhyayA same sati, dviguNitakarNasya triguNitabAhozca caturguNitakoTezca rajossaMyogasaGkhyAM dviguNIkRtya taddiguNitasaGkhyayA kSetraphale samAne sati, ityevamAdInA kSetrANAM koTibhujAkarNakSetraphalarajjuSu iSTarAzidvayasAmyasya ceSTarAzidvayasyAnyonyamiSTaguNakAraguNitaphalavatkSetrasya bhujAkoTisaGgakhyAnayanasya sUtram svaguNeSTena vibhaktAskheSTAnAM gaNaka gaNitaguNitena / guNitA bhujA bhujAH syuH samacaturazrAdijanyAnAm // 112 / / atroddezakaH / rajjurgaNitena samA samacatura zrasya kA tu mujasaGkhyA / aparasya bAhusadRzaM gaNitaM tasyApi me kathaya // 113 // For Private and Personal Use Only

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126 gaNitasArasaGgahaH. koM gaNitena samaH samacaturazrasya ko bhavedvAhuH / rajjuDhiguNo'nyasya kSetrasya dhanAcca me kathaya // 114 // Ayatacaturazrasya kSetrasya ca rajjutulyamiha gaNitam / gaNitaM karNena samaM kSetrasyAnyasya ko bAhuH // 1153 / / kasyApi kSetrasya triguNo bAhurdhanAJca ko bAhuH / karNazcaturguNo'nyaH samacaturazrasya gaNitaphalAt // 116 // Ayatacaturazrasya zravaNaM dviguNaM trisaGguNo bAhuH / / koTizcaturguNA tai rajjuyutaidiguNitaM gaNitam // 1173 / / Ayatacatura zrasya kSetrasya ca rajjuratra rUpasamaH / koTiH ko bAhurvA zIghraM vigaNayya me kathaya / / 118 // karNo dviguNo bAhustriguNaH koTizcaturguNA mizraH / rajjvA saha tatkSetrasyAyatacaturazrakasya rUpasamaH / / 119 // punarapi janyAyatacaturazrakSetrasya bIjasakhyAnayane karaNasUtramkotyUnakarNadalatatkarNAntaramubhayayozca pade / Ayatacaturazrasya kSetrasyeyaM kriyA janye // 120 // ___atroddeshkH| Ayatacatura zrasya ca koTiH paJcAzadadhikapaJca bhujA / sASTAcatvAriMzatrisaptatiH zrutirathAtra ke bIje // 121 / / iSTakalpitasaGkhyApramANavatkarNasahitakSetrAnayanasUtram--- padyatkSetraM jAtaM bIjaissaMsthApya tasya karNena / iSTaM karNa vibhajellAbhaguNAH koTidoHkarNAH // 122 // atrodezakaH / ekahikAhikatrikacatuSkasaptaikasASTakAnAM ca / gaNaka caturNA zIghaM bIjairutthApya koThibhujAH // 123 // For Private and Personal Use Only

Page #161
kSetragaNitavyavahAraH. AyatacaturazrANAM kSetrANAM viSamabAhukAnAM ca / karNo'tra paJcaSaSTiH kSetrANyAcakSva kAni syuH // 124 // iSTajanyAyatacaturazrakSetrasya rajjusayAM ca karNasaGkhyAM ca jJAtvA tajjanyAyatacaturazrakSetrasya bhujakoTisaGkhyAnayanasUtram - karNakRtau dviguNAyAM rajvardhakRtiM vizodhya tanmUlam / rajjvarthe saGkramaNIkRte bhujA koTirapi bhavati / / 1293 // atroddezakaH / paridhiH sa catustriMzat karNazcAtra trayodazo dRSTaH / janyakSetrasyAsya pragaNayyAcakSva koTi bhujau // 126 // kSetraphalaM karNasaGkhyAM ca jJAtvA bhujakoTisaGkhyAnayanasUtramkarNakRtau dviguNIkRtagaNitaM hInAdhikaM kRtvA / mUlaM koTibhujau hi jyeSThe ikhena saGkramaNe // 127 // atroddezakaH / 127 Ayatacaturazrasya hi gaNitaM SaSTistrayodazAsyApi / karNastu koTibhujayoH parimANe zrotumicchAmi // 128 // kSetraphalasaGkhyAM rajjusaGkhyAM ca jJAtvA Ayatacaturazrasya bhujakoTisaGkhyAnayanasUtram - rajjvardha vargarAzeNitaM caturAhataM vizodhyAtha | mUlana hi rajvardhe saGkramaNe sati bhujAkoThI // 1293 // atroddezakaH / saptatizataM tu rajjuH pazcazatottarasahasramiSTadhanam / nanyAyatacaturazre koThibhujau me samAcakSva // 130 // For Private and Personal Use Only

Page #162
18 . gaNitasArasaGgrahaH. AyatacaturazrakSetradvaye rajjasaGkhyAyAM sadRkSAyAM satyAM dvitIyakSetraphalAt prathamakSetraphale dviguNite sati, athavA kSetradvaye'pi kSetraphale saDaze sati prathamakSetraspa rajjusaGkhyAyA api dvitIyakSetrarajusaGkhyAyAM dviguNAyAM satyam, athavA kSetradvaye prathamakSetrarajjusaGkhyAyA api dvitIyakSetrasya rajjusaGkhyAyAM dviguNAyAM satyAM dvitIyakSetraphalAdapi prathamakSetraphale dviguNe sati, tattatkSetradvayasyAnayanasUtram svAlpahatarajjudhanahatakRtiriSTapnava koTissyAt / vyekA dostulyaphale'nyatrAdhikagaNitaguNiteSTam // 131 // vyekaM tadUnakoTiH triguNA doH syAdathAnyasya / rajjvardhavargarAzeriti pUrvoktena sUtreNa / tadgaNitarajjamititaH samAnayettagujAkoTI // 133 // atroddezakaH / asamavyAsAyAmakSetre dve dvAvatheSTaguNakAraH / prathamaM gaNitaM dviguNaM rajjU tulye kimatra koTibhuje // 134 // Ayatacaturazre dve kSetre dvayameva guNakAraH / gaNitaM sadRzaM rajjuDhiguNA prathamAt dvitIyasya // 135 / / Ayatacaturazre dve kSetre prathamasya dhanamiha dviguNam / dviguNA dvitIyarajjustayorbhujAM koThimapi kathaya // 136 / / hisamatribhujakSetrayoH paraspararajjadhanasamAnasaGkhyayoriSTaguNakaguNitarajuSanavatorvA dvisamatribhujakSetradvayAnayanasUtram rajjukRtinAnyonyadhanAlpAptaM SaDvinamalpamekonam / taccheSaM dviguNAlpaM bIje tajjanyayorbhujAdayaH prAgvat // 137 / / atroddeshkH| dvisamatribhujakSetradvayaM tayoH kSetrayossamaM gaNitam / rajjU same tayossyAt ko bAhuH kA bhavedrUmiH // 138 / For Private and Personal Use Only

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kSetragaNitadhyavahAraH. 129 dvisamatribhujakSetre prathamasya dhanaM dvisaGguNitam / rajjuH samA dvayorapi ko bAhuH kA bhavedbhUmiH // 139 // dvisamatribhujakSetre he rajardviguNitA dvitIyasya / gaNite dvayossamAne ko bAhuH kA bhavedbhUmiH // 140 / / dvisamatribhujakSetre prathamasya dhanaM dvisaGguNitam / dviguNA dvitIyarajjuH ko bAhuH kA bhavedbhUmiH // 141 / / ekadvayAdigaNanAtItasaGkhyAsu iSTasaGkhyAmiSTavastuno bhAga - sayAM parikalpya tadiSTavastubhAgasaGkhyAyAH sakAzAt samacaturazrakSetrAnayanasya ca samavRttakSetrAnayanasya ca samAtribhujakSetrAnayanasya cAyatacatura zrakSetrAnayanasya ca sUtrama---- khasamIkRtAvadhRtihatadhanaM caturtI hi vRttasamacatura zravyAsaH / SaNitaM tribhujAyatacatura zrabhujArdhamapi koTiH // 142 // atroddezakaH / svAntaHpure narendraH prAsAdatale nijAGganAmadhye / divyaM sa ranakambalamapIpatattaJca samavRttam // 143 // tAbhirdevIbhi(tamebhirbhujayozca muSTibhirlabdham / paJcadazaikasyAH syuH kati vanitAH ko'tra viSkambhaH // 14 4 // samacatura zrabhujAH ke samatribAhI bhujAzcAtra / Ayatacaturazrasya hi tatkoTibhujau sarave kathaya / / 145 // kSetraphalasaGkhyAM jJAtvA samacaturazrakSetrAnayanasya cAyatacaturazrakSetrAnayanasya ca sUtram sUkSmagaNitasya mUlaM samacaturazrasya bAhuriSTahatam / dhanamiSTaphale syAtAmAyatacaturazrakoTibhujau // 146 // For Private and Personal Use Only

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130 gaNitasArasAhaH. atroddezakaH / .. kasya hi samacaturazrakSetrasya phalaM catuHSaSTiH / phalamAyatasya sUkSmaM pRSTiH ke vAtra koTibhuje // 147 / / iSTadvisamacatura kSetrasya sUkSmaphalasaGkhyAM jJAtvA, iSTasaGkhyAM guNakaM parikalpya, iSTasaGkhyAGkabIjAbhyAM janyAyatacatura kSetraM parikalpya, tadiSTadvisamacaturazrakSetraphalavAdiSTadvisabhacaturazrAnayanasUtram tadanaguNiteSTakRtirjanyadhanonA bhujAhRtA muravaM koThiH / dviguNA samuravA bhUdaulamvaH karNau bhuje tadiSTahRtAH // 148 // atroddeshkH|| sUkSmadhanaM sapteSTaM trikaM hi bIje dvike trike dRSTe / dvisamacaturazrabAhU mukhabhUmyavalambakAn brUhi // 149 // iSTasUkSmagaNitaphalavatrisamacaturazrakSetrAnayanasUtramiSTaghana bhaktadhanakRtiriSTayutArdha bhujA dviguNiteSTama / vibhujaM muravamiSTAptaM gaNitaM yavalambakaM trisamajanye // 15 // atroddezakaH / kasyApi kSetrasya trisamacaturbAhukasya sUkSmadhanam / SaNNavatiriSTamaSTau bhUbAhumuravAvalambakAni vada // 11 // sUkSmaphalasaGkhyAM jJAtvA caturbhiriSTacchedaizca viSamacaturazrakSetrasya muravabhUmujApramANasaGkhyAnayanasUtram---- dhanakRtiriSTacchedaizcatubhirAptaiva labdhAnAm / yutidalacatuSTayaM tairUnA viSamAkhyacatura zrabhujasaGkhyA // 152 // atroddeshkH| navatirhi sUkSmagaNitaM chedaH paJcaiva navaguNaH / dazadhRtiviMzatiSatihataH krmaadvissmcturshre|| . muravabhUmibhunAsayA vigaNayya mamAzu saGkathaya // 153 // For Private and Personal Use Only

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* kSetragaNitavyavahAraH. ii sUkSmagaNitaphalaM jJAtvA tatsUkSmagaNitaphalavatsamatribAhukSetrasya bAhusaGkhyAnapanasUtram gaNitaM tu caturguNitaM vargIkRtvA' majet tribhirlabdham / tribhujasya kSetrasya ca samasya bAhoH kRtervargam / / 154 // __ atroddezakaH / kasyApi samaya kSetrasya ca gaNitamuddiSTam / rUpANi trINyeva brUhi pragaNayya meM bAhum / / 155 / / sUkSmagaNitaphalasaGkhyAM jJAtvA tatsUkSmagaNitaphalavaTvisamatribAhu. kSetrasya mujabhUmyavalambakasakhyAnayanasUtram icchAptadhanecchAkRtiyutimUlaM doH kSitirdviguNitecchA / icchAptadhanaM lambaH kSetre dvisamatribAhujanye syAt / / 156 // atroddezakaH / kasyApi kSetrasya dvisamatrimujamya sUkSmagaNitaminAH / trINIcchA kathaya sarave bhujabhUmyavalambakAnAzu / / 157 // sUkSmagaNitaphalasaGkhyAM jJAtvA tatsUkSmagaNitaphalavadviSamatribhujAnayanasya sUtram--- bhaSTaguNiteSTakRtiyutadhanapadaghanamiSTapadahadiSTArdham / bhUH syAbUnaM dvipadAhRteSTavarge bhuje ca saGkamaNam / / 158 // aloddezakaH / kasyApi viSamabAhosvyazrakSetrasya sUkSmagaNitamidam / he rUpe nirdiSTe trINISTaM bhUmibAhavaH ke syuH // 1593 // punarapi sUkSmagaNitaphalasaGkhyAM jJAtvA tatphalavadviSamatribhujAnayanasUtram 'vargIkRtvA ought to be vargIkRtya: but this form will not suit the require. ments of the metre. For Private and Personal Use Only

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132 gaNitasArasagrahaH. khASTahatAtseSTakRteH kRtimUlaM ceSTamitaraditarahRtam / jyeSThaM svalpA?naM svalpArdha tatpadena ceSTena // 160 // kramazo hatvA ca tayoH saGkamaNe bhUbhujau bhavataH / iSTArdhamitaradoH syAdviSamatrakoNake kSetre // 1613 // atroddezakaH / he rUpe sUkSmaphalaM viSamAtribhujasya rUpANi / trINISThaM bhUdoSo kathaya sarave gaNitatattvajJa // 1623 // sUkSmagaNitaphalaM jJAtvA tatsUkSmagaNitaphalavatsamaTattakSetrAnayanastram gaNitaM caturabhyastaM dazapadabhaktaM pade bhavedyAsaH / sUkSmaM samavRttasya kSetrasya ca pUrvavatphalaM paridhiH // 163 / / bhatroddezakaH / samavRttakSetrasya ca sUkSmaphalaM paJca nirdiSTam / viSkambhaH ko vAsya pragaNayya mamAzu taM kathaya // 164 / / vyAvahArikagaNitaphalaM ca sUkSmaphalaM ca jJAtvA tavyAvahArikaphalavatatsUkSmagaNitaphalavAvisamacaturazrakSetrAnayanasya trisamacaturazrakSetrAnayanasya ca sUtram dhanavargAntarapadayutiviyutISTaM bhUmurave bhuje sthUlam / dvisame sapadasthUlAtpadayutiviyutISTapadahRtaM trisame // 1653 // ___ atroddezakaH / gaNitaM sUkSmaM paJca trayodaza vyAvahArikaM gaNitam / dvisamacaturazrabhamuravadoSaH ke SoDazecchA ca // 1663 // For Private and Personal Use Only

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133 kSetragaNitavyavahAraH. 138 trisamacatura asyodaahrnnm| gaNitaM sUkSmaM paJca trayodaza vyAvahArikaM gaNitam / trisamacaturazrabAhUn saJcintya sarave mamAcakSva // 1673 // vyAvahArikasthUlaphalaM sUkSmaphalaM ca jJAtvA tavyAvahArikasthUlaphalavatsUkSmagaNitaphalavatsamatribhujAnayanasya ca samavRttakSetravyAsAnayanasya ca sUtram dhanavargAntaramUlaM yattanmUlADvisaGguNitam / / bAhustrisamatribhuje samasya vRttasya viSkambhaH // 1986 // atroddezakaH / sthUlaM dhanamaSTAdaza sUkSmaM trighano navAhataH karaNiH / vigaNayya sarave kathaya trisamatribhujapramANaM me // 169 // paJcakRterva! dazaguNitaH karaNirbhavedidaM sUkSmam / sthUlamapi paJcasaptatiretatko vRttaviSkambhaH // 1706 // vyAvahArikasthUlaphalaM ca sUkSmagaNitaphalaM ca jJAtvA tavyAvahArikaphalavattatsUkSmaphalavaddisamatribhujakSetrasya bhabhujApramANasaGkhyayorAnayanasya sUtram phalavargAntaramUlaM dviguNaM bhUrvyAvahArikaM bAhuH / mamyardhamUlabhakte dvisamatribhujasya karaNamidam / / 171 // atroddezakaH / sUkSmadhanaM SaSTiriha sthUladhanaM padhaSaSTiruddiSTam / gaNayitvA hi sarave hisamatribhujasya mujasaGkhyAm // 172 / / For Private and Personal Use Only

Page #168
184 gANatasArasabahaH. iSTasaGkhyAvaTvisamacatura zrakSetraM jJAtvA taDvisamacaturazrakSetrasya sU. kSmagaNitaphalasamAnasUkSmaphalavadanyadvisamacaturazrakSetrasya bhabhujamuravasA / khyAnayanasUtramlambakRtAviSTenAsamasaGkamaNIkate bhujA jyeSThA / tasvayutiviyuti muravabhUyutidalitaM talamurave dvisamacaturazre // 173 // bhatroddezakaH / bhUrindrA dorvizve vakraM gatayo'valambako ravayaH / / iSTaM dik sUkSmaM tatphalavaTvisamacaturazramanyat kim // 1743 // dvisamacaturazrakSetravyAvahArikasthUlaphalasaGkhyAM jJAtvA tavyAvahArikasthUlaphale iSTasaGkhyAvibhAge kRte sati taTvisamacaturazrakSetramadhye tattadvAgasya bhUmisakhyAnayane'pi tattatsthAnAvalambakasaGkhyAnayane'pi sU. tramravaNDayutibhaktatalamuvikRtyantaraguNitarakhaNDamukhavargayutam / mUlamadhastalamukhayutadalahatalabdhaM ca lambakaH kramazaH // 175 // __ atroddeshkH| vadanaM saptoktamadhaH kSitispayoviMzatiH punastriMzat / vAhU dvAbhyAM bhaktaM caikaikaM labdhamatra kA bhUmiH // 1763 // mamirdviSaSThizatamatha cASThAdaza vadanamatra sandRSTam / lambazcatuzzatIMdaM kSetraM bhaktaM naraizcaturbhizca // 177 / / ekatikatrikacatuHkhaNDAnyekaikapuruSalabdhAni / prakSepatayA gaNitaM talamapyavalambakaM brUhi // 178 // bhUmirazItirvadanaM catvAriMzaJcaturguNA SaSTiH / . avalambakapramANaM trINyaSTau pazca ravaNDAni // 179 // For Private and Personal Use Only

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kSetragaNitavyavahAraH. 135 stambhayapramANasaGkhyAM jJAtvA tatstambhadvayAgre sUtradvayaM bar3hA tatsU. tradvayaM karNAkAreNa itaretarastambhamUlaM vA tatstambhamUlamatikramya vA saMspR. zya tatkarNAkArasUtradayasparzanasthAnAdArabhya adhaHsthitabhUmiparyantaM tanmadhye ekaM sUtraM prasArya tatsUtrapramANasaGkhyaiva antarAvalambakasaMjJA bhavati / antarAvalambakasparzanasthAnAdArabhya tasyAM bhUmyAmubhayapArzvayoH karNAkArasUtradvayasparzanaparyantamAvAdhAsaMjJA syAt / tadantarAvalamvakasaGyAnaya. nasya AbAdhAsaGkhyAnayanasya ca sUtram stambhau rajvantarabhUhatau khayogAhRtau ca bhUguNitI / AbAdhe te vAmaprakSepaguNo'ntaravalambaH // 1803 // __ atroddezakaH / SoDazahastocchrAyau stambhAvavanizca SoDazoddiSTau / AbAdhAntarasaGkhyAmatrApyavalambakaM brUhi // 1813 // stammaikasyocchrAyaH SaTtriMzadizatidvitIyasya / / mami daza hastAH kAbAdhA ko'yamavalambaH // 182 // hAdaza ca pazcadaza ca sambhAntara bhAmarApi ca catvAraH / dvAdazakastambhAgrAdranuH patitAnyato mUlAt // 183 // Akramya caturhastAtparasya mUlaM tathaikahastAcca / patitApAtkAbAdhA ko'sminnavalambako bhavati // 1843 / bAhupratibAhU dvau trayodazAvaniriyaM caturdaza ca / vadane'pi caturhastAH kAbAdhA ko'ntarAvalambazca / / 1853 // kSetramidaM murava bhUmyorekaikonaM parasparAgrAcca / rajjuH patitA mUlAcvaM brUyavalambakAbAdhe // 1863 // For Private and Personal Use Only

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136 gaNitasArasabahaH. bAhupayodazaikaH paJcadaza pratibhujA muravaM sapta / bhUmiriyamekaviMzatirasminnavalambakAbAdhe // 1876 // samacaturazrakSetraM viMzatihastAyataM tasya / koNebhyo'tha catubhyoM vinirgatA rajjavastatra // 188 // bhujamadhyaM dviyugabhuje' rajjuH kA syaatsusNviitaa| ko vAvalambakaH syAdAbAdhe ke'ntare tasmin // 1896 // stambhasyonnatapramANasaGkhyA jJAtvA tasmin stambhe yenakenacitkAra. Nena bhagne patite sati tatstambhAramUlayormadhye sthitI bhUsaGkhyAM jJAtvA tatstambhamUlAdArabhya sthitaparimANasaGkhyAnayanasya sUtram- -- nirgamavargAntaramitivargavizeSasya yadbhavedardham / nirgamanena vibhaktaM tAvasthitvAtha bhamaH syAt // 1906 // atroddeshkH| stambhasya paJcaviMzatirucchrAyaH kazridantare manaH / sambhAramUlamadhye pazca sa gatvA kiyAn bhanaH // 1916 // veNUcchrAye hastAH saptakRtiH kazcidantare bhanaH / bhUmizca saikaviMzatirasya sa gatvA kiyAn bhamaH // 1923 // vRkSocchrAyo vizatiraprasthaH ko'pi tatphalaM puruSaH / / karNAkRtyA vyakSipadatha tarumUlasthitaH puruSaH // 193 / / tasya phalasyAbhimuvaM pratibhujarUpeNa gatvA ca / phalamanahIca tatphalanarayorgatiyogasaGkhyaiva // 1943 // paJcAzadabhUttatphalagatirUpA karNasaGkhyA kaa| takSamUlagatanaragatirUpA pratibhujApi kiyatI syAt // 199 / / bhajacatuSaM ca is the reading found in the MS.., but it is not, correot. * The Handhi in Sa is gramatically inoorrect; but the author seems to have intended the phonetic fusion for the take of the amete; vide stanza 2047 of this chapter. For Private and Personal Use Only

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kSetragaNitavyavahAraH. 1187 jyeSThastambhasaGkhyAM ca alpastambhasaGkhyAM ca jJAvA ubhayasta. mbhAntarabhUmisaGkhyAM jJAtvA tajyeSThasakhye bhagne sati jyeSThasambhAgre aspastambhAnaM sTazati sati jyeSThastambhasya manasaGkhyAnayanasya sthitazeSaptayAnayanasya ca sUtram jyeSThastambhasya kRterhasvAvanivargayutimapodyArtham / stambhavizeSeNa hRtaM labdhaM bhagronnatirbhavati // 1963 // ___atroddezakaH / stambhaH paJcocchrAyaH parastrayoviMzatistathA jyeSThaH / madhyaM dvAdaza agrajyeSThAgraM patitamitarAye // 197 // AyatacaturazrakSetrakoThisaGkhyAyAstRtIyAMzadvayaM parvatotsedhaM pari. kalpya tatparvatotsedhasaGkhyAyAH sakAzAt tadAyattacaturazrakSetrasya bhuja. saGkhyAnayanasya karNasaGkhyAnayanasya ca sUtram giryutsedho dviguNo giripuramadhyakSitigirerardham / gagane tatrotpatitaM giryardhavyAsasaMyutiH karNaH // 19 // atroddeshkH| SaDyojanordhvaziravariNi yatIzvarau tiSThatastatra / eko'GgicaryayAgAttatrApyAkAzacAryaparaH // 199 // zrutivazamutpatya puraM giriziravarAnmUlamavaruhyAnyaH / samagatiko saJjAtau nagaravyAsaH kimutpatitam // 200 // DolAkArakSetre sambhadvayasya vA giridvayasya vA utsedhaparimANa. saGkhyAmeva AyatacaturazrakSetradvaye bhujadvayaM parikalpya taddiridvayAntarabhamyAM vA tatstambhadvayAntarabhUmyAM vA AbAdhAdvayaM parikalpya tadAbAdhA. 12 For Private and Personal Use Only

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138 : gaNita sArasaGgrahaH. iyaM vyutkrameNa nikSipya tadyutkramaM nyastAbAdhAdvayameva AyatacaturazrakSetra. dvaye koTidvayaM parikalpya tatkarNadvayasya samAnasaGkhyAnayanasUtramDolAkArakSetra stambhadvitayordhvasaGkhye vA / Acharya Shri Kailassagarsuri Gyanmandir zikharidvayordhvasaGkhye parikalpya bhujadvayaM trikoNasya || 2013 // taddordvitayAntaragata bhUsaGkhyAyAstadAbAce / AnIya prAgvatte vyutkramataH sthApya te koTI || 2023 // syAtAM tasminnAyatacaturazrakSetrayozca taddorbhyAm / koTibhyAM karNau dvau prAgvatsyAtAM samAnasaGkhyau tI // 203 // atroddezakaH / stambhastrayodazakaH pacadazAnyazcaturdazAntaritaH / rajjurbaddhA zikhare bhUmIpatitA ka' AbAdhe || te rajjU samasaGkhye syAtAM tadrajjumAnamapi kathaya // 209 // dvAviMzatirutsedho girestathASTAdazAnyazailasya / viMzatirubhayormadhye tayozca zivayossthitau sAdhU // 206 // AkAzacAriNau tau samAgatau nagaramatra bhikSAyai / samagatiko saJjAtau tatrAvAdhe kiyatsaGkhye || samagatisaGkhyA kiyatI DolAkAre'tra gaNitajJa // 207 // viMzatirekasyonnatiradrezva jinAstathAnyasya / tanmadhyaM dvAviMzatiranayoradyozca zRGgayoH sthitvA // 208 // AkAzacAriNI hI tanmadhyapuraM samAyAtau / bhikSAyai samagatikau syAtAM tanmadhyaziravarimadhyaM kim // 209 // viSamatrikoNakSetrarUpeNa hInAdhika gatimatornarayoH samAgamadina. sakhyAnayanasUtram -- 1. ka AbAdhe is grammatically incorrect since there oan be no sandhi between ke in the dual number and AbAdhe ; vide footnote on page 136. For Private and Personal Use Only

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kSetragaNitavyavahAraH. 139 dinagatikRtisaMyogaM dinagatikRtyantareNa hatvAtha / hatvodaggatidivasaistallabdhadine samAgamaH syAnnoH / / 2103 // atroddezakaH / he yojane prayAti hi pUrvagatistrINi yojanAnyaparaH / uttarato gacchati yo gatvAsau tadinAni paJcAtha // 2113 // gacchan karNAkRtyA katibhirdivasairnara samAnoti / ubhayoyugapadmanaM prasthAnadinAni sadRzAni // 2123 // paJcavidhacaturazrakSetrANAM ca trividhatrikoNakSetrANAM cetyaSTavidhabAhyasattavyAsasakhyAnayanasUtram zrutirabalambakabhaktA pArzvabhujaghnA caturbhuje tribhuje / bhujaghAto lambahRto bhavedvahirvRttaviSkambhaH // 213 // atroddezakaH / samacatura asya trikabAhupratibAhukasyaM cAnyasya / koTiH paJca dvAdaza bhujAsya kiM vA bahirvRttam // 2143 / / bAhU trayodaza muravaM catvAri dharA caturdaza proktaa| dvisamacaturazrabAhiraviSkambhaH ko bhavedatra / / 215 / / paJcakRtirvadana mujAzcatvAriMzaJca bhuumirekonaa|| trisamacatura zrabAhiravRttavyAsaM mamAcakSva // 2113 // vyekA catvAriMzadAhuH pratibAhuko dvipaJcAzat / SaSTibhUmirvadanaM pazcakRtiH ko'tra viSkambhaH // 217 // trisamasya ca SaT bAhustrayodaza dvisamabAhukasyApi / bhUmirdaza viSkammAvanayoH ko bAhyavattayoH kathaya / / 218 // For Private and Personal Use Only

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140 gaNitasArasagrahaH. bAhU paJcavyuttaradazakau bhUmizcaturdazo viSame / tribhujakSetre bAhiravRttavyAsaM mamAcakSva / / 2196 // dvikabAhuSaDazrasya kSetrasya bhavedvicintya kathaya tvam / bAhiraviSkambha me paizAcikamatra yadi vetsi / / 220 // iSTasaGakhyAvyAsavatsamavRttakSetramadhye samacaturazrAdyaSTakSetrANAM mukha. bhamujasakhyAnayanasUtram ----- labdhavyAseneSTavyAso vRttasya tasya bhaktazca / labdhena bhujA guNayedbhavecca jAtasya bhujasaGkhyA / / 2213 // atroddezakaH / vRttakSetravyAsastrayodazAbhyantare'tra saJcintya / samacaturazrAdyaSTakSetrANi sarave mamAcakSva // 222 / / AyatacaturazraM vinA pUrvakalpitacaturazrAdikSetrANAM sUkSmagaNitaM ca rajjusakhyAM ca jJAtvA tattatkSetrAbhyantarAvasthitavRttakSetraviSkammAnayanasUtram -- paridheH pAdena bhajedanAyatakSetrasUkSmagaNitaM tat / kSetrAbhyantaravRtte viSkambho'yaM vinirdiSTaH // 223 / / atroddeshkH| samacatura zrAdInAM kSetrANAM pUrvakalpitAnAM ca / kRtvAbhyantaravRttaM brUhyadhunA gaNitatattvajJa // 2246 / / samavRttavyAsasaGkhyAyAmiSTasaGkhyAM bANaM parikalpya tadvANaparimANasya jyAsayAnayanasUtrama--- For Private and Personal Use Only

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kSetra gaNitavyavahAraH. vyAsAdhigamonassa ca caturguNitAdhigamena saGguNitaH / yattasya vargamUlaM jyArUpaM nirdizetprAjJaH // 229 // atroddezakaH / vyAso daza vRttasya dvAbhyAM chinno hi rUpAbhyAm / chinnasya jyA kA syAtpragaNadhyAcakSva tAM gaNaka / / 2263 // samavRtta kSetra vyAsasya ca maurvyAzca saGkhyAM jJAtvA bANasakhyAnayanasUtram - vyAsajyArUpakayorvargavizeSasya bhavati yanmUlam / tadviSkambhAcchobhyaM zeSArdhamiSuM vijAnIyAt // 227 // atroddezakaH / daza vRttasya viSkambhaH ziJjinyabhyantare sarakhe / dRSTASTau hi punastasyAH kaH syAdadhigamo vada / / 228 // jyAsaGkhyAM ca bANasaGkhyAM ca jJAtvA samavRttakSetrasya madhyavyAsa. sayAnayanasUtram - bhaktazcaturguNena ca zareNa guNavargarAziriSusahitaH / samavRttamadhyamasthitaviSkambho'yaM vinirdiSTaH // 229 // atroddezakaH / kasyApi ca samavRttakSetrasyAbhyantarAdhigamanaM he / jyA dRSTASTau daNDA madhyavyAso bhavetko'tra // 2103 // 141 For Private and Personal Use Only samavRttadvayasaMyoge ekA matsyAkRtirbhavati / tanmatsyasya mukhapuccha. vinirgatarekhA kartavyA / tathA rekhayA anyonyAbhimukhadhanurdvayAkRtirbha

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142 gaNitasArasahaH. vati / tanmukhapucchavinirgatareravaiva tadanuIyasyApi jyAkatirmavati / taddhanuIyasya zaradvayameva vRttaparasparasampAtazarau jJeyo / samavRttadvayasaMyoge tayoH sampAtazarayorAnayanasya sUtram grAsonavyAsAbhyAM grAse prakSepakaH prakartavyaH / vRtte ca parasparataH sampAtazarau vinirdiSTau // 231 // ___ atroddezakaH / samavRttayoIyorhi dvAtriMzadazItihastavistRtayoH / grAse'STau ko bANAvanyonyabhavau samAcakSva // 232 // iti paizAcikavyavahAraH samAptaH / / iti sArasabahe gaNitazAstre mahAvIrAcAryasya kRtau kSetragaNitaM nAma SaSThavyavahAraH samAptaH // For Private and Personal Use Only

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saptamaH khAtavyavahAraH. sarvAmarendramakuTArcitapAdapIThaM sarvajJamavyayamacintyamanantarUpam / bhavyaprajAsarasijAkarabAlabhAnu bhaktyA namAmi zirasA jinavardhamAnam // 1 // kSetrANi yAni vividhAni puroditAni teSAM phalAni guNitAnyavagAhanAni (nen)| karmAntikauNDUphalasUkSmavikalpitAni vakSyAmi saptamamidaM vyavahAraravAtam // 2 // sUkSmagaNitam. atra paribhASAzlokaH hastaghane pAMsUnAM dvAtriMzatpalazatAni pUryANi / utkIryante tasmAt SaTtriMzatpalazatAnIMha // 3 // svAtagaNitaphalAnayanasUtram--- kSetraphalaM vedhaguNaM samaravAte vyAvahArikaM gaNitam / muravatalayutidalamatha satsaGkhyAptaM syAtsamIkaraNam // 4 // atroddezakaH. samacatura zrasyASTI bAhuH pratibAhukazca vedhazca / kSetrasya vAtagaNitaM samaravAte kiM bhavedatra // 5 // tribhujasya kSetrasya dvAtriMzadvAhukasya vedhe tu / SaTtriMzadRSTAste SaDaGgulAnyasya kiM gaNitam // 6 // sASTazatavyAsasya kSetrasya hi paJcaSaSTisahitazatam / vedho vRttasya tvaM samaravAte kiM phalaM kathaya // 7 // Ayatacaturazrastha vyAsaH paJcAyaviMzatirbAhuH / SaSTivaidho'STazataM kathayAzu samasya vAtasya // 8 // For Private and Personal Use Only

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144 gaNitasArasaGgrahaH. masmin svAtagaNite karmAntikasaMjJaphalaM ca auNDUsaMjJaphalaM ca jJAtvA tAbhyAM karmAntikoNDUsaMjJaphalAbhyAM sUkSmakhAtaphalAnayanasUtram -- bAhyAbhyantarasaMsthitatattatkSetrasthabAhukoThabhuvaH / khapratibAhusametA bhaktAstatkSetragaNanayAnyonyam // 9 // guNitAzca vedhaguNitAH karmAntikasaMjJagaNitaM syAt / tadvAhyAntarasaMsthitatattatkSetre phalaM samAnIya // 10 // saMyojya saGkhyayAptaM kSetrANAM vedhaguNitaM ca / auNDraphalaM tatphalayorvizeSakasya tribhAgena / / saMyuktaM karmAntikaphalameva hi bhavati sUkSmaphalam // 11 // atroddeshkH| samacaturazrA vApI viMzatiruparIha SoDazaiva tle| vedho nava kiM gaNitaM gaNitavidAcakSva me zIghram / / 12 / / vApI samatribAhurviMzatiruparIha SoDazaiva tale / vedho nava kiM gaNitaM karmAntikamauNDUmApi ca sUkSmaphalam // 13 // samavRttAsau vApI viMzatiruparIha SoDazaiva tale / veSo dvAdaza daNDAH kiM syAtkarmAntikauNDUsUkSmaphalam / / 14 // Ayatacaturazrasya tvAyAmappaSTireva vistAraH / dvAdaza murave tale'dhaM vedho'STau kiM phalaM bhavati // 15 // navatirazItiH saptatirAyAmazcordhvamadhyamUleSu / vistAro dvAviMzat SoDaza daza sapta vedho'yam // 16 // vyAsaH SaSTirvadane madhye triMzattale tu pnycdsh| . samavRttasya ca vedhaH SoDaza kiM tasya gaNitaphalam // 17 // For Private and Personal Use Only

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khAtavyavahAraH. 146 tribhujasya murave'zItiH SaSTimadhye tale ca paJcAzat / bAhutraye'pi vedho nava kiM tasyApi bhavati gaNitaphalam // 18 // ravAtikAyAH ravAtagaNitaphalAnayanasya ca vAtikAyA madhye sUcImuravAkAravat utsedhe sati khAtagaNitaphalAnayanasya ca sUtram pariravAmuravena sahito viSkambhastribhujavRttayostriguNAt / AyAmazcaturazre caturguNo vyAsasaGguNitaH // 19 // sUcImukhavadvedhe pariravA madhye tu pariravArdham / muravasahitamatho karaNaM prAgvattalasUcivedhe ca // 20 // atroddezakaH. tribhujacaturbhujavattaM puroditaM pariravayA parikSiptam / daNDAzItyA vyAsaH parivAzcatururvikAstrivedhAH syuH / / 21 / / AyatacaturAyAmo viMzatyuttarazataM punarvyAsaH / / catvAriMzat pariravA catururvIkA trivedhA syAt // 22 // utsedhe bahuprakAravati sati khAtaphalAnayanasya ca, yasya kasyacita svAtaphalaM jJAtvA tatvAtaphalAt anyakSetrasya khAtaphalAnayanasya ca sUtram vedhayutiH sthAnahatA vedho mukhaphalaguNaH svaravAtaphalam / tricaturbhujavRttAnAM phalamanyakSetraphalahRtaM vedhaH // 23 // atroddezakaH / samacatura zrakSetre bhUmicaturhastamAtra vistAre / tatraikadvitricaturhastanikhAte kiyAn hi samavedhaH // 24 // 14-A For Private and Personal Use Only

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Shri Mahavir Jain Aradhana Kendra 146 www.kobatirth.org gaNitasArasaGgrahaH. sUtram - samacaturazrASTAdazahastabhujA vApikA caturvedhA | vApI tajjalapUrNAnyA navabAhAtra ko vedhaH // 253 // yasya kasyacitkhAtasya UrdhvasthitabhujAsaGkhyAM ca adhassthita - bhujAsaGkhyAM ca utsedhapramANaM ca jJAtvA, tatkhAte iSTotsedhasaGkhyAyAH bhujAsaGkhyAnayanasya, aghassUcivedhasya ca saGkhyAnayanasya sUtram - mukhaguNavedho mukhatalazeSahRto'traiva sUcivedhaH syAt / viparItavedhaguNamurakhatalayutyavalambayAsaH // 26 // atroddezakaH / samacaturazrA vApI viMzatirUrdhvaM caturdazAdhazca / vedho mukhe navAdhastrayo bhujAH ke'tra sUcivedhaH kaH // 273 // golakAkArakSetrasya phalAnayanasUtram - vyAsArthaghanArdhaguNA nava golavyAvahArikaM gaNitam / taddazamAMzaM navaguNamazeSasUkSmaM phalaM bhavati // 28 // atroddezakaH / Acharya Shri Kailassagarsuri Gyanmandir SoDazaviSkambhasya ca golakavRttasya vigaNayya / kiM vyAvahArikaphalaM sUkSmaphalaM cApi me kathaya // 293 // zRGgATakakSetrasya khAtavyAvahArika phalasya khAtasUkSmaphalasya va bhuja kRtidalaghanaguNadazapadanavahaya / vahArikaM gaNitam / triguNaM dazapadabhaktaM zRGgATakasUkSmaghanagaNitam // 103 // atroddezakaH / *yazrasya ca zRGgATakaSaDUbAhughanasya gaNapitvA / kiM vyAvahArikaphalaM gaNitaM sUkSmaM bhavetkathaya // 313 // For Private and Personal Use Only

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khAtavyavahAraH. 147 vApIpraNAlikAnAM vimocane tattadiSTapraNAlikAsaMyoge tajjalena vApyAM pUrNAyAM satyAM tattatkAlAnayanasUtram --- vApIpraNAlikAH svasvakAlabhaktAH savarNavicchedAH / tadyutibhaktaM rUpaM dinAMzakaH syAtpraNAlikAyutyA // tadinabhAgahatAste tajjalagatayo bhavanti tadvApyAm // 33 // atroddezakaH / catastraH praNAlikAH syustatraikaikA prapUrayati vApIm / dvitricatuHpaJcAMzerdinasya katibhirdinAMzaistAH // 34 // trairAzikAkhyacaturthagaNitavyavahAre sUcanAmAtrodAharaNameva ; atra samyagvistArya pravakSyate - samacaturazrA vApI navahastadhanA nagasya tale / tacchiravarAjaladhArA caturazrAGgulasamAnaviSkambhA // 35 // patitAgre vicchinnA tayA ghanA sAntarAlajalapUrNA / zailotsedhaM vApyAM jalapramANaM ca me brUhi // 31 // vApI samacaturazrA navahastaghanA nagasya tale / aGgalasamavRttaghanA jaladhArA nipatitA ca tacchikharAt // 37 // agre vicchinnAbhUttasyA vApyA mukhaM praviSTA hi / sA pUrNAntaragatajaladhArotsedhena zailasya / utsedhaM kathaya sarave jalapramANaM ca vigaNayya // 38 // samacaturazrA vApI navahataghanA nagasya tle| tacchikharAjjaladhArA patitAGgulavanatrikoNA sA // 39 // vApImuravapraviSThA sAgre chinnAntarAlajalapUrNA / kathaya sarave vigaNayya ca giryutsedhaM jalapramANaM ca // 40 // For Private and Personal Use Only

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148 gaNitasArasaGgrahaH. samacaturazrA vApA navahastaghanA nagasya tale / aGgulavistArAGgularavAtAGgulayugaladIrghajaladhArA // 416 // patitAye vicchinnA vApImuravasaMsthitAntarAlajalaiH / sampUrNA syAdvApI giryutsedho jalapramANaM kim / / 42 // iti vAtavyavahAre sUkSmagaNitaM sampUrNam / citigaNitama. itaH paraM vAtavyavahAre citigaNitamudAhariSyAmaH / atra paribhASA--- hasto dI| vyAsastadardhamaGgulacatuSkamutsedhaH / dRSTastatheSTakAyAstAbhiH karmANi kAryANi / / 43 // iSTakSetrasya khAtaphalAnayane ca tasya khAtaphalasya iSTakAnayane ca sUtram-- mukhaphalamudayena guNaM tadiSTakAgaNitabhaktalabdhaM yat / citigaNitaM tadvidyAttadeva bhavatISTakAsaGkhyA // 44 // atrodezakaH / vediH samacaturazrA sASTabhujA hastanavakamutsedhaH / ghaTitA tadiSTakAbhiH katISTakAH kathaya gaNitajJa // 45 // aSTakarasamatrikoNanavahastotsedhavedikA rcitaa| pUrveSTakAbhirasyAM katISTakAH kathaya vigaNayya // 46 // samavRttAkRtivedirnavahato; kraassttkvyaasaa|| ghaTiteSTakAbhirasyAM katISTakAH kathaya gaNitajJa / / 475 / / Ayatacatura zrasya tvAyAmaH SaSTireva vistAraH / paJcakRtiH Sa vedharatadiSTakAcitimihAcakSva // 48 // For Private and Personal Use Only

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khAtavyavahAraH. 149 prAkArasya vyAsaH sapta caturvizatistadAyAmaH / ghaTiteSTakAH kati syuzcocchrAyo viMzatistasya // 49 // vyAsaH prAkArasyorce SaDadho'thASTa tIrthakA dIrghaH / ghaTiteSTakAH kati syucocchrAyo viMzatistasya // 50 // dvAdaza SoDaza viMzatirutsedhAH sapta SaT ca paJcAdhaH / vyAsA murave catustrihikAzcaturviMzatirdIrghaH // 11 // iSTavedikAyAM patitAyAM satyAM sthitasthAne iSTakAsaGkhyAnayanasya ca patitasthAne iSTakAsaGkhyAnayanasya ca sUtram muravatalazeSaH patitotsedhaguNaH sakalavedhahatsamuravaH / mukhabhUmyobhUmimukhe pUrvoktaM karaNamavaziSTam // 12 // __ atroddeshkH| dvAdaza dairdhya vyAsaH paJcAdhazcordhvamekamutsedhaH / daza tasmin paJca karA bhagnAstatreSTakAH kati syustAH // 13 // prAkAre karNAkAreNa bhagne sati sthiteSTakAnayanasya ca patiteSTakAnayanasya ca sUtram bhUmimurave dviguNe muravabhUmiyate'bhamabhUdayayutone / daiyodayaSaSThAMzapne sthita patiteSTakAH krameNa syuH // 14 // atroddezakaH / prAkAro'yaM mUlAnmadhyAvartena vAyunA vidvaH / / karNAkRtyA bhagnastatsthitapatiteSTakAH ki tyaH syuH // 55 // prAkAro'yaM mUlAnmadhyAvartena caikahastaM gatvA / karNAkRtyA bhagnaH katoSTakAH syuH sthitAzca patitAH kaaH|| 56 / / For Private and Personal Use Only

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150 gaNitasArasaGgrahaH. prAkAramadhyapradezotsedhe taravRddhacAnayanasya prAkArasya ubhayapArzvayoH tarahAnerAnayanasya ca sUtram iSTeSTakodayahRto vedhazca tarapramAName konam / mukhatalazeSeNa hRtaM phalameva hi bhavati tarahAniH // 57 // atroddezakaH / prAkArasya vyAsaH sapta tale viMzatistadutsedhaH / / ekenAne ghaTitastarabRddhayUne karodayeSTakayA / / 586 // samavRttAyAM vApyAM vyAsacatuSke'rdhayuktakarabhUmiH / ghaTiteSTakAbhirabhitastasyAM vedhastrayaH kAH syuH / ghaTiteSTakAH sarakhe me vigaNayya brahi yadi vetsi // 6 // iSTakAghaTitasthale adhassalavyAse sati UrdhvatalavyAse sati ca gaNitanyAyasUtram dviguNanivezo vyAsAyAmayuto dviguNitastadAyAmaH / Ayatacatura zre syAdutsedhavyAsasaGgaNitaH // 61 // atroddezakaH / vidyAdharanagarasya vyAso'STau dvAdazaiva cAyAmaH / paJca prAkAratale murave tadekaM dazotsedhaH // 62 / / iti ravAtavyavahAre citigaNitaM samAptam / krakacikAvyavahAraH. itaH paraM krakacikAvyavahAramudAhariSyAmaH / tatra paribhASAhastadvayaM SaDaGgalahInaM kiSkvAhvayaM bhavati / iSTAdyantacchedanasaGkhyaiva hi mArgasaMjJA syAt / / 63 // atha zAkAkhyayAdidrumasamudAyeSu vakSyamANeSu / vyAsodayamArgANAmaGgulasaMkhyA parasparannAptA // 14 // For Private and Personal Use Only

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vAtavyavahAraH. hastAGgulavargeNa krAkacike paTTikApramANa syAt / zAkAhayadvamAdidrumeSu pariNAha dairdhya hastAnAm || 61 // saMkhyA parasparannA mArgANAM saMkhyayA guNitA / tatpaTTikAsamAptA krakacakRta karmasaMkhyA syAt // 66 // zAkArjunAmlavetasa saralAsita sarja DuNDukAkhyeSu / zrIparNI kSAkhyameSvamISvekamArgasya / SaNNavatiraGgulAnAmAyAmaH ki kureva vistAraH // 67 // atroddezakaH / 11 For Private and Personal Use Only 151 zAkAkhyatarau dIrghaH SoDaza hastAzca vistAraH / sArdhatrayazca mArgAzcASTau kAnyatra karmANi // 68 // iti svAtavyavahAre kacikAvyavahAraH samAptaH // iti sArasaGgrahe gaNitazAstre mahAvIrAcAryasya kRtau saptamaH khAtavyavahAraH samAptaH //

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aSTamaH. chAyAvyavahAraH. zAntirjinaH zAntikaraH prajAnAM jagatprabhutisamastabhAvaH' / yaH prAtihAryASTavivardhamAno namAmi taM nirjitazatrusaGgam // 1 // AdI prAcyAdyaSTadiksAdhanaM pravakSyAmaH saliloparitalavasthitasamabhUmitale liravedRttam / bimbaM svecchAzaGkuhiguNitapariNAhasUtreNa // 2 // tavRttamadhyasthatadiSTazaGkozchAyA dinAdau ca dinAntakAle / tavRttareravAM spRzati krameNa pazcAtpurastAcca kakup pradiSTA // 3 // taddigdvayAntargatatantunA liravenmatsyAlati yAmyakuberadisthAm / tatkoNamadhye vidizaH prasAdhyAzchAyaiva yAmyottaradigdizArdhajAH // 4 // ajadhaTharavisaGkamaNadyudalaja bhaikyArdhameva viSuvadA // 4 // laGkAyAM yavakoTyAM siddhapurIromakApuryoH / viSuvadbhA nAstyeva triMzaddhaThikaM dinaM bhavettasmAt // 5 // dezeSvitareSu dinaM triMzannADyAdhikonaM syAt / / meSadhaTAyanadinayostriMzadaThikaM dinaM hi sarvatra // 6 // dinamAnaM dinadalamA jyotizzAstroktamArgeNa / / jJAtvA chAyAgaNitaM vidyAdiha vakSyamANasUtraughaiH // 7 // 1 M reads tattva:. For Private and Personal Use Only

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chAyAvyavahAraH. viSuvacchAyA yatrayatra deze nAsti tatratatra deze iSTazaGkAriSTakAla cchAyAM jJAtvA tatkAlAnayanasUtrama chAyA saiA dviguNA tathA hRtaM dinamitaM ca pUrvAhNe | aparAhNe taccheSaM vijJethaM sArasaGgrahe gaNite // 8 // atroddezakaH / pUrvA pauruSI chAyAtriguNA vada kiM gatam / aparAhne'vazeSaM ca dinasyAMzaM vada priya // 93 // dinAMze jAte sati ghaTikAnayanasUtram Acharya Shri Kailassagarsuri Gyanmandir aMzahataM dinamAnaM chedavibhaktaM dinAMzake jAte / pUrvAhNe gatanAjyastvaparAhNe zeSanADyastu // 103 // atroddezakaH / viSuvacchAyAvirahitadeze'STAMza dinasya gataH zeSazcASTAMzaH kA ghaTikAH syuH khAgninADyo'hnaH // 113 // mallayuddha kAlAnayanasUtram kAlAnayanAddinagatazeSasamAsonitaH kAlaH / stambhacchAyA stambhapramANabhaktaiva pauruSI chAyA // 123 // atroddezakaH / pUrvAhNe zaGkusamacchAyAyAM mallayudramArabdham / aparAhNe dviguNAyAM samAptirAsIcca yuddhakAlaH kaH aparArdhasyodAharaNam / For Private and Personal Use Only 15% 133 // dvAdazahastastambhacchAyA caturuttaraiva viMzatikA / tatkAle pauruSikacchAyA kiyatI bhavedgaNaka || 143 //

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154 gaNitasArasaGgrahaH. viSuvacchAyAyukte deze iSTacchAyAM jJAtvA kAlAnayanasya sUtramzakyuteSTacchAyA madhyacchAyonitA dviguNA / / tadavAptA zaGkamitiH pUrvAparayodinAMzaH syAt // 11 // atroddezakaH / dvaadshaanggulshngkocudlcchaayaangguldvyii| iSTacchAyASTAGgulikA dinAMzaH ko gataH sthitH| . tryaMzo dinAMzo ghaTikAH kAstriMzannADikaM dinam // 17 // iSTanADikAnAM chAyAnayanasUtram --- dviguNitadinabhAgahRtA zaGkumitiH zaGkhamAnonA / dhudalacchAyAyuktA chAyA tatsveSTakAlikA bhavati // 18 // achoddeshkH| dvaadshaanggulshngkoryudlcchaayaangguldvyii| dazAnAM ghaTikAnAM mA kA chiMzannADikaM dinam // 19 // pAdacchAyAlakSaNe puruSasya pAdapramANasya paribhASAsUtram puruSonnatisaptAMzastatpuruSAGgrestu deyaM syAt / yadyevaM cetpuruSaH sa bhAgyavAnAbhA spaSTA // 20 // ArUDhacchAyAyAH saGkhyAnayanasUtramnRcchAyAhatazaGkabhittistambhAntaronito bhaktaH / nRcchAyayaiva labdhaM zaGkobhittyAzritacchAyA // 21 // achoddezakaH / viMzatihastaH stambho bhittistambhAntaraM karA aSTau / puruSacchAyA dvinnA bhittigatA sammamA kiM syAt // 22 // 1 Not found in any of the MSS. For Private and Personal Use Only

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chAyAvyavahAraH. 166 stambhapramANaM ca bhicyArUDhastambhacchAyAsaGkhyAM ca jJAtvA bhittistambhAntarasaGkhyAnayanasUtram puruSacchAyAnighnaM stambhArUDhAntaraM tayormadhyam / stambhArUDhAntarahRtatadantaraM pauruSI trAyA // 23 // atroddezakaH / vizatihastaH stambhaH SoDaza bhittyaashritcchaayaa| dviguNA puruSacchAyA bhittistambhAntaraM kiM syAt // 24 // aparArdhasyodAharaNam / viMzatihasaH stambhaH SoDaza mityAzritacchAyA / kiyatI puruSacchAyA bhittistambhAntaraM cASTau / / 25 // ArUDhacchAyAyAH saGkhyAM ca bhittistambhAntarabhUmisayAM ca puruSacchAyAyAH saGkhyAM ca jJAtvA stambhapramANasaGkhyAnayanasUtram-- nRcchAyAnArUDhA bhittistambhAntareNa sNyuktaa| pauruSabhAhRtalabdhaM viduH pramANa budhAH stambhe // 26 // atroddeshkH| SoDaza bhicyArUDhacchAyA dviguNaiva pauruSI chaayaa| stambhotsedhaH kaH syAdbhittistambhAntaraM cASTau // 27 / / zaGkapramANazaGkacchAyAmizravibhaktasUtram zaGkapramANazaGkacchAyAmizraM tu saikpaurussyaa| bhaktaM zaGkamitiH syAcchaGkucchAyA tadUnamizraM hi // 28 // atroddeshkH| zaGkupramANazaGkucchAyAmizraM tu paJcAzat / zaGkatsedhaH kaH syAJcaturguNA pauruSI chAyA // 29 / / For Private and Personal Use Only

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156 gaNitasArasanAhaH. ~ zaGkucchAyApuruSacchAyAmizravibhaktasUtram zaGkanaracchAyayutirvibhAjitA zaGkasaikamAnena / labdhaM puruSacchAyA zaGkucchAyA tadUnamizraM syAt // 30 // atroddezakaH / zaGkorutsedho daza nRcchAyAzaGkubhAmizram / paJcottarapaJcAzanRcchAyA bhavati kiyatI ca // 31 // stambhasya avanatisaGkhyAnayanasUtram--- chAyAvargAcchodhyA naramAkatiguNitazaGkukRtiH / saikanaracchAyAkRtiguNitA chAyAkRteH zodhyA // 32 // tanmUlaM chAyAyAM zodhyaM nrbhaanvrgruupenn| bhAgaM hRtvA labdhaM stambhasyAvanatireva syAt // 33 // atrodezakaH / dviguNA puruSacchAyA vyuttaradazahastazaGko / ekonatriMzatsA stambhAvanatizca kA tatra / / 34 // kazcidrAjakumAraH prAsAdAbhyantarasthassan / pUrvAhne jijJAsurdinagatakAlaM naracchAyAm // 35 // dvAtriMzaddhasto jAle prAgbhittimadhya AyAtA / ravibhA pazcAdittI vyakatriMzatkarordhvadezasthA // 36 // tadbhittidvayamadhyaM caturuttaraviMzatiH karAstAsman / kAle dinagatakAlaM nRcchAyAM gaNaka vigaNayya / kathayacchAyAgaNite yadyasti parizramasava cet // 37 // samacatura zrAyAM dazahastaghanAyAM nAcchAthA / puruSotsedhAdviguNA pUrvAhne prAktaThAyA // 38 // is the reading given in the MSS, for narabhAna; but it is metrically incorrect. For Private and Personal Use Only

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chAyAvyavahAraH. 157 tasmin kAle pazcAttaThAzritA kA bhavedgaNaka / ArUDhacchAyAyA AnayanaM vetsi cetkathaya // 39 // zaGkordIpacchAyAnayanasUtram zaGkanitadIponnatirAptA zaGkapramANena / tallabdhahRtaM zaGkoH pradIpazakuntaraM chAyA // 40 // atroddezakaH. zaGkupradIpayormadhyaM SaNNavatyaGgulAni hi| dvAdazAGgulazakostu dIpacchAyAM vadAzu me / SaSTirdIpazikhotsedho gaNitArNavapAraga // 42 // dIpazakuntarAnayanasUtram zaGkanitadIponnatirAptA zaGkuprAmANena / tallabdhahatA zaGkucchAyA zaGkupradIpamadhyaM syAt // 43 // __ atroddezakaH / zaGkucchAyAGgulAnyaSTau SaSTirdIpaziravodayaH / zaGkadIpAntaraM brUhi gaNitArNavapAraga // 44 // dIponnatisaGkhyAnayanamUtram zaGkucchAyAmaktaM pradIpazavantaraM saikam / zahupramANaguNitaM labdhaM dIponnatirbhavati // 15 // bhatroddezakaH / zaGkucchAyA dvinimnaiva dvizataM zaGkudIpayoH / antaraM yamulAnyatra kA dIpasya samunnatiH // 46 // For Private and Personal Use Only

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158 gaNitasArasAhaH. zahupramANamatrApi hAdazAGgulakaM gate / jJAtvodAharaNe samyagvidyAtsutrArthapaDatim // 47 // puruSasya pAdacchAyAM ca tatpAdapramANena vRkSacchAyAM ca jJAtvA vRkSo. nateH saGkhyAnayanasya ca, vRkSonnatisaGkhyAM ca puruSasya pAdacchAyAyAH sA ca jJAtvA tatpAdapramANenaiva vRkSacchAyAyAH saGkhyAnayanasya ca sUtram khacchAyayA bhaktanijeSTavRkSacchAyA punassaptabhirAhatA sA / vRkSonnatiH sAdrihRtA svapAdacchAyAhatA syAdrumabhaiva nanam // 48 // , atroddeshkH| AtmacchAyA catuHpAdA vRkSacchAyA zataM padAm / vRkSocchrAyaH ko bhavetsvapAdamAnena taM vada // 49 // vRkSacchAyAyAH saGkhyAnayanodAharaNam / AtmacchAyA catuHpAdA paJcasaptatibhiryutam / zataM vRkSonnativRkSacchAyA syAtkiyatI tadA // 10 // purato pojanAnyaSTau gatvA zailo dazodayaH / sthitaH pure ca gatvAnyo yojanAzItitastataH // 11 // tadagrasthAH pradRzyante dIpA rAtrau pure sthitaiH / puramadhyasthazailasyacchAyA pUrvAgamUlayuk / asya zailasya vedhaH ko gaNakAzu prakathyatAm // 52 / / iti sArasaGgrahe gaNitazAstre mahAvIrAcAryasya kRtau chAyAvyavahAro nAma aSTamaH samAptaH / samApto'yaM sArasaGgrahaH // For Private and Personal Use Only

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ENGLISH TRANSLATION AND NOTES. For Private and Personal Use Only

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CONTENTS. CHAPTER I. PAGE diction do. Do. TERMINOLOGY Salutation and Benediction An appreciation of the science of calculation Terminology relating to the measurement of space Do. do. time Do. grain do. gold Do. do. silver other metals ... Names of the operations in Arithmeti) ... ... ... .. General rules in regard to zero and positive and negative quantities. Words denoting nambers ... Tho names of notational placos ... ... Qualities of an arithmetician Do. do. CHAPTER II. ... ARITHMETICAL OPERATIONS Multiplication Division Squaring Square root Cubing Cube root ... Summation ... Vyutkalita (subtraction) CHAPTER III. FRACTIONS Multiplication of fractions : Division of fractions ... ... ... Squaring, Square-root, Cu bing and Cube-root of fractions Summation of fractional series in progression ... Vyutkalita of fractions in series ... ... Six varieties of fractions ... ... Simple fractions (addition and subtraction) Componnd and complex fractions Bhaganubandha fractions... Bhaga pavaha fractions ... Bhagamatr fractions For Private and Personal Use Only

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CONTENT&. CHAPTER IV. PAGE MISCELLANEOUS PROBLEMS (on fraccions)... Bhaga and Sega varieties Mula variety Segamula variety Seramula variety involving two known quantities A vngamula variety ... ... ... Bhagasariwarga variety .. Ambavarga variety charaoterized by the subtraction or addition of known quantities Mulamiera variety ... Bhinnadrsya variety : : : CHAPTER V. RULE OF THREE Inverse double and treble rule of three ... Inverse quadruple rule of three ... Problems bearing on forward and backward movement Double, treble and quadruple rule of three ... ... CHAPTER VI. 110 MIXED PROBLEMS Sankramana and Visamasankramana Double rule of three Problems bearing on interest Proportionate division Vallika-Kuttikara Visama-Kutfikara Sakala-Kuttikara Suvarna-Kutfikara ... Visitra-Kuttikara .. Summation of series 117 125 126 138 149 168 CHAPTER VII. 184 .. 187 CALCULATIONS RELATING TO THE MEASUREMENT OF AREAS Calculation relating to approximate measurement of areas The minutely accurate oalonlation of the measure of areas Subject of treatment known as the Janya operation ... Do. do. Paisacika or devilishly problems ... ... ... ... . .. 197 209 . . difficult . ... 220 For Private and Personal Use Only

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CONTENTS. CHAPTER VIII. CALCULATIONS REGARDING EXCAVATION ... .. Calculations relating to piles of bricks ... ... ... ... Operations relating to the work done with saws in sawi.ng wood PAGE 258 268 273 ... CHAPTER IX. CALCULATIONS RELATING TO SHADOWS ... ... ... . .. 275 For Private and Personal Use Only

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GANITA-SARA-SANGRAHA. ENGLISH TRANSLATION. CHAPTER I. ON TERMINOLOGY. Salutation and Benediction, 1. Salutation to Mahavira, the Lord of the Jinas, the protector (of the faithful), whose four * infinite attributes, worthy to be esteemed in (all) the three worlds, are unsurpassable (in excellence). 2. I bow to that highly glorious Lord of the Jinas, by whom, as forming the shining lamp of the knowledge of numbers, the whole of the universe has been made to shine. 3. That blessed Amoghavarsa (i.e., one who showers down truly useful rain), who (ever) wishes to do good to those whom he loves, and by whom the whole body of animals and vegetables. having been freed from the effects of) pests and drought, has been made to feel delighted : 4. He, in whose mental operations, conceived as fire, the enemies in the form of sins have all been turned into the condition of ashes, and who in consequence has become one whose anger is not futile : 5. He, who, having brought all the world under his control and being himself independent, has not been overcome by (any) opponents, and is therefore au absolute lord (liko) a new God of Love: 6. He, to whom the work of service) is rendered by a circle of kings, who have been overpowered by the progress of (his) heroism, and who, being Cakrikabhanjana by name, is in reality a enkrikabhanjana (i.e., the destroyer of the cycle of recurring re-births): * These four attributes of Jina Mahavira are said to be his faith, understand. ing, blissfulness and power. For Private and Personal Use Only

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GANITASARASANGRAHA. 7. He, who, being the receptacle of the (numerons) rivers of learning, is characterised by the adamantine bank of propriety and holds the gems (of Jainism) within, and (80) is appropriately famous as the great ocean of moral excellence: 8. May (his rule)-the rule of that covereign lord who has destroyed in philosophical controversy the position of single conclusions and propounds the logic of the syadrada*--(may the rule) of that Nrpatunga prosper! An Appreciation of the Science of Calculation. 9. In all those transactions which relate to worldly, Vedio or (other) similarly religious affairs, calculation is of use, 10. In the science of love, in the science of wealth, in music and in the drama, in the art of cooking, and similarly in medicine and in things like the knowledge of architecture : 11. In prosody, in ,,poetics and poetry, in logic and grammar and such other things, and in relation to all that constitutes the peculiar value of (all) the (various) arts : the science of computation is held in high esteem. 12. In relation to the movements of the sun and other heavenly bodies, in connection with eclipses and the conjunctions of planets, and in connection with the triprasna + and the course of the moon-indeed in all these (connections) it is utilised. 13-14. The number, the diameter and the perimeter of islands, oceans and mountains; the extensive dimensions of the rows of habitations and halls belonging to the inhabitants of the * The syddvdda in a process of reasoning adopted by the Jainas in relation to the question of the reality or otherwise of the totality of the perceptible objects found in the phenomenal universe. The word is translatable as the may-be-argument; and. this muy-be-argument declares that the phenomenal nniverse (1) may be real, (2) may not be real, (3) may and may not be real, (4) may be indescribable, (5) may be real and indescribable, (6) may be nnreal and indescribable, and (7) may be real and unreal and indescribable. The position represented by this argument is not, therefore, one of a single conolusion. + The triprasna is the name of a chapter in Sanskrit astronomical works and the fact that it deals with three questions is responsible for that name The questione dealt with are Dik (direction), Disa (position) and Kala (time) a appertaining to the planets and other heavenly bodies. For Private and Personal Use Only

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CHAPTER 1-TERMINOLOGY. (earthly) world, of the interspace (between the worlds), of the world of light, and of the world of the gods; (as also the dimensions of those belonging) to the dwellers in hell : and (other) miscellaneous measurements of all sorts--all these are made out by means of computation. 15. The configuration of living beings therein, the length of their lives, their eight attributes and other similar things, their progress and other such things, their staying together and such other things-all these are dependent upon computation (for their due measurement and comprehension). 16. What is the good of saying much in vain? Whatever there is in all the three worlds, which are possessed of moving and non-moving beings-all that indeed cannot exist as apart from measurement. 17-19. With the help of the accomplished holy sages, who are worthy to be worshipped by the lords of the world, and of their disciples and disciples' disciples, who constitute the well-known jointed series of preceptors, I glean from the great ocean of the knowledge of numbers a little of its essence, in the manuer in which gems are (picked up from the sea, gold is from the stony rock and the pearl from the oyster shell ; and give out, according to the power of my intelligence, the Sarasangraha, a small work on arithmetic, which is (however) not small in value. 20-23. Accordingly, from this ocean of Sarasangraha, which is filled with the water of terminology and has the (eight) arithmetical operations for its bank; which (again) is full of the bold rolling fish represented by the operations relating to fractions, and is characterised by the great crocodile represented by the chapter of miscellaneous examples; which (again) is possessed of the waves represented by the chapter on the rule-of-three, and is variegated in splendour through the lustre of the gems represented by the excellent language relating to the chapter on mixed problems; and which (again) possesses the extensive bottom represented by the chapter on" area-problems, and has the sands represented by the chapter on the cubic contents of exoavations; and wherein (finally) shines forth the advancing tide represented by the chapter on For Private and Personal Use Only

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GANITASARASANGRAHA. shadows, which is related to the department of practioal calculation in astronomy--(from this ocean) arithmeticians possessing the necessary qualifications in abundance will, through the instrumentality of calculation, obtain such pure gems as they desire. 24. For the reason that it is not possible to know without (proper) terminology the import of anything, at the (very) commencement of this science the required terminology is mentioned. Terminology relating to (the measurement of) Space. 25-27. That infinitely minute (quantity of) matter, which is not destroyed by water, by fire and by other such things, is called a paramanu. An endless number of them makes an anu, which is the first (measure) here. The trasarenu which is derived therefrom, the ratharenu, thence (derived), the hair-measure, the louse-measure, the sesamum-measure, which (last) is the same as the mustard-measure, then the barley-measure and (then) the angula are (all) in the case of (all) those who are born in the worlds of enjoyment and the worlds of work, which are (all) differentiated as superior, middling and inferior-eight-fold (as ineasured in relation to what immediately precedes each of them), in the order (in which they are mentioned). This angula is known as vyavaharangula. 28. Those, who are acquainted with the processes of measure ment, say that five-hundred of this (wyavaharangula) constitutes (another angula known as) pramana. The finger measure of men now existing forms their own anyula. 29. They hold that in the established usage of the world the angula is of three kinds, vyavahara and pramana constituting two (of them), and then there being) one's own angula; and six angulas make the foot-measure as measured across. 30. Two (such) feet make a viiasti; and twice that is a hasta. Four hastas make a danda, and two thousands of that make a krosa. 31. Those who are well versed in the measurement of space (or surface-area) say that four krasas form a yojana. After this, I mention in due order the terminology relating to the measurement of) time. For Private and Personal Use Only

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CHAPTER TERMINOLOGY. Terminology relating to (the measurement of) Time. 32. The time in which an atom (moving) goes beyond another atom (immediately next to it) is a samaya; innumerable samayas make an avali. 33. A measured number of avalis makes an ucchrasa; seven ucchvasas make one stoka ; seven stokas make one lava, and with thirty-eight and a half of this the ghati is formed. 34. Two ghatis make one muhurta; thirty muhurtas make one day; fifteen days make one paksa; and two paksas are taken to be a month. 35. I'wo montbs make one rtu; three of these are understood to make one ayana ; two of these form one year, Next, I give the grain-measure. Terminology relating to the measurement of) Grain. 36. Know that four sodasikas form here one kudaha ; four kudahas one prastha ; and four prasthas one adhaka. . 37. Four adhakas mako one drona, and four times one drona make one mani; four manis make one khara; five kharis make one pravartika. 38. Four times that same (pravartika) is a vaha; five provartikas make one kumbha. After this the terminology relating to the measurement of gold is described. Terminology relating to the measurement of) Gold. 39. Four gandakas make one gunja ; five gunjas make one pana, and eight of this (pana) make one dharana ; two dharana's make one karsa, and four karsas make one pala. Terminology relating to the measurement of) Silver. 40. Two grains make one gunja; two gunjas make one masa ; sixteen masas are said here to make one dharana. 41. Two and a half of that (dharana) make one karsa; four puranas (or karsas) make one pala--so say persons well versed in calculation in respect of the measurement of silver according to the standard current in Magalba. For Private and Personal Use Only

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GANITASARASANGRAHA. Tube Terminology relating to (the measurement of) Other Metals. 42. What is known as a kala consists of four padas : six and a quarter kalas make one yava; four yavas make one amea ; four amsas make one bhaga. 43. Six bhagas make one druksuna; twice that (draksuna) is one dinara; two dinaras make one satera. Thus say the learned men in regard to the (measurement of other) metals. 44. Twelve and a half palas make one prastha; two hundred palas make one tula; ten tulas make one bhara. Thus say thoso who are clever in calculation 45. In this matter of measurement) twenty pairs of cloths, of jewels or of canes (are called) a kotika. Next I give the names of the (principal) operations (in arithmetic). Names of the Operations in Arithmetic. 46. The first among these (operations) is gunakara (multiplication), and it is also called) pratyutpanna; the second is what is known as bhagahara (division); and krti (squaring) is said to be the third. 47. The fourth, as a matter of course, is varga-mula (square root), and the fifth is said to he ghana (cubing); then ghanamula (cube root) is the sixth, and the seventh is known as citi (summation). 48. This is also spoken of as saikalita. Then the eighth is vyutkalita (the subtraction of a part of a series, taken from the beginning, from the whole series), and this is also spoken of as sesa. All these eight (operations) appertain to fractions also. General rules in regard to zero and positive and negative quantities. 49. A number multiplied by zero is zero, and that (number remains unchanged when it is divided by," combined with (or) . It can be easily seen here that a number when divided by zero does not really remain unchanged. Bhaskara calls the quotient of euch nero-divisions khahara and rightly assigns to it the value of infinity. Mahaviracarya obvionsly thinks that a division by zero is no division at all. For Private and Personal Use Only

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CHAPTER TERMINOLOGY. diminished by zero. Multiplication and other operations in relation to zero (give rise to) zero; and in the operation of addition, the zero becomes the same as what is added to it. 50. In multiplying as well as dividing two negative (or) two positive (quantities, one by the other), the result is a positive (quantity). But it is a negative quantity in relation to two (quantities), one (of which is) positive and the other negative. In adding a positive and a negative (quantity, the result) is (their) difference. 51. The addition of two negative (quantities or) of two positive (quantities gives rise to) a negative or positive (quantity) in order. A positive (quantity) which has to be subtracted from & (given) number becomes negative, and a negative (quantity) which has to be (80) subtracted becomes positive. 52. The square of a positive as well as of a negative (quuntity) is positive; and the square roots of those (square quantities) are positive and negative in order. As in the nature of things a negative (quantity) is not a square (quantity), it has therefore no square root. 53-62. [These stanzas give certain names of certain things, which names are frequently used to denote figures and numbers in arithmetical notation. They are not therefore translated here ; but the reader is referred to the appendix wherein an alphabetical list of such of these names as occur in this work is given with their ordinary and numerical meanings.] The names of Notational Places. 63. The first place is what is known as eka (unit); the second place is named dasa (ten); the third they call as sata (hundred), while the fourth is sahasra (thousand). 64. The fifth is dasa-sahasra (ten-thousand) and the sixth is no other than laksa (lakh). The seventh is dasa-laksa (ten-lakh) and the eighth is said to be koti (crore). For Private and Personal Use Only

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Shri Mahavir Jain Aradhana Kendra 8 GANITASARASANGRAHA. 65. The ninth is dasa-koti (ten-crore) and the tenth is satakoti (hundred-crore). The (place) characterised by eleven is arbuda and the twelfth (place) is nyarbuda. 66. The thirteenth place is kharva and the fourteenth is mahakharva. Similarly the fifteenth is padma and the sixteenth mahapadma. 67. Again the seventeenth is ksoni, the eighteenth mahd-kseni. The nineteenth place is sankha and the twentieth is maha-sankha. 68. The twenty-first place is ksitya, the twenty-second mahaksitya. Then the twenty-third is ksobha and the twenty-fourth maha-ksobha. 69. By means of the (following) eight qualities, viz., quick method in working, forethought as to whether a desirable result may be arrived at, or as to whether an undesirable result will be produced, freedom from dullness, correct comprehension, power of retention, and the devising of new means in working, along with getting at those numbers which make (unknown) quantities known-(by means of these qualities) an arithmetician is to be known as such. 70. Great sages have briefly stated the terminology thus. What has to be further said (about it) in detail must be learnt from (a study of) the science (itself). Thus ends the chapter on Terminology in Sarasangraha, which is a work on arithmetic by Mahaviracarya. For Private and Personal Use Only

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CHAPTER II-ARITHMETICAL OPERATIONS. CHAPTER II. ARITHMETICAL OPERATIONS. Acharya Shri Kailassagarsuri Gyanmandir .9 The First Subject of Treatment. Hereafter we shall expound the first subject of treatment, which is named Parikarman. Multiplication. The rule of work in relation to the operation of multiplication, which is the first (among the par ikarman operations), is as follows: 1. After placing (the multiplicand and the multiplier one below the other) in the manner of the hinges of a door, the multiplicand should be multiplied by the multiplier, in accordance with (either of) the two methods of normal (or) reverse working, by adopting the process of (i) dividing the multiplicand and multiplying the multiplier by a factor of the multiplicand, (ii) of dividing the multiplier and multiplying the multiplicand 1. Symbolically expressed, this rule works out thus: In multiplying ab by cd, the product is (i) abx (a x cd); or (ii) (ab x c) x a For Private and Personal Use Only cd ; or (iii) ab x cd. Obviously the object of the first two devices here is to facilitate working through the choice of suitable factors. C The anuloma or normal method of working is the one that is generally followed. The viloma or the reverse method of working is as follows: To multiply 1998 by 27: 1998.

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10 GANITASARASANGRAHA. in the by a factor of the multiplier, or (iii) of using them multiplioation) as they are (in themselves). Examples in illustration thereof. 2. Lotuses were given away in offering)-eight of them to each Jina temple. How many (were given away) to 114 temples ? 3. Nine padmaraga gems are seen to have been offered in worship in a single Jina temple. How many will they be (at that same rate) in relation to 288 temples ? 4. One hundred and thirty-nine pusyaraga gems have to be offered in worship in a single Jina temple. Say, how many gems (have to be so offered) in 109 temples. 5. Twenty-seven lotuses have been given away in offering to a single Jina temple. Say, how many they are (which have been at that rate given away) to 1998 (temples). 6. (At the rate of) 108 golden lotuses to each temple, how many will they be in relation to 85697481 (temples)? 7. If the number represented by the group (of figures) consisting of 1, 8, 6, 4, 9, 9, 7 and 2 (in order from the units' place upwards) is written down and multiplied by 441, what is the value of the (resulting) quantity ? 8. In this (problem), write down (the number represented by). the group (of figures) consisting of 1, 4, 4, 1, 3 and 5 (in order from the units' place upwards), and multiply it by 81; and then tell me the (resulting) number. 9. In this problem), write down the number 157683 and multiply it by 9, and then tell me, friend, the value of the resulting) quantity. 10. In this (problem), 12345679 multiplied by 9 is to be written down; this (product) has been declared by the holy preceptor Mahavira to constitute the necklace of Narapala. 4. Here, 139 is mentioned in the original as 40 + 100 - 1. 5. Here, 1998 is mentioned in the original as 1098 + 900. 10. Here as well as in the following stanzas, certain numbers are said to constitute different kinds of recklaces on account of the symmetrical arrangement of similar figures which is readily noticeable in relation to them. For Private and Personal Use Only

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CHAPTER II-ARITHMETICAL OPERATIONS. 11 11. Six 3's, five 6's, and (one) 7, which is at the end, are put down in the descending order down to the units' place); and this (number) multiplied by 33 has (alsu) been declared to be a (kind of) necklace. 12. In this (problem), write down 3,4, 1, 7, 8, 2, 4, and 1 (in order from the units' place upwards), and multiply the resulting number) by 7; and then say that it is the neoklace of precious goms. 13. Write down (the number) 142857143, and multiply it by 7; and then say that it is the royal necklace. 14. Similarly 37037037 is multiplied by 3. Find out (the result) obtained by multiplying (this product) again to get such multiples (thereof) as have one as the first and nine as the last (of the multipliers in order). 15. The (figures) 7,0, 2, 2, 5 and 1 are put down (in order from the units' place upwards); and then this number) which is to be multiplied by 73, should also be called a necklace (wben so multiplied). 16. Write down (the number represented by) the group (of figures) consisting of 4, 4, 1, 2, 6 and 2 (in order from the units' place upwards); and when this is) multiplied by 64, you, who know arithmetic, tell me what the (resulting) number is: 17. In this (problemn) put down in order (from the units' place upwards) 1,1,0,1,1, (), 1 and 1, which (figures so placed) give the measure of a (particular) number; and (then) if this (nnmber) is multiplied by 91, there results that necklace which is worthy of a prince. Thus ends multiplication, the first of the operations known as Parikarman. 11. The multiplicand here is 333333666667. 14. This problem rednces itself to, this: multiply 37037037 * 3 by 1, 2, 3, 4, 5, 6, 7, 8, and 9 in order. For Private and Personal Use Only

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12 GANITASABASANGRAHA. Division, The role of work in relation to the operation of division, which is the second (among the parikarman operations), is as follows: 18. Put down the dividend and divide it, in accordance with the process of removing common factors, by the divisor, which is placed below that (dividend), and then give out the resulting (quotient). Or: 19. The dividend should be divided in the reverse way (i.e., from left to right) by the divisor placed below, after performing in relation to (both of) them the operation of removing the common factors, if that be possible. Examples in illustration thereof. 20. Dinaras (amounting to) 8192 have been divided between 64 men. What is the share of one man ? 21. Tell me the share of one person when 2701 pieces of gold are divided among 37 persons. 22. Dinaras (amounting to) 10349 have been divided between 79 persons. What is it that is obtained by one (person) ? 23. Gold pieces (amounting to) 14141 are given to 79 temples. What is the money (given) to each (temple) ? 24. Jambu fruits (amounting to) 31317 have been divided between die persons. Tell me the share of each. 25. Jambu fruits (amounting to) 31313 have been divided between 181 persons. Give out the share of each. 26. Gems amounting to 36261 (in number) are given to 9 persons (equally). What does one man obtain here? 27. O friend, gold pieces to the value of the number wherein the figures in order from the units' place upwards are) such as 20. Here, 8192 is mentioned in the original as 8000 +92 + 100. 22. In the original, 10349 is giveu as 10000 + 300 + 7*. 23. Here, 14141 is given as 10000 + (40 + 4000 + 1 + 100). 24. Here, 31317 is given as 17 +300+ 31000. 25. llere, 31313 is given as 13 +800+ 31000. 26. Here, 36261 is given as 30000 + 1 + (60 + 200 + 6000). 27. Here, the given dividend is obviously 12345654321. For Private and Personal Use Only

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CHAPTER II-ARITHMETICAL OPERATIONS. begin with 1 and end with 6, and then become gradually diminished, are divided between 441 persons. What is the share of each? 13 28. Gems (amounting to) 28483 (in number) are given (in offering) to 13 Jina temples. Give out the share of each (temple). Thus ends division, the second of the operations known as Parikarman. Squaring. The rule of work in relation to the operation of squaring, which is the third (among the parikarman operations), is as follows: 29. The multiplication of two equal quantities: or the multiplication of the two quantities obtained (from the given quantity) by the subtraction (therefrom), and the addition (thereunto), of any chosen quantity, together with the addition of the square of that chosen quantity (to that product): or the sum of a series in arithmetical progression, of which 1 is the first term, 2 is the common difference, and the number of terms wherein is that (of which the square is) required: gives rise to the (required) square. 30. The square of numbers consisting of two or more places is (equal to) the sum of the squares of all the numbers (in all the places) combined with twice the product of those (numbers) taken (two at a time) in order. 28. Here, 28483 is given as 83+ 400+ (4000 x 7). 29. The rule given herein, expressed algebraically, comes out thus: (i) axaa2; (ii) (a + x) (a-x) +2 a2; (iii) 1+3+5+7+ For Private and Personal Use Only . up to a terms a2. 30. The word translated by place here is ; it obviously means a place in notation. Here, as a commentary interprets it, it may also denote the component parts of a sum, as each such part has a place in the sum. According to both these interpretations the rule works out correctly. For instance, (1234)2 (1002+2002+302 +42) + 2 x 1000 x 200+ 2 x 1000 x 30 +2x 1000 x 4+2 x 200 x 30+2 x 200 x 4+2 x 30 x 4. Similarly (1+2+3+4)=(1+2+32 +42) + 2(1 x 2+1x3+1x4+2x3+2x4 + 3 x 4).

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GANITASARASANGRAHA. 31. Get the square of the last fignre (in the number, the order of counting the figures being from the right to the left, and then multiply this last (figure), after it is doubled and pushed on (to the right by one notational place), by (the figures found in) the remaining places. Each of the remaining figures in the number) is to be pushed on (by one place) and then dealt with similarly. This is the method of squaring. Examples in illustration thereof. 32. Give out the squares of (the numbers from 1 to 9, of 15, 16, 25, 36 and 75. 33. What will 338, 4661 and 256 become when squared ? 34. O arithmetician, give out, if you know, the squares of 65536, 12345 and 3333. 35. (Each of the numbers) 6387, and then 7135, and (then) 1022 is squared. O clever arithmetician, tell me, after multiplying well, the value of those three (squares). Thus ends squaring, the third of the operations known as Parikarman. 31. The pushing on to the right mentioned herein will become clear from the following worked out examples: To square 131. To square 192. To square 555. 50 6 12 - 2x1x3= 2x1x1= 32 = 2x3x1 1 = 2x1x3= 2x1x2 32 2 x 3x2= 22 2 x 5x5 = 2 x 5 x 5= 52 = 2 x 5x5= 125 12 50 25 7 (5)) (8) (5) (2) K(1) 1 7 1.6 1 7 4 2 4 30 8 0 2 5 33. Here, 4661 is given as 4000+61 +600. . 35. Here, 7135 is given as 135+ (1000 x 7). For Private and Personal Use Only

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CHAPTER 11-ARITHMETICAL OPERATIONS. 15 Square Root. The rule of work in relation to the operation of (extracting) the square root, which is the fourth (of the parikarman operations) is as follows: 36. From the (number represented by the figures up to the) last odd place (of notation counted from the right), subtract the (highest possible) square number; then multiply the root (of this number) by two, and divide with this (product the number represented by taking into position the figure belonging to) the (next) even place; and then the square of the quotient (80 obtained) is to be subtracted from the (number represented by taking into position the figure belonging to the next) odd place. (If it is so continued till the end), the half of the (last) doubled quantity (comes to be) the resulting square root. Examples in illustration thereof. 37. O friend, tell me quickly the roots of the squares of the numbers from 1 to 9, and of 256 and 576. 38. Find out the square root of 6561 and of 65536. 39. What are the square roots of 4294967296 and 622521 ? 40. What are the square roots of 63664441 and 1771561 ? 41. Tell me, friend, after considering well, the square roots of 1296 and 625. 36. To illustrate the rule, the following example is worked out below:-- To extraot the square root of 65536 9-915665538 exemple 2 x 2 = 4)25(5 20 - 5% = 55 25 25 x 2 = 50)803(6 300 36 36 62 = 256 x 2 = 512) 0 (0 Square root required - 2-256. For Private and Personal Use Only

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16 GANITASARASANGRAHA. 42. Tell me, O leading arithmetician, phe square roots of 110889, 12321, and 844561. Thus ends square root, the fourth of the operations known as Parikarman. Cubing The rule of work in relation to the operation of cubing, which is the fifth (of the parikarman operations), is as follows : 43. The product of (any) three equal quantities : or the product obtained by the multiplication of any (given) quantity by that (given quantity) as diminished by a chosen quantity and (then again) by that (given quantity) as increased by the (same) chosen quantity, when combined with the square of the chosen quantity as multiplied by the least (of the above three quantities) and combined) also with the cube of the chosen quantity: gives rise to a cubic quantity 44. Or, the summing up of a series in arithmetical progression, of which the first term is the quantity (the cube whereof is) required, the common difference is twice this quantity, and the number of terms is (equal to) this (same given) quantity, (gives rise to the cube of the given quantity). Or, the square of the quantity (the cube whereof is required), when combined with the product (obtained by the multiplication) of this given quantity diminished by one by the sum of a series in arithmetical progression in which the first term is one, the common difference is two and the number of terms is (equal to) the given quantity, (gives rise to the cube of the given quantity). 43. Symbolically expresved, this rule works out thus: (i) axaxa=a:: (ii) a (a + b) (4-6) + b2 (a - b) + 38 = 28. 44, Algebraically, this role means (i) a = a +3a + 5a +70+.........to a terms, (ii) a= a + (a-1) (1+3+5+7+ .........to a terms). For Private and Personal Use Only

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CHAPTER II-ARITHMETICAL OPERATIONS. 17 45. In an arithmetically progressive series, wherein one is the first term as well as the common difference, and the number of terms is (cqual to the given number, multiply the preceding terms by the immediately following ones. The sum of the products (80 obtained), when multiplied by three and combined with the last term in the above series in arithmetical progression), becomes the cube (of the given quantity). 46. (In a given quantity, the squares of the number represented by the figures in) the last place as also (by those in) the other *(remaining places) are taken ; and each of theso (squares) is multiplied by the number of the other place and also by three; the sum of the two (quantities resulting thus), when combined again with the cubes of the numbers corresponding to all the (optional) places, (gives rise to) the cube (of the given quantity). 47. Or, the cube of the last figure (in the number counted from right to left is to be obtained); and thrice the square (of that last figure) is to be pushed on to the right by one notational place) and multiplied by (the number represented by the figures found in the remaining (places); then the square of this number represented by the figures found in the remaining (places) is to be pushed on (as above) and multiplied by thrice the last figure (above-mentioned). These three quantities) are then to be placed in position (and then summed up). Such is the rule (to be carried out) here. Examples in illustration thereof. 48. Give out the cubes of the numbers from 1 to 9 and of 15, 25, 36, 77 and 96. 49. Give out the cubes of 101, 172, 516, 717 and 1344. 45. 3 } 1 x 2 + 2 x 3 + 3 * 4+ * 5+ ... +a-Txa+a=al. 46. 3a2b + 3ab" + a3 + b3 = (a + b)3. To make the rule general and applicable to numbers having more than two places, it is clearly implied here that 3a* (b + c) + 30 (b + c)*+as+- (b + c) = (a + b + c); and it is obvions that any number may be represented as the sum of two other suitably chosen numbers. 47. The pushing on of a figure here referred to is similar to what is exhibited in the note under stanza 31 in this chapter. For Private and Personal Use Only

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GANITASARASANGRAHA. 50. The number 213 is cubed; and twice, thrice, four times and five times that (number are) also (cubed ; find out the corresponding quantities) 51. It is seen that 168 multiplied by all the numbers from 1 to 8 is related (as base) to the required cubes. Give out those cnbes quickly. 52. O you, who have seen the other shore of the deep and excellent ocean of the practice of arithmetical) operations, write down the figures 4, 0,6, 0,5, and 9 in order (from right to left), and work out the cube of the number (represented by those figures), and mention the result at once. Thus ends cabing, the fifth of the operations known as Parikarman. Cube Root. The rule of work in relation to the operation of extracting the cube root, which is the sixth (among the parikarman operations), is as follows: 53. From (the number represented by the figures up to) the last ghana place, subtract the (highest possible) cube ; then divide the (number represented by the next) bhajya place (after it is taken into position) by three times the square of the root (of that cube); then subtract from the number represented by the next) lodhya place (after it is taken into position) tho square of the (above) quotient as multiplied by three and by the alreads mentioned (root of the highest possible cube); and then (subtract) from 53 and 54. The figures in any given number, the cabe-root whereof is required, are conceived in these rules to be divided into gronps, each of which consiste ag far as possible of three figures, Danied, in the order from right to ieft, as ghana or that which ia cubio, that is, from which the cube is to be subtracted, as sodhya or that which is to be subtracted from, and as bhajya or that which is to be divided. The bhajya and sodhya are also known as aghana or non-cubic. The last group on the left need not always consist of all these three figures ; it may For Private and Personal Use Only

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CHAPTER II - ARITHMETICAL OPERATIONS, 19 the number represented by the figure in the next) ghana place (after it is taken into position) the cube (of this same quotient). 54. One (figure in the various groups of three figures) is cubio : two are non-cubic. Divide (the non-cubio figure) by three times the square of the cube root. From the (next) non-cubic (figuro) subtract the syuare of the quotient (obtained as above and) multiplied by three times the previously mentioned (cube-root of the highest cube that can be subtracted from the previous cubic figure) and (then subtract) the cube of the above) quotient (from the next cubic figure as taken into position). With the help of the cube-root-figures (so) obtained (and taken into position, the proccdure is) as before. Examples in illustration thereof. 55. What is the cube root of the numbers beginning with 1 and ending with 9, all cubed; and of 4913; and of 18608677 56. Extract the cube root of 13824, 36926037 and 618470208. consist of one or two or three figures, as the case may be. The rule mentioned will be clear from the following worked out example. To extract the cube root of 77308776 : $. gh. bh. $. yh. th, s, yh. 7 7 3 0 8 1 7 7 6 ... ... 42 * 3 48)133(2 96 ... ... 22 3 X 4 = 370 48 3228 ... ... 42 x 3 5292)322070 31752 4557 ... 6deg X 3 X 42 -- 4536 *. ... 216 216 Cube root = 126. The rule does not state what figures constitate the cube root; but it is meant that the cube root is the number made up of the figures which are cubed in this operation, written down in the order from above from left to right For Private and Personal Use Only

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20 GANITASABASANGRAHA. 57. Give the cube roots of 270087225344 and 76332940488. 58. Give the cube roots of 77308776 and also of 260917119. 59. Give the cube roots of 2427715584 and of 1626379776. 60. O arithmetician, who are clever in calculation, give out after examination the root of 859011369945948864, which is a ouhic quantity. Thus cnds onbe root, the sixth of the operations known as Parikarman. Summation The rule of work in relation to the operation of summation of series, which is the seventh (among the parikarman operations), is as follows: 61. The number of terms in the series is (first) diminished by one and is then) halved and multiplied by the common difference; this when combined with the first term in the series and (then) multiplied by the number of terms (therein) becomes the sum of all (the terms in the series in arithmetical progression). The rule for obtaining the sum of the series in another manner : 62. The number of terms in the series) as diminished by one and (then) multiplied by the common difference is combined with twice the first term in the series, and when this combined sum) i: multiplied by the number of terms in the series and is then divided by two, it becomes the sum of the series in all cases. 01. This rule comes out thus when expressed algebraically : n=S, where a is the first term, b the common difference, the number of terms, and S the sum of the whole scries. ( -+ a)" 62. Similarly, For Private and Personal Use Only

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CHAPTER II-ARITHMETICAL OPERATIONS. The rule for finding out the adidhana, the uttaradhana and the sarvadhana: 63. The adidhana is the first term multiplied by the number of terms (in the series). The uttaradhana is (the product of) the number of terms multiplied by the common difference (and again) multiplied by the half of the number of terms less by one. The sum of these two (gives) the sarvadhana, i.e., the sum of all the terms in the series; and (this sum will be the same as that of a scries which is) characterised by a negative common difference, when (the order of the terms in the series is reversed so that) the last term is made to be the first term. The rule for finding the antyadhana, the madhyadhana and the sartadhana :-- 64. The number of terms (in the scries) lessened by one and multiplied by the common difference and (then) combined with the first term (gives) the antyadhana. Half of the sum of (1) Adidhanan x a. 63-64. In these rules, each of the terms in an arithmetically progressive series is supposed to be obtained by adding to the first term thereof a multiple of the common difference, the nature of this multiple being determined by the position which any specified term holds in the series. According to this conception we have to find in every term of the series the first term along with a multiple of the common difference. The sum of all such first terms so found is what is here called the adidhana; the sum of all such multiples of the common difference constitutes the uttaradhana; and the sarvadhana which is obtained by adding these two sums is of course the sum of the whole series. The expression antyadhana denotes the value of the last term in an arithmetically progressive series. And madhyadhana means the value of the middle term which value, however, corresponds to the arithmetical mean of the first and the last terms in the series, so that when there are 2n + 1 terms in the series, the value of the (n + 1)th term is the madhyadhana, but when there are 2n terms in the series the arithmetical mean of the value of the nth term and of that of the (n + 1)th term becomes the madhyadhana. Accordingly we have (2) Uttaradhana == (3) Antyadhana Acharya Shri Kailassagarsuri Gyanmandir n-1 2 X N x b. (n-1) x b + a. (n - 1) b + 0 2 (4) Madhyadhana (5) Sarvadhana - (1) + (2) = (nx a) + ( 22-1 2 x { (~ ~ 1) 6 + a] + a (22 a} 2 or = (4) x n = n x For Private and Personal Use Only + a 21 x n x );

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22 GANITASARASANGRAHA. this (antyadhana) and the first term (gives the madhyadhana. The product of this (madhyadhana) and the number of terms (in the series gives) the desired sum of all the terms therein. Examples in illustration thereof. 65. (Each of) ten merchants gives away money in an arithmetically progressive series) as a religious offering, the first terms of the (ten) series being from 1 to 10, the common difference in each of these series) being of the same value (as the first terms thereof), and the number of terms being 10 (in every one of the series). Calculate the sums of those (series). 66. A certain excellent sravaka gave gems in offering to 5 temples (one after another) commencing (the offering) with 2 (gems), and then increasing (it successively by 3 (gems). O you who know how to calculate, mention what their total number is. 67. The first term is 3; the common difference is 8; and the number of terms is 12. All thesc three (quantities) are (gradually) increased by 1, until (there are) 7 (series). O arithmetician, give out the sams of all (those series). 68. O you who possess enough strength of arms to cross the ocean of arithmetic, give out the total value of the offerings made in relation to 1000 citios, commencing (the offering) with 4 and increasing it successively by 8. The rule for finding out the number of terms (in a series in arithmetical progression) : 69. When, to the square root of the quantity obtained by the addition of the square of the difference between twice the first It is quite obvious that an arithmetically progressive series having a negative common difference becomes changed into one with a positive common difference when the order of the terms is reversed throughout so as to make the last of them become the first. 66. A fravaka is a lay follower of the Jaina religion, who merely hears, q.e., listens to and learns the dharmas or duties, as opposed to the ascetics who are entitled to teach those religious duties. 69. Algebraically this rule works out thus (2a - b) + 8 6S + b For Private and Personal Use Only

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CHAPTER IIARITHMETICAL OPERATIONS. term and the common difference to 8 times the common differcuce multiplied by the sum of the series, the common difference is added, and the resulting quantity is halved; and when (again) this is diminished by the first term and then divided by the common difference, we get the number of terms in the series. The rule for finding out the number of terms (stated) in another manner :-- 70. When, from the square root of (the quantity obtained by) the addition of the square of the difference between twice the first term and the common difference to 8 times the common difference multiplied by the sum of the series, the ksepapada is subtracted, and (the resulting quantity) is halved; and (when again this is) divided by the common difference, (we get) the number of terms in the series. Examples in illustration thereof. 71. The first term is 2, the common difference 8; these two are increased successively by 1 till three (series are so made up). The sums of the three series are 90, 276 and 1110, in order. What is the number of terms in each series? Acharya Shri Kailassagarsuri Gyanmandir 72. The first term is 5, the common difference 8, and the sum of the series 333. What is the number of terms? The first term (of another series) is 6, the common difference 8, and the sum 420. What is the number of terms? }= 23 The rule for finding out the common difference as well as the first term: 73. The sum (of the series) diminished by the adidhana, and (then) divided by half (the quantity represented by) the square S-na n2 - n 70. Kepapada is half of the difference between twice the first term and the 2a-b common difference, i.e., It is obvious that this stanza varies the rule 2 mentioned in the previous stanza only to the extent necessitated by the introduction of this kaipapada therein. 73. For adidhana and uttaradhana, see note under stanzas 63 and 64 in this chapter. Symbolically expressed this stanza works out thus: - n (n 1) 2 and a= 3 S n For Private and Personal Use Only

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24 GANITASARASANGRAHA. of the number of terms as lessened by the number of terms, (gives) the common difference. The sum of the series) diminished by uttaradhana and (then) divided by the number of terms, (gives) the first term of the series. The rule for finding out the first term as well as the common difference :-- 74. The sum of the series divided by the number of terms (therein), when diminished by the product of the common difference multiplied by the half of the number of terms less by one, gives the first term of the series. The common difference is obtained, when the sum, divided by the number of terms and then diminished by the first term, is divided by the half of the number of terms less by one. Two rules for finding out, in another way, the common difference and the first term : 75. Understand that the common difference is (obtained, when) the sum of the series, multiplied by two and divided by the number of terms (therein), is diminished by twioe the first term, and is (then) divided by the number of terms lessened by one. 76. Twice the sum of the series divided by the number of terms therein, and (then) diminished by the number of terms as lessened by one and multiplied by the common difference, when divided by two, (gives the first term of the series. Escamples in illustration thereof. 77. The first term is 9; the number of terms is 7; and the sum of the series is 105. Of what value is the common difference ? - 1 -a 74. Algebraically, a = -. b; and ten 28 - 75. Symbolically, b = --- 2 a 2 -1 2 S - (n-1) 1 n 76. Algebraioally, a = " For Private and Personal Use Only

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CHAPTER II-ARITHMETICAL OPERATIONS. 25 The common difference (in respect of another series) is 5, the number of terms is 8, and the sum is 156. Tell me the first term. The rule for finding out how (when the sum is given) the first term, the common difference, and the number of terms may, as desired, be arrived at : 78. When the sum is divided by any chosen number, the divisor becomes the number of terms (in the series); when the quotient here is diminished by any number chosen (again), this subtracted number becomes the first term (in the series); and the remainder (got after this subtraction) when divided by the half of the number of terms lessened by one becomes the common difference. Example in illustration thereof. 79. The sum given in this problem is 540. O crest-jewel of arithmeticians, tell me the number of terms, the common difference, and the first term. Three rule-giving stanzas for splitting up (into the component. elements) such a sum of a series (in arithmetical progression) as is combined with the first term, or with the common difference, or with the number of terms, or with all these. 80. O crest-jewel of calculators, understand that the misradhana diminished by the uttaradhana, and (then) divided by the number of terms to which one has been added, gives rise to the first term. 81. The misradhana, diminished by the adidhana, and (then) divided by the (quantity obtained by the) addition of one to the (product of the) uumber of terms multiplied by the half of the number of terms lessened by one, (gives rise to) the common For Private and Personal Use Only 78. Symbolically, the problem herein is to find out b, when 8 is given, and a and n are allowed to be chosen at option. Naturally, there may be in relation to any given value of S any values of b, which depend upon the chosen values of a and . When the values of a and n are definitely chosen, the rule herein given for finding out 6 turns out to be the same as that given in stanza 74.above. 80-82. The expression misradhana means a mixed sum. It is used here to denote the quantity which may be obtained by adding the first term or the common difference or the number of terms or all three of these to the sum of a

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26 GANITASABASANGRAHA. difference. (In splitting up the number of terms from the misradhana), the (required) number of terms (is obtained) in accordance with the rule for obtaining the number of terms, provided that the first term is taken to be increased by one (so as to cause a corresponding increase in all the terms). 82. The misradhana is diminished by the first term and the number of terms, both (of these) being optionally chosen; (then) that quantity, which is obtained (from this difference) by applying the rule for (splitting up) the uttora-misradhana, happens to be the common difference (required here). This is the method of work in (splitting np) the all-combined (misradhana). Examples in illustration thereof. 83. Forty exceeded by 2, 3, 5 and 10, ropresents in order) the adi-misradhana and the other (misradhanas). Tell me what (respectively) happens in these cases to be the first term, the common difference, the number of terms and all (these three). series in arithmetical progression. There are accordingly four different kinds of misradhana mentioned here, and they are respectively adi-migradhana, uttaramieradhana, gaccha-misradhana and sarva-misradhana. For ddidhana and uttara. dhana see note ander stanzas 63 and 64 in this chapter. n(n - 1) Sa - 6 Algebraically, stanza 80 works out thus : a =- Sa is the ddi-misradhana, i.e., $ + a. 86 - na where Sy is the uttara-misradhana, +1_ where And stanza 81 gives b=n(n-1) + 1 i.e., S + b: and further points out that the value of may be found out, when the valae of En, which, being tbe gaccha-misradhana, is equal to S + n, is given, from the fact that, when S= a + (a + b) + (a + 2) + ... up to n terms, Sn = (a + 1) + (a +1+b) (a +1 +26) +..... up to the same n terms. Since, in stanza 82, the choice of a and n are left to our option, the problem of finding out a, n, and b from the given value of Sano which, being the rarva misradhana, is equal to 8 + a + n + b, resolver itself easily to the finding out of b from any given value of Sy in the manner above explained. 83. The problem expressed in plainer terms is :-(1) Find out a when Sa= 42 b=3 and n =5. (2) Find out b, when 8b =43, a= 2 and n = 5. (3) Find out * when $ + n = 45, a= 2 and b = 3. And (4) find out a, b, and n wher 8 + + + n = 50, For Private and Personal Use Only

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CHAPTER II-ARITHMETICAL OPERATIONS. 27 The rule for finding out, from the known sum, first term, and common difference (of a given series in arithmetical progression), the first term and the common difference (of another series), the optionally chosen sum (whereof) is twice, three times, balf, onethird, or some such (multiple or fraction of the known sum of the given series): 84. Put down in two places (for facility of working) the chosen sum as divided by the known (i.e., the given) sum; this (quotient) when multiplied by the (known) common difference gives the (required) common difference; and that (same) quotient when multiplied by the (known) first term gives the required) first term of (the series of which the sum is either a multiple or a fraction (of the known sum of the given series). Examples in illustration thereof. 85. Sixty is the (known) first term, and the (known) common difference is twice that, and the number of terms is the same, i.e., 4 (in the given series as well as in all the required series). Give out the first terms and the common differences of these required (series, the sums whereof are) represented by that (known sum) as multiplied or divided by the (numbers) heginning with 2. The rule for finding out, in relation to two (series), the number of terms wherein are optionally chosen, their mutually interchanged first term and common difference, as also their stms which may be equal, or (one of which may be) twice, thrice, half, or onethird, or any such (multiple or fraction of the other) : 86. The number of terms (in one series), multiplied by itself as lessened by one, and then multiplied by the chosen (ratio between the sums of the two series), and then diminished by 84. Symbolically, aj = b, where 31, 4, and by are the sum, the first term and the common difference, in order, of the series whose sum is chosen. Given the sums of two series, the ratio between the two first terms and that between the two common differences need not always be The solution here given is hence applicable only to certain particular cases. 86. Algebraically, a = + (n - 1) * p - 2ni, and b= (nu) -n-2 pn, where a, b and n are the first term, the common difference and the number of For Private and Personal Use Only

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28 GANITASARASANGRAHA. twice the number of terms in the other series (gives rise to the interchangeable) first term of one (of the series). The square of the number of terms in the other (series), diminished by that (number of terms) itself, and (then) diminished (again) by the product of two (times the) chosen (ratio) and the number of terme (in the first series gives rise to the interchangeable) common difference (of that series). Examples in illustration thereof. 87. In relation to two men, (whose wealth is measured respectively by the sums of two series in arithmetical progression) having 5 and 8 for the number of terms, the first term and the common difference of both these series being interchangeable (in relation to each other); the sums (of the series) being equal or the sum of one of them) being twice, thrice, or any such (multiple of that of the other)-0 arithmetician give out the value of these). sums and the interchangeable first term and common difference after calculating (them all) well. 88. In relation to two series (in arithmetical progression), having 12 and 16 for their number of terms, the first term and the common difference are interchangeable. The sums (of the series) are equal, or the sum of one of them) is twice or any such multiple, or half or any such fraction (of that of the other). You, who are versed in the science of calculation, give out (the value of these sums and the interohangeable first term and common difference). The rule for finding out the first terms in relation to such (series in arithmetical progression) as are characterised by varying common differences, equal numbers of terms and equal sums : 89. Of that (series) which has the largest common difference, one is taken to be the first term. The difference between this terms in the first series, n, the number of terms in the second series, and p the ratio between the two sums : a and b being thus found out, the first term and the common difference of the second series are 0 and a respeotively in value. 89. The solution herein given is only a partioular case of the general rule a = " (b-b) + a, where a and a, are the first terms of two series, and For Private and Personal Use Only

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CHAPTER 11--ARITHMETICAL OPERATIONS. 29 largest common difference and any other) remaining common difference is multiplied by the half of the number of terms lessened by one ; and when this (product) is combined with one, (we get,) O friend, the first terms of (the various series having) the remaining (smaller) common differences. Examples in illustration thereof. 90. Give out quickly, O friend, the first terms of all the series found in two sets of) such (series) as have equal sums (in relation to each set) and are characterised by 9 as the number of terms in each (series), when those (series belonging to the first and second sets) have (respectively) common differences beginning with I and ending with 6 (in one oase) and have 1, 3, 5 and 7 as the common differences in the other case). The rule for finding out the common difference in relation to such (series in arithmetical progression) as are characterised by varying first terms, equal numbers of terms and equal sums : 91. Of that (series) which has the largest first term, one is taken to be the common difference. The difference between this largest first term and each of the remaining (smaller) first terms is divided by the half of the number of' terms lessened by one ; and when this (quotient in each case) is combined with one (we get the common differences of the various series having) the remaining (smaller) first terms. An example in illustration thereof. 92. O arithmetician, who have seen the other shore of calculation, give out the common differences of (all) those (series) which are characterised by equal sums and have 1, 3, 5, 7, 9 and 11 for their first terms and 5 for the number of terms in each. d and by their corresponding common differences. It is obvious that in this formula, when b, b, and n are given, a, is determined by choosing any value for a; and one is chosen as the value of a in the rule here. 91. The general formula in this case is b = 4 41 + b, wherein also the value of b is taken to be one in the role n - 1 ' given above. For Private and Personal Use Only

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GANITASARASANGRAHA The rule for finding out the gunadhana and the sum of a series in geometrical progression : 93. The first term (of a series in geometrical progression), when multiplied by that self-multiplied product of the common ratio in which (product the frequency of the occurrence of the common ratio is) measured by the number of terms (in the series), gives rise to the gunadhana. And it has to be understood that this gunadhana, when diminished by the first term, and (then) divided by the common ratio lessened by one, becomes the sum of the series in geometrical progression. Another rule also for finding out the sum of a series in geometrical progression : . 94. The number of terms in the series is caused to be marked in a separate column) by zero and by one (respectively) corresponding to the even (value) which is halved and to the uneven (value from which one is subtracted till by continuing these processes zero is ultimately reached); then this (representative series made up of zero and one is used in order from the last one therein, so that this one multiplied by the common ratio is again) multiplied by the common ratio (wherever one happens to be the denoting item), and multiplied so as to obtain the square (whereever gero bappens to be the denoting item). When the result 93. The guradhana of a series of n terms in geometrical progression corre. sponds in value to the (n + 1)th term thereof, when the series in continued. The value of this gunadhana algebraioally stated is Txr xr..... up to n such factors x a, i.e., arn. Compare this with the uttaradhana. This rule for finding out the sum may be algebraically expressed thus:S ara , where a is the first term, - 1 the common ratio, and n the number of terms. 94. This rale differs from the previous one in so far as it gives a new method for finding out on by using the processes of squaring and ordinary multiplication; and this niethod will become clear from the following example: Let n in yn be equal to 12. 12 is even ; it has therefore to be divided by 2, and to be denoted by 0: >> 90: $ = 3 is odd ; 1 is , subtracted from it, and it is 3-1=2 is even; it has , , divided by 2, and to be = 1 is odd; 1 is subtracted from it, and it is 1-1=0, which concludes this part of the operation. For Private and Personal Use Only

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CHAPTER II --ARITHMETICAL OPERATIONS. 31 of) this operation) is diminished by one, and (is then) multiplied by the first term, and (is then) divided by the common ratio lessened by one, it becomes the sum of the series). The rule for finding out the last term in a geometrically progressive series as also the sum of that (series) : . 95. The antyadhana or the last term of a series in geometrical progression is the gunadhana (of another series) wherein the number of terms is less by one. This (antyadhana), when multiplied by the common ratio, and (then) diminished by the first term, and (then) divided by the common ratio lessened by one, gives rise to the sum (of the series). An example in illustration thereof. 96. Having (first) obtained 2 golden coins (in some city), a man goes on from city to city, earning (everywhere) three times (of what he earned immediately before). Say how much he will make in the eighth city. Now, in the representative colamn of figures so derived and given in the margin o the lowest 1 is multiplied by r, which gives r: since this lowest 1 has 0 O above it, the r obtained as before is squared, which gives 12: since this 0 has 1 above it, the qui now obtained is multiplied by r, which gives 78 ; since this 1 has 0 above it, this 18 is squared, which gives p(r): and since again this 0 has another O above it, this is squared, which gives 2 Thus the value of r may be arrived at by using as few times as possible the processes of squaring and simple multiplication. The object of the method is to facilitate the determination of the value of g; and it is easily seen that the method holds true for all positive and integral values of n. 95. Expressed algebraically, S = S arn 1xr - a 9 -4. The antyadhana is the value of the last term in a series in geometrical progression; for the meaning and value of guradhana, see stanza 93 above in this chapter. The antyadhana of a geometrically progressive series of n terms is ari-1, while the gunadhana of the same series is arn. Similarly the antyad hana of a geometrically progressive series of n-1 terms is ar -2, while the gunadhana thereof is arn-1, Here it is evident that the antyadhana of the series of >> termy is the same 48 the gunadhana of the series of n l terms. For Private and Personal Use Only

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GANITASABASANGRAHA. The rule for finding out the first term and the common ratio in relation to a (given) gunadhana : 97. The gunadhana when divided by the first term becomes equal to the (self-multiplied) product of a certain quantity in which (product) that (quantity) occurs as often as the number of terms (in the series); and this (quantity) is the (required) common ratio. The gunadhana, when divided by that (self-multiplied) product of the common ratio in which (product the frequency of the occurrence of this common ratio) is measured by the number of terms (in the series, gives rise to the first term. The rule for finding out in relation to a given gunadhana the number of terms (in the corresponding geometrically progressive series): 98. Divide the gunadhana (of the series) by the first term (thereof). Then divide this (quotient) by the common ratio (time after time) so that there is nothing left (to carry out such a division any further); whatever happens (here) to be the number of vertical strokes, (each representing a single such division), so much is (the value of the number of terms in relation to the (given) gunadhana. 'Examples in illustration thereof. 99. A certain man (in going from city to city) earned money (in a geometrically progressive series) having 5 dinaras for the first term (thereof) and 2 for the common ratio. He (thus) entered 8 cities. How many are the dinaras (in) bis (possession)? 100. What is the value of) the wealth owned by a merchant (when it is measured by the sum of a geon: etrically progressive series), the first term whereof is 7, the common ratio 3, and the number of terms (wherein) is 9: and again (when it is measured by the sum of another geometrically progressive series), the first 97 and 93. It is clear that ain, when divided by a gives 7'; and this is dirisible by r as many times as n, which is accordingly the measure of the number of terms in the series. Similarly rxrx ...... up to times gives y; and the gunadhana i.e., ara divided by this gives a, which is the required first term of the series, For Private and Personal Use Only

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CHAPTER II-ARITHMETICAL OPERATIONS. 33 term, the common ratio and the number of terms thereof being 3,5 and 15 (respectively)? The rule for finding out the common ratio and the first term in relation to the (given) sam of a series in geometrical progression : 101. That (quantity) by which the sum of the series divided by the first term and (then) lessened by one is divisible throughout (when this process of division after the subtraction of one is carried on in relation to all the successive quotients) time after time-(that quantity) is the common ratio. The sum, multiplied by the common ratio lessened by one, and (then) divided by that self-multiplied product of the common ratio in which product) that (common ratio) occurs as frequently as the number of terms (in the series), after this (same self-multiplied product of the common ratio) is diminished by one, gives rise to the first term. Examples in illustration thereof. 102. When the first term is 3, the number of terms is 6, and the sum is 4095 (in relation to a series in geometrical progression), what is the value of the common ratio ? The common ratio is 6, the number of terms is 5, and the sum is 3110 (in relation to another series in geometrical.progression). What is the first term here? 140 101. The first part of the rule will become clear from the following example: The sum of the series is 4095, the first term 3, and the number of terms 6. Here, dividing 4095 by 3 we get 1365. Now, 1365 - 1=1364. Choosing by trial 4, we have 1564 = 341; 341 - 1 = 340; 340 = 85; 85 - 1 =- 84 ; 84 * = 21; 21 - 1=20; * = 5; 5 - 1=4; 1= 1. Hence 4 is the common ratio. The principle on which this method is based will be clear from the following: - a (ra-1). h is obviously divisible 7-1 by r. a(rn -1) r-1 The second part expressed algebraically is a r-1 -10 . Hand For Private and Personal Use Only

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34 GANITASARASANGRAHA. The rule for finding out the number of terms in a goometrioally progressive series : 103. Multiply the sum of the given series in geometrical progression) by the common ratio lessened by one ; (ther) divide this (product) by. the first term and (then) add one to this (quotient). The number of times that this (resulting quantity) is (successively) divisible by the common ratio--that gives the measure of the number of terms (in the series). Examples in illuxtration thereof. 104. O my excellently atte mathematical friend, tell me of what value the number of terms is in relation to (a series, whereof) the first term is 3, the common ratio is 6, and the sum is 777. 105. What is the value of the number of terms in those (series) which (respectively) have 5 for the first term, 2 for the common ratio, 1275 for the sum : 7 for the first term, 3 for the common ratio, 68887 for the sum : and 3 for the first term, 5 for the common ratio and 22888183593 for the sum ? Thus ends summation, the seventh of the operations known as Parikarman. Vyutkalita. The rule of work in relation to the operation of Vyutkalita, * which is the eighth (of the Parikarman.operations), is as follows: 106. (Take) the chosen-off number of terms as combined with the total number of terms (in the series), and (take) also your own chosen-off number of terms (simply); diminish (each of) * In a given series, any portion chosen off from the beginning is called i ta or the chosen-off part, and the rest of the series is oalled sea, and it contains the remaining terms and forms the remainder-series. It is the sum of these sa terms which is called vyutkalita. 106. Algebraically, vyutkalita or Sv == { "*"-20 + a}(n - d), and the sum of the isla or Si= 1.6+ a) d; where d is the number of terms in the chosen-off part of the series. For Private and Personal Use Only

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CRAPTER II - ARITHMETICAL OPERATIONS. 35 these (quantities) by one and (then) halve it and multiply it by the common difference; and (then) add the first term to (each of) these resulting products). And these resulting quantities), when multiplied by the remaining number of terms and the chosen-off number of terms (respectively), give rise to the sum of the remainder-series and to the sum of the chosen-off part of the series in order). The rule for obtaining in another manner the sum of the remainder-series and also the sum of the chosen-off part of the given series : 107. (Take) the chosen-off number of terms as combined with the total number of terms (in the series), and (take) also the chosen-off number of terms (simply); diminish (each of) these by one, and (then) multiply by the common difference, and (then) add to (each of) these resulting products) twice the first term. These (resulting quantities), when multiplied by the half of the remaining number of terms and by the half of the chosen-off number of terms (respectively), give rise to the sum of the remainder-series and to the sum of the chosen-off part of the series in order). The rule for finding out the sum of the remainder-series in respect of an arithmetically progressive as well as a geometrically progressive series, as also for finding out the remaining number of terms (belonging to the remainder-series) : 108. The sum (of the given series) diminished by the sum of the chosen-off part (of the series) gives rise to the sum of the remainder-series in respect of the arithmetically progressive as well as the geometrically progressive series ; and when the difference between the total number of terms and the chosen-off number of terms in the series) is obtained, it becomes the remaining number of terms belonging to that (remainder-series). 107. Again, $c = {*+ d - 1) 6+2a)"mod, and $i= {(1- 1) b+2a) For Private and Personal Use Only

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36 GANITASARASANGRAHA. The rule for finding out the first term in relation to the remaining number of terms (belonging to the remainder-eeries) : 109. The chosen-off number of terms multiplied by the common difference and then combined with the first term (of the given series) gives rise to the first term in relation to the remaining terms (belonging to the remainder-series) The already mentioned common difference is the common difference in relation to these (remaining terms also); and in relation to the chosen-off number of terms (also both the first term and the common difference) are exactly those (which are found in the given series). The rule for finding out the first term in relation to the remaining number of terms belonging to the remainder-series in a geometrically progressive series : 110. Even in respect of a geometrically progressive series, the common ratio and the first term are exactly alike in the given series and in the chosen-off part thereof). There is (however) this difference here in respect of (the first term in relation to the remaining number of terms in the remainder-series), viz., that the first term of the (given) series multiplied by that self-multiplied product of the common ratio, in which (product) the frequency of the occurrence of the common ratio is measured by the chosen-off number of terms, gives rise to the first term of the remainder-series). Examples in illustration thereof. 111. Calculate what the sums of the remainder-series are in respect of a series in arithmetical progression, the first term of which is 2, the common difference is 3, and the number of terms is 14, when the chosen-off numbers of the terms are 7, 8, 9, 6 and 5 (respectively). 112. (In connection with a series in arithmetical progression here (given), the first term is 6, the common difference is 8, the number of terms is 36, and the chosen-off numbers of terms are 10, 109. The first term of the remainder series = db + a. The series dealt with in this rule is obviously in arithmetical progression. 110. The first term of the remainder series is ard. For Private and Personal Use Only

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CHAPTER 11-ARITHMETICAL OPERATIONS. 37 12 and 16 (respectively). In connection with another (similar series), the first term and the other things are 5, 5, 200 and 100 (in order). Say what the sums are of the corresponding) remainder-serie's. 113. The number of terms (in a series in arithmetical progression) is 216; the common difference is 8; the first term is 14; 37 is the chosen-off number of terms (to be removed). Find the sums both of the remainder-series and of the chosen-off part (of the given series). 114. The first term (in a given series in arithmetical progression) is, in this (problem), 64; the common difference is minus 4; the number of terms is 16. What are the sums of the remainderseries when the chosen-off numbers of terms are 7, 9, 11 and 12? Examples on vyutkalita in respect of a geometrically progressive series. 115. Where (in the process of reckoning of the fruits on trees in serial bunches), 4 happens to be the first term, 2 the common ratio, and 16 the number of terms, while the chosen-off number of terms (removed) are 10, 9, 8, 7, 6, 5 and 4 (respectively) - there, say, O you who know arithmetic and bave penetrated into the interior of the forest of practical mathematical operations, (the interior) wherein wild elephants sport-(there say) what the total of the remaining fruits is on the tops of the various good trees (dealt with therein). Thus ends vyutkalita, the eighth of the operations known as Parikarman. Thus ends the first subject of treatment known as Parikar man in Sarasangraha, which is a work on arithmetic by Mahaviracarya. 115. In this problem, there are given 7 different fruit trees, each of which has 16 bunches of fruits. The lowest bunch on each tree has 4 fruits; the fruits in the higher banches are geometrically progressive in number, the common ratio being 2; and 10, 9, 8, 7, 6, 5 and 4 represent the numbers of the bunches removed from below in order from the 7 trees. We have to find out "the total of the remaining fruits on the top of the various good trees". Mattebhavikridita, as it occurs in this stanza, is the name of the metre in which it is composed, at the same time that it means the sporting of wild elephants. For Private and Personal Use Only

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Shri Mahavir Jain Aradhana Kendra 38 www.kobatirth.org GANITASARASANGRAHA. Acharya Shri Kailassagarsuri Gyanmandir CHAPTER III. FRACTIONS. The Second Subject of Treatment. 1. Unto that excellent Lord of the Jinas, by whom the tree of karman has been completely uprooted, and whose lotus-like feet are enveloped in the halo of splendour proceeding from the tops of the crowns belonging to the chief sovereigns in all the three worlds -(unto that Lord of the Jinas), I bow in devotion. Hereafter, we shall expound the second subject of treatment known as Kalasavarna (t.e., fractions). * Multiplication of Fractions. The rule of work here, in relation to the multiplication of fractions, is as follows:-- 2. In the multiplication of fractions, the numerators are to be multiplied by the numerators and the denominators by the denominators, after carrying out the process of cross-roduction, if that be possible in relation to them. Examples in illustration thereof. 3. Tell me, friend, dried ginger, if he gets what a person will got for of a pala of of a pana for 1 pala of such ginger. 4. Where the price of 1 pala of pepper is of a pana, there, say, what the price will be of g of a pala. 5. A person gets of a pala of long pepper for 1 pana. O arithmetician, mention, after multiplying, what (he gets) for panas. 6. Where a merchant buys of a pala of cumin seeds for 1 pana, there, O you who possess complete knowledge, mention what (he buys) for panas. 7. The numerators of the given fractions begin with 2 and go on increasing gradually by 2; again their denominators begin *Kalasavarna literally means parts resembling, since kala denotes the sixteenth part. Hence the term Kalasavarna has come to signify fractions in general. 2. When is reduced asx, the process of cross-reduction is applied. 7. The fractions herein mentioned are: 3,,, &o. For Private and Personal Use Only

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CHAPTER III - FRACTIONS. 39 with 3 and go on increasing by 2; those (numerators and denominators) are, in both (the cases), 10 in number. Mention, of what value the products here will be, when those (fractions) are multiplied, thoy being taken two by two. Thus ends multiplication of fractions. Division of Fractions. The rule of work, in relation to the division of fractions, is as follows: 8. After making the denominator of the divisor its numerator (and vice versa), the operation to be conducted then is as in the multiplication (of fractions). Or, whon (the fractions conscituting) the divisor and the dividend are multiplied by the denominators of each other and (these two products) are (thus reduced so as to be) without denominators, (the operation to be conducted is as in the division of whole numbers. Examples in illustration thereof. 9. When the cost of half a palu of asafoetida is of a pana, what does a person get if he sells 1 pala at that (same) rate ? 10. In case a person gets 20 of a pana for of a pala of red sandalwood, what will ho get for 1 pala (of the same wood) ? 11. When 5,2 palas of the perfume nakha is obtainable for of a pana, what (will be obtainable) for 1 pana at that (same rate) ? 12. The numerators (of the given fractions) begin with 3 and go on increasing gradually by 1, till they are 8 in number; the denominators begin with 2 and are tthroughout) less by one (than the corresponding numerators). Tell me what the result is when the succeeding (fractions here) are divided in order by the preceding ones). Thus ends the division of fractions. 8. (i) + 8 = $; (ii) = a = ad ;-bc. For Private and Personal Use Only

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40 GANITASARASANGRAHA. The Squaring, Square-Root, Cubing, and Cube Root of Fractions. In connection with the squaring, the square root, the cubing, and the cabe root of fractions, the role of operation is as follows: 13. If, after getting the square, the square root, the cube (or) the cube root of the (simplified) denominator and numerator (of the given fraction), the (new) numerator (so obtained) is divided by the (similarly new) denominator, there arises the result of the operation of squaring or of any of the other above-mentioned (operations as the case may be) iu relation to fractions. Examples in illustration thereof. 14. V arithmetician, tell me the squares of 1, 1, , 10, 199 and 200. 15. The numerators of the given fractions) begin with 3 and (gradualls) rise by 2; the denominators begin with 2 and (gradually) rise by 1; the number of these (fractions) is known to be 12. Tell me quickly their squares, you who are foremost among arithmeticians. 16. Tell me quickly, O arithmetician, the square roots of 1, to the n's and 36 17. O clever man, tell me what the square roots are of the squared quantities which are found in the examples bearing on the) squaring of fractions and also of 67 18. The following quantities, wamely, }, }, }, }, , }, } and }, are given. Tell me their cubes separately. 19. The numerators (of the given fractions) begin with 3, and (gradually rise by 4; the denominators begin with 2 and (gradually rise by 2; the number of such (fractional) terms is 10. Tell me their cubes quickly, O friend who are possessed of keen intelligence in calculation. 17. Here 676 is given in the original as 700** 8. For Private and Personal Use Only

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CHAPTER INFRACTIONS. 20. Give the oube roots of 185 and 42. 21. O friend of prominent intelligence, give the cube roots of the cubed quantities found in the examples on) the onbing of fractions and (give also the cube root) of 242. Thus end the squaring, square-root, cubing and cube-root of fractions. Summation of fractional series in progression. In regard to the summation of fractional series, the rule of work is as follows: 22. The optional number of terms (making up the fractional series in arithmetical progression) is multiplied by the common difference, and then it) is combined with twice the first term and diminished by the common difference. And when this (resulting quantity) is multiplied by the balf of the number of terms, it gives rise to the sum in relation to a fractional series (in arithmetical progression) Examples in illustration thereof. * 23. Tell me what the sum is (in relation to a series) of which , } and are the first term, the common difference and the number of terms (in order); as also in relation to another of which }, { and f (constitute these elements). 24. The first term, the common difference and the pumber of terms are }, } and in order in relation to a given series in arithmetical progression). The numerators and denominators of all (these fractional quantities) are (successively) increased by 2 and 3 (respectively) until seven (series are so made up). What is the sum of each of these) ? 22 Algebraically S=(nb + 2a-6) . Cf. note under 62, Chap. JI. 23. Whenever the number of terms in a series is given as a fraction, as here, it is evident that such a series cannot generally be formed actually number of terms. But the intention seenis to be to show that the rule holds good even in such cases. For Private and Personal Use Only

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GANITASARASANGRAHA. The rule for arriving, in relation to (a series made up of any) optional number of terms, at the first term, the common difference and the (related) sum, which is equivalent firstly to the square and secondly to the cube (of the number of terms) : 25. Whatever is (80) chosen is the number of terms, and one is the first term. The pumber of terms diminished by the first term, and (then) divided by the half of the number of terms diminished by one, becomes the common difference. The sum of the series) in relation to these is the square of the number of terms. This multiplied by the number of terms becomes the cube thereof. Examples in illustration thereof. 26. The optional number of terms (in a given series) is (taken to be) 4; and the numerator as well as the denominator (of this fraction) is (successively) increased by one till ten (such different fractional terms) are obtained. In relation to these (fractions taken as the number of terms of corresponding arithmetically progressive series), give out the first term, the common difference and the square and the cube (values of the sums in the manner explained above). The rule for finding out the first term, the common difference and the number of terms, in relation to the sum of a series in arithmetioal progression) which (sum) bappens to be the cube of (any) chosen quantity : 27. One-fourth of the chosen quantity is the first term; and from this first term, when it is multiplied by two, results the 25. It is obvious that, in the formula S= (2a + n-1.6), the value of s becomes equivalent to no when a = 1, and b=n 24 24. In the molt In the multiplication of nthis gum by n, there is necessarily involved the multiplication of a as well as of b by n, so that, when a=n and b=(n=9) 2n, S=n8 A little consideration will show how the value of b as an 87-1" - makes it possible to arrive at nag the value of S whatever may be the value of a, whether fractional or integral. 27. This rule gives only a particular case of what may be generally applied. The rule as given here works out thns: +*+*+ ......up to 2x terms 1- 1 For Private and Personal Use Only

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CHAPTER III-FRACTIONS. 43 common difference. The common difference multiplied by four is the number of terms (in the required series). The sum as related to these is the cube (of the chosen quantity). Examples in illustration thereof. 28. The numerators begin with 2 and are successively increased by 1; the denominators begin with 3 and are (alxo) successively increased by l; and both these kinds of terms (namely, the numerators and the denominators) are severally) five (in number). In relation to these (chosen fractional quantities), give out, o friend, the cubic sum and the corresponding) first term, common difference, and number of terms. The rule for finding out, from the known sum, first term and common difference (of a given series in arithmetical progression), the first term and the common difference (of a series), the optionally chosen sum (whereof) is twice, three times, half, one-third, or some such (multiple or fraction of the known sum of the given series): 29. Put down in two places (for facility of working) the chosen sum as divided by the known sum. This (quotient), when multiplied by the (kuown) common difference, gives the (required) common difference--and that (same quotient), when multiplied by the (known) first term, gives the (required) first term--of (the series of which the sum is either a multiple or a fraction (of the known sum of the given series). Examples in illustration thereof. 30. The first term (of a series) is i, the common difference is 1, and the number of terms common (to the given as well as the = (2x) = <3. The general applicability of this process can be at once made out from the equality, (px)2 = 2, 80 that in all such cases the number of terms in the series is obtained by multiplying by p3 the first term, which is representable and the common difference is of course taken to be twice this first term in every case. 29. See note ander 84, Chap. II, For Private and Personal Use Only

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44 GANITASARASANGRABA. required series) is taken to be) . The sum of the required series is of the same value . Find out, O friend, the first term and the common difference (of the required series). 31. The first term is twice the common difference (which is taken to be 1); the number of terms is (taken to be) * The sum of the required series is 67. Find out the first term and the common difference. 32. The first term is 1, the common difference is and the number of terms common (to both the given as well as the required series) is taken to be) . The sum of the required series is Give out the first term and the common difference (of the required series) The rule for finding out the number of terms (in a series in arithmetical progression) : 33. When, to the square root of (the quantity obtained by) the addition, of the square of the difference between the half of the common difference and the first term, to twice the common difference multiplied by the sum of the series, half the common difference is added, and when (this sum is) diminished by the first term, and then divided by the common difference, (we get) the number of terms in the series. He (the author) states in another way (the rule for finding out) the same number of terms): 34. When, from the square root of (the quantity obtained by) the addition, of the square of the difference between the half of the common difference and the first term, to twice the common difference multiplied by the sum of the series, the ksepapada is subtracted, and when (this resulting quantity is divided by the common difference, (we get the number of terms in the series. N 88. Symbolically expresoa, 2.0+(88) * +3. 05. nove and - CJ. note ander 33. Symbolioally expressed, n = 69, in Chap. II. . 34. For ksipapada, see note under 70 in Chap. II. For Private and Personal Use Only

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CHAPTER 111--FRACTIONS. 45 Examples in illustration thereof. 35. In relation to this (given) series, the first term is }, the common difference is , and the sum given is ; again in relation to another series), the common difference is the value of the first term is , and the sum is to In respect of these two (series), O friend, give out the number of terms quickly. The rule for finding out the first term as well as the common difference : 36. The sum (of the series) divided by the number of terms (therein), when diminished by the product of) the common difference multiplied by the half of the number of terms less by one, (gives) the first term (of the series). The common difference is (obtained when the sum, divided by the number of terms and (then) diminished by the first term, is divided by the half of the number of terms less by one. Examples in illustration thereof. 37. Give out the first term and the common difference (respectively) in relation to (the two series characterised by) 3 as the sum, and having in one case) as the common difference and f as the number of terms, and (in the other case) as the first term and as the number of terms. The rule for finding out in relation to two (series), the number of terms wherein is optionally chosen, their mutually interchanged first term and the common difference, as also their sums which may be equal, or (one of which may be) twice, thrice, half or onethird (of the other) : 38. The number of terms (in one series), multiplied by itself as lessoned by one, and then multiplied by the chosen (ratio between the sums of the two series), and then diminished by twice the number of terms in the other series, gives rise to the interchangeable) first term (of one of the series). The square of the 36. See note under 74, Chap. II. 88. See note ander 86, Chap. II. For Private and Personal Use Only

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Shri Mahavir Jain Aradhana Kendra 46 www.kobatirth.org GANITA SARASANGRAH. (number of terms in the) other (series), diminished by that (number of terms) itself, and (then) diminished (again) by the product of two (times) the chosen (ratio) and the number of terms (in the first series, gives rise to the interchangeable) common difference (of that series). Acharya Shri Kailassagarsuri Gyanmandir Examples in illustration thereof. 39. In relation to two series, having 103 and 93 to (respectively) represent their number of terms, the first term and the common difference are interchangeable, the sum of one (of the series) is either a multiple or a fraction (of that of the other, this multiple or fraction being the result of the multiplication or the division as the case may be) by means of (the natural numbers) commencing with 1. O friend, give out (these) sumns, the first, terms and the common differences. The rule for finding out the gunadhana and the sum of a series in geometrical progression 40. The first term (of a series in geometrical progression), when multiplied by that self-multiplied product of the common ratio, in which (product) the frequency of the occurrence of the common ratio is measured by the number of terms (in the series), gives rise to the gunadhana. And it has to be understood that this (gunadhana), when diminished by the first term and (then) divided by the common ratio lessened by one, becomes the sum of the series in geometrical progression. The rule for finding out the last term in a geometrically progressive series as well as the sum of that (series) : 41. The antyadhana or the last term of a series in geometrical progression is the gunadhana of (another series) wherein the number of terms is less by one. This (antyadhana), when multiplied by the common ratio and (then) diminished by the first term and (then) 40. See note under 93, Chap. II. 41. See note under 95, Chap. II. For Private and Personal Use Only

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CHAPTER III-FRACTIONS. 47 divided by the common ratio lessened by one, gives rise to the sum (of the series). An example in illustration thereof. 42. In relation to a series in geometrical progression, the first term is , the common ratio is and the number of terms is here 5. Tell me quickly the sum and the last term of that (series). The first term, the common ratio and the number of terms, in relation to the gunadhana and the sum of a series in geometrical progression, should also be found out by means of the rules stated already (in the last chapter).* The rule for finding out the common) first term of two series having the same sum, one of them being in arithmetical progression and the other in geometrical progression, their optionally chosen number of terms being equal and the similarly chosen common difference and common ratio also being equal in value. 43. One is taken as the first term, the number of terms and the common ratio as well as the common difference (which is equal to it) are optionally chosen. The uttaradhana (here), divided by the sum of this geometrically progressive series as diminished by the adidhana (thereof), and (then) multiplied by whatever is taken as the first term, gives rise to the (required common) first term in relation to the two series, (one of which is in geometrical progression and the other in arithmetical progression, and both of) which are characterised by sums of the same value. For these rules, see 97, 98, 101 and 103, Chap. II. 43. For adidhana and uttaradhana, see note under 63 and 64, Chap. II. This m (n-1) o rule, symbolically expressed, works out thus: a= 2 - --- Where 0 =r. (P-1)1.-hx1 For facility of working, 1 is chosen as the provisional first term, but it is obvious that any quantity may be so provisionally chosen. The use of the provisional first term is seen in facilitating the statement of the rule by means of the expressions adidhana and utturadhana. The formula bere given is obtained by equating the formulae giving the sums of the geometrical and the arithmetical series. It is worth noting that the word caya is used here to denote both the common difference in an arithmetical and the common ratio in a geometrical series. For Private and Personal Use Only

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GANITASARASANGRAHA. Examples in illustration thereof. 44. The number of terms are 5, 4 and 3 (respectively) and the common ratios as well as the (equal) common differences are and (in order). What is the value of the (corresponding) first terms in relation to these (sets of two series, one in geometrical progression and the other in arithmetical progression), which are characterised by sums of the same value ? Thus ends the summation of fractions in series. Vyutkalita of fractions in series. The rule for performing the operation of vyutkalita is as follows: 45. Take) the chosen-off number of terms as combined with the total number of terms (in the series), and (take) also your chosen-off number of terms (separately). Multiply each of these quantities by the common difference and diminish (the products) by the common difference ; (then) multiply by two; and these (resulting quantities), when multiplied by the half of the remaining number of terms and by the half of the chosen-off number of terms (respectively), give rise to the sum of the remainder-series and to the sum of the chosen-off part of the (given) series (in order) The rule for finding out the first term in relation to the remaining number of terms (making up the remainder-series) :-- 46. The first term (of the series), diminished by the half of the common difference, and combined with the chosen-off number of terms as multiplied by the common difference, as also with the half of the common difference, (gives) the first term of the remaining number of terms (making up the remainder-series). And the common difference (of the remainder-series) is the same as what is found in the given series. 45. Cf. note under 106, Chap. II. 46. Cf. note under 109, Chap. II. For Private and Personal Use Only

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CHAPTER III-FRACTIONS. 46 47. Even in respect of a geometrically progressive series, the common ratio and the first term are exactly alike in the given series and in the chosen-off part thereof). There is (however) this difference here in respect of the first term among) the remaining number of terms (constituting the remainder-series), viz., that the first term of the (given) series multiplied by that self-multipliod product of the common ratio, in which (product) the frequency of occurrence of the common ratio is measured by the chosen-off number of terms, gives rise to the first term (of the remainder-series). Examples in illustration thereof. . 48. Calculate what the sum of the remainder-series is in relation to that (series) of which is the common difference, the first term, and is (taken to be) the number of terms, when the chosen-off number of terms (to be removed) is (taken as) 1. 49. In relation to a series in arithmetical progression, the first term is 1, the common difference is t, and the number of terms is (taken to be) 3. When the chosen-off number of terms (to be removed) is taken as $, give out, you who know calculation, the sum of the remainder-series. 50. What is the value of the sum of the remainder-series in relation to a series of which the first term is i, the common difference is }, and the number of terms is (taken to be) }, when the chosen-off number of terms is ty? 51. The first term is , the common difference is }, and the number of terms is (takou as) A, and the chosen-off number of terms is taken to be or} O you, who, being the abode of kalas *, are the moon shining with the moonlight of wisdom, tell me the sum of the remaining number of terms. 52, Calculate the sum of the remaining number of terms in relation to a series of which the number of terms is 12, the common difference is minus }, and the first term is 41, the chosenoff number of terms being 3, 4, 5 or 8. 47. See note under 110, Chap. II. Kala is here used in the double sensc of moon'. learning' and 'the digits of the For Private and Personal Use Only

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GANITASARASANG RAHA. Example in illustration of vyutkalita in relation to a series in geometrical progression. 53. The first torm is 71, the common ratio is j, and the number of terms is 8; and the chosen-off number of terms is 3, 4 or 5. What are the first term, the sum and the number of terms in relation to the respective) remainder-series? Thus ends the vyutkalita of fractions, The six varieties of fractions. Hereafter we shall expound the six varieties of fractions. 54. Bhaga (or simple fractions), Prabhaga (or fractions of fractions), then Bhagabhaga (or complex fractions), then Bhaganubandha (or fractions in association), Bhagapavaha (or fractions in dissociation), together with Bhagamatr (or fractions consisting of two or more of the above-mentioned fractions)--these are here said to be the six varieties of fractions. Simple fractions: (addition and subtraction) The rule of operation in connection with simple fractions therein : 55. If, in the operations relating to simple fractions, the numerator and the denominator (of each of two given simple fractions) are multiplied in alternation by the quotients obtained 55. The methid of reducing fractions to commion denominators described in this rule applies only to pairs of fractions. The rule will be clear from the following worked out example :-- To simplify + llere, a andry are to be multiplied by a which is z the quotient obtained by dividing yz, the denominator of the other fraction, by y which is the common factor of the denominators. Thns we get 42. Similarly in the second fraction, by multiplying band yz by a which is the qnotient obtained by dividing the first denominator xy by y the common factor, we get ox Now a + b az + br khu xyz - aua For Private and Personal Use Only

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CHAPTER 11-FRACTIONS. 51 by dividing the denominators by means of a common factor thereof, (the quotient derived from the denominator of either of the fractions being used in the multiplication of the numerator and the denominator of the other fraction), those (fractions) become so reduced as to have equal denominators. (Then) removing one of these (equal) denominators, the numerators are to be added (to one another) or to be subtracted from one another, so that the result may be the namerator in relation to the other equal denominator). Another rule for arriving at the common denominator in another manner : 56. The niruddha (or the least common multiple) is obtained by means of the continued multiplication of (all) the (possible) comioon factors of the denominators and (all) their (ultimate) quotients. In the case of (all) such multiples of the denominators and the numerators of the given fractions), as are obtained by inultiplying those (denominators and numerators) by means of the quotients derived from the division of the niruddha by the (rospective) denominators, the denominators become equal (in value). Examples in illustration thereof. 57 and 58. A ravaka purchased, for the worship of Jina, jambu fruits, limes, oranges, cocoanuts, plantains, mangoes and pomegranates for i, , 's, zo zo zlo, and of the golden coin in order : tell me what the result is when these (fractions) are added together. 59. Add together 8 do 16 dj and ot. 60. (There are 3 scts of fractions), the denominators whereof begin with 1, 2 and 3, (respectively) and go on increasing gradually by one till the last (of such denominators) becomes 9, 10 and 00. The resulting problems are to find the values of - (i) x2+2x3+3 34+ ..: +8$+ + ikot ikot :::::+9x10+a (ii) Sa + X6+ oxot.....+16 % 18 + in For Private and Personal Use Only

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Shri Mahavir Jain Aradhana Kendra 52 GANITASARASANGRAHA. 16 (in order in the respective sets); the numerators (of these sets of fractions) aro of the same value as the first number (in these sets of denominators), and every one of these (above-mentioned denominators in each set) is multiplied by the next or (the last denominator, however, remaining in each case unchanged for want of a further multiplying denominator). What is the sum of (each of) these (finally resulting sets of fractions)? 61 and 62. (There are 4 sets of fractions), the denominators whereof begin with 1, 2, 3 and 4 (respectively) and rise successively in value by 1 until 20, 42, 25 and 36 become the last (denominators in the several sets) in order; the numerators of these (sets of fractions) are of the same value as the first number (in these sets of denominators). And every one of these (denominators in each set) is multiplied in order by the next one, (the last denominator, however, remaining unchanged in each case). What is the sum on adding these (finally resulting sets of fractions)? 63. A man purchased on account of a Jina-festival sandalwood, camphor, agaru and saffron for,,2% and of a golden coin. What is the remainder (left thereof)? 64. A worthy sravaka gave me two golden coins and told me that I should bring, for the purpose of worshipping in the temple of Jina, blossomed white lotuses, thick curds, ghee, milk and sandal-wood for,,, and of a golden coin, (respectively, out of the given amount). Now tell me, O arithmetician, what remains after subtracting the (various) parts (so spent). 65 and 66. (There are two sets of fractions) the denominators whereof begin with 8 and 5 (respectively) and rise in both cases successively in value by 1, until 30 becomes (in both cases) the last (denominator). The numerators of these (sets of fractions) are of the same value as the first term in each (of these sets of denominators). And every one of the denominators (in each sot) is multiplied by the next one, the last (denominator) being (in each case) multiplied by 4. After subtracting from 1, (each of) these two (sums obtained by the addition of the sets of fractions finally resulting as above), tell me, O friend, who have gone over to the other shore of the ocean of simple fractions, what it is that remains. For Private and Personal Use Only

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CHAPTER III-FRACTIONS. 67 to 71. The denominators of certain given fractions) are stated to be 19, 23, 62, 29, 123, 35, 188, 37, 98, 47, 140, 141, 116, 31, 92, 57, 73, 55, 110, 49, 74, 219, (in order); and the numeratore begin with 1 and rise successively in value by 1 (in order). Add (all) these (fractions) and give the result, O you who have reached the other shore of the ocean of simple fractions. ... Iero, the rule for arriving at the numerators, when the denoininators and the sum of a number of fractions are given, is as follows) : 72. Make one the numerator (in relation to all the given denominators) ; then, multiply by means of such (numbers) as are optionally chosen, those numerators which are derived from these fractions so as to have a common denominator. (Here), those (numbers) turn out to be the required numerators, the sum of the products whereof, obtained by multiplying them with the numera - tors (derived as above), is equal to the numerator of) the given sum of the fractions concerned). The rule for arriving at the numerators, (the denominators and the sum being given as before), in relation to such (fractional) quantities as have their nunerators (successively) rising in value by one, when, in the (given) sum of these fractions), the denominator is higher in value than the numerator: 73. The quotient obtained by dividing the (given) sum of the fractions concerned) by the sum of those (tentative fractions) 72. This role will become clear from the working of the example in stanza No. 74, wherein we assume 1 to be the provisional no merator in relation to each of the given denominators ; thus we get 5, 7, and 1, which, being reduced so as to have a coinmon denominator, becoine %, 0% and 99%. When the numerators are multiplied by 2, 3 and 4 in order, the suin of the products thus obtained becomes equal to the numerator of the given sum, namely, 877. Hence, 2, 3, and 4 are the requirerl namorators. Here it may be pointed out that this given sum also must be understood to have the same denominator as the common denominator of the fractions. 73. To work ont the sum given under 74 below, according to this rule :-- Reducing to the same denominator the fractions formed by assuming 1 to be the namorator in relation to each of the giren denominators, we get 138,% and %. Dividing the giren eum 357 by the sum of these fractions 69, we get the quotient %, which is the numerator in relation to the first denominator. The remainder 279 For Private and Personal Use Only

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54 GANITASARASANGRAUA. which, (while having the given denominators), have one for the numorators and (are then reduced so as to) have a common denominator, becomes the (first required) numerator among those which (successively rise in value by one (and are to be found out). On the remainder (obtained in this division) being divided by the sum of the other numerators having the common denominator as above), it, (.e., the resulting quotieut), becomes another (viz., the second required) numerator (if added to the first one already obtained). In this manner the problem has to be worked out) to the end. In example in illustration thereof. 74. The sum of (certain numbers which are divided (respectively) by 9, 10 and 11 is 877 as divided by 990. Give out what the numerators are in this operation of adding fractions), The rule for arriving at the required) denominators (is as follows) : 75. When the sum of the different fractional) quantities having one for their numerators is one, the required) denominators are such as, beginning with one, are in order multiplied (successively) by obtained in this division is then divided by the sum of the romaining provisional numerators, i.e., 189, giving the qbotient 1, which, combined with the numerator of the first fraction, namely 2, becomes the Duinerator in relation to the second denominator. The renainder in this second division, viz., 90, is divided by the provisional numerator 90 of the last fraction, and the quotient 1, when corubined with the bumerator of the previous fruction, namely 3, gives rise to the numerator in relation to the last denominator. Hence the fractions, of which is the sum, are $, *. and tr. It is noticeable here that the numerators successively found out thas become the required numerators in relation to the given denominators in the order in which they are given. Algebraically also, given the denominator's a, b & c, in respect of 3 fractions bc+ (+ 2) ac + (x + 2) ab , whose sum is the numerators , 1 + abc 1 and * + 2 are easily found out by the method as given above. 75. In working out an example according to the method stated herein, it will be found that when there are fractions, there are, after leaving out the first and the last fractions, * - 2 terms in geometrical progression with 3 as the first tormi and as the common ratio. The 11 of these "-2 tome is 1 1 .. . , which when reduced beconies - which is the same For Private and Personal Use Only

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CHAPTER III-FRACTIONS. 55 three, the first and the last (denominators so obtained) being (however) multiplied (again) by 2 and 3 (respectively). Examples in illustration thereof. 70. The sum of five or six or seven (different fractional) quantities, having 1 for (each of) their numerators, is 1 (in each case). O you, who know arithinetic, say what the required) denominators are, The rule for finding out the denominators in the case of an meven munber of fractions): 77. When the sum of the different fractional, quantities, having one for each of their numerators, is one, the required) denominators are such as, beginning with two, go on (successively) rising in value by one, each (such denominator) being (further) multiplied by that as . From this it is clear that, when the first fraction - X 3n and the last fraction - are added to this last result, the sum becomes 1. . 3. 3n-) are added to In this connection it may be noted that, in a series in geometrical progression consisting of n terms, having as the first term and as the common ratio, the sum is, for all positive integral values of a, less than - by X the (n + 1)th term in the series. geometrical progression 1 Therefore, if we add to the sum of the series in x the (n + 1)th term, which is the last fraction according to the rule stated in this stauza, we get a To this ***a - ), we Q - 2 a - 2 have to add in order to get l as the sam. This is mentioned in the rule as the first fraction, and so 3 is the value chosen for a, since the numerator of all the fractions has to be 1. 77. Here note: 2 X 3 X 3 X 4 X 4 X 5 X3 ****** (n-1), x 1 42 X = 2{2x3+xzetekto+ .... + mom'- 11 + int} 3) + ( - 1 ) + ( - ) + ... + n For Private and Personal Use Only

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Shri Mahavir Jain Aradhana Kendra 56 (number) which is (immediately) next to it (in value) and then halved. www.kobatirth.org The rule for arriving at the (required) denominators (in the case of certain intended fractions), when their numerators are (each) one or other than one, and when the (fraction constituting their) sum has one for its numerator:-- GANITASARASANGRAHA. 78. When the sum (of certain intended fractions) has one for its numerator, then (their required denominators are arrived at by taking) the denominator of the sum to be that of the first (quantity); and (by taking) this (denominator) combined with its own (related) numerator to be (the denominator) of the next (quantity) and so on, and then by multiplying (further each such denominator in order) by that which is (immediately) next to it, the last (denominator) being (however multiplied) by its own (related) numerator. Examples in illustration thereof. 79. The sums (of certain intended fractions) having for their numerators 7, 9, 3 and 13 (respectively) are (firstly) 1, (secondly) and (thirdly). Say what the denominators (of those fractional quantities) are. d (n + The rule for arriving at the denominators (of certain intended fractions) having one for their numerators, when the sum (of those fractions) has one or (any quantity) other than one for its numerator:--- a n (n + a) 1 78. Algebraically, if the sum is, and a, b, c, and d are the given numerators, the fractions summed up are as below:-- d Acharya Shri Kailassagarsuri Gyanmandir a + b + c) b C + + + (n + a) (n + a + b) (n + a + b) (n + a + b + c) + a (n + a + b) + bu n (n + a) (n + a + b) (n + a) (a + b) 12 (n + a) (n + a + b) a + b + n 1 n (n + a + b) n + c+ n + a + b (n + a + b) (n + a + b + c) 1 nta + b For Private and Personal Use Only

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CHAPTER III-TRACTIONS. 57 80. The denominator (of the given sum), when combined with an optionally chosen quantity and then divided by the numerator of that sum so as to loave no romainder, becomes the denominator related to the first numerator in the intended series of fractions) ; and the (above) optionally chosen quantity, when divided by this (denominator of the first fraction) and by the denominator of the (given) sum, gives rise to the sum of) the remaining (fractions in the series). From this (known sum of the remaining fractions in the series, the determination) of the other (denominators is to be carried out) in this very manner. Examples in illustration thereof. 81. Of three (different) fractional quantities having 1 for each of their numerators, the sum is ; and of 4 (such other quantities, the sum is) 4. Say what the denominators are. The rule for arriving at the denominators (of certain intended fractions) having either one or (any number) other than one for their numerators, when the sum of those fractions) has a numerator other than one : 82. When the known numerators are multiplied by (certain) chosen quantities, so that the sum of these (products) is equal to the numerator of the (given) sum (of the intended fractions), then, if the denominator of the sum of the intended fractions) is divided by the multiplier (with which a given numerator has) itself (been multiplied as above), it gives rise to the required denominator in relation to that (numerator). 80. Algebraically, it is the sum, the first fraction is nt and the sum of the remaining fractions is mentioned in the role to be in + Pn pa -- 'where p is the whorenie p optionally chosen quantity. This - is obtained obviously by simplifying " n + P __1 n + P. a We must here give such a value to p that to not a valo to that not p boones erant + p becomes exactly divisible by a. For Private and Personal Use Only

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Shri Mahavir Jain Aradhana Kendra 58 www.kobatirth.org GANITASARASANGRAHA. Examples in illustration thereof. 83. Say what the denominators are of three (different fractional) quantities each of which has 1 for its numerator, when the sum (of those quantities) is . 84. Say what the denominators are of three (fractional quantities) which have 3, 7 and 9 (respectively) for their numerators, when the sum (of those quantities) is 1. The rule for arriving at the denominators of two (fractional) quantities which have one for each of their numerators, when the sum (of those quantities) has one for its numerator: 85. The denominator of the (given) sum multiplied by any chosen number is the denominator (of one of the intended fractional quantities); and this (denominator) divided by the (previously) chosen (number) as lessened by one gives rise to the other (required denominator). Or, when in relation to the denominator of the (given) sum (any chosen) divisor (thereof) and the quotient (obtained therewith) are (each) multiplied by their sum, they give rise to the two (required) denominators. Examples in illustration thereof. 86. Tell me, O you who know the principles of arithmetic, what the denominators of the two (intended fractional) quantities are when their sum is either or Acharya Shri Kailassagarsuri Gyanmandir The first rule for arriving at the denominators of two (intended fractions) which have either one or (any number) other according to this rule are 1 85. Algebraically, when is the sum of two intended fractions, the fractions n 1 pn 1 b(a+b) 1 and, where p is any chosen quantity. It will pn p-1 1 be seen at once that the sum of these two fractions is n 1 Or, when the sum is ggeulhi the fractions may be taken to be For Private and Personal Use Only 1 a(a+b) and

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CHAPTER III--FRACTIONS, than one for their numerators, when the sum (of those fractions) has either one or (any number) other than one for its numerator : 87. (Either) numerator mulitiplied by a chosen (number), then combined with the other numerator, then divided by the numerator of the (given) sum of the intended fractions) so as to leave no remainder, and then divided by the above) chosen number and mulitiplied by the denominator of the (above) sum (of the intended fractions), gives rise to a (required) denominator. The denominator of the other (fraction), however, is this (denominator) multiplied by the (above) chosen (quantity). Examples in illustration thereof. 88. Say what the denominators are of two (intended fractional) quantities which have 1 for each of their numerators, when the sum (of those fractional quantities) is either or }; as also of two (other fractional quantities) which have 7 and 9 (respectively) for (their) numerators. The second rule (is as follows) : 89. The numerator (of one of the intended fractions as multiplied by the denominator of the sum of the intended fractions), when combined with the other numerator and then divided by the numerator of the sum of the intended fractions), gives rise to the denominator of one of the fractions). This (denominator), when multiplied by the denominator of the sum (of the intended fractions), becomes the denominator of the other (fraction). 87. Algebraically, if " is the sum of two intended fractions with a and b n t * Khr a as their namerators, then the fractions are -----, where p ap +b m P is avy number #o chosen that ap + b is divisible by m. The sum of these fractions. it will be found, is : 89. This rule is only a particular case of the rule given in stanza No. 87, as the denominator of the sum of the intended fractions is itself substituted in this role for the quantity to be chosen in the previous rule. For Private and Personal Use Only

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GANITASARASANGRAHA. Examples in illustration thereof. 90. O friend, tell me what the denominators are of two (fractional) quantities which have 1 for each of their numerators, when the sum of those intended fractions) is ; as also of two other intended fractions which have 6 and 8 (respectively) for (their) numerators, 91. The sum of , b and is 1. When is left out here, what two (fractions) having 1 (for each of their numerators) have to be added (instead so as to give the same total)? 92. The sum of 1, 1, and to is 1. If z% is left out here, what two (fractions) having 7 and 11 for their numerators should be added (instead so as to give the same total) ? The rule for arriving at the denominators (of a number of intended fractions by taking (them) in pairs 93. After splitting up the sum (of all the intended fractions) into as many parts having one for each of their numerators as there are (numbers of) pairs (among the given numerators), these (parts) are taken (severally) as the sums of the pairs; (and from them) the (required) denominators are to be found out in accordance with the rule relating to two (such component fractional quantities). Examples in illustration thereof. 94. What are the denominators of those intended) fractions whose numerators are 3, 5, 13, 7, 9 and 11, when the sum of (those fractional) quantities is 1 or ? The rule for arriving at a number of) denominators, with the help of the denominators that have one as their corresponding) numerators and are arrived at according to one of the (already given) rules (for finding out the denominators), as also with the help of the denominators that have one as their corresponding) numerators and are arrived at according to any other of those 93. The rules relating to two fractional quantities have been given in stanzas 85, 87 and 89. For Private and Personal Use Only

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CHAPTER 111-FRACTIONS. rules, when the sum (of all the intended fractions) is one; and also (the rule) for getting at (the value of) the part that is left out : 95. The denominators derived in accordance with any) chosen rule, when (severally) multiplied by the denominators derived in accordance with another rule, become the (reqnired) denominators. The sum of all the fractions), diminished by the sum of the specified part (thereof), gives the measure of the optionally left-. out part. Examples in illustration thereof. . 96. The number of fractions (obtained) hy rule No. 77 is 13, and 4 (is obtained) by rule No. 78. When the sum of the fractions arrived at with the help of these rules) is 1, how many are the component) fractions ? 97. The number of fractions (obtained) by rule No.78 is 7, and 3 (is obtained) by rule No. 77. When the sum of the fractions arrived at) with the help of these (rules) is 1, how many are the (component) fractions ? 98. Certain fractions are given with 1 for each of their numerators, and 2, 6, 12 and 20 for their respective denominators. The (fifth fractional) quantity is here left out. The sum of all these five) being 1, what is that (fractional) quantity (which is left out) ? Here end Simple Fractions. Compound and Complex Fractions The rule for (simplifying) compound and complex fractions : 99. In (simplifying) compound fractions, the multiplication of the numerators (among themselves) as well as of the denominators (among themselves) shall be (the operation). In the operation (of simplification) relating to complex fractions, the denominator of (the fraction forming) the denominator (becomes) the multiplier of the number forming the numerator (of the given fraction). 99. The complex fraction here dealt with is of tho sort which has an integer for the numerator and a fraction for the denominator, For Private and Personal Use Only

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Shri Mahavir Jain Aradhana Kendra 62 www.kobatirth.org GANITASARASANGRAHA. Examples in compound fractions. 100 to 102. To offer in worship at the feet of Jina, lotuses, jasamines, ketakis and lilies were purchased in return for the payment of of 1, of, of of, of of, of of, of ofofofofofofofof, of of of andof, of a pana. Sum up these (paid quantities) and give out the result. Acharya Shri Kailassagarsuri Gyanmandir 103 and 104. A certain person gave (to a vendor) of 1, of of, of, of, and of, (of a pana) out of the 2 panas (in his possession), and brought fresh ghee for (lighting) the lamps in a Jina temple. O friend, give out what the remaining balance is. 105 and 106. If you have taken pains in connection with compound fractions, give out (the resulting sum) after adding these (following fractions):- of, of, 13 of 158, 1 of 1, 1 of and of of. The rule for finding out the one unknown (element common to each of a set of compound fractions whose sum is given): 107. The given sum, when divided by whatever happens to be the sum arrived at in accordance with the rule (mentioned) before by putting down one in the place of the unknown (element in the compound fractions), gives rise to the (required) unknown (element) in (the summing up of) compound fractions. An example in illustration thereof. of, of of, of a certain What is this unknown (quantity)? 108. The sum of, of, quantity is. The rule for finding out more than one unknown (clement, one such occurring in each of a set of compound fractions whose sum is given) : 109. Make the unknown (values of the various partially known compound fractions) to be (equivalent to) such optionally chosen 109. This rule will be clear from the following working of the problem given in stanza No. 110: Splitting up, the sum of the intended fractions, into 3 fractions according to rule No. 78, we get, and. Making these the values of the three For Private and Personal Use Only

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CHAPTER III-FRACTIONS. 3 quantities, as, (being equal in'number to the given compound fractions), have their sam equal to the given sum of the partially given compound fractions) : then, divide these (optionally chosen) values of the unknown (compound-fractional) quantities by (their) known (elements) respectively. An example in illustration thereof. 110. (The following partially known compound fractions, viz.,) 1 of a certain quantity, tof of another (quantity), and of of (yet) another (quantity give rise to) i as (their) sum. What are the unknown (elements here in respect of these compound fractions)? Examples in complex fractions. 111. (Given) }, }, { and }; say what the sum is when these are added. 112. After subtracting }, }, and also and }, from 9, give out the remainder. Thus end Compound and Complex Fractions. Bhaganubandha Fractions. The rule in respect of the (simplification of) Bhaganubandha or associated fractions : 113. In the operation concerning (the simplification of) the Bkaganubandha class (of fractions), add the numerator to the partially known compound fractions, we divide them in order by , t of , and off respectively. The fraetious thus obtained, viz, 5, 8, and /, are the quantities to be found out. 113. Bhaganubandha literally means an associated fraction. This rule contemplates two kinds of associated fractions. The first is what is known as a mixed number, i.e., a fraction associated with an integer. The second kind consists of fractions associated with fractions, e.g., associated with }, associated with its own and with 1 of this associated quantity. The expressione associated with " means 1 + $ of. The meaning of the other example here is + of + of H+ of }). This kind of relationship is what is denoted by ##sociation in additive consecution. For Private and Personal Use Only

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Shri Mahavir Jain Aradhana Kendra 64 GANITASARASANGRAHA. (product of the associated) whole number multiplied by the denominator. (When, however, the associated quantity is not integral, but is fractional), multiply (respectively) the numerator and denominator of the first (fraction, to which the other fraction is attached) by the denominator combined with the numerator, and by the denominator (itself, of this other fraction). Examples on Bhaganubandha fractions containing associated integers. 114. Niskas 2, 3, 6 and 8 in number are (respectively) associated with,, and . O friend, subtract (the sum of these) from 20. 115. Lotuses were saffron for 2 (niskas). purchased for 1, camphor for 10, and What is (their total) value when added? 116. O friend, subtract from 20 (the following) :--81, 67, 271/2 and 3g. 117. A person, after paying 71, 81, 91 and 101 masas, offered in worship in a Jina temple, garlands of blooming kuravaka, kunda, jati and malli flowers. O arithmetician, tell me quickly (the sum of those masas) after adding them. Examples on Bhaganubandha fractions containing associated fractions. 118. (Here) is associated with its own and with (of this associated quantity); and also (is similarly associated); is asso-ciated with its own and with (of this associated quantity). What is the value when these are (all) added ? 119. For the purpose of worshipping the exalted Jinas a certain person brings-flowers (purchased) for (niska) associated (in additive consecution) with fractions (thereof) commencing with and ending with (in order); and scents (purchased) for (niska) associated (similarly) with,, and (thereof); and incense (purchased) for (niska) associated (similarly) with , and (thereof): what is the sum when these (niskas) are added? For Private and Personal Use Only

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CHAPTER III-FRACTIONS. 85 120. O friend, subtract (the following) from 3 : 1 associated with of itself and with 1 of this (associated quantity), & associated with 5, and 4 of itself (in additive consecution), s (similarly) associated with (fractions thereof) commencing with ; and ending with }, and associated with of itself. 121. O friend, you, who have a thorough knowledge of Bhaganubandha, give out the result) after adding associated with of itself, to associated with of itself, i associated with of itself, 1 associated with of itself, and associated with of itself. Now the rule for finding out the one unknown (element) at the beginning (in each of a number of associated fractions, their sum being given): . 122. The optionally split up parts of the (given) sum, which are equal (in number) to the intended) component elements (thereof), when divided in order by the resulting quantities arrived at by taking one to be the associated quantity (in relation to these component elements), give rise to the value of the required) unknown (quantities in association). Examples in illustration thereof. 123. A certain fraction is associated with }, } and of itself (in additive consecution); another (is similarly associated) with }, i, and ss of itself; and another (again is similarly associated) with , and of itself; the sum of these (three fractions so associated) is 1 : what are these fractions ? 124. A certain fraction, when associated (as above) with 1, }, and ss of itself, becomes 1 Tell me, friend, quickly the measure of this unknown (fraction). 122. This rule will be clear from the working of example No. 123 as explained below: There are three sets of fractions given; and splitting up the sum, 1, into three fractions according to rule No. 75, we get, and . By dividing these fractions by the quantities obtained by simplifying the three given sets of fractions wherein 1 is assumed as the unknown quantity, we obtain , 1 and 16, which are the required quantities. For Private and Personal Use Only

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66 GANITABARASANGRAHA. The rule for finding out any unknown fraction in other required places (than the beginning): 125. The optionally split up parts of the (given) sum, when divided in order by the simplified known quantities (in the intended Bhaganubandha fractions), and (then) diminished by one, become the unknown (fractional quantities) in the required places of our choice. Thus ends the Bhaganubandha class (of fractions). Bhagapavaha Fractions, Then (comes the rule for the (simplification of) Bhagapavaka (or the dissociated) variety (in fractions) : 126. In the operation concerning (the simplification of) the Bhagapavaha class (of fractions), subtract the numerator from the (product of the dissociated) whole number as multiplied by the denominator. (When, however, the dissociated quantity is not integral, but is fractional,) mulitiply (respectively) the numerator and the denominator of the first (fraction to which the other fraction is negatively attached) by the denominator diminished by the numerator, and by the denominator (itself, of this other fraction). Examples on Bhaga pavaha fractions containing dissociated integers. 127. Karsus 3, 8, 4 and 10, diminished by 1, 1, tz and of a karsa, are offered by certain men for the worship of tirthankaras. What is the sum) when they are added ? 125. The method given in this rule is similar to what is explained under stanza No. 122: only the results thus obtained have to be, in this case, each diminished by one, 126. Bhaga pavaha literally means fractional dissociation. As in Bhaganubandha, there are two varieties here also. When an integer and a fraction are in Bhagapavaha relation, the fraction is simply subtracted from the integer. Two or more fractions may also be in such relation, as for example, dissoci. ated from of itself or dissociated from & g, and of itself. It is meant here that of is to be subtraoted from $ in the first example; and tbe second example comes to 1-3 of 3 - of (84 of ) - } of 4-3 of 1-1 of ($ - of )}. For Private and Personal Use Only

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CHAPTER III-FRACTIONS. 62 128. Tell me, friend, quickly the amount of the money remain. ing after subtracting from 6 x 4 of it, (the quantities) 9,7 and 9 as diminished in order by 4, and . Examples on Bhagapavaba fractions containing dissociated fractions. 129. Add }, }, }, } and } which are (respectively) diminished by , , }, } and of themselves in order; and (then) give out (the result). 130. (Given) of a pana diminished by b and 1 of itself (in consecution); ' (similarly) diminished by 1, and of itself; similarly) diminished by, and lof itself; and another (quantity), viz., diminished by of itself-when these are (all) added, what is the result ? 131. If you have taken pains, O friend, in relation to Bhagapavala fractions, give out the remainder after subtracting from 1} (the following quantities) : } diminished (in consecution) by , i and ss of itself; also } (similarly) diminished by , and 1 of itself ; and (also) } (similarly) diminished by 1 and of itself. Here, the rule for finding out the (one) unknown element at the beginning (in each of a number of dissociated fractions, their sum being given): 132. The optionally split up parts of the (given) sum which are equal (in number) to the intended) component elements (thereof), when divided in order by the resulting quantities arrived at by taking one to be the dissociated quantity (in relation to these component elements), give rise to the value of the required) unknown (quantities in dissociation). Examples in illustration thereof. 133. A certain fraction is diminished in consecution) by 1, and 1 of itself; another fraction is similarly) diminished by 1, 7 and } of itself; and (yet) another is (similarly) diminished by }, 132. The working is similar to what has been explained under stanza No. 122. For Private and Personal Use Only

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68 GANITASARASANGRAHA. 7 and 4 of itself. The sum of these (quantities so diminished) is 1. What are the unknown fractions here? 134. A certain fraction, diminished (in consecution) by 1, 5, }; * and of itself, becomes . O you, who know the principles of arithmetic, what is that (unknown) fraction ? The rule for finding out any unknown fraction in other required places (than the beginning): 135. The optionally split up parts derived from the (given) sum, when divided in order by the simplified known quantities (in the intended Bhagapavaha fractions), and (then) subtracted from one (severally), become the unknown (fractional quantities) in the (required) places of our choice. Thus ends the Bhagapavaha variety of fractions. The rule for finding out the unknown fractions in all the places in relation to a Bhaganubandha or Bhagapavaha variety of fractions (when their ultimate value is known) : 136. Optionally choose your own desired fractions in relation to all unknown places, excepting (any) one. Then by means of the rules mentioned before, arrive at that (one unknown) fraction with the help of these (optionally chosen fractional quantities). Examples in illustration thereof. 137. A certain fraction combined with five other fractions of itself in additive consecution) becomes }; and a certain (other) fraction diminished (by five other fractions of itself in consecution) becomes 1. O friend, give out (all) those fractions. 135. This rule is similar to the rule already given in stanza No. 125. 136. The previous rules here intended are those given in stanzas 122, 125, 132 and 135. 137. In working out the first case in this example, choose the fractions , , }, # and ; in places other than the beginning; and then find out, by the rule given in stanza 122, the first fraotion which comes to be . Or choosing . . . . and .. find out the fraction left out in a place other than the beginning in accordance with the rule given in stanza 125; the result arrived at is }. Similarly, the second case which involves fractions in dissociation can be worked out with the help of the rules given in stanzas 132 and 135. For Private and Personal Use Only

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CHAPTER III-FRACTIONS. 69 Bhagamat? Fractions. The rule for (the simplification of) that class of fractions which contains all the foregoing varieties of fractions : 138. In the case of the Bhagamatr class of fractions (or that class of fractions which contains all the foregoing varieties), the respectivo rules portaining to the different) varieties beginning with simple fractions (hold good). It, i.e., Bhagamaty, is of twenty-six kinds. One is (taken to be) the denominator (in the case) of a quantity which has no denominator. Examples in illustration thereof. 139 and 140. (Given) }; }); } of } ; & of $; }; }; 13; 1}; } associated with lof itself; then associated with } of itself; 1 diminished by ); 1 diminished by ti diminished by of itself; and diminished by } of itself : after adding these according to the rules which are strung together in the manner of a garland of blue lotuses made up of fractions, give out, o friend, (what the result is). Thus ends the Bhagamotr variety of fractions.. Thus ends the second subject of treatment known as Fractions in Sarasangraha which is a work on arithmetic by Mahaviracarya. - ------- ---- - - --- 138. The twenty-six varieties here mentioned are Bhaga, Prabhaga, Bhagabhaga, Bhaganubandha, hnd Bhagapavala, in combinations of two, three, four or five of these at a time; such as, the variety in which Bhaga and Prabhaga are mixed, or Bhaga and Bhagabhaga are mixed, and so on. The number of varieties obtained by mixing two of them at a time is 10, by mixing three of them at a time is 10, and by mixing four of them at a time is 5, and by mixing all of them at a time is 1; so there are 26 varieties. The example given in stanza 139 belongs to this last-mentioned variety of Bhagamats in which all the five simple varieties are found. 139. The word utpalamalika, which occurs in this stanza, means a garland of blue lotuses, at the same time that it happens to be the name of the metre in which the stanza is composed. For Private and Personal Use Only

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GANITASARASANGRABA. CHAPTER IV. MISCELLANEOUS PROBLEMS (ON FRACTIONS). The Third Subject of Treatment. 1. After saluting the Lord Jina, Mahavira, whose collection of infinite attributes is highly praiseworthy, and who vouchsafes boons to (all) the three worlds that worship (him), I shall treat of miscellaneous problems (on fractions). 2. May Jina, who has destroyed the darkness of unrighteousness, and is the authoritative exponent of the syadvada, and is the joy of learning, and is the great disputant and the best of sages, be (ever) victorious. Hereafter we shall expound the third subject of treatment, viz., miscellaneous problems (on fractions). 3. There are these ten (varieties in miscellaneous probleras on fractions, namely!, Bhaga, Sesa, Mula, Sesamula, the two varieties 3. The Bhaga variety consists of problems wherein is given the numerical value of the portion remaining after removing certain specified fractional parts of the total quantity to be found out. The fractional parts removed are each of them called a bhaga, and the numerical value of the known remainder is termed drsya. The 8 a variety consists of problems wherein the numerical value is given of the portion remaining after removing a known fractional part of the total quantity to be found out as also after removing certain known fractional parts of the successive 87 as or remainders. The Mula variety consists of problems wherein the numerical value is given of the portion remaining after subtracting from the total quantity certain fractional parts thereof as also a multiple of the square root of that total quantity. The Sesumulu variety is the same as the mula variety with this difference, viz.. the square root here is of the remainder after subtracting the given fractional parts, instead of being of the whole. The Dviragra-s ramula variety consists of problems wherein a known number of things is first removed, then some fractional parts of the suocessive remaindera and then some multiple of the square root of the further remainder are removed, and lastly the numerical value of the remaining portion is given. The known number first removed is called purvagra. In the Ameamula variety, a multiple of the square root of a fractional part of the total number is supposed to be first removed, and then the numerical value of the remaining portion is given. For Private and Personal Use Only

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CHAPTER IV-MISCELLANEOUS PROBLEMS (ON FRACTIONS). 71 Dviragrafesamula and Amsamula, and then Bhagabhyasa, then Arsararga, Mulamiera and Bhinnad, sya. The rule relating to the Bhaga and the Sesa varieties therein, (i.e., in miscellaneous problems on fractions). 4. In the operation relating to the Bhaga variety, the (required) result is obtained by dividing the given quantity by one as diminished by the (known) fractions. In the operation relating to the sesa variety, (the required result) is the given quantity divided by the product of (the quantities obtained respectively by) subtracting the (known) fractions from one. Examples in the Bhaga variety. 5. Of a pillar, t part was seen by me to be (buried) under the ground, } in water, in moss, and 7 hastas (thereof was free) in the air. What is the length of the pillar ? In the Bhagribhyasa or Bhagasanvarga variety, the numerical value is given of the portion remaining after removing from the whole the product or products of certain fractional parts of the whole taken two by two. The Avisavarga variety consists of problems wherein the numerical value is given of the remainder after removing from the whole the square of a fractional part thereof, this fractional part being at the same time increased or decreased by a given number. The Mulamiera variety consists of problems wherein is given the numerical value of the sum of the square root of the whole when added to the square root of the whole as increased or diminished by a given number of things. In the Bhinnadraya variety, a fractional part of the whole as multiplied by another fractional part thereof is removed from it, and the remaining portion is expressed as a fraction of the whole. Here it will be seen that unlike in the other varieties the numerical value of the last remaining portion is not actually given, but is expressed as a fraction of the whole. 4. Algebraically, the rule relating to the Bhaga variety is x=, where x is the unknown collective quantity to be found out, a is the drsya or agra, and b is the bhaga or the fractional part or the sum of the fractional parts given, It is obvious that this is derivable from the equation & -baera. The rule relating to the Besu variety, when algebraically expressed, comes to (1-6,) (1-6) (1-63) * &c. where bz, ba, ba, &c, are fractional parts of the " successive remainders. This formula also is derivable from the equation 2-61-b2 ( - 6zx) -63, -612-b, (x to. a. For Private and Personal Use Only

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72 GANITASARASANGRAHA. 6. Out of a collection of excellent bees, I took delight in patali trees, } in kadamba tree, 1 in mango trees, f in a campaka tree with blossoms fully opened ; ab in a collection of full-blown lotuses, opened by the rays of the sun; and (finally), a single intoxicated bee has been circling in the sky. What is the number (of bees) in that collection? 7. A certain sravaka, having gathered lotuses, and loudly uttering hundreds of prayers, offered those (lotuses) in worship, of those lotuses and }, and of this (%) respectively to four terthankaras commencing with the excellent Jina Vrsabha; then to Sumati ) as well as it of this (same of the lotuses); (thereafter) he offered in worship to the remaining (19) tirthankaras 2 lotuses each with a mind well-pleased. What is the numerical value of (all) those (lotuses)? 8 to 11. There was seen a collection of pious men, who had brought their senses under control, who had driven away the poison-like sin of karma, who were adorned with righteous conduct and virtuous qualities and whose bodies had been cmbraced by the Lady Mercy. Of that (collection), was made up of logicians; this (1) diminished by lof itself was made up of the teachers of the true religion; the difference between these two (, namely, ia and i1-} of 12) was made up of those that knew the Vedas; this (last proportional quantity) multiplied by 6 was made up of the preachers of the rules of conduct, and this very same (quantity) diminished by of itself was made up of astrologers; the difference between these two (last mentioned quantities) was made up of controversialists; this (quantity) multiplied by 6 was made up of penitent ascetics; and 9x 8 leading ascetics were (further) seen hy me near the top of a mountain with their shining bodies highly heated by the rays of the sun. Tell me quickly (the measure of this collection of prominent sages. 12 to 16. (A number of) parrots descended on a paddy-field beautiful with (the crops) bent down through the weight of the ripe corn. Beiny scared away by men, all of them suddenly flew up. One-half of them went to the east, and I went to the south-east; the difference between those that went For Private and Personal Use Only

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CHAPTER IV-MISCELLANEOUS PROBLEMS (ON PRACTIONS). 73 to the cast and those that went to the south-east, diminished by half of itself and (again) diminished by the half of this (resulting difference), went to the south; the difference between those that went to the south and those that went to the south-east, diminished by of itself, went to the south-west ; the difference between those that went to the south and those that went to the south-west, went to the west; the difference between those that went to the south-west and those that went to the west, together with of itself, went to the north-west ; the difference between those that went to the north-west and those that went to the west. together with of itself, went to the north ; the sum of those that went to the north-west and those that went to the north, diminished hy of itself, went to the north-east; and 280 parrots were found to remain in the sky (above). How many were the parrots (in all)? 17 to 22. One night, in a month of the spring season, a certain young lady ... was lovingly happy along with her husband on ... the floor of a big mansion, white like the moon, and situated in a pleasure-garden with trees bent down with the load of the bunches of flowers and fruits, and resonant with the sweet sounds of parrots, cuckoos and bees which were all intoxicated with the honcy obtained from the flowers therein. Then on a love-quarrel arising between the husband and the wife, that lady's necklace made up of pearls became sundered and fell on the floor. Oue-third of that necklace of pearls reached the maid-servant there ; & fell on the bed; then of what remained (and one-haif of what remained thereafter and again of what remained thereafter) and so on, counting six times (in all), fell all of them everywhere; and there were found to remain (unscattered) 1,161 pearls; and if you know how to work) miscellaneous problems (on fractions), give out the numerical) measure of the pearls (in that necklace). 23 to 27. A collection of bees characterized by the blue color of the shining indranila gem was seen in a flowering pleasure 17. Certain epithets here have not been considered fit for translation. 10 For Private and Personal Use Only

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Shri Mahavir Jain Aradhana Kendra 74 GANITASARASANGRAHA, garden. One-eighth of that (collection) became hidden in asoka trees, in kutaja trees. The difference between those that hid themselves in the kutaja trees and the asoka trees, respectively, multiplied by 6, became hidden in a crowd of big patali trees. The difference between those that hid themselves in the patali trees and the asoka trees, diminished by of itself became hidden in an extensive forest of sala trees. The same difference, together with of itself, became hidden in a forest of madhuka trees; of that whole collection of bees was seen hidden in the vakula trees with well-blossomed flower-buds; and that same part was found hidden in tilaka, kuravaka, sarata and mango trees, and on collections of lotuses, and at the base of the temples of forest elephants; and 33 (remaining) bees were seen in a crowd of lotuses, that were variegated in color on account of the large quantity of (their) filaments. Give out, O you arithmetician, the (numerical) measure of that collection of bees. 28. Of a herd of cattle, is on a mountain; of that is at the base of the mountain; and 6 more parts, each being in value half of what precedes it, are found together in an extensive forest, and there are (the remaining) 32 cows seen in the neighbourhood of a city. Tell me O you my friend, the (numerical) measure of that herd of cattle. Here end the examples in the Bhaga variety. Examples in the Sesa variety. 29-30. Of a collection of mango fruits, the king (took) }; the queen (took) of the remainder, and three chief princes took and (of that same remainder); and the youngest child took the remaining three mangoes. O you, who are clever in (working) miscellaneous problems on fractions, give out the measure of that (collection of mangoes). 31. One-seventh of (a herd of) elephants is moving on a mountain; portions of the herd, measuring from in order up to , in the end, of every successive remainder, wander about in a forest; and the remaining 6 (of them) are seen near a lake. How many are those elephants? For Private and Personal Use Only

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CHAPTER 1V-MISCELLANEOUS PROBLEMS (ON FRACTIONS). 75 32. Of (the contents of) a treasury, one man obtained part; others obtained from in order to , in the end, of the successive remainders; and (at last) 12 puranas were seen by me (to remain). What is the (numerical) measure of the puranas contained in the treasury) ? Here end examples in the Sesa variety. The rule relating to the Mula variety of miscellaneous problems on fractions) :-- 33. Half of the coefficient of the square root of the unknown quantity) and (then) the known remainder should be (each) divided by one as diminished by the fractional (coefficient of the unknown) quantity. The square root of the (sum of the) known remainder (80 treated), as combined with the square (of the coefficient of the square root (of the unknown quantity dealt with as above), and (then) associated with the similarly treated coefficient of) the square root (of the unknown quantity), and (thereafter) squared (as a whole), gives rise to the required unknown) quantity in this mula variety of miscellaneous problems on fractions). Exampies in illustration thereof. 34. One-fourth of a herd of camels was seen in the forest; twice the square root (of that herd) had gone on to mountainslopes; and 3 times 5 camels (were), however, (found) to remain on the bank of a river. What is the (numerical) measure of that herd of camels ? 35. After listening to the distinct sound caused by the drum made up of the series of clouds in the rainy season, and (of a collection) of peacocks, together with of the remainder and of the remainder (thereafter), gladdened with joy, kept on dancing on 33. Algebraically expressed, this rule comes to 2 }; this is easily obtained from tho 1l-- bu equation - (bw + CW +a)=0. This equation is the algebraical expression of problems of this variety. Here c stands for the coefficient of the square root of the unknown quantity to be found out. For Private and Personal Use Only

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Shri Mahavir Jain Aradhana Kendra 76 www.kobatirth.org GANITASARASANGRAHA. the big theatre of the mountain top; and 5 times the square root (of that collection) stayed in an excellent forest of vakula trees: and (the remaining) 25 were seen on a punnaga tree. O arithmetician, give ont after calculation (the numerical measure of) the collection of peacocks. 36. One-fourth (of an unknown number) of sarasa birds is moving in the midst of a cluster of lotuses; and parts. (thereof) as well as 7 times the square root (thereof) move on a mountain; (then) in the midst of (some) blossomed vakula trees (the remainder) is (found to be) 56 in number. O you clever friend, tell me exactly how many birds there aro altogether. 37. No fractional part of a collection of monkeys (is distributed anywhere); three times its square root are on a mountain; and 40 (remaining) monkeys are seen in a forest. What is the measure of that collection (of monkeys) ? 38. Half (the number) of cuckoos were found on the blossomed branch of a mango tree; and 18 (were found) on a tilaka trec. No (multiple of the) square root (of their number was to be found. anywhere). Give out (the numerical valuc of) the collection of cuckoos. 39. Half of a collection of swans was found in the midst of vakula trees; five times the square root (of that collection was found) on the top of tamala trees; and here nothing was scen (to remain thereafter). O friend, give out quickly the numerical measure of that (collection). Here ends the Mula variety (of miscellaneous problems on fractions). The rule relating to the Sesamula variety (of miscellaneous problems on fractions). 40. (Take) the square of half (the coefficient) of the square root (of the remaining part of the unknown collective quantity), and Acharya Shri Kailassagarsuri Gyanmandir 2 40. Algebraically, xbx (+() From this the value / of a is to be found out according to rule 4 given in this chapter. This value of - ba is obtained easily from the equation xbx+ (c x - bx + a) = 0. + a a} For Private and Personal Use Only 2

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CHAPTER IV-MISCELLANEOUS PROBLEMS (ON FRACTIONS). 77 combine it with the known number remaining, and (then extract) the square root (of this sum, and make that square root become) combined with half of the previously mentioned (coefficient of the) square root (of the remaining part of the unknown collective quantity). The square of this (last sum) will here be the required result, when the remaining part (of the unknown collective quantity) is taken as the original (collective quantity itself). But when that remaining part (of the unknown collective quantity) is treated merely as a part, the rule relating to the bhaga variety (of miscellaneous problems on fractions) is to be applied. Examples in illustration thereof. square 41. One-third of a herd of elephants and three times the root of the remaining part (of the herd) were seen on a mountainslope; and in a lake was seen a male elephant along with three female elephants (constituting the ultimate remainder). many were the clephants here? How 42 to 45. In a garden beautified by groves of various kinds of trecs, in a place free from all living animals, many ascetics were scated. Of them the number equivalent to the square root of the whole collection were practising yoga at the foot of the trees. One-tenth of the remainder, the square root (of the remainder after deducting this), (of the remainder after deducting this), then the square root (of the remainder after deducting this), (of the remainder after deducting this), the square root (of the remainder after deducting this), (of the remainder after deducting this), the square root (of the remainder after deducting this), (of the remainder after deducting this), the square root (of the remainder after deducting this), (of the remainder after deducting this), the square root (of the remainder after deducting this)-these parts consisted of those who were learned in the teaching of literature, in religious law, in logic, and in politics, as also of those who were versed in controversy, prosody, astronomy, magic, rhetoric and grammar and of those who possessed the power derived from the 12 kinds of austerities, as well as of those who possessed an intelligent knowledge of the twelve varieties of the anga-sastra; and For Private and Personal Use Only

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78 GANITASARASANG RAHA. at last 12 ascetics were seen (to remain without being included among those mentioned before). O (you) excellent ascetic, of what numerical value was (this) collection of ascetics? 46. Five and one-fourth times the square root (of a herd) of elephants are sporting on a mountain slope ; of the remainder sport on the top of the mountain ; five times the square root of the remainder (after deducting this) sport in a forest of lotuses; and there are 6 elephants then (left) on the bank of a river. How many are (all) the elephants here? Here ends the Sesamula variety (of miscellaneous problems on fractions). The rule relating to the Sesamula variety involving two known (quantities constituting the remainders : 47. The (coefficient of the) square root (of the unknown collective quantity), and the (final) quantity known (to remain), should (both) be divided by the product of the fractional (proportional) quantities, as subtracted from one (in each case) ; then the first known quantity should be added to the other) known quantity (treated as above). Thereafter the operation relating to the Sesamula variety (of miscellaneous problems on fractions is to be adopted). 47. Algebraically, this rule enables us to arrive at the expressions ... + an, which are required to (1-6) (1-0) x &c. (1---) (1.-) * &o. be substitnted for c and a respectively in the formola for sesamula, which is 2 + + a . In applying this formula tho valac of becomes zero, as the mula or square root involved in the dvirajra-mumula is that of the total collective quantity and not of a fractional part of that quantity. Substitating as desired, we get *=12 (1--10) xde + s 2 . W (21 - 6,) (1 - ba) * &c.) + (1 - 63) (1-bg) * &c. + uns. may easily be obtained from the equation x -- a, -- bz (-)-- bas --- -- -.... -CN- ag=0, where bi, b, &c., are, the various fractional parts of the successive remainders; and a, and an are the first known quantity and the final known quantity respectively. For Private and Personal Use Only

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CHAPTER IV-MISCELLANEOUS PROBLEMS (ON FRACTIONS). 79 Examples in illustration thereof. 48. A single bee (out of a swarm of bees) was seen in the sky; of the remainder (of the swarm), and of the remainder (left thereafter), and (again) of the remainder (left thereafter), and (a number of becs equal to) the square root (of the numercial value of the swarm, were seen) in lotuses; and two (bees remaining at last were seen) on a mango tree. How many are those (bees in the swarm)? 49. Four (out of a collection of) lions were seen on a mountain; and fractional parts commencing with and ending with of the successive remainders (of the collection), and (lions equivalent in number to) twice the square root (of the numerical value of the collection), as also (the finally remaining) four (lions), were seen in a forest. How many are those (lions in the collection)? 50. (Out of a herd of deer) two pairs of young female deer were seen in a forest; fractional parts commencing with and ending with of the (successive) remainders (of the herd were seen) near a mountain; (a number) of them (equivalent to) 3 times the square root (of the numerical value of the herd) were seen in an extensive paddy field; and (ultimately) only ten remained on the bank of a lotus-lake. What is the (numerical) measure of the herd? Thus ends the Sesamula variety quantities. Acharya Shri Kailassagarsuri Gyanmandir 2 The rule relating to the Amsamula variety (of miscellaneous problems on fractions). 51. Write down (the coefficient of) the square root (of the given fraction of the unknown collective quantity) and the known quantity (ultimately remaining, both of these) having been + a involving two known 50. The word harini occurring in this stanza not only means a female deer ' but is also the name of the metre in which the stanza is composed. 51. Algebraically stated, this rule helps us to arrive at cb and ab, which are required to be substituted for c and b respectively in the formula - b { + (2 2 , as in the & samalla variety. As pointed out in the note For Private and Personal Use Only

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Shri Mahavir Jain Aradhana Kendra 80 www.kobatirth.org GANITASARASANGRAHA. multiplied by the (given proportional) fraction; then that result which is arrived at by means of the operation of finding out (the unknown quantity) in the Sesamula variety (of miscellaneous problems on fractions), when divided by the (given proportional) fraction, becomes the required quantity in the Amsamula variety (of miscellaneous problems on fractions). Another rule relating to the Amsamula variety. 52. The known quantity given as the (ultimate) remainder is divided by the (given proportional) fraction and multiplied by four; to this the square (of the coefficient) of the square root (of the given fraction of the unknown collective quantity) is added ; then the square root (of this sum), combined with (the above mentioned coefficient of) the square root (of the fractional unknown quantity), and (then) halved, and (then) squared, and (then) multiplied by the (given proportional) fraction, becomes the required result. Examples in illustration thereof. 53. Eight times the square root of part of the stalk of a lotus is within water, and 16 angulas (thereof are) in the air (above water); give out the height of the water (above the bed) as well as of the stalk (of the lotus). 54-55. (Out of a herd of elephants), nine times the square root of part of their number, and six times the square root of of the remainder (left thereafter), and (finally) 24 (remaining) elephants with their broad temples wetted with the stream of the exuding ichor, were seen by me in a forest. How many are (all) the elephants? c+ under stanza 47, xbx becomes x here also. After substituting as desired, and dividing the result by b, we get x= + cl (ch)2+ab}/b. This value of 2 a may be easily arrived at from the equation x-cbx-a= 0. 52. Algebraically stated, a 2 the equation given in the note under the previous stanza. Acharya Shri Kailassagarsuri Gyanmandir c2 + 4a 2 b r For Private and Personal Use Only x b. This is obvions from

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CHAPTER IV-MISCELLANEOUS PROBLEMS (ON FRACTIONS). 81 56. Four times the square root of the number of a collection of boars went to a forest wherein tigers were at play; 8 times the square root of its of the remainder (of the collection) went to a mountain; and 9 times the square root of i of the (further) remainder (left thereafter) went to the bank of a river; and boars equivalent in (numerical) measure to 56 were seen (ultimately) to remain (where they were) in the forest. Give out the (numerical) measure of (all) those (boars). Thus ends the Amsamula variety. The rule relating to the Bhagasamvarga variety (of miscellaneous problems on fractions) : 57. From the simplified) denominator of the specified compound fractional part of the unknown collective quantity), divided by its own (related) numerator (also simplified), subtract four times the given known part (of the quantity), then multiply this (resulting difference) by that same (simplified) denominator (dealt with as above). The square root of this product is to be added to as well as subtracted from that (same) denominator (so dealt with); (then) the half (of either) of these two quantities resulting as sum or difference is the unknown) collective quantity (required to be found out). E.camples in illustration thereof. 58. A cultivator obtained (first) of a heap of paddy as multiplied by to (of that same heap); and (then) he had 24 vahas (left in addition). Give out what the measure of the heap is. 59. One-sixteenth part of a collection of peacocks as multiplied by itself, (i.e., by the same to part of the collection), was found 56. The word sardularikrilita in this stanza means tigers at play,' and at the same time happens to be the name of the metre in which the stanza is coin posed. 57. Algebraically stated <== and this value of x may m -u=0, where and are easily be obtained from the equation <--xxx the fractions contemplates in the rule. 11 For Private and Personal Use Only

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82 GANITASARASANGRAHA. ou a mango tree; } of the remainder as multiplied by that same (t part of that same remainder), as also (the remaining) fourteen (peacocks were found in a grove of tamala trees. How many are they (in all)? 60. One-twelfth part of a pillar, as multiplied by a part thereof, was to be found under water; o of the remainder, as multiplied by thereof, was found (buried) in the mire (below); and 20 hastas of the pillar were found in the air (above the water). O friend, you give out the measure of the length of the pillar. Here ends the Bhagasarivarga variety. The rule relating to the Avrsararga variety (of miscellaneous problems on fractions), characterised by the subtraction or addition (of known quantities) :-- 61. (Take) the half of the denominator (of the specified fractional part of the unknown collective quantity), as divided by its own (related) numerator, and as increased or diminished by the (given) known quantity which is subtracted from or added to (the specified fractional part of the unknown collective quantity). The square root of the square of this (resulting quantity), as diminished by the square of the above known) quantity to be subtracted or to be added and also by the known romainder (of the collective quantity), when added to or subtracted from the square root (of the square quantity mentioned above) and then divided by the (specified) fractional part (of the unknown collective quantity), gives the (required) value (of the unknown collective quantity). Examples of the minus rariety. 62. (A number) of buffaloes (equivalent to) the square of (of the whole herd) minus 1 is sporting in the forest. The 61. Algebraically, *={EN ( + a) --a + ( a + ) } =*. Thin vulno 1 m is obtained from the equation - = 0, where d is the given known quantity, For Private and Personal Use Only

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CHAPTER IV-MISCELLANEOUS PROBLEMS (ON FRACTIONS). 83 (remaining) 15 (of them) are seen grazing grass on a mountain. How many are they (in all)? 63. (A number) of elephants (equivalent to) of the herd minus 2, as multiplied by that same (% of the herd minus 2), is found playing in a forest of sallaki trees. The (remaining) elephants of the herd measurable in number by the square of 6 are moving on a mountain. How many (together) are (all) these elephants here? Acharya Shri Kailassagarsuri Gyanmandir An example of the plus variety. 64. (A number of peacocks equivalent to) 15 of their whole collection plus 2, multiplied by that same (s of the collection plus 2), are playing on a jambu tree. The other (remaining) proud peacocks (of the collection), numbering 22 x 5, are playing on a mango tree. O friend, give out the numerical measure of (all) these (peacocks in the collection). Here ends the Amsavarga variety characterised by plus or minus quantities. The rule relating to the Mulamisra variety (of miscellaneous problems on fractions). 65. To the square of the (known) combined sum (of the square roots of the specified unknown quantities), the (given) minus quantity is added, or the (given) plus quantity is subtracted (therefrom); (then) the quantity (thus resulting) is divided by twice the combined sum (referred to above); (this) when squared gives rise to the required value (of the unknown collection). In relation to the working out of the Mulamisra variety of problems, this is the rule of operation. 64. The word mattamayura occurring in the stanza means a proud peacock' and is also the name of the metre in which the stanza is composed. { m2 + d 2m This is easily derived from the equa. 65. Algebraically a tion ++- d = m. mentioned in the rule. The quantity m is here the known combined sum [d] deg For Private and Personal Use Only

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GANITASARASANGRAHA. Examples of the minus variety. 66. On adding together (a number of pigeons equivalent to) the square root of the (whole) collection of pigeons and another number equivalent to) the square root of the (whole) collection as diminished by 12, (exactly) 6 pigeons are seen (to be the result). What is the numerical value of) that collection (of pigeons)? 67. The sum of two (quantities, which are respectively equivalent to the square roots of the (whole) collection of pigeons and of that same) collection as diminished by the cube of 4, amounts to 16. How many are the birds in that collection? An example of the plus variety. 68. The sum of the two (quantities, which are respectively equivalent to the square root (of the numerical value) of a collection of superior swans and (the square root of that same collection) as combined with 68, amounts to 1" - 2. Give out how many swans there are in that collection, Here ends the Mulamiera variety. The rule relating to the Bhinnadraya variety of miscellaneous problems on fractions) : 69. When one, diminished by the (given) fractional remainder (related to the anknown quantity), is divided by the product of the specified) fractional parts (related thereto), the result which is (thus) arrived at becomes the required) answer in working out the Bhinnalisya variety of problems on fractions). Examples in illustration thereof. 70. One-eighth part of a pillar, as multiplied by the part (of that same pillar), was found to be buried) in the sands; of the pillar was visible (above). Say how much the (vertically measured) length of the pillar is. 69. Algebraically stated, 2 = mp. This is obvious from the ng equation << - .* --*=0. For Private and Personal Use Only

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CHAPTER IV-MISCELLANEOUS PROBLEMS (ON FRACTIONS). 85 71. (Elephants equivalent in number to) part of the whole herd of elephants, as multiplied by (of that same herd) as divided by 2, are in a happy condition on a plain. The remaining (ones forming) } (of the herd), resembling exceedingly dark masses of clouds in form, are playing on a mountain. O friend, you tell me now the numerical measure of the herd of elephants. 72. (Ascetics equivalent in number to) e part of a collection of ascetics, as divided by 3 and as multiplied by that same Ge part divided by 3), are living in the interior of a forest; (the remaining ones forming) 1 part of that collection) are living on a mountain. O you, who have crossed over to the other shore of the ocean-like miscellaneous problems on fractions, tell mne quickly the (numerical) value of that (collection of ascetics). Here ends the Bhinnadr-ya variety. Thus ends the third subject of treatment known as Prakarnaku in Sarasangraha which is a work on arithmetic by Mahaviracarya. 71. The word prthvi occurring in this stanza means the earth', and is also the name of the metre in which the stanza is composed. For Private and Personal Use Only

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86 GANITASARASANG RATA. CHAPTER V. RULE-OF-THREE. The fourth subject of treatment. 1. Salutation to that blessed Vardhamana, who is like a (helpful) relation to (all) the three worlds, and is (resplendent) like the sun in the matter of absolute knowledge, and has cast off (the taint of) all the karmas. Next we shall expound the fourth subject of treatment, viz., rule-of-three. The rule of operation in respect thereof is as follows: 2. Here, in the rule-of-three, Phala multiplied by Iccha and divided by Pramana, becomes the (required) answer, when the Iccha and the Pramana are similar, i.e., in direct proportion); and in the case of this (proportion) being inverse, this operation (in volving multiplication and division) is reversed, (so as to have division in the place of multiplication and multiplication in the place of division Examples relating to the former half of the above rule, i.e., on the direct rule-of-three. 3. The man who in 3 days goes over 53 yojanas--give out what (distance) he (goes over) in a year and a day. 4. A lame man walks over of a krosa together with (thereof) in 7 days. Say what (distance) ho (goes over) in 3 years (at this rate). 5. A worm goes in of a day over of an angulu. In how many days will it reach the top of the Meru mountain from its bottom ? 6. The man who in 3 days uses up 11 karsapanas- in what time (will) he (use up) 100 puranas along with 1 pana ? 2. Pramana and Phala together give the rate, in which Phala is a quantity of the same kind as the required answer and Promana is of the same kind as Iccha. This Iccha is the quantity about which something is required to be found out at the given rate. For instance in the problem in stanza 3 here, 3 days is the Pramana, 57 yojanas is the Phala, and 1 year and I duy is the Iccha. 5. The height of the Meru mountain is supposed to be 99,000 yojanus or 76,032,000,000 angulas, For Private and Personal Use Only

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CHAPTER V--RULE-OF-THREE. 87 7. A good piece of krsna garu, 12 hastas in length and 3 hastas in. diameter, is consumed (at the rate of) 1 cubic angula a day. What is the time required for the (complete) consumption of this cylinder ? 3. (If) a vaha of very superior black gram, along with 1 droao, Iadhaka and l kudava (thereof), has been purchased by means of 10.4 svarnas, what measure (may we purchase of it) by means of 1001 svarnas? 9. Where 1} pala of kunkuma is obtainable by means of 3.1 puranas, what measure (of it) may (we obtuin) tliere by means of 100 puranas? 10. By means of 7 palas of ginger, 13. panas were obtained ; say, O friend, what (may be obtained) in return for 321 palas of ginger? 11. By means of 4 karsapanas, a man obtains 163 palas of silver; what (weight does he obtain thereof) by means of 10,000 karsas? 12. By means of 7% palas of camphor, a man obtains 5 dinaras along with 1 bhaga, 1 arsa and 1 kala. What does he obtain) here by means of 1,000 palas (thereof) ? 13. The man who purchases here 5} palas of ghee by means of 3! panas--what (measure of it does he purchase) by means of 100% karsas? 14. By means of 5puranas, 16 pairs of cloths were obtained. O friend, say what number of them may be obtained) by means of 61 karsas ? 15-16. There is a square well without water, (cubically) measuring 512 hastas. A hill rises on its bank; from the top 7. Here the process of finding out, from the given diameter, the area of the cross-section of a cylinder is supposed to be known. This is given in the sixth Vyavahara, in the 19th stanza, where the area of a circle is said to be approximately equal to the diameter squared and then divided by 4 and multiplied by 3. Krsnagaru is a kind of fragrant wood burnt in fire as incense. 15-16. In this problem, the stream of water is as long as the mountain is high, so that as soon as it reaches the bottom of the mountain, it is supposed to cease to flow at the summit. For finding out the quantity of water in Vahas, etc., the relation between cubical measure and liquid measure should have been given. The Sanskrit commentary in P and the Kanarese tika in B state that 1 cabic angula of water is equal to 1 karsa thereof in liquid measure. For Private and Personal Use Only

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88 GANITASARASANGRAHA. thereof flows down, (to the bottom) a crystal-clear stream of water having 1 aigula for the diameter of its circular section, and the well becomes quite filled with water within. What is the height of the bill, and (what) the numerical value of the liquid-measure) of water? 17. A king gave, on (the occasion of) the sankranti, to 6 Brahmins, 2 dronas of kidney-bean, 9 kudabas of ghee, 6 dronus of rice, 8 pairs of cloths, 6 cows with calves and 3 svarnas. Give out quickly, O friend, what (the measure) is (of) the kidney-bean and the other things given by him (at that rate) to 336 Brahmins. Here ends the (direct) rule-of-three. Examples on inverse rule-of-three as explained in the fourth poda * of the rule given above). 18. How much is the gold of 9 varnas for 90 of pure gold, as also for 100 gold (Dharanas) along with a gunija thereof made up of gold of 10 varnas ? 19. There are 300 pieces of China silk of 6 hastas in breadth as well as in length; give out, () yon who know the method of inverse proportion, how many pieces of that same silk) there are (in them, each) measuring 5 by 3 hastas. Here ends the inverse rule-of-three. An example on inverse double rule-of-three. 20. Say how many pieces of that famous clothing, cach measuring 2 hastas in breadth and 3 hastas in length, are to be found in 70 (pieces) of China silk, (each) measuring 5 hastas in breadth and 9 hastas in length. An example on inverse treble rule-of-three. 21. Say how many images of Tirthaikaras, (each) measuring 2 by 6 by 1 hastas, there may be in a big gem, which is 4 hastas in breadth, 9 hastas in length and 8 hastas in height. 17. Sankranti is the passage of the sun from one zodiacal sign to another. 18. Pure gold is here taken to be of 16 varnas. * The reference here is to the fourth quarter of the second stanza in this chapter. For Private and Personal Use Only

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CHAPTER V-RULE-OF-THREE. An example on inverse quadruple rule-of-three. 22. There is a block of stone (suited for building purposes), which measures 6 hastas in breadth, 30 hastas in length and 8 hastas in height, and (it is) 9 in worth. By means of this (given in exchange), how many (blocks) of such stone, fit to be used in building a Jina temple, (may be obtained, each) measuring 2 by 6 by 1 (hastas), and being 5 in worth ? Thus ends the inverse double, treble and quadruple rule-ofthree. The rule in regard to (problems bearing on associated) forward and backward movement. 23. Write down the net daily movement, as derived from the difference of the given rates of) forward and backward movements, each of these rates) being (first) divided by its own (specified) time; and then in relation to this (net daily movement), carry out the operation of the rule-of-three. Examples in illustration thereof. 24-25. In the course of of a day, a ship goes over } of a krasa in the ocean ; being opposed by the wind she goes back (during the same time) 4 of a krasa. Give out, O you who have powerful arms in crossing over the ocean of numbers well, in what time that (ship) will have gone over 99% yojanas. 26. A man earning (at the rate of) 11 of a gold coin in 3 days, spends in 41 days of the gold coin as also } of that (1) itself; by what time will he own 70 (of those gold coins as his net earnings)? : 27. That excellent elephant, which, with temples that are Attacked by the feet of bees greedy of the (flowing) ichor, goes over { as well as of a yojana in 54 days, and moves back in 31 days over of a krosa : say in what time he will have gone over (a net distance of) 100 yojanas legs by krosa. 28-30. A well completely filled with water is 10 dandas in depth; a lotus sprouting up therein grows from the bottom 28-30. The depth' of the well is mentioned in the original as 'height' measured from the bottom of it. 12 For Private and Personal Use Only

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90 GANITASARASANGRAHA. (at the rate of) 24 angulas in a day and half; the water (thereof) flows out through a pump at the rate of) 24 angulas (of the well in depth) in 14 days; 1angulas of water in depth) are lost in a day by evaporation owing to the (heating) rays of the sun; a tortoise below pulls down 51 angulas of the stalk of the lotus plant in 3 days. By what time will the lotus be on the same level with the water (in the well) ? 31. A powerful unvanquished excellent black snake, which is 32 hastas in length, enters into a hole (at the rate of) 7] angulas in of a day; and in the course of 1 of a day its tail grows by 2 of an angula. O ornament of arithmeticians, tell me by what time this same (serpent) enters fully into the hole. Thus end the problems bearing on associated) forward and baokward movements. The rule of operation relating to double, treble and quadruple rule-of-three. 32. Transpose the Phala from its own place to the other place (wherein a similar concrete quantity would occur); (then, for the purpose of arriving at the required result), the row consisting of the larger number (of different quantities) should be, (after they are all multiplied together), divided by the row consisting of the 32. The transference of the Phala and the other operations herein mentioned will be clear from the following worked out example. The data in the problem in stanza No. 36 are to be first represented thus :9 Manis. 1 Vaha + 1 Kumbha. 3 Yojanas. 10 Yajanas. 60 Panas. When the Phala here, viz., 69 panas, is transferred to the other row we have 9 Manis. 1 Vaha + 1 Kumbha=1} Vaha. 3 Yojanas. 10 Yojanas. 60 Panas. Now the right hand row, consisting of a larger number of different quantities, should be, after they are all multiplied together, divided by the smaller left hand row similarly dealt with. Then we Lave 17 x 10 x 60 9 x 3 The result here gives the number of panas to be found out. For Private and Personal Use Only

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CHAPTER V-RULE-OF-THREE. 91 smaller number (of different quantities, after these are also similarly thrown together and multiplied); but in the matter of the buying and selling of living animals (the transposition is to take place) only (in relation to the numbers representing) them. Examples in illustration thereof. 33. At the rate of 2, 3 and 4 per cent. (per month), 50, 60 and 70 Puranas were (respectively) put to interest by a person desiring profit. How much interest does he obtain in ten months ? 34. The interest on 801 gold coins for l of a month is 14. How much (will it be) on 90 gold coins for 57 months ? 35. He who obtains 20 gems in return for 100 gold pieces of 16 varnas--what (will he obtain) in return for 288 gold pieces of 10 varnas ? 36. A man, by carrying 9 manis of wheat over 3 yojanas, obtained 60 panas. How much (would he obtain) by carrying one kumbha along with one vaha (thereof) over 10 yojanas ? Examples on barter, 37. A man obtains 3 karsas of musk for i0 gold coins and 2 karsas of camphor for 8 gold coins. How many (karsas of camphor does he obtain) in return for 300 karsas of musk? 38. In return for 8 (masas in weight of silver), a man obtains 60 jack fruits; and in return for 10 masas (in weight of silver he obtains) 80 pomegranates. How many pomegranates (does he obtain) in return for 900 jack fruits ? Examples of problems bearing on) the buying and selling of animals. 39. Twenty horses, (each) of 16 years (of age), are worth 100,000 gold coins. O leading arithmetician, say how much 70 horses, (each) of 10 years (of age), will be (wortb) at this (rate). 40. Three hundred gold coins form the price of 9 damsels, (each) of 10 years (of age). What is the price of 36 damsels, (each) of 16 years (of age) ? For Private and Personal Use Only

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92 GANITASA RASANGRABA. 41. What is the interest for 10 months on 90, invested at the rate of 6 per 100 (per month)? O you, who are a mirror to the face of arithmeticians, say, with the aid of the two other requisite) known quantities, what the time in relation to that (interest) is, and what the capital is (in relation to that interest and time). An example on treble rule-nt-three. 42. Two pieces of sandal-wood, measuring 3 and 4 hastas in diameter and length respectively, are worth 8 gold coins. At this (rate) how much will be the worth of 14 (pieces of sandal-wood, each) measuring 6 and 9 hastas in diameter and length (respectively)? Thus ends treble rule-of-three. An example on quadruple rule-of-three. 43. A household well, measuring 5, 8 and 3 hastas in breadth, length and height (from the bottom, respectively), contain 6 vahas of water; O you, who are learned, give out how much (water) 9 wells, (each being) 7 hastas in breadth, 60 in length and 5 in height (from the bottom, will contain). Thus ends the fourth subject of treatment known as Rule-of-three in Sarasangraha which is a work on arithmetic by Mahaviracarya. 43. The word salini occurring in this stanza indicates the name of the metre in which the stanza is composed, at the same time that it means 'belonging to a house.' For Private and Personal Use Only

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CHAPTER VI--MIXED PROBLEMS. 93 CHAPTER VI. MIXED PROBLEMS. The Fifth Subject of Treatment. 1. For attaining the supreme good, we worshipfully salute the holy Jinas, who are in possession of the fourfold infinite attributes, who are the makers of tirthas, who have attained self-conquest, are pure, are honoured in all the three worlds and are also excellent preceptors--the Jinas who have gone over to the (other) shore of the ocean of the Jaina doctrines, and are the guides and teachers of (all) born beings, and who, being the abode of all good qualities, are good in themselves and do good to others. Hereafter we shall expound the fifth subjeot of treatment known as mixed problems. It is as follows: Statement of the meaning of the technical terms sankramana and visama-sankramana : 2. Those who have gone to the end of the ocean of calculation say that the halving of the sum and of the difference of any two quantities) is (known as) sankramana, and that the sankramona of two quantities which are (respectively) the divisor and the quoties is that which is visama (i.e., visama-sankramana). Examples in illustration thereof. 3. What is the sankramana where the number 12 (is associ: with 2; and what is the divisional visama-sankramana of (same) number (12 in relation to 2)? . 1 Tirtha is interpreted to mean a ford intended to cross the river oi existence which is subject to karma and reincarnation. The Jinas are c to be capable of enabling the souls of men to get out of the stream of 8 or the recurring cycle of embodied existence. The Jinas are therefore tarthankaras. 2. Algebraically the sankramana of any two quantities a und b is fii out a 10 and abo; their visama-sankramana is arriving at * 6+ and For Private and Personal Use Only

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Shri Mahavir Jain Aradhana Kendra 94 www.kobatirth.org GANITASARASANGRAHA. Double Rule-of-three. The rule for arriving at (the value of) the interest which (operation) is of the nature of double rule-of-three :-- 4. The number representing the Iccha, i.e., the amount the interest whereon is desired to be found out, is multiplied by the time connected with itself and is then multiplied by (the number representing) the (given) rate of interest for the given capital; (then the resulting product) is divided by the time and the capital quantity (connected with the rate of interest); this (quotient) is, in arithmetic, the interest of the desired amount. Examples in illustration thereof. 5. Puranas, 50, 60, and 70 (in amount) were lent out on interest at the rate of 3, 5 and 6 per cent (per mensem respectively); what is the interest for 6 months? 6. (A sum of) 30 karsapanas and 8 panas were lent out on interest at the rate of 7 per cent (per month); what is the interest produced in exactly 7 months? 7. The interest on 60 for 2 months is seen to be 5 puranas with 3 panas; what would be the interest on 100 for 1 year? 8. The interest for 1 month and a half on lending out 150 15. What would be the interest obtained at this rate on 300 10 months? . A merchant lent out 63 karsapanas at the rate of 8 for 108 month). What (is the interest) for 7 months ? cxtx I Tx0 Acharya Shri Kailassagarsuri Gyanmandir The rule for finding out the capital lent out: ). The capital quantity (involved in the rate of interest) is lied by the time connected with itself and is then divided mbolically i= where T, C and I are respectively the time, and interest of the pramana or the rate, and t, c and i are respectively time, capital and interest of the iccha. For an explanation of pramina, , &c., see note under Ch. V. 2. Unless otherwise mentioned, the rate of interest is for 1 month. V. Symbolically CxTxi Ixt = c. For Private and Personal Use Only

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CHAPTER VI-MIXED PROBLEMS. by the interest connected with itself. (Then) this (quotient) has to be divided by the time connected with the capital lent out; (this last) quotient when multiplied by the interest (that has accrued) becomes the capital giving rise to that (interest). Examples in illustration thereof. 11. In lending out at the rate of 21 per cent (per mensem), a month and a half (is the time for which interest has accrued), and a certain person thus obtains 5 puranas as the interest. Tell me what the capital is in relation to that (interest). 12. The interest on 70 for 1 months is exactly 2. When the interest is 2 for 7 months what is the capital lent out? 13. In lending out at the rates of 3, 5 and 6 per cent (per mensem), the interest has so accrued in 6 months as to be 9, 18 and 25 (respectively); what are the capital amounts lent out? Acharya Shri Kailassagarsuri Gyanmandir The rule for finding out the time (during which interest has accrued): 14. Take the capital amount (involved in the given rate of interest) as multiplied by the time (connected therewith); then cause this to be divided by its own (connected) rate-interest and by the capital lent out; then this (quotient) here is multiplied by the interest that has accrued on the capital lent out. Wise men say that the resulting (product) is the time (for which the interest has accrued). 14. Symbolically, 95 Examples in illustration thereof. 15. O friend, mention, after calculating the time, by what time 28 will be obtained as interest on 80, lent out at the rate of 3 per cent (per mensem). 16. The capital amount lent out at the rate of 20 per 600 (per mensem) is 420. The interest also is 84. O friend, you tell me quickly the time (for which the interest has accrued). Ox Txi Ix c =t. For Private and Personal Use Only

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Shri Mahavir Jain Aradhana Kendra 96 www.kobatirth.org GANITASARASANGRAHA. 17. It is 96 that is lent out at the rate of 6 per cent (pe mensem); the interest thereon is seen to be 573. What is the time (for which interest has accrued)? The rule regarding barter or exchange of commodities:-- 18. The quantity of the commodity taken in exchange is divided by its own price as well as by the quantity of the commodity given in exchange. (It is then) multiplied by the price of the commodity given in exchange, and thereafter multiplied by the quantity of the commodity intended to be exchanged. This (resulting) product is the required quantity corresponding to the prices of the commodity given in exchange as well as of the commodity taken in exchange. An example in illustration thereof. 19 and 20. Palas 8 of dried ginger were purchased for fil panas and palas 5 of long pepper for 8 panas. Think out and tell me quickly, O you who know arithmetic, how many palas of long pepper have been purchased by oue (at the above rate) by means of 80 palas of dried ginger. Acharya Shri Kailassagarsuri Gyanmandir Thus end the problems on double rule-of-three in this chapter on mixed problems. Problems bearing on interest. Next, in the chapter on mixed problems, we shall expound problems bearing on interest. The rule for the separation of the capital and interest from their mixed sum: 21. Symbolically, c=. 21. The result arrived at by carrying out the operation of division in relation to the given mixed sum of capital and interest 1+ m 1x tx I' TxC where m ci; hence im c. For Private and Personal Use Only

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CHAPTER VI-MIXED PROBLEMS. 97 by means of one, to which the interest thereon for the (given) time is added, (happens to be the required) capital; and the interest required is the combined sum minus this capital. An example in illustration thereof. 22. If one lends out money at the rate of 5 per cent (per month), the combined sum of interest and capital becomes 48 in 12 months. What are the capital and the interest therein ? Again another rule for the separation of the capital and the interest from their combined sum : 23. The product of the given time and the rate-interest, divided by the rate-time and the rate-capital and then combined with one, is the divisor of the combined sum of the capital and interest; the resulting quotient has to be understood as the (required) capital. An example in illustration thereof. 24. Having given out on interest some money at the rate of 2 per cent (per mensem), one obtains 33 in 4 months as the combined sum of the capital and the interest). What may be the capital (therein) ? The rule for the separation of the time and the interest from their combined sum : 25. Take the rate-capital multiplied by the rate-time and divided by the rate-interest and by the given capital, and then combine this (resulting quantity) with one ; then the quotient obtained by dividing the combined sum (of the time and interest) hy this (resulting sum) indeed becomes the (required) interest. Examples in illustration thereof. 26. Money amounting to 60 exactly was lent out at the rate of 5 per cent (per month) by one desirous of obtaining interest. 23. Symbolically c=m= 7*6+1}. It is evident that this is very much the same as the formula given under 21. CxT 25. Symbolically i=m +1 =i, where m=i + t. 13 For Private and Personal Use Only

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98 GANITASARASANGRAHA. The time (for which the interest has accrued) combined with the interest therefor is 20. What is the time here? 27. The capital put to interest at the rate of 1} per 703 (per mensem) is 705. The mixed sum of its time and interest is 80. (What is the value of the time and of the interest ?) 28. The capital put to interest at the rate of 31 per 80 for 27 months is 400, and the mixed sum of time and interest is 36. (What is the time and what the interest ?) The rule for arriving at the separation of the capital and the time of interest from their mixed sum :-. 29. From the square of the given mixed gum (of the capital and the time), the rate-capital divided by its rate-interest and multiplied by the rate-time and by four times the given interest is to be subtracted. The square root of this (resulting remainder) is then used in relation to the given mixed sum so as to carry out the process of sankramana. Examples in illustration thereof. 30. This, viz., 4 Puranas is the interest on 70 (per month). The interest (obtained on the whole) is 25. The mixed sum of the capital used and the time of interest) is 454. What is the capital lent out? 31. By lending out what capital for what time at the rate of 3 per 60 (per mensem) would a man obtain 18 as interest, 66 being the mixed sum of that time and that capital ? 32. It has been ascertained that the interest for 11 months on 60 is only 27. The interest here (in the given instance) is 24, and 29. Symbolically, - x 4i Im - = c or t as the case may be, where m= C + t. The value of the quantity under the root, as given in the rule, is (c ); and the square root of this and the misra have the operation of sankramana performed in relation to them. For the explanation of sankramana see Ch. VI. 2. For Private and Personal Use Only

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CHAPTER VI-MIXED PROBLEMS. 99 60 is (the value of) the time combined with the capital lent out. (What is the time and what the capital ?) The role for arriving at the separation of the rate-interest and the required time from their sum : 33. The rate-capital is multiplied by its own rate-time, by the given interest and by four, and is then divided by the other (that is, the given) capital. The square root of the remainder (obtained by subtracting this resulting quotient) from the square of the given mixed sum is then used in relation to the mixed sum so as to carry out the process of sankramana. An example in illustration thereof. 34. The mixed sum of the rate-interest and of the time (for which interest has accrued) at the rate of the quantity to be found out per 100 per month and a half is 123, the capital lent out being 30 and the interest accruing thereon being 5. (What is the rate of interest and what the time for which it has accrued ?) The rule for arriving separately at the capital, time, and the interest from their mixed sum : 35. Any (optionally chosen) quantity subtracted from the given mixed sum may happen to be the time required. By means of the interest on one for that same time, to which interest one is added, (the quantity remaining after the optionally chosen time is subtracted from the given mixed sum) is to be divided. (The resulting quotient) is the required capital. The mixed sum diminished by its own corresponding time and capital becomes the (required) interest. An example in illustration thereof. 36. In a loan transaction at the rate of 5 per cent (per mensem), the quantities representing the time, the capital and the interest 33. Symbolically, Nm _ *T***4 is used with m in carrying out the required sankramana, m being equal to I+t. 35. Bere, of the three unknown quantities, the value of the time is to be optionally chogen, and the other two quantities are arrived at in accordance with rule in Ch. VI. 21. For Private and Personal Use Only

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100 SANITASARABAANG RATA. (connected with the loan) are not known. Their sum however is 82. What is the capital, what is the time, and what the interest ? The rule for arriving separately at the various amounts of interest accruing on various capitals for various periods of time from the mixed sum of (those) amounts of interest : 37. Let each capital amount, multiplied by the corresponding) time and multiplied (also) by the (given) total (of the various amounts) of interest, be separately divided by the sum of the products obtained by multiplying each of the capital amounts by its corresponding time, and let the interest (of the capital so dealt with) be (thus) declared. An example in illustration thereof. 38. In this problem), the given) capitals are 40, 30, 20 and 50; and the months are 5, 4, 3 and 6 (respectively). The sum of the amounts of interest is 34. (Find out each of these amounts.) The rule for separating the various capital amounts from their mixed sum : 39. Let the quantity representing the mixed sum of the various capitals lent out be divided by the sum of those (quotients) which are obtained by dividing the various amounts of interest by their corresponding periods of time, and let the resulting) quotient be multiplied (respectively) by (the various) quotients obtained by 37. Symbolically, cit, m City + C2 t2 + cgts+.. . and C, to m C =ig: where m = 11+ is + ig+.. ti + Ce sig + ca ta +. . * and cu, cg, Cs, etc., are the various capitals, and ti, tg, ts, etc., are the various periods of time. 39. Symbolically, 1-ci _11 + 12 + ig t t y ti : te . . and 19 + . + ditats . . where in = 6 + C + 3+ For Private and Personal Use Only

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CHAPTER VI- MIXED PROBLEMS. 101 dividing the various amounts of interest by their corresponding periods of time. Thus the various capital amounts happen to be found out. Examples in illustration thereof. 40. (Sums represented by) 10, 6, 3 and 15 are the (various given) amounts of interest, and 5, 4, 3 and 6 are the corresponding) months (for which those amounts of interest have accrued); the mixed sum of the corresponding) capital amounts is seen to be 140. (Find out these capital amounts.) 41. The (various) amounts of interest are 5, 6, 101, 16 and 50; (the corresponding periods of time are) 5, 6, 7, 8 and 10 months ; 80 is the mixed sum of the various capital amounts lent out. What are these amounts respectively ?) The rule for arriving separately at the various periods of time from their given mixed sun: 42. Let the quantity representing the mixed sum of the (various) periods of time be divided by the sum of those (various quotients) obtained by dividing the various amounts of interest by their corresponding capital amounts; and (then) let the (resulting) quotient be multiplied (separately by each of the abovementioned quotients). (Thus) the (various) periods of time happen to be found out. An example in illustration thereof. 43. Here, (in this problem,) the (given) capital amounts are 40, 30, 20 and 50 ; and 10, 6, 3 and 15 are the corresponding) amounts of interest; 18 is the quantity representing the mixed sum of the respective) periods of time (for which interest has accrued. Find out these periods of time separately). 42. Symbolioally, =t, where m=t+tz+tz+ &c. iz + 18+ . " Ct C2 C Similarly ta, ts, etc., may be found out. For Private and Personal Use Only

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102 GANITASABASANGRAHA. The rule for arriving separately at the rate-interest of the ratecapital from the quantity representing the mixed sum obtained by adding together the capital amount lent out, which is itself equal to the rate-interest, and the interest on such capital lent ont: 44. The rate-capital as multiplied by the rate-time is divided by the other time (for which interest has accrued); the square root of this (resulting quotient) as multiplied by the (given) mixed sum once, and (then) as combined with the square of half of that (above-mentioned) quotient, when diminished by the half of this (same) quotient, becomes the (required) rate-interest (which is also equal to the capital lent out). E.camples in illustration thereof. 45. The rate-interest per 100 per 4 months is unknown. That (unknown quantity) is the capital lent out ; this, when combined with its own interest, happens to be 12; and 25 months is the time for (which) this interest has accrued. Find out the rate-interest equal to the capital lent out). 46. The rate-interest per 80 per 3 months is unknown ; 74 in the mixed sum of that (unknown quantity taken as the capital lent out and of the interest thereon for 1 year. What is the capital here and what the interest ? The rule for separating the capital, which is of the same value in all cases, and the interest (thereon for varying periods of time), from their mixed sum :-- 47. Know that, when the difference between (any two of) the (given) mixed sums as multiplied by each other's period * (of CT2 CT 44, Symbolically, -*m + = I which is equal to c. 47. Symbolicalls mit 5 mg - * By " the period of interest " here is meant the time for which interest hay accrued in connection with any of the given mixed sums of capital and interest. For Private and Personal Use Only

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CHAPTER VIMIXED PROBLEMS. 103 interest) is divided by the difference between those periods, what happens to be the quotient is the required capital in relation to (all) those (given mixed sums). Examples in illustration thereof. 48. The mixed sums are 50, 58 and 66, and the months (during which interest has accrued respectively) are 5, 7 and 9. Find out what the interest is (in each case). 49 and 50. O arithmetician, a certain man paid out to 4 persons 30, 31, 33 and 35, (these) heing the mixed sums (of the same capital and the interest due thereon) at the end of 3, 4, 5 and 6 months (respectively). Tell me quickly, what may be the capital here? The rule for separating the capital, which is of the same value in all cases, and the time (during which interest has accrued), from their mixed sum: 51. Wise men say that that is the (required) capital, which is obtained as the quotient of the difference between (any two of) the (given) mixed sums as multiplied by each other's interest, when this difference is divided by the difference between the two chosen) amounts of interest. Examples in illustration thereof. 52. The given) mixed sums of the capital and the periods of interest are 21, 23 and 25; here, (in this problem,) the amounts of interest are 6, 10 and 14. What may be the capital of equal valne here? 53. The (given) mixed sums are 35, 37 and 39; and the amounts of interest are 20, 28 and 36. (What is the common capital ?) 51. Symbolically, "? miem, " -- c, where my, my, etc., are the various misras or mixed sums. For Private and Personal Use Only

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104 GANITASARASANGRAHA, The rule for arriving at the capital dealt out at two different rates of interest : 54. Let the balance quantity (i.e., the difference between the two amounts of interest,) be divided by the difference between those (two quantities) which form the interest on one for the given periods of time; (this quotient) becomes the capital thought of by one's self before. Eramples in illustration thereof. 55. Borrowing at the rate of 6 per cent, and then lending out at the rate of 9 per cent, one obtains in the way of the differential gain 81 duly at the end of 3 months. What is the capital (utilized here)? 56. Borrowed at the rate of 3 per cent per mensem, a certain capital amount is put out to interest at the rate of 8 per cent per mensem. The differential gain is 80 at the end of 2 months. How much is the capital (so used)? The rule for arriving at the time when both capital and interest will become paid up (by instalments) :-- 57. The capital lent out is multiplied by its time (of instalment) and is again multiplied by the rate-interest; this product, when divided by the rate-capital and the rate-time, becomes the interest in relation to the instalment. The capital (in the instalment) and the time of discharge of the debt are to be made out) as before from (this) interest. Examples in illustration thereof. 58. The rate of interest is 5 for 70 per mensem ; the amount of the instalment to be paid is 18 in (every) 2 months; the capital lent out is 84. What is the time of discharge ? 54. Symbolically, 1x tix1 1x tz * I, TxC T, X 0, cxp XI 57. Symbolically, - -=interest in the instalment, where p is the 'CXT time of each instalment. For Private and Personal Use Only

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CHAPTER VIMIXED PROBLEMS. 105 59. The monthly interest on 60 is exactly 5. The capital lent out is 35; the amount of the) instalment (to be paid) is 15 in (every) 3 months. What is the time of discharge) of that (debt)? The rule for separating various capital amounts, on which the same interest has accrued, from their mixed sum : 60. Let the (given) mixed sum multiplied by the time (given) in relation to it be divided by the sum of that quantity, wherein are combined the various rate-capitals as multiplied by their respective rate times and as divided by their respective rate-interests. The interest (is thus arrived at); and (from this the capital amounts are arrived at an before. Examples in illustration thereof. 61. The mixed sum (of the capital amounts lent out) at the rates of 2, 6 and 4 per cent per mensem is 4,400. Here the capital amounts are such as have equal amounts of interest accruing after 2 months. What are the capital amounts lent, and what is the equal interest) ? 62. An amount represented (on the whole) by 1,900 was lent out at the rates of 3 per cent, 5 per 70, and 3) per 60 (per mensem); the interest (accrued) in 3 months on the various lent parts of this capital amount) is the same (in each case). (What are these amounts lent out and what is the interest ?) The rule for arriving at the lent oat capital in relation to the known time of discharge by instalments: 63. Let the amount of the instalment as divided by the time thereof and as multiplied by the time of discharge be divided by this, the capitals 60. Symbolically, C, T -= i; from + C2 * 13 + &c. are found ont by the rnle in Ch. VI. 10. --- = 1xt XI , where s= amount of instalment. 1+ TRC p the time of an iustalment, and t = the tinie of discharge. For Private and Personal Use Only

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106 GANITASARASANGRAHA. that interest on one for the time of discharge to which one is added ; the capital lent out is (thus arrived at). Examples in illustration thereof. 64. In accordance with the rate of 5 per cent (per mensem), 2 months is the time for each instalment; and paying the instalment of 8 (on each occasion), a man bere became free (from debt) in 60 months. What is the capital (borrowed by him) ? 65. A certain person gives once in 12 days an instalment of 23, the rate of interest being 3 per cent (per mensem). What is the capital amount of the debt discharged in 10 months ? The rule for arriving separately at the various capital-amounts which, when combined with or diminished by their respective interests, are equal to one another, from their mixed sun, (the interests being either added to the capital amounts in all the given cases or subtracted from them similarly in all the given cases) : 66. One is to be either combined with or diminished by the interest (accruing) thereon for the (given period of time (in cach case in accordance with the respectively given rate of interest; then again in each case,) one is divided respectively by these (combined or diminished quantities arrived at as before). Thereafter the (given) mixed surn (of the various capital amounts lent out) is divided by the sum of these (resulting quotients), and in relation to the mixed sum (so treated) the process of multiplication is to be conducted (separately in each case by multiplying it) by (the corresponding) proportionate part (of the aboveinentioned sum of the quotients). This gives rise to the capital 66. Symbolically, m + &c. 1+ -+ 1 x t xl - TxC Ixt x1 I1;xci 1 + 1 xt x1 T. XC. Similarly, to. do. = + Con Ixt x I. TX C And so on for C3, C4, &c. For Private and Personal Use Only

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CHAPTER VI-MIXED PROBLEMS. 10% amounts lent out, which on being combined with or diminished by their respective amounes of interest are equal (in value). Escamples in illustration thereof. 67. The total capital represented by 8,520 is invested (in parts) at the respective) rates of 3, d and 8 per cent (per month). Then, in this investment, in 5 months the capital amounts lent out arc, on being diminished by the (respective) amounts of interest, (seen to be) equal in value. (What are the respective amounts invested thus ?) 68. The total capital represented by 4,250 is invested (in parts) at the respective) rates of 3, 6 and 8 for 60 for 2 months; then, in this investment, in 8 months the capital amounts lent out are, on being diminished by the respective) amounts of interest, (seen to be) equal in value. (What are the respective amounts invested thus ?) 69. The total capital represented by 13,7 40 is invested (in parts) at the respective) rates of 2,5 and 9 per cent (per month); then, in this investment, in 4 months the capital amounts lent out are, on being combined with the (respective) amounts of interest, (scen to be) equal in value. (What are the respective amounts invested thus?) 70. The total capital represented by 3,646 is invested (in parts) at the respective) rates of 1$, and for 80 (per month); then, in this investment, in 8 months (the capital amounts lent out are, on being combined with the respective amounts of interest, seen to be equal in value. What are the respective amounts invested thus?) The rule for arriving at the capital, the interest, and the time of discharge of the debt) in relation to the debt-amount (paid up) in instalments in arithmetical progression : 71. (The required capital amount in the due debt) is that capital amount (which results) by adding the product of the 71. The rule is very elliptical and will become clear from the following working of the example contained in stanzas 72-73 : Here the mula or the maximum available amount of an instalment is 60 ; this, when divided by 7, the amount of the first instalment, gives 40 or 84, of which For Private and Personal Use Only

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108 GANITASARASANGRAHA. optionally chosen (maximum available amount of an instalment) by (whatever happens to be) the outstanding (fractional part of the number of terms in the series), to the amount of the (first) instalment as multiplied by the sum of that series in arithmetioal progression, which has (one for the first term, one for the common difference, and has for the number of terms the integral value of) the quotient obtained by dividing (the above optionally chosen maximum) amount of debt (discharged at an instalment) hy the (above amount of the first) instalment. The interest thereon is that which accrues for the period of an instalment. The time (of an instalment) divided by the amount of the (first) instalment and multiplied by the (optionally chosen maximum) amount of debt (discharged at an instalment) gives rise to the time (which is the time of the discharge of the whole debt). Examples in illustration thereof. 72 and 73). A certain man utilised, (for the discharge of a debt) bearing interest at 5 per cent (per month), 60 (as the available maximum amount) with 7 as the first instalment amount, increasing it by 7 in successive instalments due every of a month. He thus gave in discharge of the debt the sum of a scries in arithmetical progression consisting of be terms, and gave also the interest accruing on those multiples of 7. What is the debt anwunt corresponding to the sum of the series, what is that interest (which he paid), and (what is) the time of discharge of that debt ? 731 to 76. A certain inan utilised for the discharge of a debt, bearing interest at 5 per cent (per mensem), E0 (as the available maximum amount) with 8 as the first instalment amount, increasing it by 8 in successive instalments due every I of a month. IIe thus 8 represents the number of terms of the series in arithmetical progression, which has 1 for the first term and 1 for the common difference, and is the agra or the outstauding fractional part. The sum of the above-mentioned series, viz., 36, multiplied by 7, the amount of the first instalment, is added to the product of : und 60, which latter is the maximum available amount of an instalment. Thus, we get 36 X 7+ X 60 = 204*, which is the required capital amount in the due debt. The interest on 2004 for of a month at the rote of 5 per cent per mensem will be the interest paid on the whole. The time of discharge will be ( 7) X 20 = 4 months. For Private and Personal Use Only

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Shri Mahavir Jain Aradhana Kendra CHAPTER VI-MIXED PROBLEMS. gave in discharge of the debt the sum of a series in arithmetical progression consisting of 80 terms and gave also the interest accruing on those multiples of 8. The debt amount (corresponding to the sum of the series), the interest (which he paid), and the time of discharge (of that debt)-tell me, friend, after calculating, what the (respective) value of these quantities is. The rule for arriving at the average common interest :-- 77 and 77. Divide the sum of the (various accruing) interests by the sum of the (various corresponding) interests due for a month; the resulting quotient is the required time. The product of the (assumed) rate-time and the rate-capital is divided by this required time, then multiplied by the sum of the (various accruing) interests and then divided again by the sum of the (various given) capital amounts. This gives rise to the (required) rate-interest. www.kobatirth.org An example in illustration thereof. 7. In this problem, four hundreds were (separately) invested at the (respective) rates of 2, 3, 5 and 4 per cent (per mensem) for 5, 4, 2 and 3 months (respectively). What is the average common time of investment, and what the average common rate of interest? and Thus end the problems bearing on interest in this chapter on mixed problems. Symbolically, C C1 x 1 x 1, 77 and 77. The various accruing interests are the various amounts of interests accruing on the several amounts at the various rates for their respective periods. Tx C ta C1 x 1, x 1, Tx C + Acharya Shri Kailassagarsuri Gyanmandir Cox to x I Tx C C2 x 1 x 12 Tx C Co X to x I 1 x C + + +. 109 +. c1 I * } + { x 1 x + = } Tx C ta or average time; .} / (0) + c2+ . . .) - ia or average interest. For Private and Personal Use Only

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110 GANITASARASANGRAHA. Proportionate Division, Hereafter we shall expound in this chapter on mixed problems the working of proportionate division : 79. The operation of proportionate division is that wherein the (given) collective quantity , to be divided) is first divided by the sum of the numerators of the cominon-denoininator-fractions (representing the various proportionate parts), the denominators of which fractions are struck off out of consideration ; and (then it) has to be multiplied (respectively in each case) by (these) proportional numerators. This is called kuttikara by the learned. Examples in illustration thereof. 801. Here, (in this problem,) 120 gold picces are divided among 4 servants in the (respective) proportional parts of , and a. 0 arithmetician, tell me quickly what they obtained 811. (The sum of) 363 dinaras was divided among five, the first one (among them) getting 3 parts, and 3 being the common ratio successively (in relation to the shares of the others). What was the share of cach? 821 to 85). A certain faithful sravaka took a number of lotus flowers, and going into the Jina temple conducted (therein) with devotion the worship of the chief Jinas that were worthy of worship. He offered 1 part to Vrsabha, to worthy Parsva, and to Jinapati, and l to sage Suvrata; he dovotedly gave to Aristanemi who destroyed all the eight kinds of karmas and who was beloved by the world ; and of I to Jinasanti : 480 lotuscs were brought (for this purpose.) By adopting the operation known 79. In working the exaniple in stanza 80+ according to this rule we get: %, 3, E12, 12, in, 12. After removing the denominators here, we have 6, 4, 3 and 2. These are also called praksepus or proportional numerators. The sum of these is 15, by which the amount to be distributed, viz., 120, is divided ; and the resulting quotient 8 is separately multiplied by the proportional numerators 6, 4, 3 and 2. Then the amounts thus obtained are 6 x 8 or 48, 4 x 8 or 32, 3 x 8 or 24,2 x 8 or 16. It is worthy of note that praksepa means both the operation of proportionate division and a proportional numerator, For Private and Personal Use Only

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CHAPTER VIMIXED PROBLEMS. 111 as praksepaka, give out the proportionate distribution of the flowers. 861. (A sum of) 480 was divided among five men in the proportion of 2, 3, 4, 5 and 6; 0 friend, give out the share of each). The rule for arriving at (certain) results in required proportions :-- 873. The (number representing the) rate-price is divided by (the number representing) the thing purchasable therewith ; (it) is (then) multiplied by the (given) proportional number; by means of this, (we get at the sum of the proportionate parts, (through the process of addition. Then the given amount multiplied by the (respective) proportionate parts and then divided by this sum of) the proportionate parts gives rise to the value of the various things in the required proportion). Another rule for this same) purpose :-- 88.). Multiply the numbers representing the rate-prices (respectively) by the numbers representing the (given) proportions of the various things (to be purchased); then divide (the result) by the (respective) numbers measuring the things purchasable for the rate-price ; the resulting quantities happen to be the (requisite) multipliers in the operation of praksepaka. The intelligent man may (then) give out the required answer by adopting the rule-of-three. Again a rule for this (same) purpose : 89.1. The numbers representing the various) rate-prices are respectively divided by their own related (numbers representing the) things purchasable therefor and are (then) multiplied by their related proportional numbers. With the help of these, the remainder (of the operation shonld be carried out) as before. 87 to 89}. In working the example in stanza 90% and 91} according to these rnles 2, 3 and 5 are divided by 3, 5 and 7 respectively and are similarly multiplied hy 6, 3 and 1. Thus we have x 6,3 x 3, x1 = 4, , . These are the proportional parts. The rules in stanzas $8! and 89: require thereafter the operation of prak: pa to be applied in relation to these proportional parts ; but the rule in stunza 87} expressly describes this operation. For Private and Personal Use Only

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112 GANITASARASANGRAHA. The required result is well arrived at by going through the process of the rule-of-three. Examples in illustration thereof. 901 and 911. Pomegranates, mangoes and woodapples are obtainable at the respective) rates of 3 for 2, 5 for 3, and 7 for 5 panas. O you friend, who know the principles of calculation, come quickly having purchased fruits for 76 panas, so that the mangoes may be three times as the woodapples, and the pomegranates six times as much. 921 to 94. A follower of Jina had the image of Jina bathed in potfols of curds, ghee and milk. Three pots became filled with 72 palas (of these); 32 palas were found in the first pot and 24 in the second pot and 16 in the third pot. From these (potfuls of mixedup) curds, ghee and milk, find out cach of those ingredients) separately and give them ont, there being altogether 24 palas of ghee, 16 palas of nilk and 32 palas of curds. 95% and 96). Three puranas formod the pay of one man who is a mounted soldier; and at that rate there were 65 men in all. Some among them) broke down, and the amount of their pay was given to those that remained in the field. Of this, each man obtained 10 puranas. You tell me, after thinking weli, how many remained in the field and how many broke down. The rule for the operation of proportionate division, wherein there is the addition or the subtraction of certain optionally chosen integral quantities : 973. The given total quantity is diminished by the integral quantities that are to be added, or is combined with the positive integral quantities that are to be subtracted; then with the help of this resulting quantity the operation of proportionate division is to be conducted, and the resulting proportionate parts are respectively combined with those (integral quantities that are to be added to them), or they are diminished (respectively) by those integral quantities that are to be subtracted). 971. The operation of proportionate division to be conducted here is according to any of the rules in stanzas 874 to 89). For Private and Personal Use Only

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CHAPTER VI-MIXED PROBLEMS. 113 Examples in illustration thereof. 981. Four men obtained their shares in sucoessively doubled proportions and with successively doubled differences in addition, the first man obtaining one share : 67 (is the quantity 80 to be distributed) here. What is the share of each ? 991. (A sum of) 78 is divided by these four (among themselves) in proportions which are successively from the first 11 times (what precedes) and with differences (in addition, which,) commencing with 1, (go on increasing three-fold. Give out the (valne of the parts obtained (by each.) 100%. The shares of) five (persons) are (successively) from the first 11 times (what goes before), and the differences in addition are quantities which are (successively) 24 times (the preceding difference) 51is (the total quantity) to be divided. (Find out the values of the portions obtained by each.) 101}. (A sum of) 400 minus 15 is divided by four men (among themselves) in proportions which from the first are 27 times (what precedes), and which (besides) are loss by differences which are (successively) 4 times (the preceding difference). (Find out the values of the various portions obtained.) The rule for arriving at the value of the prices producing equal sale-proceeds and at the value of the highest capital (invested in the transactions concerned): 102]. The largest capital invested) combined with one becomes the vending rate of the commodity (to be sold). That (same vending rate), multiplied by the (given) price at which the rempant is to be sold, and diminished by one, gives rise to the 98. The difference quantity to be added to the shares here is 1 in the case of the second man, and twice the preco ting difference in the case of each of the remaining two men ; and this difference in the case of the second man is nct expressly mentioned as l in this example as well as in the example in stanza 1017. 102. The examples bearing on this rule contemplate the purchase of a commodity at a certain common rate for various capital amounts; then the commodity go purchased is to be sold at a certain other common rate. That quantity of the commodity which is left over, owing to its not being enoagh to be sold for a unit of the kind of money employed in the transaction, is here 15. For Private and Personal Use Only

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114 GANITASARASANGRAHA. purchasing rate. By reversing the processes, one may arrive at the valuation of the highest capital (invested in the transaction). D.camples in illustration thereof. 1031. The capital amounts invested by (three) men are (respectively) 2, 8 and 36; 6 is the price at which the remnants of the commodity are to be sold. Having purchased and sold at the same rates, they became possessors of equal wealth. (Find out the buying and selling prices.) 1043 Those three persons took up 12, i and 21 (as their respective capital amounts) and conducted the operations of buying and selling (in relation to the same commodity at the same rates of price); by selling the remnaut (in the end) at a price represented by 6, they became possessors of cqual wealth. Find out their buying and selling prices.) 1054. The quantity measuring the equal wealth is 41, and the price at which the remnants of the commodity are sold is 6. () arithmetician, tell me quickly what the highest capital invested) is, and what the (various) capitals are. 106. In the case where 35 dinaras give the numerical measure of the equal wealth, and 4 is the price at which the remnant is to be sold, you tell me, O arithmetician, what the highest capital (invested) is. spoken of as the remnant, and the price at which this remnant is sold is the remnant-price. Symbolically, let a, a + b and a + b + c be the capitals, where the last is the IT or the largest capital, and let p be the fare or the remnantprice ; then, according to the rule, a + b + c + 1 = the vending rate ; and (a + b + c + 1) P-1= the purchasing rate. From these, it can be easily shown that the sum of the amonnts realised by selling the commodity at the vending rate and the remnant at the remnantprice turus out to be the same in each case. It may be noted that the purchasing rate happens in problems bearing on this rule to be the same in value as the F o r the equal sale-proceeds. 1054. It may be noted here that, according to the rule, it is only the largest sapital that is found out; while the other capitals required in the problem are optionally chosen, so as to be less than the largest capital. For Private and Personal Use Only

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CHAPTER VIMIXED PROBLEM. 115 The rule for arriving at the value of the prices producing equal sale-proceeds when the price of the remnant is fractional in character: 107}. When the remnant-price is fractional in character, the selling and the buying rates are to be derived as before with (the data consisting of) the (invested) capitals and the remnant-price reduced to the same denominator, which is (however) ignored (for the time being); these selling and buying rates are (then respectively) to be multiplied by (this) denominator and the square of (this) denominator (for arriving at the required selling and buying rates). The value of the equal sale-proceeds is (then obtained) by means of the rule-of-threa. An example in illustration thereof. 1081. (In a transaction) { }, are the capital amounts (invested respectively by three persons); the remnant-price is s. By purchasing and selling at the same prices, they became possessed of cqual sale-proceeds. (What is the buying price, what the selling price, and what the equal sale-amount ?) Again, another rule for arriving at the value of the equal saleproceeds, when the remnant-price is fractional : 1091. The continued product of the highest numerator, of tuo, and of (all) the denominators (to be found in the values of the capital amounts invested), when combined with the last) denominator belonging to the value of the remnant-price, gives rise to the selling rate. This multiplied by the remnant-price, and then diminished by one, and then multiplied (successively) by two and all the denominators, becomes the purchasing rate. Then the rule-of-three (is to be used for arriving at the common value of the sale-amounts). An example in illustration thereof. 1103. Having invested i, j, (respectively), and having bought and sold (the same commodity), and with as the remnantprice, three merchants became possessors of equal sale-proceeds For Private and Personal Use Only

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Shri Mahavir Jain Aradhana Kendra 116 GANITASARASANGRAHA. (in the end. What is the buying price, what is the selling price, and what the equal sale-amount?) The rule for arriving at (the solution of a problem wherein) optionally chosen quantities (are) bestowed in optionally chosen multiples for an optionally chosen number of times : 111. Let the penultimate quantity be added to the ultimate quantity as divided by its own corresponding multiple number, and let the result of this operation be divided by that (multiple number which is associated with this) penultimate quantity (given in the problem). What results (from carrying out this operation throughout in relation to all the various quantities bestowed) happens to be the (required) original quantity. Examples in illustration thereof. 112 and 113. A certain lay follower of Jainism went to a Jina temple with four gate-ways, and having taken (with him) fragrant flowers offered them (thus) in worship with devotion :-- At the four gate-ways, they became doubled, then trebled, then quadrupled and then quintupled (respectively in order.) The number of flowers offered by him was five at every (gate-way). How many were the lotuses (originally taken by him)? 114. Flowers were obtained and offered in worship by devotees with devotion, the flowers (so offered) being (succossively) 3, 5, 7 and 8; (their corresponding) multiple quantities being, and (in order. Find out the original number of flowers). Thus ends proportionate division in this chapter on mixed problems. For Private and Personal Use Only

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CHAPTBR VI-MIXED PROBLEMS. 117 Vallika-kuttikara. Hereafter we shall explain the process of calculation known as Vallika-kuttikara*:-- The rule underlying the process of calculation known as Vallika in relation to Kuttikara (which is a special kind of division or distribution): 1151. Divide the (given) group-liumber by the (given) divisor; discard the first quotient; then put down one below the other the (various) quotients obtained by the successive division (of the various resulting divisors by the various resulting remainders ; again), put down below this the optionally choson number, * It is so called because the method of kuttikara explained in the rule is banod upon a creeper.like chain of figures. 115$. The rule will become olear from the following working of the problem in stanza No. 1174. Here it is stated that 63 heaps of plantains together with 7 separate fruits are exactly divisible among 23 persons ; it is required to find out the number of fruits in a heap. Here the 63 is called the group-number, and the numerical value of the fruits contained in each heap is called the 'group-value'; and it is this latter which has to be found out. Now, according to the rule, we divide first the rise, or group-number 63, by the cheda or the divisor 23; and then we continue the process of division as in finding out H.C.F. of two numbers :23) 632 46 Here, the first quotient 2 is discarded; the other 17)23 (1 quotients are written down in a line one below the other as in the wargin; then we have to choose such a number as, when multiplied by the last remainder 1, and then combined with 7, (the 12 number of separate fruits given in the problem, 5)61 will be divisible by the last divisor 1. We accordingly choose 1, which is written down below the last figure in the chain; and below 1)5(4 this chosen number, again, is written down the quotient obtained in the above division with the help of the chosen number. Here we stop the division with the fifth remainder as it is the least remainder in the odd position of order in the series of divisions carried out here, 17 6)17(2 For Private and Personal Use Only

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118 GANITASARASANGRAHA. with which the least remainder in the odd position of order (in the above-meutioned process of successive division is to be multiplied; and (then put down) below (this again) this product increased or decreased (as the case may be by the given known number) and then divided by the last divisor in the above mentioned process of successive division. Thus the Vallika or the creeper-like chain of figures is obtained. In this) the sum obtained by adding (the lowormost number in the chain) to the product obtained by multiplying the number above it with the number (immediately) above (this upper number, this process of addition being in the same way continued till the whole chain is exhausted, this sum, is to be divided by the (originally Thus wo get the chain or Vallika noted in the first colunin of figures in the margin. Then we multiply the penaltimate figure below in 1-51 the chain, viz., 1, by 4, which is above it, and add 8, the last 2--38 number in the chain; the resulting 12 is written down so as 1-13 to be in the place corresponding to 4; then multiplying this 4-12 12 by 1 which is the figure above it in the crecper chain, and adding 1, the figure similarly below it, we got 13 in the place of 1; proceeding in the same manner 38 and 51 are obtained in the places of 2 and I respectively. This 51 is divided by 23, the divisor in the problem ; and the remainder 5 is seen to be the least number of fruits in a bunch. The rationale of the rule will be clear from the following algebraical representation :B + b = y (an integer) = 114 + Di, where p = (B- Aq) + b A : * = Aps --, (where r, = B - Aqi the first remainder) = pi + P2, where pe = 'P , and q. is the second quotient and r, the second remainder. Hence, p = "123 + = 9apa + Pa, where pa = *sp + and qo is the third quotient and rs the third remainder. Similarly, Pa = ": -b = 94 P8 + pa, whero Pa = " P3S; Tapi -- b P2 = P3 P4 + b P3 = 8 PA + Ps, where P = Thus we have, q2 DI + P2; Di=93 P2 + Ps; P 94 Ps + P4; Ps=9804 + Ps. For Private and Personal Use Only

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CHAPTER VI--MIXED PROBLEMS. 119 given) divisor. (The remainder in this last division becomes the multiplier with which the originally given group-number is to be multiplied for the purpose of arriving at the quantity which is to be divided or distributed in the manner indicated in the problem. Where, however, the given group-numbers, increased or decreased in more than one way, are to be divided or distributed in more than one proportion,) the divisor related to the larger group-value, (arrived at as explained abovo in relation to either of two specified distributions), is to be divided (as above) by the divisor (related to --- - By choosing a value for P, such that 5 P4 T -, which is, as shown above, the res which is, as she value of ps becomes an integer, and by arranging in a chain 92, 93, 94, 95, P4 and po we get at the value of x by proceeding as stated in the rule, that is, by the processes of multiplication by the upper quantity and the addition of the lower quantity in the chain, which are carried up to the topmost quantity. The value of so obtained is divided by A, and the remainder represents the least value of x; for the values of x which satisfy the equation, bu Bx + b =an integer, are all in an arithmetical progression wherein the common difference is 4. This same rule contemplates problems where two or more conditions are given, suoh as the problems given in stanzas 1214 to 129. The problem in 1213 may be thas worked out according to the rule :-It is given that a heap of fruits when diminished by 7 is exactly divisible among 8 men, and the same heap when diminished by 3 is exactly divisible among 18 mon. Now, according to the method already given, find out first the least number of fruits that will satisfy the first condition, and then find out the number of fruits that will satisfy the second condition. Thus we get tlie group-valuos 15 and 16 reppe otively. Now, the divisor related to the larger group-value is divided as before by that related to the smaller group-value to obtain a fresh walliki chain. Thus dividing 13 by 8 and continuing the division, we have 8)13(1 From this the Vallika chain comes out thus:-Ohoosing 1 as the inati, and adding the ditference between the two groap-values already arrived at, that is, 5)8(1 1 10--15, or 1, to the product. of the mati and the last divisor, and dividing this sum by the last 3)5(1 divisor, we have 2, which is to be written down below the mati in the Vallika shain. Then pro ceeding as before with the vallika, we get 11, 2)3(1 which, when divided by the first divisor 8, leaves the remainder 3. This is multiplied by 1)2(1 the divisor related to the larger group-value, viz., 13, and then is combined with the larger group-value. Thus 55 is the number of fruits in the heap. For Private and Personal Use Only

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120 GANITASARASANGRAHA. the smaller group-value, so that a creeper-like chain of successive quotients may be obtained in this case also as before. Below the lowermost quotient in this chain, the optionally chosen multiplier of the least remainder in the odd position of order in this last successive division is to be put down; and below this agaiu is to be put down the number which is obtained by adding the difference between the two group-values (already referred to) to the product (of the least remainder in the last odd position of order multiplied by the above optionally chosen multiplier thereof, and then by dividing the resulting sum by the last divisor in the The rationale of this process will be clear from the following considerations : B2 * + b2 is an integer; and 1 is an integer; (ii) 52 We have 6) Bi + b A1 A, (iii) Byc 03 is an integer. In (i) Let the lowest valne of a = 31. Ag In (ii) 2 3 = R2. In (iii) ? (iv) When both (1) and (ii) are to be satisfied, dA, + 8 has to be equal t. kA, + 89, so that 8, 8, KA, - 84. That is, 41a (81 -- 89) = k. LA From (iv), which is an indeterminate equation with the values of d and k unknown, we arrive, according to what has been already proved, at the lowest positive integral value of d. This value of d multiplied by Aj, and then increased by 81, gives the value of a which will satisfy (i) and (ii). Let this be ti; and let the next higher value of x which will satisfy both these equations be ts. (v) Now, to + n4=ta; (vi) and ti + m 2 = t. . ..^=". Thus 4, = mp, and Ag = np, where p is the highest common factor between A, and A2. .. m = , and >>=4. Substituting in (v) or (vi), we have pP 4, A2 = to. From this it is obvious that the next higher value of a satisfying the two equations is obtained by adding the least common multiple of A and A, to the lower value. Now again, let v be the value of a which satisfies all the three equations. Then v=t+ + 44, 19 x r, (where is a positive integer) = (say) 4. + lr ; 1P and v= 33 + cAg = t + Ir. .. r=cAg + $g - t For Private and Personal Use Only

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CHAPTER VI - MIXED PROBLEMS. 121 above division chain. Thus the creeper-like chain of figures required for the solution of this latter combined problem is obtained. This chain is to be dealt with as before from below upwards, and the resulting number is to be divided as before by the first divisor in this last division chain. The remainder obtained in this operation is then) to be multiplied by the divisor (related to the larger group-value, and to the resulting product, this) larger group-value is to be addel. (Thus the value of the required inulti plier of the given group-number is obtained ; and this will satisfy both the specified distributions takeu together into consideration). Exumplez in illustration thereof. 1161. Into the bright and refreshing outskirts of a forest, which were full of numerous trees with their branches bent down with the weight of flowers and fruits, trees such as jambu trees, limo trees, plantains, areca palms, jack trees, date-palms, hintala trecs, palmyras, punnaga trees and mango trees-- (into the ontskirts, the various quarters whereof were filled with the many sounds of crowds of parrots and cackoos found near springe containing lotuses with bees roaming about them-into such forest outskirts) a number of weary travellers entered with joy. 117]. (There were) 63 (numerically equal) heaps of plantain fruits put together and combined with 7 (more) of those same fruits; and these were (equally) distributed among 23 travellers so as to leave no remainder. You tell me now) the (numerical) measure of a heap (of plantains.) 118). Again, in relation to 12 (numerically equal) heaps of pomegranntes, which, after having been put together and By applying the principle of vallika-kuttikara in the last equation, the value of c is obtained, and thence the value of v can be easily arrived at. It is seen from this that, when, in order to find out v, we deal with t, and og in accordance with the louttikora method, the cheda or the divisor to be taken in -, or the least common multiple of the divisors in the first two equations. 16 relation to ti is 41 42 PS For Private and Personal Use Only

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122 GANITASARASANGRAHA, combined with 5 of those (same fruits), were distributed similarly among 19 travellers. Give out the (numerical) measure of (any) one (heap). 1191. A traveller sees heaps of mangoes (equal in numerical value), and makes 31 heaps less by 3 (fruits); and when the remainder (of these 31 heaps) is cqually) divided among 73 men, there is no remainder. Give out the numerical value of one (of these heaps). 1207. In the forest 37 heaps of wood-apples were seen by the travellers. After 17 fruits were removed (therefrom, the remainder) was (equally) divided among 79 persons (so as to leave no remainder). What is the share obtained by each ? 1211. When, after seeing a heap of mangoes in the forest and renoving 7 fruits (therefrom), it was divided equally among 8 of the travellers; and when again after removing 3 (fruits) from that (same) heap it was (equally) divided among 13 of them ; it left no remainder (in both cases). O arithmetician, tell me (the numerical measure of this) single heap. 1221. A single heap of wood-apples divided among 2, 3, 4, or 5 (persons) leaves 1 as remainder (in each case). O you who know arithmetic, tell me the (numerical) measure of that (heap). 1231. When (divided) by 2, the remainder is 1; when by 3, it is 2; when by 4, it is 3; when by 5, it is 4. Tell me, O friend, what this heap is. 1241. When (divided) by 2, the remainder is 1; when by 3, there is no remainder; when by 4, it is 3 ; when by 5, it is 4. Tell me, O friend, what the heap is (in numerical value). 1251. When divided by 2, there is no remainder; when by 3, there is 1 as remainder; when by 4, there is no remainder; and when by 5, there is one as remainder. What is this quantity ? 1261. When divided by 2 (the remainder is) 1; when by 3, there is no renainder; when by 4, (the remainder is) 3; and when divided by 5, there is no remainder Tell me now what (this) quantity is. 1271. The travellers saw on the way certain (equal) heaps of jambu fruits. Of them, 2 (heaps) were equally divided among 9 For Private and Personal Use Only

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CHAPTER VI-MIXED PROBLEMS. 123 ascetics and left 3 (fruits) as remainder. Again 3 (heaps) were (similarly divided among 11 persons, and the remainder was 5 fruits ; then again 5 of those heaps were similarly divided among 7, and there were 4 more fruits (left out) of them. O you arithmetician who know the meaning of the kuttikara process of distribution, tell me after thinking out well the numerical measure of a heap (hero). 1281. In the interior of the forest, 3 heaps (equal in value) of pomegranates were divided (equally) among 7 travellers, leaving 1 (fruit) as remainder; 7 (of such beaps) were divided (similarly) among 9, leaving a remainder of 3 (fruits ; again) 5 (of such heaps) were (similarly) divided among 8, leaving 2 fruits as remainder. O arithmetician, what is the numerical value of a heap here). 129). There were 5 (heaps of fruits equal in numerical value), which after being combined with 2 (fruits of the same kind) were (equally) divided among 9 travellers (and left no remainder); 6 (heaps) combined with 4 (fruits) were (similarly) divided among 8 of them; and 4 (heaps) combined with 1 (fruit) were also similarly) divided among 7 of them. Give out the numerical measure (of a heap here). The rule for arriving at the original quantity distributed (as desired), after obtaining the remainder due to the removal of certain specified) known quantities :. 1301. (Ohtain) the product of the (given) known quantity (to be removed), as multiplied by the fractional proportion of what is jeft (after a specified fractional part of what remains on the removal of the given known quantity has been given away). The next quantity is (obtained by means of) this (product), to which 1309. Here the known quantity to be removed is called the agra. What remains after the removal of the agra is the remainder. That fraction of this remainder which is given or taken away is the agravea, and what is left of the remainder after the agramsa is given or taken away is the sera mea or the remaining fractional proportion of the remainder. For example, where a is the quantity to be found out, and a is the agra in relation to the first distribution with $ as the fractional proportiou distributed," "happens to be the age a visa, and (x - a) - a to be the sesa visa. For Private and Personal Use Only

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124 GANITAS.BASANGRAHA. the specified known quantity which is to be taken away (from the previous remainder) is added; and this resulting sum) is multiplied by that (same kind of) remaining fractional proportion (of the remainder as has been mentioned above). This is to be done as many times as there are distributions to be made. Then these quantities so obtained should be deprived of their denominators; and these denominator-less quantities and the successive products of the above-mentioned remaining fractional proportious of the remainder) are (to be used as the known quantity and the (other elements, viz., the cocfficient) multiple (of the unknown quantity and the divisor, required in relation to a problem on Vallika-kuttikara). Examples in illustration thereof. 1311. On a certain mau bringing mango fruits (home, bis) elder son took one fruit first and then half of what remained. (On the elder son going away after doing this, the younger (son) did similarly (with what was left there. He further took half The rule will be clear from the following working of the problem in 132133 : Here 1 is the first agra, and is the first agri visa; therefore 1- or in the sesu visa. Now, obtain the product of agra and sesa visa or 1 x 3 or $. 'Write it down iu 2 places. Repeat the quantities { }; add the second ayra 1 (to one of the quantities) Then we have multiply both by the next sesa ma l or, so that you get 11 Take these figures and add the third agra 1 as before ; and you bave { }; multiply by the next sesanisu 1- or f and by the last arsa or }; and you have {11} Ill The denominators of the first fractions in these three sets of fractions marked I, II, III, are dropped, and the numerators represent negative agras in & problem on Vallika-kuttikara wherein the numerator and the denominator of each of the second fractions in those sets represent respectively the dividend coefficients and the divisor. Thus we have 2.8 - 2. is an integer ; a 8a - 38 is an integer. 42 - 10 is an integer; and --81 The value of satisfying these three conditions gives the number of Howers, For Private and Personal Use Only

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CHAPTER VIMIXED PROBLEMS. 125 of what was thereafter left); and the other (son) took the other half. (Find the number of fruits brought by the father.) 1323 and 133}. A certain person went (with flowers) into a Jina temple which was (in height) three times the height of a man. At first he offered one (out of those flowers) in worship at the foot of the Jina and then offered in worship) one-third of the remaining number (of flowers) to the first height-measure of the Jina). Out of the remaining two-thirds of the number of flowers, he conducted worship) in the same manner in relation to the second heightmeasure; and (then he did) the same thing in relation to the third height-measure also. The two-thirds which remained at last were also made into 3 equal parts (by him); and having worshipped the 24 tirthankaras (with these parts at the rate of eight tirthankaras for each part), he went away with no (flower) on hand. (Find out the number of flowers taken by him.) Thus ends simple Kuttikaru in this chapter on mixed problems. *Visama-kuttikara. Hereafter we shall expound complex kutlakara. The rule relating to complex kuttikara :--- 1344. The (given) divisor, (written down in two (places), is to be multiplied (in each place) by an optionally chosen number; and the (known) quantity given in the problem) for the purpose of being added is to be subtracted (from the product in one of these places); and the quantity given in the problem) for the purpose of being subtracted is to be added to the product noted down in the other place. The two quantities thus obtained are to be divided by the known (coefficient) multiplier (of the unknown * The words Visama and Bhinna here used in relation to Kuttikara have obviously the same meaning and refer to the fractional character of the dividend quantities occurring in the problems contemplated by the rule. For Private and Personal Use Only

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126 GANITASARASANGRABA. quantities to be distributed in accordance with the problem). Each (of the quotients so obtained) happens to be the required (quantity which is to be multiplied by the given) multiplier in the process of Bhinnakuttikara.* An example in illustration thereof. 135. A certain quantity multiplied by 6, (then) increased by 10 and (then) divided by 9 leaves no remainder. Similarly, (a certain other quantity multiplied by 6, then) diminished by 10 (and then divided by 9 leaves no remainder). Tell me quickly what those two quantities are (which are thus multiplied by the given multiplier bere). Sakala-kuttikara. The role in relation to sakala-kuttikaru. 1361. The quotient in the first among the divisions, carried on by means of the dividend-coefficient (of the known quantity to be distributed), as well as by means of the divisor and the successively) resulting remainders, is to be discarded. The other quotients obtained by means of this mutual division (carried on till the divisor and the remainder become equal) are to be written down (in a vertical chain along with the ultimately equal remainder and divisor); to the lowermost figure in this chain), the remainder (obtained by dividing the given known quantity in the problem by the divisor therein), is to be added. (Then by means of these numbers in the chain), the sum, (which has to be obtained by adding (successively to the lowermost number) the product of the two 1364. This rule will become clear from the following working of the problem given in 1373 :The problem is, when 177x + 240 U is an integer, to find out the values of 2. 201 Removing the common factors, we have is an integer. For Private and Personal Use Only

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CHAPTER VI-MIXED PROBLEMS. 127 numbers immediately above it, (till the topmost figure in the chain becomes included in the operation), is to be arrived at. Thereafter) this resulting sum and the divisor in the problem (give rise), in the shape of two remainders, (to the two values of) the unknown quantity (which is to be multiplied by the given dividend-coefficient in the problern), which (values are related either to the known given quantity that is to be added or to the known given quantity that is to be subtracted, according as the nuraber of figure-links in the above-mentioned chain of quotients is even or odd. (Where, however, the given groups, increased or decreased in more than one way, are to be divided or distributed in more than one proportion), the divisor related to the larger group-value, (arrived at as explained above in relation to either of two specified distributions), is to be divided over and over (as above by the divisor Carry out the required process of continued division :07)59(0 59 59)67(1 After discarding the first quotient, the others are written down in a chain thus:-- 8)5907 Below this are next written down 1 and 1, 56 the last equal divisor and remainder. Here also, as in Vallika-kutsikara, it is worthy 3)8(2 of note that in the last division there can be 1 really no remainder, as 2 is fully divisible by 1. But since the last remainder is 2)3(1 1+13 = 14 wanted for the chain, it is allowed to occur by making the last quotient smaller than possible. And to the last number 1 here, ada 1)2(1 13, which is the remainder obtained by dividing 80 by 67; the 14 so obtained in also written down at the bottom of the chain, which now becomes complete. Now, by the continned process of multiplying and adding the figures in this chain, ay already explained in the note under stanza No. 115), 1-392 we arrive at 592. This is then divided by 67; and the remain7-345 der 57 is one of the values of a, when 80 is taken as negative 2-47 1--16 owing to the number of figures in the chain being odd. When 1-15 80 is taken as positive, the value of x is 67 - 57 or 10. If the number of figures in the chain happen to be even, then the value of tiret arrived at is in relation to the positive agra ; if this value be sa tracted from the divisor, the value of x in relation to a negative agra is arrived at.. For Private and Personal Use Only

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Shri Mahavir Jain Aradhana Kendra 128 www.kobatirth.org GANITASARASANGRAHA. related to the smaller group-value obtained as above so that a creeper-like chain of successive quotients may be obtained in this case also. Below the lowermost quotient in this chain the optionally chosen multiplier of the least remainder in the odd position of order in this last successive division is to be put down The principle underlying the process given in the rule is the same as that explained in the rule regarding Vallika-kuttikara-but with this difference, namely, that the last two figures in the chain here are obtained in a different way. Again, from the rationale given in the footnote to rale in 115, Ch. VI, it will be seen that the agra, b, associated with the remainder in the odd position of order, has the same algebraical sign as is given to it in the problem ; while the sign of the agra, b, associated with the remainder in the even position of order is opposite to its sign as given in the problem. Hence, when the continued division is carried up to a remainder in the odd position of order, the value of a arrived at therefrom is in relation to such an agra as has its sign unchanged; on the other hand, when the continued division is carried up to a remainder in the even position of order, the value of a arrived at therefrom is in relation to an agra that has its sign changed. When the number of remainders obtained is odd, the number of quotients in the chain is even; and when the remainders are even, the quotients are odd in number. As the agra associated with the last remainder is in this rule always taken to be positive, the value of a arrived at is in relation to the positive agra, if the last remainder happens to be in the odd position of order. And it is in relation to the negative agra, if the last remainder happens to be in the even position of order. In other words, if the number of quotients be even, the value is in relation to the positive agra; and if the number of quotients be odd, it is in relation to the negative agra. AB B The value of x in relation to the positive or the negative agra being thus found ont, the other value is arrived at by subtracting this value from the divisor in the problem. How this turns out will be clear from the following representation: Ax + b B know that = an integer. Here let x = c; then Acharya Shri Kailassagarsuri Gyanmandir is also an integer. Hence Ac + b B For Private and Personal Use Only an integer. We AB Ac + b A(B c) b or B B B is an integer. It has to be noted here that the common factor, if any, of the three given numerical quantities is to be removed before the operation of continued division is begun. The last divisor and the last remainder being required to be equal it will invariably happen that these come to be 1. The mati, required to be chosen in the rule relating to the Vallika kuttikara and required to be written below the chain of quotients, is in this rule always 1, the last divisor being 1. Therefore the last divisor here takes the place of the mati in the Vallika-kuttikara. It will be seen further that the last figure of the chain obtained according to this rule, i.e., agra, is the same as the last figure in the chain obtained in the Vallikd-kettihara by dividing by the last divisor the sum of the agra and the product of the mati as multiplied by the last remainder.

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CHAPTER VI-MIXED PROBLEMS 129 as before ; and below this again is to be put down the number which is obtained by adding the difference between the two group-values, (already referred to, to the product of the least remainder in the odd position of order multiplied by the above optionally chosen multiplier thereof, and then hy dividing this resulting sum by the last divisor in the above division chain. Thus the cree per-like chain of figures required for the solution of this latter kind of problem is obtained. This chain is to be dealt with as before from below upwards, and the resulting number is to be divided as before by the first divisor in this last division chain. The remainder obtained in this operation is then to he) multiplied by the divisor (related to the larger group-value); and to the resulting product this larger groap-value is to be added. (Thus the value of the required multiplier of the given group number is obtained so as to satisfy the two specified distributions taken into consideration.) Examples in illustration thereof. 1371. One hundred and seventy-seven (is the dividend-coefficient of the unknown factor), 240 is the known quantity associated with the product so as to be added to or subtracted from it); the whole is divided by 201 (and leaves no remainder). What is the (unknown factor here (with which the given dividend. coefficient is to be multiplied)? 1381. Thirty-fivo and other quantities, 16 in number, rising (thence successively in value) by 3, (are the given dividend-coefficients). The given divisors are 32 (and others) as successively increased by 2. And 1 successively increased by 3 gives rise to the associated known (positive and negative) quantities. What are the valuos of the (unknown) factors (of the known dividend. coefficients), according as they are additively associated with positive or negative (known) numbers ? 17 For Private and Personal Use Only

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130 GANITASARASANGRAHA. The rule for separating the prices of (an interchangeable) larger and (a similar) smaller number of two different things from the given mixed sums of the prices of these things : 1397. From the higher price-sum, as multiplied by the corresponding larger number of one of the two kinds of things, subtract the lower price-number as multiplied by the smaller number relating to the other of the two kinds of things. Then divide the result by the difference between the squares of the numbers relating to these things. This gives rise to the price of the thing which is larger in number. The other, that is, the price of the thing which is smaller in number, is obtained by interchanging the multipliers. An example in illustration thereof. 1402 to 1421. The mixed price of 9 citrons and 7 fragrant woud-apples is 107; again the mixed price of 7 citrons and 9 fragrant wood-apples is 101, O you arithmetician, tell me quickly the price of a citron and of a wood-apple here, having distinctly separated those prices well. The rule for separating the prices and the numbers of differont mixed quantities of different kinds of things from their given mixed price and given mixed values : 143). The different) given (mixed) quantities (of the different things) are to be multiplied by an optionally chosen uumber; the given (mixed price of these mixed quantities) is to be diminished (by the value of these products separately). The resulting quantities 1391. Algebraically, if ax + by m and bac + ay = n, then a ae + aby = am and ba9s + aby = bn. .. (al-ba)=am - bn. um - bn a -02 143. The rule will become clear by the following working of the problem in stanzas 1447 and 145:The total number of fruits in the first homp is 21. Do. do. second do. 22. Do. do. third do. 23. For Private and Personal Use Only

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CHAPTER VI- MIXED PROBLEMS. 131 are to be divided (one after another) by an optionally chosen number (and the remainders again are to be divided by an optionally chosen number, this process being repeated) over and over again. The given (mixed) quantities of the different things are to be (successively) diminished by the corresponding quotients in the above process. (In this manner the numerical values of the various things in the mixed sums are arrived at). The optionally chosen divisors (in the above processes of continued division) combined with the optionally chosen multiplier as also that multiplier constitute (respectively) the prices of a single thing in each of the varieties of the given different things). Choose any optional number, say 2, and multiply with it these total numbers; we get 42, 44, 46. Subtract these from 73, the price of the respectiva heaps. The remainders are 31, 29, and 27. These are to be divided by another optionally chosen number, say 8. The quotients are 3, 3, 3, and the remainders are 7,5 and 3. These remainders are again divided by a third optionally chosen number say 2. The quotients are 3, 2, 1, and the remainders are 1, 1, 1. These last remainders are in their turn divided by a fourth optionally chosen number which is 1 here. The quotients are 1, 1, 1 with no remainders. The quotients derived in relation to the first total number are to be subtracted from it. Thus we got 21-(3 + 3 + 1) = 14; this number and the quotients 3, 3, 1 represent the number of fruits of the different sorts in the first heap. Similarly we get in the second group 16, 3, 2, 1, and in the third group 18, 3, 1, 1, as the number of the different sorts of fruits. The prices are the first chosen multiplier, viz., 2, and its sums with tbe other optionally cbogen multipliere. Thus we get 2, 2 + 8 or 10, 2 + 2 or 4. and 2 + 1 or 3, as the price of each of the four different kinds of fruits in order. The principle underlying this : ethod will be clear from the following algebraical representation : ad + by + cx + dw = p ... a + b + c + d = 4 ... ... ... JI Let w=8. Multiplying II by s, wo have 8 (a + b + c+d) = on ... Subtrastiny 111 from I, we get a (x -) + b (y-s) + c(2-)= p-on ... ... ... ... ... ... ... ... ... IV Dividing IV by *-*, wo get a as the quotient, and b (y-s) + c(-) as the remainder, where x - s is a suitable integer. Similarly we proceed till the end. Thus it will be seen that the successively chosen divisors - 8, y , and -, when combined with 8, give the value of the various prices, 8 by itself being the price of the first thing; and that the successive quotients a, b, c, along with n - (a + b + c) are the numbers measuring the various kinds of things. It may be noted that, in this rule, the number of divisions to be carried out is one less than the number of the kinds of things given, and that there should bo no remainder left in the last division. For Private and Personal Use Only

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132 GANITASARASANGRAHA. An example in illustration thereof. 1441 and 145]. There are here fragrant citrons, plantains, wood-apples and pomegranates mixed up (in three heaps). The number of fruits in the first (heap) is 21, in the second 22, and in the third 23. "The combined price of each of these Theaps) is 73. What is the number of the (various) fruits (in each of the heaps), and what the price (of the different varieties of fruits) ? The rule for arriving at the numerical value of the prices of dearer and cheaper things (respectively) from the given mixed value (of their total price) : 1461. Divide (the rate-quantities of the given things) by their rate-prices. Diminish (these resulting quantities separately) by the least among them. Then multiply by the least of the abovementioned quotient-quantities) the given mixed price of all the things; and subtract (this product) from the given (total number of the various) things. Then split up (this remainder optionally) into as many (bits as there are remainders of the above quotientquantitios left after subtraction); and then divide (these bits by those remainders of the quotient-quantities. Thus the prices of the various cheaper things are arrived at). These, separated from the total price, give rise to the price of the dearest article of purchase. Examples in illustrution thereof. 1471 to 149. "In accordance with the rates of 3 peacocks for 2 panas, 4 pigeons for 3 panas, 5 swans for 4 panas, and 6 sarasa 146). The rule will be clear from the following working of the problem given in 1471149: Divide the rate-quantities 3, 4, 5, 6 by the respective rato-prices 2, 4, 4, 5; thus we have 1, , , . Subtract the least of these from each of the other throe. We get to, 1, . By multiplying the given mixed price, 56, by the abovementioned least quantity, we have 56 x. Subtract this from the total number of birds, 72. Split up the remainder 4 into any three parts, say }, 6, Dividing these respectively by 18, is, zo we get the prices of the first three kinds of birds, 14, 12, 36. The price of the fourth variety of birds can be found out by subtracting all these three prices from the total 56. For Private and Personal Use Only

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CHAPTER VIMIXED PROBLEMS. 133 birds for 5 panas, purchase, O friend, for 56 panas 72 birds and bring them (to me)". So saying a man gave over the purchasemoney (to his friend). Calculate quickly and find out how many birds (of each variety he bought) for how many panas. 150. For 3 panas, 5 palas of ginger are obtained ; for 4 panas, 11 palas of long pepper; and for 8 panas, 1 pala of pepper is obtained By means of the purchase-money of 60 panas, quickly obtain 68 palas (of these drugs). The rule for arriving at the desired numerical value of certain specified objects purchased at desired rates for desired sums of money as their total price : 151. The rate-values of the various things purchased are each separately) multiplied by the total value (of the purchase-money), and the various values of the rate-money are (alike separately) 151. The following working of the problem given in 152-153 will illustrate the rule : 5 7 9 3 3 5 7 9 Write down the rate-things and the rate-prices in 500 700 900 300 two rows, one below the other. Multiply by the total 300 500 700 900 price and by the total number of things respectively. Then subtract. Remove the common factor 100. Multi0 0 0 600 200 200 200 0 ply by the chosen numbers 3, 4, 5, 6. Add the numbers in each horizontal row and remove the common factor 6. 0 0 0 6 Change the position of these figures, and write down in two rows each figure as many times as there are compo. nent elements in the corresponding sumn changed in position, 0 0 0 36 6 8 10 0 Multiply the two rows by the rate-prices and the ratethings respectively. Then remove the common factor 6. Multiply by the already chosen numbers 3, 4, 5, 6. The numbers in the two rows represent the proportions according to which the total price and the total number of things become distributed." 6 6 6 4 CD 30 42 42 54 36 12 This rule relates to a problem in indeterminate equations, and as such, there may be many sets of answers, these answers obviously depending upon the quantities chosen optionally as multipliers. It can be easily seen that, only when certain sets of numbers are chosen as optional multipliers, integral answers are obtained ; in other cases, fractional answers are obtained, which are of course not wanted. For an explanation of the rationale of the proces, see the note given at the end of the chapter. e 5 7 9 2 9 20 35 36 For Private and Personal Use Only

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184 GANITASARASANGRAHA. multiplied by the total number of things purchased ; (the latter products are subtracted in order from the former products; the positive remainders are all written down in a line below, the negative remainders in a line above; and all these are reduced to their lowest terms by the removal of the factors which are common to all of them. Then each of these reduced) differences is multiplied by (a separate) optionally chosen quantity ; (then those products which are in a line below as well as those which are so above are separately added together); and the sums are written upside down, (the sum of the lower row of numbers being written above and the sum of the upper row being written below. These sums are also reduced to the lowest terms by means of the removal of common factors, if any; and the resulting quantities are each of them written down twice, (80 as to make one be below the other, as often as there are component elements in the corresponding alternate sum. These numbers thus arranged in two rows) are multiplied by their respective rate-prices and rate-values of things, (the rate-price multiplication being conducted with one row of figures and the rate-number multiplication being in relation to the other row of figures. The products so obtained are again reduced to their lowest terms by the removal of such factors as are common to all of them. The resulting figures in each vertical row are separately) multiplied (each) by means of its corresponding originally chosen) optional multiplier. (And the products should be written down as before in two horizontal rows. The numbers in the upper row of products give the proportion in which the purchase money is distributed ; those in the lower row of products give the proportion in which the corresponding things purchased are distributed. Therefore) what remains thereafter is only the operation of praksepaka-karana (proportionate distribution in accordance with rule-of-three). An example in illustration thereof. 152 and 153. Pigeons are sold at the rate of 5 for 3 (panas), sarasa birds at the rate of 7 for 5 (panas), swans at the rate of 9 for 7 (pan...), and peacocks at the rate of 3 for 9 (panas). A certain man For Private and Personal Use Only

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CHAPTER VIMIXED PROBLEMS. 135 was told to bring at these rates 100 birds for 100 panas for the amusement of the king's son, and was sent to do so. What (amount) does he give for each of the various kinds of birds that he buys)? The rule for arriving at the measure of two given commodities whose prices are interchanged: 154. Let (the numerical value of) the sum of the (total selling and buying) money-prices (of the two given commodities) be divided by the numerical measure of) the sum (of the commodities put together); then let the difference between the above-mentioned buying and selling prices) be divided by the (numerical measure of any such difference as may be obtained by subtracting any optionally chosen commodity-quantity from the given measure of the sum of the given commodities. If the operation of sankramana is conducted in relation to these, (viz., the quotient obtained in the first operation above and any one of the many quotients that may be obtained in the second operation), the rates at which those commodities are purchased is obtained. Then if the same operation of sankramana as relating to the sum of the commodities and to their difference is carried out, it of course gives rise to the numerical measure of) the commodities (in question). The alternation of these above-mentioned purchase-rates) gives rise to the sale-rates. This is the solution of this kind of) problems as propounded by the learned ; and the rule (itself) has been declared by the great Jina. 17 154. The algebraical representation of the method described in the rule may be given thus in relation to the problem proposed in stanzas 155 and 156 Let ax + by = 104 ... ... ... ... ... ... 1 ay + bx=116 .. . II a+b= 20 ... ... II Adding I and II, we have (a + b) (x + y)= 220 .. +y=11 ... ... .. V Again subtracting I from II, we get (a - b) (y ) = 12 Now 26 is optionally chosen to be equal to 6. a+b-20 or a b =20-6 or 14 ... ... ... . VI ..y - = 12 ... ... ... ... ... ... VII Carry out the operation of sankramana with reference to VII and V, and VI and III; and the values of ae, y, a and b are all made out. For Private and Personal Use Only

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136 GANITASARASANGRAHA. An example in illustration thereof. 155 and 156. The original price of one piece of sandalwood and one piece of agaru wood, they being together 20 palas (in weight), is 104 panas; when after a time they were sold with their prices mutually altered, 116 panas were obtained. You give out their buying and selling rates and the numerical measure of the commodities, taking 6 and 8 separately as the optional (number) needed by the rule. The rule for arriving at the distance in yojanas travelled by the horses of the sun's chariot when yoked as desired : 157. The number representing the total yojanas, divided by the total number of horses, gives the yojanas (which each has at a stage to travel) in turn These yojanas, as multiplied by the optionally chosen number of horses to be yoked, give the measure of the distance to be travelled over by each horse. An ecample in illustration thereof. 158. It is well-known that the horses belonging to the sun's chariot are 7. Four horses (have to) drag it along, being harnessed to the yoke. They have to do a journey of 70 yojanas. How many times are they unyoked and how many times yoked (again) in four ? The rule for arriving at the value of the commodity to be found in the hands of each ( of a body of joint proprietors), from the conjoint remainder left after subtracting whatever is desired from the total value of all the commodities : 159. Let the sum (of the values of the conjoint remainders) of the commodities be divided by the number of men lessened by one; the quotient will be the total value of all the commodities (owned in common). This total value as diminished by the specified values gives in the corresponding cases) the value of commodity in the Lands (of each of the proprietors in turn). An example in illustration thereof. 160 to 162. Four merchants who had invested their money in common were asked each separately by the customs officer what the value of the commodity (they were dealing in) was; and one For Private and Personal Use Only

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CHAPTER VI --MIXED PROBLEMS. 137 eminent merchant (among them), deducting his own investment, said that that (value) was in fact 22. Then another said that it was 23; then another said 24; and the fourth said that it was 27; (in saying so) each of them deducted his own invested amount (from the total value of the commodity for sale). O friend, tell me separately the valae of the (share in the) commodity owned by each. The rule for arriving at equal amounts of wealth, (as owned in precious gems, after mutually exchanging any desired number of gems : 163. The number of gems to be given away is multiplied by the totul number of men (taking part in the exchange). This product is (separately) subtracted from the number (of the gems) for sale (owned by each); the continued product of the remainders (80 obtained) gives rise to the value of the gem (in each case), provided the remainder relating to it is given up (in obtaining such a product). Examples in illustration thereof. 164. The first man had 6 azure-blue gems (of equal value), tho second man had 7 (similar) emeralds, and the other--the third man--had 8 (similar) diamonds. Each (of them), on giving to each of the others) the value of a single gem (owned by himself), became equal (in wealth-value to the others. What is the value of a gem of each variety ?) 165 and 166. The first man has 16 azure-blue gems, the gecond has 10 emeralds, and the third man has 8 diamonds. Each among them gives to each of the others 2 gems of the kind owned by himself; and then all three men come to be possessed of equal 163. Let m, n, p, be respectively the numbers of the three kinds of gems owned by three different persons, and a the number of gems mutually exchanged and let x, y, z, be the value in order of a single gem iu the three varieties concerned. Then it may be easily found out as required that = (n --- 3a) (p - 3a); y = m-3a) (p -- Sa); = (m - 3a) (n - 3a); 18 For Private and Personal Use Only

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188 JANITASARASANGRAHA. wealth. Of what nature are the prices of those azure-blue gems, emeralds, and diamonds ? The rule for arriving at the (value of the) invested capital by means of the rate of purchase, the rate of sale, and the profit obtained : 167. The buying and the selling rate-measures of the commodity are each multiplied alternately by the rate-prices; (the product obtained with the help of) the buying rate-ineasure is divided by (the other product obtained with the aid of) the selling rate-measure. The profit, divided by the resulting quotient as diminished by one, gives rise to the originally employed capital amount An example in illustration thereof. 168. A merchant buys at the rate of 7 prasthas of grain for 3 panas, and sells it at the rate of 9 prasthas for 5 panas, and makes a profit of 72 panas. What is the capital employed in this transaction ? Thus ends Sakala-kuttikara in the chapter on mixed problems. Suvarna-kuttikara, Hereafter wg shall explain that kuttikara which consists of oaloulations relating to gold.. The rule for arriving at the varna of the resulting mixed gold obtained by putting together (different component varieties of) gold of (various) desired varnas : - 169. It has to be known that the sum of the various products of the various component quantities of) gold as multiplied by (their respective) narnas, when divided by the total quantity of) 187. If the buying rate is a things for b and the selling rate is c things for d. and if m is the gain by the transaction, then the capital invested is ad -1) m+ (be For Private and Personal Use Only

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CHAPTER VI-MIXED PROBLEMS. Acharya Shri Kailassagarsuri Gyanmandir 139 the mixed gold gives rise to the (resulting) varna. (The original varna of any component part thereof), when divided by the latter resulting varna (of the mixed up whole), and multiplied by the (given) quantity of gold (in that component part), gives rise to (that) corresponding quantity of (the mixed) gold (which is equal in value to that same component part thereof). An example in illustration thereof. 170 to 171. There are 1 part (of gold) of 1 varna, 1 part of 2 varnas, 1 part of 3 varnas, 2 parts of 4 varnas, 4 parts of 5 varnas, 7 parts of 14 varnas, and 8 parts of 15 varnus. Throwing these into the fire, make them all into one (mass), and then (say) what the varna of the mixed gold is. This mixed gold is distributed among the owners of the foregoing parts. What does each of them get? The rule for arriving at the required weight of gold (of any desired varna equivalent in value to given quantities of gold) of given varnas: 172. The given quantities of gold are all (separately) multiplied by their respective varnas, and the products are added. The resulting sum is divided by the total weight of the mixed gold; the quotient is to be understood as the resulting average varna. This (above-mentioned sum of the products) is separately divided by the desired varnas (to arrive at the required equivalent weight of this gold). Examples in illustration thereof. 173. Twenty panas (in weight of gold) of 16 varnas have been exchanged for (gold of) 10 varnas in quality; you give out how many puranas (in weight) they become now. 1744. One hundred and eight. (in weight of) gold of 11 varnas is exchanged for (gold of) 14 varnas. What is the (equivalent quantity of this new) gold? For Private and Personal Use Only The rule for finding out the unknown varna : 1751. From the product obtained by multiplying the total quantity of gold by the resulting varna of the mixture, the sum of

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140 GANITASARASANGRAHA. the products obtained by multiplying the several component) quantities of gold by (their respective varnas) is to be subtraoted. The remainder, when divided by the known component quantity of gold, (the varna of which is to be found out), gives rise to the required rarna; and when divided by the difference between the resulting varna and the known varna (of an unknown component quantity of gold) gives rise to the (required weight of that) gold. Another rule in relation to the unknown varna : 1764. The sum of the products of the (various component quantities of) gold as multiplied by their respective varnas is to be subtracted from the product of the total quantity of gold as multiplied by the resulting varna. Wise people say that this remainder when divided by the weight of the gold of the unknown varna gives rise to the required varna. Examples in illustration thereof. 1774 and 178. With gold of 6, 4 and 3 (in weight),characterised respectively by 13, 8 and 6 as their varnas, 5 in weight of gold of an unknown varna is mixed. The resulting varna of the mixed gold is 11. O you, friend, who know the secrets of calculation, tell me the numerical value of this unknown varna. 179. Seven in weight (of a given specimen) of gold has exactly 14 as the measure of its varna; then 4 in weight (of another specimen of gold) is added to it. The resulting varna is 10. Give out the unknown varna (of this second specimen of gold). The rule for arriving at the unknown weight of gold : 180. Subtraot the sum, obtained by adding together the products of the various component quantities of) gold as multiplied by their respective varmas, from the product of the sum of the known weights of gold as multiplied by the now durable resulting varna ; the remainder divided by the difference between the (known) varna of the unknown quantity of gold and the resulting durable varna (of the mixed gold) gives rise to the (weight of) gold. For Private and Personal Use Only

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OKAPTER VI-MIXED PROBLEMS. 141 An example in illustration thereof. 181. Threo pieces of gold, of 3 each in weight, and of 2, 3, and 4 varnas (respectively), are added to an unknown weight of) gold of 13 varnas. The resulting varna comes to be 10. Tell me, O friend, the measure (of the unknown weight) of gold. The rule for arriving at (the weights of) gold (corresponding to two given varnas) from the known weight and varna of) the mixture of two (given specimens of) gold of (given) varnas : 182. Obtain the differences between the resulting varna (of the mixture on the one hand) and the known higher and lower varnas (of the unknown component quantities of gold on the other hand); divide one by these differences (in order); then carry out as before the operation of praksepaka (or proportionate distribution with the aid of these various quotients). In this manner it is possible to arrive oven at the value of many component quantities of gold also. Again, the rule for arriving at (the weights of) gold (corresponding to two given varnas) from (the known weight and varna of) the mixture of two (given specimens of) gold of (given) varnas :-- 183. Write down in inverse order the difference between tho resulting varna and the higher (of the two given varnas of the two oomponent quantities of gold), and also the difference between the resulting varna and the lower (of the two given varnas). The result arrived at by means of the operation of proportionate distri. bution (carried out with the aid of these inversely arranged differences),--that (result) gives the required (weights of the component quantities of) gold. An example in illustration thereof. 184. If gold of 10 varnas, on being combined with gold of 16 varnas, produces as result 100 in weight of gold of 12 varnas, give out separately (the measures in weight of the two different varieties of gold. For Private and Personal Use Only

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142 GANITASARASANGRAHA. The rule for arriving at the weights of) many component quantities of) gold (of known varnas in a mixture of known varna and weight): 185. (In relation to all the known component varnas) excepting one of them, optionally chosen weights may be adopted. Then what remains should be worked out as in relation to the previously given cases by means of the rule bearing upon the determination of an) unknown weight of gold. An example in illustration thereof. 186. The (given) varnas (of the component quantities of gold) are 5, 6, 7, 8, 11, and 13 (respectively); and the resulting varna is in fact 9; and if (the total) weight (of all the component quantities of gold be 60, what may be the several measures in weight of the various component quantities) of gold ? The rule for arriving at the unknown varnas of two (known quantities of gold when the resulting varna of the mixture is known) : 187. Divide one (separately) by the two (given weights of) gold ; multiply (separately each of the quotients thus obtained) by (the weight of) the (corresponding quantity of) gold and (also) by the resulting) varna; write down (both the products so obtained) in two different places ; (each of these in each of the two sets,) if diminished and increased alternately by one as divided by (the 185. The rule referred to here is found in stanza 180 above. 187. Tho ralo will become clear by the following working of the problema in stanza 188:-- ex 16 1 and 10 * 10 il are written down in two places thus: 11 11 11 Then and are added and mbtracted alternately in each of the two gets thus : 11 + 16 } and 11 * These give the two sets of answers. 1 (11 + For Private and Personal Use Only

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CHAPTER VIMIXED PROBLEMS. 143 known weight of) the corresponding (variety of) gold, gives rise as a matter of course, to the required varnas. . An example in illustration thereof. 188. If, the component) varnas not being known, the resulting varna obtained by means of two different kinds of) gold weighing 16 and 10 respectively happens to be 11, what would be the (respective) varnas of those two (different kinds of) gold ? Again, the rule for arriving at the unknown varnas of two (known quantities of gold, when the resulting varna of the mixture is known) : 189. Choose an optional varna in relation to one (of the two given quantities of gold); what remains (to be found out) may then be arrived at as before. In relation to (the known quantities of all) the numerous varieties of gold excepting one, the varnas are optional; then (proceed) as before. An example in illustration thereof. 190. On fusing together (two different kinds of) gold which are 12 and 14 (respectively in weight), the resulting varna is made out to be 10. Think out and say (what) the varnas of those two (kinds of gold are). An example to illustrate the latter half of the rule. 191. On fusing together 7, 9, 3, and 10 (in weight respectively of four different kinds of gold, the resulting mixture turns out to be (gold of) 12 varnas. Give out the varnas (of the various component kinds of gold) separately. The rule regarding how to arrive at (an estimate of the value of) the test sticks (of gold) : 192. The varna of every stick is to be separately divided by the (given) maximum varna, and (the quotients 90. obtained) are (all) to be added together. The resulting sum gives (the measure of) the required quantity of (pure) gold. From the summed up For Private and Personal Use Only

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144 GANITASARASANGRAHA. (weight of all the sticks, this is to be subtracted. What remains is (the quantity of) the prapuranika (that is, the quantity of the baser metal mixed). An example in illustration thereof. 193--1967. (Three) merchants, well acquainted with the varna of gold, were desirous of making test sticks of gold, and produced (such) golden sticks. The gold of the first (merchant) was of 12 varnas; (that of the second was of) 14 varnas; and that of the third was of 16 varnas. The (various specimens of the test sticks of) gold in the case of the first (merchant) were (regularly) less by 1 in varno); those of the second were less by land ; and those of the third were (in regular order) less by (The specimens of test gold) possessed by the first (merchant) began with that of (his) maximum varna and ended with that of 1 varna ; (similarly, those of the second began with that of his maximum varna and ended with that of 2 varnas; and those of the third merchant (began with that of his maximum varna and) ended with that of 3 varnas. Every test stick is 1 masa in weight. O mathematician, if you indeed know gold calculation, tell me separately and soon what the measure of pure gold here is, and what that of the baser metal mixed. The rule for arriving at the different weights of) gold obtained in exchange and characterised by two given) varnas : 1971. The two differences between, (firstly,) the product of the (given weight of) gold to be exchanged as multiplied by the (given) varna (thereof) and the product of the weight of gold obtained in exchange as multiplied by the (first of the two specified) varnas (of the exchanged gold)--(and, secondly, between the first product above-mentioned and the product of the weight of 197). This role will be clear from the following working of the problem given in stanza 198) : 700 x 16 - 1008 x 10, and 1008 x 12-700 x 16 are altered in position and written down as 896 and 1120; and these, when divided by 12-10 or 2, give rise to the answers, namely, 448 and 560 in weight of gold of 10 and 12 varras respectively For Private and Personal Use Only

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CHAPTER VIMIXED PROBLEMS. 146 gold obtained in exchange as multiplied by the second of the specified vurnas of the exchanged gold--these two differences) have to be written down. If then, they are altered in position and divided hy the difference between the two specified) varnas (of rieties) of the exchanged gold, the result happens to the two required) quantities of the two kinds) of gold (obtainod in exchange). An example in illustration thereof. 1983. Seven hundred in weight of gold characterised by 16 varnas produces, on being exchanged, 1,008 (in weight) of two kinds of gold characterised (respectively) by 12 and 10 varnas. Now, what is the weight (of each of these two varieties) of gold? The rule for finding out the (various weights of) gold obtained as the result of many (specified) kinds of exchange : 1992. If the (givon) weight of gold (to be exchanged) as multiplied by the varna (thereof) is divided by (the quantity of) the desired gold (obtained in oxchange), there arises the uniform averagu varna. On carrying out (further) operations as mentioned before, the result arrived at gives the required weights of the various kinds of gold obtained in exchange. An example in illustration thereof. 2002-201. In the case of a man exchanging 300 in weight of gold characterised by 14 varnas, the gold (obtained in exchange) is seen to be altogether 500 in weight, (the various parts whereof are respectively) characterised by 12, 10, 8 and 7 varnas. What is the weight of gold separately corresponding to each of these (different) varnas ? The rule for arriving at (the various weights of) gold obtained in exchange which are characterised by known varnas and are (definite) multiples in proportion : 202-203. The sum of the (giveni) proportional multiple numbers is to be divided by the sum of the products (obtained) by 1994. The operation which is stated here as having been mentioned before is what is given in stanza 185 above. 19 For Private and Personal Use Only

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146 GANITASARASANGRAHA, multiplying the (given proportional quantities of the various kinds of the exchanged) gold by (their respective specified) varnas. (The resulting quotient) is to be multiplied by the original varna (of the gold to be exchanged). If by this proluct as diminished by one, the increase in the weight of gold due to exchange) is divided, and the quotient (so obtained) is subtracted from the original wealth of gold, the remaining (weight of unexchanged) gold is arrived at. This (weight of the unexchanged gold) is then to be subtracted from the sum of the weight of the original gold and the increase (in weight due to exchange). Then if the resulting remainder (here) is divided by the sum of the proportional multiple numbers connected with the exchange, and is then multiplied by (each of those) proportional numbers (separately), the (various weights of) gold obtained in exchange and characterised by the specified varnas and the specified proportions are arrived at. An example in illustration thereof. 204--205. There is a certain merchant desirous of obtaining profit; and the gold (in his possession) is of 16 varnas and 200 in weight. A. portion of it is exchanged in return for (four different kinds of) gold characterised respectively by 12, 8, 9 and 10 varsi18, (so that those varieties of gold are by weight) in proportions which begin with 1 and are then (regularly) multiplied by 2. The gain (in the weight of gold resulting out of this oxchange transaction) is 102. What is the remaining (weight of the unexchanged) gold ? Tell me also the weights of gold obtained in exchange corresponding to those (above-mentioned varnas). The rule for arriving at (the weight of) the original (quantity of) gold with the aid of the gold exchanged (in part), and with the aid (of the weight) of gold seen to be in excess (in consequence of the exchange : 206. Each specified part of the original) gold (to be exchanged) is divided by the varna corresponding to its exchange. (The resulting quotient is in each case to be multiplied by the For Private and Personal Use Only

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CHAPTER VIMIXED PROBLEMS. 147 optionally chosen varma (of the originally given gold; and then all these products are to be added). From this sum, the sum of the (various) fractional (exchanged) parts (of the original gold) is to be subtracted. (If now the observed excess (in the weight of gold due to the exchange) is divided by this resulting remainder, what comes out here happens to be the original wealth of gold. An example in illustration thereof. 207-208. A certain small ball of gold of 16 varnas belonging to a merchant is taken ; and }, { and parts thereof are in order exchanged for (different kinds of) gold characterised (respectively) by 12, 10 and 9 varnas. (The weights of these exchanged varieties of gold are) added to what remains (unexchanged) of the original gold. Then 1,000 is observed to be in excess on removing from the account the weight of the original gold. What then is (the weight of this) original gold ? The rule for arriving at the desired varna with the help of the (mutual) gift of a desired fractional part of the gold (owned by the other), and also for arriving at the (weights of) gold (respectively) corresponding to those optionally gifted parts :-- 209 to 212. One divided by (the numerical measure of each of two specifically gifted) parts is to be noted down in reverse order; ard (if each of the quotients so obtained is) multiplied by an 209--212. The rule will be clear from the following working of the problem in 213-215: Dividing 1 ly and }, we get respectively 2, 3; altering their position and moltiplying them by any optionally chosen number, say 1, we get 3, 2. These two numbers represent the quantities of gold owned rompectively by the two merchants. Choosing 9 as the varna of the gold owned by the first werohant, we can easily arrive, from the exchange proposed by him, at 13 as the varna of the gold owned by the second merchant. These varnas, 9 and 13, give, in the exchange proposed by the second merchant, the average varna of , while the average varna as given in the sum has to be 12 or . Therefore the varnas 9 and 13 have to be altered. If 8 is chosen instead of 9, 13 has to be increnged to 16 in the first exchange. Using these two varnas, 8 and 16, in the second exchange, we obtain 49 as the average varna, instead of For Private and Personal Use Only

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148 GANITASARASANGRAHA. optionally chosen quantity, (itgives rise to the weights of each of the two small) balls of gold. The varna (of each) of these (little balls of gold) as also that of the gold gifted by the other person in the transaction) has to be arrived at as before with the aid of tho (given) final average varna (in each caso). If in this manner both sets of answers (arrived at) happen to tally (with the requirements of the problem), the two varnas arrived at in accordance with the previously adopted option become the verified varnas mentioned in relation to the two (given) little balls of gold. If, (however, these answers do) not (tally), the varnas belonging to the first set (of answers have to be made (as the case may be) a little less or a little more; (then the average varna corresponding to these modified component varnas has to be further obtained). Thereafter, the difference between this (average) varna and the previously obtained (untallying average) varna is written down ; (and the required proportionate quantities) are (therefrom) derived by means of the operation of the Rule of Three : and the varnas (arrived at according to the option chosen before, when respectively) diminished by one of these two quantities and increased by the other, turn out to be evidently the required varnas (here). An example in illustration thereof. 213--215. Two merchants well versed in estimating the value of gold asked cach other (for an exchange of gold). Then the first (of them) said to the other--"If you give me half (of your gold), I shall combine that small pellet of gold with my own gold and make the whole become gold of) 10 varnas." Then this other said-"If I only obtain your gold by one-third (thereof), I shall likewise make the whole (gold in my possession become Thus, in the second exchange, we see an increase of 40-35 or 5 in the sun of the products of weight and varna, while the decrease and the increase in relation to the originally chosen varnas are 9-8 or 1 and 10-13 or 3. But the required increase in the sum of the products of weight and varna in the second exchange is 36-35 or 1. Applying the Rule of Three, we get the corresponding decrease and increase in the varnas to be and Therefore, the varnas are 9- or 88 and 13 + { or 13%. For Private and Personal Use Only

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CHAPTER VI-MIXED PROBLEMS. 149 gold of) 12 varnas with the aid of the two pellets." ( you, who know the secret of calculation, if you possess cleverness in relation to calculations bearing upon gold, tell me quickly, after thinking out well, the measures of the quantities of gold possessed by both of them, and also of the varnas (of those quantities of gold). Thus ends Suvarna-kuttikara in the chapter on mixed problems Vicitra-kuttikara. Hereafter we shall expound the Vicitra-kuttikara in the chapter on mixed problems. The rule in regard to the ascertaining of the number of truthful and untruthful statements (in a situation like the one given below wherein both are simultaneously possible) :--- 216. The number of men, multiplied by the number of those liked (among them) as increased by one, and (then) diminished by twice the number of men liked, gives rise to the number of untruthful statements. The square of the number representing all the men, diminished by the number of those (untruthful statements), gives rise to the statements that are truthful. 216. The rationale of this rule will be clear from the following algebraical representation of the problem given in stanza 217 below: Let a be the total number of persons of whom b are liked. The number of utterances is a, and each statement refers to a persons. Hence the total number of statements is a x a or a. Now, of those a persona, bore liked, and a - b are not liked. When each of the b number of persons is told-" You alone are liked," the number of untruthful statements in each case is b - 1. Therefore, the total number of untruthful statements in b statements is b (b -- 1) . . . . 1. When, again, the same statouent is made to each of the a b persons, the number of untruthinl statements in each case is b + 1. Therefore, the total number of untruthful statements in a --butterances is (a - b)(b + 1) II. Adding I and 11, we get b (b) -- 1) + (a - b)(b + 1) = a (b + 1) -- 2b. This represents the total of untruthful statements; and on sobtracting it from a", which is the measure of all the statements, truthful and untruthful, we arrive obviously at the measure of the truthful statements. For Private and Personal Use Only

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150 GANITASARASANGRAHA. An example in illustration thereof. 217. There are five lustful men. Among them three are in fact liked by a public woman. She says (separately) to each (of them) "I like you (alone)". How many (of her statements, explicit as well as implicit) are true ones? The rule regarding the (possible) varieties of combinations (among given things) : 218. Beginning with one and increasing by one, let the numbers going up to the given number of things be written down in regular order and in the inverse order (respectively) in an upper and a lower (horizontal) row. (If) the product (of one, two, three, or more of the numbers in the upper row) taken from right to left (be) divided by the corresponding) product (of one, two, three, or more of the numbers in the lower row) also taken from right to left, (the quantity required in each such case of combination is (obtained ag) the result. Examples in illustration thereof. 219. Tell me) now, o mathematician, the combination varieties as also the combination quantities of the tastes, viz., the astringent, the bitter, the sour, the pungent, and the sline, together with the sweet taste (as the sixth). 220. O friend, you (tell me quickly how many varieties there may be, owing to variation in combination, of a (single string) necklace made up of diamonds, sapphiros, emeralds, corals, and pearls. 221. O (my) friend, who know the principles of calculation, tell (me) how many varieties there may be, owing to variation in combination, of a garland made up of tho (following) flowersketaki, asoka, campaka, and nalotpala. 218. This rule relates to a problem in combination. The formula viven n. (n-1). ( 2) . .. (14--T + 1). and this is obviously equal 1. 2. 3. . . . here is n. (n-1). (na Ton For Private and Personal Use Only

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CHAPTER VI-MIXED PROBLENS. 151 The rule to arrive at the unknown) capital with the aid of certain known and unknown profits (in a given transaction) : 222. By means of the operation of proportionate distribution, the (unknown) profits are to be determined from the mixed sum (of all the profits) minus the (known) profit. Then the capital of the person whose investment is unknown results from dividing his profit by that (same common factor which has been used in the process of proportionate distribution above). An example in illustration thereof. 223-225. According to agreement some three merchants carried out (the operation of) buying and selling. The capital of the first (of them) consisted of six puranas, that of the second of eight puranas, but that of the third was not known. The profit obtained by all those (three) men was 96 puranas. In fact the profit obtained by him (this third person) on the unknown capital happened to be 40 puranas. What is the amount thrown by him (into the transaction), and what is the profit (of each) of the other two merchants ? O friend, if you know the operation of proportionate distribution, tell (me this) after making the (necessary) calculation. The rule for arriving at the wages (due in kind for having carried certain given things over a part of the stipulated distance according to a given rate) : 226. From the square of the product (of the numerical value) of the weight to be carried and half of the (stipulated distance 226. Algebraically, the formula given in the rule is : aDV (D) - abd (D-d) where = -N O ) - abd (D-d) where wages to be found out, D-d a=the total weight to be carried, D=the total distance, d the distance gone over, and b the total wages promised. It may be noted here that the rate of the wages for the two stages of the journey is the same, although the amount paid for each stage of the journey is not in accordance with the promised rate for the whole journey. The formula is easily derived from the allowing equation containing the data in the problein : (-) (D-d). For Private and Personal Use Only

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152 GANITASARASANGRAHA. measured in) yojana, subtract the continued) product of (the numerical value of) the weight to be carried, (that of the stipulated) wages, the distance already gone over, and the distance still to be gone over. Then, if the fraction (viz, half of the weight to be carried over, as multiplied by the (whole of the stipulated) distance, and then as diminished by the square root of this (difference above mentioned), be divided by the distance still to be gone over, the required answer is arrived at. An exumple in illustration thereof. 227. Here is a man who is to receive, by carrying 2 jackfruits over yojana, 7% of them as waves. Ho breaks down at half the distance. What amount within the stipulated wages) is (then) due to him ? The rule for arriving at the distances in yopanas (to be travelled over) by the second or the third weight-carrier after the first or the second of them breaks down) 228. From the product of the (whole) weight to be carried as multiplied by tho (value of the stipulated) wages, subtract the square of the wages given to the first carrier. This (difference has to be used as the divisor in relation to the continued) product of the difference between the (stipulated) wages (and the wages already given away), the whole weight to be carried, and the whole) distance (over which the weight has to be carried. The resulting quotient gives rise to the distance to be travelled over by the second (person). An example in illustration thereof. 229. A man by carrying 24 jack-fruits over (a distance of) five yojanas has to obtain 9 (of them) as wages therofor. Warn 6 of these have been given away as wages to the first carrier), what is the distance the second carries has to travel over (to obtain the remainder of the stipulated wages)? 228. Algebraically D-d= (-3) up, which can beeasily found out from the equation in the last note. For Private and Personal Use Only

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CHAPTER VI-MIXED PROBLEMS. 153 The rule for arriving at the value of) the wages corresponding to the various stages (over which varying numbers of persons carry a given weight) : 230. The distance (travelled over by the various numbers of men) are (respectively) to be divided by the numbers of the men that are (doing the work of carrying) there. The quotients (80 obtained) have to be combined so that the first of them is taken at first separately and then) has (1, 2, and 3, etc., of) the following (quotients) added to it. (These quantities so resulting are to be respectively) multiplied by the numbers of the men that turn away (from the journey at the various stages. Then) by adopting (in relation to these resulting products the process of proportionate distribution (praksepaka), the wages (due to the men leaving at the different stages) may be found out. An example in illustration thereof. 231-232. Twenty men have to carry a palanquin over (a distance of) 2 yojanas, and 720 dinaras form their wages. Two men stop away after going over two krosas; after going over two (more) krasas, three others (stop away); after going over half of the remaining distance, five men stop away. What wages do they (the various bearers) obtain ? The rule for arriving at (the value of the money contents of) a purse which (when added to what is on hand with each of certain persons) becomes a specified multiple (of the sum of what is on hand with the others): - 233-235. The quantities obtained by adding one to each of the specified) multiple numbers in the problem, and then) 233-235. In the problem given in 236--237, let a, y, z represent the money on hand with the three merchants, and u the money in the purse. Then + s = a (y + 2) where a, b, c represent the multiples where u+y=b (2 + x)} 1 given in the problem. u +2 = (x + y)) Now u + 2+ y +2= (a + 1) (y + x) = (6+1) (+ 2) = (c+1) (x + y). For Private and Personal Use Only

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154 GANITASARASANGRAHA. multiplying these sums with each other, giving up in each case the sum relating to the particular specified multiple, are to be reduced to their lowest terms by the removal of common factors. (These reduced quantities are then) to be added. (Thereafter) the square root (of this resulting sum) is to be obtained, from which one is (to be subsequently) subtracted. Then the reduced quantities referred to above are to be multiplied by (this) square root as diminished by one. Then these are to be separately subtracted from the sum of those same reduced quantities. Thus the moneys on hand with each (of the several persons) are arrived at. These (quantities measuring the moneys on hand) have to be added to one another, excluding from the addition in each case the value of the money on the hand of one of the persons; and the several sums so obtained are to be written down separately. These are (then and T Then (a + 1) (6+1) (c+1) X (y+e) = (6+1) (0+1)...... 1. AT where T=u+*+ y + #. Bimilarly, at le (a + 1) (6+1) (C+1) + x (x+x) = (c+ 1) (a +1) ... II. T (a+1) (+1) (C+1) * (x+y) = (a + 1) (0+1)... III. Adding I, II and III, (a + 1) (b + 1) (c+1) 1 x 2 (a + +z) = (h+ 1) (c+1) + (c + 1) (a + 1) + (a + 1) (6+1) * S (say) .... ... ... IV Subtracting separately I, II, III, each multiplied by 2, from IV, we have(a + 1) (b + 1) (c + 1) x 24 = S- 2 (6 + 1) (c + 1), T (a + 1) (6+1 (c+1) x 2y =S - 2 (c+1) (a + 1), (@+1) (6+1) (c+1) x 2x =S - 2 fa +1) (0+1). T .:.*:y:%:: S-2 (6+1) (0 + 1): S-2 (c + 1)(a + 1): S-2 (a + 1) (0+1). By removing the common factors, if any, in the right-hand side of the proportion, we get at the smallest integral values of w, y, %. This proportion is given in the rule as the formula. It may be noted that the square root mentioned in the rule has reference only to the problem given in the stanzas 236-237. Correctly speaking, instead of "square root", we must have '3'. It oan be seen easily that this problem is possible only when the sum of 1 1 1 any two of 4 + 10 + 1 T is gr is greater than the third. For Private and Personal Use Only

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CHAPTER VI-MIXED PROBLEMS. 155 to be respectively) multiplied by (the specified) multiple quantities (mentioned above); from the several products 80 obtained the (already found out) values of the moneys on hand are to be separately subtracted). Then the same) value of the money in the purse is obtained (separately in relation to each of the several moneys on hand). An example in illustration thereof. 236-237. Three merchants saw (dropped) on the way a purse (containing money). One (of them) said (to the others), "If I secure this purse, I shall become twice as rich as both of you with your moneys on hand." Then the second (of them) said, "I shall become three times as rich." Then the other, the third), said, . "I shall become five times as rich." What is the value of the money in the purse, as also the money on hand (with each of the three merchants)? The rule to arrive at the value of the moneys on hand as also the money in the purse (when particular specified fractions of this latter, added respectively to the moneys on hand with each of a given number of persons, make their wealth become in each case) the same multiple (of the sum of what is on hand) with all (the others) : 238. The sum of (all the specified) fractions in the problem) -the denominator being ignored-is multiplied by the (specified common) multiple number. From this product, the products obtained by multiplying (each of the above-mentioned) fractional parts as reduced to a common denominator, whicb is then ignored), by the product of the number of cases of persons minus one and the specified multiple number, this last product being diminished 238. The formula given in the rule is-- # m (a + b + c)-a (2m-1), where at, y, > are the moneys on hand, m. yam (a+b+c) -b (2m - 1), the common multiple, and a, b, c, the and >=m (a + b + c)-C (2m. -1), specified fractional parts given. These values can be easily found out from the following equations:-- Pa + = m (y + %), 7 P0+ y =m (z + a), where P is the money in the purse, and Pc+ = m ( + y) For Private and Personal Use Only

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Shri Mahavir Jain Aradhana Kendra 156 www.kobatirth.org GANITASARASANGRAHA. by one, are (severally) subtracted. The resulting remainders constitute the several values of the moneys on hand. The value of the money in the purse is obtained by carrying out operations as before and then by dividing by any particular specified fractional part (mentioned in the problem). An example in illustration thereof. 239-240. Five merchants saw a purse of money. They said one after another that by obtaining,,,, and 1 (respectively) of the contents of the purse, they would each become with what he had on hand three times as wealthy as all the remaining others with what they had on hand together. O arithmetician, (you tell) me quickly what moneys these had on hand (respectively), and what the value of the money in the purse was. The rule for arriving at the measure of the money contents of a purse, when specified fractional parts (thereof added to what may be on hand with one among a number of persons) makes him a specified number of times (as rich as all the others with what they together have on hand): 241. The specified fractional parts relating to all others (than the person in view) are (reduced to a common denominator, which is ignored for practical purposes. These are severally) multiplied by the specified multiple number (relating to the person in view). To these products, the fractional part (relating to the person) in view (and treated like other fractional parts) is added. The resulting sums are (severally) divided each by its (corresponding specified) multiple quantity as increased by one. Then these quotients are also added. The several sums (so obtained in relation 241. The formula given in the rule is a + mb n + 1 a + me a + md 2 + J +1 b + nd + +1 bna b + nc + m + 1 q + 1 and so on; where x, y, fractional parts; m, n, q, r, number of persons concerned in the transaction. y= Acharya Shri Kailassagarsuri Gyanmandir + + For Private and Personal Use Only (82)a + - ( s - 2)a } / (n + 1) are moneys on hand; a, b, c, d, various multiple numbers; and the } / (m + 1)

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CHAPTER VI-MIXED PROBLEMS. 157 to the several cases) are diminished by the product of the particular specified fractional part as multiplied by the number of cases less by two. The difference is divided by the particular specified multiple quantity as increased by one. The result is the money on hand (in the particular case). Examples in illustration thereof. 242-243. Two travellers saw a purse containing money (dropped) on the way. One of them said to the other), "By securing half of this money in the purse), I shall become twice as rich (as you)." The other said, "By securing two-thirds (of the money in the purse), I shall, with the money I have on hand, have three times as much money as what you have on hand." What are the moneys on hand, and what the money in the purse ? 244-2441. Two travellers saw on the way a purse containing money; and the first of them took it up and said, that) that money along with the money that he had on hand became twice the money of the other (traveller. This) other said that that money in the purse with the aid of what he had on hand would be) three times (the money in the hand of the first traveller). What is the money on hand in the case of each of them), and what the money in the purse ? 2451-247. Four men saw on the way a purse containing money, The first among them said, "If I secure this purse, I shall with the money already on hand with me become possessed of money which will be) eight times (the money on hand with the remaining travellers)." Another said, that the money in the purse with what he had on hand) would be nine times the money on hand with the rest (among them). Another (said that similarly he) would be possessed of ten times the mouey, and another (that he would be possessed of eleven times the money. Tell me quickly, O mathematician, what the money in the purse was and how much the money in the hand of each of them was. 248. Four men saw on the way a purse containing money. (Then), with what each of them had on hand, the , 1, 5, and $ parts (respectively) of this (money in the purse) became twice, For Private and Personal Use Only

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158 GANITASARASANGRAHA. thrice, five times and four times (that money which the others together had on hand. What is the money in the purse, and what the money on hand with each of them ?) 249-2504. Three merchants saw on the way a purse containing money. The first among them said, "If I get of this money in the purse, I shall (with what I have on hand) become possessed of) twice (the money on hand with) both of you." Another said that, if he secured part of the money in the purse, he would with the money on hand with him (become possessed of) thrico (the money on hand with the others). The third man said, "If I obtain of this money in the purse), I shall become possessed of four times the money (on hand with both of rou)." Tell me quickly, O mathematician, what the money on hand with each of them was, and what was the money in the purse. The rule for arriving at the money on hand, which, with the moneya begged (of others), becomes a specified multiple (of the money on hand with the others) : 2512-2527. The sums of the moneys begged are multiplied each by its own corresponding multiple quantity as increased by one. With the aid of these (products) the moneys on hand are arrived at according to the rule given in stanza 241. These quantities (80 obtained) are reduced so as to have a common denominator. Then they are severally divided by the sum as diminished by unity of the specified multiple quantities (respectively) divided by (those same) multiple quantities as increased by one. The resulting quotients themselves should be uuderstood to be the moneys on hand with the various persons). 2511-2525. Algebraically, j (a+b) (m +1) + m (c+d) (n + 1) - { (a + b) (m+1) +m (f+9) (2+1) n +1 p+1 etc.-- (8--2) (a+b) (m+1) } + (m+1) ]+ m + P m +1 +1 P+1 Similarly for y, 2, etc. Here a, b, c, d, f, g, are sums of money begged of each other. For Private and Personal Use Only

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CHAPTER VI-MIXED PROBLEMS. 159 Examples in illustration thereof. 2531-2551. Three merchants begged money from the hands of each other. The first begged 4 from the second and 5 from the third man, and became possessed of twice the money (then on hand with both the others). The second (merchant) begged 4 from the first and 6 from the third, and (thus) got three times the money (held on hand at the time by both the others together). The third man begged 5 from the first and 6 from the second, and (thus) became O times (as rich as the other two). O mathematician, if you know the mathematical process known as citrakuttikara-miera, tell me quickly what may be the moneys they respectively had on hand. 2561-2581. There were three very clever persons. They begged money of each other. The first of them begged 12 from the second and 13 from the third, and became thus 3 times as rich as these two were then. The second of them begged 10 from the first and 13 from the third, and thus became 5 times as rich (as the other two at the time). The third man begged 12 from the second and 10 from the first, and became (similarly) 7 times as rich. Their intentions were fulfilled. Tell me, O friend, after calculating, what might be the moneys on hand with them. The rule for arriving at equal capital amounts, on the last man giving (from his own money) to the penultimate man an amoun equal to his own, (and again on this man doing the same in relation to the man who comes behind him, and so on) : 2591. One divided by the optionally chosen multiple quantity (in respect of the amount of money to be given by the one to the other) becomes the multiple in relation to the periultimate man's amount. This (multiplier) increased by one becomes the multiplier of the amounts in the hands of the others. The 2591. The rule will be clear from the following working of the problem given in st. 263 1+ or 2 is the multiple with regard to the penultimate man's amount; this 2 combined with ), i.e., 3 becomes the multiple in relation to the amounts of the others. For Private and Personal Use Only

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160 GANITASARASANGRAHA. amount of the last person (so arrived at) is to have one added to it. This is the process to be adopted. Examples in illustration thereof. . 2601-261. Three sons of a merchant, the eldest, the middle, and the youngest, were going out along a road. The eldest son gave out of his capital amount to the middle son exactly as much as the capital amount of (that same) middle son. This middle son gave (out of his amount) to the last son just as much as he had, (In the end), they all became possessed of equal amounts of money. O mathematician, think out and say wbat amounts they (respectively) had (with them) on hand to start with). 262. There were five sons of a merchant. From the eldest (of them) the one next to him obtained as much money as he himself had on hand. All others also did accordingly (each one giving to the brother next to him as much as he had on hand. In the end) they all became possessed of equal amounts of money. What were the amounts of money they (respectively) had on hand (to start with) 263. Fire merchants became possessed of equal amounts of money after each of them gave out of his own property to the one who went before him half of what he possessed. Think out and a Now ... ... ... 1,1. Multiplying the penultimate 1 by 2 and the other by 3, we get ... ... 2, 3. Adding ) to the last ... ... ... 2, 4. Write down ... ... 2, 4, 4. Multiply the penultimate 4 by 2, and the others by 3, and add 1 to the last ... ... 6, 8, 13. Again ... ... ... ... ... ... 6, 8, 13, 13. Repeating the same operations as above we get ... ... ... .. ... ... 18, 24, 26, 40. 54, 72, 78, 80, 121. The figures in the last row represent the amounts in the hands of the 5 merchants. Algebruically a-b b - c =if; where, a, b, c, d, f are the amounts on hand with the 5 merchants. For Private and Personal Use Only

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CHAPTER VI- MIXED PROBLEMS. 161 say what amounts of money they (respectively) had on hand (to start with). 2645. There were six merchants. The elder ones among them gave in order, out of what they respectively had on hand, to those who were next younger to them exactly two-thirds (of what they respectively had on hand). Afterwards, they all became possessed of equal amounts of inoney. What were the amounts of money they (severally bad on hand (to start with) ? The rule for arriving at equal amounts of money on hand, after a number of persons give each to the others among them as much as they (respectively) have (then) on hand : 2655. One is divided by the optionally chosen multiple quantity (in the problem). (To this), the number corresponding to the men (taking part in the transaction) is added. The first (man's) amount (on hand to start with is thus arrived at). This (and the results thereafter arrived at) are written down (in order), and each of them is multiplied by the optional multiple number as increased by one ; and the result is then diminished by one. (Thus the money on hand with each of the others (to start with is arrived at). Examples in illustration thereof. 2661. Each of three merchants gave to the others what each of these had on band (at the time). Then they all became possessed of equal amounts of money. What are the amounts of money wbich they (respectively) had on hand to start with)? 2653. The rule will be clear from the following working of the problem given in st. 266} : 1, divided by the optionally chosen multiple 1, and increased by the number of persons, 3, gives 4 ; this is the money in the hand of the first man. This 4, multiplied by the optionally chosen multiple, 1, as increased by 1, becomes 8; when 1 is subtracted from this, we get 7, which is the money on hand with the second person. This 7, again, treated as above, i.e., multiplied by 2 and then diminished by 1, gives 13, the money on hand with the third man. This solution can be easily arrived at from the following equations :-- 4 (a - b-c)= 2 26 -- (a - 6 - 6) - 2c = 4c - 2 (a - b -c) - {20 - ( - 6 -c) - 2c). For Private and Personal Use Only

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162 GANITASARASANGRAHA, 2671. There were four merchants. Each of them obtained from the others half of what he had on hand (at the time of the respective transfers of money). Then they all became possessed of equal amounts of money. What is the measure of the money (they respectively had) on hand (to start with)? The rule for arriving at the gain derived (equally) from success and failure (in a gambling operation) : 2687-2693. The two sums of the numerators and denominators of the two fractional multiple) quantities (given in the problem have to be written down one below the other in the regular order, and (then) in the inverse order. The (summed up) quantities (in the first of these sets of two sums) are to be multiplied according to the vajrapavartana process by the denominator, and those in the second set) by the numerator, (of the fractional quantity) corresponding to the other (summed up quantity). The results (arrived at in relation to the first set) are written down in the form of denominators, and (those arrived at in relation to the second set are written down) in the form of numerators : (and the difference between the denominator and numerator in each set is noted down). Then by means of these differences the products obtained by multiplying the sum of the numerator and the denominator (of each of the given multiple fractions in the problem with the denominator of the other are respectively) divided. These resulting quantities, multiplied by the value of the desired gain, give in the inverse order the measure of the moneys on hand (with the gamblers to stake). An example in illustration thereof. 270-2724. A great man possessing powers of magical charm and medicine saw a cock-fight going on, and spoke separately in 2681-2693. Algebraically, - (c + d) & (a + b) a (c + d). - (a + b) c* * P, and y=n (a + b) d - (c + a) a ? * and y are the moneys on hand with the gamblers, an the fractional parts taken from them, and p the gain. This follows from a Ep=y For Private and Personal Use Only

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CHAPTER VI--MIXED PROBLEMS. 163 confidential language to both the owners of the cocks. He said to one: "If your bird wins, then you give the stake-money to me. If, however, you prove unvictorious, I shall give you two-thirds of that stake-money then." He went to the owner of) the other (cock) and promised to give three-fourths (of his stake-inoney on similar conditions). From both of them the gain to him could be only 12 (gold-pieces in each case). You tell me, 0 ornament on the forehead of mathematicians, the values of the) stake-money which each of the cock-owners had on hand. The rule for separating the (unknown) dividend number, the quotient, and the divisor from their combined sum : 2731. Any (suitable optionally chosen) number (which has to be) subtracted from the (given) combinod sum happens to be the divisor (in question). On dividing, by this (divisor) as increased by one, the remainder (left after subtracting the optionally chosen number from the given combined sum), the (required) quotient is arrived at. The very same remainder (above mentioned), as dimi. nished by this quotient becomes the (required dividend) number. An example in illustration thereof. 2741. A certain unknown quantity is divided by a certain (other) unknown quantity. The quotient here as combined with the divisor and the dividend number is 53. What is that divisor, and what (that) quotient ? The rule for arriving at that number, which becomes a square either on adding a known number (to the original number), or on subtracting another) given number (from that same original number) : 275.. The sum of the quantity to be added and the quantity to be subtracted is multiplied by one as associated with whatever may happen to be the excess above the even number (nearest to 2753. Algebraically, let # be the quantity to be found out, and a, b, the respective quantities to be added to or subtracted from it; then, the forumla to represent the rule will be (a + b) + (1+1)-212-1 a 0+ 1 For Private and Personal Use Only

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164 GANITASARASANGRAHA. that sum). The resulting produot is (then) halved and (then) squared. (From this squared quantity), the above-referred-to possible) excess quantity is subtracted. The result is divided by four, and then combined with one. Then the resulting quantity is either added to or subtracted from (respectively) by the half of the difference between the two given quantities as diminished or increased by the odd-making excess quantity (above referred to) according as the original given quantity to be subtracted is greater or less tban the original given quantity to be added. The result arrived at in this manner happens to be the required) number, which (when associated as desired with the (given) quantities) surely yields the square root (exactly). Examples in illustration thereof. 2762. A certain number when increased by 10 or decreased by 17 yields an exact square root. If possible, O arithmetician, tell me quickly that number. 2773. A certain quantity either as diminished by 7, or as added to by 18, yields the square root exactly. O arithmetician, give it out after calculation. 2781. A certain quantity diminished by , or again that same (quantity) increased by $, yields the square root (exactly). Tell me that quantity quickly, O arithmetician, after thinking out what it may be. The rationale of this may be made out thus - (n + 1)2 - n = 2n + 1, an odd number; and (n + 2)2 - = 4 + 4, an even number; where n is any integer. From 2n + 1, and 4 m + 4, the rule shows how we may arrive at n2 + a when we know 2n + 1, or 4 m + 4, to be equal to a + b. 278. Since the quantities represented by b and a in the note on stanza 275$ are seen to be fractional in this problem, being actually and it is necessary to have these fractional quantities removed from the process of working out the problem in accordance with the given rule. For this purpose they are first reduced to the same denominator, and come to be represented by 1 and respectively: then these quantities are multiplied by (21), so as to yield 294 and 189, which are assumed to be the band the a in the problem. The result arrived at with these assumed values of b and a is divided by 21), and the quotient is taken to be the answer of the problen. For Private and Personal Use Only

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CHAPTER VI--MIXED PROBLEMS. 185 The rule for arriving at the square root of an unknown number as increased or diminished by a known number : 279. The known quantity which is given is first halved and (then) squared and then one is added (to it). The resulting quantity either when increased by the desired given quantity or when diminished by the (same) quantity yields the square root (exactly). An example in illustration thereof. 2803. Here is a number which, when increased by 10 or diminished by the same 10, yields an exact square root. Think out and tell me that number, O mathematician. The rule for arriving at the two required square quantities, with the aid of those required quantities as multiplied by a known number, and also with the aid of (the same known number as forming the value of the square root of the difference (between these products) :--- 2813. The given number is increased by one; and the given number is also diminished by one. The resulting quantities when halved and then squarod give rise to the two (required) quantities. Then if these be (separately) multiplied by the given quantity, the squre root of the difference between these (products) becomes the given quantity. An example in illustration thereof. 2821-283. Two unknown squared quantities are multiplied by 71. The square root of the difference between these two resulting products, is also 71. O mathematician, if you know the process of calcolation known as citra-kuttikara, calculate and tell me what (those two unknown) quantities are. 2794. This is merely a particular case of the rule given in stanza 2754 wherein a is taken to be equal to b. 2811. Algebraically, when the given number is d, are the required aquare quantities. For Private and Personal Use Only

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166 GANITASARASANGRAHA. The rule for arriving at the required increase or decrease in relation to a given multiplicand and a given multiplier (so as to arrive at a given product): - 284. The difference between the required product and the resulting product (of the given multiplicand and the multiplier) is written down in two places. To (one of) the factors of the resulting product) one is added, and to the other the required product is added. That (difference written above in two positions as desired) is (severally) divided in the inverse order by the sums (resulting thus). These give rise to the quantities that are to be added (respectively to the given multiplicand and the multiplier) or (to the quantities that are) to be respectively) subtracted from them). Examples in illustration thereof. 285. The product of 3 and 5 is 15; and the required product is 18; and it is also it. What are the quantities to be added (respectively to the multiplicand and the multiplier) here, or what to be subtracted from them)? The rule for arriving at the required result by) the process of working backwards: 286. To divide where there has been a multiplication, to multiply where there has been a division, to subtract where there has been an addition, to get at the square root where there has been a squaring, to get at the squaring where the root has been given--this is the process of working backwards. An example in illustration thereof. 287. What is that quantity which when divided by 7, (then) multiplied by 3, (then) squared, (then) increased by 5, (then) 284. The quantities !o be added or subtracted are-- dabad-ab d+b and ari: dab ab =d, where a and b are the given factors, dub and d the required multiple a + 1 For Private and Personal Use Only

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CHAPTER VIMIXED PROBLEMS. 167 divided by }, (then) halved, and then reduced to its square root, happens to be the number 5 ? The role for arriving at the number of arrows in a bundle with the aid of the even number of) arrows constituting the common circumferential layer (of the bundle): 288. Add three to the number of arrows forming the circumferential layer; then square this resulting sum) and add again three (to this square quantity). If this be further divided by 12, the quotient becomes the number of arrows to be found in the bundle. An example in illustration thereof. 289. The circumferential arrows are 18 in number. How many (in all) are the arrows to be found in the bundle) within the quiver ? O mathematician, give this out if you have taken pains in relation to the process of calculation known as vicitrakuttikara. Thus ends vicitra-kuttikara in the chapter on mixed problems. 288. The formula here given to find out the total number of arrows is (n + 3) + 3 ", where n is the number of circumferential arrows. This formula can 12 be arrived at from the following considerations. It can be proved geometrically that only six circles can be described round another circle, all of them being equal and each of them tonching its two neighbouring circles as well as the central circle ; that, round these circles again, only twelve circles of the same dimension can be described similarly; and that round these again, only 18 such circles are possible, and so on. Thus, the first round has 6 circles, the second 12, the third 18, and so on. So that the number of circles in any round, say p, is equal to 6 p. Now, the total number of circles in the given number of rounds p, calculate: from the central cirale, is 1 +1 x 6 + 2 x 6 + 3 x 6 + ... + P x6 =1 + 6 (1 + 2 + 3 + ... + p) = 1 + 6 2 2 -- 1 + 3 + (p+1). if the value of 6p is given, say, as n, the total number of circles is 1+ 3x + 1), which is easily reducible to the formula given at the beginning of this note. For Private and Personal Use Only

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Shri Mahavir Jain Aradhana Kendra 168 www.kobatirth.org GANITASARASANGRAHA. Summation of Series. Hereafter we shall expound in (this) chapter on mixed problems the summation of quantities in progressive series. Acharya Shri Kailassagarsuri Gyanmandir The rule for arriving at the sum of a series in arithmetical progression, of which the common difference is either positive or negative: 290. The first term is either decreased or increased by the product of the negative or the positive common difference and the quantity obtained by halving the number of terms in the series as diminished by one. (Then,) this is (further) multiplied by the number of terms in the series. (Thus,) the sum of a series of terms in arithmetical progression with positive or negative common difference is obtained. Examples in illustration thereof. 291. The first term is 14; the negative common difference is 3; the number of terms is 5. The first term is 2; the positive common difference is 6; and the number of terms is 8. What is the sum of the series in (each of) these cases? The rule for arriving at the first terin and the common difference in relation to the sum of a series in arithmetical progression, the common difference whereof is positive or negative :-- 292. Divide the (given) sum of the series by the number of terms (therein), and subtract (from the resulting quotient) the product obtained by multiplying the common difference by the half of the number of terms in the series as diminished by one. (Thus) the first term (in the series) is arrived at. The sum of the series is divided by the number of terms (therein). The first term is subtracted (from the resulting quotient); the remainder when divided by half of the number of terms in the series as diminished by one becomes the common difference. n 290. Algebraically, (b+-a) na, where n is the number of terms, -a the first term, b the common difference, and s the sum of the series. n-1 292. Algebraically, @= b; and b b = (~2 2 - a ) / 2 = 1. 2 n For Private and Personal Use Only

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CHAPTER VI-MIXED PROBLEMS. 169 Examples in illustration thereof. 293. The sum of the series is 40 ; the number of terms is 5; and the common difference is 3; the first term is not known now. (Find it out.) When the first term is 2, find out the common difference. The rule for arriving at the sum and the number of terms in a series in arithmetical progression (with the aid of the known labha, which is the same as the quotient obtained by dividing the sum by the unknown number of terms therein) : 294. The labha is diminished by the first term, and (then) divided by the half of the common difference; and on adding one to this same (resulting quantity), the number of terms in the series is obtained). The number of terms in the series multiplied by the labha becomes the sum of the series. An example in illustration thereof. 295. (There were a number of utpala flowers, representable as the sum of a series in arithmetical progression, whereof) 2 is the first term, and 3 the common difference. A number of women divided (thene) utpala flowers (equally among them). Each woman had 8 for her sbare. How many were the women, and how many the flowers ? The rule for arriving at the sum of the squares (of a given number of natural numbers beginning with one) : 296. The given number is increased by one, and (then) squared ; (this squared quantity is) multiplied by two, and (then) diminished by the given quantity as increased by one. The remainder thus 1- a 294. Algebraically, n = + 1, where l= wh ch is the labha. 295. The number of women in this problem is conceived to be equal to the number of terms in the series. {2 (n+1)2 - (n+1) 296. Algebraically, -=S2, which is the sum of the squares of the natural numbers up to n. 2 For Private and Personal Use Only

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170 GANITASARASANGRAITA. arrived at is) multiplied by the half of the given number. This gives rise to the combined sum of the square (of the given number), the cube (of the given number), and the sum of the natural numbers (up to the given number). This combined sum, divided by three, gives rise to the sum of the squares (of the given number of natural numbers). Examples in illustration thereof. 297. (In a number of series of natural numbers), the number of natural numbers is (in order) 8, 18, 20, 60, 81, and 36. Tell me quickly (in each case) the combined sum of the square (of the given number), the cube (of the given number), and the sum of the given number of natural numbers. (Tell me) also the sum of the squares of the natural numbers (up to the given number). The rule for arriving at the sum of the squares of a number of terms in arithmetical progression, whereof the first term, the common difference, and the number of terms are given : 298. Twice the number of terms is diminished by one, and (then) multiplied by the square of the common difference, and is (then) divided by six. (To this), the product of the first term and the common difference is added. The resulting sum is multiplied by the number of terms as diminished by one. (To the product so arrived at), the square of the first term is added. This sum multiplied by the number of terms becomes the sum of the squares of the terms in the given series. Again, another rule for arriving at the sum of the squares of a number of terms in arithmetical progression, whereof the first term, the common difference, and the number of terms are given 299. Twice the number of terms (in the series) is diminished by one, and (then) multiplied by the square of the common differerce, and (also) by the number of terms as diminished by one. This 298. n-1) a sum of the squares of the terms in B a series in arithmetical progression. For Private and Personal Use Only

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CHAPTER VI-MIXED PROBLEMS. 171 product is divided by six. (To this resulting quotient), the square of the first term and the continued product) of the number of terms as diminished by one, the first term, and the common difference, are added. The whole of this) multiplied by the number of terms becomes the required result. Examples in illustration thereof. 300. (In a series in arithmetical progression), the first term is 3, the common difference is 5, the number of terms is 5. Give out the sum of the squares (of the terms) in the series. (Similarly, in another series), 5 is the first term, 3 the common difference, and 7 the number of terms. What is the sum of the squares (of the terms in this series ? The rule for arriving at the sum of the cubes (of a given number of natural numbers) : 301. The quantity represented by the square of half the (given) number of terms is multiplied by the square of the sum of one and the number of terms. In this (science of) arithmetic, this result is said to be the sum of the cubes of the given number of natural numbers) by those who know the secret of calculation. Examples in illustration thereof. 302. Give out (in each case) the sum of the cubes of (the natural numbers up to) 6, 8, 7, 25 and 256. The rule for arriving at the sum of the cubes (of the terms in a series in arithmetical progression), the first term, the common difference, and the number of terms whereof are optionally chosen : 303. The sum of the simple terms in the given series), as multiplied by the first term (therein), is (further) multiplied by the 301. Algebraically, 2 =9g, which is the sum of the cubes of the natural numbers up to n. 303. Algebraically, Lsa (a - b) + 32b = the sum of the cubes of the terms in a series in arithmetical progression, where s=the sum of the simple terms of the series. The sign of the first terna in the formula is + or - according ar a> or Page #369
172 GANITASARASANGRAHA. differenco betwoen the first term and the common difference (in the series). (Then) the square of the sum of the series is multiplied by the common difference. If the first term is smaller than the common difference, then the first of the products obtained above is) subtracted (from the second product). If, however, the first term is greater than the common difference), then the first product above-mentioned is added (to the second product). (Thus) the (required) sum of the cubes is obtained. Examples in illustration thereof. 304. What may be the sum of the cubes when the first term is 3, the common difference 2, and the number of terms 5 ; or, when the first term is 5, the common difference 7, and the number of terms 6 ? The rule for arriving at the sum of (a number of terms in a series wherein the terms themselves are successively) the sums of the natural numbers (from 1 up to a specified limit, these limiting numbers being the terms in the given series in arithmetical progression) 305-3053. Twice the number of terms (in the given series in arithmetical progression) is diminished by one and (then) multiplied by the square of the common difference. This product is divided by six and increased by half of the common difference and also) by the product of the first term and the common difference. The sum (so obtained) is multiplied by the number of terms as diminished by one and then increased by the product obtained by multiplying the first term as increased by one by the first term itself. The quantity (so resulting) when multiplied by - half the number of terms (in the given series) gives rise to the required sum of the series wherein the terms themselves are sums (of specified series). 305--8054. Algebraically, 2n-1)621 ) + a sum of the series in arithmetical progression, wherein each term represents the sum of a series of natural numbers up to a limiting number, which is itself a member in a series in arithmetical progression. For Private and Personal Use Only

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CHAPTER VI-MIXED PROBLEMS. 173 Examples in illustration thereof. 306). It is seen that (in a given series) the first term is 6, the common difference 5, and the number of terms 18. In relation to (these) 18 terms, what is the sum of the sums of (the various) series having 1 for the first term and 1 for the common difference. The rule for arriving at the sum of the four quantities (specified below and represented by a certain given number): - 307. The given number is increased by one, and (then) halved. This is multiplied by the given number and (then) by seven. From the resulting) product, the given number is subtracted ; and the (resulting) remainder is divided by three. The quotient (thus obtained), when multiplied by the given number as increased by one, gives rise to the (required) sum of (the four specified quantities, namely,) the sum of the natural numbers (up to the given number), the sum of the sums of the natural numbers (up to the given number), the square (of the given number), and the cube (of the given number). Examples in illustration thereof. 308). The given numbers are 7, 8, 9, 10, 16, 50 and 61. Taking into consideration the required rules, separately give out in the case of each of them the sum of the four (specified) quantities. The rule for arriving at the collective sum of the four different kinds of series already dealt with) : 3094. The number of terms is combined with three ; it is (then) multiplied by the fourth part of the number of terms ; (then) one 2 * * (n + 1) 7 307. Algebraically, --- * (n + 1) is the son of the 3 four quantities specified in the role. These are (i) the sum of the natural numbers up to n; (ii) the sum of the sums of the various series of natural numbers respectively limited by the various natural numbers up to n; (iii) the square of m; and (iv) the cube of n. 3091. Algebraically, ((n + 3) + 1} (9* + n) is the collective sum of the sums, namely, of the sums of the different series dealt with in rules 290, 301, 305 to 305) above, and also of the sum of the series of natural numbers up to th. For Private and Personal Use Only

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174 GANITASA RASANGRAHA, is added (thereunto). The (resulting) quantity when multiplied by the square of the number of terms as increased by the number of terms gives rise to the (required) collective sum. Examples in illustration thereof. 3103. What wonld be the (required) collective sum in relation to the (various) series represented by (each of) 49, 66, 13, 14, and 25 ? The rule for arriving at the sum of a series of fractions in geometrical progression :-- 3111. The number of terms (in the sories) is caused to be inarked (in a separate column) by zero and by one (respectively), corresponding to the even (value) which is halved and to the uneven (value from which one is subtracted, till by continuing these processes zero is ultimately reached); then this (representative series made up of zero and one is used in order from the last one therein, so that this one multiplied by the common ratio is again) multiplied by the common ratio (wherever one happens to be the denoting item), and multiplied so as to obtain the square (wherever zero happens to be the denoting item). The result (of this operation) is written down in two positions. (In one of them, what happens to be) the numerator in the result (thus obtained) is divided (by the result itself; then) one is subtracted (from it); the (resulting) quantity is multiplied by the first term in the series) and (then) by (the quantity placed in) the other of the two positions noted above). The product (so obtained), when divided by one as diminished by the common ratio, gives rise to the required sum of the series. Examples in illustration thereof. 3123-313. In relation to 5 cities, the first term is dinara and the common ratio is . (Find out the sum of the dinaras obtained in all of them. The first term is $, the common ratio is 311. In this rule, the numerator of the fractional common ratio is taken to be always 1. See stanza 94, Ch. II and the note thereunder. For Private and Personal Use Only

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CHAPTER VI-MIXED PROBLEMS. 175 1, and 7 is the number of terms. If you are acquainted with calculation, then tell me quickly what the sum of the series of fractions in geometrical progression here is. The rule for arriving at the sum of a series in geometrical progression wherein the terms are either increased or decreased (in a specified manner by a given known quantity) : 314. The sum of the series in (pure) geometrical progression (with the given first term, given common ratio, and the given number of terms, is written down in two positions) ; one of these sums so written down) is divided by the (given) first term. From the resulting) quotient, the (given) number of terms is subtracted. The resulting) remainder is (then) multiplied by the (given) quantity which is to be added to or to be subtracted from the terms in the proposed series). The quantity (so arrived at) is (then) divided by the common ratio as diminished by one. The sum of the series in pure geometrical progression written down in) the other (position) has to be diminished by the (last) resulting quotient quantity, if the given quantity is to be subtracted (from the terms in the series). If, bowever, it is to be added, (then the sum of the series in geometrical progression written down in the other position) has to be increased by the resulting quotient (already referred to. The result in either case gives the required sum of the specified series). Examples in illustration thereof. 315. The common ratio is 5, the first term is 2, and the quantity to be added to the various terms) is 3, and the number of terms is 4. O you who know the secret of calculation, think out and tell me quickly the sum of the series in geometrical progression, wherein the terms are increased by the specified quantity in the specified manner). 314. Algebraically, (r - 1) + 3 is the sum of the series of the following form : a, ar m, (ar 3. m) Em, {(ar m)r m }o+m, and so on, For Private and Personal Use Only

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176 GANITASARASANGRAHA. 316. The first term is 3, the common ratio is 8, the quantity to be subtracted (from the terms) is 2, and the number of terms is 10. O you mathematician, think out and tell me quickly what happens to be here the sum of the series in geometrical progression, whereof the terms are diminished (by the specified quantity in the specified manner). Acharya Shri Kailassagarsuri Gyanmandir The rule for arriving at the first term, the common difference and the number of terms, from the mixed sum of the first term, the common difference, the number of terms, and the sum (of a given series in arithmetical progression) : 317. (An optionally chosen number representing) the number of terms (in the series) is subtracted from the (given) mixed sum. (Then) the sum of the natural numbers (beginning with one and going up to) one less than this optionally chosen number is combined with one. By means of this as the divisor (the remainder from the mixed sum as above obtained is divided). The quotient here happens to be the (required) common difference; and the remainder (in this operation of division) when divided by the (above optionally chosen) number of terms as increased by one gives rise to the (required) first term. An example in illustration thereof. 318. It is seen here that the sum (of a series in arithmetical progression) as combined with the first term, the common difference, and the number of terms (therein) is 50. O you who know calculation, give out quickly the first term, the common difference, the number of terms, and the sum of the series (in this case). The rule for arriving at the common limit of time when one, who is moving (with successive velocities representable) as the terms in an arithmetical progression, and, another moving with steady unchanging velocity, may meet together again (after starting at the same instant of time) : 317. See stansas 80--82 in Ch. II and the note relating to them. For Private and Personal Use Only

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CHAPTER VI--MIXED PROBLEMS. 177 319. The unchanging velocity is diminished by the first term (of the velocities in series in arithmetical progression), and is (then) divided by the half of the common difference. On adding one (to the resulting quantity), the required) time (of meeting is arrived at. (Where two persons travel in opposite directions, each with a definite velocity), twice (the average distance to be covered by either of them) is the whole) way (to be travelled). This when divided by the sum of their velocities gives rise to the time of (their) meeting. An example in illustration thereof. 320. A certain person goes with a velocity of 3 in the beginning increased (regularly) by 8 as the successive) common difference. The stoady unchangiug velocity (of another person) is 21. What may be the time of their meeting (again, if they start from the same place, at the same time, and move in the same direction)? An ea'ample in illustration of the latter half of the rule given in the stanza above). 321-321. One nian travels at the rate of 6 yojanus and another at the rate of 3 yojanas. The average) distance to be covered by either of them moving in opposite directions is 108 yojanas. O arithmetician, teil me quickly what the time of their meeting together is. The rule for arriving at the time and distance of meeting together, (when two persons start from the same place at the same timo and travel) with varying) velocities in arithmetical progression. 322). Tie difforence between the two first terms divided by the difference between the two common differences, when multiplied by two and increased by one, gives rise to the tinjo of coming together on the way by the two persons travelling simultaneously (with two series of velocities varying in arithmetical progression). b . 319. Algebraically, (v=c) = + 1 = t, where v is the unchanging velocity, itud t the time. d al x 2 + 1. 322. Algebraically, n = babi 23 For Private and Personal Use Only

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Shri Mahavir Jain Aradhana Kendra 178 www.kobatirth.org GANITASARASANGRAHA. Acharya Shri Kailassagarsuri Gyanmandir An example in illustration thereof. 323. A person travels with velocities beginning with 4, and Increasing (successively) by the common difference of 8. Again, a second person travels with velocities beginning with 10, and increasing (successively) by the common difference of 2. What is the time of their meeting? The rule for arriving at the time of meeting of two persons (starting at the same time and travelling in the same direction with varying velocities in arithmetical progression), the common difference (in the one case) being positive, and (in the other) negative: 324. The difference between the two first terms is divided by half of the sum of the numbers representing the two (given) common differences, and (then) one is added (to the resulting quantity). This becomes the time of meeting on the way by the two persons (starting at the same time and) travelling simultaneously (with velocities in arithmetical progression, the common difference in the one case being positive and in the other negative). An example in illustration thereof. 3254. The first man travels with velocities beginning with 5, and increased (successively) by 8 as the common difference. In the case of the second person, the commencing velocity is 45, and the common difference is minus 8. What is the time of meeting ? The rule for arriving at the time of meeting of two persons, (starting at different times and) travelling (respectively) with a quicker and a less quick velocity (in the same direction) - 326. He who travels less quickly and ho who travels more quickly-both move in the same direction. What happens to be the distance to be overtaken here is divided by the difference between those (two) velocities. In the course of the number of days represented by the quotient (here), the more quickly moving person goes to the less quickly moving one. 324. Compare this with the rule given in 322 above. For Private and Personal Use Only

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CHAPTER VI-MIXED PROBLEMS. 179 An example in illustration thereof. 3274. A certain person travels at the rate of 9 yojanas (a day); and 100 jojanas have already been gone over by him. Now, a messenger sent after him goes at the rate of 13 yojanas (a day). In how many days will this (messenger) meet him ? The rule for working out the circumferential number of arrows in the quiver with the aid of the (given) uneven number of arrows (contained in the quiver; and vice versa) : 3284. The number of the circumferential arrow is increased by three and (then) halved. This is squared and then divided by three. On adding one (to the resulting quantity), the number of arrows in the quiver) is obtained. When, however, the number of the circumferential arrows has to be arrived at, the reverse process is (to be adopted in relation to these operations). Examples in illustration thereof. 3294. The circumferential number of the arrows is 9. Their total number, however, is not known. (What is that ?). The total number of arrows (in the quiver) is 13. Tell me, O arithmetician, the number of the circumferential arrows also in this case. The rule for arriving at the number of bricks to be found in structures made up of layers (of bricks one over another) : 3301. The square of the number of layers is diminished by one, divided by three, and (then), multiplied by the number of layers. On adding (to the quantity so obtained) the product, obtained by multiplying the optionally chosen number (representing the bricks in the topmost layer) by the sum of the (natura! numbers beginning with one and going up to the given) number of layers, the required answer is obtained. 12--1 (n + * (n 1) 3301. Algebraically, - X1 + a2 . ) is the total number of bricks in the structure, where n is the number of layers, and a the optionally chosen number of bricks in the topmost layer. The number of bricks along the length or breadth of any layer is one less than the same in the immediately lower layer. For Private and Personal Use Only

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180 GANITASA RASANGRAHA. Examples in illustration thereof. 3311. There is constructed an equilateral quadrilateral structure consisting of 5 layers. The topmost layer is made up of 1 brick. O you who know the calculation of mixed problems, tell me how many bricks there are (here in all). 332). There is a structure built up of successive layers of bricks, which is in the form of the wamilyavarta. There are 4 layers built symmetrically with 60 (as the numerical measure of the top-bricks in single row). Tell me how many are all the bricks (here). Rules regarding the six things to be known in the science of prosody : 3331-336}. (The number of syllables in a given syllabic motre or chandas is caused to be marked in a separate column) by cero and 332). The nandyavarta figure referred to in the stanza is 333-336$. As each syllable found in a line forming a quarter of a stanza may be short or long, there arises a number of varieties corresponding io the different arrangements of long and short syllables. In arranging these varieties, a certain order is followed. The rules given lere enable us to find out (1) the number of varieties possible in a metre consisting of a specified number of syllables, (2) the manner of arrangement of the syllables in these varieties, (3) the arrangement of the syllables in a variety specified by its ordinal position, (4) the ordinal position of a specified arrangement of syllables, (5) the number of varieties containing a specified number of long or short syllables, and (6) the amount of vertical space required for exhibiting the varieties of a particular metre. The rules will become clear from the following working of the problemas given in stanza 3375:(1) There are 3 syllables in a metre; now, wo proceed thus: Now, multiplying by 2 the figures in the 3-1 2 22 right-hand chain, we obtain 0. By the process of 2 1-1 multiplication and squaring, as explained in the note to stanza 94, Ch. Il, we get 8; and this is the muniber of varieties. (2) The manner of arrangement of the syllables in each variety in arrived at tha:1st variety : 1, being odd, denotes a long syllable; so the first syllable is long. Add 1 to this 1, and divide the sum hy 2; the quotient is odd, und denotes another long syllable. Again, l is added to this quotient 1, and divided by 2; the result, For Private and Personal Use Only

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CHAPTER VI--MIXED PROBLEMS. 181 by one (respectively), corresponding to the even (value) which is halved, and the uneven (value from which one is subtracted, till by continuing these processes sero is ultimately reached. The numbers in the chain of figures so obtained are) all doubled, (and then in the process of continued multiplication from the bottom to the top of the chain, those figures which come to have a xiro above them) are squared. The resulting product (of this continuell multiplication gives the number (of the varieties of stanzas possible in that syllabic metre or chandas). The arrangement of short and long syllables in all the varieties of stanzas 80 obtained) is shown to be arrived at thus:-- (The natural numbers commencing with one and ending with the measure of the maximum number of possible stanzas in the given metre being noted down), every odd number (therein) has one added to it, and is (then) halved. (Whenever this process is gone through), a long syllable is decidedly indicated. Where again odd, denotes a third long syllable. Thus the first variety consists of three long syllables, and is indicated thug . 2nd variety : 2, beiny even, indicates a short syllable; when this 2 is divided by 2, the quotient is 1, which being udd indicates a long syllable. Add 1 to this l, and divide the sum by 2; the quotient being odd indicates a long syllable; thus we get 117. Similarly the other six varieties are to be found ont. (3) The fifth variety, for instance, may be found out as above. (4) To find out, for instance, the ordinal position of the variety, 171 we proceed thus :Below these syllables, write down the terms of a series in geometrical pro gression, having l as the first term and 2 as the common ratio. Add the 1 figures 4 and 1 ander the the short syllables, and increase the sum by l; 1 2 4 we get 6: and we, therefore, say that this is the sixth variety in the tri-syllabio metre. 15) Suppose the problem is : How many varieties contain 2 short syllables? Write down the natural numbers in the regular and in the inverse order, one 1 2 3 below the other thus: 8: 32 1 Takmg two te Taking two terms from right to left, both from above and from below, we divide the product of the former by the product of the latter. And the quotient 3 is the answer required. (6) It is prescribed that the symbols representing the long and short syllables of any variety of metre should occupy an angula of vertical space, and that the intervening space between any two varieties should also be an angula. The amount, therefore, of vertical space required for the 8 varieties of this metre is 2 x 8-1 or 15 augulas. For Private and Personal Use Only

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182 GANITASARASANGRAHLA. the number is even, it is immediately) halved and this indicates a short syllable. In this manner, the process (of halving with oi without the aldition of one as the case may be, noting down at the same time the corresponding long and short syllables as indicated), is to be regularly carried on (till the actual number of syllables in the metre is arrived at in each case). (If the number representing in the natural order any given variety of a stanza), the arrangement of the syllables wherein has to be found out, (happens to be even, it) has to be halved, and indicates a short syllable. (If it happens to be however odd), one has to be added to it, and (then) it is to be halved : and this indicates a long syllable. Thus (the long and short sylinbles have to be put down over and over again (in their respective positions), till the maximum number of syllables in the stanza is arrived at. This gives the arrangement (of long and short syllables in the required variety of the stanza). Where (a stanza of a partionlar variety is given, anil) its ordinal position (among the varieties of stanzas possible in the metre) is to be found out, the terms (of a series in geonetrical progression) commencing with one and having tiro as the common ratio are written down, (the number of terms in the series being equal to the number of syllables in the given metre. Above those terms, the corresponding long or short syllables are noted down). Then the terms (immediately) below the position of short syllables are all added ; the sum (so obtained) is increased by on. (This gives the required ordinal number.) Natural numbers commencing with one, and going np to the number (of syllables in the given metre), are written down in the regular and in the inverse (order in two rows) one below the other. When the numbers in the row are multiplied (1, 2, 3 or more at a time) from the right to the left, and the products (so obtained in relation to the upper row) are divided by the corresponding) products in relation to the lower row), the quotient represents the result of the operation intended to arrive at (the number of varieties of stanzas in the given metre, with 1, 2, 3 or more short or long syllables in the verse). For Private and Personal Use Only

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CHAPTER VI-MIXED PROBLEMS. Acharya Shri Kailassagarsuri Gyanmandir 183 The possible number (of the varieties of stanzas in the given metre) is multiplied by two and (then) diminished by one. This result gives (the measure of what is called) adhvan, (wherein an interval equivalent to a stanza is conceived to exist between every two successive varieties in the metre). Examples in illustration thereof. 337. In relation to the metre made up of 3 syllables, tell me quickly the six things to be known--viz., (1) the (maximum) number (of possible stanzas in the metre), (2) the manner of arrangement (of the syllables in those stanzas), (3) the arrangement of the syllables (in a given variety of the stanza, the ordinal position whereof among the possible varieties in the metre is known), (4) the ordinal position (of a given stanza), (5) the number (of stanzas in the given metro containing any given number) of short or long syllables, and (6) the (quantity known as) adhvan. Thus ends the process of summation of scries in the chapter on mixed problems. For Private and Personal Use Only Thus ends the fifth subject of treatment, known as Mixed Problems, in Sarasangraha, which is a work on arithmetic by Mahaviracarya.

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184 GANITASARASANGRAHA, CHAPTER VII. CALCULATION RELATING TO THE MEASUREMENT OF AREAS. 1. For the accomplishment of the object held in view, I how again and again with true earnestness to the most excellent Siddhas who have realized the knowledge of all things. Hereafter we shall expound the sixth variety of calculation forming the subject known by the name of the Measurement of Areas. Aud that is as follows:-- 2. (The measurement of) area has been taken to be of two kinds by Jina in accordance with (the nature of) the result, namely, that which is (approximate) for practical purposes and that which is minately accurate. Taking this into consideration, I shall clearly explain this subject. 3. (Mathematical) teachers, who have reached the other shore of the ocean of calculation, have given out well (the various kinds of) areas as consisting of those that are trilateral, quadrilateral and curvi-linear, being differentiated into their respective varieties. 4. A trilateral area is difforentiated in three ways; a quadrilateral one in five ways; and a curvilincar one in eight ways. All the remaining (kinds of) areas are indeed variations of the varieties of these different kinds of areas). 5. Learned men say that the trilateral area may be cynilateral, isosceles or scalone, and that the quadrilateral area also may be 5 and 6. The various kinds of enclosed areas inentioned in these illustrated below :-- anzus are Samatribhujal Equilateral trilateral ligure. Dvisaputribhuja - Isosceles trilateral figure Visaniatribhuja = Scalcue trilateral figure. For Private and Personal Use Only

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CHAPTER VII--MEASUREMENT OF AREAS. 185 equi-lateral, equi-dichastic, equi-bilateral, equi-trilateral and inequi-lateral. Samacaturasra = Equi-lateral quadrilateral. Dvid visamacaturabra Equi-dichastic quadrilateral. Dvisamacaturafra - Eqni-bilateral Trisamacaturasra ** Equi-trilateral quadrilateral. quadrilateral. Visamacaturabra = Inequi-lateral quadrilateral. Samavrtta = Circle. Ardhavrtta = Semicircle. Ayatavrtta = Ellipse. 24 For Private and Personal Use Only

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186 GANITASARASANGRAHA. 6. (The curvi-linear area may be) a circle, a semicircle, an ellipse, a conchiform area, a concave circular aroa, a convex circular area, an out-lying annulus or an in-reaching annulus. 13 Kambukavrtta=conchiform Nimnavrtta = concayo area. circular arca. Unpalavrtta = convex circular area. 16 Bahiscakravalavrita=Out-lying annulus. Antascakravalavrita = In reaching annulos. From a consideration of the rules given for the measurement of the dimersions and areas of quadrilateral figures, it has to be concluded that all the quadrilateral figures mentioned in this chapter are cyclic. Hence an equilateral quadrilateral is a square, an equidichastio quadrilateral is an oblong; and equi. bilateral and equi-trilateral quadrilaterals have their topeide parallel to the base For Private and Personal Use Only

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CHAPTER VII-MEASUREMENT OF AREAS. 187 Calculation relating to approximate measurement (of areas), The rulo for arriving at the (approximate) measure of the areas of trilateral and quadrilateral fields : 7. The product of the halves of the sums of the opposite sides becomes the (quantitative) measurement (of the area) of trilateral and quadrilateral figures. In the case of (a figure constituting a circular annalus like the rim of a wheel, half of the sum of the (inner and outer) circumferences multiplied by the measure of) the breadth of the annulus gives the quantitative measure of the area thereof). Half of this result happens to be here the area of (a figure resembling) the crescent moon. Examples in illustration thereof. 8. In the case of a trilateral figure, 8 dandas happen to be the measure of the side, the opposite side and the base ; tell me quickly, after calonlating, the practically approximate value (of the area) thereof. 9. In the case of a trilateral figure with two equal sides, the length (represented by the two sides) is 77 dandas; and the breadth (measured by the base) is 22 dandas associated with 2 hastas. (Find out the area.) 7. A trilateral figure is here conceived to be formed hy making the topside, j.e., the side opposite to the base, of a quadrilateral so small as to be neglected. Then the two lateral sides of the trilateral figure become the opposite sides, the topside being taken to be nil in value. Hence it is that the rule speaks of opposito sides even in the case of a trilateral figure. As half the sum of the two sides of a triangle is, in all cases, bigger than the altitude, the value of the area arrived at according to this rule cannot be accurate in any instance. In regard to quadrilateral figures the value of the area arrived at according to this rule can be accurate in the case of a square and an oblong, but only approximate in other cases. Nemi is the area enclosed between the circumferences of two concentric circles ; and the rule here stated for finding out the approxinate measure of the area of a Nemiksetra happens to give the accurate measur. thereof. In the case of a figure resembling the crescent moon, it is evident that the result arrived at according to the rule gives only an approximate measure of the area. For Private and Personal Use Only

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188 GANITASARASANGRAHA. 10. In the case of a scalene trilateral figure, one side is 13 dundas, the opposite side is 15 dandas; and the base is 14 dandas. So what is the quantitative measure (of the area) of this (figure) ? 11. In the case of a figure resembling (the medial longitudinal section of) the tusk of an elephant, the length of the outer curve is seen to be 88 dandas ; that of the inner curve is (seen to be) 72 dandas; the measure of the thickness at) the root of the tusk is 30 dandas. (What is the measure of the area ?) 12. In the case of an equilateral quadrilateral figure, the sides and the opposite sides (whereof) are each 60 dandas in measure, you tell me quickly, O friend, the resulting (quantitative measure (of the area thereof). 13. In the case of a longish quadrilateral figure here, the length is 61 dandas, the breadth is 32. Give out the practically approximate measure (of the area thereof). 14. In the case of a quadrilateral with two equal sides, the length (as measured along either of the equal sides) is 67 dandas, the breadth of this figure is 38 dandas (at the base) and 33 dandas (at the top. What is the measure of the area of the figure ?) 15. In the case of a quadrilateral figure with three equal sides, (each of these three sides measures 108 dandi18, the remaining side here called) mukha or top-side measures 8 dundas and 3 hastas. Accordingly, tell me, O mathematician (the measure of the area of this figure). 16. In the case of a quadrilateral the sides of which are all unequal, the side forming the base measures 38 dandas, the side forming the top is 32 landas : one of the lateral sides is 50 dandas and the other is 60 dandas. What is (the area) of this (figure) ? 17. In an annulus, the inner circular boundary neasures 30 Gandas; the outer circular boundary is seen to be 300. The breadth 11. The shape oi the figure mentioned in this stanza seems to be what is given here in the margin: it is intended that this should be treated as a trilateral figure, and that the area thereof should be found ont in accordance with the rale given in relation to trilateral figures. For Private and Personal Use Only

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CHAPTER VII-MEASUREMENT OF AREAS. 189 of the annulus is 45. What is the calculated measure of the area of (this) annulus ? 18. In the case of a figure resembling the crescent moon, the breadth is seen to be 2 hastas, the outer curve 68 hastas, and the inner curve 32 hastas. Say what the (resulting) area is. The rule for arriving at the (practically approximate value of the) area of the circle : 19. The (measure of the) diameter multiplied by three is the measure of the circumference; and the number representing the square of half the diameter, if multiplied by three, gives the (resulting area in the case of a complete circle. Teachers say that, in the case of a semicircle, half (of these) give (respectively) the measure of the circumference and of the area). Examples in illustration thereof. 20. In the case of a circle, the diameter is 18. What is the circumference, and what the (resulting) area (thereof)? In the case of a semicircle, the diameter is 18: tell me quickly what the calculated measure is (of the area as well as of the circumference). The rule for arriving at (the value of) the area of an elliptical figure : 21. The longer diameter, increased by half of the (shorter) diameter and multiplied by two, gives the measure of the circumference of the elliptical figure. One-fourth of the shorter) diameter, multiplied by the circumference, gives rise to the (measure of the area (thereof). 19. The approximate character of the measure of the circumference as well 28 of the area as given here is due to the value of being taken as 3. 21. The formula given for the circumference of an ellipse is evidently an approximation of a different kind. The area of an ellipse is . a.b, where a and b are the semi-axes. If is taken to be equal to 3, then 7. a.b, = 3 a.b. But the formula given in the stanza makes the area equal to 2ab + b2. For Private and Personal Use Only

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190 GANITASARASANGRAHA. An example in illustration thereof. 22. In the case of an elliptical figure the shorter) diameter is 12, and the longer diameter is 36. What is the circumference and what is the (resulting) area (thereof) ? The rule for arriving at the resulting) area of a conchiform curvilinear figure :-- 23. In the case of a conchiform curvilinear figuro, the measure of the greatest) breadth diminished by half the measure of the mouth and multiplied by three gives the measure of the perimeter. One-third of the square of half (this) perimeter, increased by three-fourths of the square of half the measure of the mouth, (gives the area). An example in illustration thereof. 24. In the case of a conchi-form figure the breadth is 18 hastas, and the measure of the mouth thereof is 4 (hastax). You tell me what the perimeter is and what the calculated area is. The rule for arriving at the (resulting) area of the concave and convex circular surfaces : 25. Understand that one-fourth of the circumference multiplied by the diameter gives rise to the calculated (resulting) area. Thence, in the case of concave and convex areas like that of a 23. If a is the diameter and m in the measure of the month, then 3 (a - m) is the measure of the circumference; and ( 3 ( a m) 12x +3x7m) ence; and {- 2 * 3 +7 (2) is the measure of the area. The exact shape of the figure is not clear from the description given; but from the values given for the circumference and the area, it may be conceived to consist of 2 unequal semicircles placed so that their diameters coincide in position as shown in figure 12, given in the foot-note to stanza 6, in this chapter. 25. The area here specified seems to be that of the surface of tho segment of a sphere, and the measure of the area is stated to be, when symbolically representer, equal to (c) x d, where c is the circumference of the sectional circle, and d is the diameter thereof. Bat the area of the surface of a spherical segment of this kind is equal to 2 r. r.l, where r is the radius of the sectional circle and h is the height of the spherical segment. For Private and Personal Use Only

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CHAPTER VII-MEASUREMENT OF AREAS. * 191 sacrificial fire-pit and like that of (the back of) the tortoise, (the required result is to be arrived at). As example in illustration thereof. 26. In the case of the area of a sacrificial fire-pit the measure of the diameter is 27, and the measure of the circunference is seen to be 56. What is the calculated measure of the area of that same (pit) ? An example about a convex circular surface resembling the back) of a tortoise. 27. The diameter is 15, and the circumference is seen to be 36. In the case of this area resembling the back of a) tortoise, what is the practically approximate measure as calculated ? The rule for arriving at the practically approximate value of the area of an in-lying annular figure as well as of an out-reaching atnular figure : 28. The (inner) diameter increased by the breadth (of the annular arca) when multiplied by three and by the breadth (of the annular area) gives the calculated measure of the area of the outreaching annular figure. (Similarly the measure of the calculated area of the in-lying annular figure is to be obtained from the din neter as diminished by the breadth (of the annular area). Examples in illustration thereof. 29. The diameter is 18 hastas, and the breadth of the outreaching annular area is 3 in this case : the diameter is 18 hastas and again the breadth of the in-lying annular area is 3 hastas. What may be (the area of the annular figure in each case) ? 28. The shape of the antazcakravAlavRttakSetra as well as of the bahizcakravAlavRttakSetra is identical with the shape of the HTC mentioned in the note to stanza 7 in this chapter. Eience the rule given for arriving at the area of all these figures works out to be the same practically. For Private and Personal Use Only

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192 GANITASARASANURAHA. The rule for arriving separately at the numerical measures of the circumference, of the diameter, and of the area of a circular figure, from the combined sum obtained by adding together the approximate measure of its area, the measure of its circumference and the measure of its diameter : 30. In relation to the combined sum of the three quantities) as multiplied by 12, the quantity thrown in so as to be added is 64. Of this (second) sum the square root diminished by the square root of the quantity thrown in gives rise to the measure of the circumference. An example in illustration thereof. 31. The combined sum of the measures of the circumference, of the diameter and of the area (of a circle) is 1116. Tell me what the measure of the) circumference is, what (that of) the caloulated area and what (of) the diameter is. The rule for arriving at the practically approximate value of surface-areas resembling (the longitudinal sections of the yava grain, (of) the mardula, (of) the panava, and (of) the vajra --- 32. In the case of areas shaped in the form of the yava grain, of the muraja, of the panava and of the vajra, the 30. This rule will be clear from the following algebraical representation :Let c be the circumference of the circle. Ast is taken to be equal to 3, is the diameter and is the area of the circle. If m stands for the combined sum of the circumference, the diameter and the area of the circle, then the rule given in the stanza to the effect that ca 12 m + 64 - N 64 may be easily arrived at from the quarratic equation containing the data in the problem :- C + + 3 =m. 32. Muraja means the same thing as marda la and mrdanga. The shape of the various figures mentioned in this stanza is as follows: o CRD Yavakaraknetra. Morajakaraksetra. Panavakaraksetra. Vajrakaraksotra For Private and Personal Use Only

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CHAPTER VII-MEASUREMENT OF AREAS. Acharya Shri Kailassagarsuri Gyanmandir (require dmeasurement of) area is that which results by multiplying half the sum of the end measure and the middle measure by the length. Examples in illustration thereof. 33. In the case of an area resembling the configuration of a yava grain, the length is 80 and the breadth in the middle is 40. Tell me, what may be the calculated measure of that area ? 34. Tell (me what may be the calculated measure of the area) in relation to a field which has the outline configuration of the mrdanga, and of which the length is 80 dandas, the end measure is 20 and the middle measure is 40 dandas. 35. In the case of a field having the outline of the panava, the length is 77 dandas, the measure of each of the two ends is 8 dandas, and the measure in the middle is 4 dandas. (What is the measure of the area ?) 193 36. Similarly in the case of a field having the outline of the vajra, the length is 96 dandas, in the middle there is the middle point; and at the ends the measure is 13 dandas. (What is the measure of the area ?) These are looked upon as being derived from a quadrilateral figure which is divided into four triangles by means of its diagonals crossing each other. The The rule for arriving at the measure of areas such as the ubhaya-nisedha or di-deficient area : 37. On subtracting the product of the length into half the breadth from the product of the length into the breadth, you The measures of the area arrived at according to the rule given in this stanza are approximately correct in the case of all the figures, as the rule is based on the assumption that each of the bounding curved lines may be taken to be equal to the sum of two straight lines formed by joining the ends of the curves with the middle point thereof. 37. The figures mentioned in this stanza are those given below: For Private and Personal Use Only 25

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Shri Mahavir Jain Aradhana Kendra 194 www.kobatirth.org GANITASARASANGRAHA. Acharya Shri Kailassagarsuri Gyanmandir declare the measure of the di-deficient area. That which is less (than the latter product here) by half of this (above-mentioned quantity to be subtracted) is the measure of the area of the uni-deficient figure. An example in illustration thereof. 38. The length is 36, and the breadth is only 18 dandas. What is the resulting measure of the area in the case of a dideficient area, and what in the case of the uni-deficient area? The rule for arriving at the practically approximate measure of the area of fields resembling the outline of a multiplex vajra :-- 39. One-third of the square of half the perimeter, divided by the number of sides and (then) multiplied by the number of sides as diminished by one, gives indeed in the result the value of the area of all figures made up of sides. In the case of the area di-deficient. figure is that in which any two of the opposite triangles out of the four making up the quadrilateral are left out of consideration, the uni-deficient figure being that in which only one out of the four triangles is neglected. 92 n n 39. The rule stated in this stanza gives the area of figures made up of any number of sides. If s is half the sum of the measures of the sides, and n the number of sides, the area is said to be equal to X This formula is found to give the approximate value of the area in the case of a triangle, a quadrilateral, a hexagon and a circle conceived as a figure of infinite number of sides. The other part of the rule deals with the interspace bounded by parts of circles in contact, and the value of the area arrived at according to the rule here given is also approximate. The figure below shows an interspace so bounded by four touching circles. 88 For Private and Personal Use Only

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CHAPTER VIT-MEASUREMENT OF AREAS. 196 included between circles (in contact), one-fourth thus arrived at gives the required measure). of the result Examples in illustration thereof. 40. In the case of a six-sided figure the measure of a side is 5, and in the case of another figure of 16 sides the measure of a side is 3. Give out (the measure of the area in each case). 41. In the case of a trilateral figure one of the sides is 5, the opposite (i.e., the other) side is 7, and the base is 6. In the case of another bexalateral figure the sides are in measure from 1 to 6 in order. (Find out the value of the area in each case). 12. (Give out) the value of the interspace included inside four (equal) circles (in contact) having a diameter which is 9 in measure ; and (give out) the value of the area of the interspace iuclnded inside three circles having diameters measuring 6, 5 and 4 (respectively). The rule for arriving at the practically approximate area of a field resembling a bow in outline : 43. In the case of a bow-shaped field the calculated measure (of the area) is obtained by adding together (the measure of the arrow and (that of ) the string and multiplying the sum by half (the measure) of the arrow. The square root of the square of the (measure of the) arrow as multiplied by 5 and (then) as combined with the square of the (measure of the) string gives the (measure of the bent) stick (of the bow). 43. The field resembling a bow in outline is in fact the segment of a cirole, the bow forming the arc, the bow-string forming the chord, and the arrow measuring the greatest perpendicular distance between the arc and the chord, If a, c, and p represent the lengths of these three lines, then, according to the rules given in stanzas 43 and 45 -- Area = c + v) * Length of bow =V5p2 + c , of arrow = 5 >> of bow-string av a' -50 For accurate value see stanzas 731 and 74} in this chapter. For Private and Personal Use Only

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196 GANITASARASANGRAHA. An example in illustration thereof. 44. A bow-shaped field is seen whereof the string-measure is 26, and the arrow-measure is 13. Tell me quickly, O mathematician, what the calculated measure of this area) is, and what the measure of this (bent) stick (curve). The rule for arriving at the arrow-measure as well as the string-measure (in relation to a bow-shaped field): 45. The difference between the squares of the string and of the bent bow is divided by 5. The square root (of the resulting quotient) gives the intended measure of the arrow. The square of the arrow is multiplied by o; and (this product) is subtracted from the square (of the arc) of the bow. The square root (of the resulting quantity) gives the measure corresponding to the string. Examples in illustration thereof. 46. In the case of this (already given bow-shaped) field the measure of the arrow is not known; and in the case of another (similar field) the measure of the string is not known. O you who know calculation, give out both these measures. The rule for arriving at the practically approximate value of the area of the circle which is circumscribed about or inscribed within a four-sided figure : 47. Half of three times (the measure of the area of the inscribed quadrilateral figure) gives the measure of the area of the circle in the case in which it is circumscribed outside. In the case where it is inscribed within and the quadrilateral is the other way (i.e., escribed), half of the above measure (is the required quantity). 47. The formula here given may be seen to be accurate in the case of a equare, but only approximate in the case of other quadrilaterals, if 3 be taken to be the correct value of t. For Private and Personal Use Only

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CHAPTER VIIMEASUREMENT OF AREAS. 197 An example in illustration thereof. 48. In relation to a quadrilateral figure, each of whose sides is 15 (in measure), tell me the practically approximate value of the inscribed and the escribed circles. Thus ends the calculation of practically approximate value in relation to areas. The Minutely Accurate Calculation of the Measure of Areas. Hereafter in the calculation regarding the measurement of areas we shall expound the subject of treatment known as minutely acourate calculation. And that is as follows: The rule for arriving at the measure of the perpendicular (from the vertex to the base of a given triangle) and (also) of the segments into which the base is thereby divided) : 49. The process of sankramana carried out between the base and the difference between the squares of the sides as divided by the base gives rise to the values of the two segments of the base) of the triangle. Learned teachers say that the square root of the difference between the squares of (either of) these (segments) and of the (corresponding adjacent) eide gives rise to tbe measure of the perpendicular. 49. Algebraically represented c = (c + d = *)**;. p=Va? - 6,2 or 16 - . Here a, b, c, represent the measures of the sides of a triangle, C1, C, the measures of the segments o the base whose total length is c; and p represents the length of the perpendicular. For Private and Personal Use Only

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Shri Mahavir Jain Aradhana Kendra 198 www.kobatirth.org GANITASARASANGRAHA. The rule for arriving at the minutely accurate measurement of the area (of trilateral and quadrilateral figures) : -- Acharya Shri Kailassagarsuri Gyanmandir 50. Four quantities represented (respectively) by half the sum of the sides as diminished by (each of) the sides (taken in order) are multiplied together; and the square root (of the product so obtained) gives the minutely accurate measure (of the area of the figure). Or the measure of the areas may be arrived at by multiplying by the perpendicular (from the top to the base) half the sum of the top measure and the base measure. (The latter rule does) not (hold good) in the case of an inequi-lateral quadrilateral figure. Examples in illustration thereof. 51. In the case of an equilateral triangle, 8 dandas give the measure of the base as also of each of the two sides. You, who know calculation, tell me the accurate value of the area (thereof) and also of the perpendicular (to the base) as well as of the segments (of the base caused thereby). 52. In the case of an isosceles triangle (each of the) two (equal) sides measures 13 dandas, and the base measures 10. (What is) the accurate measure of the area thereof, and of the perpendi 50. Algebraically represented : Area of a trilateral figures (8-a) (8-6) (8-c); where s is half the sum of the sides, a, b, c, the respective measures of the sides of the trilateral figure; s or= x p,where p is the perpendicular 2 distance of the vertex from the base. Area of a quadrilateral figure = (-a) (s-b) (s-c) (s-d) where s is half the sum of the sides, and a, b, c, d the measures of the respective sides of the quadrilateral figure; or = b + d 2 xp (except in the case of an inequilateral quadrilateral) where p is the measure of either of the perpendiculars drawn to the base from the extremities of the top side. The formulas here given for trilateral figures are correct; but those given for quadrilatral figures hold good only in the case of cyclic quadrilaterals, as in these formulas sight is lost of the fact that for the same measure of the sides the value of the area as well as of the perpendicular may vary. For Private and Personal Use Only

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CHAPTER VII - MEASUREMENT OF AREAS, 199 cular (to the base) as also of the segments of the base caused thereby)? 53. In the case of a scalene triangle one of the sides is 13 (in measure), the opposite side is 15, and the base is 14. What indeed is the calculated measure of the area of this figure), and what of the perpendicnlar (to the base) and of the basal segments ? Hereafter we give) the rule for arriving at the value of the diagonal of the five varieties of quadrilateral figures. 54. The two quantities obtained by multiplying the basal side by the (larger and the smaller of the right and the left sides are (respectively) combined with the two (other) qualitities obtained by multiplying the top side by the (smaller and the larger of the right and the left) sides. The resulting) two sums constitute the multiplier and the divisor as also the divisor and the multiplier in relation to the sum of the products of the opposite sides. The square roots (of the quantities so obtained) give the required measures of the diagonale. Examples in illustration thereof. 55. In the case of an equilateral quadrilateral which has all around a side measure of 5, tell me quickly, O friend who know the secret of calculation, the value of the diagonal and also the accurate value of the area. 54. Algebraically represented the measure of the diagonal of a quadrilateral figure as given bere is - (ac+bd) (ab + cd) Mac + bd) (ad + be). N ad + bc of N ab + cd These formulas alan are correct only for cyclic quadrilaterals. Bhaskar carya is aware of the futility of attempting to give the measure of the area of a quadrilateral without previously knowing the values of the perpendicular or of the diagonals. Vide the following stanza from his Lilavati : lambayoH karNayorvaikamanirdizyAparAn katham / pRcchatyaniyatatve'pi niyataM cApi tatphalam / / sa pRcchakaH pizAco vA vaktA vA nitarAM tataH / yo na vetti caturbAhukSetrasyAniyatAM sthitim / / For Private and Personal Use Only

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200 GANITASARASANGRAHA. 56. In the case of a longish quadrilateral, the (horizontal) side is 12 in measure and the perpendicular side is 5 in measure. Tell me quickly what the measure of the diagonal is and what the accurate measure of the area. 57. The basal side of an equi-bilateral quadrilateral is 36. One of the sides is 61 and the other also is the game. The top side is 14. What is the diagonal and what the accurate measure of the area ? 58. In the case of an equi-trilateral quadrilateral, the square of 13 (gives the measure of an equal side); the base, however, is 407 in measure. What is the value of the diagonal, of the basal segments, of the perpendicular and of the area ? 59. The (right and the left) sides of an inequilateral quadrilateral are 13 x 15 and 13 X 20 (respectively in measure); the top side is 53, and the side below is 300. What are all the values here beginning with that of the diagonal ? Hereafter are given the rules for arriving at the minutely accurate values relating to curvilinear figures. Among them the rule for arriving at the minutely accurate values relating to a circular figure is as follows: 60. The diameter of the circular figure multiplied by the square root of 10 becomes the circumference in measure). The circumference multiplied by one-fourth of the diameter gives the area. In the case of a semicircle this happens to be half (of what it is in the case of the circls). Examples in illustration thereof. 61. In the case of one (circular) field the diameter of the circle is 18; in the case of another it is 60; in the case of yet another it is 22. What are the circumferences and the areas ? 60. The value of given in this stanza is v10, which is equal to 3.16 ....... Compare this with the more approximate value 5280% (= 3.1416) given by Aryabhata. Bbaskaracarya also gives to it the same value, and represente it in reduced terms as 1250 For Private and Personal Use Only

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CHAPTER VIIMEASUREMENT OF AREAS. 201 62. In the case of a semicircular field of a diameter measuring 12, and of (another) field having a diameter of 36 in measure what is the ciroumference and what the area ? The rule for arriving at the minutely accurate values relating to an elliptical figure :-- 63. The square of the (shorter) diameter is multiplied by 6 and the square of twice the length (as measured by the longer diameter) is added to this. (The square root of this sum gives) the measure of the circumference. This measure of the circumference multiplied by one-fourth of the (shorter) diameter gives the minutely accurate measure of the area of an elliptical figure. An e:.ample in illustration thereof. 64. In the case of an elliptical figure, the length (as measured by the longer diameter) is 36, and the breadth (as measured by the shorter diameter) is 12. Tell me, after calculation, what the measure of the circumference is, and what the minutely accurate measure of the area. The rule for arriving at the minutely accurate values in relation to a conchiform figure :-- 65. The maximum measure of the) breadth (of the figure), diminished by half (the measure of the breadth) of the mouth, and (then) multiplied by the square root of 10, gives rise to the measure of the perimeter. The square of half the (maximum) 63. If a represents the measure of the longer dia ineter and b that of the shorter diameter of an ellipse, then, according to the rule given here, the oircumference is V 682 + 4a", and the area is bx V 662 + 1a. It may be noted that this stanza, as found in the MSS., omits to mention that the square root of the quantity is to be taken for arriving at the value of the circnmference. The formula for the area given here is only an approximation, and seems to be based on the analogy of the area of a circle as represented by ad where d is the diameter and md is the circumference. 654. Algebraically, circumference = (a - m) x V 10; 72 m 2 area=[{(a - m) * }'+4)*7x vid; where a is the measure of the 26 For Private and Personal Use Only

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202 GANITASARASANGRAHA, breadth (of the figure) as diminished by half the (breadth of the) mouth, and the square of one-fourth of the (breadth of the mouth are added together; and the resulting sum is multiplied by the square root of 10. This gives rise to the minutely accurate measure of the area in the case of the conchiform figure. An example in illustration thereof. 665. In the case of a conchiform curvilinear figure the (maxi. mum breadth is 18 dandas, and the breadth of the mouth is 4 (dandas). What is the measure of the perimeter and what the minutely accurate measure of the area as calculated ? The rule for arriving at the minutely accurate measures in relation to outreaching and inlying annular figures : 671. The inner) diameter, to which the breadth (of the annulus) is added, is multiplied by the square root of 10 and by the breadth (of the annulus). This gives rise to the value of the area of the out-reaching annulus. The (outer) diameter as diminished by the breadth of the annulus) gives rise (on being treated in the same manner as above) to the value of the area of the inlying annular figure. Examples in illustration thereof. 681. Eighteen dandas measure the (inner or the outer) diameter of the annulus (as the case may be); the breadth of the annulus is, however, 3 (dandas). You give out the minutely acourate value of the area of the outreaching as well as the inlying annular figure. 691. The (outer) diameter is 18 dandas, and the breadth of the inlying annulus is 4 dandas. You give out the minutely accurate value of the area of the inlying annular figure. maximum breadth, and m the measure of the mouth of a conchiform figure. As observed in the pote relating to stanza 23 of this chapter, the figure intended is obviously made up of two unequal semioircles. For Private and Personal Use Only

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CHAPTER VII-MEASUREMENT OF AREAS. ! 20% The rule for arriving at the minutely accurate values relating to a figure resembling (the longitudinal section of the yava grain, and also to a figure having the outline of a bow: 70. It should be known that the measure of the string (chord) multiplied by one-fourth of the measure of the arrow, and then multiplied by the square root of 10, gives rise to the (accurate) value of the area in the case of a figure having the outline of a bow as also in the case of a figure resembling the (longitudinal) section of a yava grain. Examples in illustration thereof. 713. In the case of a figure resembling (the longitudinal) section of the yava grain, the (maximum) length is 12 andas; the two ends are needle points, and the breadth in the middle is 4 dandas. What is the area ? 72. In the case of a figure having the outline of a bow, the string is 24 in measure ; and its arrow is taken to be 4 in measure. What may be the minutely accurate valne of the area ? The rule for arriving at the measure of the (bent) stick of the bow as well as of the arrow, in the case of a figure having the outline of a bow: l'he square of the arrow measure is multiplied by 6. To this is added the square of the string measure. The square 704. The figure resembling a bow is obviously the segment of a circle. The area of the segment as given here=cx?.* 10. This formula is not accurate. It seems to be based on the analogy of the rule for obtaining the area of a semi-circle, which area is evidently equal to the product of , the diameter and one-fourth of the radius, i.e., * * 27 * . The figure regembling the longitudinal section of a yava grain may be easily seen to be made up of two similar and equal segments of a circle applied to each other so as to have a common chord. It is evident that in this case the value of the arrow-line becomes doubled. Thus the same formula is made to hold good here also. 731 & 743. Algebraically, arc =V6px +ca; perpendicular = chord =Va -- 6 pa. For Private and Personal Use Only

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204 GANITASARASANGRAHA. root of that (which happens to be the resulting sum here) gives rise to the measure of the (bent) bow-stick. In the case of finding out the measure of the string and the measure of the arrow, a course converse to this is adopted. The rule relating to the process according to the converse (here mentioned) : 741. The measure of the arrow is taken to be the square root of one-sixth of the difference between the square of the string and the square of the (bent stick of the) bow. And the square root of the remainder, after subtracting six times the square of the arrow from the square of the (bent stick of the bow, gives rise to the measure of the string. An example in illustration thereof. 751. In the case of a figure having the outline of a bow, the string-measure is 12, and the arrow-measure is 6. The measure of the bent stick is not known. You (find it out), O friend. (In the case of the same figure) what will be the string-measure (when the other quantities are known), and what its arrow-moasure (when similarly the other requisite quantities are known) ? The rule for arriving at the minutely accurate result in relation to figures resembling a Mrdanga, and having the outline of a Panava, and of a Vajra: 761. To the resulting area, obtained by multiplying the (maximum) length with (the measure of the breadth of) the mouth, the value of the areas of its associated bow-shaped figures is added. The resulting sum gives the value of the area of a figure resembling (the longitudinal section of) a Mrdanga. In the case In giving the rule for the measure of the arc in terms of the chord and the largest perpendicular distance betweeen the arc and the chord, the art forming a semicircle is taken as the basis, and the formula obtained for it is utilized for arriving at the value of the aro of any segment. The semicircular aro =rX V 10 V 1079=V6rP + 4r2 : based on this is the formula for any arc; where p= the largest perpendicular distance between the aro and the chord, and c = the chord. 761. The rationale of the rule here given will be clear from the figures given in the note under stanza 32 above. For Private and Personal Use Only

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CHAPTER VII-MEASUREMENT OF AREAS. 205 of those two (other) figures which resemble (the longitudinal section of) the Panava, and (of) the Vajra, that (same resulting area, which is obtained by multiplying the maximum length with the measure of the breadth of the mouth), is diminished by the measure of the areas of the associated bow-shaped figures. (The remainder gives the requird measure of the area concerned.) Examples in illustration thereof. 771. In the case of a figure having the outline configuration of a Mrdanga, the (maximum) length is 24; the breadth of each of) the two mouths is 8; and the (maximum) breadth in the middle is 16. What is the area ? 787. In the case of a figure having the outline of a Panava, the (maximum.) length is 24 ; similarly the measure of the breadth of either of the two mouths is 8; and the central breadth is 4. What is the area ? 791. In the case of a figure having the outline of a Vajra, the (maximum) length is 24; the measure (of the breadth of either) of the two mouths is 8; and the centre is a point. Give out as before what the area is. The rule for arriving at the minutely accurate value of the areas of figures resembling (the annulus making up) the rim of a wheel, (resembling) the crescent moon and the (longitudinal) section of the tusk of an elephant : 801. In the case of (a circular annulus resembling) the rim of a wheel, the sum of the measures of the inner and the outer curves is divided by 6, multiplied by the measure of the breadth 803. The rule here given for the area of an annulus, if expressed algebraically, comes to be 41 2 * p * v 10, where a, and a, are the measures of the two circumferences, and p is the measure of the breadth of the annulus. On & comparison of this value of the area of the annulus with the approximate value of the same as given in stanza 7 above (vide note thereunder), it will be evident that the formula here does not give the accurate value, the value mentioned in the rule in stanza 7 being itself the accurate value. The mistake seems to have arisen from & wrong notion that in the determination of the value of this area, * is involved even otherwise than in the values of ai and ago anze 7 does not give them note thereunder, For Private and Personal Use Only

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206 GANITASARASANGRAHA. of the annulus, and again multiplied by the square root of 10. (The result gives the value of the required area.) Half of this is the (required) value of the area in the case of figures resembling the crescent moon or (the longitudinal section of) the tusk of an elephant. Examples in illustration thereof. 813. In the case of a field resembling (the circular annulus forming) the rim of a wheel, the outer curve is 14 in measure and the inner 8; and the (breadth in the middle is 4. (What is the area?) What is it in the case of a figure resembling the crescent moon, and in the case of a figure resembling the longitudinal section of the task of an elephant (the measures requisite for calculation being the same as above) ? The rule for arriving at the minutely accurate value of the area of a figure forming the interspace included inside four (equal) circles (touching each other) :-- 821. If the minutely accurate measure of the area of any one circle is subtracted from the quantity which forms the square of the diameter (of the circle), there results the value of the area of the interspace inoluded within four equal circles (touching each other). An example in illustration thereof. 831. What is the minutely accurate measure of the area of the interspace included within four mutually touching (equal) circles whose diameter is 4 (in value) ? 82}. The rationale of the rule will be clear from the figare below: For Private and Personal Use Only

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CHAPTER VII-MEASUREMENT OF AREAS. 207 The rule for arriving at the minutely accurate value of the figure formed in the interspace caused by three (equal) circular figures touching each other : 843. The minutely accurate measure of the area of an equilateral triangle, each side of which is equal in measure to the diameter of the circles) is diminished by half the area of any of the three equal) circles. The remainder happens to be the measure of the interspace area caused by three (mutually touching equal circles). An example in illustration thereof. 851. What is the minutely accurate calculated value of a figure forming the interspace enclosed by three mutually touching (equal) circles the diameter (of each of which is 4 in measure ? The rule for arriving at the minutely, accurate values of the diagonal, the perpendicular and the area in the case of a (regular) six-sided figure : 861. In the case of a (regular) six-sided figure, the measure of the side, the square of the side, the square of the square of the side multiplied respectively by 2, 3 and 3 give rise, in that same order, to the values of the diagonal, of the square of the perpendi. cular, and of the square of the measure of the area. 84). Similarly the figure here elucidates at once the reason of the rule : o 86. The role seems to contemplate a regular hexagon. The formula given for the value of the area of the hexagon is v3a", where a is the length of a side The correct formula, however, is a 2x Per, is a 2x 3 V3 For Private and Personal Use Only

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208 GANITASARASANGRAHA. An example in illustration thereof. 87. In the case of a (regular) six-sided figure each side is 2 dundas in measure. In relation to it, what are the squares of the measures of the diagonal, of the perpendicular and of the minutely accurate area of the figure ? The rule for arriving at the numerical measure of the sum of a number of square root quantities as well as of the remainder left after subtracting a number of square root quantities one from another in the natural order : 881. (The square root quantities are all) divided by (such) & (common) factor (as will give rise to quotients which are square quantities). The square roots (of the square quantities so obtained) are added together, or they are subtracted (one from another in the natural order). The sum and remainder (80 obtained) are (both) squared and (then) multiplied (separately) by the divisor factor (originally used). The square roots (of these resulting products) give rise to the sum and the (ultimate) difference of the quantities (given in the problem). Know this to be the process of calculation in regard to (all kinds of) square root quantities. An example in illustration thereof. 891. O my friend who know the result of calculations, tell me the sum of the square roots of the quantities consisting of 16, 36 and 100; and then (tell me) also the (ultimate) remainder in relation to the square roots (of the same quantities). Thus ends the minutely accurate calculation (of the measure of areas). 883. The word karani occurring here denotes any quantity the square root of which is to be found out, the root itself being rational or irrational as the case may be. The rule will be clear from the following working of the problemi given in stanza 89: To find the value of V18+ 36+ 100, and 100- (V36-V16). These are to be represented as vA (V4 +v9+ 25); v4 (125-(19-v4). =V 4 (2+3+5); =VA { 5 - (3 - 2). =V 4 (10); =N4* v 100; =NXV 16. = V 400; =V 64. = 8. = 20; For Private and Personal Use Only

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CHAPTER VII-MEASUREMENT OF AREAS. 209 Subject of treatment known as the Janya operation. Hereafter we shall give out the janya operation in calculations relating to measurement of areas. The rule for arriving at a longish quadrilateral figure with optionally chosen numbers as bajas :-- 902. In the case of the optionally derived longish quadrilateral figure the difference between the squares (of the bija numbers) constitutes the measure of the perpendicular-side, the product (of the baja numbers) multiplied by two becomes the (other) side, and the sum of the squares (of the bija numbers) becomes the hypotenuse. Examples in illustration thereof. 91. In relation to the geometrical figure to be derived optionally, 1 and 2 are the bejas to be noted down. Tell (me) quickly after calculation the measurements of the perpendicular-side, the other side and the hypotenuse. 927. Having noted down, O friend, 2 and 3 as the bijas in relation to a figure to be optionally derived, give out quickly, after calculating, the measurements of the perpendicular-side, the other side and the hypotenuse. Again another rule for constructing a longish quadrilateral figure with the aid of numbers denoted by the name of bijas: 93. The product of the sum and the difference of the bajas forms the measure of the perpendicular-side. The sankramana of 903. Janya literally means "arising from " or "apt to be derived"; hence it refers here to trilateral and quadrilateral figures that may be derived out of certa ir given data. The operation known as janya relates to the finding out of the length of the sides of trilateral and quadrilateral figures to be so derived. Bija, as given here, generally happens to be a positive integer. Two such are invariably given for the derivation of trilateral and quadrilateral tigures dependent on them. The rationale of the rule will be clear from the following algebraica representation : If a and b are the bija numbers, then a? - b is the measure of the perpendicular, 2 ab that of the other side, and a2 + b2 that of the hypoten uge, of an oblong. From this it is evident that the bejas are numbers with the aid of the product and the squares whereof, as forming the measures of the sides, a rightangled triangle may be constructed. 27 For Private and Personal Use Only

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210 GANITASARASANGRAHA. the squares of that (sum and the difference of the bajas) gives rise (respectively) to the measures of the other) side and of the hypotenuse. This also is a process in the operation of (constructing a geometrical) figure to be derived (from given lijas). An example in illustration thereof. 941. O friend, who know the secret of calculation, construct a derived figure with the aid of 3 and 5 as bajas, and then think out and mention quickly the numbers measuring the perpendicular-side, the other side and the hypotenuse (thereof). The role for arriving at the bija numbers relating to a given figure capable of being derived (from bejas). 954. The operation of sankramana between (an optionally chosen exact) divisor of the measure of the perpendicular-side and the resulting quotient gives rise to the (required) bejas. (An optionally chosen exact) divisor of half the measure of the (other) side and the resulting quotient (also) form the bijas (required). Those (bijas) are, (respectively), the square roots of half the sum and of half the difference of the measure of the hypotenuse and the square of a (suitably) chosen optional number. An example in illustrution thereof. 964. In relation to a certain geometrical figure, the perpendigular is 16 : what are the bijas? Or the other side is 30 : what are the bujas? The hypotenuse is 34 : what are they (the bejas)? The rule for arriving at the numerical measures of the other side and of the hypotenuse, when the numerical measure of the perpendicular-side is known; for arriving at the numerical measures of the perpendicular-side and of the hypotenuse, when the numerical measure of the other side is known; and for arriving 934. In the rule given bere, a -, 2 ab, and a + b2 are represented as (a + b)? - (a - b), (a + b)* + (a - b) la + b)(a - b), 2 953. The processes mentioned in this rule may be seen to be converse to the operations mentioned in stanza 90). For Private and Personal Use Only

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Shri Mahavir Jain Aradhana Kendra 211 at the numerical measure of the perpendicular-side and of the other side, when the numerical measure of the hypotenuse is known : 97. The operation of sankramana, conducted between (an optionally chosen exact) divisor of the square of the measure of the perpendicular-side and the resulting quotient, gives rise to the measures of the hypotenuse and of the other side (respectively). Similarly (the same operation of sankramana) in relation to the square of the measure of the other side (gives rise to the measures of the perpendicular-side and of the hypotenuse). Or, the square root of the difference between the squares of the hypotenuse and of a (suitably chosen) optional number forms, along with that chosen number, the perpendicular-side and the other side respectively. www.kobatirth.org An example in illustration thereof. 981. In the case of a certain (geometrical) figure, the perpendicular-side is 11 in measure; in the case of another figure, the (other) side is 60; and in the case of (still) another figure the hypotenuse is 61. Tell me in these cases the measures of the unmentioned elements. CHAPTER VII-MEASUREMENT OF AREAS. The rule regarding the manner of arriving at a quadrilateral figure having two equal sides (with the aid of the given bijas, : I. Acharya Shri Kailassagarsuri Gyanmandir 99. The perpendicular-side of the primary figure derived (with the aid of the given hijas), on being added to the perpendicularside (in another figure) derived with the aid of the (two optionally chosen) factors of half the base of (this original) derived 97. This rule depends on the following identities: {' (a2 62) 2 (ab)2 (2 ab)2 262 II. *{ -+-(ab)2 / 2 = a2 + b2 or 2ab as the case may be * b) 2 } / 2 = a2 + +262 } / 2 = a2 + b2 or a2 -- b2. III. (a2 + b2)2 -- (2 ab)" -- a2 -- b2. 99. The problem solved in the rule stated in this stanza is to construct with the aid of two given bijas a quadrilateral having two equal sides. The lengths of the sides, of the diagonals, of the perpendicular from the end-points of the topside to the base, and of the segments thereof caused by the perpendicular are all derived from two rectangles constructed with the aid of the given bijas. The first of these rectangles is formed according to the rule given in stanza 90 above. The second rectangle is formed according to the same rule from two optionally chosen factors of half the length of the base of the first rectangle, For Private and Personal Use Only

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212 GANITASARASANGRAHA. figure (taken as the bejas, gives rise to the measure of the base of the (required) quadrilateral with two equal sides. The difference (between the measures of these two perpendiculars) gives the top-measure of the quadrilateral). The smaller of the diagonale (relating to the two derived figures already mentioned) gives the measure of (either of the two equal) sides. The smaller of the (two) perpendicular-sides in relation to the two derived figures under reference) gives the measure of the smaller) segment (of the base formed by the perpendicular dropped thereunto from either of the end-points of the top-side). The larger of the two) diagonals (in relation to the two derived figures of reference) gives the measure of the (required) diagonal. The area of the larger (of two derived figures of reference) is the area of the (required) taken as bijas. Hence the first rectangle is called the primary figure in the translation to distinguish it from the secoud rectangle. The rationale of the rule will be clear from the following diagrams illustrating the problein given for solution in stanza 100. Here 5 and 6 are the bejas given ; and the first rectangle or the primary figure derived from the bijas is ABCD :-- A Half the length of the base in this figure is 30; and two factors of this, namely, 3 and 10 may be chosen. The rectangle constructed with the aid of these numbers as bejas is EFGH: 80 109 To construct the required quadrilateral with two equal sides, one of the two triangles into which the first rectangle is divided by its diagonal is applied to the 91 second ractangle on one side, and a portion equal to the same triangle is removed from the same second rectangle on the other side, as shown in the figure HAFO'. For Private and Personal Use Only

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CHAPTER VII-MEASUREMENT OF AREAS. 213 figure ; and the measure of the base (of either of the derived figures of reference) happens to be the measure of the perpendicular (dropped to the base from either of the end-points of the topside in the required figure). An example in illustration thereof. 1001. In relation to a quadrilateral with two equal sides constructed with the aid of 5 and 6 as bejas give out the measures of the top side, of the base, of (either of the two equal) sides, of the perpendicular (from the top to the base), of the diagonal), of the lesser) segmeat (of the base), and of the area. The rule for arriving at the measures of the top-side, of the base, of any one of) the (equal) sides, of the perpendicular (from the top to the base), of the diagonal, of the (lesser) segment (of the base) and of the area, in relation to a quadrilateral having three equal sides (with the aid of given bijas) : The process will be clear from a comparison of the diagrams: I G 111 80 Area of the reqnired quadrilateral, HAFC' 61 = area of the second 160109 rectangle, EFGH. 61 60 A' E 102 Basc AF = perpendicular-side of the first rectangle plus perpendicular side of the second rectangle, i.e., AB + EF. Top side HC =perpendicular-side of the second rectangle minua per pendicular-side of the first rectangle, i.e, GH - CD. Diagonal HF = diagonal of the second rectangle. Smaller segment of the base, 6.e., A'E= perpendicular-side of the first rectangle, i.e., AB. Perpendicular HE= base of the first or of the second rectangle, i.e., BC or FG. Each of the lateral equal sides A'H and FC'= diagonal of the first rec tangle, i.e., AC. For Private and Personal Use Only

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214 GANITASARASANGRAHA. 1013. The difference between the (given) bijas is multiplied by the square root of the base (of the quadrilateral immediately derived with the aid of those bijas). The area of (this immediately) derived (primary) quadrilateral is divided by the product so obtained). Then, with the aid of the resulting quotient and the divisor (in the operation utilized as bijas, a second derived quadrilateral of reference is constructed. A third quadrilateral of 101. If a and b represent the given bijas, the measures of the sides of the immediately derived quadrilateral are: Perpendicular-side=a? - 62 Base = 2ab Diagonal= a2 + 12 Area = 2ab x (a - b) As in the case of the construction of the quadrilateral with two equal sides (vide stanza 99$ ante), this rule proceeds to construct the required quadrilateral with three equal sides with the aid of two derived rectangles. The bujas in relation to the first of these rectangles are :2ab x (a - 82): i.e., 2ab x (a + b), and v2ab x (a - b). V 2ab + (a - b) Applying the role given in stanza 90) above, we havo for the first rec tangle : Perpendicular-side = (a + b) ? x 2ab - (a - b) ? x 2ab or 8 abo. Base = 2 x V 2ab x (a + b) x V 2ab x (a - b) or 4ab (a- b). Diagonal = (a + b)2 x 2ab + (a - b) ? x 2ab or 4a) (a + b). The bejas in the case of t.be second rectangle are: a - b and 2ab. The various elements of this rectangle are : Perpendicular-side = 4a 1o - (a? - 6%) ; Base = 4ab (a2 - 12); Diagonal = 4a2 b2 + (a? - baya or (u? + b) With the help of tbese two rectangles, the measures of the sides, diagonals, etc., of the required quadrilateral are ascertained as in the rule given in stanza 99% above. They are : Base = sum of the perpendicular sides=82b2 + 4a2b? -(a* - 12). Top-side greater perpendicular-side ninus smaller perpendicular-side = Saab% - 4a2j2 - (a? - 62)2 = (a' +62)2. Either of the lateral sides=smaller diagonala? + b2, 2. Lesser segment of the base = smaller perpendicular-sido = 4a2b2 - (a* -6?). Perpendionlar = base of either rectangle == 4ab (a? -69). Diagonal = the greater of the two diagonals =4ab (a? +62). Area area of the larger reotangle =80262 x 4ab (as - 62). It may be noted here that the measure of either of the two lateral sides is equal to the measure of the top-side. Thus is obtained the required quadrilateral with three equal sides. For Private and Personal Use Only

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CHAPTER VII-MEASUREMENT OF AREAS. reference is further constructed) with the aid of the measurements of the base and the perpendicular-side (of the immediately derived quadrilateral, above referred to, used as bijas. Then, with the aid of these two last derived secondary quadrilaterals, all the required) quantities appertaining to the quadrilateral with three equal sides are (to be obtained) as in the case of the quadrilateral with two equal sides. Acharya Shri Kailassagarsuri Gyanmandir An example in illustration thereof. 102. In relation to a quadrilateral with three equal sides and having 2 and 3 as its bijas, give out the measures of the top-side, of the base, of (any one of) the (equal) sides, of the perpendicnlar (from the top to the base), of the diagonal, of the (lesser) segment (of the base) and of the area. The rule for arriving at the measures of the top-side, of the base, of the (lateral) sides, of the perpendiculars (from the ends of the top-side to the base), of the diagonals, of the segments (of the base) and of the area, in relation to a quadrilateral the sides of which are (all) unequal : :-- Segments= 103. With the longer and the shorter diagonals (of the two derived rectangular quadrilateral figures related to the two sets 215 103. The rule will be clear from the following algebraical representation. Let a, b, and c, d, be two sets of given bijas. Then the various required clements are as follow: Lateral sides 2ab (c2 + d2) (a2+ b2) and (a2-b2)(c2 + d2) (a2 + b2). Base 2cd (a2 + b2)(a2 + b2). Top-side (c-d2)(a+b2) (a2 + b2). Diagonals = {(a2*--b3) x 2cd+ (c2 -- d2)2ab } x (a2+b2); and {(a2-b2) (c2-d2) + 4abcd} x (a2+b2) Perpendiculars = {(a2 -- b2) x 2cd + (c2 --d2) 2ab} x 2ab; and {(a2 - b ) (c3 -- d2)+4abcd} x (a2 -- b2) + 4abcd x 2ab. {(a2_b2) x 2cd + (c2-d2) x 2ab} (a -b2); and {(a2--ba) (c3--42) } For Private and Personal Use Only

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216 GANITASARASANGRAHA. of given bajas), the base and the perpendicular-side of the smaller and the larger derived figures of reference are respectively multiplied. The products (so obtained) are (separately) multiplied (again) by the shorter diagonal. The resulting products give the measures of the two (unequal) sides, of the base and of the top-side (in relation to the required quadrilateral). The perpendicular-sides (of the derived figures of reference) are multiplied by each other's bases ; and the two products (80 obtained) are added together. Then to the product of the (two) perpendicular-sides (relating to the two figures of reference), the product of the bases (of those same figures of reference) is added. The (two) sums (80 obtained), when multiplied by the shorter of the (two) diagonals (of the two figures of reference), give rise to the measures of the (required) diagonals. (Those same) sums, when multiplied by the base and the perpendicular-side (respectively) of the smaller figure (of reference), give rise to the measures of the perpendiculars (droppeu from the ends of the diagonals); and when multiplied (respectively) by the perpendicular-side and the base (of the same figure of reference), give rise to the measures of the segments of the base (caused by the perpendicrtlars). The measures of these segments, when subtracted from the measure of the base, give the values of the (other) segments (thereof). Half of the product of the diagonals (of the required figure arrived at as above) gives the measure of the area of the required figure). An example in illustration thereof. 1041. After forming two derived figures (of reference) with 1 and 2, and 2 and 3 as the requisite bejas give out, in relation to a quadrilateral figure the sides whereof are all unequal, the values of the top-side, of the base, of the lateral) sides, of the perpendiculars, of the diagonals, of the segments (of the base), and of the area. Again another rule for arriving at (the measures of the sides, etc., in relation to a quadrilateral, the sides of which are all unequal For Private and Personal Use Only

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CHAPTER VII-MEASUREMENT OF AREAS. 217 1052--1073. The square of the diagonal of the smaller (of the two derived oblongs of reference), as multiplied (separately) by the base and also by the perpendicular-side of the larger (oblong of reference, gives rise to the measures (respectively) of the base and of the top-side of the required quadrilateral having unequal sides). The base and the perpendicular-side of the smaller (oblong of reference, each) multiplied successively by the two diagonals (one of each of the oblongs of reference), give rise to the measures (respeotively) of the two (lateral) sides (of the required quadrilateral). The difference between the base and the perpendicular-side of the larger (oblong of reference) is in two positions (separately) multiplied by the base and by the perpendicular-side of the smaller (oblong of reference). The two (resulting) products (of this operation) are added (separately) to the product obtained by multiplying the sum of the base and the perpendicular-side of the smaller (oblong of reference) with the perpendicular-side of the larger (oblong of reference). The two sums (80 obtained), when multiplied by the diagonal of the smaller (oblong of reference), give rise to the values of the two diagonals (of the required quadrilateral). The diagonals (of the required quadrilateral) are (separately) divided by the diagonal of the smaller (oblong of 105-107}. The same values an are mentioned in the footnote to stanza 1031 above are given here for the measures of the sides, etc.; only they are stated in a slightly different way. Adopting the same symbols as in the note to stanza 103}, we hare: Diagonals- [{2cd=(^ -- de) } 2ab + {2ab +(02-00)} --- 4*)] *Page #415
218 GANITASARASANGRAHA. reference). The quotients (80 obtained) are multiplied respectively by the perpendicular-side and the base of the smaller (oblong of reference. The resulting) products give rise to the measures of the perpendiculars (in relation to the required quadrilateral). To these two perpendiculars), the above values of the two sides (other than the base and the top-side) are (separately) added, (the larger side being added to the larger perpendicular and the smaller side to the smaller perpendicular). The differences between these perpendiculars and sides are also obtained (in the same order). The sums (above noted) are multiplied (respectively) by (these) differences. The square roots (of the products so obtained) give rise to the values of the segments of the base in relation to the required quadrilateral). Half of the product of the diagonals (of the required quadrilateral) gives the value of (its) area. The rule for arriving at an isosceles triangle with the aid of a single derived oblong (of reference). 1081. The two diagonals (of the oblong of reference constructed with the aid of the given bijas) become the two (equal) sides of the (required) isosceles triangle. The base (of the oblong of reference), multiplied by two, becomes the base of the required triangle). The perpendicular-side (of the oblong of reference) is the perpendicular (of the required triangle from the apex to the base thereof). The area (of the required triangle) is the area (of the oblong of reference). 108. The rationale of the rule may be made out thus :- Let ABCD be an oblong and let AD be produced to E so that AD=DE. Join EC. It will be Seen that ACE is an isosceles triangle whose equal sides are equal to the diagonals of the oblong and whose area is equal to that of the ublong. For Private and Personal Use Only

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CHAPTER VII- MEASUREMENT OF AREAS. 219 An example in illustration thereof. 1091. O mathematician, calculate and tell me quickly the measures of the two (equal) sides, of the base and of the perpendicular in relation to an isosceles triangle derived with the aid of 3 and 5 as bejas. The rule regarding the manner of constructing a trilateral figure of unequal sides 1101. Half of the base of the (oblong of reference) derived (with the aid of the given bajas) is divided by an optionally chosen factor. With the aid of the divisor and the quotient (in this operation as bijas), another (oblong of reference) is derived. The sum of the perpendicular-sides belonging to these two (oblongs of reference) gives the measure of the base of the required) trilateral figure having unequal sides. The two diagonals (related to the two oblongs of reference) give the two sides (of the required triangle). The base (of either of the two oblongs of reference) gives the measure of the perpendicular (in the case of the required triangle). An eaample in illustration thereof. 1111. After constructing a second (derived oblong of reference) with the aid of half the base of the (original) figure (i.e. oblong of reference) derived with the aid of 2 and 3 as bujas, you tell (me) by means of this (operation) the values of the sides, of the base and of the perpendicular in a trilateral figure of unequal sides. Thus ends the subject of treatment known as the Janya operation. 1104. The role will be clear from the following construction :-Let ABCD and EFGH be the two B OF derived oblongs, such that the base AD = the base EH. Produce BA to Ko that Ak=EF. It can be easily shown that DK = A DE EG and that the triangle BDK has its base BK= BA + EF, called the perpendiculars of the oblongs, and has its sides equal to the diagonals of the same oblongs, For Private and Personal Use Only

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220 GANITASARASANGRAHA. Subject of treatment known as Paisacika or devilishly difficult problems. Hereafter we shall expound the subject of treatment known as Paisacika. The rule for arriving, in relation to the equilateral quadrilateral or longish quadrilateral figures, at the numerical measure of the base and the perpendicular-side, when, out of the perpendicular side, i he base, the diagonal, the area and the perimeter, any two are optionally taken to be equal, or when the area of the figure happens to be the product obtained by multiplying respectively by optionally chosen multipliers any two desired quantities (out of the elements mentioned above): that is---(the rule for arriving at the numerical values of the base and the perpendicular-side in relation to an equilateral quadrilateral or a longish quadrilateral figure,) when the area of the figure is (numerically) equal to the measure of the perimeter (thereof); or, when the area of the figure is numerically equal to the measure of the base (thereof); or, when the area of the figure is numerically equal to the measure of the diagonal (thereof); or, when the area of the figure is numerically equal to half the measure of the perimeter; or, when the area of the figure is numerically equal to one-third of the base; or, when the area of the figure is numerically equal to one-fourth of the measure of the diagonal; or, when the area of the figure is numerically equal to that doubled quantity which is obtained by doubling the quantity which is the result of adding together twice the diagonal, three times the base, four times the perpendicularside and the perimeter and so on : 112). The measure of the base (of an optionally chosen figure of the required type), on being divided by the (resulting) optional factor in relation thereto, (by multiplying with which the area 1124. The rule will be clear from the following working of the first example given in stanza 1137 :-Here the problem is to find out the measure of the side of an equilaterul quadrilateral, the numerical value of the area where. of is equal to the numerical value of the perimeter. Taking an equilateral quadrilateral of any dimension, say, with 5 as the meagare of its side. we have the perimeter equal to 20, and the area equal to 25. The factor with which For Private and Personal Use Only

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CHAPTER VII-MEASUREMENT OF AREAS. 221 of the said optionally chosen figure happens to be arrived at); or the base (of such an optionally chosen figure of the requisite type), on being multiplied by the factor with which the area (of the said figure) has to be multiplied (to give the required kind of result); gives rise to the measures of the bases of the (required) equilateral quadrilateral and other kinds of derived figures. Examples in illustration thereof. 113. In the case of an equilateral quadrilateral figure, the (numerical measure of the) perimeter is equal to (that of) the area. What then is the numerical measure of (its) base? In the case of another similar figure), the numerical measure of the) area is equal to (that of) the base. Tell me in relation to that (figure) also (the numerical measure of the base). 114. In the case of an equilateral quadrilateral figure, the (numerical) measure of the diagonal is equal to (that of) the area. What may be the measure of (its) base? And in the case of another (similar) figure, the (numerical) measure of the perimeter is twice that of the area. Tell me (what may be the measure of its base). 115. Here in the case of.a longish quadrilateral figure, the (numerical) measure of the area is equal to that of the perimeter; and in the case of another (similar); figure, the (numerical) measure of the area is equal to that of the diagonal. What is the measure of the base (in each of these cases) ? 116. In the case of a certain equilateral quadrilateral figure, the (numerical) measure of the base is three times that of the area. (In the case of) another equilateral quadrilateral figure, the (numerical) measure of the diagonal is four times that of the area. What is the measure of the base (in each of these cases) ? For Private and Personal Use Only the measure of the perimeter, viz. 20, has to be multiplied in order to make it equal to the measure of the area, viz., 25, is . If 5, the measure of a side of the optionally chosen quadrilateral is divided by this factor, the measure of the side of the required quadrilateral is arrived at. The rule gives also in another manner what is practically the same process thus: The factor with which the measure of the area, viz. 25 has to be multiplied in order to make it equal to the measure of the perimeter, viz. 20, is . If 5, the measure of a side of the optionally chosen figure is multiplied by this factor t, the measure of the side of the required figure is arrived at.

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222 GANITASARASANGRAHA. aso. 1174. In the case of a longish quadrilateral figure, (the numerical measures of) twice the diagonal, three times the base and four times the perpendicular-side heing taken, the measure of the perimeter is added to them. Twice (this sum) is the (numerical) measure of the area. (Find out the measure of the base. 118. In the case of a longish quadrilateral figure, the (numerical) measure of the perimeter is 1. Tell me quickly, after calculating, what the measure of its perpendicular side is, and what that of the base.. 119]. In the case of a longish quadrilateral figure, the (numerical measures of twice the diagonal, three times the base, and four times the perpendicular, on being added to the (numerical) measure of the perimeter, become equal to 1. (Find out the measure of the base.) Another rule regarding the process of arriving at the number representing the bejas in relation to the derived longish quadrilateral figure : 1203. The operation to arrive at the generating (bejas) in relation to a longish quadrilateral figure consists in getting at the square roots of the two quantities represented by (1) half of the diagonal as diminished by the perpendicular-side and (2) the difference between this quantity and the diagonal. An example in illustration thereof. 1213. In the case of a longish quadrilateral figure, the perpendicular-side is 55, the base is 48, and then the diagonal is 73. What are the bajas here? 120$. The rule in stan za 95} of this chapter relates to the method of arriv. ing at the bijas from the base or the perpendicular cr the diagonal of a longish amadrilateral. But the rule in this stanza gives a method for finding out the bijas from the perpendicular and the diagonal of a longish quadrilateral. The process described is based on the following identities : V az + 62-(a? - b2) = b; and N othe_a? + -- (a?-67) 2 where a+ is the measure of the diagonal, and al-ba is the measnre of the perpendioular-side of a lopgish quadrilateral, a and b being the required bejas. For Private and Personal Use Only

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CHAPTER VII-MEASUREMENT OF AREAS. 223 The rule for arriving at the (longish quadrilateral) figure associated with a diagonal having a numerical value optionally determined : Acharya Shri Kailassagarsuri Gyanmandir 122. Each of the various figures that are derived with the aid of the given (bejas) is written down; and by means (of the measure) of its diagonal the (measure of the) given diagonal is divided. The perpendicular-side, the base, and the diagonal (of this figure) as multiplied by the quotient (here) obtained, give rise to the perpendicular-side, the base and the diagonal (of the required figure). An example in illustration thereof. 123-124. O mathematician, quickly bring out with the aid of the given (bijas) the (value of the) perpendicular-sides and the bases of the four longish quadrilateral figures that have respectively 1 and 2, 2 and 3, 4 and 7, and 1 and 8, for their bijas, and are also characterised by different bases. And, (in the problem) here, the diagonal is (in value) 65. Give out (the measures of) what may be the (required) geometrical figures (in that case). The rule for arriving at the numerical values of the base and the perpendicular side of that derived longish quadrilateral figure, the numerical measures of the perimeter as also of the diagonal whereof are known :-- 125. Multiply the square of the diagonal by two; (from the resulting product), subtract the square of half the perimeter; (then) get at the square root (of the resulting difference). If (this square root be thereafter) utilized in the performance of the 122. The rule is based on the principle that the sides of a right angled triangle vary as the hypotenuse, although for the same measure of the hypotenuse there may be different sets of values for the sides. 125. If a and b represent the sides of a rectangle, then a2+ b2 is the measure of the diagonal, and 2a + 2b is the messure of the perimeter. It can be seen easily that 2a +26 2 2 2a +26 + 2 (o + 5 ) = ( 2+ + 21 2 For Private and Personal Use Only / 2 = a; and 2 2 2a 2a 20+ 26-N2 (F+). " ( 24+ 26 ) deg } + 2 2 These two formulas represent algebraically the method described in the rule here. / 2 = b.

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224 GANITASARABANGRAHA. operation of sankramana along with half the perimeter, the (required) base and also the perpendicular-side are arrived at. An example in illustration thereof. 1267. The perimeter in this case is 34; and the diagonal is geen to be 13. Give ont, after calculating, the measures of the perpendicular-side and the base in relation to this derived figure. The rnle for arriving at the numerical values of the base and the perpendicular-side when the area of the figure and the value of the diagonal are known :-- 1274. I'wice the measure of the area is subtracted from the square of the diagonal. It is also added to the square of the diagonal. The square roots (of the difference and of the sum so obtained) give rise to the measures ofthe (required) perpendicularside and the base, if the larger (of the square roots) is made to undergo the process of sankramana in relation to the smaller (square root). An example in illustration therenf. 1281. In the case of a longish quadrilateral figure, the measure of the area is 60, and the measure of its diagonal is 13. I wish to hear (from you) the measures of the perpendicular-side and the base. The rule for arriving at the numerical values of the base and the perpendicular-side in relation to a longish quadrilateral figure, when the numerical value of the area of the figure and the numerical value of the perimeter (thereof) are known : 1291. From the quantity representing the square of half the perimeter, the measure of the area as multiplied by four is to be 127. Adopting the same symbols as in the note to stanza 125, we have the following formula to represent the rule here given : 2 {N (vae+0) 4240 = N (va + c)22at ) +2=o or by +62 + 2 ab - 20 PS N m + }-2=a or b, as the case may be. 129. Here we have 2a + 2b 2a + 2b 2 4 ab +2=a or b, as the case may be. For Private and Personal Use Only

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OHAPTER VIIMEASUREMENT OF AREAS. 225 subtracted. Then, on carrying out the process of sankramana with the square root (of this resulting difference) in relation to half the measure of the perimeter, the values of the (required) base and the perpendicular-side are indeed obtained. An example in illustration thereof. 1304. In a derived longish quadrilateral figure, the measure of the perimeter is 170; the measure of the given area is 1,500. Tell me the values of the perpendicular-side and the base (thereof). The rule for arriving at the respective pairs of (required) longish quadrilateral figures, (1) when the numerical measures of the perimeter are equal, and the area of the first figure is double that of the second ; or, (2) when the areas of both the figures are equal, and the numerical measure of the perimeter of the second figure is twice the numerical measure of that of the first figure ; or, (3) (again) when, in relation to the two required figures, the numerical measure of the perimeter of the second figure is twice the numerical measure of the perimeter of the first figure, and the area of the first figure is twice the area of the second figure : 1314--133. (The larger numbers in the given ratios of) the perimeters as also (of) the areas (relating to the two required longish quadrilateral figures,) are divided by the smaller (numbers) corresponding to them. The resulting quotients) are multiplied (between themselves) and (then) squared. (This same quantity,) 1311 to 1.33. If a and y represent the two adjacent sides of the first rectangle, and a and t the two adjacent sides of the second rectangle, the conditions mentioned in the three kinds of problems proposed to be solved by this rule may be represented thus:-- (1) x+y=a+b: y = 2ab. (2) 2(x + y) = a +b : aey = ab. (3) 2 (ae+ y) = a +b: y = 2ab. The solution given in the role seems to be correot only for the particular cases given in the problems in stanzas 134 to 136. 29 For Private and Personal Use Only

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226 GANITASARASANGRAHA. on being multiplied by the given optional multiplier, gives rise to the value of the perpendicular-side. And in the case in which the areas (of the two required figures) are (held to be) equal, (this measure of) the perpendicular-side as diminished by one becomes the measure of the base. But, in the other case (wherein the areas of the required figures are not held to be equal), the larger (ratio number) relating to the areas is multiplied by the given optional multiplier, and the resulting product is) diminished by one. The measure of the perpendicular-side (arrived at as above) is diminished by the quantity (thus resulting) and is (then) multiplied by three : thus the measure of the base (is arrived at). Then, in respect of arriving at the other of the two required quadrilateral figures), its base and perpendicular are to be brought out with the aid of the (now knowable) measure of its area and perimeter in accordance with the rule already given (in stanza 129}). Examples in illustration thereof. 134. There are two (quadrilateral) figures, each of which is characterised by unequal length and breadth ; and the given multiplier is 2. The measure of the area of the first (figure) is twice that of the second), and the two perimeters are equal. What are the perpendicular-sides and the bases here (in this problem)? 135. There are two longish quadrilateral figures ; and the (given) multiplier is also 2. (Their) areas are equal, (bat) the perimeter of the second (figare) is twice that of the first. (Find out their perpendicular-sides and bases.) 136. There are two longish quadrilateral figures. The area of the first (figure) here is twice that of the second figure). The perimeter of the second (figure) is twice (that of the first). Give out the values of their bases and their perpendicular-sides. The rule for arriving at a pair of isosceles triangles, so that the two isosceles triangles are characterised either by the values of their perimeters and of their areas heing equal to each other, or by the values of their perimeters and of their areas forming multiples of each other : For Private and Personal Use Only

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Shri Mahavir Jain Aradhana Kendra CHAPTER VII-MEASUREMENT OF AREAS. II 137. The squares (of the ratio-values) of the perimeters (of the required isosceles triangles) are multiplied by (the ratio-values of) the areas (of those triangles) in alternation. (Of the two products so obtained), (the larger one is) divided by the smaller; and (the resulting quotient) is multiplied by six and (is also separately multiplied) by two. The smaller (of the two products so obtained) is diminished by one. The larger product and the diminished smaller product constitute the two bijas (in relation to the longish quardrilateral figure) from which one (of the required triangles) is to be obtained. The difference between these (two bajas above noted) and twice the smaller one (of those biijas) constitute the tijas (in relation to the longish quadrilateral figure) from which the other (required triangle) is to be obtained. (From the two longish quadrilateral figures formed with the aid of their respective bijas), the sides and the other things (relating to the required triangles) are to be arrived at as (explained) before. + 1 and 137. When a b is the ratio of the perimeters of the two isosceles triangles, and 662 c c: d the ratio of their areas, then, according to the rule, 26 c -1 and a2 d a2 d 462 c 2 are the two sets of bijas, with the help of which a2 d and 462 c a2 d the vales of the various required elements of the two isosceles triangles may be arrived at. The measures of the sides and the altitudes, calculated from these bijas according to stanza 108 in this chapter, when multiplied respectively by a and b, (the quantities occurring in the ratio of the perimeters), give the required measures of the sides and the altitudes of the two isosceles triangles. They are as follow:--- I Equal side a x Altitude a x Base= ax 2 x 2 x {(663)2 + (2,69 EUR 1)deg} c a d -1 6b2 d 8 62 c a 2 d b2 c 2 {(25) - d www.kobatirth.org Equal sideb x Altitudeb x Base b x 2 x 2 x X 4 c a d 62 (2022 - 1) d 2 4 b2 c c a2 d { (12 + 1 ) + ( 122 )"} -2 d c (2-1)} a d (1380 + 1) x (+382) {(+)-(-)"} +1 Acharya Shri Kailassagarsuri Gyanmandir c a2 d 2 227 For Private and Personal Use Only Now it may be easily proved from these values that the ratio of the perimeters is a b, and that of the areas is c: d, as taken for granted at the beginning.

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,228 GANITASARASANGRAHA. Examples in illustration thereof. 138. There are two isosceles triangles. Their area is the same. The perimeters are also equal in value. What are the values of their sides, ard what of their bases ? 139. There are two isosceles triangles. The area of the first one is twice (that of the second). The perimeter of both of them) is the same. What are the values of (theirsides, and what of (their) bases ? 140. There are two isosceles triangles. The perimeter of the second (triangle is twice (that of the first). The areas of the two (triangles) are equal. What are the values of (their) sides, and what of (their) bases ? 141. There are two isosceles triangles. The area of the first (triangle) is twice (that of the second); and the perimeter of the second (triangle is twice that of the first). What are the values of (their) sides, and what of (their) bases ? The rule for arriving at an equilateral quadrilateral figure, or for arriving at a regular circular figure, or for arriving at an equilateral triangular figure, or for arriving at a longish quadrilateral figure, with the aid of the numerical value of the proportionate part of a given suitable thing (from among these), wben any optionally chosen number from among the natural) numbers, starting with one, two, &c., and going beyond calculation, is made to give the numerical measure of that proportionate part of that given suitable thing : 142. The (given measure of the) area (of the proportionate part) is divided by the appropriately) similarised measure of the part held (in the hand). The quotient (80 obtained), if multiplied by four, gives rise to the measure of the breadth of the circle and 142. In probleme of the kind given under this rule, a circle, or a square, or an equilateral triangle,or an oblong is divided into a desired number of equal parts, each part being bounded on one side by a portion of the perimeter and bearing the same proportion to the total area of the figure as the portion of the perimeter bears to the perimeter as a whole. It will be seen that in the case of a cirole each part is a sector, in the case of a square and an oblong it is a rectangle, and in the case of an equilateral triangle it is a triangle. The area of each part and the length of the original perimeter contained in each part are both of given For Private and Personal Use Only

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CHAPTER VII-MEASUREMENT OF AREAB. 229 229 (also) of the square. (That same) quotient, if multiplied by sid, gives rise to the required measure of the base of the (equilateral) triangle as also of the longish quadrilateral figure. Half (of this is the measure of the perpendicular-side (in the case of the longish quadrilateral figure). An example in illustration thereof. 143-145. A king caused to be dropped an excellent carpet on the floor of (his) palace in the inner apartinents of his zenana amidst the ladies of his harem. That (carpet) was in shape) a regular circle. It was held in band) by those ladies. The fistfuls of both their arms made each of them) acquire 15 (dandas out of the total area of the carpet). How many are the ladies, and what is tbe diameter (of the circle) here? What are the sides of the square (if that same carpet be square in shape) ? and what the magnitude. The stanza states a rule for finding out the measure of the diameter of the circle, or of the sides of the square, or the equilateral triangle or the oblong. If m represents the area of each part and n the length of a part of the total perimeter, the formulas given in the rule are *4= diameter of the circle, or side of the square ; and x 6 = side of the equilateral triangle or of the oblong; n and half of x B= the length of the perpendicular-side in the case of the oblong. The rationale will be clear from the following equations, where a represents the number of parts into which each figare is divided, a is the length of the radius in the case of the circle, or the length of a side in the case of the other figures, and b is the vertical side of the oblong: In the case of the Circle 2 x m_ta% xn 2 a akhta 2 In the case of the Square In the case of the Equilateral Triangle = X 1 3a In the case of the Oblong x xm a xb here b is taken to be equal xn 2 ( a + b) to half of a. It has to be noted that only the approximate value of the area of the equilateral triangle, as given in stanza 7 of this chapter, is adopted here. Otherwise the formula given in the role will not hold good. 143-145. What is called fistful in this problem is equivalent to four angulas in measure. For Private and Personal Use Only

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230 GANITASARASANGRAHA. sides of the equilateral triangle (if it be equilaterally triangular in shape)? Tell (me), O friend, the measures of the perpendicular side and the base, in case the carpet happens to be) a longish quadrilateral figure in shape). The role for arriving at an equilaterally quadrilateral figure or at a longish quadrilateral figure when the numerical value of tbe area of the figure is known : 146. The square root of the accurate measure of the (given) area gives rise to the value of the side of the (required) equilateral quadrilateral figure. On dividing the (given) area with an optionally chosen quantity (other than the square root of the value of the given area, this) optionally chosen quantity and the resulting quotient constitute the values of the perpendicular-side and the base in relation to the (required) longish quadrilateral figure. An example in illustration thereof. 147. What indeed is that equilateral quadrilateral figure, the area whereof is 64 ? The accurate value of the area of the longish (quadrilateral) figure is 60. What are the values of the perpendicular-side and the base here? The rule for arriving at a quadrilateral figure with two equal sides baying the given area of such a quadrilateral figure with two equal sides, after getting at a derived longish quadrilateral figure with the aid of the given numerical bijax and also aftes utilizing a given number as the required multiplier, when the numerical value of the accurate measure of the area of the required quadrilateral figure with two equal sides is known : 148. The square of the given (multiplier) is multiplied by the that (given) area. The resulting) product is diminished by the value of the area (of the longish quadrilateral figure) derived (from the given bijas). The remainder, when divided by the base 148. The problem here is to construct a quadrilateral figure of given area and with two equal sides. For this purpose an optionally chosen number and a set of two bujas are given. The process described in the role will become clear by applying it to the problem given in the next stanza. The bijas mentioned therein are 2 and 3 ; and the given area is 7, the given optional number being 3. For Private and Personal Use Only

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CHAPTER VII-MEASUREMENT OF AREAS. (of this derived longish quadrilateral figure), gives rise to the measure of the top-side. The value of the perpendicular-side (of the derived longish quadrilateral figure), on being multiplied by two and increased by the value of the top-side (already arrived at), gives rise to the value of the base. The value of the base (of the derived longish quadrilateral figure) is (the same as that 12 The first thing we have to do is to construct a rectangle with the aid of the given bijas in accordance with the rule laid down in stanza 90 in this chapter. That rectangle comes to have 5 for the measure of its smaller side, 12 for the measure of its larger side, and 13 for the measure of its diagonal; and its area is 60 in value. Now the area, given in the problem is to be multiplied by the square of the given optional number in the problem, so that we obtain 7 x 32 63. From this 63, we have to subtract 60, which is the measure of the area of the rectangle constructed on the basis of the given bijas: and this gives 3 as the remainder. Then the thing 12 # 4 to be done is to construct a rectangle, the area whereof is equal to this 3, and one of the sides is equal to the longer side of the rectangle derived from the same bijas. Since this longer side is equal to 12 in value, the smaller side of the required rectangle has to be in value as shown in the figure here. Then the two triangles, into which the rectangle derived from the bijas may be split up by its diagonal, are added one on each side to this last rectangle, so that the sides measuring 12 in the case of these triangles coincide with the sides of the rectangle having 12 as their measure. The figure here exhibits the operation. Thus in the end we get the quadrilatera] figure having two equal sides, each of which measures 13, the value of the other two sides being and 10 respectively. From this the values of the sides of Acharya Shri Kailassagarsuri Gyanmandir 13 For Private and Personal Use Only 5 13 12 12 231 13 5 5 the quadrilateral required in the problem may be obtained by dividing by the given optional number namely 3, the values of its sides represented by 13, 1, 13 and 10.

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232 GANITASARASANGRAHA. of) the perpendicular dropped (from the ends of the top-side); and the diagonals (of the derived longish quadrilateral figure) are (equal in value to the sides. These (elements of the quadrilateral figure with two equal sides arrived at in this manner) have to be divided by the given multiplier (noted above to arrive at at the required quadrilateral figure with two equal sides). An example in illustration thereof. 149. The accurate value of the (given) area is 7; the optional given multiplier is 3; and the bejas are seen to be 2 and 3. Give out the values of the two sides of a quadrilateral figure with two equal sides and of its top-side, base, and perpendicular. The role for arriving at a quadrilateral figure with three equal sides, having an accurately measured given area, (with the aid of a given multiplier) : 150. The square of the value of the given area is divided by the cube of the given (multiplier). (Then) the given (multiplier) is added (to the resulting quotient). Half (of the sum so obtained) gives the measure of one) of the equal) sides. The given 150. It is stated in the rule here that the given area when divided by the given optional number gives rise to the value of the perpendicular in relation to the required figure. As the area is equal to the product of the perpendi. cular and half the sum of the base and the top-side, the given optional number represents the measure of half the sum of the base and the top-side. If A BCD be a quadrilateral with three equal sides, and CE the perpendicular from C on AD, then AE is half the sam of AD and BC, and is equal to the given optional number. It can be A easily shown that 2 AD. AE=CE + AE. CEP X AE2 CE+ A E _ CE AE AE +AE - ..AD= 2AE 2 AE+2= 2 Here CE X AE -- the given area of the quadrilateral. This last formula happens to be what is given in the role for finding out any of the three equal sides of the quadrilateral contemplated in the problem. (CE AE)', AE. For Private and Personal Use Only

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CHAPTER VII--MEASUREMENT OF AREAS. 233 (multiplier) as multiplied by two and (then) diminished by the value of the side (just arrived at) gives rise to the value of the top-side. And the (given) area divi led by the given (multiplier) gives rise to the value of the perpendicular (dropped from the ends of the top-side) in relation to this required quadrilateral figure with three equal sides. An example in illustration thereof. 151. In the case of a certain quadrilateral figure with three equal sides, the accurate value of the area is 96. The given multiplier is 8. Give out the values of the base, of the sides, of the top-side and of the perpendicular. The rule for arriving at the numerical measures of the topside, of the base, and of the other sides in relation to a quadrilateral figure having unequal sides, with the aid of 4 given divisors, when the accurate value of the area (of the required quadrilateral figare) is known :-- 152. The square of the given area is divided (separately) by the four given divisors; (and the four resulting quotients are separately noted down). Half of the sum of (these) quotients is (noted down) in four positions, and is (in order) diminished (respectively) by those (quotients noted down above). The remainders (80 obtained) give rise to the numerical values of the sides of a quadrilateral figure (having unequal sides and consequently) named 'unequal.' 152. The area of a quadrilateral with unequal sides has already been men. tioned to be : (-a) (8-5) (8-0) (8-d), where :=balf the perimeter, and a, b, c, and d are the measures of the sides (ide note to stanza 50 in this chapter). The rule here given requires that the numerical value of the area should be squared and then divided separately by the four optionally chosen divisory. If (8-a) (8-b) (-c) (8-d) is divided by four suitably chosen divisors so as to give as quotients 8-a, 8-5, 8-c, and s-d, then on adding these quotients and halving their sum, the result is seen to be 8. If 8 is diminished in order by s-a, s-, 8-c, and s-d, the remainders represent respectively the values of the sides of the quadrilateral with onequal sides. 30 For Private and Personal Use Only

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234 GANITASABASANGRAHA. An example in illustration thereof. 153-1531. In the the case of a quadrilateral figure with unequal sides, the (given) acourate measure of the area is 90. And the product of 5 multiplied by 9, as multiplied by 1.0, 18, 20 and 36 respectively, gives rise to the (four given) divisors. Tell me quickly, after calculating, the numerical values of the top-side, the base and (other) sides. The rule for arriving at the numerical value of the sides of an equilateral triangular figure possessing a given accurately measured area, when the value of (that) accurately measured area is known : 1541. Four times the (given) area is squared. The resulting quantity) is divided by 3. The quotient (s0) obtained happens to be the square of the square of the value of the side of an equilateral triangular figure. An example in illustration thereof. 155. In the case of a certain equilateral triangular figure, the given area is only 3. Calculate and tell me the value of its) side. After knowing the exact numerical measure of a (given) area, the rule for arriving at the numerical values of the sides, the base and the perpendicular of an isosceles triangular figure having that same accurately measured area (as its own) : 1561. In the case of the isosceles triangle (to be so) constructed, the square root of the sum of the squares of the quotient obtained by dividing the (given) area by an optionally chosen quantity, as also of (that) optionally chosen quantity, gives rise to the value of the side : twice the optionally chosen quantity gives the measure of the base; and the area divided by 1547. The rule here given may be seen to be derived from the formula for the area of an equilateral triangle, viz., area = a 2 x where a is the measure of a side. 156). In problemg of the kind contemplated in this role, the measure of the area of an isosceles triangle is given, and the valae of half the base chosen at option is also given. The measures of the perpendicnlar and the side are then easily derived from these known quantities. For Private and Personal Use Only

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CHAPTER VII--MEASUREMENT OF AREAS. 235 the optionally chosen quantity gives rise to the measure of the perpendicular. An example in illustration thereof. 1571. In the case of an isosceles triangular figure, the accurate measurement of the area is 12. The optionally chosen quantity is 3. Give out quickly, O friend, the values of (its) sides, base, and perpendicular. The rule for arriving, after knowing the exact numerical measure of a (given) area, at a triangular figure with unequal sides, having that same accurately measured area (as its own) = 1581. The giver area is multiplied by eight, and to the resulting product the square of the optionally chosen quantity is added. Then the square root (of the sum so resulting is obtained). The cube (of this square root) is (thereafter) divided by the optionally chosen number and (also) by the square root (obtained as above), Half of the optionally chosen number gives the measure of the base (of the required triangle). The quotient (obtained in the previous operation) is lessened (in value) by the (measure of this) base. (The resulting quantity) is to be used in carrying out the sankramana process in relation to the equare of the optionally chosen quantity as divided by two as well as the square root (mentioned above). (Thus) the values of the sides are arrived at. 158). If A represents the area of a triangle, and d is the optionally chosen number, then according to the role the required values are obtained thus : 2 == base ; (W 8A + da). + de & N 8A + d2 2 2 788 + da and -=sides. 2 When the area and the base of a triangle are given, the locus of the vertex is a line parallel to the base, and the sides oan have any set of values. In order to arrive at a specific set of values for the sides, it is evidently assumed here that the sum of the two sides is equal to the sum of the base and twice the altitude, i.e., equal to +24. With this assumption, the formula above 2" " " given for the measure of the sides can be derived from the general formula for the area of the triangle, Nils-a) (5-6) (8-c) given chapter. For Private and Personal Use Only

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236 GANITASI KASANGRAHA. An example in illustration thereof. 1594. In the case of a certain triangular figure with unequal sides, it has been pointed out that 2 constitutes the accurate measure of its area and 3 is the optionally chosen quantity. What is the value of the base as well as of the sides (of that triangle) ? Again, another rule for arriving, after knowing the exact numerical measure of a (given) area, at a triangular figure with unequal sides having that same (accurately measured) area (as its own) : 1607-161}. The square root of the measure of the given area as multiplied by eight and as increased by the square of an optionally chosen number is obtained. This and the optionally chosen number are divided by each other. The larger (of these quotients) is diminished by half of the smaller (quotient). The remainder (thus obtained) and (this) half of the smaller (quotient) are respectively multiplied by the above-noted square root and the optionally chosen number. On carrying out, in relation to the products (thus obtained), the process of sankramana, the values of the base and of one of the sides are arrived at. Half of the optionally chosen number happens to be the measure of the other side in a triangular figure with unequal sides. An example in illustration thereof. 1621. In the case of a triangle with unequal sides, the accurate measure of the area is 2, and the optionally chosen quantity is 3. O friend who know the secret of calculation, give out the measure of the base as well as of the sides. The rule for arriving, after knowing the accurate measure of a (given) area, at a regularly circular figure having that accurately measured area (as its own) : 1637. The accurate measure of the area is multiplied by four and is divided by the square root of ten. On getting at the square da 1635. The rule in this stanza is derived from the formula, area = **NTO, where d is the diameter of the circle. For Private and Personal Use Only

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CHAPTER VII-MEASUREMENT OF AREAS. 237 root (of the quotient resulting thus), the value of the diameter happens to result. In relation to a regular circular figure, the measure of the area and the circumference are to be made out as explained before. An example in illustration thereof. 164}. In the case of a regular circular figure, the accurate measure of the area has been pointed out to be 5. Calculate quickly and tell me what the diameter of this (circle) may be. On knowing the approximate measure as well as the accurate measure of an area, the rule for arriving at a quadrilateral figure with two equal sides as well as at a quadrilateral figure with three equal sides, having those same approximate and accurate measures (as such measures of their areas) : 1651. In the case of (the quadrilateral with) two equal sides, the square root of the difference between the squares of the (approximate and accurate measures of the area is to be obtained. On adding (this square root) to the optionally chosen quantity and on subtracting (the same square root from the same optionally chosen quantity), the base and the top-side are so obtained as to have to be divided by the square root of the optional quantity. The approximate measure of the area gives rise to the value of the sides so as to have to be divided by the square root of the optional quantity. - NRP - pt i top-side 165]. If R represents the approximate area of a quadrilateral with two equal sides, and r the accurate value thereof, and p is the optionally chosen number, then WR? - q? + P.. base = i and each of the Ne Np equal rides =- . If a, b, c and d be the measures of the sides of NP the quadrilateral with two equal sides, then it may be seen that R AN For Private and Personal Use Only

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238 GANITASARASANGRAHA In the case of the quadrilateral figure witb) three equal sides, the square root of the difference between the two area-squares above noted) is added to the approximate measure of the area. (On treating the resulting sum as the optional quantity and) on adding and subtracting (the said square root as before), the base and the top-side are obtained so as to have to be divided by the square root of (such) optional quantity. (Here also), the approximate measure of the area, on being divided by the square root of (this optional quantity, gives rise to the measure of the other sides. An example in illustration thereof. 1663. The accurate measure of the area is 5; the approximate measure of the area is 13; and the optionally chosen quantity is 16. What are the values of the base, the top-side, and the (other) side in the case of a quadrilateral figure with two equal sides ? An example relating to a quadrilateral figure with three equal sides. 1671. The accurate measure of the area is 5; and the approximate measure of the area is 13. Think out and tell me, O friend, the values of the sides of the quadrilateral figure with three equal sides. The rule for arriving, when the approximate and the accurate ineasures of an area are known, at the equilateral triangle and also at the diameter of the circle, having those same approximate and accurate measures (for their area) : 1683. That which happens to be the square root of the square root of the difference between the squares of the (approximate measure and of the accurate measure of the given) area is to be + d2 XN 2 ); and usb+d 2 R -a (b + d) a (b - d)2 2 The formulas given above for the base and the top-side can be easily verified by substituting these values of R, r and p therein. Similarly the rule may be seen to hold good in the case also of a quadrilateral figure with three equal sides. 168. For the approximate and accurate values of an equilateral triangle Bee rules in stanzas 7 and 50 of this chapter. For Private and Personal Use Only

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CHAPTER VII - MEASUREMENT OF AREAS. 239 multiplied by two. The result is the measure of the side in the (required) equilateral triangle. It is also the measure of the diameter of the (required) regular cirole. Examples in illustration thereof. 1691. The approximate area is 18. The accurate area is the square root of 38 as multiplied by 9. Tell me, O friend, after calculating, the measurement of the required) equilateral triangle. 1701. The accurate measure (of the area) is the square root of 6,250. The approximate measure (of the area) is 75. What is the measure of the diameter of the circle (having such areas) ? When the practically approximate and the accurately calculated measures of an area are known, the role for arriving at the numerical values of the base and the side of an isosceles triangle having the same approximate and docurate measures for its area :-.. 1711. Twice the square root of the difference between the squares of the (approximate and the accurate) measures of the area is to be taken as the base of a certain isosceles) triangle ; and the given approximate measure (of the area) is to be taken as the value of one of the equal sides. And on dividing (these values of the base and the side) by the square root of half (the above derived value) of the base, (the required measures of the base and the side of the required isosceles triangle are obtained). This is the rule in relation to the isosceles triangle. An example in illustration thereof. 1727. It is pointed out that here, in this case, the accurate measure of the area is 60, and the approximate measure is 65. Tell me, O friend, after calculation, the numerical measure of the sides of the (required) isosceles triangle. An optional number and a quadrilateral figure with two equal sides being given, the rule for arriving at the numerical values of the base, and the top-side, and the other) sides of another quadrilateral figure (with two equal sides) which has an accurate measure For Private and Personal Use Only

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240 GANITASARASANGRAHA. of area equal to the accurate measure of the area of the given quadrilateral figure with two equal sides : 173). If the square of the value of the perpendicular (in the given quadrilateral figure with two equal sides) is used along with the given optional number in carrying out the process of visamasankramana, then the larger (of the two results obtained) becomes the measure of either of the equal sides (in the required quadri. lateral figure with two equal sides). Half of the sum of the values of the top-side and the base (in the given quadrilateral figure with two equal sides), on being respectively increased and decreased by the smaller (of the two results in the visamasankramani process above-mentioned), gives rise to the values of the base and the top-side in the required) quadrilateral figure with two equal sides. 173). The problem contemplated in this rule is to construct a quadrilateral with two equal sides that shall be equal in area to a given quadrilateral with two equal sides and shall also have the same perpendicular distance from the topside to the base. Let a and c be the equal sides of the given quadrilateral, and h and d be the top side and the base thereof respectively; and let the value of the perpendicular distance be p. If a 1, b1, C1, d, be taken to be the corresponding sides of the required quadrilateral, then, since the area and the perpendicular are the same in the case of both the quadrilaterals, we have, di + b = d + b(I): and a? - (4:50) *=p* (II): Let as - oh = N; then an + ds 1. ; 2 =a (111); d-b .1. 01 - d--6.17 N-N and d + b di + b 17 2 27 la d, or by: (IV) Here N is what is called e or the optionally given nomber in the rule, and formulas III and IV are those that are given in the role for the solution of the problem. For Private and Personal Use Only

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CHAPTER VII---MEASUREMENT OF AREAS. 241 An example in illustration thereof. 1745. The base (of the given quadrilateral figure) is 14 ; each of the (two equal) sides is 13; the top-side is 4; the perpendicular is 12; and the optionally given number is 10. Wbat is that other quadrilateral figure with two equal sides, the accurate measure (of the area of which is the same as the accurate measure of the area of (this given quadrilateral) ? Wher an area with a given practically approximate measure is divided into any required number of parts, the rule for arriving at the numerical measure of the bases of those various parts of the quadrilateral figure with two equal sides, as also at the numerical measure of the sides as measured from the various division-points thereof, the numerical measure of the practically approximate value of the area of the quadrilateral figure with two equal sides being given :--- 175). The difference between the squares of the (numerical) values of the base and the top-side (of the given quadrilateral figure with two equal sides) is divided by the total value of the (required) proportionate parts. By the quotient (so obtained), 175$. If ABCD be a quadrilateral with two equal sides, and if EF, GH and KL divide the quadrilateral so that the divided portions A bD are in the proportion of m, n, pand o in respect of area, then according to the rule, EF = M - xm + b2; m +n + P + 9 d? - 12 - X(m + n) + 62 m +n + P + q* - X (m +n + p) +62; Ym +n + P + 9 and so on. + b . 2 m +n + + Similarly AE = ; EG = EF + AD m 1 X m+n +p+qi GH + EF d + b 2 ax = (** ) m GK + 1 + P +:and so on. 9 *978 jana no con KL + GH It can be easily shown that ma AB BC-AD EF-AD: 31 . For Private and Personal Use Only

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242 GANITASARASANGRAHA. the ratio values of the (various) parts are respectively multiplied. To each of the products (so obtained), the square of the measure of the top-side (of the given figure) is added. The square root (of the sum so obtained) gives rise to the value of the base of each of the parts). The area (of each part divided by half the sum of the values of the base and the top-side (thereof) gives in (the requisite) order the value of the perpendicular (which for purposes of approximate measurement is treated as the side). Examples in illustration thereof. 1761. The measure of the top-side is given to be 7; that of the base below is 23, and that of each of the remaining) sides is 30. The area (included within such a figure) is divided between two so that each obtains one (share). What is the value of the base (to be found out here) ? 1773-1781. The measure of the base (of a quadrilateral with two equal sides) is 162, and that of the top-side (thereof) is seen to be 18. The value of each of the (two equal) sides is 400. The area of this (figure 80 enclosed) is divided among 4 men. The parts obtained by the men are in the proportion of) 1, 2, 3, and 4 respectively. Give out, in accordance with this proportionate distribution, the values of the area, of the base, and (of either) of the two equal) sides (in each case). 1794. The measure of the base of the given quadrilateral figure) is 80, that of the top-side is 40 ; the measure (of either) of m - -72 d + A D AB (BO + AD) - BCP-AD2 ..AE (EF + AD E F2 - AD AB (B C + AD)m +n + 2 +9.. BC-AD2 Bat m +n + p + q. "AE (EF+AD) . F2--A D2 = EF- m (B 0 -A D) 2 X m + Ba; m + n + P + 9 m +n + P + 9 and E F=N Vm + + p + q* m + bo. Similarly the other formulas may also be verified. * Although the text simply states that the quotient has to be multiplied by the value of the parts, what is intended is that the quotient has to be multiplied by the number representing the value of the parts up to the top-side in each case. That is, in the figure on the previous page, to arrive at GH, for instance, - has to be multiplied by m + n and not by n merely. m +n + p + q** For Private and Personal Use Only

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CHAPTER VIIMEASUREMENT OF AREAS. 243 the two equal) sides is 4 X 60. The share parts are in the proportion of) 3, 8, and 5. (Find out the values of the areas, the bases, and the sides of the required parts). In the case of two pillars of known height, two strings are tied, one to the top of each. Each of these two strings is stretched in the form of a hypotenuse so as to touch the foot of the other pillar, or so as to go beyond the other pillar and touch (the ground). From the point where the two hypotenuse strings meet, another string is suspended (perpendicularly) till (it touches) the ground. The measure of this (last) string goes by the name antaravalambaka or the inner perpendicular. The line starting on either side from the point where (this) perpendicular string touches (the ground) and going to the points where the (abovementioned) hypotenuse strings touch the ground has the name of abadha, or the segment of the base. The rule for arriving at the values of such inner perpendicular and (such) segments of the base :--- 1801. The measurement of each of the pillars is divided by the measurement of the base covering the length between the (foot of the pillar) and the (point of contact of the hypotenuse) string (with the ground). Each of the quotients (so obtained) is 180. If a and b represent the height of the pillars in the diagram, c the distance between the two pilla rs, and m and n the respective distances of the pillars from the point where the string stretched from the top of the other pillar meets the carth, then, according to "he rule, . 4 (c + m) + b c + n) x (c + m + x); Ca = { 1 mi Lc+n . (c + m) (c + n) a (c + m) + b(c + n) 7 (c + m + n) ; where ci and c, are segments of (c + m) (c + n) * the babe as a whole ; and p=ci x --_-, or cg * - where p is the meas. * C + m c + n ure of the inner perpendicular. From a consideration of the similar triangles in the diagram it may be seen that C2 = c+ and and Gi 1 = + m. P For Private and Personal Use Only

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244 GANITASARASANGRAHA. (then) divided by their sum. The (resulting) quotients, on being multiplied by the measure of the base (as a whole) give rise to the (respective basal) segments. These (measures of the segments respectively) multiplied in the inverse order by the quotients (obtained in the first instance as above), give rise (in each case) to the value of the inner perpendicular. Examples in illustration thereof. 1811. (The given) pillars are 16 hastas in height. The base (covering the length between the points where the strings touch the ground) is pointed out to be 16 hastas. Give out, in this case, the numerical value of the segments of the base and also of the inner perpendicular. 1821. The height of one pillar is 36 hastas ; that of the second is 20 hastas. The length of the base-line is 12 hastas. What is the measure of the (basal) segments and what of the (inner) perpendicular? 1834-1841. (The two pillars are) 12 and 15 hastas (respectively); the measure of the interval between the two pillars is 4 hastas. From the top of the pillar of 12 hastas a string is stretched so as to cover 4 hastas (along the basal line) beyond the foot of the other pillar. From the top of (this) other pillar (which is 15 hastas in height) a string is (similarly stretched so as to cover 1 hasta (along the basal line) beyond the foot (of the pillar of 12 hastas in height). What is the measure of the (basal) segments here, and what of the inner perpendicular? 1853. (In the case of a quadrilateral with two equal sides), each of the two sides is 13 hastas in measure. The base here is 14 0(c + n) (c+n+n), and p=0, b From these ration we get = a (c + m) Cyb e + n) ; . . _ a c +m)__.. . - Ac+m) (C+ + ). **C,+ C a (c + m) + b(c+n)' a (c+ m) + b (c+n)' Similarly cg a(c+ m) + b(c+n) 1855. Here a quadrilateral with two equal si ies is given; in the next stanza a quadrilateral with three equal sides, and in the one next to it a quadrilateral with unequal sides are given. In all these cases the diagonale of the quadri. lateral have to be first fonnd out in accordance with the rule given in stanza n. fi c + m. For Private and Personal Use Only

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CHAPTER VII-MEASUREMENT OF AREAS. 245 hastas, and the top-side is 4 hastas. What is the measure of the (basal) segments (caused by the inner perpendicular) and what of the inner perpendicular (itself) ? 1861. In the case of the (quadrilateral) figure above-mentioned, the measures of the top-side and the base are each to be taken to be less by 1 hasta. From the top of each of the two perpendiculars, a string is stretched so as to reach the foot of the other perpendicular). You give out the measures of the inner perpendicular and of the basal segments (caused thereby). 187}. (In the case of a quadrilateral with unequal sides), one side is 13 hastas in measure, the opposite side is 15 hastas; the top-side is 7 hastas ; and the base here is 21 hastas What are the values of the inner perpendicular and of the basal segments (caused thereby)? 1883-1891. There is an equilateral quadrilateral figure, measuring 20 hastas at the side. From the four angles of that VII-54, and then the measures of the perpendiculars from the ends of the topside to the base as also the measures of the segments of the babe caused by those perpendiculars have to be arrived at by the application of the rule given in stanza VII.49. Then taking these measures of the perpendiculars to be those of the pillars, the rule given in stanza 180 above is applied to arrive at the measures of the inner perpendicular and the basal segments caused thereby. The problem given in stanza 187% is however worked in a slightly different way in the Kanarere commentary. The top-side is supposed to be parallel to the base, and the measures of the perpendicular and of the basal segments caused thereby are arrived at by constructing a triangle whose sides are the two sides of the quadrilateral, and whose base is equal to the difference between the base and the top-side of the quadrilateral. 1885-1894. The figure contemplated A av in this problem seems to be this: The inner perpendiculars referred to herein are GH and KL. To find out there, FE in first determined. FE, according to the commentary, is suid to be equal to VCMT N 2 - DM2 + DE2 + (HDM). XIKI Then with FE and BC or AD taken as pillars, the rule under reference may be applied. HEL For Private and Personal Use Only

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246 GANITASARASANGRAHA. (figure) strings are stretched out so as to reach the middle point of the (opposite) sides, (this being done) in respect of all the four sides. What may be the measure of each of the strings 80 stretched out ? In the interior of such (a quadrilateral figure with strings so stretched out), what may be the value of the inner) perpendicular and of the basal segments (caused thereby) ? The measure of the height of the pillar is known. For some reason or other that pillar gets broken and the upper part of the broken pillar) falls (to the ground, the lower end of the broken off part, however, remaining in contact with the top of the lower part). Then the basal distance between the foot of the pillar and its tup (now on the grouud) is ascertained. And (here is the rule for arriving at the numerical value of the measure of the remaining part of the pillar measured from its foot : 1903. The half of the difference between the square of the total height and the square of the (known) measure of the basal distance, when divided by the total height, gives rise to the measure of what remains unbroken. What is left thereafter (out of the total height) is the measure of the broken part. Examples in illustration thereof. 1911. The height of a pillar is 25 hastas. It is broken somewhere between (the top and the foot). The distance between the (fallen) top (on the floor) and the foot of the pillar is 5 hastas. How far away (from the foot) is it (viz., the pillar) broken? 1907. If A B C is a right-angled triangle, and if the measures of AC and of the sum of AB and BC are given, then AB and BC can be found out from the fact that BC2 = AB2+ AC%. The formula given in the rule is AB=(AB+ RC)- AC2 2 (AB + BC) and this can be easily proved to be true from the above equality. For Private and Personal Use Only

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CHAPTER VI-MEASUREMENT OF AREAS. 247 1927. There are 49 hastas in the measurement of the height of a bamboo (as it is growing). It is broken somewhere between (the top and the bottom). The distance (between the fallen top on the floor and the bottom of the bamboo) is 21 hastas. How far away (from the foot) is it broken? 1933-1951. The height of a certain tree is 20 hastas. A certain man seated on the top (of it) threw down a fruit thereof along a path forming a hypotenuse. Then another man standing at the foot of the tree went towards that fruit taking a path representing the other side (i.e., the base of the triangle in the situation) and received that fruit. The sum of the distances travelled by that fruit and this man turned out to be 50 hastas. What is the numerical value of the hypotenuse representing the path of that fruit? What may be the measure of the other side representing the path of the man who was at the foot of the tree i The numerical value (of the height) of a taller pillar as also the numerical valae (of the height of a shorter pillar is known. The numerical value (of the length) of the intervening space between the two pillars is also known. The taller (of the two pillars) gets broken and falls so that the top thereof rests ou the top of the shorter pillar, (the other end of the broken bit of the taller pillar being in contact with the top of the remaining portion thereof). And now the rule for arriving at the numerical value (of the length) of the broken part of the taller pillar as also at the numerical value of the height) of the remaining part (of the same taller pillar) : 1961. From the square of (the numerical measure of the taller (pillar), the sum of the square of the measure of the shorter a-aj 1963. If a represents the height of the taller pillar and b that of the shorter pillar, c the length of the intervening space between them, and a, the height of the standing portion of the broken pillar, then, according to the rule, di a = a - (b + c). 2 (a - b) For Private and Personal Use Only

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248 GANITASARASANGRAHA, (pillar) and the square of that of the base is subtracted. Half (of the resulting remainder) is divided by the difference between (the measures of) the two pillars. The quotient gives rise to the measure of the height of the standing part) of the broken (pillar). An example in illustration thereof. 1973. One pillar is 5 hastas in height; similarly another pillar, which is the taller, is 23 hastas (in height). The (length of the) intervening space (between the pillars) is 12 hastas. The top of the broken taller (pillar, falls on to the top of the other (pillar). (Find out the height of the standing part of the broken taller pillar.) Taking two-thirds of the numerical value of the vertical side of a longish quadrilateral as the height of a mountain, the rule for arriving, with the aid of the numerical value of the height of that mountain, at the numerical values of the horizontal side and of the diagonal of that longish quadrilateral : 1981. Twice the height of the mountain is the measure of the distance between the (foot of the mountain and the city (there). Half (the height of the mountain is the measure of the distance) of the upward flight in the sky. The diagonal is arrived at on adding together half the height of the mountain and the distance (of the city from the foot of the mountain). An example in illustration thereof. 1994-2001. On a mountain having a height of 6 yojanas there were 2 ascetics. One of them went walking on foot. The other 1994-2004. If in the marginal figure, a represents the height of the mountain, b the distance of the city from the foot of the mountain, and c the length of the hypotenuse course, then a is, according to the supposition made in the preamble to the rule in 1981, of the side AB, Therefore the height of the flight upwards i.e., EB., is a ... ... ... ... I a A As the courses of the two ascetics are oqual, c+ ta=a+b;..c=la +b ... II c = 402 + 0 + ab. But c = a + b*; .. ab = 2a .:.b = 2a ... ... ... ... ... III The three formulas marked I, II and III above are those given in the rule. For Private and Personal Use Only

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CHAPTER VII-MEASUREMENT OF AREAS. 249 was capable of moving in the sky. This ascetic flew up and then came down to the city taking the hypotenuse course. The other ascetic descended from the summit (vertically) to the foot of the mountain and walked along) to the city. (It was found that) both of them had travelled over the same distance. (What is the distance of the city (from the foot of the mountain) and what the height of the flight upwards? In an area representable by a (suspended) swing (and its vertical supports resting on the ground), the measures of the heights of either two pillars or two bill-tops are taken to be the measures of the horizontal sides of two longish quadrilateral figures. Then, (with the aid of these known horizontal sides and) in relation to the base line either between the two bills or between the two pillars, (as the case may be), the values of the two segments (caused by the meeting point of the perpendicular) are arrived at. These two segments are written down in the inverse order. The values of the two segments so written down in the inverse order are taken to be the values of the two perpendicular sides of the two longish quadrilateral figures. And, now, the rule for arriving at the equal numerical value of the diagonals of those (two longish quadrilateral figures) : 2011-2033. In relation to a figure representable by a (848pended) swing (and its vertical supports resting on the ground), the measures of the heights of either two pillars or two hills are taken to be the measures of the two sides of a triangle. Then, in relation to the value of the base (line) enclosed between those two 2011-2031. In the two quadrilaterals of the kind contemplated in this rule, let the vertical sides be represented by a, b; b let the base be c; and let C1, C2, be its seg ments and l the length of each of the equal portions of the rope. 32 For Private and Personal Use Only

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250 GANITASARASANGRAHA. sides (which has to be the same in value as the base line between the given pillars or bills), the segments (of the base caused by the meeting of the perpendicular from the vertex with the base) are arrived at in accordance with the rule laid down already. If the values of these (segments) are written down in the inverse order, they become the values of the two perpendicular sides of the two longish quadrilaterals in the required operation. Then, in accordance with the rule given already, the values of the diagonals of the two longish quadrilateral figures may be arrived at with the aid of the values of those two sides of the triangle above mentioned which are taken here as the two horizontal sides of the longish quadrilateral) and of those two perpendicular sides. These (diagonals) are of equal numerical value. Examples in illustration thereof. 2041-205. One pillar is 13 (hastas in height). The other is 15 (hastas in beight). The intervening distance (between them) is 14 (hastas). A rope (having its two ends) tied to the tops of these two pillars) hangs down so as to touch the ground some where between the two pillars). What are the values of the two segments, (so caused, of the base-lino between the pillars)? The two (hanging) parts of the rope are (in their length) of equal numerical value. Give out also the rope-measure. 206-2071. The height of (one) hill is 22 (yojanus). That of another hill is 18 (yojanas). The intervening space between the two bills is 20 (Yojanas in length). There stand two religious mendicants, (one) on the top of each, who can move along the sky. For the purpose of begging (their food), they came down 0 ..C2 = Now, a + CS=0+ cm, .. (cg + c) (c2 - C1)=0 -62; and + C2=c; 22 - 62 a? - 52 c --- and c =These values are obviously those of the segments of the base c of a triangle having the sides a and b, the segmente having been caused by the perpendicular from the vertex. This is what is stated in this rule. Vide rule given in stanza 49 above. ..C2 = 2 For Private and Personal Use Only

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CHAPTER VII-MEASUREMENT OF AREAS. 251 through the sky and) met in the city there (between the hills); and it turned out that they had travelled (along the sky) over equal distances. (Under these circumstances), of what numerical value were the segments (of the basal line between the two hills)? Of what value, O you who know calculation, is the numerical measure of the equal distance travelled in this (area) representable by a (suspended) swing. 2084-2094. The height of one hill is 20 yojanas ; and similarly, that of another (hill) is 24 yojanas. The intervening space between them is 22 yojanas (in length). Two mendicants, who stayed on the tops of these two hills, (one on each), and were able to move through the sky, came down, for the purpose of begging their food, to the city situated between those (two hills), and were found to have travelled (along the sky) over equal distances. What is the measure of the length) of the intervening space between that (city) in the middle and the hills (on either side). The rule for arriving at the value of the number of days required for the meeting together of two persons moving with unequal speed along a course representable by (the boundary of) a triangle consisting of (three) unequal sides : 2101. The sum of the squares (of the numerical values) of the daily speeds (of the two men) is divided by the difference between the squares of the values of (those same) daily speeds. 'The quotient (so obtained) is multiplied by the number of days spent (hy ne of the men) in travelling northwards (before travelling to the south-east to meet the other man). The meeting together of these two men takes place at the end of the number of days measured by this product. 210$. The course contemplated here is that along the sides of a right angled triangle. The formula given in the rale, if algebraically represented, is where ae is the number of days taken to go through the hypotenuse course, a and b the rates of journey of the two men, and d the number of days taken in going northwards. This follows from the under mentioned equation which in based on the data given in the problem: 6deg 2deg = 2 2 + (x+) x a? For Private and Personal Use Only

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252 GANITASARASANGRAHA. An example in illustration thereof. 2111-2121. The man who travels to the east moves at the rate of 2 yojanas (a day); and the other man who travels northwards moves at the rate of 3 yojanas (a day). This (latter man) having thus moved on for 5 days turns to move along the hypotenuse. In how many days will he meet the (other) man ? Both (of them) move out at the same time, and the number of days spent (by both of them) in journeying out is the same. The rule for arriving at the numerical value of the diameters of circles described about the eight kinds of figures consisting of the five kinds of quadrilateral figures and the three kinds of triangular figures (already mentioned) : 2131. In the case of a quadrilateral figure, the value of the diagonal (thereof), divided by that of the perpendicular, and (then) multiplied by that of the lateral side, gives rise to the value of the diameter of the circumscribed circle. In the case of a trilateral figure, the product of the values of the two sides (other than the base) divided by the value of the perpendicular (gives rise to the required diameter of the circumscribed circle). Examples in illustration thereof. 2141. In the case of an equilateral quadrilateral figure having 3 as the measure of each of its) sides, and also in the case of another (quadrilateral figure) of which the vertical side measures 5 and the horizontal side measures 12, what is the measure of the diameter) of the circumscribed circle ? 213). Let ABC be a triangle inscribed in a circle, AD the diameter thereof, and BE the perpendicular on AC. Join BD. Now the triangles ABD and BEC are pimilar. .. AB: AD=BE: BC ABX BC :. AD = BE This is the formula given in the rule A for the diameter of a circle circumscribed about a quadrilateral or a triangle. For Private and Personal Use Only

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CHAPTER VII---MEASUREMENT OF AREAS. 253 215]. The two lateral sides are (each) 13 in measure ; the topside is 4 ; and the base is said to be 14 in measure. In this case, what may be the diameter of the circle described about (such) a quadrilateral figure with two equal sides ? 2161. The top-side and the (two) lateral sides are cach 25 in measure. The base is 39 in measure. Tell me (here) the measure of the diameter of the circle described about such a quadrilateral figure with three equal sides. 2171. One of the lateral sides is 39 in measure ; the other lateral side is 52 in measure; the base is 60 and the top-side is 2). In relation to this (quadrilateral figure), wbat is the value of the diameter (of the circumscribed circle) ? 2181. The measure of the side of an equilateral triangle is 6 ; and that of an isosceles triangle is 13, the base (in this case) being 10 in measure. Give out what the valnes are of the diameters of the circles described about these triangles. 2197. In the case of a triangle with unequal sides the two sides are 15 and 13 in measure; the base is 14. Tell me the value of the diameter of the circle described about it. 2204. If you know the paisacika (processes of calculation), tell me after thinking well what may be the value of the diameter of the circle described about a (regular) six-sided figure having 2 as the measure of each of its) sides. The rule for arriving at the numerical values of the base, of the top-side and the other) sides of the eight (different kinds of figures beginning with the square, which are inscribed in a regular circular figure having a diameter of known numerical value: 2211. The value of the given diameter (of the circle) is divided by the value of the (hypothetically) arrived diameter of the circle (described about an optionally chosen figure belonging to 220%. The Kanarese commentary on this stanza works out this problem by pointing out that the diagonal of a regular hexagon is equal to the diameter of the circumscribed circle. 2214. The rule follows as a matter of course from the similarity of the required and the optionally chosen figures. For Private and Personal Use Only

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254 GANITASARASANGRAHA. the specified variety). The values of the sides of this optionall., chosen figure) should be multiplied by the resulting quotient (arrived at as mentioned above). Thus, the numerical values of the sides of the figure produced in the given circle) are deduced. Examples in illustration thereof. 2221. The diameter of a circular figure is 13. O friend, think out well and tell me the (various measurements relating to the) eight different kinds of figures beginning with the square which are (inscribed) in this (circle). The role for arriving at the value of the diameter of the circular figure inscribed within the various kinds of quadrilateral and trilateral) figures mentioned before, with the exception of the longish quadrilateral figure, when the accurate measure of the area and the numerical value of the perimeter are known in relation to those same) quadrilateral and other figures :-- 2231. The (known) accurate measure of the area of any of the figures other than the longish quadrilateral figure should be divided by a quarter of the numerical value of the perimeter (of that figure). The result is pointed out to be the diameter of the circle inscribed within that figure. Examples in illustration thereof. 2247. Having drawn the inscribed circle in relation to the already specified figures beginning with the square, O you who know the secret of calculation, give out now (the value of the diameter of each such inscribed circle). 223. If a represents the sum of the sides, and d the diarreter of the inscribed oircle, and A the area of the quadrilateral or the triangle in which the circle is insoribed, then X = 22 Hence the formula given in the rule in d=1; For Private and Personal Use Only

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CBAPTER VII-MEASUREMENT OF AREAS. 255 Within the (known) numerical measure of the diameter of a regular circle, any known number being taken as the measure of an arrow, the rule for arriving at the numerical value of the string of the bow) having an arrow of that same measure :-- 2251. The difference between (the given value of) the diameter and (the known value of the arrow is multiplied by four times the value of the arrow. Whatever is the square root of the resulting product), that the wise man should point out to be the (required) measure of the string (of the bow). An example in illustration thereof. 226]. The diameter of the circle is 10. It is cut off by 2. 0 mathematician, give out, after calculating well, what may be the string of the bow) in relation to (that) cut off portion (of the given diameter). The rule for arriving at the numerical value of the arrowline, when the numerical value of the diameter of a (given) regular circle and the value of a bow-string line in relation to that circle) are (both) known :-- 2277. That which happens to be the square root of the difference between the squares of the (known) values of the diameter and the bow-string line (relating to the given circle)that has to be subtracted from the value of the diameter. Half of the resulting) remainder should be understood to give (the required value of) the arrow-line. An example in illustration thereof. 2281. The diameter of the (given) circle is 10 in measure. Moreover, the bow-string line inside is known to be 8 in measure. Give out, o friend, what the value of the arrow-line may be in relation to that (bow-string). 225). The rules given in stanzas 225), 227, 229 and 231} are all based on the fact that in a circle the rectangles contained by the segments of two intersecting chords are equal, For Private and Personal Use Only

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256 GANITASARASANGRAHA. The rule for arriving at the numerical value of the diameter of a (given) circle when the numerical values of the (related) bow-string line and the arrow line are known : 2291. The quantity representing the square of the value of the bow-string line is divided by the value of the arrow line as multiplied by four. Then the value of the arrow line is added (to the resulting quotient). What is so obtained is pointed out to be the measure of the breadth of the regular circle measured through the centre. An example in illustration thereof. 230. In the case of a regular circular figure, it is known that the arrow line is 2 dandas in measure, and the bow-string line 8 dandas. What may be the value of the diameter in respect of this (circle) ? When two regular circles cut each other, there arises a fishshaped figure. In relation to that fish-shaped figure, the line going from the mouth to the tail (thereof) should be drawn. With the aid of this line, there will come into existence the outlines of two bows applied to each other face to face. The line drawn from the mouth to the tail of the fish-figure) happens to be itself the bow-string line in relation to both these bows. The two arrow lines in relation to both these bows are themselves to be understood as forming the two arrow lines connected with the mutually overlapping circles. And the rule here is to arrive at the values of the arrow lines connected with the overlapping portion when two regular circles cut each other : 2311. With the aid of the values of the two diameters (of the two cutting circles) as diminished by the value of the greatest breadth of) the overlapped portion (of the circles), the operation of praksepaka should be carried out in relation to this (known) value of the greatest breadth of) the overlapped portion (of the circles). The two results (so obtained) are in the matter 2314. The problem here contemplated may be seen to have been also solved by Aryabhata, and the rule given by him coincides with the one under reference here. For Private and Personal Use Only

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CHAPTER VII-MEASUREMENT OF AREA. 257 of (such) circles pointed out to be the values, each of the other, measuring the two arrow lines related to the overlapping (oireles). An example in illustration thereof. 232). In relation to two ciroles whose extent is measured by (diameters of) 32 and 80 hastas (in value), the (greatest breadth of the common) overlapping portion is 8 hastas. Give out what the values of the arrow lines, as related respectively to those two (circles), are (here). Thus ends the section treating of devilishly difficult problems. Thus ends the sixth subject of treatment, known as Calculations regarding Areas, in Sarasangraha, a work on arithmetic by Mahaviracarya. For Private and Personal Use Only

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258 GANITASARASANGRAHA. CHAPTER VIII. CALCULATIONS REGARDING EXCAVATIONS. 1. I bow in religious devotion with my head bent downwards) to Jina Vardhamana, whose foot-stool is honoured by the crowns worn by all the chief gods, who is omniscient, ever-enduring, unthinkable, and infinite in form, and is (further) like the young (rising) sun in relation to the lotus-lakes representing the good and worthy people that are his devotees. 2. I shall now give out the three) varieties of karmantika, aundraphala, and suksmaphala (in relation to excavations), which varieties are all derived from those various kinds of geometrical figures, mentioned before, as results obtained by multiplying them by (quantities measuring).depth. This seventh subject of treatment is the subject of excavations. A stanza regarding the conventional assumption (implied in this chapter) : 3. The quantity of earth required to fill an excavation measuring one cubio hasta is 3,200 palms. From that (same cubic volume of excavation) 3,600 palas (of earth) may be taken out. The rule for arriving at the oubical contents of excavations : 4. Area multiplied by depth gives rise to the approximate measure of the cubical contents in a regular excavation. The sums of all the various) top dimensions with the corresponding bottom dimensions are halved; and then these halved quantities of the same denomination are all added, and their sum is) divided by the number of the said (halved quantities). Such is the process of arriving at the average equivalent value. 2. The term Aundra in Aundraphala is rather strange Sanskrit and in perhaps related to the Hindi word 311g neaning 'deep.' 3. The idea in this stanza evidently is that one cubic hasta of compressed earth weighs 3,600 palas, while 3,200 palas of earth are sufficient to fill loosely the & pace of 1 cubic hasta, 4. The latter half of this stanza evidently gives the process by which we may arrive at the dimensions of a regular excavation fairly equivalent to any given irregular excavation. For Private and Personal Use Only

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CHAPTER VIII.-CALCULATIONS REGARDING EXCAVATIONS. 259 Examples in illustration thereof. 5. In relation to (an equilateral) quadrilateral area (representing the section of a regular excavation), the sides and the depth are 8 hastas (each in measure). In respect of this regular excavation, what may be the value of the cubical contents here? 6. In relation to an (equilateral) triangular area (representing the section of a regular excavation), the sides are 32 hastas each, and in the depth there are found 36 hastas and 6 angulas. What is the calculation (of the contents) here? 7. In relation to a (regular) circular area representing (the section of) a regular excavation, the diameter is 108 hastas, and the depth (of the excavation) is 165 hastas. (Now), give out what the cubical contents are. 8. In relation to a longish quadrilateral area (forming the section) of a regular excavation, the breadth is 25 hastas, the side (measuring the length) is 60 hastas and the depth (of the excavation) is 108 hastas. Quickly give out (the cubical contents of this regular excavation). The rule for arriving at the accurate value of the cubical contents in the calculation relating to excavations, after knowing the result designated karmantika as well as the result designated aundra and with the aid of these results : 9-111. The values of the base and the other sides of the figure representing the top sectional area are added respectively to the values of the base and the corresponding sides of the figure representing the bottom sectional area. The (several) sums (80 arrived at) are divided by the number of the sectional areas taken into consideration in the problem). The resulting) quantities are 9-11}. The figures dealt with in this rule are truncated pyramids with rectangular or triangular bases, or truncated cones all of which have to be conceived as turned upside down. The rule deals with three different kinds of measures of the cubical contents of excavations. Of these, two, viz., the Karmantika and Aundra measures give only the approximate values of the contents. The accurate measure is calculated with the help of these values. If K represents the Karmantika-phala and A represents the Aundra-phala then the accurate measure is said to be equal to + K, i.e., K + $ A, For Private and Personal Use Only

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260 GANITASARASANGRAHA. multiplied with each other (as required by the rules bearing upon the finding out of areas when the values of the sides are known). The area (so arrived at), when multiplied by the depth, gives rise to the cubical measure designated the karmantika result. In the case of those same figures representing the top sectional area and the bottom sectional area, the value of the area of each of these figures is (separately) arrived at. The area values (so obtained) are added together and then divided by the number of sectional) areas (taken into consideration). The quotient (so obtained) is multiplied by the yalue of the depth. This gives rise to (the cubical measure designated) the aundra result. If one-third of the difference between these two results is added to the karmantika result, it indeed becomes the accurate value of the required cubica! contents). Examples in illustration thereof. 12]. There is a well whose (sectional) area happens to be an equilateral quadrilateral. The value (of each of the sides) of the top (sectional area) is 20 (hastas), and that (of each of the sides) of the bottom (sectional area) is only 16 (hastas). The depth is 9 (hastas). O you who know calculation, tell me quickly what the cubical measure here is. 131. There is a well whose (sectional) area happens to be an equilateral triangular figure. The value (of each of the sides) of the top (sectional area) is 20 (hastas), and that (of each of the sides of the bottom (sectional area) is 16; the depth is 9 (hastas). What is the value of the karmantika cubical measure, of the If a and I be the measures of a side of the top and bottom surfaces respectively of a truncated pyramid with a square base, it can be easily shown that the acourate measure of the cubical contents is equal to h (a + b 2 + ab), where h is the height of the truncated pyramid. The formula given in the rule for the acourato measure of the cabical contents may be verified to be the same as this with the help of the following values for the Karmantika and Aundra resolts given in the rule:--- La 2 + 2 Similar verifications may be arrived at in the case of troncated pyramids having an equilateral triangle or a rectangle for the base, and also in the case of truncated cones. Kala + b 2 Ta + b \ x 1; For Private and Personal Use Only

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CHAPTER VIII-CALCULATIONS REGARDING EXCAVATIONS. 261 aundra cubical measure, and of the accurate cubical measure here? 141. There is a well whose (sectional) area happens to be regularly circular. The (diameter of the) top (sectional area) is 20 dandas, and that of the bottom (sectional arca) is only 16 dandas. The depth is 12 dandas. What may be the karmantika, the aundra, and the accurate cubical measures here? 151. In relation to an excavation whose sectional area happens to be) a longish quadrilateral figure (i.e., oblong), the length at the top is 60 (hastas), the breadth is 12 (hastas); at the bottom, these are (respectively) half of what they measure at the top). The depth is 8 (hastas). What is the cu bical measure here? 161. (Here is another well of the same kind), the lengths of whose sectional areas) at the top, at the middle, and at the bottom are (respectively) 90, 80, and 70 (hastas), and the breadths are (respectively) 32, 16, and 10 hastas. This is 7 (hastas) in depth. (Find out the required cubical measure.) 171. In relation to an excavation whose sectional area happens to be) a regular circle, the diameter at the mouth is 60 (hastas), in the middle 30 (hastas), and at the bottom 15 (hastas). The depth is 16 hastas. What is the calculated result giving its cubical measure ? 181. In relation to (an excavation whose sectional area happens to be) a triangle, each of the three sides measures 80 hastas at the top, 60 hastas in the middle, and 50 hastas at the bottom. The depth is 9 hastas. What is the calculated result giving its cubical contents ? The rule for arriving at the value of the cu bical contents of a ditch, as also for arriving at the value of the cubical contents of an excavation having in the middle (of it) a tapering projection (of solid earth) : 197-203. The breadth (of the central mass) increased by the top-breadth of the surrounding ditch, and (then) multiplied by 191-205. These stanzas deal with the measurement of the cubic contents of & ditch dug round a central mass of earth of any shape. The central mass may be in section a square, a reotangle, an equilateral triangle, or a circle ; For Private and Personal Use Only

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262 GANITASARASANGRAHA. three, gives rise to the value of the (required) perimeter in the case of triangular and circular excavations. In the case of a quadrilateral excavation, (this same value of the required perimeter results) by multiplying the quantity four (with the value of the breadth as before). In the case of excavations having central masses tapering upwards or downwards the operation (for Karmantikaphala) is (to add the value of) half the breadth of the excavation to (that of the breadth of) the central mass, and (for Aundraphala), to add (the value of) the breadth (of the excavation to the value of the breadth of the central mass); then (the procedure is) as (given) before. Examples in illustration thereof. 21. The already mentioned trilateral, quadrilateral, and circular (areas) have ditches thrown round them. The breadth measures 80 dandas, and the ditches are as much as 4 (dandas) in breadth, and 3 (dandas) in depth. (Find out the cubical contents.) Acharya Shri Kailassagarsuri Gyanmandir and the excavation may be of the same width both at the bottom and the top, or may be of diminishing or increasing width. The rule enables us to find out the total length of the ditch in all these cases. I. When the width of the ditch is uniform, the length of ditch = (a + b) x 3 in the case of an equilateral triangular or circular ditch, where d is the nieasure of a side or of the diameter of the central mass and b is the width of the ditch: but this length = (d+ b) x 4 in the case of a square excavation with a central mass, square in section. II. When the ditch is tapering to a point at the bottom or the top, the length of the ditch for finding out the Karmantika-phalad + X 3, or d + b +) 2 x 4, according as the central mass (1) is in section trilateral or circular, or (2) square. Length of ditch for finding out Aundra-phala x 4 respectively. = For Private and Personal Use Only (a + b) x 3 and (a + b) These expressions have to be multiplied by half of the width of the ditch and by its depth for finding out the respective cubical phalas. The formulas given above in relation to triangular and circular excavations give only approximate results. With the aid of the total length of the ditch so obtained, the cubical contents are found out in the case of ditches with sloping sides by applying the rule given in stanzas 9 to 11 above.

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CHAPTER VIII-CALCULATIONS REGARDING EXCAVATIONS. 263 221. The length of a longish quadrilateral is 120 (dandas) and the breadth is 40. The ditch around is as big as 4 dandas in breadth and 3 in depth. (Find out the cubical contents.) The rule for arriving at the value of the cubical contents of an excavation, when the depth of the excavation varies (at various points), and also for arriving, when the cubical contents of an excavation are known, at the depth of digging necessary in the case of another (known) area (80 that the cubical contents may be the same) :-- 231. The sum of the depths (measured in different places) is divided by the number of places; this gives rise to the average) depth. This multiplied by the top area (of the excavation) gives rise to the (required) cubical contents of the excavation in the case where that area is trilateral, quadrilateral or circular. The cubical contents (of a given excavation), when divided by the (known) value of another area, gives rise to the depth (to which there should be digging, so that the resulting cubical contents may be the same). Examples in illustration thereof. 243. In an equilateral quadrilateral field, the ground covered by which has an extent measured by 4 hastas (in length and breadth). the excavations are (in depth) 1, 2, 3, and 4 hastas (in four different cases). What is the measure of the average depth (of the excavations)? 25]. There is a well with an equilateral quadrilateral section, the sides whereof are 18 hastas in measure; its depth is hastas. With the water of this (well), another well measuring 9 hastas at each of the sides (of the section) is filled. What is the depth (of this other well)? When the measures of the sides of the top (sectional area) and also of the bottom (sectional aren) are known, and when the 22. For finding out the total length of the surrounding ditch when the central mess of earth is rectangular in section, the measnres of the sides as increased by the width or half the width of the ditch are added together, according as the Ka: mantika or the Aundra result is required. For Private and Personal Use Only

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264 GANITASARASANGRAHA. measure of the depth also is known, in relation to a certain given excavation, the rala for arriving at the value of the sides (of the resulting bottom section) at any optionally given depth, and also for arriving at the (resulting) depth (of the excavation) if the bottom is reduced to a mere point :-- 263. The product resulting from multiplying the (given) depth with (the given measure of a side at the top, when divided by the difference between the measures of the top side and the bottom side gives rise here to the (required) depth (when the bottom is) made to end in a point. The depth measured (from the pointed bottom) upwards (to the position required) multiplied by the measure of the side at) the top and (then) divided by the sum of the side measure, if any, at the pointed bottom and the (total) depth (from the top to the pointed bottom), gives rise to the side measure (of the excavation at the required depth). An example in illustration thereof. 27. There is a well with an equilateral quadrilateral section. The (side) measure at the top is 20 and at the bottom 14. The depth given in the beginning is 9. (This depth has to be) further (carried) down by 3. What will be the side value of the bottom here)? What is the measure of the depth, (if the bottom is) made to end in a point ? 26$. The problems contemplated in this stanza are (a) to find out the full latitude of an inverted pyramid or cone and (b) to find out the dimensions of the cross section thereof at a desired level, when the altitude and the dimensions of the top and bottom surfaces of a truncated pyramid or cone are given. If, in a truncated pyramid with square base, a is the measure of a side of the base and b that of a side of the top surface and h the height, then accoi ding to the rule given here, H taken as the height of the whole pyramid .. axh , and the reasure of a side of the cross section of the pyramid at any given height represented by = an 'These formulas are applicable in the case of a cone as well. In the rule the measure of the side of section forming the pointed art of the pyramid is required to be added to H, the denon inator in the second formula, for the reason that in some cases the pyramid may not actually end in a point. Where, however, it does end in a point, the value of this side has to be zero as a matter of course. For Private and Personal Use Only

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CHAPTER VIII-CALCULATIONS REGARDING EXCAVATIONS. 265 The rule for arriving at the value of the cubical contents of a spherically bounded space : 28. The half of the cube of half the diameter, multiplied by nine, gives the approximate value of the cubical contents of a sphere. This (approximate value), multiplied by nine and divided by ter. on neglecting the remainder, gives rise to the accurate value of the cubical measure. An example in illustration thereof. 29. In the case of a sphere measuring 16 in diameter, calculate and tell me what the approximate value of (its) cubical measure is, and also the accurate measure (thereof). The rule for arriving at the approximate value as well as the accurate value of the cubical contents of an excavation in the form of a triangular pyramid, (the height whereof is taken to be equal to the length of one of the sides of the equilateral triangle forming the base) : Acharya Shri Kailassagarsuri Gyanmandir 301. The cube of half the square of the side (of the basal equilateral triangle) is multiplied by ten; and the square root (of the resulting product is) divided by nine. This gives rise to the approximately calculated value (required). (This approximate) value, when multiplied by three and divided by the square root of X 28. The volume of a sphere as given here is (1) approximately= de and (2) accurately== 9 9 2 x 10' cal contents of a sphere is 4 r3; and this becomes cemparable with the 3 above value, if is taken to be 10. Both the MSS. read Hid , making it appear that the accurate value is of the approximate 10 9 X () The correct formula for the cubi a value; but the text adopted is which makes the accurate value 9 of the approximate one. It is easy to see that this gives a more accurate result in regard to the measure of the cubical contents of a sphere than the other reading. 10 30. Algebraically represented the approximate value of the cubical contents of a triangular pyramid according to the rule comes to x 1/5, as 18 as i.e., 12 as 12 XV2; where 34 20 ; and the accurate value becomes equal to 9 For Private and Personal Use Only X

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266 GANITASARASANGRAHA. ten, gives rise to the accurately calculated cabioal contents of the pyramidal excavation. An example in illustration thereof. 311. Calculate and say what the approximate value and the accurate value of the cubical measure of a triangular pyramid are, the side of the (basal) triangle whereof is 6 in length. When the pipes leading into a well are (all) open, the rule for arriving at the value of the time taken to fill the well with water, when any number of optionally chosen pipes are together (allowed to fill the well). 321-33. (The number one representing) each of the pipes is divided by the time corresponding to each of them (separately); and (the resulting quotients represented as fractions) are reduced so as to have a common denominator; one divided by the sum of these (fractions with the common denominator) gives the fraction of the day (within which the well would become filled) by all the pipes (pouring in their water) together. Those (fractions with the common denominator) multiplied by this resulting fraction of the day give rise to the measures of the flow of water (separately through each of the various pipes) into that well. An example in illustration thereof. 3. There are 4 pipes (leading into a well). Among them, each fills the well in order) in }, }, }, } of a day. In how much of a day will all of them (together fill the well, and each of them to what extent)? In the Fourth Subject of Treatment named Rule of Three, an example (like this) has already) been given merely as a hint; the a gives the measure of the altitude of the pyramid as also of a side of the basal equilateral triangle. It may be easily seen that both these values are somewhat wide of the mark, and that the given approximate value is nearer the correct value than the 80-called acourate value, For Private and Personal Use Only

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CHAPTER VIII-CALCULATIONS REGARDING EXCAVATIONS. 267 subject (of that example) is expanded here and is given out in detail. 35-36. There is at the foot of a hill a well of an equilaterally quadrilateral section measuring 9 hastas in each of the (three) dimensions. From the top of the hill there runs a water channel, the section whereof is (uniformly) an equilateral quadrilateral having 1 angula for the measure of a side. (As soon as the water flowing through that channel begins to fall into the well), the stream is broken off at the top; and (yet), with it (that well) becomes filled in with water. Tell me the height of the bill and also the measure of the water in the well. 37-381. There is at the foot of a hill a well of an equilaterally quadrilateral section measuring 9 hastas in each of the (three) dimensions. From the top of the hill, there runs a water channel, (the section whereof is throughout)a circle of 1 angula in diameter. As soon as the water (flowing through the channel) begins to fall into the well, the stream is broken off at the top. With the water filling the whole of the channel, that well becomes filled. O friend, calculate and tell me the height of the mountain and also tbe measure of the water. 391-401. There is at the foot of a bill a well of an equilaterally quadrilateral section measuring 9 hastas in each of the) three dimensions. From the top of the hill there runs a water obannel: (the section whereof is throughout) triangular, each side measuring 1 angula. As soon as the water (lowing through that channel) begins to fall into the well, the stream is broken off at the top. With the water (filling the whole of the channel) that (well) becomes filled. O friend, calculate and tell (me) the height of the mountain and the measure of the water. 35 to 42. The reference here is to the example given in stanzas 15-16 of ohapter V-ride also the footnote thereunder. The volume of the water is probably intended to be expressed in vahas. (Vide the table relating to this kind of volume measure in stanzas 36 to 38, chapter I.) It is stated in the Kanarese commentary that 1 cabio angula of water is equal to 1 karsa. Then according to the table given in stanza 41 of chapter I, 4 kargas make one pala ; according to stanza 44 in the same chapter, 12palas make one prastha; and stanzas 36 to 37 therein give the relation of the prastho to the vaha. For Private and Personal Use Only

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268 GANITASARABANGRAHA. 414-42). There is at the foot of a bill a well of an equilaterally quadrilateral section measuring 9, hastas in each of the three dimensions. (From the top of the hill) there runs a water channel, (the section whereof is uniformly) 1 angula broad at the bottom, 1 angula at (each of) the dug (side slopes), and 2 angulas in length (at the top). As soon as the water (flowing through that channel) begins to fall into the well, the stream is broken off at the top. With the water (filling the whole of the channel) that well becomes filled. What is the height of the hill and (what) the measure of the water ? Thus ends the section on accurate measurements in the caloulations relating to excavations. Calculations Relating to Piles (of Bricks). Hereafter, in (this) chapter treating of operations relating to excavations, we will expound calculations relating to (brick) piles. Here there is this convention (regarding the unit brick). 431. The (unit) brick is 1 hasta in length, half of that in breadth, and 4 angulas in thickness. With such (bricks all) operations are to be carried out. The rule for arriving at the cubical contents of a given excavation in a field and also at the number of bricks corresponding to the above cubical contents. 441. The area at the mouth (of the excavation) is multiplied by the depth; this (resulting product) is divided by the cubis measure of the (unit) brick. The quotient so obtained is to be understood as the (cubical) measure of a (brick) pile; that same (quotient) also happens to be the measure of the number of the bricks. Examples in illustration thereof. 451. There is a raised platform equilaterally quadrilateral (in section) having a side measure of 8 hastas and a height of 9 44}. The oubioal measure of the brick pile here is evidently in terms of the unit briok. For Private and Personal Use Only

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Shri Mahavir Jain Aradhana Kendra CHAPTER VIII-CALCULATIONS REGARDING EXCAVATIONS. 269 hastas. That (platform) is built up of bricks. O you who know calculation, say how many bricks there are (in it).. 46. A raised platform, equilaterally triangular (in section), having 8 hastas (as its side measure), and 9 hastas as height, has been constructed with the aforesaid bricks. Calculate and say how many bricks there are in this (structure). 47. A raised platform, circular in section, having a diameter of 8 hastas and a height of 9 hastas is built up with (the same aforesaid) bricks. O you who know calculation, say how many bricks there are in it. 48. In the case of (a raised platform having) an oblong (section), the length is 60 hastas, the breadth 25 hastas, and the height is 6 hastas. Give out in this case the measure of that brick pile. 49. A boundary wall is 7 hastas in thickness, 24 hastasin length and 20 hastas in height. How many are the bricks used in building it ? 20 www.kobatirth.org 50. The thickness of a boundary wall is 6 hastas at the top and 8 hastas at the bottom; its length is 24 hastas and height 20 hastas. How many are the bricks used in building it? 51. (In the case of a raised sloping platform), the heights are (respectively) 12, 16 and 20 hastas (at three different points); 2 60-61. In finding opt the cubical contents of the wall, the average breadth calculated accor ding to the rule, given in the latter half of stanza 4 above, is used; so the Karman. tika value is taken into consideration here. 51. This sloping platform is bounded at its two ends by two vertical planes, the top and the side surfaces alone being sloping. The top forms an inclined 16 24. 3 Acharya Shri Kailassagarsuri Gyanmandir 4 12 For Private and Personal Use Only 7

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270 GANITASABASANGRAHA. the measures of the breadth at the bottom are (respectively) 7, 6 and 5, (the sa ne) at the top being 4, 3 and 2 hastas; the length is 24 hastas. (Find out the number of bricks in the pile. The rule for arriving, in relation to a given raised platform (part of) which has fallen down, at the number of bricks found. (intact) in the unfallen (part) and also at the number of bricks found in the fallen (part) : 521. The difference between the top (breadth) and the bottom (breadth) is multiplied by the height of the fallen (portion) and divided by the whole height. (To the resulting quotient) the value of the top (breadth) is added. This gives rise to the measure of the basal breadth in relation to the upper (fallen portion) as well as to the top breadth in relation to the lower (intact portion). The remaining operation has been already described. An example in illustration thereof. 53}. (In relation to a raised platform), the length is 12 hastas the breadth at the bottom is 5 hastas, (the breadth) at the top is 1 hasta, and the height of through is 10 hastas. (A measure of) 5 hastas (in height? i that (platform) gets broken down and falls. How many are those (unit) bricks there in the broken and the un broken parts of the platform) ? When a (high) fort-wall is broken down obliquely, the rule for arriving at the number of bricks which remain intact and of the bricks that have fallen down plane, the breadth of which is 2 hastas at the raised end and 4 hastas at the other plane, side diagram in the marginaath of the standing par of the latformis algebraically (a - b) d. 52]. The measure of the top.breadth of the standing par of the platform which is the same as the bottom-breadth of the fallen part of the platform--is + b; where a is the bottom-breadth, 8 is the topbreadth, h the total height and d the height of the fallen part of the raised platform. This formula can be easily shown to be correct by applying the properties of similar triangles. The operation referred to in the rule as having been already described is what in given in stanza 4 above. For Private and Personal Use Only

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CHAPTER VIII-CALCULATIONS REGARDING EXCAVATIONS. 271 541. The bottom (breadth) and the top (breadth) are (each) doubled. To these are added (respectively) the top (breadth) and the bottom (breadth). The (resulting) quantities are respectively) increased and decreased by the height (above the ground) of the unbroken (part of the wall); und (then the quantities so obtained) are multiplied by the length and also by the sixth part of the (total) height. (Thus) the number of bricks intact and the number of bricks fallen off may be obtained in order. Examples in illustration thereof. 551. This high fort-wall (of measurements already given, struck by a cyclonic wind) bas been (obliquely) from the bottom, broken down along the diagonal section. In relation thereto, how many are the bricks intact and the bricks fallen down? 564. The same high fort-wall has been broken down by the cyclone obliquely after leaving over 1 hasta from the bottoni. How many are the bricks that remain intact and how many the brioks that have fallen down ? The rule for arriving at the growing number of layers of bricks) in relation to the central height of a fort-wall, and (algo) for arriving at the rate of the) diminution of layers 54. If a be the breadth at the bottom, b the breadth at the top, h the total B height and I the > length of the wall, and d the height above the ground of the unbroken part of the wall thens 20+ 0 + d), and (26 + a-d) represent the number of bricks intaot and the number of bricks fallen 12 - off. The figure in the margin shows the wall mentioned in stanza 587; and ABCD ndicate the plane along which the wall fractured when it broke. For Private and Personal Use Only

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272 GANITASABABANGRAHA. (happening to be the diminution in breadth) on both the sides (of the wall in passing from below upwards) :-- 57}. The height (of the central section) divided by the height of the given brick gives rise to the (required) measure of the layers (of bricks). This (number) is diminished by one and (then) divided by the difference between the top (breadth) and the bottom (breadth). The resulting quotient gives (in itself) the value of the (rate of the) diminution (in breadth) measured in terms of the layers. Examples in illustration thereof. 581. The breadth of a high fort-wall is 7 hastas at the bottom Its height is 20 hastas. It is built so as to have 1 hasta (as its breadth) at the top. With the aid of bricks of 1 hasta in height, (find out) the (measure of the growth of the central) layers and of the (rate of) diminution (in the breadth). 591-60. In a regularly circular well, 4 hastas in diameter, & wall of 11 hastas in thickness is built all round by means or tho-- already mentioned typical) bricks. The depth of that (well) is 3 hastas. If you know, calculate and tell me, O friend, how many are the bricks used in the building. In relation to a structure built of bricks (around a place), the rule for arriving at the value of the cubical contents (of that structure), when the breadth at the bottom (of the structure) is given and also the breadth at the top:-- 61. Twice the average) thickness of the structure has added to it the given length and the breadth (of the place). The sum (so obtained) is doubled, and the result is the (total) length (of the structure when it is) ip (the form of) an oblong. This (resulting quantity), multiplied by the (given) height and the (already mentioned average) thickness, gives rise to the (required) cubical measure). 59-60. The bricks contemplated here is the unit brick mentioned in stanza 43} above. This problem does not illustrate the rule given above in stanza 571. but it has to be worked according to the rules given in stanzas 19-204 and 443 of this chapter For Private and Personal Use Only

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CHAPTER VIII-CALCULATIONS REGARDING EXCAVATIONS. 273 An example in illustration thereof. 62. In relation to the place known) as vidyadhara-naghara, the breadth is 8, and the length is 12. The thickness of the surrounding wall is 5 at the bottom and 1 at the top. Its height is 10. (What is the cubic measure of this wall ?) Thus ends (the section on) the measurement of (brick) piles in the operations relating to excavations. Hereafter, we shall expound the operations relating to the work done with saws (in sawing wood). The definitions of terms in relation thereto are as follow : 63. Two hastas less by six angulas is what is called a kisku, The number measuring the courses of cutting from the beginning to the end of a given (log of wood) has the name of marga (or way). 64-66. Then, in relation to collections (of logs) of wood of not less than two varieties, consisting of teak logs and other such logs hereafter to be mentioned, the number of angulas measuring the breadth, and those measuring the length, and the number of margas are (all three) multiplied together. The resulting product is divided by the square of the number of angulas found in a hasta. In operations relating to saw-work, this gives rise to a valuation of the work as measured) in what is known as pattikas. In relation to logs (of wood) consisting of teak logs and other such logs, the number of hastas measuring the breadth and of those measuring the length are multiplied with each other, and (then) multiplied by the number of margas, and (thereafter) divided by the pattikas as above determined ; this gives rise to the numerical measure of the work done by means of the saw. 63 to 674. Kisku =1 hasta. Marga is the name given to any desired course or line of sawing in a log of wood. The extent of the cut surface in & log of wood measures ordinarily the work done in sawing it provided that the wood is of a definite hardnes assumed to be of unit valae. This extent of the cut surface is measured by means of a special unit area which is called a pattika and is 96 angulas in length and one kisku or 42 angul asin breadth. It is easy to see that a pattika is thus equal to seven square hastas. 36 For Private and Personal Use Only

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Shri Mahavir Jain Aradhana Kendra 274 GANITASARASANGRAHA. 67-67. In relation to (logs of wood obtained from) trees named saka, arjuna, amla-vetasa, sarala, asita, sarja and dunduka, and also (in relation to varieties of wood) named griparni and plaksa, the marga is 1 in each case, the length is 96 angulas, and the breadth is 1 kisku (for arriving at the measure of a pattika). An example in illustration thereof. 68. In relation to a log of teak wood, the length is 16 hastas, the breadth is 31 hastas and the margas (or saw-courses) are 8 in number. How many are (the units of saw-work) done here ? # Thus ends the section on saw-work in the (chapter on) operations relating to excavations. Thus ends the seventh subject of treatment known as Operations relating to Excavations in Sarasangraha, which is a work on arithmetic by Mahaviracarya. For Private and Personal Use Only

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CHAPTER IX-CALCULATIONS RELATING TO SHADOWS. 275 CHAPTER IX. CALCULATIONS RELATING TO SHADOWS. 1. That Jina, santi, who bestows peace upon people, is the lord of the world, knows all beings, and is (ever) growing in influence through his eight miraculous powers--to him, who has vanquished the hosts of his enemies, I bow in salutation. In the beginning, we shall give out the means of determining the eight directions oommencing with the east. 2. On an even ground-surface which is (a horizontal plane) like the upper surface of water, a (perfectly) round circle should be drawn with the aid of a looped string having twice the length of an optionally chosen style (fixed in the centre). 3. The shadow of that optionally chosen style fixed in the centre of that circle touches the (circumferential) line of that circle at the beginning of the day as also at the time forming the close of the day. By this, the western direction and the eastern direction are pointed out in order. 4. By means of the string running in the line of these two (ascertained) directions, a fish-shaped figure (or lune) should be 4. The string with the aid of wbioh the fish-shaped figure is drawn should be longer than the radius of the circle drawn according to star.za 2 above. If OE and OW in the annexed diagram represent the eastern and the western directions, WL NPSR will be the love drawn by desoribing two ciroles with centres respectively at E and W and with ER and WP 28 equal radii. The line NS outting the angles of the lune marks the northern and the southern directions. -UE For Private and Personal Use Only

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276 GANITASARASANGRAHA. drawn which will extend from north to south. The straight line ranning through the middle of the angles of this (fish-shaped figure) represents of itself the northern and the southern directions. The intermediate directions have to be ascertained as being derivable from half the interspace between these) directions. 47. The (measure of the) equinoctial shadow is indeed half of the sum of the measures of the shadows obtained at the middle of the day-time (or noon) on days when the sun enters the sign of Aries as also the sign of Libra. 52. In Larka, Yavakati, Siddhapuri, and Rumakapuri, there is no (such) equinoctial shadow at all; and, therefore, the day-time is of 30 ghatas. 63. In other regions, the day-time happens to be longer or shorter hy 30 ghatis. On the days of the entrance of the sun) into Aries and into Libra, the day-time is everywhere of 30 ghatis (in duration). 77. Having understood the measure of the duration of the day. time and also of the shadow at (noon or) the middle of the day according to the way described in astronomy, one should learn herein the calculations regarding shadows by means of the collection of rules hereafter to be given. The rule for arriving at the time of day, on knowing the measure of the shadow of a given style at a given time in the forenoon or afternoon) in relation to a place where there is no equinoctial shadow :-- 8}. One is added to the measure of the shadow (expressed in terms of the height of an object), and the sum 80 resulting) 8. If a be the height of the object and , the length of its shadow, then the time of the day that has elapsed or has to elapse is, &coording to the rule given here, equal to 1 , where A is the angle repre. 2 (cot. A + 1) senting the altitude of the sun at the time. It may be seen that this formula gives only the approximate value of the time of the day in all cases except wben the altitude is 45', and that the approximation is very rough only in the case of large altitudes, pearing 90deg. The forma la seems to be based on the fact that for small values the angles in a right-angled triangle are approximately proportional to the opposite sides. For Private and Personal Use Only

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CHAPTER X-CALCULATIONS RELATING TO SEADOWS, 277 is doubled ; with the resulting) quantity the measure of the (whole) day-time is divided. It should be made out that this gives rise, according to the mathematical work (known as) Sarasangraba, to the portion of the day elapsed in the forenoon and also to the portion of the day remaining in the afternoon. An example in illustration thereof. 91. The length of the) shadow of a man is 3 times (bis height). Say, dear friend, what portion of the day has gone in the forenoon, or what portion of the day remains in the afternoon. The rule for arriving at the corresponding number of) ghatas, when the portion of the day (elapsed or to elapse) has been arrived at (already). 101. The (known) measure of (the duration of) the day inultiplied by the numerator and divided by the denominator of the fraction representing the (already arrived at) portion of the day (elapsed or to elapse) gives rise to the ghatis elapsed in relation to the forenoon, and to the ghatis to elapse in relation to the afternoon. An example in illustration thereof. 113. In a region without the equinoctical shadow, $ part of the day has elapsed; (or in relation to the afternoon), the remaining portion (of the day which has to elapse) is also s. What are the ghatis (corresponding to this portion)? There are, it may be taken), 30 ghatis in a day. The rule for arriving at the time taken up by a prize-fight between gymnasts. 121. The day diminished by the sum of the portion of the day elapsed and of the portion (thereof) remaining to elapse, when brought into the form of time (measured by ghatis), gives rise to the (required duration of) time. The measure of the shadow of a pillar divided by the measure of (the height of) the pillar gives rise to the measure of the shadow of a man (in torms of his own height). For Private and Personal Use Only

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278 GANITASA RASANGRAHA. An example in illustration thereof. 13]. A prize-fight between gymnasts began in the forenoon, when the shadow was equal in measure to the style. (Its) oonclusion took place in the afternoon, when the measure of the shadow was) twice (that of the style). What is the duration of the fight? An example in illustration (of the rule) in the latter half (of the stanza). 145. The shadow of a pillar, 12 hastas (in height), is 24 hastas in measure. At that time, O arithmetician, of what measure will the human shadow be? The rule for arriviug, at the period (of the day elapsed or to elapse), in places having the equinoctial shadow, wheu the measure of the shadow at any time is known : 15%. To the measure of the known shadow (of the style) the measure of the style is added ; (this sum is) diminished by the measure of the equinoctial shadow, and (the resulting difference is) doubled. The measure of the style divided by the quantity (so arrived at) gives rise to the value of the portion of the day (elapeed) in the forenoon, or (to elapse) in the afternoon, (as the case may be). An example in illustration thereof. 164-17. In the case of a style of 12 angulas, the (equinoctial) noon-shadow is 2 angulas, and the known shadow (at the time of observation) is 8 angulas. What portion of the day is gone, or what portion (yet) remains? If the portion of the day (elapsed or to elapse) happens to be }, what are the ghatis (corresponding to it), the duration of the day being 30 ghatis ? 153. Algebraically the formula given here for the measure of the time of the day is 2 (8 + a-e) where e is the length of the equinoctial shadow of the style. The formula is obviously based on the formula given in the note to the rule in stanza 8} above. For Private and Personal Use Only

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CHAPTER IX-CALCULATIONS RELATING TO SHADOWS. 279 The rule for arriving at the measure of the shadow correspond. ing to a time (of day) given in ghatis. 18. The measure of the style is divided by twice the measure of the given) portion of the day ; (from the resulting quotient) the measure of the style is subtracted, and (to it) the (equinoctial) noon-shadow is added. This gives rise to the measure of the shadow at the required time of day. An example in illustration thereof. 19. If, in the case of a style of 12 angulas, the (equinoctial) shadow is 2 angulas, what is the measure of the shodow (of the style) at a time when i0 ghatis have elapsed or have to elapse, the duration of the day-time being 30 ghatis ? The definition of the measure of a man's foot in relation to measurements carried out by means of the foot-measure as involved in the shadow. 20. One seventh of the height of a person happens to be the length of that person's foot. If this be. so, that person sball be fortunate. (Thus the measure of the shadow in terms of the footmeasure is obvious. The rule for arriving at the numerical measure of the shadow which has ascended up (a perpendicular wall). 21. (The height of) the style is multiplied by the measure of the human shadow (in terms of the man's height). The resulting) 82y 18. Algebraically - a + e, where g is the measure of the time of the day in ghutia. This formula may be seen to follow fron that given in the note to stanza 15} above. axb --C 21. Aigebraically, h= . where a is the altitude of the shadowwb casting style, h the height of the shadow on the wall, b the measure of the human shadow in terms of the man's height, and the distanee be. tween the pillar and the wall. The diagram here given eluci. dates the rule. It has to be noted here that the distance between the pillar and the wal! has to be measured along the line of the shadow which is cast in sunlight, C For Private and Personal Use Only

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Shri Mahavir Jain Aradhana Kendra 280 GANITASARASANG RAHA. product is diminished by the measure of the interval between the wall and the style. The difference (so obtained) is divided by the very measure of the human shadow (referred to above). The quotient so obtained happens to be the measure of (that portion of) the style's shadow which is on the wall. An example in illustration thereof. 22. A pillar is 20 hastas (in height); the interval between (this) pillar and the wall (on which its shadow falls) is 8 hastas. The human shadow (at the time) is twice (the man's height). What is the measure of (that portion of) the pillar-shadow which is on the wall ? The rule for arriving at the numerical value of the measurement of the interspace between a wall and a pillar, when the height of the pillar and the numerical value of (that portion of) the shadow thereof which has fallen upon the wall are (both) known. 23. The difference between the height of a pillar and that (of its shadow) cast on (a wall), multiplied by the measure of the human shadow (in terms of the man's height), gives rise to the measure of the interspace between that (pillar) and that (wall). This value of the interspace divided by the difference between the height of the pillar and that of (the portion of) the shadow thereof cast on (the wall), gives rise to the measure of the human shadow (in terms of the man's height). An example in illustration thereof. 24. A pillar is 20 hastas (in height); and the (portion of its) shadow on a wall is 16 (hastas in height). The human shadow (at the time) is twice (the human height). What may be the measure of the interspace between the pillar and the wall? 23. This rule and the one in stanza 26 following give the converse cases of the rule in stanza 21 above, For Private and Personal Use Only

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CHAPTER IX.-CALCULATIONS RELATING TO SHADOWS. An example in illustration of the (rule in the) latter half (of the stanza). 25. A pillar is 20 hastas (in height) and the (portion of its) shadow on a wall is 16 (hastas in height). The measure of the interspace between the wall and the pillar is 8 (hastas). What is the measure of the human shadow (in terms of the man's height) ? 281 The rule for arriving at the numerical value of the height of a pillar, when the numerical measure of the (portion of its) shadow cast on (a wall and the measure of the interspace between (that) pillar and (that) wall, and also the human shadow (in terms of the human height) are known. 26. The measure of the (pillar-shadow) cast on (the wall) is multiplied by the measure of the human shadow (in terms of the human height); and to this product the measure of the interspace between the pillar and the wall is added The quotient obtained by dividing (the sum so resulting) by the measure of the human shadow (in terms of the human height) is made out by the wise to be the measure (of the height) relating to the pillar. An example in illustration thereof. 27. The measure of (the height of the portion of) the pillarshadow cast on the wall is 16 (hastas). The value of the human shadow (at the time) is only twice (the human height). The measure of the interspace between the wall and the pillar being 8 (hastas), what is the height of the pillar? The rule for separating the measure (of the height) of the style and the measure of (the length of) the shadow of the style from (their given) combined sum : 28. The combined sum of the measure of the style and the measure of the shadow (thereof), when divided by the measure of the human shadow (in terms of the human height) as increased by one, gives rise to the measure of the height of the style. The measure of the shadow of the style is of course the (given) combined sum diminished by this (measure of the style). For Private and Personal Use Only 26. Vide note under stanza 23 above. 28 and 80. The rules here given are based on the rule stated in the latter half of the stanza 12 above. 36

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282 GANITASARASANGRABA. An example in illustration thereof. 29. The combined sum of the (height) measure of the style and the (length) measure of its shadow is 50. What may be the height of the style, the human shadow being (at the time) 4 times (the human height)? . The rule for separating the length) measure of the shadow of the style and the measure of the human shadow (in terms of the human height) from (their) combined sum : 30. The combined sum of the measures of the shadows of a style and of a man is divided by the (known height) measure of the style as increased by one. The quotient (so obtained) is the measure of the human shadow (in terms of the human height). The combined sum (above-mentioned) as diminished by this (measure of the human shadow) gives rise to the length-moasure of) the shadow of the style. An example in illustration thereof. 31. The height of a style is 10. The sum of the human shadow (in terms of the human height) and (the length of) the shadow of the style is 55. How much is the measure of the human shadow (in terms of the human height and how much is the length of the shadow of the style) ? The rule for arriving at the measure of the inclination of a pillar (or vertical style) : 32-33. The product of the square of the human shadow and the square (of the height of the style is to be subtracted from the 32-33. Let AB represent the position of a planting pillar, and AC its shadow : and let AD be the same pillar in the vertical D position and AE its shadow. Then All AE AD G is equal to the ratio of the shadow of a man to his height at the time, and let this ratio ber. BG, the perpendicular from Bon AD, represents the amount of slanting of the pillar, AB. It can be easily AD shown that N AB? - BG2 AC - BG A = From this it can be seen that - AC - NAC (AC-AB?) (2 + 1). ? +1 same formula. C BG A r A F E s The rulo here gives this For Private and Personal Use Only

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CHAPTER IX-CALOULATIONS RELATING TO SHADOWS. 283 square of the (given) shadow. This (remainder) is to be multiplied by the sum of the square of the human shadow and one. (The quantity so arrived at) is to be subtracted from the square of the (given) shadow. The square root of this (resulting remainder) is to be subtracted from the (given) measure of the shadow; and, when (the quantity thus obtained is) divided by (the sum of) one and the square of the human shadow, there results exactly the measure of the inclination of the pillar. An example in illustration thereof. 34. The human shadow (at the time) is twice (the human height). The shadow of a pillar, 13 hastas in height, is 29 (hastas). What is the measure of the slanting of the pillar here? (General Examples). 35-37. A certain prince, staying in the interior of a palace, was, (at a certain moment) in the course of a forenoon, desirous of knowing the time elapsed in the course of the day, as also the measure of the human shadow (in terms of the human height). Then, the light of the sun coming through a window at a height of 32 hastas in the middle of the eastern wall fell at a place on the western wall at the height of 29 hastas. The distance between those two walls is 24 hastas. O mathematician, if you have taken pains (to acquaint yourself) with shadow-problems, calculate and give out the measure of the time elapsed then, on that day, and also the measure of the human shadow (at that time in terms of the human height). 381-39. At the time when, in the course of a forenoon, the human shadow is twice the human height, what, in relation to a (vertical excavation of) square (section) measuring 10 hastas in each dimension, will be the height of the shadow on the western wall caused by the eastern wall (thereof)? O mathematician, give out, if you know, how you may arrive at the value of the shadow that has ascended up (a perpendicular wall). 35-37. This example bears on the rules given in stanzas 8 and 23 above. 38-39. This example has to be worked out according to the rule given in stanza 21 above. For Private and Personal Use Only

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Shri Mahavir Jain Aradhana Kendra 284 www.kobatirth.org GANITASARASANGRAHA. The rule for arriving at the shadow of a style due to (the light of) a lamp: 40. The height of the lamp as diminished by the height of the style is divided by the height of the style. If, by means of the quotient so obtained, the (horizontal) distance between the lamp and the style is divided, the measure of the shadow of the style is arrived at. An example in illustration thereof. 413-42. The (horizontal) distance between a style and a lamp is in fact 96 angulas. The height of the flame of the lamp (above the floor) is 60 (angulas). O you who have gone to the other shore of the ocean of calculation, tell me quickly the measure of the shadow due to the flame of the lamp, in relation to a style which is 12 angulas (in height). Acharya Shri Kailassagarsuri Gyanmandir The rule for arriving at the (horizontal) distance between the lamp and the style : 43. The height of the lamp (above the floor) is diminished by the height of the style. The (resulting) quantity is divided by the height of the style. The measure of the shadow of the style, on being multiplied by the quotient so obtained, gives rise to the (horizontal) measure of the intervening distance between the style and the lamp. An example in illustration thereof. 44. The shadow of the style is 8 angulas (in length). The height of the flame of the lamp (above the floor) is 60 (angulas). 40. Algebraically stated the rule. is b-a 8=0- ; where s is the length a of the shadow of the style whose height is represented by a, b is the height of the lamp above the ground, and c the horizontal distance between the lamp and the style. The formula may be seen to be correct by means of the diagram here given. 43. Using the same symbols, c = 8 x b b-a a 44. The given measure of the height of the style is 12 angulas, vide stanzas 46-47 below. For Private and Personal Use Only

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CHAPTER 1X.-CALCULATIONS RELATING TO SHADOWS. 286 O you who have gone to the other shore of the ocean of calculation, say what (the measure of) the intervening horizontal distance 18 between the style and the lamp. The rule for arriving at the numerical measure of the height of the lamp (above the floor) : 45. The measure of the (horizontal) distance between the lamp and the style is divided by the shadow of the style. (Then) one is added (to the resulting quotient). The quantity so obtained, on being multiplied by the measure of the height of the style, gives rise to the measure of the height of the lamp (above the floor). An example in illustration thereof. 46-47. The (length of the) sbadow of the style is exactly twice (its height). The measure of the intervening (horizontal) distance between the style and the lamp is 200 angulas. What is the measure of the height of the lamp (above the floor) in this case ? Here and also in the foregoing example, the measure of the height of the style has to be understood as consisting of 12 angulas, and then the way in which the meaning of the rule works out is to be learnt well. The rule for arriving at the numerical measure of the height of a tree, when the measure (of the length) of the shadow of a man in terms of (his) foot and the measure of the length of the shadow of the tree in terms of the measure of that same foot are known; as also for arriving at the numerical measure (of the length) of the shadow of the tree in terms of that same foot-measure, when the numerical measure of the height of the tree and the numerical measure (of the length) of the shadow of a man in terms of (his) foot are known : 48. T'he measure (of the length) of the shadow of the tree chosen hy a person is divided by (the foot-measure of the length 45. Similarly, b =(.+ 1) a. 48. This deals with a converse case of the rule given in the latter half of stanza 12' above. The relation between the height of a man and his foot-measure is atilized in the statement of the rule as given here. For Private and Personal Use Only

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286 GANITABABABANGRAHA. of) his own shadow, and then it is multiplied by seven : this gives rise to the height of the tree. This (height of the tree) divided by seven and multiplied by the foot-measure of his shadow surely gives rise to the moasure (of the length) of the shadow of the tree exactly. An example in illustration thereof. 49. The foot-measur, (of the length) of one's own shadow is 4. The (length of the) shadow of a tree is 100 in terms of the (same) foot-measure. Say what the height of that tree is in terms of the measure of one's own foot. An example for arriving at the numerical measure of the shadow of a tree. 50. The measure (of the length) of one's own shadow (at the time) is 4 times the measure of (one's own) foot. The height of a tree is 175 (in terms of such a foot-measure). What is the measure of the shadow of the treo then ? 51-523. After going over (a distance of) 8 yojanas (to the east) of a city, there is a hill of 10 yojanas in height. In the city also there is a hill of 10 yojanas in height. After going over (a distance of) 80 3ojanas (from the eastern hill to the west, there is another hill. Lights on the top of this (last mentioned hill) are seen at nights by the inbabitants of the city. The shadow of the hill lying at the centre of the city touches the base of the eastern hill. Give out quickly, O mathematician, what the height of this (western) hill is. Thus ends the eighth subject of treatment, known as Calculations relating to shadows, in Sarasangraha, which is a work on arithmetic by Mahaviracarya. SO ENDS THIS SARA SANGRAHA. 51-52. This example is intended to illustrate the rule given in stanga 45 above. For Private and Personal Use Only

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287 APPENDIX I. SANSKRIT WORDS DENO SING NUMBERS WITH THEIR ORDINARY AND NUMERICAL SIGNIFICATIONS. aDa akSi ... 'The eye ... 2 Men have two eyes. agni ... Fire ... ... 3 The number of sacrificial tires is three, _viz., gArhapatya, AhavanIya, and dakSiNa. aGka ... Number 9 There are only nine numerical figures excluding the zero. .. An auxiliary divi 6 There are six auxiliary departments of sion or depart study in relation to the Vedas, viz., ment of science. zikSA, kalpa, vyAkaraNa, nirukta, chandasa, jyautiSa. acala .... A mountain ... 7 Seven principal monntains called Kuld calas are recognized in the geography of the Puranas, viz.,mahendra, malaya sahya, zaktimat, RkSa, vindhya, pAriyAtra. Adi ... A mountain ... 7 Vide acala. ananta ... The sky ... 0 The sky is considered to be void. anala ... Fire ... ... 3 Vide Agna. anIka ... An army ... 8 'There are eight kinds of army mentioned in Sanskrit, viz., patti, senAmukha, gulma gaNa, vAhinI, pRtanA, cam, anIkinI antarikSa ... The sky ___... 0 Vide ananta. abdhi .. The ocean 4 It is held that there are four oceans, viz., eastern, southern, western and northern. ambaka ... The eye ... ... 2 Vide akSi. ambara ... The sky ... 0 Vide ananta. ambIdha ... The ocean Vide abdhi . ambhodhi ... The ocean ... 4 Vide abdhi . For Private and Personal Use Only

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288 GANITASARASANGRA A A. inda ... In azva ... A horse ... ... 7 The horses of the sun's chariot are sup posed to be seven. Azvina ... Consisting of horse. 7 Vide azva. AkAza ... The sky ... ... o Vide ananta. ina ... The sun ... ... 12 The number of suns is reckoned to be 1: corresponding to the 12 months of { year, viz., dhAta, mitra, aryaman, rudra. varuNa, sUrya, bhaga, vivasvat, pUSan savitR, tvaSTha and viSNu. They are called the twelve Adityas. The moon ... 1 We have only one moon. FK ... The god Indra ... 14 Fourteen Indras are usually reckoned at the rate of one Indra for each of the fourteen manvantaras. indraya ... An organ of 5 There are five organs of sense, viz., nose. Bense. tongue, eye, skin and ear. ibha ... An elephant ... 8 Eight elephants are said to guard the eight cardinal points of the world. They are airAvata, puNDarIka, vAmana, kumuda, aJjana, puSpadanta, sArvabhauma, and supratIka. ... An arrow ... 5 The arrows of Manmatha or the Indian Cupid are declared to be five, viz. aravinda, azoka, cUta, navamallikA, and nIlotpala. ikSaNa ... The eye ... . Vide akSi. udAdhe .. The ocean ... 4 Vide abdhi . upendra . God Vinu 9 There are snid to be nine Vinnos corre. sponding to the nine past incarnations of Visnu. Rtu ... A season There are, according to Sanskrit liter. atnre, six Beasons in a year, viz. vasanta, grISma, varSa, zarad, hemanta, and zizira. kara ... The hand ... 2 Human beings have two hands. TUTT. That which has to 5 There are 5 vratas or austerities to be be done : an aot observed according to the Jaina reliof devotion or gion, vis., ahiMsA, santa, asteya, austerity. brahmacaryA, and aparigraha. For Private and Personal Use Only

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APPENDIX I. 289 karin ... An elephant ... karman ... Action : the effect of aotion as its 8 8 karma. kalAdhara ... The moon ... kaSAya ... Attachment to wordly objects. 1 4 Vide ibha. According to Jainas there are eight kinds of karma, viz. jJAnAvaraNIya, darzanAvaraNIya, mohanIya, antarAya, vedanIya, nAmika, gotrika and Ayaka, Vide indu. According to the Jaina religion there are four causes for such attachment, viz., krodha, mAna,mAyA, lobha. This War-god is supposed to have six faces. f. SaNmukha. kumAravadana The faces of Ku- 6 mara or the Hindu war-god. kezava ... A name of Vispu. 9 kSapAkara ... The moon ... 1 kha ... sky ... .... 0 khara gagana ... Sky ... ... gaja ... Elephant ... 8 gati Passage : pnsgage 4 into re.birth. Vide upendra. Vide indu. Vide ananta. giri ... Monntain ... Quality ... ... 7 3 guNa graha ... A planet ... 9 Vide ananta. Vide ibha. According to the Jaina religion souls may have four kinds of embodiment, viz., as deva, tiryaka, manuSya, naraka. Vide acala. Primordial matter is said to have three 'qualities', viz., sattva, rajas, tamas. Nine planets are recognised in Hindu astronomy, viz., Mars, Mercury, Jupi. ter, Venus, Saturn, Rahu, Kstu the Sun and the Moon. Vide akSi. Vode indu. Vide indu. Vide ananta. Vide abdhi . vide abdhi . ... ... cakSusa ... The eye candra ... The moon candramasa ... The moon jaladharapatha Sky jalAdha ... Ocean ... jalanidhi.. Ocean ... 2 1 1 0 4 4 ... ... 37 For Private and Personal Use Only

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290 GANITASARASANGRAHA. Esta ... Name of a Jaina 24 According to the Jainas there are saint. tirthankaras or saints. 54257 ... Fire ... ... 3 Vide 3fa. ATT ... Elementary prin. 7 'The Jainas recognise seven such princiciples. ples, viz., sia, astia, 119, , #at, fast, ATT 77 ... Body ... ... 8 Siva is considered to have his bods made ap of eight things, viz., gfrai, 37, -- AAE, aty, 311977, at, 4*, 4 . Thi ... Evidence ... 6 The six kinds of evidence are RH, 347917, 3447A, 7C, 342f9fat, and Vide 340. Vide fort. ar 757. Vinpu ... ... 9 aret ... Tirthankar: or 24 Jina. Cratae ... An elephant ... 8 GITA ... Worldly action ... 8 Name of a mani. 9 festation of Par. vati or Durga. Quarter or a 8 cardinal point of the universe. Do. 10 Vide $4. Vide DHT. Nine separate manifestations of Durga are recognized. There are eight cardinal points of the univerge, Sky ... ... The eye ... ... The eye ... ... Elementary sub. 2 2 Ten directions are recognized, namely the eight cardinal points of the universe, the upward and the downward directions. Vide 377. vide 37. Vide 377. Asocording to the Jainas there are six varieties of elementary substance, vir., sia, , 375, 16, , TTTET. 6 stanoe. For Private and Personal Use Only

Page #488
APPENDIX I. 291 rate Pero aty ... An elephant ... 8 ... An elephant ... 8 ... A Parapio insular 7 Vide H. Vide T. There are seven such divisions, viz., ta, H, AST, 7, ma, #, 0166T. division of the ... VODOU These are said to be seven, viz., chyle, blood, flesh, fat, bone, marrow, semen. ... Mount terrestrial world. Constituent prin- 7 ciples of the body. Name of a kind 18 of metre. Mountain Name of a dynasty 9 of kings. .. Sky ... ... ... Method of com- 2 prehending things from particular stand-points. ... The eye ... ... An elephant ... 8 ... Treasure ... 9 Each line of a stanza in this metre contains 18 syllables. Vide 3790. Vine Nanda kings are said to have reigned in Magadha. Vide 379781. A coording to Jainas there are two Nayas : Cory and preferat TETE TO 797 Ar AN ... E TERS ... The eye .. Category of things. ... The serpent 3 9 Vide sifa. Vide H. Nine famous treasures are said to belony to Kabera, the god of wenlth, viz., 77, HETTA, Tel, HA, 77939, Te , fio, e Vide 3fe. The Jainas recognize nine categories of things. Sometimes eight and sometimes seven principal serpents are reckoned in Hindu mythology. Vide 37102. Pide 3 fot. Pide affra. TET ... 7 rife ... Ocean ... Terate... Ocean ... 979 ... Fire ... ... 4 ... 4 ... 8 For Private and Personal Use Only

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Shri Mahavir Jain Aradhana Kendra 292 pura bANa puSkarin ... Elephant prAleyAMzu ... The moon bandha w bhaya bhAva bhAskara bhuvana bhUta bhUdhra 13 mada mahIdhra mAtRkA muni mRgAGka mRDa ... yati rajanIkara ... ... *** ... City *** Bondage Arrow A constellation Fear ... Elements The sun The world Element Mountair Passion Mountain A goddess Sage Sage The moon GANITASARABANGBAHA. ... ... ... ... ... ... The moon ... A name of Siva or Rudra. *** ... www.kobatirth.org ... 3 8 1 4 5 27 7 5 12 3 6 7 8 7 7 2 1 11 7 1 Three cities representing three Asuras are said in the Puranas to have caused great havoo to the gods, and Siva is said to have destroyed them. cf. tripurAntaka. Vide ibha. Vide indu. The Jainas recognize four kinds of spiritual bondage, viz., prakRti, sthiti, anubhAva and pradeza. Vide iSu. The Hindu astronomers count 27 chief stelllar constellations or lunar mansions around the ecliptic. Acharya Shri Kailassagarsuri Gyanmandir Five elements are recognized, viz., pRthivI, a, tejas, vAyu, AkAza. Vide ina. 63 The number of worlds ordinarily counted are three, viz., the upper, the lower and the middle worlds. Vide bhAva. Vide acala. Vide acala. Generally seven of these goddesses are enumerated. Seven chief sages are usually mentioned; they are, kazyapa, atri, vizvAmitra, gautama, jamadagni, vasiSTha. bharadvAja, Vide indu. It is held that there are eleven Rudras. Vide mAne. Vide indu. For Private and Personal Use Only

Page #490
APPENDIX I. :93 ... Excellent thing... 3 ... A precious gem... 9 ... Opening ... 9 ___ ... Taste ... ... 6 There are three excellent things for Jaina.s, via., samyagdazarna, samyagjJAna, samyakcAritra. Nine gems are usually recokned, viz., vajra, vaiDUrya, gomeda, puSparAga, padmarAga, marataka, nAla, muktA, pravAla. There are nine chief openings in the human body. The six principal tastes are madhura, amla , lavaNa, kaTuka, tikta, kaSAya. Vide maDa. Everything has its one only shape. The nine powers to be attained are anantadarzana, anantajJAna, kSAyakasamya, ktvakSAyakacAritra, anantadAna, anantalAbha, anantabhoga, anantopabhoga and anantavIrya. Vide labdha. Name of a deity 11 Form or shape ... 1 Attainment; 9 attainment of the nine powers. .. 9 labdhi ... Attainment lekhya loka ... World ... locana ... The eye ... 3 2 Vide bhavana. Vide akSi. varNa A class of Vedic 8 These deities are considered to be eight in deities. number. vahi ... Fire ... ... 3 Vide agni . vAraNa Elephant Vode ibha. vArdha Ocean ... 4 Vide abdhi . vidhu ... The moon ... 1 Vide indu. viSadhi ... Ocean ... ... Vide abdhi . viSanidhi ... Ocean ... ... 4 Vide abdhi . viSaya ... Object of sense... 6 The objects cognizable by the five organs of sense are five, viz., gandha, rasa, rUpa, sparza, zabda. viyat ... sky ... ... o Vide ananta. For Private and Personal Use Only

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Shri Mahavir Jain Aradhana Kendra 294 vizva vaizvAnara vyasana viSNupAda... Sky veda vyAma vrata ... zazAGka zazin *** ... ... *** ... zaila zveta salilAkara sAgara sAyaka sindhura sUrya soma stambrema svara ... ... zaGkara zara zazadhara The moon zazalAJchana The moon The moon The moon zastra Arrow zikhin Fire zilImukhapada The legs of bee. Mountain *** ... !::: ... ... A group of Vedic 13. deities. The Vedas addiction. Fire 3 An unwholesome 7 ... Sky Act of devotion or austerity. Name of Rudra... 11 Arrow 5 1 1 1 ... *** ...... GANITA8ARABAAGRAHA. Ocean Ocean Arrow Elephant The sun ... The moon ... www *** ... *** www.kobatirth.org ... ... 0 4 0.0 204 0 5 1 5 3 6 1 4 4 5 8 4 Elephant 8 A note of the 7 musical soale. 12 This group of deities is said to consist of 13 members. Vid: ananta. There are four Vedas, Rk, yajus, sAma, atharva. Vide agni. Seven such addictions are prohibited in the case of kings. Vide ananta. vide karaNIya. Acharya Shri Kailassagarsuri Gyanmandir Vide mUDa. Vide iSu. Vide indu. Vide indu. Vide indu. Vide indu. Vide iSu. Vide agni. The legs of a bee are held to be six. Vide acala. Vide abdhi. Vide obdha. Vide iSu. Vide ibha. Vide ina. Vide indu. Vide ibha. Seven notes are recognized in the Hindu musical scale, vis., sa, ri, ga, ma, pa, dha, ni. For Private and Personal Use Only

Page #492
Shri Mahavir Jain Aradhana Kendra haya hara haranetra Siva's eyes hutavaha hutAzana himakara himagu himAMzu ... ... ... ::::: Horse 7 Name of Rudra... 11 3 www Fire Fire The moon The moon The moon 446 ... *** 466 www.kobatirth.org ... APPENDIX I. 3 8 1 1 1 Vide agni, Vide agni. Vide indu. Vide indu. Vide indu. Acharya Shri Kailassagarsuri Gyanmandir Vide azva. Vido mRDa. Siva is said to have one extra eye in the middle of the forehead making up three in all. For Private and Personal Use Only 295

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Shri Mahavir Jain Aradhana Kendra Abadha Adhaka Adhvan Adidhana Adimisradhana Agaru Amla-vetasa Amoghavarsa Amsa SANSKRIT WORDS USED IN THE TRANSLATION AND THEIR EXPLANATION. A meamula Amsavarga... Anga-Sastra Angula *** Antaravalambaka... www.kobatirth.org ... 296 APPENDIX II. Acharya Shri Kailassagarsuri Gyanmandir Segment of a straight line forming the base of a triangle or a quadrilateral. A measure of grain. Vide Table 3, Appendix IV. The vertical space required for presenting the long and the short syllables of all the possible varieties of metre with any given number of syllables, the space required for the symbol of a short or a long syllable being one angula and the intervening space between each variety being also an angula. See note to VI-333 to 336. Each term of a series in arithmetical progression is conceived to consist of the sum of the first term and a multiple of the common difference. The sum of all the first terms is called the Adidhana. See note to II-63 and 64. The sum of a series in arithmetical progression combined with the first term thereof. See note to II-80 to 82. A kind of fragrant wood; Amyris agallocha. A kind of sorrel; Rumer vesicarius. Name of a king; lit: one who showers down truly. useful rain. A measure of weight in relation to metals. See Table 6, Appendix IV. Square root of a fractional part. See note to IV-3. Square of a fractional part. See note to IV-3. An auxiliary science. A measure of length; finger measure. See I-25 to 29 also Table 1, Appendix IV. Inner perpendicular; the measure of a string suspended from the point of intersection of two strings stretched from the top of two pillars to a point in the line passing through the bottom of both the pillars, For Private and Personal Use Only

Page #494
APPENDIX II. 297 Antyadhana Anu Ariptanemi Arbuia Arjuna Asita Asoka Aundra - Aundraphala Avali Ayana Bija do. ... ... ... ... Bhaga ... The last term of a series in arithmetical or geometrical progression. Atom or particle. Sed stanzas 25 to 27, Chapter I and Table 1, Appendix IV. Name of a Jaina saint ; one of the 24 Tirthankaras. Name of the eleventh place in notation. Name of a tree; Terminalia Arjuna, W. & A. Name of a tree ; Grislea Tomentosa... Name of a tree ; Jonesia Asoka Roxb. ... A kind of approximate measure of the cubical con tents of an exoavation or of a solid. See note to VIII-2. This kind of approximate measure is called Aattra by Drahmagupta. ... A measure of time. Vide Table 2, Appendix IV. ... Do. ... Literally seed; here it is used to donote a set of two positive integers with the aid of the product and the squares whereof, as forming the measure of the sides, a right angled triangle may be cons tructed. Vide note to VII--204. ... A measure of baser metals. Vide Table 6, appendix, IV. A simple fraction. A variety of miscellaneous problems on fractions, See note to IV-3. ... A complex fraction. ... A variety of miscellaneous problems on fractions. See note to IV-3. Division. Fractions consisting of two or more of the varieties of Bhaga, Prabhd ga, Bhagabhaga, Bhaganubandha, and Bhagapavaha fractions. See note to III-138. Fractions in association. Vide note to III--113. Dissociated fractions. See note to III-126. A variety of miscellaneous problems on fractions. See note to IV-3. ... The middle one of the three places forming the onbe root group ; that which has to be divided. See note to II-53 and 54. ... A meagare of baser metals. Vide Table 6, Appendir IV. ... A variety of miscellaneous problems on fractions. See note to I-3. 38 Bhagabhaga Bhagabhya sa Bha gahara... Bhaga ma tr Bhaganubandha Bhaga pavdha Bha ga sa varga Bhd jya Bhara Bhinna driya For Private and Personal Use Only

Page #495
298 GANITASARASANGRAHA. Bhinnakuttikara ... Cakrikabhanjana ... Campaka ... Chanda: ... Citi .. Citra-kuttikara Citra-huttikaramisra Danda Dasa Dasa-koti Dasa-laksa *Dasasahasra Dharana ... Proportionate distributiou involving fractional quantities. See footnote in page 125. Whe destroyer of the cycle of recurring rebirths; also the name of a king of the Rastrakuta dynasty. Name of a tree bearing a yellow fragrant flower Michelia Chumpaka. A syllabic metre. Summation of series. Curious and interesting problems involving propor. tionate division. Mixed problems of a curious and interesting nature involving the application of the operation of proportionate division, A measure of distance. Vide Table 1 of Appendix IV. Terth place. Ten crore. ... "Ten lakhs or one million. Ten thousann. A weight measure of gold or silver; Vide Tables 4 and 5 of Appendix IV. .. A weight measure of baser metals. Vide Table 6 of Appendix IV. Also used as the name of a coin, ... A weight measure of baser metals. Vide table 6 of Appendix IV. A measure of capacity in relation to grain. Vide Table 3 of Appendix IV. Name of a tree. A variety of miscellaneous problems un fractions. Unit place. A weight measure of gold. Vide Table 4, Appendix Dinara ... Draksuna... Drona ... Dunduka ... Dviragrasesa mula Eka Gandaka ... : IV. Ghana : Ghanamula Ghati Gunakara ... Gunadhana ... Cubing; the first figure on the right, among the three digits forming a group of figures into which a numerical quantity whose cube root is to be found out has to be divided. See note to II--53, 54. Cube root. ... A measure of time. Vide table 2 of Appendix IV. Multiplication. The produot of the common ratio taken as many times as the uumber of terms in a geometri. cally progressive series multiplied by the first term, See note to 11-93, For Private and Personal Use Only

Page #496
Shri Mahavir Jain Aradhana Kendra Gunja Hasta Hintala Iccha Indranila Jambu Janya Jinas Jinapati Jina-santi Kadamba Kala Jina-Vardhamana... :: Kala savarna Karmas Karsa Karmantika Karsa pana Ketaki Khari Kharva Kisku Koti Kotika ... Krti Krepapada Krosa Krana garu. ... :: 2 APPENDIX II. A weight measure of gold or silver. and 5 of Appendix IV. A measure of length. Vide Table 1 of Appendix IV. Name of a tree; Phaenix or Elate Paludosa. That quantity in a problem on Rule-of-Three in relation to which something is required to be found out according to the given rate. Sapphire. 299 Name of a tree; Eugenia Jambalona. Trilateral and quadrilateral figures that may be derived out of certain given data called bajas. The great teachers of the Jaina religion; the Jaina Tirthankaras, A weight measure of baser, metals. Appendix IV. Vide Tables 4 The Chief of the Jinas, Vardhamana. Name of a Jaina saint; a Tirthankara. Vardhamana, the great propagator of the Jaina religion and the last of the Tirthankaras. Name of a tree; Nauclea Cadamba. Vide Table 6, Fraction. See footnote on page 38. Consequence of acts done in previous births. Ac. cording to Jainas the Karmas are of eight kinds. See under in Appendix I. A kind of approximate measure of the cubical contents of an excavation or of a solid, See note to Chapter VIII-9. A weight measure of gold or silver. Vide Tables 4 and 5, Appendix IV. A Karsa. For Private and Personal Use Only Name of a tree; Pandanus Odoratissimus. A measure of capacity in relation to grain. The 13th place in notation. A measure of length in relation to the sawing of wood. Crore, the 8th place in notation. A numerical measure of cloths, jewels and canes. Vide Table 7, Appendix IV. An easure of length. Vide Table 1 of Appendix IV. A kind of fragrant wood; a black variety of Agallo chum. Squaring. Half of the difference between twice the first term and the common difference in a series in arithmeti. oal progression.

Page #497
300 GANITASARASANGRAHA. Ksitya Krobha Kg'dai Kudaha or Kudaba The 21st place in notation. The 23rd place in notation. The 17th place in notation. A measure of capacity in relation to gruin. Vide Table 3 of Appendix IV. Do. do. Kumbha ... Kunkuma ... Kuravaka ... Kutaja Kutfikara ... Labha Laksa Lanka Lava Madhuka... Madhyadhana Mahakharva Mahaksitya Mahaksobha Maha keoni Mahapadma Maha sankha Mahavira ... Mani The pollen and filaments of the flowers of saffron, Croeus sativus. Name of a tree; the Amaranth or the Barleria. Name of a tree ; Wrightia Antidysenterica, Proportionate division. See VI -791. Quntient or share. Lakh, the 6th place in notation. The place where the meridian pagsing through Ujjain meets the equator. A measure of time. Vide Table 2 of Appendix IV. Name of a tree, Bassia Latifolia. The middle term of a series in arithmetical progres sion. See note to IL-63. The 14th place in notation. The 22nd place in notation. The 24th place in notation. The 18th place in notation. The 16th place in notation. The 20th place in notatiou, A name of Vardhamana. ... A measure of capacity in relation to grain. Vide Table 3 of Appendix IV. A kind of drum; for a longitudinal section, see note to VI). -32. Section; the line along which a piece of wood is cut by a saw A weight measure of silver. See Tables 5, Appendix IV. Name of a fabulous mountain forming the centre of Jambda pipa, all planets revolving round it. Mixed sum. See note to II--80 to 82. A kind of drum ; for a longitudinal section, see note to VIII-32, A measure of time. Vide Table 2, Appendix IV. ... The topside of a quadrilateral. ... Square root; a variety of miscellaneous problems on fractions. Vide note to IV-3. Involving square root; a variety of miscellaneous problems on fractions. Vide note to 1V-3 Mardala ... Marga Masa Morr Mieradhana Mrdanga ... .. Muhurta Mukha Mula Mulamiera ... For Private and Personal Use Only

Page #498
Shri Mahavir Jain Aradhana Kendra Muraja Nandya varta Narapala Nilotpala Niruddha Niska Nyarbuda Pada Padma Padmaraga Paisa cika Paksa Paal Pana Panava Paramam Parikarman Parsva Patali Pattika Phala Plaksa Prabhaga Prakirnaka... Praksepaka Praksepaka-karana Pramana Prapuranika Prastha Pratyutpanna APPENDIX ir. A kind of drum; same as Mrdanga Name of a palace built in a particular form. See note to VI-332. 301 King; probably name of a king. Blue water-lily. Least common multiple. A golden coin. The 12th place in notation. A measure of length. Vide Table 1, Appendix IV. The 15th place in notation. A kind of gem or precious stone. Relating to the devil; hence very difficult or com. plex. A measure of time. Vide Table 2 of Appendix IV. A weight measure of gold, silver and other metals. Vide Tables 4, 5, 6 of Appendix IV. A weight measure of gold; vide table 4 of Appendix IV; also a golden coin. A kind of dram; for longitudinal section see note to VII-32. Smallest particle. Vide Table 1, Appendix IV. Arithmetical operation. A Jaina saint; one of the Tirthankaras. A tree with sweet-scented blossoms; Bignonia Suaveolens. A measure of saw-work. IV; also note to VIII-63 to 674. A given quantity corresponding to what has to be found out in a problem on the Rule-of-Three. See note to V-2. Name of a tree; the waved-leaf fig-tree, Ficus In fectoria or Religiosa. Fraction of a fraction. Miscellaneous problems. Proportionate distribution. For Private and Personal Use Only Vide Table 10, Appendix An operation of proportionate distribution. A measure of length. Vide Table 1 of Appendix IV. The given quantity corresponding to Iecha, in a problem on Rule-of-Three. See note to V-2. Literally, that which completes or fills; here, baser metals mixed with gold; dross. A measure of capacity in relation to grain. Vide Tables 3 and 6, Appendix IV. Multiplication.

Page #499
302 GANITASARASANGRAHA. Pravartika Punnaga Purana ... Pusyarigs Ratharenu Rumaka puri Rtu ... Sahasra ... Saka Sakola kuttikara Sala Sallaki ... Samaya Sankalita .. Sankha Sankramana Sankranti ... A measure of capacity in relation to grain. ... Name of a tree ; Rottleria Tinctoria. A weight measure of silver. Vide Table 5, Appendix TV; probably also a coin. A kind of gem or precious stone. ... A particle. Vide Table 1, Appendix IV. A place 90deg to the west of Lanka. Season, here used as a measure of time. Vide Table 2, Appendix IV. Thousand. The teak tree. Proprrtionate distribution, in which fractions are not involved. The sal tree; Shorea Robusta or Valeria Robusta. Name of a tree ; Boswellia Thurifera. A mesure of time. Vide Tablo 2, Appendix IV. Summation of series. The 19th place in notation. An operation involving the halves of the sum and the difference of any two quantities. See note to VI-2. The passage of the sun from one zodiacal sign to another. Name of a Jaina saint. See Jina-Santi, Name of a tree; Pinus Longifolia. A kind of bird. Literally, a brief exposition of the essentials or principles of a sabjeot ; here, the name of this work on arithmetic. Name of a tree; same as the sal tree. The sum of a series in arithmetical progression. See note to II-63 and 64. A hundred. A hundred crores. ... A weight measure of baser metals. Vide Table 6, Appendix IV. The terms that remain in a series after a portion of it from the beginning is taken away. See note or page 34. A variety of miscellaneous problems on fractions. See note to IV-3. ... A variety of miscellaneous problems on fractions. See note to IV-3. ... The antipodes of Lanka. Banti Sarala Sarasa Sara sangraha Saria Sarradhana Sata Satakoti Satera Seed Seramula ... .. Siddhapuri .. For Private and Personal Use Only

Page #500
APPENDIX IL. 803 Siddhas. ... ... Sodasika ... Sodhya Bravaka .... Sriparni ... Stoka Suksmaphala Suvarna-kutfikara... Swvrata Svarna Syadvada Tamala Tilaka Tirtha Tirthankaras Trasaranu ... Tripraina ... Those who have attained to the highest position in regard to spiritual knowledge. A moasare of capacity in relation to grain. Vide Table 3 of Appendix IV. ... One of the three figures of a cubic root group. See note to II-53 and 54. A lay follower of Jainism. Name of a tree; Premna Spinosa. A measure of time. Vide Table 2, Appendix IV. Acou rate tneasure of the area or of the cubical contents. Proportionate distribution as applied to problems relating to gold. Name of a Jaina saint ; one of the Tirtharkaras. A gold coin, The argument of may be. See footnote on page 2 Name of a tree ; Xanthoc hymus Picturirs.. Name of a tree with beautiful flowers. Ford. See note to VI---1. The 24 famous Jaina saints and teachers. See note to VI-1. A particle. Vide Table 1, Appendix IV. Name of a chapter in Sanskrit astronomical works. See footnote on page 2. A weight measure of baser metals... A di-deficient quadrilateral. See note to VII-37. A measure of time. Vide Table 2, Appendix IV. The water-lily flower The sum of all the multiples of the common differ. enoe found in a series in arithmetical progression, See note to il -63 and 64. ... A mixed sum obtained by adding together the common difference of a series in arithmetical progression and the son thereof. See note to II -80 to 82. A measure of capacity in relation to grain. ... A weapon of Indra ; for longitudinal section see note to Chapter VII-32. Cross reduction in multiplication of fractions. See note to III-2. Name of a tree ; Mimusops Elengi. Proportionate distribution based on a creeper like ... } chain of figures. Ses note to VI-1151 Tula Ubhayanisedha Ucchvasa Utpala ... Uttaradhana Uttaramisradhana Vaha Vajra Vajra pavartana ... Vakula Vallika Vallikakutsikara ... For Private and Personal Use Only

Page #501
304 GANITASA BASANGRAHA, Vardhamana ... Vargamula Varna Vicitra-kutsikara ... Vidyadhara-nagara Visamakulfikara . Pisamasaikramana ... Name of the chief of the Jinas; vide Jina-Vardha mana, ... quare root. Literally colour; here denotes the proportion of pure gold in any given piece of gold, pure gold being taken to be of 16 sarnas. ... Curious and interesting problems involving propor. tionate division. A rectangular town is what seems to be intended here. Proportionate distribution involving fractional quantities. Vide footnote on p. 125. An operation involving the halves of the sum and the difference of the two quantities represented by the divisor and the quotient of any two given quantities. See note to VI--2. ... A measure of length. Vide Table 1 of Appendix IV. Name of a Jaina saint; one of the Tirtharkaras. A measure of length. Vide Table 1 of Appendix IV. Subtraction of part of a series from the whole series in arithmetical progression. See note on page 34. A kind of grain ; a measure of length. Vide Table 1, Appendix IV. Longitudinal section of a grain; for diagram see note to VII-32. ... A place 90deg to the East of Lanka. Penance; practice of meditation and mental con. centration. ... A measure of length. Vide Table 1, Appendix VI. Vitaati .. Vrsabha .. Vyavaha rangula ... Vyutkalita Kana Yavakoti Yoga Yojana For Private and Personal Use Only

Page #502
305 APPENDIX III. ANSWERS TO PROBLEMS CHAPTER II. (2) 1152 lotuses. (3) 2592 gems. (4) 15151 geme. (5) 53948 lotuses. (6) 9255827948 lotuses. (7) 12345654321. (8) 43046721. (9) 1419147. (10) 111111111. (11) 11000011000011. (12) 200010001. (13) 1000000001. (14) 111111111 ; 222222222; 333333333; 444444444; 555555555; 886066666; 777777777; 888888888; 999999999. (16) 11111111. (16) 16777216. (17) 1002002001. (20) 128 Dinaras. (21) 73 pieces of gold. (22) 131 Dinaras. (23) 179 pieces of gold. (24) 803 fruits. (25) 173 fruits. (26) 4029 gems. (27) 27994681 gold pieces. (28) 2191 gems. (32) 1; 4; 9; 16; 25; 36; 49 ; 14; 81; 225; 256; 625; 1296 ; 5625. (33) 114244 ; 21724921; 65536. (34) 4294967296; 152399025; 11108889. (35) 40793769; 50908225; 1014484. (37) 1; 2; 3; 4; 5; 6; 7; 8; 9; 18; 24. (38) 81 ; 256. (39) 65536 ; 789. (40) 7979; 1331, For Private and Personal Use Only

Page #503
306 GANITABABASANGRAHA, (41) 36; 25; (42) 333; 111; 919. (48) 1; 8; 27; 64; 125; 216 ; 343; 512 ;729 ; 3875; 15625; 46656 ; 466638 ; 884736. (49) 1030301 ; 5088448; 137388096 ; 368601813; 2497715584. (50) 9663597; 77308778; 260917119; 618470208 ; 1207949625. (51) 4741632; 37933056; 128024064 ; 303464448; 692704000 ; 1024192612; 1626379776; 2427715584. (52) 859011369945948864. (55) 1; 2; 3; 4; 5; 6; 7; 8; 9; 17; 123. (56) 24 ; 333; 852, (57) 6464; 4242. (58) 428; 639. (59) 1344; 1176. (60) 950804. (65) 55 ; 110; 165; 220; 275; 330; 385; 440; 405; 550. (66) 40. (67) 564 ; 754; 980; 1245 ; 1552; 1904 ; 2304. (58) 4000000. (71) 5; 8; 15. (72) 9; 10. (77) 2 ; 2. (79) 2 ; 520 ; 10; when the chosen numbers are 2 end 10. (83) 2; 3; 5: 2; 3; 5. (85) 120 ; 24; when the sum of the required series is twice the known eum : 30 ; 10; when the sum of the required series is half of the known num. (87) 48 ; 4; when the sums are equal: 36; 24; when one of the samg is twice the other: 44; 26; when one of the sums is thrioe the other. (38) 100; 216; when the sums are equal : 232; 102 ; when one of the sums is twice the other : 34; 223 ; when one of the sums is half of the other. (90) 21 ; 17; 13; 9; 5; 1: 25; 17; 9; 1. (92) 6; 5; 4; 3; 2; 1. (96) 4374 coins. (99) 1275 dinaras. (100) 68887; 22888183593. (102, 4 ; 2. (104) 4. (105) 8; 9 ; 15. (111) 224; 201 ; 175; 244; 261. (112) 4836; 4658 ; 4200 ; 75250. (113) 182938 ; 5846. (114) 180, 112; 60; 40. (115) 4092 ; 2014 ; 1020 ; 508 ; 252 ; 124 ; 60. For Private and Personal Use Only

Page #504
APPENDIX 111, 307 CHAPTER III. (3) pana. (4) 1. panas. (5) 2 to panas. (8) 27 palas. W 16: 21 143 * 255 : 133 (9) popas. (10) 174 panas. (11) 14 palas (12): So (18) (17) See examples 14 and 15 in this chapter ; 26 Hei 1 1 1 1 (18) & ; 27; 64: 126 ; 218 : 343 : 512; 729 1 1 20 27 343 1331 3375 6859 12187 19683 29791 42875 (18) 8 64 218 * 612 1000; 1728 2744 : 4096 ; 5832; 59319 8000 (20) go (21) 1. 1. 1. 1. 1. 1 1 1 3 7 11 16 10 23 27 > 3 zi : 5; 6; 7; gi gigi 2; 7; 8; loi 12i: 20 35 927 3377 1391 8535 (24) 1802088: 2388 8704 : 3810 22284 (26) In each of the series the first term is 1, and the common difference is 2. 9 The more of the ama are 4 16 25 36 49 64 81 are : 18 25 36:49 6481100, To The cubes of the sums are 27, 84, 125, 226, 349, 517, 191 799 100 523 729 1000' 1000 1881 1881, 1981. For Private and Personal Use Only

Page #505
808 GANITABARASANGRARA. (28) The eubio sums are inte 225; the first terms are * the common differences are $; the numbers of terms are i (31) color elever en swh y yh (35) 3512 4108 45 75 the interchangeable first term and common difference when the sums are equal; and - a 1551488, 225 in the equal gum. When the sums are in the ratio of 1 to 2, 7852 and 45 am we are the first term and the common difference; and the double sum is 3 : 3102976 When the sums are in the proportion of 1 tn 4, the first term and the common difference are and 4883 76; and the halved gum is 6519 225 (42) 341 . 1 (42) 2048 2048 (48) (49) 127 (**) 5760 (50) 100 (51) 4367 553 9367 (11) 12000 1440 23520 (62) 90; 21; 35; 17 (53) The first terms are 176. 352 704 ; the gums are 31 : 243 : 729 ! 22880 13376 the pumbers of terms are 5; 4; 4. 6561 6561 (57 & 58) 1. (59) 1. (60) 1;1;1. 37186 4581 For Private and Personal Use Only

Page #506
APPENDIX III. 309 309 (61 & 62) (63) 0. 1;1; 1; 1. : (65 & 68) (87 to 71) 4. (74) 2 ; 3; 4. (78) (a) 2 ; 3 ; 9; 27; 54. (8) 2, 3; 9 ; 27;, 81; 162. (c) 2 ; 3 ; 9 ; 27; 81 ; 243 ; 486. (78) (1) 8; 136; 340 ; 260. (2) 44 ; 220; 460; 299. (3) 78; 286 ; 550 ; 325. (81) (1) 5; 21; 420; when the optionally chosen quantity is 1 throughont; (2) 3; 11 ; 232 ; 53592 ; when the optionally chosen quantities are 2,1,1. (83) 2; ; *; when the chosen quantites are 6, 8, 9. (84) 8; 12; 16; when the chosen quantites are 6, 4, 3. (86) (1) 18; 9; when the chosen number is 3. (2) 30; 15; when again the chosen number is 3. (84) (1) 6; 12; the chosen number being 2. (2) 3 ; 15 do (3) 46; 92 do 2. (4) 22; 110 5. (90) (1) 4 ; 28; (2) 25 ; 175. (91) 18, 240. (92) 151, 3020. (94) (1) 22 , 44, 33, 66, 58, 116; when the sum is split up into 5,and and the chosen number is 2. (2) 11; 22 ; 69 ; 236 ; 191 ; 38 ; 20; when the sum is split up into , 1 1 . (96) 52. (97) 21. (98) (100 to 102) 1. (103 & 104) 1. (105 & 106) 1. (108) (110) }, ; it and are the optionally chosen quantities. (111) 7 (112) (114) O. For Private and Personal Use Only

Page #507
310 GANITASARABANGRAHA. (115) 14 nigkas. (116) 0. (117) 2 dronas and 3 mapes. (118) 1 (119) : a niskas. (120) 1. (121) 17 (123) ; To his i are the optionally split ap parta. (124) (127) 24 kargas. (128) (129) 1. (130) 1. (131) 1. (183) de ; when se and are the optionally oplit op parts. (134) (187) 5 when $ g are the optionally chosen fractions in places other than the beginning ; so whene, pro similar fractions. (139 & 140) 8 16 OHAPTER IV. (5) 24 hastas. (6) 60 bees. (7) 108 lotuses. (8 to 11) 288 sages, (12 to 18) 2520 parrots, (17 to 22) 3456 pearle. (23 to 27) 7560 bees. (28) 8192 cows. (29 and 30) 18 mangoes. (31) 42 elephants. (32) 108 puranas, (84) 36 camels. (35) 144 peacocks. For Private and Personal Use Only

Page #508
APPENDIX 1. 311 (88) 678 birde. (87) 84 monkey.. (88) 36 onokoos. (39) 100 swang. (41) 34 elephants. (42 to 45) 100 ascetics. (48) 144 elephants. (48) 16 bees. (49) 196 lions. (50) 324 door. (53) 48 angular. (54 & 55) 150 elephants. (56) 200 boars. (58) 96 or 32 vdhas. (59) 144 or 112 peacocks. (60) 240 or 120 hastas. (62) 84 or 16 baffaloes. (63) 100 or 40 elephants. (64) 120 or 45 peacocks. (66) 16 pigeons. (67) 100 pigeong. (68) 256 swans. (70) 72, (71) 324 elephants. (72) 1798 ascetico. CHAPTER V. (8) 638 15 yojanas. (4) 6 : yojanas. (6) 105600000. (6) 10-10 days. (7) 8110 $ years. (8) 9 98086 vahas. (0) 82 palus. (10) 67** papas. (11) 196-bharas. (12) 666 29. dindras. For Private and Personal Use Only

Page #509
312 GANITASARASANGRAHA. 169 (18) 2380 $ palas.. . (14) 163 pairs. (16 & 18) 12 in yojanas ; 27 1 vahas. (17) 112 dronas of kidney bean; 604 kudabas of ghee; 336 dronas of rice 448 pairs of cloth ; 336 cowe ; 168 svarnas. (18) 160 ; 112 320 dharanas, (19) 720 pieces. (20) 525 pieces. (21) 24 Tirthankaras. (22) 216 blocks. (24 & 26) 5 years and 117 days. (26) 2184 days. (27) 10 years and 246 1 days. (28 to 30) 351 days. (31) 76 days. (33) 10 puranas; 18 puranas ; 28 puranas. (34) 29 1120 gold coins. (35) 36 gems. (36) 4000 panas, (37) 250 karsas. (38) 960 pomegranates. (39) 560000 gold coins. (40) 750 gold coins. (41) 54. (42) 262 gold coins. (43) 945 vahas. 19 CHAPTER VI (3) 7; 5: 4; 5. (5) 9; 18; and 25. puranas. (6) 17. karsa panas. (7) 51 puranas and 14 panai. (8) 200. (9) 33 karpa panas. (11) 133 } puranas. For Private and Personal Use Only

Page #510
APPENDIX 111. 313 (12) 14. (18) 60; 60; 70. (16) 10 months. (16) 6 months. (17) 10 months. (19 & 20) 35 palas. (22) 30; 18. (24) 30. (26) 5 months. (27) 5 months; 75. (98) 4 months; 31). (30) 315 (31) 60; 6 months. (32) 24 months ; 36. (34) 10; 2} months. (36) 48; 10 months ; 24. (38) 10; 6; 3 ; 15. (40) 40 ; 30; 20; 50. (41) 5; 10; 15; 20; 30. (43) 5 months ; 4 months ; 3 months ; 6 months. (45) 8. (46) 6:13. (48) 20; 28; 36. (49 & 50) 25. (52) 18. (63) 30. (55) 900. (56) 800. (58) 28 months. (59) 18 months. (81) 2400 ; 800; 1200 ; 96. (62) 1000 ; 420 ; 480 ; 90. (64) 60. (65) 50.(67) 2400 ; 2720 ; 3400. (68) 1050; 1400 ; 1800. (89) 5100; 4590; 4050, (70) 1300 ; 1198; 1150. oli 2004 (72 and months. (73% to 78) 440,11; 5 months. 36 1733 ;8 (784) 4 months ; (801) 48; 32; 24; 16. 40 For Private and Personal Use Only

Page #511
314 GANITASARASANGRABA. (814) 3; 9; 27; 81; 243. (82; to 851) 120; 80 ; 40; 160; 60 ; 20. (86}) 48; 72; 96 ; 120 ; 144. (90% and 9:}) 70 pomegranatos; 35 mangoes; 35 wood appler. (92} to 941). Curd. Ghee. Milk. I pot 128 32 64 pe 98 II pot 32 8 16 por 3 3 III DO 64 16 32 por 9 O 9 (95$ and 963) 15 men ; 50 mev. (98%) 4; 9; 18; 36. (99) 8; 13; 21; 36. (100) 2; 4; 7; 13; 257. (1013) 16; 39 ; 96 ; 234. (103) 220; 37. (1041) 20; 5. (1053) 6; 4; 3. (the latter two having been optionally choosen). (1061) 8. (1081) 8031600; 1860; 2231. (1101) 148 ; 35328; 184. (1124 and 113})flowers. (1141) 10+ Howers. (117) 5. (118) 17. (1191) 26. (1201) 9. (1214) 55. (1224) 61. (123) 59. (1245) 30. (125) 16. (126) 15. (127_) 537. (1281) 138, (129) 184. (131) 11. (132 and 133.) 25. (1854) + (137) 10 ;57. (1881) In the case of positive associated numbers : 21 ; 18; 13; 11 ; 21 ; 19; 37; 7; 37; 6; ; i} ; 13; 5; 12; 1 ; 25. For Private and Personal Use Only

Page #512
APPENDIX III. 315 83 I heap II >> 18 4.5 In the case of negative associated numbers : 11 ; 18; 23, 27 ; 19 ; 23; 7; 39 ; 11 ; 44; 3; 41 ; 51 ; 46 ; 59 ; 37. (140% to 142}) 8; 5. (144) and 1455)-- Citrons. Plantains. Wood-apples. Pomegranates. 14 16 III, Price (147 to 149). Peacooks. Pigeons. Swans. Sarasa-birds. Number 7 16 Price in panas 12 36 10 (150) Ginger. Long pepper. Pepper. Quantity 20 44 Price in panas 12 16 32 (152 and 153) Panas 9 ; 20; 35; 36. (155 and 156) When the optional number is 6 ; ;a; 3; When the optional number is 8; 5; 6; 16; 4. (158) Length of a stage 10 Yojanas; each horse has to travel 40 Yojanas. (160 to 162) 10; 9; 8; 5. (164) 20; 15 and 12. (165 and 166) 8; 20; 40. (168) 243 panas. (170 to 1711) 104 ; 21; 2 2 4 2 16 40 28 80 7 : 21' 21' 3 7 (1737) 32 (1745) 871. (177 and 178) 14. (179) 3. (181) 21. 83 71 (184) 200. 100 (186) 20; 4; 4 ; 4 ; 4; 24. (188) 117 200 ; o 176 H10 18:10 16 (190) 18; 13. (191) 8 ; 18; 10; (193 to 1967) (a) : 10:192; (6) 2::02 (1981) 560 ; 448. (2003 to 201) 200, 100; 1800 ; 800 (204 and 205) 47; 17; 34; 68; 136. (207 and 208) 2400. For Private and Personal Use Only

Page #513
316 GANITASARASANGRAHA. (313 to 215) (217) 11. (219) 6; 16; 20; 15; 6; 1: 63. (320) 5; 10; 10; 5; 1 : 31. (221) 4; 6; 4; 1; 15. (223 to 225) 10; 24; 32. (227) 4 jaok fruits, (229) 2 Yojanas. (231 and 232) Dinaras 18; 57; 155 ; 400. (236 and 237) 15; 1 ; 3 ; 5. (239 and 240) 281; 921; 1416; 1801; 2109 ; 110880. (242 and 243) 11 ; 13 ; 30. (244 and 2441) 3; 4; 5. (245) and 247) 5177 ; 103 ; 169 ; 223 ; 239. (248) 14760; 356 ; 585 ; 445 ; 624. (240 to 250) 55 ; 71; 66 ; 876. (2531 to 255$) 7; 8; 9. (256} .to 2581) 11; 17; 20. (2604 and 2811) 7;3; 2. (2621) 8; 12; 14, 15; 31. (2631) 54 ; 72; 78 ; 80 ; 121. (2641) 1875; 2625; 2925; 3045; 30935187. (2661) 4; 7; 13. (2074) 12; 16 ; 22 ; 31. (270 to 2723) 421; 40. (274) 5; 8. (276) 186. (277) 151. TORON58375 (278) 441 (2901) 26. (2824 to 283) 1296 ; 1225. (285) (a); < ; (287) 35. (289) 37, (291) 40 ; 184. (293) 2; 3. (295) 5 women ; 40 flowers. (297) 204 ; 2109; 2870; 73810; 180441 ; 16206(300) 1095; 1624. (302) 441 : 1296 ; 784 ; 105625; 1082146818. (304) 2555'; 126225. (3061) 27663. (308t) 504 ; 732; 1020 ; 1375; 5304 ; 150875; 27 2304, For Private and Personal Use Only

Page #514
APPENDIX III. 317 (3104) 1563100, 5038889 ; 9646 ; 12705 ; '114400. (8127--313) 121 5461 1621 12288 (315) 426. (316) 416348873. (318) 2; 3; 5; 40. (320) 11. (320) I (321 to 321) 24 days. (3236) 3. (325) 6. (327) 25 days. (329) 13;19. (3311) 55. (332) 620. (337) For answer see footnote in the translation. . CHAPTER VII. (8) 32 sq. dandas. (9) 866 sq. dandas and 4 sq. hastas, (10) 98 sq. dandas. (11) 1200 sq. dondas. (12) 3600 sq. dandas, (13) 1952 sq. dandas. (14) 2378 sq. dandas. (15) 63041 sq. dandas. (16) 1925 sq. dandas. (17) 7425 sq. dandas. (18) 60 eq. hastas. (20) (i) 54 ; 243. (ii) 27"; 121. (22) 84 ; 252. (24) 48 hastas; 195 sq. hastas. * (26) 378. (27) 135. (29) 189 sq. hastas ; 135 sq. hastas, (31) 108; 972; 26. (33) 1600. (34) 2,400 sq. dandas. (35) 462 sq. dandas. (36) 840 sq. dandas. (38) 324 mq. dandas); 486 sq. dardas. (40) ; 180. (41) 18; 804 For Private and Personal Use Only

Page #515
318 GANITASARASANGRAHA. (42) 201; 35. (44) 253}; 39. (46) 13; 26. M 675 675 (51) V 768 sq. dandas ; dandas 48; 4; 4. (82) 60 sq. dandas ; dandas 12; 5; 5. (53) 84; 12; 5; 9. (55) V50; 25. (56) 13; 60. (57) 65; 1500. (58) 312; 288; 119; 120; 34560. (69) 315; 280; 48; 252; 132; 188; 224 ; 189; 44100. (61) V3240; V65610; v 36000; V 8100000 ; 4840 ; v146410. (62) V380; V 3240 ; 13240 ; 262440. (64) V 6048 ; V54432. (664) V2560 dandas; V 42250 sq. dandax, (68%) V 39690 Bq.dandas; 20250 sq. dandas. (691) v31360 sq. dandas. (714) 71440 sq. dandas. (72}) 5760. (75) V380 ; 12; 6. (773) 192+ V 23040. (78}) 102-V6760. (791) 192-123040. (814) 19360, 4840, V4840 99. (834) 16-V180. (851) V 48 - V20 (871) 16, 12; 48. (891) 20; 8. (914) 3; 4; 5. (92) 5; 12; 13. (943) 16; 30; 34, (961) 5; 3; for the three cases. 1981) (i) 60; 61. (ii) 11; 61. (iii) 11 ; 60. (100}) 80; 102; 61 ; 60; 109; 11 ; 5460. (1021) 169; 407 ; 189; 120; 312; 119; 34560. (1045) 125; 300; 260; 195 ; 224 ; 189; 48; 252; 168 ; 132; 44100. (1091) 34 ; 60; 16. For Private and Personal Use Only

Page #516
APPENDIX III. 319 (1111) 13; 16; 14; 12. (1131) 4; 1. (1143) V2; 3. . (1151) 6; 3. (1164) vi. (1171) 32; (perpendicular 24). (1181) it in : (1197) izi (perpendionlar 16). (1214) 3; 8. (1234 and 1244) 39; 52; 25; 60; 33; 56 ; 13; 16. (1264) 5, 12. (1283) 5; 12. (180) 25; 60. (134) 8; 15; 3; 20. (135) 8; 7; 2; 28. (136) 32; 87; 6; 232. (138) 37 ; 24 ; 28 ; 40. (139) 17 ; 16; 13; 24. (140) 625 ; 672; 970 ; 1904. (141) 281 ; 320 ; 442; 880. (143 to 145) Circle : 25920 ladies; 720 dandas. aquare : 34660 ladies; 720 dandas. Equilateral triangle: 38880 ladies; 1080 dandas. Longich quadrilateral : 38880 ladies ; 1080 dandas ; 540 dandas. (147) (i) Side 8 (u) Base 12; perpendicular 5. 13 13 1. 41 (149) gigiz'i; 4. (151) 18; 13; 13; 3; 12. (153 to 1535) 3 ; 16; 11; 12. (1557) W28 (1571) 6; 6; 4. 46 89 116 (1601) 30: 80 116 83 45. (1831) 30 30 30 13 (1844) VAO. (1684) 7; 1; (187) 87 13 13 13 (1894) 6. (1705) 10. For Private and Personal Use Only

Page #517
320 GANITASARASANGRAHA. 30 39 (1721) 10; 13. (174}) sides ; top-side ; base 56 (176) 17. (177} to 1781) (a) 3600; 7200; 10800 ; 14400 ; (b) 54 ; 80; 126; 162 ; (c) 100 ; 100; 100; 100. (179}) (a) 2700; 7200; 4500; (6) 50 : 70; 80; (c) 60; 120; 60. (1811) 8 hastas; 8 hastas. 54 (1825) * hastas ; *hastas ; 99 hastas. (1831 and 1845) 3 hastas ; 6 hastas ; 9 hastas. (1854) 7 hastas ; 7 hastao ; ens hastas. (1861) i hastas ; hastas ; hastas. (1871) 9 hastas ; 12 hastas ; 9 hastas, (1884 and 189}) 8 hastas; 2 hastas ; 4 hastas. (1915) 13 hastas. (1921) 29 hastas. (1934 to 1951) 29 hastas ; 21 hastas. (197) 10 hastas. (1994 to 200}) 12 yojanas; 3 yojanas. (2044 to 205) 9 hastas; 5 hastas; V 250 hastas, (206 to 2075) 6 yojanas; 14 yojanas ; V 520 yojanas. (2084 to 2091) 15 yojanas ; 7 yojanas. (2111 to 212}) 13 days. (214)) V18; 13. (2151) (2167) 125 (2171) 65. (2181) V18: 169 (2194) (2204) 4. (2224). Squaro: N1. Oblong: 0; 12. Quadrilateral with two equaloides : vides a top-side 1.9; baso . Quadrilatorul with three squel nidos : 39 1521 39 52 sides ; base 125 Inequilateral quadrilateral: wides 126 ; top-sido 507 5; base 12. Equilateral triangle: N . Isosceles triangle: sides 12 : bare 120. Soaleno trianglo: sider, 12; 0,2; bado este For Private and Personal Use Only

Page #518
APPENDIX III. 321 108 (2245) Square, 3. Quadrilateral with two equal sides : 1. Quadrilateral with three equal sides : 512. Inequilateral quadrilateral: *11. Equilateral triangle : Viz. Isoscelos triangle :. Scalene triangle : 8. Hexagon : 116 N , if the area of the same is taken as v 48 in accordance with the rule given in stanza 86} of this chapter. (2261) 8. (2281) 2. (230%) 10. (232) 6; 2. CHAPTER VIII. (161) (5) 612 cubio hastas. (6) 18560 cubic hastas. (7) 144320 cubic hastas. (8) 162000 cubic hastas. (121) 2928 cubio hastas. (131) 1458 cubio hastas; 1476 cabic hastas ; 1464 cubic hastas. (141) 2916 cabic hastas ; 2952 cubic hastus ; 2928 cubic hastas. (151) 3360 cubic hastas. 98980 cabio hastas. 9 (171) 16100 cubic hastas. 1187) 18283} cubic hastas. (811) (i) 3024 cubio dandas ; 3024 cubic dandaa ; 4032 cubic dandaa; (ii) Central mask is tapering ; 1488; 1488; 1984 cubiu dandas (221) 4032; 1984 cubio dandas. (241). 40 oubio hastas. (253). 16 hastas. (273). 12; 30. (294). 2304; 2073 . (311). ^ 720; 1648. 2 3 4 5 14deg 4 day. 14' 14' 14' iz of the well. (35 and 36). 13 yajanas, and 976 dandas ; 39 14 vahes, (37 to 881). 17 yojanas, 1 krosa and 1968 dandas. (301 and 401). 26 yojanas and 1952 dandas. (41 and 42}). 6 yojanas, 2 ks veas and 488 dandas (455). 6912 unit bricko. (467). 3456 unit bricks. For Private and Personal Use Only

Page #519
Shri Mahavir Jain Aradhana Kendra 822 (47). 5184 unit bricks. (48). 108000 unit bricks. (49). 40320 unit bricks. (50). 40320 unit bricks. (51). 20738 unit bricks. 19 (58). 20; 6 (53). 1440 unit bricks; 2880 unit bricks. (55). 2640 unit bricks; 1680 unit bricks. (561). 2880 unit bricks; 1440 unit bricks. www.kobatirth.org 1 of a day. (91). 100 (11). 3 ghatis (13). of a day. 7 12 (14). 2. GANITASARASANGRAHA. (59-60). 891 unit bricks. (62). 18,720 unit bricks. (68). 64 pattikas. (19). 8 angulas. (22). 16 hastas. (24). 8 hastas. (25). 2. (27). 20 hastas. (29). 10. (31). 5; 50. (34). 5 hastas. 1 (16 to 17). of a day; 10 ghafis. 1 (35 to 37). 18 of a day; 8. CHAPTER IX. (38 and 39). 5 hastas, (41 to 42). 24 angulas. (44). 32 angulas. (46 and 47). 112 angulas. (49). 175 foot-measures. (50). 100 foot-measures. (51 to 52). 100 yojanas.

Page #520
323 APPENDIX IV. TABLES OF MEASURES. 1. LINEAR MEASURE. Infinity of Paramanus 8 Anus 8 Trasarinus 8 Ratharenus 8 hair-measures 8 louse-measures 8 segamum-measures 8 barley-measures 500 Vyavaharangulas 6 Angulas (finger-measure) 2 feet 2 Vitastis 4 Hastas 2000 Dandas 4 Krvas. = 1 Aru. = 1 Trasarenu. = 1 Ratharenu. = 1 hair-measure. = 1 louse-meastre. = 1 sesamum-measure or mustard measuro. = 1 barley-measure. = 1 angula or Vyavaharanguta. = 1 Pramana or Pramanangul'a. = 1 foot-measure (measured across). = 1 Vitasti. = 1 Hasta. = 1 Danda. = 1 Krosa. =1 Yojana, 2. TIME MEASURE. Infinity of Samayas A number of Avalis 7 Ucchra sas 7 Sotkas 384 Lavas 2 Ghatis 30 MWhurtas 15 days 2 Paksas 2 months 3 Btus 2 Auany = 1 Avali. = 1 Ucchva sa. = 1 Stoka. = 1 Lava. = 1 Ghafi. = 1 Muhurta. = 1 day, = 1 Paksa. = ) month. = 1 Btu. = 1 Ayana. * 1 year. For Private and Personal Use Only

Page #521
324 GANITASARASANGRAHA. 3. MEASURES OF CAPACITY (GRAIN MEASUREMENT). 4. Sodasika 8 4 Kudahas 4 Prasthas 4 Adhakas 4 Dronas 4 Manis 5 Kharis 4 Pravartikas 5 Pravartikas El Kudaha, = 1 Prastha. = 1 Adhaka. = 1 Drona. =1 Mani. = 1 Khari. = 1 Pravartika. El Vaha. = 1 Kumbha. 4. MEASURES OF WEIGHT-GOLD. 4 Gandakas 5 Gunjas 8 Panas 2 Dharanas 4 Karsas = 1 Gunja. = 1 Pana. = 1 Dharana. = 1 Karsa. = 1 Pala. 5. MEASURES OF WEIGHT-SILVER. % Grains 2 Gunjas 16 Maras 21 Dharanas 4 Karpas or Puranas = 1 Gunja. = 1 Masa. = 1 Dharana. = 1 Karsa or Purana. -- 1 Pala. 6. MEASURES OF WEIGHT-OTHER METALS. 4 Padas 64 Kalas 4 Yavas 4. Anias 6 Bhagas 2 Drakgunas 2 Dinaras 124 Palas 200 Palas 10 Tulas = 1 Kala. = 1 Yava. = 1 Ansa. = 1 Bhaga. = 1 Draksuna. = 1 Dinara. = 1 Satera. = 1 Prastha. = 1 Tula. = 1 Bhara. 7. MEASUREMENT OF CLOTHES, JEWELS AND CAN BS. 20 pairs = 1 Kofika. For Private and Personal Use Only

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APPENDIX IV. 8. EARTH MEASUREMENT. -3800 Palas, 3200 Palas. 1 cubic Hasta of compressed earth 1 cabic Hasta of loose earth of a man's height: 9. BRICK MEASUREMENT. Brick of 1 hasta x Hasta x 4 Angulas 1 Hasta and 18 angulas Work done in cutting along by means of a saw a piece of wood 96 Angulas long and i Kisku broad Unit brick, 10. WOOD MEASUREMENT. =1 Kisku. = 1 Pattika. 11. SHADOW MEASUREMENT. his foot measure. 325

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kuo asnjeuosjed pue areaud 104 AGENTS FOR THE SALE OF MADRAS GOVERNMENT PUBLICATIONS, IN INDIA. . R. OAMRRAY & Co., Onlautta. COMBRIDER & Co., Marras. 1. COOPOOSAW ME NAIKER *00. Maciras. HIGGIN BOTHAN & Co.. Mount Road, Madras. V. KALYANARAMA TYRE & Co, Isplando, M G. O. LOGANADILA BRO13s, Macias S. MURTHY & Co., Kapales Press, Madras. G. 1. NATEAN & Co., Madras. P. R. RAMA TYAR & Co., Madras. D. B. TABAPOR KVALA Box & Co., Bombas TEMTII Co., Georgetown, ras, THACIBE to Chimited), Bombay Tha0 LL, SPINE & Co., Valeutta. THOMON & Co., Madras IN ENGLAND B. H. BHACKWELL, 30 and broad Street, Oxford PONDTABLE Co., 10, Orange Sirett. Leirster Squatu, London, W.O. DEIGHIEN, BET & Co., On bridge 1. ETS HE U S , 1. Adelphi Termice, lozion, W.O. GRINDLAT & Co., 3, Parlisment Street, London, B.W. BEGAN TRENCH, THUBNIT & CO., Carter Lane, London, E., HENRY KING LO , Cornbill, Loulon, .O. PS Son, 2 , Gront Szoitit Street, Wet ter, London, S. LUZA Russ SEO . B. Q . Gaton Buret, 2 W , W. W. TILACIER & Club Creen T , London, ON THE CONTINENT. FRIEDLANDER & SOHN, 11, Oil Sista OTTO HARILS KARL W. HE Serving JinShasan LORNET LER NAUTIKUS NI 093090 gyamai dirokobanti org J!puewues unsjebessejley uus eleyov bio"Chitseqoy"Mmm ejpu@y eueypey uier eyew uys