Ganitasara Sangraha of Mahavira

Author(s): Rangacharya
Publisher: Rangacharya
Catalog link: https://jainqq.org/explore/011112/1

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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 COMPARATITE PHILOLOGY, PRERNENT COLLEGE, AND CURATON, GOVIENNENT (KNTAL MANUSCRIPT LIHRANY, MARRA.. Published under the Orders of the Government of Madras. MADRAS: PRINTED BY TIE SU PARINTENDENT, GOVERNMINT PRESU, 1912.

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12499

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mahAvIrAcAryapraNItaH gaNita sA ra sa da ya ha : 9919 ma. ragAcAryeNa garizacana: AgalabhASAnanAda TIkAbhyAM ma.gajakA yAjJAnamAraNa prkaashitdh| annapUrvA rAjakIya mudrAkSaravAlAyo (mparinTeDaNyAphayama) anivAraNa mhitH| 2212.

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TABLE OF TRANSLITERATION. Conremunten Vowelve Diphthongs. Cutturala ..k, kb,g.gb, ii, h, h. .. ka, sva . ga, gha, Ga, ha.: ..a .. ) A (6) ai. Palatale . .. e . c, ch, j, jh, ii, .. ca, cha. ja. sa. na. ya. za .... / .. I Linguals ... ..tth, a, dh, II, I, Ta. Tha, Da, Dha, Na, ra. pa ...,! .. .. ka, ka Dentals ! t.th, d,dh,n, I, .. ta, tha, da, gha.na, la, sa .. . Labials .. pph, b, bh, m, v pa. pha., gha, ma. ma. va .. u, opan. ..u u .. o. zrI.

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GENERAL CONTENIS. Proton . . ... ... ... Introdnotion by Dr. David Engeue Smith ... ... ... ... ... ... ... Pago ... Vii-xill ... xix-xxiv Contents of the text in Sanskrit Text ..... ... .. .... ... .. ... .. .. 1-108 Contents of the translation in English Translation . .. ... ... ... ... ... ... ... ... ... ... .. - 1-288 APPENDIX 1.- Babakrit words denoting numbers with their ordinary and numerical signification ... ... ...... ... ... ... 287-245 APTEKDIX 11.- Banskrit words used in the traslation and their explanation ... .. ... .. .. ... ... ....... 200-404 APPENDIXIB- owers to problemi ... ... ... ... ... 806-892 APPENDILIV.- Tables of measures .... ... ... ... .. ... 329-*

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ADDENDA ET CORRIGENDA TU TILE GANITASARASANGRAHA. Linc Page X (Preface) XIV XX do. For Read wiksu mokat. Nakalya-rrimihitu Sakalya-8ahitd. Mali piracary Mahaviricirya. Do. owit much of' before kusirakuta Aryabhula Arynlibutan TIPS gazi vAlakA a . maGalita Afra 90 XXI 20 / 5 Tert * 14 1s T3 41774 A1041. 17 3:. Trundlotion. 25 foutaito 3 id nombro" at "any" 11 sociated with ... ked with of wist & of what remained them frer of thin letration, 27, 28 of that whole collection of Ixen. fra order to sin the ead 26 of this lant. ! (of that whole collection of bee). frun under to in the end. . 3

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Page 78 79 footnote do. Lido 8 4 5 4 For 8 pamula gabstituted for cand i Ksamila after known quantity ao Read sa amula. substituted for cand a. Sepa maila. add. wbich is subtracted from or added to this precitied fruotioual part of the unknown collect ive quantity.' 32 palar. 100,000 kary(18. 5) puranas. 27 angulas. 36 60 32pala: 10,000 kars8 5} puran 13 3 angulas 35 90 20 106 108 footnote 111 - do, 113 do. 881 88 example in statzn 1013 examples in stanzas 100% and 1015 Suvauna (c-d). 138 217 223 footnote do. 19 Sunctional 5 (c--) ter a?. -12 7 add before - 2 3 (a+) 224 do. (a+b) ./20. 11 20 265 o N 287 ambIdha wordly 289 292 mani 298 nali kSAyakasamya, ktva ambudhi worldly. TA. ata. kSAyakasamyaktva. A. FAHETA. Pala. cube, chose'll. Srokar. 294 301 stambrema Paal oobic choosni Sotkas 303 314 393

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PREFACE. 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 now 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 mathequatical manu. scripts 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 manu 8 made the value of this work evident in relation to the history of Hindu Mathematica. 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 scripts ma

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vii GANITABIRABANGRARA. - 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 samo 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 denota them by K. Another manuscript, denoted by M, belongs to the Gov. ernment Oriental Library at Mysore, and was received on loan from Mr. A. Mahadeva Sastri, B.A., the Curator of that institution. This inanuscript 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 Kanarose language by one Vallabha, who claims to be the author of also a Telugu commentary on the same

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PRBAD. 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 Jains monastery at Mudbidri in South Canara, and was obtained through the kind offort of Mr. R: Krishnamacharyar, M.A., the Sub-assistant Inspector of Sanskrit Schools in Madras, and Mr. U. B. Venkataramanniya of Mudbidri. This manuscript also contains the whole work, and gives, like K, in Kanarese a brief statement of the problems and their angwers. The endeavour to secure more manuscripts having proved fruitless, the work has had to be brought out with the aid of these fivo mauuscripts ; and owing to the technical character of the work and its elliptical and often riddle-like language and the inaccuracy of the manuscripts, the labour involved in bringing it out with the translation and the roquisite 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 circunstance about the Ganita -sara. sangraha 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 tbe two introductory stanzas of salutation to Jina Mahavira, six stanzas describing the greatness of a king, whose name is said to bave been Cakrika-bhasijana, and who apipears to have been commonly known by the title of Amoghavarsa Nnpatnnga; and in the last of these

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GANITAB RABANGRAHA six stanzug there is a benediction wishing progressive prosperity to the rule of this king. The results of modern Indian epigraphical research show that this king Amo. ghavarsa Nrpatunga reigned from A.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 Nrpatanga, 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, Muhavisacarya 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 Inacription of the time of Amoghavara I, A.D. 888; edited by J. P. Pleot, PH.D., C.I.v., in Epigraphia Indica, rol. VI, pp. 48-108.

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PRIVACE. Brahmagupta is caloulated to lead to the conclusion that, in all probability, Mabaviraoarya was familiar with the work of Brahmagupta and endeavoured to improve upon it to the extent to which the scope of his Gunita-sara. sangraha permitted such improvement. Mahaviracarya's classification of arithmetical operations is simpler, his rales are fuller and he gives a large number of examples for illustration and exercise. Prthudnkasvainin, the wellknown commentator on the Brahmasphuta-siddhanta, could not have been chronologically far romoved from Mahaviracarya, and the similarity of some of the exam. ples 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 Mabaviracarya wrote his Ganita-sara-sangraha, Brahmagupta must have been widely recognized as a writer of authority in the field of Hindu astronomy and mathematicy. Whether Bhaskaracarya was at all acquainted with the Ganitasara-sangraha of Mahaviracarya, it is not quite easy to gay. 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 Bhag. karacarya's Siddhanta-firomani, including his Lilavati and Bijagaaita, was intended to be an improvement in the main upon the Brahmasphutamsiddhanta 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, soems to have been widely known and appreciated in Southern India. So early as in the course of the eleventh century and perhaps

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GANITARARABAKGRAHA. ander 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 suita able attention to the historical value of Mahaviracarya's Ganita-sara-sangraha, I could not do better than seek 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 therefore due to him for bis, kindness in having readily complied with my request; and I feel no doubt that his introduction will be road 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-bara-sangraha twenty-four notational places are mentioned, commencing with the units place and ending with the place called mahaksobhain

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PRBPACE. and that the value of each succeeding place is taken to be ten times the value of the immediately precediug place. Although certain words forming the namos 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 moun, eye, fire, and sky to represent respectively 1, 2, 3 and 0. as their Sanskrit oquivalants are understood in this work, then, for instance, fire-sky-mrun-eye would denote the pomber 2103, and noon-eye-sky.fire would denoto 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 age 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 tho 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 Brabmagupta's writings also there is evidence to show that he was acquaintod with the use of nominal pomeruls and the decimal system of notation. Both Aryabbata and Brahmagupta claim that their astronomical works

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GANITARIRASANGRAHA. 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 Governmect 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 game meaning as the Sanskrit sunya, we may safely arrive at the conolusion 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

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PRBLADE. tions with the sero cannot be carried on-not to any 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 beforo 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 gymbol to represent it in written calculations. That even Aryabhata knew guch a symbol is not at all inprobable. It is worthy of note in this connection that in enumerating the norr.inal numerals in the first chapter of his work, . Mahaviracarya mentions the namos donoting the nine figures from 1 to 9, and then gives in the end the names denoting zero, calling all the ton by the naine of sankhya: and from this fact also, the inferenco,may well bo drawn that the zero had a symbol, and that it wns well known that with the aid of the ten digits and the decimal Ayrtom of notation namerical quantities of all values may be definitely and accurately expressed. What this known sero-symbol was, is, however, a different question. The labour and attention bestowed upon the study and translation and annotation of the Ganila-sara-sangraba

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Xvi GANITASIRALLIGRAHA. have made it clear to me that I was justified in thinking that its publication might prove useful in elucidating the condition of mathematical studies as they flourished in South India among the Jainas in the ninth century of the Christian eru ; 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 timo 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 Ganita-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 siyangar, 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. Sesha 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

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PRIPAORT willingness in allowing me thus to take advantage of his , export knowledge of mathematics. My thanks are, I. have to say in conclusion, very partioularly due to Mr. P. Varadacharyar, B.A., Librarian of the Government Oriental Manuscripts Library at Madras, but for whoge zoalous 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 Ganila-sara-sangraha. February 1918, 1 Madras. M. RANGACHARYA.

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21 OCT.!! -

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* INTRODUCTION BY , DAVID EUGENE SMITH, PROVESBOR OF MATHEMATICS IN TEACHERS' COLLEGE, COLUMBIA UNIVERSITY, New York, We have so long been accustomed to think of Patalipatra on the Ganges and of Ujjain over towards the western coast of India as the ancient habitats of Hindu mathemation, that we experience a kind of sarprise at the idea that other centres equally important existed among the multitude of oities of that great empire. In the same way we have known for a century, chiefly through the labours of such soholars as Colebrooke and Taylor, the works of Aryabhata, Brahmagupta, and Bhaskara, and have come to feel that to these men alone are due the poteworthy oontributions to * be found in native Hindu mathematice. Of course a little refloor tion shows this conolusion to be an incorreot one. Other great schools, partioularly of astronomy, did exist, and other mobolans 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 sobolans under the English supremaoy have done to 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 cortainly been owing to the absence of material, for Sanskrit matbematical mannsoripts are known, as are also Persian, Arabic, Chinese, and Japanese ; and many of these are well worth translating from the historioal standpoint. It has rather boon owing to the fact that it is hard to find a man with the requisito soholarship, who can afford to give his time to what is necessarily a labour of love.

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GANTTASIBASANGRAHA. 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 mathematios, no known in another part of India and at a time about midway between that of Aryabhata and Bhaskara, and two centuries later than Brabmagupta. The learned scholar, Professor M. Rangacarya of Madras, somo years ago became interested in the work of Mahaviracarsa, and has now completed its travelation, thus making the mathematical world his perpetual debtor; and I esteem it a high honour to be requested to write an introduotion to so noteworthy & work. Mabaviracarya 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 Mysoro 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-ganit of Bhaskara; it bas one more chapter than the Kut. taka of Brahma-gupta. There is, however, no significance in this nurnber, for the chapters are not at all parallel, although certain of the topios of Brahmagupta's Ganita and Bbaskara's Lilavati are included in the Ganita-sara-sangraha. In considering the work, the reader naturally repeats to bim. solf the great questions that aro so often raised :--How muob of this Hindu treatment is original ? What evidences are there here of Groek influence? What relation was there between the great mathematical centros of India ? What is the distinctive feature, if any, of the Hindu algebraic theory P Suoh 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

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INTRODUCTION. Chinese and Japanese mathematical works' mado' more or less familiar to the West; and the more important drab treatises are now quite satisfactorily known. Various editions of Bhaskara have appeared in India ; and in general the great trontises of the Orient have begun to be subjeoted to oritical study. It would be strange, therefore, if we were not in a position to woigh up, with more certainty than before, the claims of the Hindu algebra. Certainly the persovering work of Professor Rangiicarya has made this more possible than evor before. As to the relation het ween the East and the West, we should now be in a position to say rather definitely that thero is vo evidence of any considerable influence of Greuk algebra upon that of India. The two subjects wero radically different. It is true that Diophantus lived about two conturice before the first Aryabhata, that the paths of trade were open from the Weat to the East, and that the itinerant scholar undoubtedly carriod learning from place to place. But the spirit of Diophautus, showing itself in a dawning symbolism and in a prculiar ty po of equation, is not soon 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 havo u style and a range of topios peculiarly their own. Their probleme lack the cold, clour, geometric precision of the Wost; thoy itro clothod in that poetio language which distiuguishes the East, and thoy relate to subjeots that find no place in the sciuntific books of the Grecka. With perhaps the single exception of Metrodorus, it is only when we come to the puzzle problems doubtfully attributed to Alouin that we find anything in the West which resom bler, even in a slight degree, the work of Alouin's Iudian contemporary, the anthor of this treatise. It therefore secms only fuir to say that, although some knowlodge of the scientific work of any one nation would, even in those romote timen, naturally bave 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.

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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, no 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, somo years ago became interested in the work of Mahaviracarya, and bas now completed its translation, thus making the mathematical world his perpetual debtor; and I csteem it a high honour to be requested to write an introduction to so noteworthy a work. 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 contury A.D., so that we may roughly fix the date of the treatise in question is about 850. The work itself consists, as will be seen, of nine chapters, like the Bija-yanits of Bhaskara; it has one more chapter than the Kut. talu of Brahma-gupta. There is, however, no significance in this nuinber, for the chapters are not at all parallel, although certain of the topics of Brahmagupta's Ganita and Bhaskara's Lilavati aro included in the Ganita-sara-sangraha. In cousidering the work, the ruador naturally repoats to him. self tho great questions that aro so often raised :--How much of this lindu 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

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INTRODUCTION. xxi Chinese and Japanese mathematical works'inado more or less familiar to the West; and the more important Aral treatine's Are now quite satisfactorily known. Various editions of Bhaskara 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 wero not in a position to woigh up, with more certainty than before, the claims of the linelu algebra. Certainly the persovering work of Professor Rangacarya has made this more possible than ever before. As to the relation between the East and the Went, we should now be in a position to say rather definitely that there is no evidence of any considerable influence of Creek algebra upon that of India. The two subjects were radically different. It is true thut Diophantus lived about two centuries before the first Aryabhata, that the path of traile were open from the Wrist 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 ognation, is not soon at all in the works of the Ent. 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 cuntrary, the Hindu works have it stylo and a range of topics peculiarly their own. Their probleme lack the cold, clear, geometric precision of the West; they are clothed in that portio language which distinguishes the last, and they relato lo subjects that find no place in the scientific boks of the Greeks. With perhaps the singlo exception of Metrodorus, it is only when we come to the puzzle problems doubtfully attributed to Alouin that we find anything in the West which resombles, even in a slight degree, the work of Alenin's Indian contenporary, the author of this treatise. It therefore focmis only fuir to say that, although some knowlcdge of the scientific work of any one nation would, even in those remote times, naturally bave been carried to other pooples by somo wandering savant, we have nothing in tho writings of the Hindu algebraists to show any direct influence of the Wost upon their problems or their theories.

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xxu . GANTTASARASANGRAHA. 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 Arsabhata wrote, and that of Ujjain, where both Brahmagupta and Bhaskura lived and taught? And what was the relation of each of these to the school down in South India, which produced 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 Wost 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 Chiushang, Li Yeh, and Chu Shih-chieh, we focl 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. l'ho answor to the guestions as to the rolation between the schools of India oaunot yet be easily giveu. At first it would soem & simple matter to compare the teratives 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, Mahuviracarya, and Bhaskara may be desoribed as similar in spirit but entirely different in detail. For example, all of theso writers treat of the areas of polygons, but Mahaviriparya is the only one to make any point of those that are re-entrant. All of thoin touch upon the area of a segment of a circle, but all give difforent 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 gnomonics, suggest a similarity among these three great writers, and yet those of Mahavirkoarya are much better than the one to be found in either Brahmagupta or Bhaskara, and no questions are duplicated.

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INTRODUCTION. xxu In the way of similarity, both Brabmagupta and Mabavintdrys give the formula for the area of a quadrilateral, Wir-a) (1-0) (A-c)(x-2) --but neither one observee that it holds only for & oyolio figure. A few problems also show somo similarity such as that of the broken tree, the one about the anaborites, and the common one relating to tho lotur in the pond, but these provo 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 evidouce of any oloso following of one writer by another. * When it comes to geomotry there is naturally more evidence of Western in Huence. Indin seems never to have indopondently developed anything that was specially worthy in this sievoe. Brahmagupta and Mahaviriciryn both use the saino incorroot rules for the area of a triangle and quadrilateral thates fonod in tho Egyptian treatise of Ahmes. So while thoy scom to have been influenced by Western learning, this loarning as it reached India could have been only the simplest. There rules had long since been shown by Greek scholars to be incorrect, and it seeins not unlikely that a primitive geometry of Mesopotamia reushed out hoth to Egypt and to India with the result of perpetuating these errors. It has to be borne in mind, however, that Mahaviracirya gives correot rules also for the area of a triangle as well as of a quadrilateral without indicating that the quadrilatoral has to be oyclic. As to the ratio of the circumference to tho diameter, both Brahmagupta and Mahi vincarya usod the old Semitic value 3, both giving also V10 as a closer approximation, and neither one was awaro of the works of Archimedes or of Heron. That Aryabhata gave 3.1410 as the vuluo of this ratio is well known, although it seems doubtful how far be used it himself. On the whole the geometry of India seenis rather Babylonian than Greek. This, at any rate, is the inference that ono 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 mattor of

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Xxiv GANITASABASANGRAHA. 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 commonly in the West, and we must conclude that the algebraio taste, the purpose, and the method were all distinct in the two great divisions of tho 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 Indin.' Leonardo Fibonacci, for example, shows much more of the Oriental influence than Blaskara, who was practionlly his contemporary, shows of the Ocoidontal. Professor Rangacarya bay, therefore, by his groat contribution to the history of mathematics confirmed the view already taking rather concreto forin, tiat 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 influencod Europe in the matter of algebra, more than it was influenced in return; that there was no native geometry renlly worthy of the name; that trigonometry was practically non-existent savo us imported from the Greek astronoiners ; 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 Europoan scholars, and a work that is in many respocts tho most scholarly of any to be found in Indian mathometical literature. They have given us further ovidence of the fact that Oriental mathomatics lacks the cold logio, the consecutivo arrangement, an l the abstract character of Greek mathematios, 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 Wost, although abounding in the works of China and Japan. If, now, his labours shall lead others to bring to light and set forth inore and more of the classics of the East, and in particular those of early and medieval China, the world will be to a still larger extent his debtor.

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gaNita sArasa du ya ha :. .

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* saMjJAdhikAra : maGgalAcaraNam gaNitazAstraprazaMsA saMjJA kSetraparibhASA.. kAlaparibhASA dhAnyaparibhASA suvarNaparibhASA rajata paribhASA lohaparibhASA parimAni vanaNezatyaviSayakamAmA-yaniyamAH maDugAmaMjJA sthAnanAmAni gaNakagaNAnarUpaNama prathamaH parikarmavyavahAraH pratyutpanna bhAgahAraH varga: vargamUlama ghanaz ... CONTENTS. ghanamUlam maGkalitam cyutkalitam ... dvitIyaH kAlAsa varNavyavahAraH bhitrapratyutpanna: bhinnabhAgahAra: ::::::: :: Page I 2 3 es 6 fi ==255555 12 13 16 17 45 28

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CONTENTS. Page 330 bhinavargavargamUlaghanaghanamUlAni bhinnasaGkalitam bhinnavyutkAlatam kalAsavarNaghaDajAti: bhAgajAti: ... prabhAgabhAgabhAgajAtI bhAgAnubandhajAtiH mAgApavAhajAtiH bhAgamAtRjAti: tRtIyaH prakIrNakavyavahAraH - bhAgajAtizeSajAtI mala jAtiH zeSamalajAtiH dviragrazeSamUlajAti: aMzamalajAtiH bhAgasaMvargajAti: unAdhikAMzavargajAti: malamizrajAti: bhinnadRzya jAtiH caturthaH |raashikvyvhaarH trairAzikaH ... gatinivRttiH pazcasaptanavarAzikAH ... pazcamaH mizrakavyavahAraH-- sakramaNasUtram pazcagazikavidhiH vRddhividhAnam prakSepakuTIkAraH vallikAkuTIkAraH viSamakukiAraH

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OONTENTS. Page 101 110 makalakuTIkAraH suvarNakuTIkAraH . ... vicitrakuTTIkAra: zreAbaddhamaDAlitam paSTaH kSetragaNitavyavahAraH --.. vyAvahArika gaNitam sUkSmagaNitam janyavyavahAra: paizAcikavyavahAraH saptamaH khAtavyavahAra :--- khAtagaNitama nitigANitam krakacikAvyavahAra: aSTamaH chAyAvyavahAraH 110 122 126 148 160 168

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gaNita sArasaha : mahAvIrAcAryapraNIta / saMjJAdhikAraH / maGgalAcaraNam / alakSyaM trijagatsAraM yasyAnantacatuSTayam / namastasmai jinendrAya mahAvIrAya tAyine // 1 // saGkhyAjJAnapradIpena jainendreNa mahA tviSA / prakAzitaM jagatsarvaM yena taM praNamAmyaham // 2 // prINitaH prANisasyotro nirItirniravagrahaH / zrImatAmoghavarSeNa yena veSTahitaiSiNA // 3 // pAparUpAH parA yasya cittavRttihavirbhuja / . bhasmasA'dbhAvamIyuste'vandhyakopo'bhavattataH // 4 // vazIkurvena jagatsarvaM svayaM nAnuvazaH paraiH I nAbhibhUtaH prabhustasmAdapUrva makaradhvajaH // 5 // yo vikramakramAkrAntavaki cakakRtakriyaH / cakrikAbhaJjana) nAmnA cakrikAbhaJjano'asA // 6 // yo vidyAdyadhiSThAna maryAdAvantravedikaH / ratnagarbho yathAkhyAta cAritranaladhirmahAn // 7 // vidhvastaikAntapakSasya syAdvAdanyAyavAdinaH / devasya nRpatuGgasya vardhatAM tasya zAsanam // 8 // 1 M and B 4deg. *M and K HI. kI. * a praNItaH. * K, P and B bhavet. * M and B T. sa * B yAM'yaM. * P vedina:.

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gaNitasArasaGgrahaH gnnitshaastrprshNsaa| laukike vaidike vApi tathA sAmAyike'pi yH| vyApAratatra sarvatra saGkhyAnamupayujyate // 9 // kAmatantre'rthazAstre ca gAndharve nATake'pi vaa| . sUpazAstre tathA vaidye vAstuvidyAdivastuSu // 10 // chando'laGkArakAvyeSu trkvyaakrnnaadissu| . kalAguNeSu sarveSu prastutaM gaNitaM param // 11 // sUryAdigrahacAreSu grahaNe grhsNyutau| triprazne candravattau 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 / yatkizcidvastu tatsarvaM gaNitena vinA na hi // 16 // tIrthakadbhayaH kRtArthebhyaH pUjyebhyo jagadIzvaraiH / teSAM ziSyapraziSyebhyaH prasiddhAdguruparvataH // 17 // jaladheriva ratnAni pASANAdiva kAthanan / zukrmuktAphalAnIva saGkhyAjJAna mahodadhe : // 18 // 14 syAt ; B cApi. * Mand B daNDA". 1K,M and B se 'B ca. IM and R purA. Kand x mahA . R.HT kSipA. K and PT for . *M vasu.

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saMjJAdhikAraH kizciduitya tatsAraM vakSye'haM matizaktitaH / alpa granthamanalpArtha gaNitaM sArasaGgahama // 19 // saMjJAmbhobhirayoM pUrNe parikarmosa vedike| kalAsavarNasaMrUDhaluThatpAThInasaGkaleM // 20 // prakIrNakamahAmAhe vairAzikataraGgiANi / mizrakavyavahArodyatsUktiranAzapiare // 21 // kSetravistIrNapAtAle khAtArUpa sikatAkule / karaNaskandhasambandhacchAyAvelAvirAjite // 22 // gaNakairguNasampUrNastadarthamaNa yo'malAH / gRhyante karaNopAyaissArasaGgahavAridhI // 23 // atha saMjJA / na zakyate'yoM boDe yatsarvasmina saMzayA vinA / AdAvato'sya zAstrasya paribhASAbhidhArayate / / 24 // tatra tAvat kssetrpribhaassaa| jalAnalAdibhinAza yo na yAni sa pudgalaH / paramANuranantastairaNusso'trAdirucyate // 25 // trasareNuratastasmAdrayareNu: ziroruhaH / paramadhyajaghanyAkhyA bhogabhUkarmabhUbhuvAma // 26 // lIkSA tilassa eveha sarSapo'tha' yavo'Ggalam / krameNASTaguNAnyetadvyavahArAGgalaM matam || 27 / / Hand B alpa'. K saMjJAtoyasamA". ME (Probably a scrild's mistake for ttha) ad, maTe. P. Kaod l' . 'Mud Ba". .P and B-sya. P.N.

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gaNitasArasaGgrahaH tatpazcazataM proktaM pramANaM mAnavedibhiH / vartamAnanarANAmaGgulamAtmAGgulaM bhavet // 28 // vyavahArapramANe dve' rAddhAnte laukike viduH / AtmAGgulamiti tredhA tiryakpAdaH SaDaGgulaiH // 29 // . 9 pAdadvayaM vitastissyAttato hasto dvisaGgaNaH / daNDo hastacatuSeNa krozastadvisahasrakam // 30 // yojanaM caturaH krozAnprAhuH kSetravicakSaNAH / vakSyate'taH paraM kAlaparibhASA yathAkramam / / 31 // atha kAlaparibhASA | araNvantarakAle vyatikrAmati yAvati / sa kAlastamayo'saGkhyaissamayairAvalirbhavet // 32 // saGkhyAtAvalirucchrAsaH stokastUcchAsasaptakaH / stokAstapta lavasneSAM sAdhISTAtriMzatA ghaTI // 33 // ghaTI muhUrto'tra muhUrtestriMzatA dinam / pazvanaustrAdinaiH naiH pakSaH pakSau dvau mAsa iSyate // 34 // RturmAsadvayena syAtribhistairayanaM matam / tadduyaM vatsaro vakSye dhAnyamAnamataH param // 35 // atha dhAnyaparibhASA | viddhi SoDazikAstatra catasraH kuDaho' bhavet / kuset caturaH prasthazvatuH prasthAnathAdakam // 36 // caturbhidakaioNo mAnI droNezvaturguNaiH / khArI mAnIcatuSeNa vAryaH paJca pravartikA // 37 // * K and B vAM. 'Kai. M Smye.

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saMjJAdhikAraH . seyaM caturguNA vAhaH kumbhaH paJca pravartikAH / . .. .. ita : paraM suvarNasya paribhASA vibhASyate ' // 38 // atha suvarNaparibhASA / caturmirgaNDakairguJjA gumAH paJca paNo'STa te / dharaNaM dharaNe karSaH palaM karSacatuSTayam // 39 // atha rjtpribhaassaa| dhAnyadvayena gujaikA guJjAyugmena mApakaH / mASaSoDazakenAtra dharaNaM paribhApyate // 40 // taddayaM sArdhakaM karSaH purANAMcaturaH palam / rUpye mAgadhamAnena prAhussakhyAnakovidAH // 41 // atha lohpribhaassaa| kalA nAma catuSpAdAH sapAdApSaTulA yavaH / yavaizcaturmiraMzassyAvAgo'zAnAM catuSTayam // 42 // drANo mAgaSaTrena dInAro'smAdisaGgaNaH / dvau dInArI saneraM syAtmAhulauhe'tra sUrayaH / / 43 // I For the whole of dhAnyaparibhASA, P and Bald what is given below ava sakhed ing and M has it in the original with the variations which are color in teu. AdyA pozikA tatra kura(du)va: prastha AThakaH / proNo mAnI tataH khArI krameNa (maza: * ) catugahatAH / / (sahastraizca tribhiSparizzataizca zrIhibhissamam / yassampUrNo'bhavatso'yaM kulaH paribhASyate // ) pratikAtra tAH patra vAhastasyAvaturguNaH / .. kumbhassapAdavAhassyAt (paJca pravatikA: kumbhaH) svarNasaMjJAya varyate // .ysaterAvyama. *In Balno.

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gaNitasArasaGgrahaH palaiAdazabhissAdhaiH prasthaH palazatadvayama / tulA daza tulA mAra': saGkhyAdakSAH pracakSate // 44 // vastrAbharaNavetrANAM yugaLAnyatra viMzatiH / korTa kAnantaraM bhASye* parikarmANi nAmata : // 45 // ____ atha prikrmnaamaani| AdimaM guNakAro'tra pratyutpanno'pi tdbhvet| dvitIyaM mAgahArAkhyaM tRtIyaM kRtirucyate // 16 // caturtha vargamUlaM hi bhApyate pazcamaM ghanaH / ghanamUlaM tataSSaSThaM saptamaM ca citissmRtam // 47 // . tatsaGkalitamapyuktaM vyutkalitamato'STamam / taJca zeSamiti proktaM bhinnAnyaSTAvamUnyapi // 48 // __ atha dhanarNazUnyaviSayakasAmAnyaniyamAH / tADitaH vena rAziH va so'vikArI hato yutaH / hIno'pi vavadhAdiH khaM yoge khaM yojyarUpakam // 49 // aNayordhanayo_te bhajane ca phalaM dhanama / RNaM dhanarNayostu syAtsvarNayorvivaraM yutau // 50 // RNayodhanayoogo yathAsaGkhyamaNaM dhanam / zodhyaM dhanamRNaM rAzeH RNaM zodhyaM dhanaM bhavet // 51 // dhanaM dhanarNayorvargo mUle svarNe tayoH kramAt / RNaM svarUpato'vargoM yatastasmAnna tatpadaya // 59 // atha sngkhyaasNjnyaaH| 'zazI somazca candrendU bhAleyAMza rajanIkaraH / zvetaM himagu rUpaJca mRgAGkazca kalAdharaH / 13 // vidyAkalAsavarNasya. IN Di. * Stanu 63 to 68 ooour only in M, and are con horo, though atonood her and there, w found in the original. an.and.anhanamination adisa-nine

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saMjJAdhikAraH hi ve DAvubhau yugalayugmaM ca locanaM dvayam / raSTinetrAmbakaM dvandvamakSicakSurnayaM dazau // 54 // haranetraM puraM lokaM trai(tri)ralaM bhuvanatrayam / guNo vahniH ziravI jvalanaH pAvakazca hutAzanaH // 55 // ambudhiSidhirvAdhiH payodhissAgaro gatiH / naladhirbandha caturvedaH kaSAyassalilAkaraH // 56 // iSurbANaM zaraM zastra bhUtamindriyasAyakam / paca vratAni viSayaH karaNIyastantusAyakaH // 17 // atujIvo raso lekhyA dravyA SaTukaM gvarana / kumAravadanaM varNa zilImuravapadAni ca // 18 // zailamadriyaM madho nagAcalamanirgiri : / . azvAzvipannagA dvIpa dhAturyasanamAtRkam // 59 / / aSTI tanurgajaH karma vasu vAraNapuSaram / dviradaM dantI digduritaM nAgAnIkaM karI yathA // 10 // nava nandaM ca randhrazca padArtha lbdhkeshvii| nighiranaM grahANAM ca durganAma ca saGkhyayA // 11 // AkAza gaganaM zUnyamambaraM vaM nabho viyat / anantamantarikSaM ca viSNapAdaM divi. smaret // 62 // matha sthAnanAmAni / ekaM tu prathamasthAnaM dvitIyaM dazasaMkSikam / tRtIyaM zatamityAhuH caturtha tu sahasrakam // 13 // pacamaM dazasAhasraM SaSThaM syAlakSameva c| saptamaM dazalakSaM tu maSTamaM koThirucyate // 15 //

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navamaM dazakotyastu dazamaM zatakoThayaH / arbudaM rudrasaMyuktaM nyarbudaM dvAdazaM bhavet / / 65 / ravarva trayodazasthAnaM mahAravarvaM caturdazam / . padmaM paJcadazaM caiva mahApamaM tu SoDazam // 16 // ' kSoNI saptadazaM caiva mahAkSoNI dazASTakam / zaGkha navadazaM sthAnaM mahAzajhaM tu vizakam // 6 // kSityaikaviMzatisthAnaM mahAkSityA dviviMzakam / triviMzakamatha kSobhaM mahAkSobhaM caturnayam // 8 // matha gaNakaguNanirUpaNam / laghakaraNohApohAnAlasyagrahaNadhAraNopAyaiH / vyaktikarAGkaviziSTeNako'STAbhirguNai yaH // 69 // iti saMjJA samAsena bhASitA munipuGgavaiH / vistareNAgamAdvedyaM vaktavyaM yaditaH param // 70 // iti sArasahe gaNitazAstra mahAvIrAcAryasya kRtI saMjJAdhikArasamAptaH //

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prathamaH parikarmavyavahAraH. itaH paraM parikarmAbhidhAnaM prathamavyavahAramudAhariSyAmaH / pratyutpannaH / tatra' prathame pratyutpannaparikarmaNi karaNasUtraM yathA 'guNayadguNena guNyaM kavATasandhikrameNa saMsthApya / rAzyarghakhaNDatatsthairanulomavilomamArgAbhyAm // 1 // atroddezakaH / dattAnyekaikasmai jinabhavanA yAmbujAni tAnyaSTau / vasatInAM caturuttaracatvAriMzacchatA ya kati // 2 // nava padmarAgamaNayassamarcitA ekajinagRhe dRSTAH / sASTAzItidvizatImitavasatiSu te kiyantassyuH // 3 // catvAriMzazcaikonazatAdhikapupyarAgamaNayA aaH / ekasmina jinabhavane sanavazane bruhi kati maNayaH // 4 // padmAni saptaviMzatire kasmina jinagRhe pradattAni / mASTAnavatisahastre "sanavazate nAni kati kathaya // 5 // "ekaikasyAM vasatAvaSTottarazatasuvarNapadmAni / ekASTacatussaptakanavaSaTuvASTakAnAM kim // 6 // KIca. Kand B vinyasyobhI gazI. Kand BsANayeta. *B yahi. nasyA. 'zatasya kati bhavanAnAm . Nand B catvAriMzazpakA zatAdhikA. M cchA.. * 4 te kiyantassyu:. M ekaikajinAlayAya itAni. prayatanavazatagRhANAM kim / Thinatana is found only in M and B.

Page #48
gaNitasArasaGgrahaH zazivasuravarajalanidhinavapadArthabhayanayasamUhamAsthApya / himakaraviSanidhigatibhirguNite kiM' rAziparimANam // 7 // himagupayonidhigatizazivahnivratanicayamatra saMsthApya / saikAzItyA tvaM me guNayitvAcakSva tatsaGkhyAm // 8 // anivasuravarabhayendriyazazalAJchanarAzimatra saMsthApya' / randhairguNayitvA me kathaya marave rAziparimANam // 9 // nandAyUtuzaracatustridvandvaikaM sthApya matra navaguNitam / AcAryamahAvIraiH kathitaM narapAlakaNThikAbharaNam // 10 // SaTvikaM pazcaSaTuzca sapta cAdau pratiSThitam / trayastriMzatsaGgaNitaM kaNThAbharaNamAdi sht|| 11 // hutavahagatizazimunibhirvasunayagaticandramatra saMsthApya / zailena tu guNayitvA kathayedaM ratnakaNThikAbharaNam // 12 // analAbdhihimagumunizaraduritAkSipayodhisomamAsthApya / zailena tu guNayitvA kathaya tvaM rAjakaNThikAbharaNam // 13 // giriguNadivigiriguNadivigiriguNanikaraM tathaiva guNaguNitam / punarevaM guNaguNitama ekAdinavottaraM viddhi // 14 // sapta zUnyaM 'dvayaM inheM pacaikazca pratiSThitam / vayaHsaptatisaGgaNyaM "kaNThAbharaNamAdizet // 15 // jalanidhipayodhizazadharanayanadravyAkSinikaramAsthApya / guNite tu catuSSaSTayA kA saGkhyA gaNitavihi // 16 // In and B kintatya. * // pyam. aho. * me zIghram. .B PURI. Stanwas from 10 to 15 are found only in N and B. All the Mes. road sthApya tatra. 'B ze. BnayaM. lo All the Msg. give the metrimls erroneous reading kaNThAbharaNaM vinirdizeta /

Page #49
parikarmavyavahAraH zazAvenduvaikenduzUnyaikarUpaM nidhAya krameNAtra rAzipramANam / himAMzvaprarandhaiH prasantADite'smin bhavetkaNThikA rAjaputrasya yogyA // 17 // * iti parikarmavidhau prathamaH pratyutpannaH samAptaH // bhaaghaarH| dvitIye bhAgahAraparikarmaNi karaNasUtraM yathA 'vinyasya bhAjyamAnaM tasyAdhasthena bhAgahAreNa / sahazApavatavidhinA bhAgaM kRtvA phalaM pravadet // 18 // atha vA pratilomapathena majedAjyamadhassthana bhAgahAreNa / sahazApavartanavidhiryadyasti vidhAya tamapi tayoH // 19 // atrodezakaH / dInArASTasahasraM vAnavatiyutaM zatena saMyuktam / caturuttaraSaSTinarairbhaktaM ko zo nurekasya // 20 // rUpAgrasaptaviMzatizatAni kanakAni yatra bhAjyante / saptatriMzanpuruSairekasyAMzaM mamAcakSva // 21 // dInAradazasahasraM trizatayutaM saptadhargasammizram / navasaptatyA purubhaktaM kiM labdhamekasya // 22 // 'ayutaM catvAriMzazcatussahastraikazatayutaM henAm / navasaptativasatInA dattaM vittaM kimekasyAH // 23 // K kozI naMgakaraNa. | This stanon is not found in P This stanza is not found in P. . Band K hemam.

Page #50
gaNitasArasAhaH 'saptadazatrizatayutAnyekAtriMzatsahasrajamyUni / bhaktAni navatriMzannarairvadaikasya bhAgaM tvam // 24 // 'dhyadhikadazatrizatayutAnyekatriMzatsahasrajambUni / saikAzItizatena prahRtAni naraidaikAMzam // 25 // * tridazasahasrI saikA ssssttidvishtiishsrssttyutaa| rabAnAM navapuMsAM dattaikanaro'tra kiM labhate // 26 // ekAdiSaDantAni krameNa hInAni hATakAni sarave / vidhujaladhibandhasaGkhyairnarairhatAnyekabhAgaH kaH // 27 // vyazItimizrANi catuzzatAni catussahasrananagAnvitAni / ratnAni dattAni jinAlayAnAM trayodazAnAM kathayaikabhAgam // 28 // iti parikarmavidhau dvitIyo bhAgahAraH samAptaH // vargaH / tRtIye vargaparikarmaNi karaNasUtraM yathA dvisamavadho ghAto vA kheSTonayutadvayasya sessttkRtiH| ekAdihicayecchAgacchayutirvA bhavedvargaH // 29 // . M rends the problem contained in this stanzu thus:-- trizatayutaikatriMzatsahasrayuktA dazAdhikA: sakSa / bhaktAzcatvAriMzatpuruSarekonaistatra dInAram // * This stanza is found only in M. " ekadvitricatuHpaJcaSaTUhInA: krameNa sambhaktAH / saikacatuHzatasaMyutacatvAriMzagjinAlayAnA kim //

Page #51
parikarmavyavahAraH 18 dvisthAnaprabhRtInAM rAzInAM sarvavargasaMyogaH / teSAM kramaghAtena dviguNena vimizrito vargaH // 30 // kRtvAntyakati hanyAccheSapadaidviguNamantyamutsArya / zeSAnutsAyaivaM karaNIyo vidhirayaM varge // 31 // atroddezakaH / ekAdinavAntAnAM paJcadazAnAM dvisaGgaNASTAnAma / vratayugayozca rasAgnyozzaranagayorvargamAcakSva // 32 // sASTAviMzatrizatI catussahaskaSaSTiSaTachatikA / dvizatI SaTpazvAzanmizrA vargIkRtA kiM syAt // 33 // lekhyAguNeSubANadravyANAM zaragatitrisUryANAma / guNaratnAmipurANAM varga bhaNa gaNaka yadi vetsi-|| 15 // saptAzItitrizatasahita padasahasraM punazca pazcatriMzacchatasamadhikaM saptanighnaM sahasrama / hAviMzatyA yutadazazataM 'vargitaM tatrayANAM brUhi tvaM meM gaNaka guNavansaGgaNayya pramANam // 35 // iti parikarmavidhau tRtIyo vargassamAptaH / / vargamUlam / caturthe vargamUlaparikarmaNi karaNasUtraM yathA antyaujAdapahRtakatimUlena dviguNitena yugmahato / labdhakRtistyAjyone dviguNadalaM vargamUlaphalam // 3 // IP, Kand B rAzirenaskRtInAma.

Page #52
gaNitasArasAhaH atroddezakaH / 'ekAdinavAntAnAM vargagatAnAM vadAzu me mUlam / atuviSayalocanAnAM dravyamahIdhendriyANAJca / / 37 / / ekAgraSaSTisamadhikapaJcazatopetaSaTsahasrANAm / SaDargapaJcapaJcakaSaNNAmapi mulamAkalaya // 38 // ' dravyapadArthanayAcalalekhyAlabadhyabdhi nidhinayAbdhInAm / zazinetrendriyayuganayajIvAnAzcApi kiM mUlam // 39 // candrAbdhigatikaSAyadravyartahutAzanatarAzInAm / vidhulekhyendriyahimakaramunigirizazinAM ca mUlaM kim // 40 // dvAdazazatasya mUlaM SaNNavatiyatasya kathaya sazcintya / zataSaTkasyApi sarave pazcakavargaNa yuktasya / / 41 // bhAbhakarmAmbarazaGkarANAM somAkSivaizvAnarabhAskarANAm / candra bANAbdhigatihipAnAmAcakSva mUlaM gaNakAgraNIstvam // 42 // iti parikarmavidhau caturtha vargamUlaM samAptam / / ghana: / pazcame ghanaparikarmaNi karaNasUtraM yathA --- trisamAhatirghanassyAdiSTonayutAnyarAzighAto vA / alpaguNitaSTakRtyA kalito bRndena ceSTasya // 43 // iSTAdihigaNeSTapracayeSTapadAnvayo'tha vessttkRtiH| ThayekeSTahataikAdidvicayeSTapadai yayuktA vA // 44 // 'ekAdicayeSTapade pUrva rAzi pareNa saGgaNayet / guNitasamAsastriguNazcarameNa yuto ghano bhavati // 45 // Pand M vargagatAno zIghraM rUpAdinavAvasAnarAzInAm / mUlaM kathaya sakhe tvaM. 'M naba. This stanza is not found in P.

Page #53
parikarmavyavahAraH antyAnyasthAnakRtiH parasparasthAnasaGgaNA trihatA / punarevaM tadyogassarvapadadhanAnvito bRndama / / 46 / / 'antyasya ghanaH kRtirapi sA trihatotsArya zeSaguNitA vA / zeSakRtitryantyahatA sthApyatsAyaivamatra vidhiH || 47 atroddezakaH / 'ekAdinavAntAnAM paJcadazAnAM zarekSaNasyApi / rasavahnayogirinagayoH kathaya ghanaM dravyadhyozca // 48 // pikaragaganendUnAM nayagirizazinAM varenduvANAnAm / vada municandra vRndaM caturudaviguNazazinAn // 49 // rAzinIkRto'yaM zayaM mizrita trayodazabhiH / taddiguNo'smAtriguNazcaturguNaH paJcaguNitazca // 50 // zatamaSTaSaSTayuktaM dRSTamabhISTaM ghane viziSTatamaiH / ekAdibhiraSTAnyairguNitaM vada taddhanaM zIghra // 11 // banyAmbartRgaganendriya kezavAnAM saGkhyAH krameNa vinidhAya ghana gRhItvA / AcakSva labdhamadhunA karaNAnuyogagambharisArataramAgarapAradazvana / / 52 / / iti parikarmavidha pazcamo ghanassamAptaH || PM rapi. * M gAM vA. This stanza is omitted in M. Che following stanza is found as a C in P, Kand B; though not quite explioit, it mentions two of the processes thoro desoribed : raise stadvargatriguNitAM hatazeSaH / utsArya zeSakRtiratha niSTA triguNA ghanastathA vA // 15 Instead of stanzas 48 and 49, M reads ekAdinavAntAnAM rudrANAM himakAMndUnAm 1 vada municandrayatInAM vRndaM caturudadhiguNazazinAm //

Page #54
16 gaNitasArasaGgrahaH ghanamUlam / SaSThe ghanamUlaparikarmANi karaNasUtraM yathA antyaghanAda pahRtaghanamUlakRtitrihatibhAjite bhAjye / prAhitAptasya kRtizzodhyA zodhye ghane'tha dhanam // 51 // 'ghanamekaM dve aghane ghanapadakRtyA bhajetriguNayAghanataH / pUrvatriguNApta kRtistyAjyAptaghanazca pUrvavallabdhapadaiH // 54 // atroddezaH / ekAdinavAntAnAM ghanAtmanAM ratnazazinavAbdhInAm / 'nagarasava suvarta gajakSapAkarANAzca mUlaM kim // 95 // gatinayamadazivizazinAM muniguNaravatvIkSanava' kharAnInAm / 'vasu 'vayugavAdrigatikaricandrartanAM gRhANa padama // 56 // catuH payodhyazirAkSidRSTi ye bhavyomabhayekSaNasya / dASTakarmAbdhiraghAtibhAva dvivahniratnartunagasya mUlam // 57 // dravyAzvazaila duritakhavahnayadribhayasya vadata ghanamUlam / navacandra himagu munizazilabdhyambarakharayugasyApi // 98 // 'gatigaja viSayeSuvidhutvarAdrikaragatiyugasya maNa mUlam / lekhyAzvanaganavAcalapurakharanayajIvacandramasAm // 59 // gatirakharaduritebhAmbhodhitArkSyadhvajAkSadvikRti navapadArthadra vyavahInducandra / jaladharapatharandhravaSTakAnAM ghanAnAM gaNaka gaNitadakSAcakSva mUlaM parIkSya // 60 // iti parikarmavidhau SaSThaM ghanamUlaM samAptam // M giri. M rasA. This stana is not found in M This stanza is not found in M. M vidhupurakharasvarartujvalanadharArNA.

Page #55
17 parikarmavyavahAraH saGkalitam / saptame saGkalitaparikarmaNi karaNasUtraM yathA -- rUpeNono gaccho dalIkRtaH pracayatADito mizraH / prabhaveNa padAbhyastassaGkakritaM bhavati sarveSAm // 1 // prakArAntareNa dhanAnayanasUtramekavihIno gacchaH pracayaguNo dviguNitAdisaMyuktaH / gacchAbhyasto dvihataH prabhavetsarvatra saGkalitam // 12 // AdhuttarasarvadhanAnayanasUtram--- padahatamuvamAdidhanaM vyekapadArdhanacayaguNo gacchaH / uttaradhanaM 'tayoogo dhanamUnottaraM mukhe'ntyadhane // 13 // antyadhanamadhyadhanasarvadhanAnayanasUtram-.. cayaguNitaikonapadaM sAdyantyadhanaM tadAdiyogArtham / madhyadhanaM tatpadavadhamuddiSTaM sarvasaGkalitam // 64 // ____ atroddezakaH / ekAdidazAntAdyAstAvatpracavAssamarcayanti dhanam / vaNijo daza daza gacchAsteSAM saGkalitamAkalaya // 15 // dvimuravatricayairmaNimiH prAnarca zrAvakottamaH kazcit / paJcavasatIramISAM kA saGkhyA dhUhi gaNitajJa // 6 // Adistrayazcayo'STau dvAdaza gacchastrayo'pi rUpeNa / A saptakAtmavRddhAssarveSAM gaNaka bhaNa gaNitam // 17 // vikRtirmuravaM cayo'STI nagarasahasre sarcitaM gaNitam / gaNitAbdhisamuttaraNe bAhubalin tvaM samAcakSva // 6 // / tadnA saika(va !)padAptA yutiH prabhavaH / This stanu is omitted in K. .. lI. 12499.

Page #56
gaNitasArasaGgrahaH gacchAnayanasUtramaSTottaraguNarAzerdviguNAdyuttaravizeSakRtisahitAt / mUlaM cayayutamardhitamAghUnaM cayahataM gacchaH // 19 // prakArAntareNa gacchAnayanasUtram aSTottaraguNarAzerTiguNAdyuttaravizeSakRtisahitAt / mUlaM kSepapadonaM dalitaM cayabhAjitaM gacchaH // 70 // atroddezakaH // AdiauM pracayo'STau hau rUpeNA yAtkramAbadau / vAko rasAdrinetraM ravenduharA vittamatra ko gacchaH / / 71 // AdiH patra yo'STau guNalAnidhanamatra ko gacchaH / SaT prabhavazva cayo'STau svadvicanustvaM padaM kiM syAt // 72 // sora uttarAdyAnayanasUtram-- AdidhanonaM gaNitaM padonapadakRtidalena sambhajitam / pracayastadanahInaM gaNitaM padabhAjitaM prabhavaH // 73 // AdhuttarAnayanasUtramprabhavo gacchApa.(dhanaM vigataikapadArdhaguNitacayahInam / padahAdhanamAyanaM nirakapadadalahata pracayaH / / 74 // prakArAntareNottarAdyAnayanasUtradvayam -- dvihataM saGkalitadhanaM gacchahataM dviguNitAdinA rahitam / vigataikapadavibhaktaM pracayassyAditi vijAnIhi / / 75 // dviguNitasaGkalitadhanaM gacchahataM rUparahitagacchena / tADitacayena rahitaM. dvayena sambhAjitaM prabhavaH // 76 //

Page #57
parikarmavyavahAraH 19 atroddezakaH / nava vadanaM tattvapadaM bhAvAdhikazanadhanaM kiyAnpracayaH / pazca cayo'STa padaM SaTpanAzacchatadhanaM muravaM kathaya // 77 // kheSTAyuttaragacchAnayanasUtram - saGkalite sveSTahate hAro gaccho'tra labdha iSTone / UnitamAdizzaMSe vyekapadArthoddhRte pracayaH / / 78 // atroddezakaH / catvAriMzansahinA pazatI gaNitamatra sandRSTama / gacchapracayaprabhavAna gaNitajJaziromaNe kathaya // 79 // AyuttaragacchasarvamizradhanavilepaNe sUtratrayama ---. uttaradhanena rahitaM gacchenaikana saMyatena hatam / mizradhanaM prabhavarasyAditi gaNakaziromaNe vinhi / / 80 // AdidhanonaM mizraM rUponapadArthagaNitagacchena / saikena haMta pracayo gacchavidhAnAtpadaM murave saike // 81 // mizrAdapanItepTI muravagacchau pracamizravidhilabdhaH / yo rAzissa cayasyAtkaraNamidaM sarvasayoge // 82 // bhatroddezakaH / dvitrikapaNadazAgrA catvAriMzanmugvAdimizradhanam / tatra pramavaM pracayaM gacchaM sarva ca me behi // 83 // 11 vigaNayya sakhe mamAcakSna. padonapadakRtidalana sakena / ma. pratyo'tra padaM gacchavidhAnAnmukhe sake /

Page #58
gaNitasArasaGgrahaH ___ dRSThadhanAyuttarato dviguNatriguNavibhAgavibhAgAdISTadhanAdyuttarAnayana sUtram iSTavibhakteSTadhanaM dviSThaM tatpracayatADitaM pracayaH / tatprabhavaguNaM prabhavo 'guNabhAgasyeSTavittasya // 84 // atrodeshkH| samagacchazcatvAraHSaSTirmukhamuttaraM tato dviguNam / tayAdihatavibhaktasveSTasyAyuttare brUhi // 85 // iSTagacchayorvyastAdyuttarasamadhanadviguNatriguNadvibhAgatribhAgAdidhanAnayanasUtram - vyekAtmahato gacchasveSTano dviguNitAnyapadahInaH / muravamAtmonAnyakRtidikeSTapadaghAtavarjitA pracayaH // 86 // ___ atroddeshkH| pazcASTagacchapuMso yastaprabhavottare samAnadhanam / dvitriguNAdidhanaM vA brUhi tvaM gaNaka vigaNayya // 87 // dvAdazaSoDazapadayorvyastaprabhavottare samAnadhanam / yAdiguNabhAgadhanamapi kathaya tvaM gaNitazAstrajJa // 8 // asamAnottarasamagacchasamadhanasyAdyuttarAnayanasUtram-- adhikacayasyaikAdizcAdhikacayazeSacayavizeSo guNitaH / vigataikapadArdhena sarUpazca muravAni mitra zeSacayAnAm // 89 // atrodezakaH / ekAdiSaDantacayAnAmekatritayapazcasaptacayAnAm / navanavagacchAnAM samavittAnAM cAzu vada mukhAni sakhe // 90 // 1 x guNabhAgAyuttarecchAyAH. ' guNa. ' gaNakamukhatilaka /

Page #59
parikarmavyavahAraH za visadRzAdisaDazagacchasamadhanAnAmuttarAnayanamUtram adhikamukhasyaikacayazcAdhikamakhazeSamukhavizeSo bhaktaH / vigataikapadArdhana sarUpazca cA bhavanti zeSamugvAnAma // 11 // atroddeshkH| ekatripazcasaptanavaikAdazavadanapatrapatripadAnAm / samavittAnAM kathayottarANi gaNitAbdhipAradazvana gaNaka // 92 / / atha guNadhanaguNasaGkalitadhanayostutram ---- padamitaguNahatiguNitaprabhavasmyAdguNadhanaM nAdyanam / ekonaguNavibhaktaM guNasaGkalitaM vijAnIyAt // 93 // guNasaGkalite anyadapi sutram -- samadalaviSamakharUpo guNagaNito varganADino gaccha / rUponaH prabhavano vyekottarabhAjitasmA ma // 94 / / guNasaGkalitAntyadhanAnayane matsaliAdhanAnAne va sUtram -- guNasaGkalitAntyadhanaM vigatakapadagya gaNadhanaM bhavati / tadguNaguNaM mukhonaM vyakottarabhAjita mAma // 95 // guNadhanasyodAharaNam / varNadvayaM gRhItvA triguNadhanaM pratipura samA yati / yaH puruSo'pTanagaryA tasya kiryAhattamAMcakSva // 96 // guNadhanasyAyuttarAnayanasUtram - guNadhanamAdivibhaktaM yatpadamitavaghasamaM sa eva cayaH / gacchapramaguNaghAtAhataM guNitaM bhavetprabhavaH // 97 // IN samayati.

Page #60
9.2 gaNitasArasaGgrahaH guNadhanastha gacchAnayanasUtram -- mukhabhakte guNAvate yathA niragraM tathA guNena hte| yAvayo'tra zalAkAstAvAna gaccho guNadhanasya // 9 // guNasaGkalitodAharaNam / / dInApanakAdidviguNaM dhanamarjayannaraH kazcit / prAvikSa TanagarIH kati jAtAstasya dInArAH // 99 // saptamukhAtriguNa vayatrivargagacchasya kiM dhanaM vnnijH| , tripakSakapa-zaprabhavaguNottara parasyApi // 10 // guNasaGkAlinottarAdhAnayanasUtram - asakRyekaM mukhahatavittaM yenodbhutaM bhavetsa cayaH / vyekaguNaguNinagaNitaM nokapada-mAtraguNavayAtaM prabhavaH / / 101 // atrodezakaH / trimurava (gaccha yAsAGkAmbara galanidhiyane ki pracayaH / SaGgaNavApa pAvarazazi himagutrittabhatra mukhaM kim / / 102 // * guNasaGkalinagacchAnavanasUtrama - ekonaguNAbhyarata prabhavahanaM rUpasaMyu vittam / yAvatkRtvo aktaM guNena tadvArasammitirgacchaH // 103 // atraaNddeshkH| triprabhavaM paTuguNaM sAraM spttyupettptshtii| saptAmA brUhi sarave kiyatpadaM gaNaka guNanipuNa // 10 // pacAdidviguNottare zaragiridhyekapramANe dhanesaptAdi triguNe nageaduritastamberamartuprame /

Page #61
parikarmavyavahAraH tryAsye pazcaguNAdhike hutavahopendrAkSavahidvipa zvetAzudviradebhakarmakaradRDpAne'pi gaccha: kivAna // 105 // iti parikarmavidhau saptamaM saGkalitaM samAptama / / . vyutklitm| aSTame vyutkalitaparikaNi kagasUtraM yathA ... sapadeSTaM veSTamapi vyekaM dalinaM cAhataM samatvam / zeSeSTagacchaguNitaM vyutkalitaM khaMTa vittaM ca // 106 // prakArAntareNa vyutkalitadhanasveSTadhanAnayana sUtraH ... gacchasahiteSTamiSTa cekonaM cayahata vihAdiyutam / / zeSeSTapadArdhaguNaM vyatkAla svaSTavittApi // 107 // cagaguNabhavavyutkalinadhanAnayane vyutkalitadhanasya zeSeSTagacchAna yane ca sUtrama iSTadhanonaM gaNita vyavakalitaM ca yabhavaM gaNotthaM ca / sarveSTagacchazeSe zeSapadaM jAyate |my // 108 // zeSagacchasyAdyAnayanasUtram pracayaguNineSTagacchasmAdiH prabhavaH padasya, zeSasya / prAktana eva cayassyAdgacchasvaSTasya mAveva :: 109 // guNavyutkalitazeSagacchasthAdyAnayanasUtram--- guNaguNite'pi cayAdI tathaiva bhado'yamatrazepapade / iSTapadamitiguNAhatiguNitaprabhavo bhavedakam / / 111 // IM gaNitaM.

Page #62
gaNitasArasagrahaH atroddezakaH / dvimugyastricayo gacchazcaturdaza khepsitaM padaM sapta / aSTanavaSaTrapana ca kiMvyutkAlataM samAkalaya // 111 // SaDAdiraSTau pracayo'tra paTutiH padaM daza dvAdaza ssoddshepsitm|| mukhAdiranyasya tu paJcapaJcakaM zatadvayaM brUhi zataM vyayaH kiyAn // 112 // SaDDanamAno gacchaH pracayo'STau dviguNasaptakaM vakram / saptatriMzatkheSTaM padaM samAvi phalamubhayama / / 113 / / aSTakRtirAdiruttaramUnaM catvAri SoDazAtra padam / iSTAni tattvakezavarudrArkapadAni ki zeSam // 114 // guNavyutkalinasyodAharaNamcaturAdidviguNAtmakottarayuto gaccha caturNA kRtidaza vAJchApadamasindhuragiridravyendriyAmbhodhayaH / kathaya vyatkalitaM phalaM sakalasanjAgrima 'vyAptavAna karaNaskandhavanAntaraM gaNitavinmattebhavikrIDitam // 115 // iti parikarmavidhAvaSTamaM vyutkalitaM samAptam / / iti sArasanAhe gaNitazAstre mahAvIrAcAryasya kRtau parikarmanAmA prathamo vyavahAraH samAptaH // 18.

Page #63
atha dvitIyaH kalAsavarNavyavahAraH / 'trilokarAjendrakirIThakoTiprabhAbhirAlIdapadAravindam / nirmUlamunmUlitakarmavRkSaM jinendracandraM praNamAmi bhaktyA // 1 // itaH paraM kalAsavarNaM dvitIyavyavahAramRdAhariSyAmaH / / minapratyutpannaH / tatra bhinna pratyutpanne karaNasUtraM yathA -- guNayedaMzAnazaihArAna hArairghaTeta yadi teSAma / va cApavartanavidhividhAya taM bhinnagaNakAre // 2 // atroddezakaH / zuNThyAH palena labhate caturnaziM paNasya yaH puruSaH / kimasau brUhi save tvaM trigaNena palASTabhAgena // 3 // maricasya palasyArghaH paNasya saptASTamAMzako yatra / tatra mavalki mUlyaM palapaTapaJcAMzakasya vada // 4 // kazcitpaNena labhate tripaJcabhAgaM palasya pippalyAH / navabhiH paDhibhaktaiH kiM gaNakAcakSva guNayitvA // 5 // krINAti paNena vaNigrajIrakapalanavadazAMzakaM yatra / tatra paNaiH pazcAH kathaya tvaM ki samagramate // 6 // yAdayo dvitayavRddhayo'zakArUyAdayo dvayacayA hagaH punaH / te dvaye dazapadAH kiyatphalaM bRhi tatra. guNane dvayoIyoH // 7 // iti bhinngunnkaarH| . mI. This stanza inomitted in P.

Page #64
26 gaNitasArasAhaH bhinnbhaaghaarH| bhinna bhAgahAre karaNasUtraM yathA - aMzIlatyacchedaM pramANarAzestataH kriyA guNavat / pramitaphale'nyaharane vicchidi vA sakalavacca bhAgahatau // / atroddeshkH| hiGgoH palArdhamaulyaM paNatripAdAMzako avedyatra / tatrArdhe vikrINana palamekaM kiM naro labhate // 9 // agaroH palASTamena triguNena paNasya viMzativyaMzAn / upalabhate yatra pumAnekena palena kiM tatra / / 10 / / paNapaJcamaizcatubhirnaravasya palasaptamo yazItigaNaH / saMprApyo yatra syAdekena paNena kiM tatra / / 11 // dhyAdirUpaparivaddhiyujoDazA vAvadaSTapadamekavihInAH / hArakAsata iha dvitayAdyaiH kiM phalaM vada pareSu hateSu // 12 // iti bhinnabhAgahAraH / bhinnavargavargamUlaghanaghanamUlAni // 'bhinnavargavargamUlaghanaghanamUleSu karaNasUtraM yathA kRtvAcchedAMzakayoH kRtikRtimUle ghanaM ca ghanamUlam / tacchedairaMzahatI vargAdiphalaM bhavezine // 13 // 1 4 bhinnavargabhimavargamUlabhimaghanatanmUleSu.

Page #65
kalAsavarNavyavahAraH. atroddezakaH / paJcakasaptanavAnAM dalitAnAM kathaya gaNaka varga tvm| SoDazaviMzatizatakadvizatAnAM ca tribhaktAnAm // 14 // trikAdirUpadvayavRddho'zA dvikAdirUpottarakA harAzca / padaM mataM dvAdaza vargameSAM vadAza me tvaM gaNakAgragaNya // 15 // pAdanavAMzakapoDazabhAgAnAM paJcaviMzatitamasya / SatriMzadAgasya ca kRtimUlaM gaNaka bhaNa zIghram / / 16 // bhinne varga rAzayo varginA ye. teSAM mUlaM saptazatyAzca kiM syAt / tryaSTonAyAH paJcavarNoddhatAyA behi vaM me vargamUlaM pravINa // 17 // ardhatribhAgapAdAH paJcAMzakaSaSThasaptamASTAMzAH / dRSTA navamazcaiSAM pRthaka pRthagahi gaNaka ghanam // 18 // tritayAdicatazcayako zagaNo himukhadvicayo'tra harapracayaH / .. dazakaM padamAzu tadIyaghanaM kathaya priya sUkSmamate gaNite // 19 // zatakasya paJcaviMzasyASTavibhaktasya kathaya ghanamUlam / 'navayutasaptazatAnAM vizAnAmaSTabhaktAnAm // 20 // 14 saptazatasyApi sarva vyekAnazikASTakAptasya //

Page #66
gaNitasArasahaH bhinnaghane paridRSTaghanAnAM mUlamudagramate vada mitra / tryUnazatadvayayugdvisahasyAzrApi navaprahatatrihatAyAH // 21 // iti bhinnavargavargamUlaghanaghanamUlAni / / - bhinnasaGkalitam / bhinnasaGkalite karaNasUtraM yathA ----- padamiSTaM pracayahataM dviguNaprabhavAnvitaM cayenonam / gacchArthenAbhyastaM bhavani phalaM bhinnasaGkalite // 22 // __ atroddezakaH / dvidhyazaSSahabhAgastricaraNabhAgo muravaM cayo gacchaH / dvau pacamau tripAdo dvivyaMzo'nyasya kathaya ki vittam // 23 // AdiH pracayo gacchastripaJcamaH paJcamastripAdAMzaH / sarvAMzaharau vRddhau dvitribhirA saptakAJca kA citiH // 24 / / iSTagacchasyAdyuttaravargarUpaghanarUpadhanAnayanasUtram -- padamiSTamekamAdivya'keSTadalodbhutaM muravAnapadam / pracayo vitta teSAM vargo gacchAhataM vRndam // 25 // atroddeshkH| 'padamiSTaM dvivyaMzo rUpeNAMzo harazca saMvRddhaH / yAvaddazapadameSAM vada muvacayavargavRndAni // 26 // This stanza is not fourd in N.

Page #67
20 kalAsavarNavyavahAraH. iSTaghanadhanAdyuttaragacchAnayanasUtram ---- iSTacaturthaH prabhavaH prabhavAtpracayo bhaveddisaGgaNitaH / pracayazcaturabhyastA macchasteSAM yutivRndam // 27 // atroddezakaH / dvimugvaikacayA aMzAstriprabhavakottaga haga ubhaye / pacapadA vada teSAM ghanadhanamuravacayapadAni savaM // 28 // 'dRSTayanAdyuttarato dvigaNatrigarNAdrabhAgatribhAgAdISTadhanAdyattarAnayana sUtrama - dRSTavibhakteSTadhanaM dviSThaM tatpracayatADitaM pracayaH / tprabhavaguNaM prabhavo guNabhAgasyeSTavittasya // 29 // _atroddezakaH / prabhavarUpI rUpaM pravayaH paJcASTamasamAnapadam / icchAdhanamapi nAvatkathaya saba ko mukhapracayAM // 30 // pracayAdAdigiNastrayAMdazApTAdazaM padaM vessttm|| vittaM tu saptapaSTiH pacanabhaktA vadAdicayau // 31 // 'mukhamakaM dvivyaMzaH pracanA gaccharasamazcaturnavamaH / / dhanamiSTaM dvAviMzatirakAzItyA vadAdicayau // 32 // 'gacchAnayanasUtram ---- dviguNacayaguNitavittAduttaradalamakhAvazeSakatirmAhatAt / malaM pracayArdhayutaM prabhavAnaM cayahataM gacchaH // 33 // '5 guNabhAgAdyunarAnayanasUtram / 5 // pracagana. : M guNabhAgAguttaracchAyA:. This stanza takes the place of stanza No.31 in Maul is viitted in B. Insterd of the following two stanzar Mod. aAttagguNagazAnyAdinA i. dhanagaccha AnaMtavyaH and tepent stanza So. 70 give m... parikarmavyavahAra,

Page #68
gaNitasArasaGgrahaH prakArAntareNa tadevAha - dviguNacayaguNitavittAduttaradalamukhavizeSakRtisahitAt / mUlaM kSepapadonaM pracayena hRtaM ca gacchassyAt // 34 // atroddezakaH / dvipaJcAMzo vakaM triguNacaraNassyAdiha cayaH paDaMzastaptannastrikativihato vitsamuditam / * cayaH pazcASTAMzaH punarapi mukhaM vyaSTamamiti tricatvAriMzAssvaM priya vada padaM zIghramanayoH // 35 // AdhuttarAnayanasUtram -- 'gacchAptagaNitamAdivigataikapadArdhaguNitacayahInam / padahanadhanamAna nireka padadalahataM pracayaH // 36 // atroddezakaH / tricaturthacatuHpaJcamacayagacche kheSuzazihataikatriMzad- / vitte vyaMzacatuHpaJcamamukhagacche ca vada mukhaM pracayaM ca // 37 // iSTagacchayorvyastAdyuttarasamadhanadviguNatriguNavibhAgatribhAgadhanAnayanasUtram - vyekAtmahato gacchassveSTano dviguNitAnyapadahInaH / mukhamAtmonAnyatirdikeSTapadaghAtavarjitA pracayaH // 38 // atroddezakaH / ekAdiguNavibhAgastvaM vyastAdyattare hi vada mitra / dvivyazonaikAdazapazcAMzakamizranavapadayoH // 39 // - - IK and B prabhavo gacchagaptaznama,

Page #69
81 kalAsavarNavyavahAraH. guNadhanaguNasaGkalita dhanayoH sUtrampadamitaguNahatiguNitaprabhavaH syAdguNadhanaM tadAdhUnam / ekonaguNavibhaktaM guNasaGkalita vijAnIyAt // 40 // * guNasaGkalitAntyadhanAnayane tatsakalitAnayane ca sUtram-- guNasajhalitAntyadhanaM vigataikapadasya guNadhanaM bhavati / tadguNaguNaM muravonaM vyakottarabhAjitaM sAram // 41 / / atrodezakaH / prabhavo'STamazcaturthaH pracaSaH paJca padamatra guNaguNitam / guNasaGkalitaM nasyAnyadhanaM cAcakSva meM zIghram // 42 // 'guNadhanasakalitadhanayorAgRttarapadAnyapi pUrvoktasUtrairAnayet / samAneSTottaragaccha saGkalinaguNasaGkalitamamadhanasyAdyAnayanasUtram-- mukhamekaM cayagacchAviSTau mukhavittarahitaguNacityA / hatacayadhanamAdiguNaM mukhaM bhavedicitidhanasAmye // 43 // ___ atroddezakaH / bhAvavAdhibhuvanAni padAnyambhodhipazcamunayastrihatAsta / uttarANi vadanAni kati syu. yugmasaGkalitavittasameSu // 44 // iti bhinnasaGkalitaM samAptam // bhinnavyutkalitam / bhinnavyutkaline karaNasUtraM yathA - gacchAdhikaSTamiSTaM cayahatamunottaraM bihAdiyutam / zeSeSTapadArdhaguNaM vyutkalitaM kheSTavittaM ca // 15 // Pound only in B,

Page #70
32 gaNitasArasagrahaH zeSagacchasyAdyAnayanasUtram - 'pracayA?naH prabhavo yutazcayaneSTapadacayArdhAbhyAm / zeSasya padasyAdizcayastu pUrvokta eva bhavet // 46 // guNaguNite'pi cayAdI tathaiva bhedo'yamatra shesspde| . iSTapadamitaguNAhatiguNitaprabhavo bhavedvakram / / 47 / / atrodezakaH / . pAdottaraM dalAsya padaM tripAdAMzakassamuddiSTaH / kheSTaM caturthabhAgaH kiM vyutkalitaM samAkalaya // 58 / prabhavo'dhaM paJcAMzaH pracayo dvivyaMzako bhavedgacchaH / pazcASTAMzasveSTaM padamaNamAcakSva gaNitajJa / / 49 // AdizcaturthabhAgaH pracayaH paJcAMzakAstripaJcAzaH / gaccho vAJchAgaccho dazamo vyavakalitamAnaM kim // 50 tribhAgau hau vakaM paJcamAMzazcayassyAt padaM trinaH pAdaH paJcamasveSTagacchaH / SaDazassaptAMzo vA vyayaH ko vada tvam kalAvAsa prajJAcandrikAbhAsvadindo // 51 // dvAdazapadaM caturtharNottaramadhaunapaJcakaM vadanam / tricatuHpaJcASTaSTapadAni vyutkAlatamAkalaya // 52 // 1 pracayaguNiteSTagacchatsAdiH prabhavaH padasya zeSasya / pUrvoktaH pracayassyAdiSTasya prAktanAdeva // * ' ca caturbhAgaH. 'M kiM vyutkalitaM samAkalaya.

Page #71
38 kalAsavarNavyavahAraH. guNasaGkalitavyutkalitodAharaNam / dvitribhAgarahitASTamukhaM dvitryaMzako guNacayo'STa padaM bhoH| mitra ratnagatipazcapadAnISTAni zeSamukhavittapadaM kim // 53 // iti bhinnavyutkAlitaM samAptam // kalAsavarNaSar3ajAtiH / / itaH paraM kalAsavaNe par3ajAtimudAhariSyAmaH --- bhAgaprabhAgAvatha bhAgabhAgo bhAgAnubandhaH parikIrtito'taH / bhAgApavAhassaha bhAgamAtrA Sar3ajAtayo mutra kalAsavaNe // 54 // bhAgajAtiH / tatra bhAgajAtau karaNasUtraM yathAsadRzahatacchedahato miyoM'zahArau samacchidAvaMzI / luptaikaharau yojyau tyAjyau vA bhAgajAtividhau // 55 // 'prakArAntareNa samAnacchedamudrAvayitumuttarasUtrama - chedApavartakAnAM labdhAnAM cAhatI niruddhaH syAt / harahataniruddhaguNite hArAMzaguNe samo hAraH // 56 // Kand M add after this iti sAramaGgaDe mahAvIrAcArgasya kRtI dvitIyacyA. rassAmata:. This, howerer, seems to be a nistake. * This and the stanze following are not found in M.

Page #72
84 gaNitasArasaGgrahaH atroddezakaH / 'jambUjambIranAraGgacocamocAmradADimam / ajhaiSIddaLaSaDbhAgadvAdazAMzakaviMzakaiH // 57 // / hemrastriMzacaturvizenASTamena yathAkramam / zrAvako jinapUjAye tadyoge kiM phalaM vada / / 58 // aSTapaJcadazaM vizaM saptaSaTtriMzadaMzakam / ekAdazatriSaSTayaMzamekaviMzaM ca sakSipa // 59 // 'ekatikatrikAyekottaranavadazakaSoDazAntyaharAH / nijanijamukhapramAMzAstvaparAbhyastAzca kiM phalaM teSAm // 6 // ekadikatrikAdyAzraturAdyAzcaikavRddhikA hArAH / nijanijamukhapramAMzAH svAsannaparAhatAH kramazaH // 61 // viMzatyantAH SaDguNasaptAntAH pazcavargapazcimakAH / SaTatriMzatpAzcAtyAH saGkSepe kiM phalaM teSAm // 62 // 'candanaghanasArAgarukuGkamamakeSTa jinamahAya nrH| caraNadaLaviMzapabhamabhAgaiH kanakasya kiM zeSam // 63 // pAdaM pazvAMzamadhe triguNitadazamaM saptaviMzAMzakazca svarNadvandvaM pradAya smitasitakamalaM styAnadadhyAjyadugdham / Stapaus Nos. 67 and 68 are omitted in P. . This stanza is found in K and B. SANENO Bio. 63 and 64 are found in K and B.

Page #73
kalAsavarNavyavahAraH. zrIkhaNDaM tvaM gRhItvAnaya jinasadana prArthanAyAbravInmA mityadya zrAvakAryo bhaNagaNaka kiyaccheSamaMzAnvizodhya // 64 // 'aSTapaJcamukhau hArAvubhaye'pyekavRddhikAH / 'triMzadantAH parAbhyastAzcaturguNita pazcimAH // 65 // " 'svasvavakra pramANAMzA rUpAtsaMzodhya taddvayam / zeSaM sarakhe samAcakSva prottIrNagaNitArNava || 66 || ekonaviMzatiratha kramAt trayoviMzatidviSaSTizca / rUpavihInA triMzattatastrayoviMzanizataM syAt // 67 // pazcatriMzattasmAdaSTAzItikazataM vinirdiSTam / saptatriMzadamuSmAdaSTAnavatitrikonapaJcAzat // 68 // catvAriMzacchatikA saikA ca punazzataM saSAMDazakam / ekatriMzadatarasyAdadvAnavatiH saptapaJcAzat // 69 // tryadhikA saptatirasmAtsapa pacAzadapi ca sA dviguNA / saptakRtiH sacatuSA saptatirekonaviMzanidvizatam // 70 // hArA nirUpitA aMzA ekAdyekottarA amUna | prakSipya phalamAcakSva bhAganAtyabdhipAraga || 71 // azotpattau sUtram - ekaM parikalpyAMzaM tairiSTaissamaharAMzakAn hanyAt / yadguNitAMzasamAsaH phalasaDharzo'zAsta eveSTAH || 72 || This stanza is omitted in M. This stanza is not found in M. B prItIrNagaNitArNava. * B viMzatya. * K and B bhAgajAtyandhipArana. 36

Page #74
gaNitasArasaGgahaH 'ekAMzaddhInAM rAzInAM yutAvaMzAdvArasyAdhikye satyaMzotpAdaka sUtram-- samahArakAMzakayutihatayutyaMzo'za ekavaDInAm / zeSamitarAMzayutihatamanyAMzo'styevamA caramAt / / 73 // .. atroddezakaH / navakadazaikAdazahatarAzInAM navatinavazatIbhaktA / yUnAzItyaSTazatI saMyogaH keM'zakAH kathaya // 74 // chedotpattau sUtram-- rUpAMzakarAzInAM rUpAdyAstriguNitA harAH kramazaH / dvidivyaMzAbhyastAvAdimacaramau phale rUpaM // 75 // atroddeshkH| pazcAnAM rAzInAM rUpAMzAnAM yutirbhavedrUpam / SaNNAM saptAnAM vA ke hArAH kathaya gaNitajJa // 76 // viSamasthAnAnAM chedotpattA sUtram--- ekAMzakarAzInAM vyAdyA rUpottarA bhavanti harAH / svAsanaparAbhyastAssarve dalitAH phale rUpe // 77 // ekAzAnAmanekAMzAnAM caikAMze phale chedotpattau sUtralabdhaharaH prathamasyacchedaH sasvAMzako'yamaparasya / prAk svapareNa hato'ntyaH svAMzenaikAMzaka yoge // 78 // atroddeshkH| saptakanavakatritayatrayodazAMzaprayuktarAzInAm / / rUpaM pAdaH SaSThaH saMyogAH ke harAH kathaya // 79 // 1 B sazasaMzarAzInA aMzotpAdakasUtram /

Page #75
37 kalAsavarNavyavahAraH. ekAMzakAnAmekAMze'nekAMze ca phale chedotpattI sUtramseSTo hAro bhaktaH svAMzena niragramAdimAMzaharaH / tadyutihArApteSTaH zeSo'smAdityamitareSAm // 8 // atroddezakaH / trayANAM rUpakAMzAnAM rAzInAM ke harA vada / phalaM caturthabhAgassyAJcaturNA ca trisaptamam / / 81 // aikAMzAnAmanekAMzAnAM cAnekAMze phale chedotpattA sUtramiSTahatA dRSTAMzAH phalAMzasahazA yathA hi tadyogaH / nijaguNahataphalahArastaddhAro bhavati nirdiSTaH / / 82 !! atrodezakaH / 'ekakAMzena rAzInAM trayANAM ke harA vada / hAdazAptA trayoviMzatyazaMkA ca yutirbhavet / / 83 // trisaptakanavAMzAnAM trayANAM ke harA vada / yyUnapazcAzadAtA trisaptatyaMzA yutirbhavet // 84 / / ekAMzakayo rAzyorekAMze phale chedotpattI sUtramvAmchAhatayutihArazchedaH sa vyaMkavAJchyApto'nyaH / phalahArahAralabdha skhayogaguNite harau vA staH // 85 !! atrAMddezakaH / . rAzyorekAMzayozchedI ko mavaMtA. tayoyutiH / pazo dazamAgo vA brUhi tvaM gaNitArthavit / / 86 // Stadzus 88 and 84 are omitted in B.

Page #76
38 gaNitasArasaGghaH ekAMzakayoranekAMzayozca ekAMze'nekAMze'pi phale chedotpattau prathama sUtram -- 'iSTaguNAMzo'nyAMzaprayutaH zuddhaM hRtaH phalAMzena / iSTatayutiharano haraH parasya tu tadiSTahatiH // 87 // atroddezaH / rUpAMzako rAzyoH kau syAtAM hArakau yutiH pAdaH / zo vA dvihatastaptakanavakAMzayozca vada // 88 // dvitIyasUtram - phalahAratADitAMzaH parAMzasahitaH phalAMzakena hRtaH / syAdekasyacchedaH phalahara guNito'yamanyasya // 89 // atroddezakaH / rAzidvayasya ko hArAkAMzasyAsya saMyutiH / dvisaptAMzo bhavedbrUhi SaDaSTAMzasya ca priya // 90 // ardhazyaMzadazAMzakapacadazAMzakayutirbhavepam / T tyakte pazcadazAMza rUpAMzAvatra kau yojyau // 91 // dalapAdapazvamAMzaka vizAnAM bhavati saMyutI rUpam / saptaikAdazakAMzI kau yojyAviha vinA viMzam // 92 // yugmAnyAzrityacchedotpattau sUtram -- yugmapramitAn bhAgAnekaikAMzAn prakalpya phalarAzeH / tebhyaH phalAtmakebhyo dvirAzividhinA harAssAdhyAH // 93 // 1P and B add as another reading. zuddhaM phalAMzabhaktaH svAnyazayuto nijeSTaguNitAMzaH /

Page #77
80 kalAsavarNavyavahAraH. atroddezakaH / trikapa cakatrayodazasaptanavaikAdazAMzarAzInAm / ke hArAH phalamekaM paJcAzo vA caturguNitaH // 94 / / ekasUtrotpannarUpAMzahAraissUtrAntarotpannarUpAMzahAraizca phale rUpe chedopattau naSTabhAgAnayane ca sUtrama-- vAJchitasUtrajahArA harA bhavantyanyasUtrajaharamAH / dRSTAMzaikyonaM phalamabhISTanaSTAMzamAnaM syAt / / 95 // atroddezakaH / parahatidalanavidhAnAzrayodaza svaparasaGgaNavidhAnAt / . bhAgAzcatvAro'taH kati bhAgAsyuH phale rUpe // 96 // prAkasvaparahatavidhAnAtsaptasvAsanaparaguNArdhavidhAnAt / bhAgAstritayazcAtaH kati bhAgAssyuH phale rUpa / / 97 // rUpAMzakA dviSaTakahAdazavitiharA vinaSTo'tra / paJcamarAzI rUpaM marvasamAsassa rAziH kaH // 98 / / - iti bhaagjaatiH| prabhAgabhAgabhAgajAtyossUtrama aMzAnAM saGgaNanaM hArANAM ca prabhAgajAtau syAt / guNakAroM'zakarAzehariharo bhAgabhAgajAtividhI // 99 // prabhAgajAtAvuddezakaH / rUpArdha tryaMzArgha tryaMzArSi dalArdhapAMzam / pAzArdhatryaMzaM tRtIyabhAgArthasaptazim // 10 //

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gaNitasArasagrahaH 'daladaladalasaptAMzaM tryaMzavyaMzakadalArdhadalabhAgam / ardhadhyaMzavyaMzakapazcAzaM pazmazidalam // 101 // krItaM paNasya davA kokanadaM kundaketakIkumudam / jinacaraNaM prArcayituM prakSipyaitAn phalaM brUhi // 102 // " / rUpArdha tryaMzakA pAdasaptanavAMzakam / dvitribhAgadvisaptAzaM dvisaptAMzanavAMzakam // 103 // dayA paNadvayaM kazcidAnaiSInnUtanaM ghRtam / jinAlayasya dIpArtha zeSaM kiM kathaya priya // 104 // tryaMzAdvipazcamAMzastRtIyabhAgAt trayAdezaSaDazaH / pazcASTAdazabhAgAt trayodazAMzo'STamAnavamaH // 105 // navamAJcatustrayodazabhAgaH paJcAMzakAt tripAdArdham / sakSipyAcakSvaitAna prabhAgajAtau zramo'sti yadi // 106 // atraikAvyaktAnayanasUtram-- rUpaM nyasyAvyakte prAvidhinA yatphalaM bhavattena / bhaktaM paridRSTaphalaM prabhAgajAtau nadajJAtam // 107 // __ atroddezakaH / rAzeH kutazcidaSTAMzasyazapAdo'rdhapaJcamaH / SaSThatripAdapazcAzaH kimavyaktaM phalaM dalam // 10 // anekAvyaktAnayanasUtramkRtvAjJAtAniSTAn phalasahazI tabutiryayA bhavati / vibhajeta prathagvyaktairaviditarAzipramANAni // 109 //

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kalAsavarNavyavahAraH. atroddeshkH| rAzeH kutazcidardha kutazcidaSTAMzaka tripazcAzaH / kasmAddiyaMzAdhU phalamadhU ke syurajJAtAH // 110 // bhAgabhAgajAtAvuddezakaH / SaTsaptabhAgabhAgasyaSTAMzAMzazcaturnavAMzAMzaH / tricaturthabhAgabhAgaH kiM phalametadyutI brUhi // 111 // divyaMzAptaM rUpaM tripAdabhaktaM dvikaM vayaM cApi / dvivyaMzodRtamekaM navakAtsaMzodhya vada zeSam // 112 // iti prabhAgabhAgabhAgajAtI / bhAgAnubandhajAtau sUtram --- harahatarUpeSvaMzAn sakSipa bhAgAnubandhajAtividhau / 'guNayAnAMzacchedAvaMzayutacchedahArAbhyAm // 113 // rUpabhAgAnuvandha uddezakaH / "dvitriSaTrASTaniSkANi dvAdazASTaSaDaMzakaiH / pazcASTamaissametAni viMzatezzodhaya priya // 111 // sArdhenakena pake sASTAMzairdazamihimama / . sArdhAbhyAM kukarma dvAbhyAM krIta yoge kiyadbhavet // 115 // 'sASTamASTau SaDaMzAna SaDadvAdazAMzayutaM dvayam / trayaM pacASTamopetaM vizanazodhaya priya // 116 // B roads guNayedaprAMzaharau mahitAMzamaMda. Thisrtansa is not found in P. dat. This stanga i found only in P,

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gaNitasArasaGgrahaH. saptASTau navadazamASakAn sapAdAn dattvA nA jinanilaye cakAra pUjAm / unmIlatkuravakakundajAtimallImAlAbhirgaNaka vadAzu tAn samasya // 117 // . . __ bhAgabhAgAnubandha uddezakaH / svayaMzapAdasaMyuktaM dalaM paJcAMzako'pi ca / vyaMzasvakIyaSaSThArdhasahitasAtau kiyat // 118 // vyaMzAdhezakasaptamAMzacaramaisvairanvitAdardhataH puSpANyardhatarIyapazcanavamasvIyairyutAtsaptamAt / gandhaM pazcamabhAgano'rdhacaraNavyaMzAMzakaimizritAddhUpaM cAcayituM naro jinavarAnAneSTa kiM tadyutau // 119 // svadalasahitaM pAdaM svayaMzakena samanvita dviguNanavamaM svASTAMzavyaMzakAvimizritam / navamamapi ca svASTAMzAdyapazcimasaMyutaM nijadalayutaM vyaMzaM saMzodhaya tritayAtpriya // 120 // svadalasahitapAdaM sasvapAdaM dazAMzaM nijadalayataSaSThaM sasvakavyaMzamardham / caraNamapi sametasvatribhAgaM samasya priya kathaya samagraprajJa bhAgAnubandhe // 121 // atrAgrAvyaktAnayanasUtramlabdhAtkalpitabhAgA rUpAnItAnubandhaphalabhaktAH / kramazaH ravaNDamamAnAste'jJAtAMzapramANAni // 122 // 'B svacaraNAvardhAntimaiH.

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kalAsavarNavyavahAraH. 49 atroddezakaH / kAzritvakairardhatRtIyapAdairaMzo'paraH pnyccturnvaaNshaiH| anpatripaJcAzanavAMzakAyuto yutI rUpamihAzakAH ke // 123 // ko'pyaMzasvArdhapaJcAMzatripAdanavamairyataH / ardha prajAyate zIghraM vadAvyaktapramA priya // 124 // zeSeSTasthAnAvyaktabhAgAnayanasUtrama -- labdhAtkalpitabhAgAssavarNitairvyaktarAzibhibhaktAH / 'kramazo rUpavihInAssveSTapadepvaviditAMzAssyuH // 125 // iti bhAgAnubandhajAniH / atha bhAgApavAhanAtau sUtram harahatarUpepvaMzAnapanaya bhAgApavAhajAnividhau / 'guNayAgrAMzacchedAvaMzonacchedahAgabhyAm // 126 // rUpabhAgApavAha uddezakaH / myaSTacaturdazakarSAH pAdAdviAdazAMzaSaSThonAH / savanAya narairdattAsIrthakatAM nAtI kiM syAt // 127 // triguNapAdadalatrihatASTamaivirahitA nava sapta nava kramAt / guNayada prAMzaharI hitAMzadahAgabhyAm / 6-A

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gaNitasArasaGgrahaH. priya vizodhya caturguNaSaTutaH . kathaya zeSadhanapramiti drutam // 128 // bhAgamAgApavAha uddezakaH / dviguNitapaJcamanavamatryaMzASTAMzadvisaptamAna kramazaH / khaSaDaMzapAdacaraNavyaMzASTamavarjitAn samasya vada // 129 // SaTsaptAMzasvaSaSThASTamanavamadazAMzairviyuktaH paNasya syAtpazcadvAdazAMzasvakacaraNatRtIyAMzapazcAMzakonaH / khadvivyaMzadvipaJcAMzakadalaviyutaH paJcaSaDbhAgarAzidivyaMzo'nyasvapazcASTamaparirahitastatsamAse phalaM kim // 130 / ardha vyaSTamabhAgapAdanavamaisvIyavihInaM punaH . khairaSTAMzakasaptamAMzacaraNairUnaM tRtIyAMzakam / adhyardhAtparizodhya saptamamapi svASTAMzaSaSThonitaM zeSaM brUhi parizramo'sti yadi te bhAgApavAhe sakhe // 131 // atrAgrAvyaktabhAgAnayanasUtramlabdhAtkalpitabhAgA rUpAnItApavAhaphalabhaktAH / kramazaH khaNDasamAnAste'jJAtAMzapramANAni // 132 // atroddezakaH / kazcitsvakai caraNapaJcamabhAgaSaSThaiH ko'pyaMzako dalaSaDaMzakapaJcamAMzaiH / hIno'paro higuNapazcamapAdaSaSThaiH tatsaMyutirdalamihAviditAMzakAH ke // 133 //

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kalAsavarNavyavahAraH. ko'pyazasvArdhaSaDbhAgapaJcamASTamasaptamaiH / vihIno 'jAyate SaSThassa ko'zo gaNitArthavit // 134 / / zeSeSTasthAnAvyaktabhAgAnayanasUtram* labdhAtkalpitabhAgAstavarNitaiphaktarAzibhirbhaktAH / rUpAtpTathagapanItArakheSTapadepvaviditAMzAssyuH // 135 // iti bhAgApavAhajAtiH / bhAgAnubandhabhAgApavAhanAtyossarvAvyaktabhAgAnayanasUtram-- tyaktvaikaM sveSTAMzAn prakalpayaMdaviditaMSu sarveSu / aitaistaM punazaM prAguktairAnayetsUtraiH / / 136 // atrAMddezakaH / kazcidaMzoMzakaiH kaizcitpazcabhimsvairyato dalam / viyuktI vA bhavetpAdastAnaMzAn kathaya priya // 137 // mAgamAtRjAtI sUtram - bhAgAdimajAtInAM svasvavidhirbhAgamAtRjAtI syAt / sA SaDriMzatibhedA rUpaM chado-cchidA rAzaH // 138 / atroddezakaH / yaMzaH pAdo'rdhA paJcamaSaSThastripAdaha mekam / pazcArdhahataM rUpaM saSaSThamekaM sapaJcamaM rUpama // 139 // khIyatRtIyayugdalamato nijaSaSThayuto dvisaptamo hInanavazimekamapanItadazAMzakarUpamaSTamaH / 'P, K And B nayuti: for jAyate.

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gaNitasArasaGgrahaH. khena navAMzakena rahitazcaraNassvakapazcamojjhito brUhi samasya tAn priya kalAsamakotpalamAlikAvidhau // 140 / . iti bhAgamAtRjAtiH / iti sArasaGghahe gaNitazAstre mahAvIrAcAryasya kRtau kalAsavarNoM nAma dvitIyavyavahArassamAptaH //

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tRtIyaH prakIrNakavyavahAraH. * praNutAnantagaNaughaM praNipatya jinezvaraM mahAvIram / praNatajagatrayavaradaM prakIrNaka gaNitamabhidhAsye // 1 // 'vidhvastadurnayadhvAntaH siddhaH syAhAdazAsanaH / vidyAnando jino jIyAdvAdIndro munipuGgavaH // 2 // itaH paraM prakIrNakaM tRtIyavyavahAramRdAharipyAmaH--- bhAgazzeSo mUlakaM zAmalaM syAtAM jAnI de dviraprAMzamUle / bhAgAbhyAso'nA'zavargIya mUla mizraM tasmAdinnadRzyaM dazAmUH // 3 // tatra bhAganAnizeSajAtyo mUtram --- bhAgonarUpabhaktaM dRzyaM phalamatra bhAgajAtividhau / aMzonitarUpAhatihatamagraM zeSajAtividho // 4 // ___ bhAgajAtAvuddezakaH / dRSTA'STamaM pRthivyAM mambhamya 'yaMzako payA toye| pAdAMzaH zaivAlaM kaH stambhaH mapta hastAH gvaM // 5 // SaDDAgaH pATalISu bhramaravaratatastatribhAgaH kadamba pAdazrUtadrumaMSu pradalitakusume campake pazcamAMzaH / protphuSThAmbhojaSaNDe ravikaradalite triMzadaMzo-mirema tatraiko mattabhRGgo bhramati nabhasi kA tasya bRndasya saLyA // 6 // Rund Monit this planza

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// 10 // gaNitasArasaGgrahaH. AdAyAmbhoruhANi stutizatamukharaH zrAvakastIrthakadbhayaH pUjAM cakre catubhyo dRSabhajinavarAta yaMzameSAmamuSya / tryaMzaM turya SaDazaM tadanu sumataye tannavadvAdazAMzI zeSebhyo dvidvipadmaM pramuditamanasAdatta kiM tatpramANam // 7 // svavazIkRtendriyANAM dUrIkRtaviSakaSAyadoSANAm / zIlaguNAbharaNAnAM dayAGganAliGgitAGgAnAm // 8 // sAdhUnAM sadvandaM sandRSTaM dvAdazo'sya tarkajJaH / khatryaMzavarjito'yaM saiddhAntazcAndasastayozzeSaH // 9 // SaDano'yaM dharmakathI sa eva naimittikaH svapAdonaH / vAdI tayovizeSaH SaguNito'yaM tapasvI syAt / / 10 // giriziravarataThe mayopadRSTA yatipatayo navasaGguNASTa saGkhyAH / ravikaraparitApitojjvalAGgAH kathaya munIndrasamUhamAzu me tvam // 11 // phalabhAranamrako zAlikSetre zukAssamupaviSTAH / sahasotthitA manuSyaiH sarve santrAsitAssantaH // 12 // teSAmadhaM prAcImAmeyI prati jagAma SaDbhAgaH / pUrvAneyIzeSaH svadalonaH svArdhavarjito yAmIm // 13 // yAmyAneyIzeSaH sa nairRtiM svadvipaJcabhAgonaH / yAmInairmatyazakaparizeSo vAruNImAzAm // 14 // nairmatyaparavizeSo vAyavyAM sasvakatrisaptaziH / vAyavyaparavizeSo yutasvasaptASTamaH saumIm // 11 //

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prakIrNakavyavahAraH. vAyavyuttarayoyutirezAnI khatribhAgayugahInA / dazaguNitASTAviMzatiravaziSTA vyoni kati kIrAH // 16 // kAcidvasantamAse prasUnaphalagucchabhAranamrodyAne / kusumAsavarasaraJjitazakakokilamadhupamadhurAnisvananicite // 17 // himakaradhavale pRthule saudhanale sAndrarundra mRdutalpe / phaNiphaNanitambabimbA kanadamalAmaraNazobhAGgI // 18 // pAThInajaTharanayanA ktthinstnhaarnmrtnumdhyaa| saha nijapatinA yuvatI rAtrau prItyAnaramamANA // 19 // praNayakalahe samatthaM muktAmayakaNThikA tadabalAyAH / chinnAvanau nipatitA tayazazceTikAM prApat // 20 // SaDbhAgaH shyyaayaamnntgnntraardhmitibhaagaaH| ' SaTamaGkhyAnAstasyAH sarve sarvatra sampatitAH // 21 // ekAgraSaSTizatayutasahavamuktAphalAni dRSTAni / tanmauktikapramANaM prakIrNakaM vesi can kathaya // 22 // 'sphuradindranIlavarNa SaTpadabRndaM praphulchinodyAne / dRSTaM tasyASTAMzo'zoke kuTaje SaDaMzako lInaH // 23 // kuTajAzokavizeSaH SaDguNitA vipula pATalISaNDaM / pATalyazokazeSaH vanavAMzono vizAlasAlabane // 24 // pAThalyazokazeSA yutasvasaptAMzakena madhukavane / pazcazissandRSTo vakuleSutphullamukuleSu // 25 // tilakeSu kuravakaMSu ca saraleSAmeSu padmaSaNDaMSu / vanakarikapolamUlapi santasthe sa evAMzaH / / 26 // 'M reads sphu tiMndra'.

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50 gaNitasArasaGgrahaH. kiJjalka paJjarakaJjava madhukarAstrayastriMzat / dRSTA bhramarakulasya pramANamAcakSva gaNaka tvam // 27 // goyUthasya kSitibhUti dalaM taddalaM zailamUle pada tasyAMzA vipulavipine pUrva pUrvArdhamAnAH / santiSThante nagaranikaTaM dhenavo dRzyamAnA dvAtriMzat tvaM vada mama sakhe gokulasya pramANan || 28 // iti bhAgajAtyuddezakaH // zeSajAtA vuddezakaH / bar3abhAgamA mrarAzaM rAjA zeSasya paJcamaM rAjJI | turyatryaMzadalAni trayo'grahISuH kumAravarAH // 29 // zeSANi trINi cUtAni kaniSThAM dArako grahIt / tasya pramANamAcakSva prakIrNakavizArada // 30 // carati girau saptAMzaH kariNAM SaSThAdimArdhapAzcAtyAH / pratizeSAMzA vipine SadRSTAssarasi kAne te syuH // 31 // koSThasya lebhe navamAMzamekaH pare'STabhAgAdidalAntimAMzAna / zeSasya zeSasya punaH purANA dRSTA mayA dvAdaza tatpramA kA // 32 // iti zeSajAtyuddezakaH // atha mUlajAto sUtrama-- mUlArthA chindyAdaMzAMnaikena yuktamUlakRteH / dRzyasya padaM sapadaM vargitAmaha mUlajAtau svam // 33 //

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prakIrNakavyavahAraH. atroddezakaH / dRSTayA yUthasya pAdo mUle ca dve zailasAnau niviSTe / 'uSTrAstrinnAH paJca nadyAstu tIre kiM tasya syAduSTrakasya pramANam / / 34 / / zrutvA varSAbhramAlA paTapaTukhaM zailazRGgAruraGga nATyaM cakre pramodapramuditazikhinAM SoDazAMzo'STamazca / tryaMzaH zeSasya SaSTho vakhakulavane pana malAni tasthuH punnAga pakha dRSTA bhaNa gaNaka gaNaM barhiNAM saGgaNayya / / 35 / / carati kamalaSaNDe sArasAnAM caturtho navamaraNabhAga sapta mUlAni cAdrI / vikacavakalamadhye saptaniprASTamAnAH kati kathaya sa tvaM pakSiNo dakSa sAkSAt // 36 // na bhAgaH kapivRndasya trINi mUlAni parvata / catvAriMzadvane dRSTA vAnarAstadraNaH kiyAna // 37 // 5 kalakaNThAnAmadhaM sahakArataroH praphullazAkhAyAm / tilake'STAdaza tasthunI mulaM kathaya pikanikareMma // 38 // haMsakulasya dalaM vakule sthAna pazca padAni tamAlakujAgre / Brends hasti. * Breads nAgAH 51 * Bds kiM syAne kRgaNAM pramANam /

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gaNitasArasaGgrahaH. atra na kiJcidapi pratidRSTaM tatpramitiM kathaya priya zIghram / / 39 // iti mUlajAtiH // atha zeSamUlajAtI sUtrampadadalavargayutAgrAnmUlaM saprAkpadArdhamasya kRtiH / dRzye mUlaM prApte phalamiha bhAgaM tu bhAgajAtividhiH // 40 // atroddezakaH // gajayUthasya vyaMza'zzeSapadaM ca trisaGgaNaM sAnau / sarasi trihastinIbhirnAgo dRSTaH katIha gajAH // 41 // nirjantukapradeze nAnAdruma SaNDamaNDitodyAne / 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 dvAdazabhedAGgazAstrakuzaladhiyaH / dvAdaza munayA dRSTAH kiyatI manicandra yatisa mitiH // 45 / mUlAni paJca caraNena yutAni sAnI zeSasya pazcanavamaH kariNAM nagAgre / mUlAni paJca sarasIjavane ramante nadyAstaThe SaDiha te dviradAH kiyantaH // 16 // iti zeSamUlajAtiH // | B reads zeSasya padaM visaMgaNaM.

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58 prakIrNakavyavahAraH. atha dviramazeSamUlajAtau sUtrammUlaM dRzyaM ca bhajedaMzakaparihANarUpaghAnena / pUrvAgramagrarAzau kSipedatazzeSamUlavidhiH / / 47 // atroddezakaH / madhukara eko dRSTaH ve padme zeSapazcamacaturthI / zeSavyaMzo mUlaM dvAvAne te kiyantaH syuH // 18 // siMhAzcatvAro'drau pratizeSaSaDaMzakAdimArdhAntAH / mUle catvAro'pi ca vipine dRSTAH kiyantaste // 49 // taruNahariNIyugmaM dRSTaM dvisaGgaNita vane kudharanikaTe zeSAH pnaaNshkaadidlaantimaaH| . vipulakalamakSetre tAsAM padaM tribhirAhataM kamalasarasItIre nasyurdazaiva gaNaH ki yAn // 50 // iti dviragrazeSamUlajAtiH // athAMzamUlajAtau sUtrambhAgaguNa mUlAgre nyasya padaprAptadRzyakaraNaMna / yallabdhaM bhAgahataM dhanaM bhavedaMzamUlavidhau // 51. // anyadapi mUtrama - dRzyAdezakabhaktAzcaturguNAnmUlakRtiyutAnmUlam / sapadaM dalitaM vargitamaMzAbhyastaM bhavet sAram // 52 // | B reads dvau cAmra.

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gaNitasArasaGgrahaH. atroddezakaH / padmanAlatribhAgasya jale mUlASTakaM sthitam / SoDazAGgalamAkAze jalanAlodayaM vada // 53 // dvitribhAgasya yanmUlaM navapnaM hastinAM punaH / zeSatripazcamAMzasya mUlaM pahissamAhatam // 54 // vigaladdAna'dhArArdragaNDamaNDaladantinaH / caturvizatirAdRSTA mayATavyAM kati dvipAH / / 55 // kroDaughArdhacatuHpadAni vipinaM zArdUlavikrIDitaM prApuzzeSadazAMzamUlayugalaM zailaM catustADitam / zaSArdhasya padaM trivargaguNitaM va varAhA vane dRSTAssaptaguNASTakapramita yasaiSAM pramANaM vada / / 56 / / ityaMzamalajAtiH // atha bhAgasaMvargajAtau sUtramsvAMzAptaharAdanAccaturgaNAgreNa tahareNa hatAt / mUlaM yojyaM tyA tacchede taddalaM vittam // 57 / / atroddezakaH / aSTamaM SoDazAMzaghnaM zAlirAzeH kRSIvalaH / caturvizativAhAMzca lebhe rAziH kiyAn vada // 58 // 1 B rends vArA. After this stanzn all the MSS. have the following stanza; but it is simply aparaphrase of stanza No. 57: anyacca cturhtdRssttenonaadbhaagaahnyNshhthaaraat|| taDedana hatAnmUlaM yojya nyAjyaM tacchede tadarthaM vittam //

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prakIrNakavyavahAraH. zikhinAM SoDazabhAgaH svaguNabhUte tamAlaSaNDe'sthAt / zeSanavAMzaH svahatazcaturagradazApi kati te syuH // 59 // jale triMzadazAhato dvAdazAMzaH sthitazzeSavizeo hataH SoDazena / trinina pa karA viMzatiH khe sarakhe stambhadairghyasya mAnaM vada tvam // 60 // iti bhAgasaMvarga jAtiH || athonAdhikAMzavargajAnau sUtram-svAMzakabhaktaharArdhaM nyUnayugaMdhikonitaM ca tadvargAt / vyUnAdhikavargAgrAnmUlaM svarNa phalaM pareM zahatam / / 61 / / 'hInAlApa udAharaNam / mahiSINAmaSTAMzI vyeka vargIkRta vanaM ramate / pazcadazAdrau dRSTAstRNaM carantyaH kiyantyastAH / / 62 / / anekapAnAM dazamI dvivarjitaH svaGgaNaH krIDati sallakIvane / u caranti SaDurgamitA gajA giMgai kiyanta ete bhavanti dantinaH / / 63 / / 'adhikAlApa udAharaNam / jambUvRkSe padazAMza kiyuktaH svenAbhyastaH kekikalasya dvikRtighrAH / Momite hIna, 55 [+ Momats this as well as the following stanz

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56 Fan gaNitasArasaGgrahaH. pazcApyanye mattamayUrAssahakAre raMramyante mitra vadeSAM parimANam // 64 // ityUnAdhikAMzavargajAtiH // atha mUlamizrajAtau sUtram - mizrakRtirUnayuktA vyadhikA ca dviguNamizrasammaktA / vargIkRtA phalaM syAtkaraNamidaM mUlamizravidhau // 65 // hInAlA uddezakaH / mUlaM kapotavRndara dvAdazonasya cApi yat / tayoryogaM kapotA SaD dRSTAstannikaraH kiyAn // 66 // pArAvatasaGgha caturdhano'patatra yanmUlam / yogaH SoDaza tadvRnde kati vihaGgAH syuH / / 67 / / adhikAlApa uddezakaH / rAjahaMsanikarasya yatpadaM sASTaSaSTisahitasya caitayoH / saMyukavihInaSaTkRti kati marAlakA vada || 68 || iti mUlamizrajAtiH // atha bhinnadRzyajAtau sUtram- dRzyAMzane rUpe bhAgAbhyAsena bhAjite tatra / labdhaM tatsAraM prajAyate bhinnadRzyavidhau // 69 // 1 B reacts yoga:.

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prakIrNaka vyavahAraH atroddezakaH / sikatAyAmaSTazastandRSTo'STAdazAMzasaGgaNitaH / 1 stambhasyArdha ' dRSTaM stambhAyAmaH kiyAn kathaya // 70 // dvibhaktanavamAMzakaprahatasaptaviMzAMzakaH pramodamavatiSThate karikulasya pRthvItale / vinIlajaladAkRtirviharati tribhAgo naMge vada tvamadhunA sarave karikulapramANaM mama / / 71 / / sAdhUtkRtornivasati SoDazAMzaka vibhAjitaH svakaguNito vanAntare / pAdo girau mama kathayAzu tanmita prottIrNavAn jaladhisamaM prakIrNakam / / 02 / / 57 iti bhinnadRzyajAtiH // iti sArasaGghahe gaNitazAstre mahAvIrAcAryasya kRte prakIrNako nAma tRtIyavyavahAraH samAptaH // B, M and K read gagane.

Page #96
trilokabandhave tasmai kevalajJAnabhAnave / namaH zrIvardhamAnAya nirdhUtAkhilakarmaNe // 1 // caturthaH trairAzika vyavahAraH / itaH paraM trairAzikaM caturthavyavahAramudAhariSyAmaH / tatra karaNasUtraM yathA - trairAzike'tra sAraM phalamicchAsaGgaNaM pramANAptam / u icchApramayossAmye viparIteyaM kriyA vyaste // 2 // pUrvArdhoddezakaH / divase stribhistapAdairyojana SaTuM caturthabhAgonama / gacchati yaH puruSo'sau dinayutavarSeNa kiM kathaya / / 3 / / G vyardhASTAbhirahobhiH krozASTAMzaM svapaJcamaM yAti / paGgasapazca mAgairvarSaistribhiratra kiM brUhi // 4 // aGgalacaturthabhAgaM prayAti kITo dinASTabhAgena / merormUlAcchikharaM katibhirahobhistamApnoti // 5 // kArSApaNaM sapAda nirvizati tribhirahobhirardhayutaiH / yo nA purANazatakaM sapaNaM kAlena kenAsau // 6 // "kRSNAgaru satkhaNDaM dvAdazahastAyataM trivistAram / kSayametyaGgalamahnaH kSayakAlaH ko'sya vRttasya || 7 || svarNairdazabhissArdherdroNADhakakuDaba mizritaH krItaH / vararAjamASavAhaH kiM hemazatena sArdhena // 8 // JP, Kand M road sa for sva. G *B reade satkRSNAgaru khaNDaM.

Page #97
69 trairAzikavyavahAraH sAstribhiH purANaiH kuGkumapalamaSTabhAgasaMyuktam / saMprApyaM yatra sthAt purANazatakena kiM tatra // 9 // sArdhAkasaptapalaizcaturdazAonitAH paNA ' labdhAH / dvAtriMzadAkapalaissapaJcamaiH kiM sarave brUhi // 10 // kArSApaNaizcatubhiH paJcAMzayunaH palAni rajatasya / SoDaza sArdhAni naro labhate kiM karSaniyutena // 11 // karpUrasyApTapalaikhyaMzonairnAtra pabha dInArAn / bhAgAMzakalAyuktAn labhate kiM palasahasreNa / / 12 / / sAthUstribhiH paNairiha ghRtasya palapaJcakaM sapanAMzam / krINAti yo naro'yaM kiM sASTamakarSazatakena // 13 // . sArdheH paJcapurANaiH SoDaza sadalAni vastrayugalAni / labdhAni saikaSaSTayA karSANAM kiM sarave kathaya / / 14 / / vApI samacaturazrA salilaviyuktApTahastaghanamAnA / zailastasyAstIre samutthitAzzakharatastasya / / 15 / / vRttAGgalaviSambhA jaladhArA sphaTikanirmalA patitA / vApyantarajalapUrNA nagocchRitiH kA ca jalasaGkhyA // 16 // 'mundroNayugaM navAjyakuDavAn SaT taNDuladroNakAnaSTau vastrayugAni vatsasahitA gAppaTa suvarNatrayam / 'M and B read labhyAH . Breads samutthitA zi'. * Band K read the following for this stanza : duravadroNayugaM navAjyakuDavAn SaT zarkaMgadroNakAnaSTI cocaphalAni mAndradadhikhAryaSaT purANatrayam / zrIkhaNDa dadatA nRpeNa savanAthai pajinAgAraka patriMzatizateSu mitra vara me tattadugdhAdikam // 7-A

Page #98
60 gaNitasArasanAhaH saskrAntau dadatA narAdhipatinA SaDbhyo dvijebhyassakhe SaTtriMzatrizatebhya Azu vada kiM taddattamudgAdikam // 17 // iti trairAzikaH // vyastatrairAzike turIyapAdasyoddezakaH / / kalyANakanakanavateH kiyanti navavarNakAni kanakAni / sASTAMzakadazavarNakasaguJjahemnAM zatasyApi // 18 // vyAsena dairyeNa ca SaTkarANAM cInAmbarANAM trizatAni tAni / tripaJcahastAni kiyanti santi vyastAnupAtakramAvidvada tvam // 19 // iti vyastatrairAzikaH // vyastapazcarAzika uddezakaH / pazcanavahastavistRtadAyAM cInavastrasaptatyAm / dvitrikaravyAsAyati tacchutavastrANi kati kathaya // 20 // vyastasaptarAzika uddezakaH / vyAsAyAmodayato bahumANikye cturnvaassttkre| dviSaDekahastamitayaH pratimAH kati kathaya tIrthakRtAm // 21 // vyastanavarAzika uddezakaH / vistAradeyodayataH karasya SaTtriMzadaSTapramitA navArNa / zilA tayA tu dviSaDekamAnAstAH paJcakAryAH kati caityayogyAH // 21 // iti vyastapaJcasaptanavarAzikAH //

Page #99
trairAzikavyavahAraH gatinivRttau sUtram-- nijanijakAloddhRta yorgamananivRttyorvizeSaNAjAtAm / dinazuddhagatiM nyasya trairAzikavidhimataH kuryAt // 23 // atroddezakaH / krozasya paJcabhAgaM nauryAti dinatrisaptabhAgena / 'vArdhI vAtAvidyA pratyeti krozanavamAMzam // 24 // kAlena kena gacchet tripaJcabhAgonayojanazataM sA / saGkhyAbdhisamuttaraNe bAhubaliMstvaM samAcakSva / / 25 / / sapAdahema tridinaissapaJcamainaro'rjayan vyeti suvarNaturyakam / . ninASTamaM paJcadinaidalonitaiH sa kena kAlena labheta saptatim // 26 // gandhebho madalubdhaSaTpadapadaprodbhinnagaNDasthalaH / sArdha yojanapaJcamaM vrajati yaSSabhirdalonaidinaiH / pratyAyAti dinaistribhizca sadalaiH krozadvipaJcAMzakaM brUhi krozadalonayojanazataM kAlena kenAmuyAt // 27 / / vApI payaHprapUrNA dazadaNDasamucchritAbjamiha jAtam / agalayugalaM sadalaM pravardhate sArdhadivasena // 28 // nissarati yantrato'mbhaH sArdhenAhArale saviMze he| zupyati dinena salilaM sapaJcamAGgulakaminakigNaiH // 29 // krmoM nAlamadhastAt sapAdapaJcAGgalAni cAkRSati / sAkhidinaiH padmaM toyasamaM kena kAlena // 30 // IBand K read tasminkAle vAdhI.

Page #100
62 gaNitasArasagrahaH dvAtriMzaddhastadIrghaH pravizati vivare pazcabhissaptamAdhaiH kRSNAhandrio 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 plyaa| guNayitvA jIvAnAM krayavikrayayostu tAneva // 32 // ___atroddezakaH / dvitricatuzzanayoge paJcAzatSaSTisaptatipurANAH / lAbhArthinA prayuktA dazamAseSvasya kA vRddhiH / / 33 / / hemrAM sArdhAzItesivyaMzena vRddhiradhyardhA / satricaturthanavatyAH kiyatI pAdonaSaNmAsaiH // 34 / / SoDazavarNakakAJcanazatena yo ralaviMzatiM labhate / dazavarNasvarNAnAmaSTAzItidvizatyA kim // 35 // preads as variations the following: prakArAntareNa satram sar3akramya phalaM chindyAlaghapaGaktyAnekarAzikA pddaaktm| svaguNAmazvAdInAM krayavikrayayostu tAneva // anyadapi sUtramsaGakramya phalaM chindyAt pRthupayatyabhyAsamalpayA pngktyaa| azvAdInAM krayavikrayayorazvAdikAMzca saGkramya // B gives only the latter of these stanzas with the following variation in the second uarter: pathapatyabhyAsamalpapaGktyAhatyA.

Page #101
trairAzikavyavahAraH godhUmAnAM mAnIva nayatA yojanatrayaM labdhAH / SaSTiH paNAH saghAhaM kumbhaM dazayojanAni kati // 36 // bhANDapratibhANDasyoddezakaH / kastUrIkarSatrayamupalabhate dazabhiraSTabhiH kanakaiH / karSadvayakarpUraM mRganAbhitrizatakarSakaiH kati nA // 37 // panasAni SaSTimaSTabhirupalabhate'zItimAtuluGgAni / dazabhirmASairnavazatapanamaiH kati mAtuluGgAni // 38 // jIvakrayavikrayayoruddezakaH / SoDazavarSAsturagA viMzatirarhanti niyutakanakAni / ' dazavarSasaptisaptatiriha kati gaNakAgraNIH kathaya // 39 // svarNatrizatI mUlyaM dazavarSANAM navAganAnAM syAt / SaTtriMzannArINAM SoDazasaMvatsarANAM kim // 40 // SaTkazatayuktanavaterdazamAsaivRddhiratra kA tasyAH / - kaH kAlaH kiM vittaM viditAbhyAM bhaNa gaNakamukhamukura / / 41 / / saptarAzika uddezakaH / tricatuLasAyAmau zrIkhaNDAvahato'STahemAni / SaNNavavistRtidairdhyA hastena caturdazAtra kati // 42 // iti saptarAzikaH || Badde 71 at the end K, M and B read hamako : nA.

Page #102
gaNitasArasaGgrahaH navarAzika uddezakaH / 'paJcASTatrivyAsadairyodayAmbho dhatte vApI zAlinI vAhaSaTkam / saptavyAsA hastataH SaSTidairdhyAH pAsadhoH kiM navAcakSva vidvan // 43 // iti sArasaGghahe gaNitazAstre mahAvIrAcAryasya kRtau trairAziko nAma caturthavyavahAraH. The following stanza is found in Kanit B in addition to stanza No. 48. dvathaSTAzItivyAsadaiyonatAmbho patte vApI zAlinI saardhvaahau| hastAdaSTAyAmakAH SoDazAMcchAH paTakavyAsA: ki catanA vara lam /

Page #103
pacamaH mizrakavyavahAraH. prAptAnantacatuSTayAn bhagavatastIrthasya kartRn jinAn siddhAn zuddhaguNAMstrilokamahitAnAcAryavaryAnapi / siddhAntArNavapAragAn bhavabhRtAM netanupAdhyAyakAn sAghUn sarvaguNAkarAn hitakarAn vandAmahe zreyase // 1 // itaH paraM mizragaNitaM nAma paJcamavyavahAramadAhariSyAmaH / tadyathAsaskramaNasaMjJAyA viSamasaGkramaNasaMjJAyAzca sutram yutivitidalanakaraNaM sakramaNaM chedalabdhayo rAzyoH / sakramaNaM viSamamidaM prAhugaNitArNavAntagatAH // 2 // atroddezakaH / dvAdazasaGkhyArAzeAbhyAM sakramaNamatra kiM bhavati / tasmAdrAzermaktaM viSamaM vA kiM tu sahakramaNam // 3 // pazcarAzikavidhiH // pazcarAzikavarUpavRddhyAnayanasUtramicchArAziH svasya hi kAlena guNaH prapANaphalaguNitaH / kAlapramANabhakto bhavati tadicchAphalaM gaNite // 4 // atroddezaka / trikapazcakapadRzate pacAzatvaSTisaptatipurANAH / lAbhArthataH prayuktAH kA dirmAsaSaTsya // 5 //

Page #104
66 gaNitasArasaGgrahaH vyardhASTakazatayuktAstriMzatkArSApaNAH paNAzcASTau // . mAsASTakena jAtA dalahInenaiva kA vRddhiH // 6 // SaSTyA vAdvidRSTA paJca purANAH paNatrayavimizrAH / mAsadvayena labdhA zatavRddhiH kA tu varSasya // 7 // sArdhazatakaprayoge sArdhakamAsena paJcadaza lAmaH / mAsadazakena labdhA zatatrayasyAtra kA vRddhiH // 8 // sASTazatakASTayoge triSaSTikArSApaNA vizA dattAH / saptAnAM mAsAnAM paJcamabhAgAnvitAnAM kim // 9 // mUlAnayanasUtram-- mUlaM khakAlaguNitaM svaphalena vibhAjitaM tdicchaayaaH| * kAlena bhajellabdhaM phalena guNitaM tadicchA syAt // 10 // atroddezakaH / pazcArdhakazatayoge paJca purANAndalonamAsau dvau / vRddhiM labhate kazcit kiM mUlaM tasya me kathaya // 11 // saptatyAH sArdhamAsena phalaM pazcArdhameva ca / vyardhASTamAse mUlaM kiM phalayossArdhayoIyoH // 12 // trikapazvakaSaTUzate yathA navASTAdazAtha pnyckRtiH| pazcAMzakena mizrA SaTsu hi mAseSu kAni mUlAni // 13 // kAlAnayanasUtram-- kAlaguNitapramANaM svaphalecchAbhyAM hRtaM tataH kRtvA / tadihecchAphalaguNitaM labdhaM kAlaM budhAH prAhuH // 15 //

Page #105
mizrakavyavahAraH 67 atroddezakaH / saptArdhazatakayoge vRddhistvaSTAyaviMzatirazItyA / kAlena kena labdhA kAlaM vigaNayya kathaya sarave // 15 // viMzatiSaTzatakasya prayogataH saptagaNaSaSTiH / vRddhirapi caturazItiH kathaya sarave kAlamAzu tvam // 16 // SaTUzatena hi yuktAH SaNNavatirdRddhiratra sandRSTA / saptottarapazcAzat tripaJcabhAgazca kaH kAlaH // 17 // bhANDapratibhANDasUtram - bhANDasvamUlyabhaktaM pratibhANDaM bhANDamUlyasaGgaNitam / svecchAmANDAbhyastaM mANDapratibhANDamUlyaphalametat // 18 // __ atroddezakaH / kItAnyaSTau zuNThyAH palAni SabhiH paNaiH sapAdAMzaiH / pippalyAH palapaJcakamatha pAdonaiH paNaivabhiH // 19 // zuNThyAH palaizca kenacidazItibhiH kati palAni pippalyAH / krItAni vicintya tvaM gaNinavidAcakSva me zIghram // 20 // iti mizrakavyavahAre pazcarAzikavidhiH samAptaH // dRddhividhAnam // itaH paraM mizrakavyavahAre DividhAnaM vyAkhyAsyAmaH / mUladdhimizravibhAgAnayanasUtram-- rUpeNa kAlavRddhyA yutena mizrasya mAgahAravidhim / kRtvA labdhaM mUlyaM vRddhibhlonamizradhanam // 21 // Both Mand Bhave the rroneoun reading kathita svazAtibhiH sara palAni pippatyA:.

Page #106
gaNitasArasahiH atroddeshkH| pazcakazataprayoge dvAdazamAsairdhanaM prayukta cet / sASTA catvAriMzanmizraM tanmUlavRddhI ke // 22 // punarapi mUlAdimizrAvibhAgasUtramicchAkAlaphalanaM svakAlamUlena bhAjitaM saikam / sammizrasya vimaktaM labdhaM mUlaM vijAnIyAt // 21 // atroddezakaH / sArdhadvizatakayoge mAsacatuSaNa kimapi dhanamekaH / datvA mizraM labhate ki mUlyaM syAt trayastriMzat // 24 // kAladdhimizravibhAgAnayanasUtrammUlaM svakAlaguNitaM svaphalecchAbhyAM hRtaM tataH kRtvA / sakaM 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 zobhyam / mizrakRtastanmUlaM mizre kriyate tu saGkramaNam // 29 //

Page #107
mibhakanyavahAraH 89 matroddezakaH / saptatyA vRddhiriyaM catuHpurANAH phalaM ca paJcakatiH / mizra nava paJcaguNAH pAdena yutAstu kiM mUlam // 30 // trikaSazyA datvaikaH kiM mUlaM kena kAlena / prApto'STAdazavRddhi SaTSaSTiH kAlamUlamizraM hi // 31 // adhyardhamAsikaphalaM SaSTyAH pazcArdhameva sandRSTam / vRddhistu caturviMzatiratha SaSTirmUlayuktakAlazca // 32 // pramANaphalecchAkAlamizravibhAgAnayanasUtram mUlaM svakAlavAddhihikatiguNaM chinnamitaramUlena / mizrakRtizeSamUlaM mizre kriyate tu saGkramaNam // 15 // atroddezakaH / madhyardhamAsakasya ca zatasya phalakAlayona mizradhanam / dvAdaza dalasaMmizraM mUlaM triMzatphalaM pazca / / 34 // . mUlakAlavRddhimizravibhAgAnayanasUtramamizrAdunitarAziH kAlastasyaiva rUpalAbhena / saikena bhajenmUlaM svakAlamUlAnitaM phalaM mizrama // 35 // matrodezakaH / pacakazataprayoge na jJAtaH kaalmuulphlraashiH| tanmizra 'hAzItirmUlaM kiM kAlavRddhI ke // 31 // This wrong form in the roading found in the M88.; and the correct for don not satisfy the oxigencies of the metro. Um

Page #108
gaNitasArasahaH bahumUlakAlAdimizrAvamAgAnayanasUtramvimajetsvakAlatADitamUlasamAsena phalasamAsahatam / kAlAbhyastaM mUlaM pRthak pRthak cAdizedRddhim / / 37 // atroddezakaH / catvAriMzatriMzaviMzatipaJcAzadatra mUlAni / mAsAH pazcacatustrikaSaT phalapiNDazcatustriMzat // 38 // bahumUlamizravibhAgAnayanasUtramsvaphalaissvakAlabhaktastadyutyA mUlamizradhanarAzim / 'chindyAdazaM guNayet samAgamo bhavati mUlAnAm / / 39 // atroddeshkH| dazaSatripaJcadazakA raddhaya iSavazcatustriSaNmAsAH / mUlasamAso dRSTazcatvAriMzacchatena saMmizrA // 40 // pazcArdhaSaDdazApi ca sArdhAH SoDaza phalAni ca triMzat / mAsAstu pazca SaT khalu saptASTa dazApyazItiratha piNDaH // 41 bahukAlAmazrAvibhAgAnayanasUtram--- svaphalaiH svamUlabhaktaistadyutyA kAlamizradhanarAzim / 'chindyAdazaM guNayet samAgamo bhavati kAlAnAm // 42 // atroddezakaH / catvAriMzatriMzadizatipaJcAzadatra mUlAni / ' dazaTvipaJcadaza phalamaSTAdaza kAlamizradhanarAziH // 43 // pramANarAzI phalena tulyamicchArAzimUlaM ca tadicchArAzau vRddhi The M99. road fpoort wbiob does not seem to be corroot.

Page #109
mibhakavyavahAraH ca saMpIjya tanmizrarAzau pramANarAzeH vRddhivimAgAnayanasUtram - kAlaguNitapramANaM parakAlahRtaM tadekaguNamizradhanAt / itarArdhakRtiyutAt padamitarArdhAnaM pramANaphalam // 4 4 // atroddezakaH / mAsacatuSkazatasya pranaSTaddhiH prayogamUlaM tat / svaphalena yutaM dvAdaza paJcakRtistasya kAlo'pi / / 45 // mAsatritayAzItyAH pranaSTaddhiH svamUlaphalarAzeH / pazcamabhAgenonAthASTI varSeNa mUlaba ddhI ke // 46 // samAnamUladRddhimizravibhAgasUtramanyonyakAlavinihatamizravizeSasya tasya bhAgAkhyam / kAlavizeSeNa hRte teSAM mUlaM vijAnIyAt // 47 // atroddezakaH / pazcAzadaSTapazcAzanmizraM SaTSaSTireva ca / pazca sapteva nava hi mAsAH kiM phalamAnaya // 18 // trizacakatriMzahitryaMzAH syuH punastrayastriMzat / satryaMzA mizradhanaM pazcatriMzazca gaNakAdAt // 19 // kazcinnarazcaturNA tribhizcaturmizca pathabhiH SabhiH / mAsairlabdhaM kiMsyAnmUlaM zIghraM mamAcakSva // 50 // samAnamUlakAlamizravibhAgasUtramanyonyaddhisaGgaNamizravizeSasya tasya bhAgAkhyam / idivizeSeNa ite labdhaM mUlaM budhAH prAhuH // 51 / /

Page #110
gaNitasArasagrahaH matroddezakaH / ekatripaJcamizritaviMzatiriha kAlamUlayormizram / SaD daza caturdaza syurlAbhAH 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 // trivaddhacaiva zate mAse prayukta zvASTabhizzate / lAbho'zItiH kiyanmUlaM bhavettanmAsayoIyoH // 5 // iddhimUlavimocanakAlAnayanasUtrammUlaM svakAlagaNitaM phalaguNitaM tatpramANakAlAbhyAm / bhaktaM skandhasya phalaM mUlaM kAlaM phalAtprAgvat // 57 // bhatroddezakaH / mAse hi pazcaiva ca saptatInAM mAsadaye'STAdazakaM pradeyam / This carno rule is Binewhat delectively stated again with modification in ronding than : punarapyubhayaprayogamalAnayanasUtram -- icchAkAlAdubhayaprayogavRddhiM samAnIya / tapantarabhaktaM labha mUlaM vijAnIyAt //

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* mizrakavyavahAraH skandhaM caturbhissahitA tvazItiH mUlaM bhavetko nu vimuktikAlaH // 58 // SaSTyA mAsikavRddhiH pazcaiva hi mUlamapi ca paJcatriMzat / mAsatritaye skandhaM tripaJcakaM tasya kaH kAlaH / / 59 / / samAna vRddhimUla mizra vibhAga sUtram -- mUlaiH svakAlaguNitaiIddhivibhaktairasamAsakairvibhajet / mizraM svakAlanighnaM vRddhirmUlAni ca prAgvat // 60 // atroddezakaH H / dvikaSaTTu catuzzatake caturasahastraM catuzzataM mizram / mAsadvayena vRddhayA samAni kAnyatra mUlAni // 61 // trikazatapaJcakasaptatipAdonacanuSkaSaSTiyAMgaMSu / navazatasahasrasaGkhyA mAsatritaye samAyuktA // 62 // vimuktakAlasya mUlAnayanasUtrama - skandhaM svakAlabhaktaM vimuktakAlena tADitaM vibhajet / nirmuktakAlavRddhayA rUpamya hi sakayA mUlam || 63 || 8 atroddezakaH / pazvakazataprayoge mAsau dvau skandhamaSTakaM datvA / mAsaiSSaSTibhiriha vai nirmuktaH kiM bhavenmUlam || 64 || satribhAgI skandhaM dvAdazadinairdadAtyekaH / 78

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gaNitasArasAhaH trikazatayoge dazabhirmAsairmuktaM hi mUlaM kim / / 65 // haDiyuktahInasamAnamUlamizravibhAgasUtramkAlasvaphalonAdhikarUpoddhRtarUpayogahatamizre / prakSepo guNakAraH svaphalonAdhikasamAnamUlAni // 66 // bhatroddezakaH / trikapazcakASTakazataiH prayogato'STAsahasrapaJcazatam / vizatisahitaM vRddhibhiruddhRtya samAni paJcabhirmAsaiH // 6 // trikaSaTrASTakaSaSTyA mAsadvitaye catussahasrANi / padhAzadvizatayutAnyato'STamAsakaphalADhate sahazAni // 68 // dvikapaJcakanavakazate mAsacatuSke trayodazasahasram / saptazatena ca mizrA catvAriMzatsAhisamamUlAni // 69 // saikArdhakapazcArdhakaSaDarghakAzItiyogayuktAstu / mAsASTake SaDadhikA catvAriMzacca SaTutizatAni // 70 // saGkalitaskandhamUlasya mUlavRddhivimuktikAlAnayanasUtramskandhAptamUlacitiguNitaskandhecchAgraghAtiyutamUlaM syAt / skandhe kAlena phalaM skandhoitakAlamUlahatakAlaH // 1 // atroddeshkH| kenApi saMprayuktA SaSTiH pazcakazataprayogeNa / mAptatripaJcabhAgAt saptottaratazca saptAdiH // 72 // tatSaSTisaptamAMzakapadamitisaGkalitadhanameva / datvA tatsaptAMzakavRddhi prAdAca citimUlam / / PN: in the reading found in the MS.; is adopted as being more mtalsstory from matioally

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mizrakavyavahAraH ____78 kiM tadRddhiH kA syAt kAlastadRNasya maukSiko bhavati / 73, / kenApi saMprayuktAzItiH paJcakazataprayogeNa // aSTAdyaSTottaratasadazItyaSTAMzagacchena / mUladhanaM datvASTAdyaSTottarato dhanasya mAsArdhAt / / 75 // vRddhi prAdAnmUlaM saddhizca vimuktikAlazca / eSAM parimANaM kiM vigaNayya sarakhe mamAcakSva // 7 // ekIkaraNasUtram-- vRddhisamAsa vibhajenmAsaphalaikyena labdhamiSThaH kAlaH / kAlapramANaguNitastAdaSTakAlena sambhaktaH // dRddhisamAsena hato mUlasamAsena bhAjito vRddhiH / 77 / atrAdezakaH / yuktA catuzzatIha dvikatrikapanakacatuSazatena / mAsAH paJca caturditrayaH prayogaikakAlaH kaH // 70 // iti mizrakavyavahAre vRddhividhAnaM samAptam / / prakSepakakuTIkAraH // ___ itaH paraM mizrakavyavahAra prakSepakakuTTIkAragaNitaM vyAkhyA. yAmaH / prakSepakakaraNami: svrgvicchednaaNshgutihRtmishrH| . prakSepakaguNakAraH kuTTIkAro vRdhaimsamuddiSTam // 79 // atrAMddezakaH / dvitricatuSpadabhAgaivibhAjyate dviguNaSaSTighi hemrAma / bhRtyebhyo hi catubhyoM gaNakAcakSvAzu maM bhAgAna // 8 // 8-A

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gaNitasArasaGgrahaH prathamasyAMzatritayaM triguNottaratazca pnycbhirbhktm|| dInArANAM trizataM triSaSTisahitaM ka ekAMzaH // 816 // AdAya cAmbujAni pravizya sacchrAvako'tha jinanilayam / pUjAM cakAra bhaktyA pUjAhebhyo jinendrebhyaH // 823 // vRSabhAya caturthAzaM SaSThAMzaM ziSTapArthAya / dvAdazamatha jinapataye vyaMzaM munisuvratAya dadau // 83 / / naSTASTakarmaNe jagadiSTAyAriSTanemaye'STAMzam / SaSThaghnacaturbhAgaM bhaktyA jinazAntaye pradadau // 84 // kamalAnyazItimizrANyAyAtAnyatha zatAni catvAri / kusumAnAM bhAgAkhyaM kathaya prakSepakAkhyakaraNena // 85 // catvAri zatAni sarave yutAnyazItyA naraivibhaktAni / paJcabhirAcakSva tvaM dvitricatuHpa caSaDaNitaiH // 86 // iSTaguNaphalAnayanasUtrambhaktaM zeSairmUlaM guNaguNitaM tena yojitaM prakSepam / tadravyaM mUlapaghnaM kSepavibhaktaM hi mUlyaM syAt / / 87 // asminnarthe punarapi sUtram-- phalaguNakArairhatvA paNAn phalaireva bhAgamAdAya / prakSepake guNAssyustrairAzikataH phalaM vadenmatimAn // 8 // asminnarthe punarapi sUtramsvaphalahatAH svaguNanAH paNAstu tairbhavati pUrvavaccheSaH / iSTaphalaM nirdiSTaM trairAzikasAdhitaM samyak // 89 //

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mizrakavyavahAraH atroddezakaH / dvAbhyAM trINi tribhiH paJca paJcabhissapta mAnakaiH / dADimAmrakapitthAnAM phalAni gaNitArthavit // 90 // kapitthAt triguNaM hyAnaM dADimaM SaDguNaM bhavet / krItvAnaya sarave zIghraM tvaM SaTsaptatibhiH paNaiH // 91 // dadhyAjyakSIraghaTairjinabimbasyAbhiSecanaM kRtavAn / jinapuruSo dvAsaptatipalaistrayaH pUritAH kalazAH // 92 // dvAtriMzatprathamaghaTe punazcaturviMzatiDhitIpaghaTe / SoDaza tRtIyakalaze pRthaka pRthaka kathaya ma kRtvA // 93 // teSAM dadhighRtapayasAM tatazcaturviMzatighRtasya palAni / SoDaza payaHpalAni dvAtriMzada dadhipalAnIha // 94 // vRttistrayaH purANAH puMsavArohakasya natrApi / sarve'pi paJcaSaSTiH kacidganA dhanaM teSAm // 95 // sannihitAnAM dattaM labdhaM paMsA dazaiva caikamya / ke sannihitA bhanAH ke mama sacintya kathaya tvam / / 96 // iSTarUpAdhikahInaprakSepakakaraNamUtram-- . piNDo'dhikarU pono hInottararUpasaMgunaH zeSAt / prakSepakakaraNamanaH kartavyaM tayutA hInAH // 97 / / atroddezakaH / prathamasyaikAMzo'to dviguNadviguNottagarananti nagaH / catvArA zaH kassyAdekasya hi mataSTiriha // 98 //

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7R gaNitasArasAhaH prathamAdadhyardhaguNAt triguNAdrUpottarAdvibhAjyante / sASTA saptatirebhizcaturbhirAptAMzakAn brUhi // 99 // prathamAdadhyardhaguNAH pazcArdhaguNottarANi rUpANi / paJcAnAM paJcAzatsaikA caraNatrayAbhyaSikA // 10 // prathamAtpazcArdhaguNAzcaturguNottaravihInabhAgena / bhaktaM naraizcaturbhiH paJcadazonaM zatacatuSkam // 1016 // samadhanAryAnayanatajjyeSThadhanasaGkhyAnayanasUtram--- jyeSThadhanaM saikaM syAt svavikraye'ntyArghaguNamapaikaM tat / krayaNe jyeSThAnayanaM samAnayet karaNaviparItAt // 102 // atroddezakaH / hAvaSTau SaTrizanmUlaM nRNAM SaDeva caramArghaH / ekAdhaiMNa krItvA vikrIya ca samadhanA jAtAH // 1036 / / sAdhaikamardhamardhadvayaM ca sagRhya te trayaH puruSAH / krayavikrayau ca kRtvA SaDDiH paznArdhAtsamadhanA jAtAH // 10 catvAriMzat saikA samadhanasaGkhyA SaDeva crmaarghH| . AcakSva gaNaka zIghaM jyeSThadhanaM kiM ca kAni mUlAni // 105 // samadhanasaGkhyA pazcatriMzadbhavanti yatra dInArAH / catvArazcaramA? jyeSThadhanaM kiM ca gaNaka kathaya tvam // 10 // caramArghabhinnajAtI samadhanA_nayanasUtramtukhyApacchedadhanAntyAIbhyAM vikrayakrayA! prAgvat / chedacchedakRtighAvanupAtAt samadhanAni bhinne'ntyApeM // 10 //

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mizrakavyavahAraH ardhatripAdabhAgA dhanAni SaTpaJcamAMzakAzcaramArghaH / ekArgheNa krItvA vikrIya va samadhanA jAtAH // 108 // punarApa antyArdhe bhinne sati samadhanAnayanasUtramjyeSThAMzadviharahatiH sAntyaharA vikrayo 'ntyamUlyannaH / naiko dvyakhilaharaghnaH syAtkrayasaGkhyAnupAto'tha // 109 // atroddezakaH / artha hau dhyaMzau ca trIn pAdAMzAMzca saGgRhya | vikrIya krItvAnte pacabhirakGgyaMzakaissamAnadhanAH // 110 // iSTaguNeSTasaGkhyAyAmiSTasaGkhyAsamarpaNAnayanasUtram- antyapade svaguNahate kSipedpAntyaM ca tasyAntam / tenopAntyena bhajedyalabdhaM tadbhavenmUlam // 111 // atroddezakaH / kacicchrAvaka puruSazcaturmukhaM jinagRhaM samAsAdya / pUjAM cakAra bhaktyA surabhINyAdAya kusumAni // 112 // dviguNamabhUdAdya mukhe triguNaM ca caturguNaM ca pavaguNan / sarvatra pazJca paJca ca tatsaGkhyAmbhoruhANi kAni syuH / / 113 / / caturbhAgaguNa pathArdhaguNAstri saptASTau / ' bhaktairbhaktyAbhyo dattAnyAdAya kusumAni // / 1943 / / iti mizrakavyavahAre prakSepaka kukAraH samAptaH // 79 The following stanza is added in M after stanza No. 119g, but it is not found B: resurebhAgA dhanAni papazvamAMzakAntyArthaH / ekArgheNa krItvA vikrIya na mamadhanA jAtAH //

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80 gaNitasArasaGgrahaH kAkukAraH // itaH paraM vallikAkuTTIkAragaNitaM vyAkhyAsyAmaH / kuTTIkAre balchikAgaNitanyAyasUtram- chitvA chedena rAzi prathamaphalama pohyAptamanyonyabhaktaM sthApyordhvAdharyato'dho matiguNamayujAlape'vaziSTe dhanarNam / chitvAdhaH svoparino pariyutahara bhAgo'dhikAgrasya hAraM chitvA chedena sAgrAntaraphalamadhikAgrAnvitaM hAraghAtam // 119 // atroddezakaH / 5: jambUjambIrarambhAkamukapanasarakharjUra hintAlatAlIpunAgAmrAdyanekadramakusumaphalairnanazAsvAdhirUDham / bhrAmyaGgAnavApIzukapikakulanAnAdhvanivyAptadikaM pAnthAH zrAntA vanAntaM zramanudamamalaM te praviSTAH prahRSTAH // 116 11 rAzitriSaSTiH kadalIphalAnAM sampIDya saMkSipya ca saptabhistaiH / pAnthaistrayoviMzatibhirvizuddhA rAzestvamekasya vada pramANam // 117 // rAzIn punardvAdaza dADimAnAM samasya saMkSipya ca paJcabhistaiH / pAnyairnaraiviMzatibhirnirekai - bhaktAMstathaikasya vada pramANam | 1183 // vAzI pathiko yaka triMzatsamUhaM kurute trihInam /

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mizrakavyavahAraH 81 zeSe hRte saptatibhistrimitrai narairvizuddhaM kathayaikasaGkhyAm // 119 // dRSTAssaptatriMzatkapitthaphalarAzayo vane pathikaiH / saptadazApohya hate vyekAzItyAMzakapramANaM kim // 1203 // dRSTvAmrarAzimapahAya ca sapta pazcAdakte'STabhiH punarapi pravihAya tasmAt / trINi trayodazabhiruddalite vizaddhaH pAnthairvane gaNaka me kathathaikarAzim // 121 // dvAbhyAM tribhizcaturbhiH panAbhirekaH kapitthaphalarAziH / bhakto rUpAgrastatpramANamAcakSva gaNitajJa // 122 // dvAbhyAmekastribhiTThauM ca caturbhirbhAjine trayaH / catvAri paJcabhizzeSaH ko rAzirvada meM priya // 123 / / dvAbhyAmekastribhizzuddhazcatubhijite trayaH / catvAri paJcabhizzeSaH ko rAzirvada meM priya // 124 // dvAbhyAM niragra ekAgrastribhirnAgro vibhAjitaH / caturbhiH paJcabhirbhako rUpAyo rAzireSa kaH // 125 // dvAbhyAmekastribhizzaddhazcatubhirbhAjite trayaH / / niragraH paJcabhirbhaktaH ko rAziH kathayAdhunA || 126 // dRSTA jambUphalAnAM pathi pathikajanai rAzayastatra gazI dvau vyagrau nau navAnAM traya iti punakAdazAnAM vibhaktAH / pazcAgrAste yatInAM cataradhikatarAH pana te saptakAnAM kuTTIkArArthavinme kathaya gaNaka saJcintya gazipramANam // 127 //

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80 gaNitasArasagrahaH banAntare dADimarAzayaste pAnyaistrayassaptabhirekazeSAH / sapta trizeSA navabhirvibhaktAH paJcASTabhiH ke gaNaka hiragrAH / / 128 // bhaktA dviyuktA navabhistu pazca yuktAzcaturbhizca SaDaSTabhistaiH / pAnthairjanaissaptabhirekayuktA catvAra ete kathaya pramANam / / 129 // agrazeSavibhAgamUlAnayanasUtramzeSAMzAgravadho yuk svAgreNAnyastadaMzakena guNaH / pAvadAgAstAvadvicchedAH syustadagraguNAH // 130 // atroddezakaH / . mAnItavatyAmraphalAni puMsi prAgekamAdAya punastadardham / gate'graputre ca tathA jaghanyastatrAvazeSArdhamatho tamanyaH // 11 // pravizya jainaM bhavanaM tripUruSaM prAgekamabhyarcya jinasya paade'| zeSatribhAgaM prathame'numAne tathA dvitIye ca tRtIyake tathA // 12 // zeSatribhAgaha yatazca zeSa'yaMzadvayaM cApi tatasvibhAgAn / kRtvA caturvizatitIrthanAthAn . samarcayitvA gatavAn vizuddhaH // 133 // iti mizrakavyavahAre sAdhAraNakuTIkAraH samAptaH // The M88. giro on, which does not seem to be correct bero. B roade sur pAre.

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mizrakavyavahAraH 8 viSamakuTIkAraH // itaH paraM viSamakuTTIkAraM vyAkhyAsyAmaH / viSamakuTTIkArasya sUtram-- * matisaGgANitI chedI yojyonatyAjyasaMyutau rAzihatau / bhinne kuTTIkAre guNakAro'yaM samuddiSTaH // 1343 // atroddezakaH / rAziH SaTrena hato dazAnvito navahato niravazeSaH / dazabhitnazca tathA tadguNako' ko mamAzu saGkathaya / / 135 // mklkuttttiikaarH|| sakalakuTTIkArasya sUtrambhAjyacchedAgrazeSaiH prathamahatiphalaM tyAjyamanyAMnyabhaktaM nyasyAnte sAgramUva~rupariguNayutaM taissamAnAsamAne / varNanaM vyAptahArau guNadhanamRNayozcAdhikAgrasya hAra hatyA hatvA tu sAmAntaradhanamAdhikAgrAnvitaM hAraghAtama / / 131 // bhatroddezakaH / saptottarasaptatyA yutaM zataM yojyamAnamaSTatriMzat / saikazatadvayabhaktaM ko guNakAro bhavedatra || 137 // pazcatriMzat vyuttaraSoDazapadAnyeva hArAzca / / dvAtriMzayAdhikakA vyuttarato'grANi ke dhanarNaguNAH // 13 // adhikAlparAzyormUlamizravibhAgasUtram-- jyeSThanamahArAzerjaghanyaphalanADitAnamapanIya / phalavargazeSabhAgo jyeSThArtho'nyo guNasya viparItam // 139 // 'B guNakArI.

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84 gaNitasArasaGgahaH atroddezakaH / navAnAM mAtuluGgAnAM kapitthAnAM sugandhinAm / saptAnAM mUlyasammizraM saptottarazataM punaH // 1403 // saptAnAM mAtuluGgAnAM kapitthAnAM sugandhinAm / navAnAM mUlyasammizramekottarazataM punaH // 14 1 // mUlye te vada me zIghraM mAtuluGgakapitthayoH / anayorgaNaka tvaM me kRtvA samyak pRthak pRthak // 142 // bahurAzimizratanmUlyamizravibhAgasUtramiSTapraphalairUnitalAbhAdiSTAptaphalamasakRt / tairUnitaphaLapiNDasacchedA guNayutAstadarghAssyuH // 1 4 3 // atroddezakaH / atha mAtuluGgakadalIkapitthadADimaphalAni mizrANi / prathamasya saikaviMzatiratha dviragrA dvitIyasya / / 1443 // viMzatiratha surabhINi ca punastrayoviMzatistRtIyasya / teSAM mUlyasamAsastrisaptatiH kiM phalaM ko'rghaH // 1453 // jaghanyonamilitarAzyAnayanasUtram-- paNyahRtAlpaphalonaizchindyAdalpanamUlyahIneSTam / kRtvA tAvatkhaNDaM tadUnamUlyaM jaghanyapaNyaM syAt // 146 // atroddezakaH / dvAbhyAM trayo mayUrAstribhizca pArAvatAzca catvAraH / haMsAH paJca caturbhiH paJcabhiratha sArasASaTa ca // 147 //

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85 mizrakavyavahAraH yatrAstatra sarave SaTpaJcAzatpaNaiH ravagAn kriitvaa| dvAsaptatimAnayatAmityuktvA mUlamevAdAt / katibhiH paNaistu vihagAH kati vigaNayyAzu jaaniiyaaH|| 149 // tribhiH paNaiH zuNThipalAni paJca caturbhirekAdaza pippalAnAm / aSTAbhirekaM maricasya mUlyaM SaSTyAnayASTottaraSaSTimAza // 150 // iSTAMrghariSTamUlyauraSTavastupramANAnayanasUtrammUlyanaphalecchAguNapaNAntareSTanayutiviparyAsaH / dviSThaH svadhaneSTaguNaH prakSepakakaraNamavaziSTam / / 151 // atroddezakaH / tribhiH pArAvatAH paJca paJcabhissapta sArasAH / saptabhinava haMsAna navanizzirivanastrayaH // 152 / / krIDArtha nRpaputrasya zatena zatamAnaya / ityuktaH prahitaH kazcit tena kiM kasya dIyate // 15 // vyastArthapaNyapramANAnayanasUtram'paNyaikyena paNaikyamantaramataH paNyeSTapaNyAnarezchindyAtsaGkamaNe kRte tadubhayorabhavatAM punaH / paNye te khalu pa rayogavivare vyastaM tayorghayoH / praznAnAM viduSAM prasAdanamidaM sUtraM jinendrAditam / / 154 / / atroddezakaH / AdyamUlyaM yadekasyaM candanasyAgarostathA / palAni viMzatimizraM caturagrazataM paNAH // 155 // INot found in any of the Mgs. consuited.

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gaNitasArasahaH kAlena vyatyayArghassyAtsaSoDazazataM paNAH / tayorarghaphale dhUhi tvaM SaDaSTa pRthak pRthak // 15 // sUrparathAzveSTayogayojanAnayanasUtram arivalAptArivalayAjanasaGkhyAparyAyayojanAni syuH / tAnISTayogasaGkhyAninAnyekaikagamanamAnAni // 157 // mnoddeshkH| ravirathaturagAssapta hi catvAro'zvA vahanti dhUryuktAH / yojanasaptatigatayaH ke vyUDhAH ke caturyogAH // 158 // sarvadhaneSTahInazeSapiNDAt svasvahastaga dhanAnayanasUtramrUponanaraivibhajet piNDIkRtabhANDasAramupalabdham / sarvadhanaM syAttasmAduktavihInaM tu hastagatam // 159 // atroddezakaH / vaNijaste catvAraH pRthak pRthaka zaulkikena paripRSTAH / kiM bhANDasAramiti khalu tatrAhaiko vaNikaccheThaH // 160 // AtmadhanaM vinigRhya dvAviMzatiriti tataH paro'vocat / tribhiruttarA tu viMzatiratha caturadhikaiva viMzatisturyaH !! 161 / / saptottaraviMzatiriti samAnasArA nigRhya sarve'pi / UcuH kiM brUhi sarakhe pRthaka pRthagbhANDasAraM me // 162 // bhanyo'nyamiSTaratnasaGkhayAM dattvA samadhanAnayanasUtram-- puruSasamAsena guNaM dAtavyaM tadvizodhya paNyebhyaH / zeSaparasparaguNitaM khaM khaM hitvA maNermUlyam // 163 / /

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mibhakavyavahAraH atroddeshkH| prathamasya zakranIlAH SaT sapta ca marakatA dvitiiysy| vajANyaparasyASTAvekaikArgha pradAya samAH // 164 // prathamasya zakranIlAH SoDaza daza marakatA dvitIyasya / vajAstRtIyapuruSasyASTau hau tatra datvaiva // 165 // teSekaiko'nyAbhyAM samadhanatAM yAnti te trayaH puruSAH / tacchakanIlamarakatavajrANAM kiMvidhA arghAH // 166 // kayavikrayalAbhaiH mUlAnayanasUtram--- anyo'nyamUlyaguNite vikrayabhakte kayaM yadapa kabdham / / tenaikonena hato lAbhaH pUrvAhRtaM mUlyam // 167 / / atroddezakaH / tribhiH krINAti saptava vikrINAti ca pazmabhiH / nava prasthAn vaNik kiM syAllAbho hAsaptatirdhanam // 16 // iti mizrakavyavahAre sakalakuTIkA': samAptaH // suvarNakuTIkAraH // itaH paraM suvarNagaNitarUpakuTIkAraM syAmyAmaH / samasteSTavagairekIkaraNena saGkaravarNAnayanasUtram--- kanakakSayasaMvargoM mizrasvarNAhataH kSayo jJeyaH / paravarNabhavimakaM suvarNaguNitaM phalaM hemnaH // 169 //

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88 gaNitasArasaGgrahaH atroddezakaH / ekakSayamekaM ca dvikSayamekaM trivarNamekaM ca / varNacatuSke ca he paJcakSayikAzca catvAraH // 170 // sapta caturdazavarNAstriguNitapaJcakSayAzcASTau / etAnekIkRtya jvalane kSiptvaiva mizravarNa kim / etamizrasuvarNa pUrvebhaktaM ca kiM kimekasya // 171 // iSTavarNAnAmiSTasvavarNAnayanasUtramsvaissvairvarNahamizraM svarNamizreNa bhAjitam / labdhaM varNa vijAnIyAttadiSTAptaM pRthak pRthak // 1723 // atroddeshkH| viMzatipaNAstu SoDaza varNA dazavarNaparimANaiH / parivartitA vada tvaM kati hi purANA bhavantyadhunA // 173 // aSTottarazatakanakaM varNASTAMzatrayeNa saMyuktam / / ekAdazavarNaM caturuttaradazavarNakaiH kRtaM ca kiM hema / / 174 // ajJAtavarNAnaganasUtramkanakakSayasaMvarga mizraM varNanamizrataH zodhyam / svarNena hRta varNa varNavizeSeNa kanakaM syAt // 175 // ajJAtavarNasya punarapi sUtramsvasvarNavarNavinihatayogaM svarNaikyadRDhahatAcchodhyam / ajJAtavarNahenA bhaktaM varNaM budhAH prAhuH // 1763 / / atraaddeshkH| 'SaDjaladhivahikanakaistrayodazASTartuvarNakaiH krmshH| Hore vahi is ruletituted for ranala, and tuvarNakaiH for TAvRtukSayaH, us thereby ___ bhe reading will be better grainmatically,

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89 mizrakavyavahAraH ajJAtavAhana : paJca vimizrakSayaM ca saikadaza / ajJAtavarNasaGkhayAM brUhi sarave gaNitatacvajJa / / 178 / / caturdazaiva varNAni sapta svarNAni ttkssye'| caturavaNe dazotpannamajJAtakSayakaM vada // 179 // ajJAtavarNAnayanamUtramkhasvarNavarNavinihatayogaM svarNekyaguNitadRDhavarNAt / tyaktvAjJAtasvarNakSayadRDhavarNAntarAhRtaM kanakam / / 180 / / atroddezakaH / dvitricatuHkSayamAnAstristriH kanakAstrayodazakSayikaH ! varNayutirdaza jAtA bahi samve kanakaparimANam // 181 // gugmavarNamizrasuvarNAnaganasUtram- . jyeSThAlpakSayazodhitapakvavizeSAptarUpakaiH prAgvat / prakSepamataH kuryAdevaM bahuzo'pi vA sAdhyama // 182 // panarapi yugmavarNamizrasvarNAnayanasUtram iSTAdhikAntaraM caiva hIneSTAntarameva c| ubhe te sthApagaDyastaM svarNa prazaMpataH phalam / / 183 // atraaddeshkH| dazavarNasuvarNaM gata SoDazavarNana saMyutaM pakvam / dvAdaza cetkanakazataM dvimaMdakanakaM pRthaka pRthagyahi / / 184 // bahusuvarNAnaganamUtram vyekapadAnAM kramazaH svarNAnASTAni kalpagacchapam / avyaktakanakavidhinA prasAdhayaMta prAktanAyava / / 185 // The reading in the M88 is 1979, which is obviously run ou.

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gaNitasArasaGgrahaH atroddeshkH| varNAzzaratunagavasumRDavizve nava ca pakvavarNaM hi| kanakAnAM SaSTizcet pRthaka pRthaka kanakamA kiM syAt / / 186 // dvayanaSTavarNAnayanasUtram-- svarNAbhyAM hatarUpe suvarNavarNAhate dviSThe / svasvarNahRtaikena ca hInayute vyastato hi varNaphalam // 187 // atroddeshkH| SoDazadazakanakAbhyAM varNaM na jJAyate' pakvam / varNa caikAdaza cedvau~ tatkanakayorbhavetAM ko // 18 // punarapi dvayanaSTavarNAnayanasUtram -- ekasya kSayamiSTaM prakalpya zeSaM prasAdhayet prAgvat / bahukanakAnAmiSTaM vyekapadAnAM tataH prAgvat // 189 // __ atroddeshkH| dvAdazacaturdazAnAM svarNAnA samarasIkate jAtam / varNAnAM dazakaM syAt tadvau~ brUhi sshcinty|| 190 // aparArdhasyodAharaNam / saptanavazividazAnAM kanakAnAM saMyute pakvam / dvAdazavarNa jAtaM kiM brUhi pRthak pRthagvarNam // 191 // parIkSaNazalAkAnayanasUtram paramakSayAptavarNAH sarvazalAkAH pRthak pRthagyojyAH / varNaphalaM tacchodhyaM zalAkapiNDAt prapUrANakA // 192 // Budda hers yte|

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91 mizrakavyavahAraH atroddeshkH| vaizyAH svarNazalAkAnikIrSavaH svarNavarNajJAH / cakruH svarNazalAkA dvAdazavarNaM ta dAdyasya // 193 // caturuttaradazavarNa SoDazavarNaM tRtIyasya / kanakaM cAsti prathamasyaikonaM ca dvitIyasya / / 194 // ardhArdhanyUnamatha tRtIyapuruSasya pAdonam / paravarNAdArabhya prathamasyaikAntyameva ca dyantyam // 195 // uyantyaM tRtIyavaNijaH sarvazalAkAstu mApamitAH / zuddhaM kanakaM kiM syAt prapUraNI kA pRthak pRthak tvaM me| AcakSva gaNaka zIghraM suvarNagaNitaM hi yadi vetsi // 196 // vinimayavarNasuvarNAnayanasUtram - krayaguNasvarNavinimayavarNeSTanAntaraM punaH sthaapym| vyastaM bhavati hi vinimayavarNAntarahRtphalaM kanaka : / 1976 // ___ atroddeshkH| poDazavarNa kanakaM saptazataM vinimayaM kRtaM labhate / dvAdazadazavarNAbhyAM sASTasahavaM tu kanakaM kim // 198 // bahupadavinimayamuvarNakaraNasUtram --- varNanakanakamiSTasvarNanAptaM dRDhakSayAM bhavati / prAgvatprasAdha labdhaM vinimayabahupadasuvarNAnAma // 199, // atroddeshkH| varNacaturdazakanakaM zatatrayaM vinimayaM prakurvantaH / vargAdazadazavasunagazca zatapacakaM svarNam / . 9-A

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92 gaNitasArasaGgrahaH eteSAM varNAnAM pRthak pRthaka varNamAnaM kim // 201 // vinimayaguNavarNakanakalAbhAnayanasUtram vrnnnvrnnyutihRtgunnytimuulkssynruuponen| AptaM labdhaM zodhyaM mUladhanAccheSavittaM syAt / / 202 / / tallabdhamUlayogAdvinimayaguNayogabhAjitaM lbdhm| prakSepakeNa guNitaM vinimayaguNavarNakanakaM syAt / / 203 / / atroddeshkH| kazcidvaNik phalArthI SoDazavarNaM zatadvayaM kanakam / yatkicidvinimayakRtamekAdyaM dviguNitaM yathA krmshH|| 204 // dvAdazavasunavadazakakSayakaM lAbho dvirgrshtm| zeSaM kiM syAdvinimayakAMsteSAM cApi me kathaya / 205 // dRzyasuvarNavinimayasuvarNemUlAnayanasUtram --- vinimayavarNenAptaM svAMzaM veSTakSayanasaMmizrAt / aMzaikyonenAptaM dRzyaM phalamatra bhavati mUladhanam / / 206 // atroddezakaH / vaNijaH kaMcit SoDazavarNakasauvarNagulakamAhRtya / tricatuHpaJcamabhAgAn krameNa tasyaiva vinimayaM kRtvA / / 207 // dvAdazadazanavavarNaiH saMyujya ca pUrvazeSeNa / mUlena vinA dRSTaM svarNasahasraM tu kiM mUlam // 208 / / iSTAMzadAnena iSTavarNAnayanasya tadiSTAMzakayoH suvarNAnayanasya ca sUtrama aMzAptakaM vyastaM kSiptveSTannaM bhavet suvarNamayI / sA gulikA tasyA api parasparAMzAptakanakasya // 209 //

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02 mizrakavyavahAraH khadRDhakSayeNa varNI prakalpayetprAgvadeva ythaa| evaM tavayayorapyubhayaM sAmyaM phalaM bhavedyadi cet / / 210 / / prAkkalpaneSTavo gulikAbhyAM nizcayau bhavataH / no cetprathamasya tadA kiJcinnyUnAdhiko kSayau kRtvA // 211 / / tatkSayapUrvakSayayorantarite zeSamatra saMsthApya / trairAzikavidhilabdhaM varNI tenonitAdhiko spssttau|| 2 12 // atroddezakaH / varNaparIkSakavaNijau parasparaM yAcitI tataH prathamaH / ardhaM prAdAt tAmapi gulikAM svasuvarNa Ayojya // 213 / ' varNadazakaM karomItyaparo'vAdIt tribhaagmaatrtyaa| labdhe tathaiva pUrNa dvAdazavarNa karomi gulikAbhyAm / / 214 / / ubhayoH suvarNamAne varNI savintya gaNitatattvajJa / sauvarNagaNitakuzalaM yadi te'si nigadyatAmAzu // 215 / / iti mizrakavyavahAre suvarNakuTTIkAraH samAptaH // vicitrkuttttiikaarH| itaH paraM mizrakavyavahAra vicitrakuTIkAraM vyAkhyAsyAmaH / satyAnRtasUtram puruSAH saikeSTaguNA dviguNeSTonA bhavantyamatyAni / puruSakRtistairUnA satyAni bhavanti vacanAni / / 216 // atroddeshkH| kAmukapuruSAH paJca hi vezyAyAzca priyAstrayastatra / pratyekaM sA bRte tvamiSTa iti kAni satyAni // 217 //

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gaNitasArasaGgrahaH prastArayogabhedasya sUtrama ekAdyakottarataH padamUrdhvAdharyataH kramotkramazaH / sthApya pratilomannaM pratilomannena bhAjitaM sAram // 218 // atroddezakaH / varNAzcApi rasAnAM kaSAyatiktAmlakaTu kalavaNAnAm / madhurarasena yutAnAM bhedAn kathayAdhunA gaNaka // 219 // vajendranIlamarakatavidrumamuktAphalasta racitamAlAyAH / kati bhedA yutibhedAt kathaya sarave samyagAzu tvam / / 220 // ketakyazokacampakanIlotpalakusumaracitamAlAyAH / kati bhedA yutibhedAtkathaya sakhe gaNitatattvajJa // 221 / / jJAtAjJAtalAbhairmUlAnayanasUtram--- lAbhonAmazrarAzeH prakSepakataH phalAni saMsAdhya / tena hRtaM tallabdhaM mUlyaM tvajJAtapuruSasya // 222 / / atroddezakaH / samaye kecidvaNijastrayaH krayaM vikrayaM ca kurin / prathamasya SaT purANA aSTau mUlyaM dvitIyasya // 223 / / na jJAyate tRtIyasya vyAptistai raistu SaNNavatiH / ajJAtasyaiva phalaM catvAriMzaddhi tenAptam // 224 // kastasya prakSepo vaNijorubhayorbhavecca ko lAbhaH / pragaNayyAcakSva sarave prakSepaM yadi vijAnAsi // 225 // bhATakAnayanasUtram bharabhRtigatagamyahatiM tyaktvA yojanadalaghamArakRteH / tanmUlonaM gamyacchinnaM gantavyabhAjitaM sAram / / 226 / Mund Budd ta here; metrically it is faulty.

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95 mizrakavyavahAraH atroddezakaH / panasAni dvAtriMzannItvA yojanamasau dalonASTau / gRhAtyantarbhATakamadhai bhano'sya kiM deyam // 227 // * dvitIyatRtIyayojanAnayanasya sUtram bharabhAThakasaMvargo'dvitIyabhatikRtivivarjitazchedaH / tatyantarabharagatihatergatiH syAda dvitIyasya // 228 // atroddezakaH / panasAni caturviMzatimA nItvA paJcayojanAni naraH / lamate tatimiha nava SaDbhRtiviyute dvitIyanRgatiH kaa|| 229 // bahupada bhATakAnayanasya sUtram sannihitanarahateSu prAguttaramizriteSu mArgeSu / vyAvRttanaraguNeSu prakSepakasAdhita mUlyama / / 230 // atroddeshkH| zibikA nayAnta puruSA viMzatiratha yojanadvayaM naSAm / vRtti-nArANAM viMzatyadhikaM ca saptazatam // 231 // krozadvaye nivRttI vAvubhayoH krozayostrayazcAnye / pazca naraH zeSArdhAdyAvRttAH kA bhRtistapAma // 232 // iSTaguNitapoTTalakAnayanasUtram---- saikaguNA svasveSTaM hitvAnyAnyavazeSamitiH / apavartya yojya mUlaM(viSNoH) kRtvA DhaMgakena mUlena / / 233 / / pUrvApavartarAzIn hatvA pUrvApavartagaziyuteH / pRthageva pRthak tyaktvA hastagatAH svadhanasaGkhyAH syuH / / 23 4 / / tAH svasvaM hitvaiva tvazeSayogaM pRthaka pRthaka sthApya / svagaNanAH svakaragatairUnAH poTalakasamayAH syuH / / 235 / / 'Bonits pada here.

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gaNitasArasaGgrahaH atroddezakaH / mArge tribhirvaNigbhiH podRlakaM dRSTamAha tatraikaH / pAlakamidaM prApya dviguNadhano'haM bhaviSyAmi // 236 // hastagatAbhyAM yuvayA striguNadhano'haM dvitIya Aheti / ' paguNo'haM tvaparaH polaha stasthamAnaM kim // 237 // sarvatulyaguNakapoTTala kAnayanahastagatAnayanasUtram- 96 vyekapadavyeka guNeSTAMzavaghonitAMzayutiguNaghAtaH ! 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 // iSTaziSTaguNapoTTalakAnayanasUtrama - iSTaguNannAnyAMzAH seSTAMzAH saikanijaguNahatA yuktAH / padaneSTazinyUnAH saikeSTa guNahatA hastagatAH // 241 // atroddezakaH / dvAbhyAM pathi pathikAbhyAM poTTalakaM dRSTamAha tatraikaH / asyArdha samprApya dviguNacano'haM bhaviSyAmi // 242 // aparastryaMzadvitayaM triguNadhanastvatkarasthadhanAt / matkaradhanena sahitaM hastagataM kiM ca poTTalakam // 243 // dRSTaM pathi pathikAbhyAM poTTalakaM tadgRhItvA ca / dviguNamabhUdAdyastu svakarasthadhanena cAnyasya //

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mizrakavyavahAraH 97 hastasthadhanAdanyastriguNaM kiM karagataM ca pohalakam / 244 // mArge naraizcaturbhiH poTTalakaM dRSTamAha tatrAdyaH / poTTalakAmada labdhvA hyaSTaguNo'haM bhaviSyAmi // 245 // * svakarasthadhanenAnyo navasaGgaNitaM ca zeSadhanAt / dazaguNadhanavAnaparastvekAdazaguNitadhanavAna syAt / poTTalakaM kiM karagatadhanaM kiyahi gaNakAzu // 247 / / mArge naraiH poTTalakaM catubhirdaSTaM hi tasyaiva tadA babhUvuH / paJcAMzapAdArdhatRtIyabhAgAstadvitripananacaturguNA'zca / / 248 // mArge tribhirvaNigbhiH poTTalakaM dRSTamAha tatrAdyaH / yadyasya caturbhAgaM labhe'hamityAha sa yuvayorDiguNaH // 249 // Aha tribhAgamapara: svahastadhanasahitameva ca trigaNaH / asyAdhaM prApyAhaM tRtIyapuruSazcatudhanavAn syAm / AcakSva gaNaka zIghraM kiM hastagataM ca poTTalakam // 250 / / yAcitarUpairiSTaguNakahastagatAnayanasya sUtramyAcitarUpaikyAni svasaikaguNavardhitAni taiH prAgvat / hastagatAnA nItvA ceSTaguNagneti sUtreNa // 251 // sadRzacchedaM kRtvA saikeSTaguNAhanaSTaguNayutyA / rUmenitayA bhaktAna tAneva karasthitAn vijAnIyAt // 252 // bhatroddezakaH / vaizyastribhiH parasparahastagataM yAcitaM dhanaM prathamaH / catvAryatha dvitIyaM pazca tRtIyaM naraM prArthya // 253 // 'M and Bread : ; and it is obviously inappropriu in.

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98 gaNitasArasaGgrahaH dviguNo'bhavavitIyaH prathamaM catvAri SaT tRtIyamagAt / triguNaM nRtIthapuruSaH prathamaM paJca dvitIyaM ca // 2543 // SaT prArthyAbhUtpaJcakaguNaH svahastasthitAni kAni syuH / kathayAzu citrakuTTImizraM jAnAsi yadi gaNaka // 255 // puruSAstrayo'tikuzalAzcAnyonyaM yAcitaM dhanaM prathamaH / sa dvAdaza dvitIyaM trayodaza prArthya tatriguNaH // 256 / / prathamaM daza trayodaza tRtIyamabhyarthya ca dvitIyo'bhUt / pazcaguNito dvitIyaM dvAdaza daza yAcayitvAdyam // 2573 saptaguNitastRtIyo'bhavannaro vAJchitAni labdhAni / kathaya sakhe vigaNayya ca teSAM hastasthitAni kAni syuH / / 258 antyasyopAntyatulyadhanaM davA samadhanAnayanasUtramvAJchAbhaktaM rUpaM sa upAntyaguNaH sarUpasaMyuktaH / zeSANAM guNakAraH saiko'ntyaH karaNametatsyAt / / 259 // atroddezakaH / vaizyAtmajAstrayaste mArgagatA jyeSThamadhyamakaniSThAH / svadhane jyeSTho madhyamadhanamAtraM madhyamAya dadau // 2603 // sa tu madhyamo jaghanyajadhanamAtraM yacchati smAsya / samadhanikAH syusteSAM hastagataM brUhi gaNaka saMzcintya // 261, vaizyAtmajAzca paJca jyeSThAdanujaH svakIyadhanamAtram / lebhe sarve'pyevaM samavittAH kiM tu hastagatam // 262 / /

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99 mizraka vyavahAraH vaNijaH pazca svasvAdarthaM pUrvasya dattvA tu / samavittAH saJcintya ca kiM teSAM brUhi hastagatam / / 263 3 / / vaNijaSTra svadhanAdddditribhAgamAtraM krameNa tajjyeSThAH / svasvAnujAya dattvA samavittAH kiM ca hastagatam || 264 // parasparahastagatadhanasaGkhyAmAtradhanaM dattvA samadhanAnayanasUtramvAJchAbhaktaM rUpaM padayutamAdAvuparyuparyetat / saMsthApya saikavAJchAguNitaM rUponamitareSAm // 265 // atroMdezakaH / vaNijastrayaH parasparakarasthadhanamekato'nyonyam / dattvA samavittAH syuH kiM syAdvastasthitaM dravyam / / 266 / / vaNijazcatvAraste'pyanyonyadhanArdhamAtramanyasmAt / svIkRtya parasparataH samavittAH syuH kiyatkarasthavanam || 267 jayApajayayolI mAnayanasUtram- svasvacchedAMzayutI sthApyodhvadhiryataH kramotkramazaH / anyonyacchedAMzakaguNitau vajrApavartanakramazaH || 268 // chedAMzakramavatsthitatadantarAbhyAM krameNa mambhaktau / svAMzaharaghnAnyaharau vAJchAghnau vyastataH karasthamitiH || 269 // atroddezakaH / dRSTvA kukuTayuddhaM pratyekaM tau ca kukkuTako / uktau rahasyavAkyairmantrauSadhazaktimanmahApuruSeNa // 270 //

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100 gaNitasArasagrahaH jayati hi pakSI te me dehi svarNa hyavijayo'si dadyAM te / taDviyaMzakamadyetyaparaM ca punaH sa saMsRtya / / 2713 // tricaturtha prativAJchatyubhayasmAddvAdazaiva lAbhaH syAt / . tatkukuTikakarasthaM brUhi tvaM gaNakamuvatilaka !! 272 // rAzilabdhacchedamizravibhAgasUtram--- mizrAdUnitasaGkhyA chedaH saikena tena zeSasya / bhAgaM hatvA labdhaM lAbhonitazeSa eva rAziH syAt / / 273, atroddezakaH / kenApi kimapi bhaktaM sacchedo rAzimizrito lAbhaH / pazcAzatribhiradhikA tacchedaH kiM bhavellabdham // 2743 // iSTasaGkhyAyojyatyAjyavargamUlarAzyAnayanasUtram--- yojyatyAjyayutiH sarUpaviSamAgranArdhitA vargitA vyagrA bandhahatA ca rUpasahitA tyAjyaikyazeSAgrayoH / zeSaikyArdhayutonitA phalamidaM rAzirbhavedvAJchayostyAjyAtyAjyamahattvayoratha kRtermUlaM dadAtyeva saH / / 275 // atroddezakaH / rAziH kazcidazabhiH saMyuktaH saptadazabhirapi hInaH / mUlaM dadAti zuddhaM taM rAzi syAnmamAzu vada gaNaka // 276 / / rAzissaptabhirUno yaH so'STAdazabhiranvitaH kazcit / mUlaM yacchati zuddhaM vigaNayyAcakSva taM gaNaka // 277 //

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mizrakavyavahAraH 101 rAzidivyazonastrisaptabhAgAnvitassa eva punaH / mUlaM yacchati ko'sau kathaya vicintyAza taM gaNaka // 278 // iSTasaGkhyAhInayuktavargamUlAnayanasUtramuddiSTo yo rAzistva(latavargito'tha rUpayutaH / yacchati mUlaM kheSTAtsaMyukta cApanIte ca // 279 // atroddezakaH / dazamissammizro'yaM dazabhistairvarjitastu saMzuddham / yacchati mUlaM gaNaka prakathaya sazcintya gazi meM / / 280 // iSTavargIkRtarAzidvayAdiSTanAdantaramUlAdiSTAnayana sUtram--- saikeSTavyekaSTAvarSIkRtyAtha vargitau raashii| etAviSTanAvatha tahizleSasya mUlamiSTaM syAt / / 281 // atroddezakaH / yaukaucidvargIkRtarAzI guNitI tu saikasaptatyA / sahizleSapadaM syAdekottarasaptatizca gazI ko / / vigaNaypa citrakuTTikagaNitaM yadi vetti gaNaka me behi // 283 / / yutahInaprakSepakaguNakArAnayanasUtram-- . saMvargiteSTazeSaM dviSThaM rUpeSTayutaguNAbhyAM nat / viparItAbhyAM vimajetprakSepau tatra hInI vA // 285 // atroddezakaH / trika paJcakasaMvargaH paJcadazASTAdazaiva caMSTamapi / iSTaM caturdazAtra prakSepaH ko'tra hAnirvA // 285 / /

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102 gaNitasArasaGgrahaH viparItakaraNAnayanasUtrampratyutpanne bhAgo bhAge guNito'dhike punazzodhyaH / varge mUlaM mUle vargoM viparItakaraNamidam // 286 // . atroddezakaH / saptahate ko rAzistriguNo vargIkRtaH zarairyuktaH / triguNitapaJcAMzahatastvardhitamUlaM ca paJcarUpANi // 287 // sAdhAraNazaraparidhyAnayanasUtram zaraparidhitrikamilanaM vargitametatpunastribhissahitam / dvAdazahate'pi labdhaM zarasaGkhyA syAtkalApakAviSTA // 28 // atroddezakaH / paridhizarA aSTAdaza tUNIrasthAH zarAH ke syuH / gaNitajJa yadi vicitre kuTThIkAre zramo'sti te kathaya / / 289 / / iti mizrakavyavahAre vicitrakuTTIkAraH samAptaH // zreDhIbaddhasaGkalitam / itaHparaM mizrakagaNite zreDhIbaddhasaGkalitaM vyAkhyAsyAmaH / hInAdhikacayasaGkalitadhanAnayanasUtram ... vyekArdhapadonAdhikacayaghA nonAnvitaH punaH prabhavaH / gacchAbhyasto hInAdhikacayasamudAyasaGkalitam // 290 // / atroddezakaH / caturuttaradaza cAdirhAnacayastrINi paJca gacchaH kim / dvAvAdiddhicayaH SaT padamaSTau dhanaM bhavedatra // 291 //

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103 mizrakavyavahAraH. adhikahInottarasaGkalitadhane AdyuttarAnayanasUtram - gacchavibhakte gaNite rUponapadArdhaguNitacayahIne / AdiH padahatavittaM cAcUnaM vyekapadadalahataH pracayaH // 292 / / atroddezakaH / catvAriMzadANitaM gacchaH paJca trayaH pracayaH / na jJAyate'dhunAdiH prabhavo dviH pracayamAcakSva // 293 / / zreDhIsaGkalitagacchAnayanasUtram---- AdivihIno lAbhaH pracayArdhahatassa eva rUpayataH / gaccho lAbhena guNo gacchassaGkalitadhanaM ca sambhavati / / 294 / / atroddezakaH / trINyuttaramAdi vanitAbhizcotpalAni bhaktAni / ekasyA bhAgo'STau kati vanitAH kati ca kusumAni // 291 // vargasaGkalitAnayanasUtram saikaSTakatirdinA saikeSToneSTadalaguNitA / kRtighanacitisaGghAtastrikabhakto vargasaGkalitam // 296 // atroddezakaH / aSTASTAdazaviMzatiSaSTayekAzItiSadakanInAM ca / kRtighanacitisaGkalitaM vargaciti cAza me kAya // 297 // iSTAdhuttarapadavargasaGkalinadhanAnayanasUtram .. higuNaikonapadottarakRtihatiSaSThAMzamukhacAhatayutiH / vyekapadAMnA muravakRtisahinA padatADitaSTakRticitikA // 298 //

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104 gaNitasArasahaH . punarapi iSTAdyuttarapadavargasaGkalitAnayanasUtram-- dviguNaikonapadottarakRtihatirekonapadahatAGgahatA / vyekapadAdicayAhatimukhakRtiyuktA padAhatA sAram // 299 // atroddezakaH / trINyAdiH paJca cayo gaccha: paJcAsya kathaya kRticitikAm / pazcAdistrINi cayo gacchaH saptAsya kA ca kRticitikA // 300 // ghanasaGkalitAnayanasUtram ---- gacchArdhavargarAzI rUpAdhikagacchavargasaGgaNitaH / ghanasaGkalitaM proktaM gaNite'smin gaNitatatvajJaiH // 301 // atroddezakaH / SaNNAmaSTAnAmapi saptAnAM paJcaviMzatInAM ca / SaTpaJcAzanmizritazataddayasyApi kathaya ghanapiNDama // 302 / / iSTAdyuttaragacchaghanasaGkalitAnayanasUtrama ... cityAdihatimaravacayazeSaghnA pracayanimnacitivarge / Adau pracayAdUne vidyutA yuktAdhike tu ghanacitikA // 303 // atroddezakaH / Adistrayazcayo dvau gaccha: paJcAsya ghanacitikA / pazcAdissaptacayo gacchappaTa kA bhavecca ghanacitikA // 304 // saGkalitasaGkalitAnayanasUtram - dviguNaikonapadottarakRtihatiraGgAhatA cayArdhayutA / AdicayAhatiyuktA vyekapadaghnAdiguNitena || saikaprabhavena yutA padadalaguNitaiva citicitikA / / 3056 //

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mizraka vyavahAraH atroddezakaH / AdiSaT patha cayaH padamapyaSTAdazAtha sandRSTam / ekAdyekottaracigsiGkalitaM kiM padASTadazakasya // 306 // caturasaGkalitAnayanasUtram - saika padArthapadAtirazvairnihatA padAMnitA vyAptA / saikapadannA citiciticitikRtighanasaMyutirbhavati // 307 // atroddezakaH / saptASTanavadazAnAM SoDazapazcAzadekaSaSTInAm / brUhi catuHsaGkalitaM sUtrANi pRthaka pRthak kRtvA / / 108 / / saGghAtasaGkalitAnayanasUtram - gacchastrirUpasahito gacchacaturbhAgatADitassaikaH / sapadapadakRtivinighno bharvAta hi saGghAtasaGkalitam // 109 // atroddezakaH / saptakRteH paTSaSTyAstrayodazAnAM caturdazAnAM ca / pazcAgraviMzatInAM kiM syAt saGghAtasaGkalitam // 110 // 105 bhinnaguNasaGkalitAnayanasUtram --- samadalaviSamagvarUpaM guNaguNitaM vargatADitaM dviSTam / aMzAptaM vyekaM phalamAdyannaghnaM guNona rUpahRtam // 3116 // atroddezakaH / dInArArdha pazcasu nagareSu cayastribhAgo'bhUt / AdisvayaMzaH pAdo guNottaraM saptabhinnaguNacitikA / 10

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106 gaNita sArasaGgrahaH kA bhavati kathaya zIghraM yadi te'sti parizramo gaNite // 313 // adhika hInaguNasaGkalitAnayanasUtram- guNacitiranyAdihatA vipadAdhikahI nasaguNA bhaktA / vye guNenAnyA phalarahitA hIne'dhike tu phalayuktA // 314 // atroddezakaH / patha guNottaramAdi trINyadhikaM padaM hi catvAraH / adhikaguNottaracitikA kathaya vicintyAzu gaNitatattvajJa // 311 // AdistrINi guNottaramaSTI hInaM dvayaM ca daza gacchaH / guNottaracitikA kA bhavati vicintya kathaya gaNakAzu // 311 / / AdyuttaragacchadhanamizrAdyuttaragacchAnayanasUtram mizrAduddhRtya padaM rUponecchAdhanena saikena / labdhaM pracayaH zeSaH sarUpapadabhAjitaH prabhavaH // 317 // atroddezakaH / AdyuttarapadamizraM paJcAzaddhanamihaiva sandRSTam / gaNitajJAcakSva tvaM prabhavottarapadadhanAnyAzu || 398 // saGkalitagatidhruvagatibhyAM samAnakAlAnayanasUtram dhruvagatirAdivihanizcayadalabhaktassarUpakaH kAlaH / dviguNo mArgastadgatiyogahRto yogakAlassyAt // 319 // atroddezakaH / kacinnaraH prayAti tribhirAdA uttaraistathASTAmiH / niyatagatirekaviMzatiranayoH kaH prAptakAlaH syAt // 320 //

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107 mizrakavyavahAraH 107 apraa|daahrnnm| par3a yojanAni kazcitpuruSastvaparaH prayAti ca triinni| . ubhayorabhimukhagatyorapTottarazatakayojanaM gamyam / pratyekaM ca tayoH syAtkAlaH kiM gaNaka kathaya me zIghram / / 321 // saGkalitasamAgamakAlayojanAnayanasUtram-- ubhayorAdyozzeSazcayazeSahato dvisaGgaNaH saikaH / .. yugapatprayANayossyAnmArge tu samAgamaH kAlaH // 322 // atroddezakaH / catvAryAdyaSTottarameko gacchatyatho dvitIyo naa| ho pracayazca dazAdiH samAgama kastayoH kAlaH / 323 // hRdyuttarahInottarayossamAgamakAlAnayanasUtram zeSazrAdyorubhayozcagayutadalabhaktarUpayutaH / yugapatprayANakRtayormArge saMyogakAlaH syAt // 324 // atroddezakaH / pazcAdyaSTottaranaH prathamo nAtha dvitIyanaraH / AdiH panaghnanava pracanA hInA Ta yogakAlaH kaH // 325 // zaghigatimandagatyossamAgamakAlAnayanasutram mandagatizIghragatyorekAzAgamanamatra gamyaM yat / tadatyanarabhaktaM labdhadinainaH prayAti zIghrolpam // 326 / / atroddezakaH / navayojanAni kazcitprayAti yojanazanaM gataM tena / pratidUto vanani punastrayodazAmAMti kaidivasaH // 327 / / 10-A

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108 gaNitasArasaGgrahaH viSamabANaissUNIrabANaparidhikaraNasUtram pariNAhastribhiradhiko dalilo vargIkRtastribhirbhaktaH / saikazzarAstu paridherAnayane 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 // 330 // atroddezakaH / paJcataraikenAgraM vyavaghaTitA gaNitavinmizre / samacaturazrazreDhI katISTakAssyurmamAcakSva // 3316 // nandyAvartAkAraM catustarAH SaSTisamaghaTitAH / sarveSTakAH kati syuH zreDhIbaddhaM mamAcakSva // 332 // chandazzAstroktaSaTpratyayAnAM sUtrANi --- samadalaviSamasvarUpaM dviguNaM vargIkRtaM ca padamaGkhyA / saGkhyA viSamA saikA dalano gurureva samadalataH // 333, // syAlaghurevaM krapazaH prastAro'yaM vinirdiSTaH / naSTAGkA laghuratha tatsaikadale guruH punaH punaH sthAnam // 334 // rUpAhiguNottaratastaddiSTe lAGkasaMyutiH saikA / ekAdyakottarataH padamUrdhvAdharyataH kramotkramazaH // 335 //

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mizrakavyavahAraH sthApya pratilomanaM pratilomannena bhAjitaM sAram / syAllaghugurukriyeyaM saGkhyA dviguNaikavarjitA sAdhvA // 336 // atroddezakaH / saGkhyAM prastAravidhiM naSToddiSTe lagakriyAdhvAnau / SaTpratyayAMzca zIghraM vyakSaravRttasya me kathaya // 3373 // 100 iti mizrakavyavahAre zreDhIbaddhasaGgatiM samAptam // iti sArasaGghahe gaNitazAstre mahAvIrAyAsya ko mizrakagaNitaM nAma pazcamavyavahAraH samAptaH //

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SaSThaH kSetragaNitavyavahAraH. siddhebhyo niSThitArthebhyo variSThebhyaH kRtAdaraH / abhipretArthasiddhayartha namaskurve punaH punaH // 1 // itaH paraM kSetragaNitaM nAma SaSThagaNitamudAhariSyAmaH / tadyathA kSetraM ninapraNItaM phalAzrayAdvyAvahArikaM sUkSmamiti / bhedAd dvidhA vicintya vyavahAraM spaSTametadabhidhAsye // 2 // tribhujacaturbhujavattakSetrANi svasvabhedabhinnAni / gaNitArNavapAragatairAcAryaissamyaguktAni // 3 // tribhuja tridhA vibhinnaM caturbhujaM paJcadhASTadhA vRttam / avazeSakSetrANi hyeteSAM bhedabhinnAni // 4 // tribhujaM tu samaM dvisamaM viSamaM caturazramApa samaM bhavati / dvidvisamaM dvisamaM syAtrisamaM viSamaM budhAH prAhuH // 5 // samavRttamardhavRttaM cAyatavRttaM ca kambukAvRttam / ninonnataM ca vRttaM bahirantazcakravAlavRttaM ca // 1 // vyAvahArikagaNitam / tribhujacaturbhujakSetraphalAnayanasUtramtribhujacaturbhujabAhupratibAhusamAsadalahataM gaNitam / neme jayutyarthaM vyAsaguNaM tatphalArdhamiha bAlendoH // 7 // atroddezakaH / tribhujakSetrasyASTau bAhupratibAhubhUmayo daNDAH / tayAvahArikaphalaM gaNayitvAcakSva me zIgham // 8 //

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kSetragaNitavyavahAraH. 111 dvisamatribhunakSetrasyAyAmaH saptasaptatirdaNDAH / visAro dvAviMzatiratha hastAbhyAM ca sammizrAH // 9 // tribhunakSetrasya bhujastrayodaza pratibhujasya pazcadaza / ' bhUmizcaturdazAsya hi daNDA viSamasya kiM gaNitam // 10 // gajadantakSetrasya ca pRSThe'STAzItiratra sandRSTAH / dvAsaptatirudare tanmUle'pi triMzadiha' daNDAH // 11 // kSetrasya daNDaSaSTirbAhapratibAhakasya gaNayitvA / samacaturazrasya tvaM kathaya samve gaNitaphalamAzu // 12 // Ayatacatura zrarupa vyAyAmaH saikaSaSTiriha daNDAH / vistAro dvAtriMzadyavahAraM gaNitamAcakSva // 13 // daNDAstu saptaSaSTirdisamacaturbAdakasya cAyAmaH / - vyAsazcASTatriMzat kSetrasyAsya trayastriMzat // 14 // kSetrasyASTottarazatadaNDA bAhavayaM marave cASTau / hastaistribhiryutAmtatrisamacaturbAhukasya vada gaNaka | // 15 // viSamakSetrasyApTatriMzaddaNDAH kSitirmurave dvAtriMzat / pabhAzatprati bAhuH paTimvanyaH kimasya caturazre // 16 // parighAMdarastu daNDAstriMzanTaSThaM zatatrayaM dRSTam / navapazcaguNAM vyAsA nemikSetrasya kiM gaNitam // 17 // hastI hI viSkambhaH pRSThe'STApaTiriha ca sandRSTAH / udare tu dvAtriMzadvAlendoH kiM phalaM kathaya // 18 // JThe reading in both BandMin vizAtaH utnetha in erronsomitin porrooted into a man to out the requirements of the metro aldo. * Broads deka for prati

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112 gaNitasArasanhaH uttakSetraphalAnayanasUtram-- triguNIkRtaviSkambhaH paridhiAsArdhavargarAzirayam / triguNaH phalaM same'rdhe vRtte'dhaiM prAhurAcAryAH // 19 // atroddezakaH / vyAso'STAdaza uttasya paridhiH kaH phalaM ca kim / vyAso'STAdaza sattAdhe gaNitaM kiM vadAzu me // 20 // AyatavRttakSetraphalAnayanasUtramvyAsArdhayuto dviguNita Ayatattasya paridhirAyAmaH / viSkambhacaturmAgaH pariveSahato bhavetsAram // 21 // atroddezakaH / kSetrasyAyatavRttasya viSkambho dvAdazaiva tu / mAyAmastatra SaTtriMzat paridhiH kaH phalaM ca kim // 22 // zalAkAravRttasya phalAnayanasUtram ----- vadanA|no vyAsastriguNaH paridhistu kambukAvRtte / valayArdhakRtitryaMzo muravArdhavargatripAdayutaH / / 23 // atroddezakaH / vyAso'STAdaza hastA muravavistAro'yamapi ca catvAraH / kaH paridhiH kiM gaNitaM kathaya tvaM kambukAitte // 24 // nimnonnatattayoH phalAnayanasUtram paridhezca caturbhAgo viSkambhaguNaH sa viddhi gaNitaphalam / catvAle kUrmanime kSetre nimnonnate tasmAt // 21 //

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118 kSetragaNitavyavahAraH atroddezakaH / catvAlakSetrasya vyAsastu bhasayakaH paridhiH / SaTpaJcAzadaSTaM gaNitaM tasyaiva kiM bhavati // 26 // kUrmanibhasyonnatavRttasyodAharaNamviSkambhaH pazcadaza dRSTaH paridhizca patriMzat / kUrmanibhe kSetre kiM tasmin vyavahArajaM gaNitam // 27 // antazcakravAlavRttakSetrasya bahizcakravAlavRttadotrasya ca vyavahAraphalAnayanasUtram nirgamasahito vyAsastriguNo nirgamaguNo bahiNatam / rahitAdhigamavyAsAdabhyantaracakravAlavRttasya // 28 // atroddezakaH / vyAso'STAdaza hatAH punarbahinirgatAstra yastatra / vyAso'STAdaza hamtAzcAnaH punaradhigatAstrayaH kiM syAt / / 29 / / samavattakSetrasya vyAvahArikaphalaM ca paridhipramANaM ca vyAsapramANaM ca saMpojya etatsaMyogasaGkhyAmeva svIkRtya tatsaMyogapramANarAzeH sakAzAta pathak paridhivyAsaphalAnAM saGyAnayanasUtram .... gaNite dvAdazaguNite mizraprakSepakaM catuHSaSTiH / tasya ca mulaM kRtvA paridhiH prakSepakapadonaH // 30 // atrodezakaH / paridhivyAsaphalAno mizra poDazazataM sahasrayutam / kaH paridhiH kiM gaNitaM vyAsaH ko vA mamAcakSva // 1 //

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114 gaNitasArasAhahaH yavAkAramardalAkArapaNavAkAravajAkArANAM kSetrANAM vyAvahArika phalAnayanasUtram yavamurajapaNavazakrAyudhasaMsthAnapratiSThitAnAM tu / muravamadhyasamAsArdhaM tvAyAmaguNaM phalaM bhavati // 32 // " atroddezakaH / yavasaMsthAnakSetrasyAyAmo'zItirasya viSkambhaH / mabhyazcatvAriMzatphalaM bhavatkiM mamAcakSva // 33 // AyAmo'zIriyaM daNDA mukhamasya vizatimadhye / catvAriMzatkSetre mRdaGgasaMsthAnake brUhi // 34 // paNavAkArakSetrasyAyAmaH saptasaptatirdaNDAH / muravayorvistAro'STI madhye daNDAstu catvAraH // 35 // bajAkatestathAsya kSetrasya SaDagranavatirAyAmaH / madhye sUcimuravayostrayodaza vyaMzasaMyutA daNDAH / / 31 // ubhayaniSedhAdikSetraphalAnayanasUtram vyAsAtkhAyAmaguNAdikambhArdhapadIrghamutsRjya / tvaM vada niSedhamubhayostadardhaparihINamekasya // 37 // atroddezakaH / AyAmaH SaTtriMzadvistAro'STAdazaiva daNDAstu / ubhayaniSedhe kiM phalamekaniSedhe ca kiM gaNitam // 38 // bahuvidhavajAkArANAM kSetrANAM vyAvahArikaphalAnayanasUtram-. rajjvardhakativyazo bAhuvibhakto nirekabAhuguNaH / sarveSAmazravA phalaM hi bimbAntare caturthAzaH // 39 / '

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kSetragaNitavyavahAraH 115 atrodezakaH / SaDbAhukasya bAhoviSkambhaH paJca cAnyasya / vyAsastrayo bhunasya tvaM SoDazabAhukasya vada // 40 // tribhunakSetrasya bhujaH paJca pratibAharapi ca sapta dharA SaT / anyasya SaDaasya TekAdiSaDantavistAraH // 41 // maNDalacatuSTayasya hi navaviSkambhasya madhyaphalam / SaTpazcacatuvyosA vRttatritayasya madhyaphalam // 42 // dhanurAkArakSetrasya vyAvahArikaphalAnayanasutram ..... kRtveSuguNasamAsaM bANArdhaguNaM zarAsanaM gaNitam / zaravargAtpazcaguNAjyAvargayutAtpadaM kASTham // 43 // __ bhatrodezakaH / jyA SAviMzatireSA trayodazeSazca kArmukaM dRSTam / kiM gaNitamasya kAThaM kiM vAcakSvAzu me gaNaka || 44 // bANaguNapramANAnayanasUtram --- guNacApakRtivizeSAt pAhatAtpadamiSuH samudiSTaH / zaravargAtpazcaguNAdanA dhanuSaH kRtiH padaM nIvA // 15 // atrodezakaH / bhasya dhanuHkSetrasya zaro'tra na jJAyate parasyApi / na jJAyate ca maurvI tadyamAcakSva gaNitajJa // 11 // bahirantazcatura akasattasya vyAvahArikaphalAnayanasUtram bAho vRttasyedaM kSetrasya phalaM trisaMguNaM dalitam / abhyantare tadardha viparIte tatra catura zre // 17 //

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116 gaNitasArasaGgahaH atroddezakaH / pazcadazabAhukasya kSetrasyAbhyantaraM bahirgaNitam / caturazrasya ca vRttavyavahAraphalaM mamAcakSva // 48 // iti vyAvahArikagaNitaM samAptam / atha sUkSmagaNitam. itaH paraM kSetragaNite sUkSmagaNitavyavahAramudAhariSyAmaH / tadyathA' -- AbAdhAvalambakAnayanasUtram ----- mujakRtyantarabhUtabhUsaGkamaNaM tribAhukAbAdhe / tadbhunavargAntarapadamavalambakamAhurAcAryAH // 49 // sUkSmagaNitAnayanasUtrambhujayutyardhacatuSkADhujahInAddhAtitAtpadaM sUkSmam / athavA muravatalayutidalamavalambaguNaM na viSamacaturazre // 50 // atroddezakaH / tribhujakSetrasyASTau daNDA mUrbAhako samasya tvam / sUkSmaM vada gaNitaM me gaNitavidavalambakAbAdhe // 51 // dvisamatribhujakSetre trayodaza syurbhujadvaye daNDAH / daza bhUrasyAbAce athAvalamba ca sUkSmaphalam // 12 // viSamatribhujasya bhujA trayodaza pratibhujA tu pazcadaza / bhUmizcaturdazAsya hi kiM gaNitaM cAvalambakAbAdhe // 13 // I after this M adds the following :-tribhujakSetrasya apasthitabhAmisaMspRSTharavAyA nAma avalampaka: syAt / bhujadvayasaMyogasthAnamArabhya

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kSetragaNitAvahAraH 117 itaH paraM pazcaprakArANAM caturazrakSetrANAM karNAnayanasUtram--- kSitihataviparItabhujau muravaguNa mujamizrito guNacchedo / chedaguNau pratibhujayoH saMvargayuneH padaM karNau // 14 // atroddezakaH / samacaturazrasya svaM samantataH paJcabAhukasyAzu / karNa ca sUkSmaphalamapi kathaya saravaM gaNitatattvajJa // 15 // Ayatacaturazrasya dvAdaza bAhazca koTirapi pazca / karNaH kaH sakSmaM kiM gaNita cAcakSva meM zIrAma // 16 // dvimamacatura abhUmiH ptrishdvaahrekssssttishr| so'nyazcatardazAsyaM karNaH kaH sUkSmagaNitaM kim // 55 // vargastrayodazAnAM trisamacaturbAhakaM pnbhuumiH| " sapta catuzzatayuktaM karNAvAdhAvalambagaNitaM kim // 58 // viSamacatara zrabAhU tryodshaabhystpshcdviNshtikii| pakSaghanA vadanamadhastrizataM kAnyana karNamukhaphalAni // 59 // itaH paraM vRttakSetrANAM sukSmaphalAnaganamatrANi / tatra samavRsakSetrasya sUkSmaphalAnayana sUtrama-- vRttakSetravyAso dazapadaNatA mavetparikSapaH / vyAsacaturbhAgaguNaH paridhiH phalamardhamardhe nat // 6 // atrAMddezakaH / samavRttavyAso'STAdaza viSkammazca paSTignyasya / zAMtiparasta kssetr| hi ke ca paridhiphalaM // 61 // bAdazaviSkambhastha kSetramA hi cArdhavRttamya / SaTuziyAsasya kaH paridhiH ki phalaM pati // // 12 //

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gaNitasArasajjandaH AyatavRttakSetrasya sUkSmaphalAnayanasUtramvyAsakRtiSSaNitA dvisaGgaNAyAmakRtiyutA (padaM) paridhiH / vyAsa caturbhAga guNazcAyatavRttasya sUkSmaphalam // 13 // atroddezakaH / AyatavRttAyAmaH SaTtriMzaGkhAdazAsya viSkambhaH / kaH paridhiH kiM gaNitaM sUkSmaM vigaNayya me kathaya || 14 | 118 zaGkhAkArakSetrasya sUkSmaphalAnayanasUtram vadanAno vyAso dazapadaguNito bhavetparikSepaH / mukhadalarahitavyAsArdha vargamukhacaraNakRtiyogaH || 15 || dazapadaguNitaH kSetre kambunibhe sUkSmaphalametat // 65 // atroddezakaH / vyAso'STAdaza daNDA mukhavistAro'yamapi ca catvAraH / kaH paridhiH kiM gaNitaM sUkSmaM tatkambukAvRtte // 6663 // bahizcakravAlavRttakSetrasya cAntazcakravAlavRttakSetrasya ca sUkSmaphalAnaya nasUtram-- 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 // "

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kSetragaNitavyavahAraH 110 yavAkArakSetrasya ca dhanarAkArakSetrasya ca sUkSmaphalAnayanasUtramiSupAdaguNazca guNo dazapadaguNitazca bhavati gaNitaphalam / yavasaMsthAnakSetre dhanurAkAre ca vijJeyam // 70 // atroddezakaH / dvAdazadaNDAyAmo mukhadvayaM sUcirapi ca visAraH / catvAro madhye'pi ca yavasaMsthAnasya kiM tu phalam // 71, / dhanurAkArasaMsthAne jyA caturvizatiH punaH / catvAro'syepuruddiSTasUkSmaM kiM nu phalaM bhavet // 72 // panAkArakSetrasya dhanaHkASThavANapramANAnayanamUtram zaravargaH SaDguNito jyAvargasamanvitastu yastamya / mUlaM dhanurguNeSuprasAdhane tatra viparItam // 73, / / viparItakriyAyA sUtram-- guNacApakRtivizaMpAttakahanAtpadamipuH samuddiSTaH / zaravargAt paNitAdUnaM' dhanuSaH kRteH padaM jIvA / / 74 // atroiMzakaH / dhanurAkArakSetre japA dvAdaza SaT zara: kASTham / na jJAgate sarave tvaM kA jIvA kazzarastasya / / 75 // mRdaGganibhakSetrasya ca paNavAkArale sya ca vatrAkArakSetrasya ca sUkSmaphalAMnayanasUtram -- mugvaguNitAyAmaphalaM svadhanuHphalamaMyutaM mRdaGganibha / tatpaNavavajanipayordhanuHphalonaM tayorubhayoH // 6 // | The reading in thern IB and Mir ki vivah in ro : Int pahANatAdanAyA panuSkRta: wfat given the required morning

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120 gaNitasArasahaH phalAna atroddezakaH / caturvizatirAyAmo vistAro'STI muravadvaye / kSetre mRdaGgasaMsthAne madhye SoDaza kiM phalam // 70 // caturvizatirAyAmastathASTau muravayoIyoH / catvAro madhyaviSkambhaH kiM phalaM paNavAkatau // 786 // caturviMzatirAyAmastathASTau muravayoIyoH / madhye sUcistathAcakSva vajrAkArasya kiM phalam // 79 // nemikSetrasya ca bAlenDAkArakSetrasya ca ibhadantAkArakSetrasya ca sUkSmaphalAnayanasUtram pRSThodarasaMkSepaH SaDbhakto vyAsarUpasaGgaNitaH / / dazamUlaguNo nemerbAlendribhadantayozca tasyArdham // 8 // atroddezakaH / pRSThaM caturdazodaramaSTau nemyAkRtI bhUmau / madhye catvAri ca tadvAlendoH kimibhadantasya // 816 // caturmaNDalamadhyasthitakSetrasya sUkSmaphalAnayana sUtram-- viSkambhavargarAzervRttasyaikasya sUkSmaphalam / tyaktvA samavRttAnAmantarajaphalaM caturNA syAt // 82 // atroddezakaH / golakacatuSTayasya hi parasparasparzakasya mdhysy| / sakSma gaNitaM kiM syAJcatuSkavi'kambhayuktasya // 3 //

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191 kSetragaNitavyavahAraH. vRttakSetratrayasthAnyo'nyasparzanAjAtasyAntarAsthitakSetrasya sUkSmaphalAnapanasUtrama viSkammamAnasamakatribhujakSetrasya sUkSmaphalam / vRtaphalArdhavihInaM phalamannarajaM trayANAM syAt // 84 // bhatroddezakaH / viSkambhacatuSkANAM vRttakSetratrayANAM ca / anyo'nyasdRSTAnAmantaranakSetrasUkSmagaNitaM kim // 856 // paraprakSetrasya karNAvalambakasUkSmaphalAnapanasUtram -- mujabhujakRtikRtivargA dvitritriguNA yathAkrameNaiva / zrutyavalambakatidhanakRtayazca SaDake kSetre // 8 // atroddeshkH| mujaSaTakSetre dvau hau daNDau pratibhujaM syAtAm / asmin zrutyavalambakamUkSmaphalAnAM ca vargAH ke // 7 // vargasvarUpakarANirAzInAM yutitakyAnayanasya ca neSAM vargasvarUpa. karaNirAzInAM yathAkrameNa parampaviyutinaH zeSasAkhyAnayanasya / sUtram kenApyapavartitaphalapadayogaviyogakRtihatAcchedAt / mUlaM padayutiviyutI rAzInAM viddhi karagigaNitamidam // 8 // atroddeshkH| SoDazaSaTtriMzacchatakaraNInAM vargamUlapiNDaM me / matha caitatpadazeSaM kathaya sarave gaNitatasyajJa // 89 // hAta sUkSmagaNitaM samAptam //

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122 gaNitasArasAhA. janyavyavahAraH. itaH paraM kSetragaNite janyavyavahAramudAharipyAmaH / iSTasajAya bIjAbhyAmAyatacaturazrakSetrAnayanasUtram vargavizeSaH koTisaMvargo dviguNito bhavedbAhuH / vargasamAsaH karNazcAyatacaturazrajanyasya // 90 // atroddezakaH / ekahike tu bIje kSetre janye tu saMsthApya / kathaya vigaNayya zIghraM koTibhujAkarNamAnAni // 91 // bIje he trINi sareva kSetre janye tu saMsthApya / kathaya vigaNayya zIghraM koTibhujAkarNamAnAni // 92 // punarapi bIjasaMjJAbhyAmAyatacatura zrakSetrakalpanAyAH sUtrambIjayutiviyutighAtaH koThistadvargayozca saGkramaNe / bAhuzrutI bhavetAM janyavidhI karaNametadapi // 93 // atroddezakaH / trikapaJcakabIjAbhyAM janyakSetraM sarave samutthApya / koThibhujAzrutisaGkhyAH kathaya vicintyAzu gaNitatattvajJa // 94 / iSTajanyakSetrAdvIjamaMjJasaGkhyayorAnayanasUtram - koThicchedAvAptyorasaGkramaNe bAhudalaphalacchedau / bIne zrutISTakRtyoryogaviyogArdhamUle te // 95 // atroddezakaH / kasyApi kSetrasya ca SoDaza koTina bIje ke| triMzadathavAnyabAhubarSIne ke te zrutizcatustriMzat / / 96 //

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128 kSetragaNitavyavahAraH. koTisaGkhyA jJAtvA mujAkarNasaGkhyAnayanasya ca bhujasaGkhyA jJAtvA koTikarNasaGkhyAnadhanasya ca karNasaGkhyAM jJAtvA koThibhujAsalyAnayanasya ca sUtram -- , koTikRtezchedApyossaGkamaNe zrutibhujau bhujalatervA / / athavA zrutISTakRtyorantarapadamiSTamapi ca koTibhuje // 97 // bhatroddezakaH / kasyApi koTirekAdaza bAhuSpaSTiranyasya / zrutirekaSaSTiranyasyAnuktAnyatra me kathaya // 98 // dvisamacatura kSetrasyAnayanaprakArasya sUtram janyakSetrabhujArdhahAraphalajaprAgjanyakovyoryatibhUrAsyaM viyunirbhujA zrutisthAlpAlpA hi koTirbhavet / AbAdhA mahatI zrutiH zrutirabhUjyeSThaM phalaM syA phalaM bAhussyAdavalambako dvisamakakSetra caturbAhake / / 99 // atroddezakaH / caturazrakSetrasya dvisamasya ca para SaTUbIjasya / muravabhUbhujAvalambakakarNAvAdhAdhanAni vada // 10 // trisamacaturazrakSetramA muravabhUbhunAvalambakakarNAvAdhAdhanAnaganas tram mujapadahatabIjAntarahatajanyadhanAtabhAgahArAbhyAm / tanakoTibhyA ca hisama iva trisamacatura aM // 101 / / atroddezakaH / catura prakSetrasya trisamasyAsya dvikatrikasvabIjasya / murava mubhujAvalambakakarNAbAdhAdhanAni vada // 102 / 11-A

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124 124 gaNitasArasAhA. viSamacatura kSetrasya muravabhUmujAvalambakakarNAbAdhAdhanAnapanasU . tram jyeSThAlpAnyonyahInazrutihatabhujakoThI muje bhumurave te koyoranyonyadobhyA hatayutiratha dorghAtayukkoThighAtaH / karNAvalpazrutighrAvanadhikabhujakoTyAhatau lambako tAvAbAdhe koTido vavanivivarake karNaghAtArdhamarthaH // 103 // atroddezakaH / ekahikahikatrikajanye cotthApya vissmcturshre| murava pujAbalambakakarNAbAdhAdhanAni vada // 104 // punarapi viSamacaturazrAnayanasUtram---- dvasva zrutikRtiguNito jyeSThabhujaH koTirapi dharA vadanam / karNAbhyAM saGguNitAvubhayabhujAvalpabhujakoTI // 1056 // jyeSThabhujakoTiviyunidhiAlpamujakoTitADitA yuktA / hasvabhujakoThiyutiguNapRthukovyAlpazrutinako karNau // 10 // alpazrutihatakarNAlpakoTibhujasaMhatI pRthaglambau / tagujayutiviyutiguNAtpadamAbAdhe phalaM zrutiguNArdham // 107 / / ekasmAjanyAgatacaturazrAhisamAtrabhujAnaganasUtram -- kaNe mujadvayaM syAdvAhuDhiguNIkRto bhavabhUmiH / koTiravalambako'yaM dvisamatribhuje dhanaM gaNitam // 10 // atroddezakaH / trikapazcakabIjotthadvisamatribhujasya gaNaka jAhU do| bhUmimavalambakaM ca pragaNayyAcakSva me zIghram / / 109 / /

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kSetragaNitavyavahAraH. viSamatribhujakSetrasya kalpanAprakArasya sUtram-janyabhujArthaM chitvA kenApicchedalabdhajaM cAbhyAm / koTiyutirbhUH karNau bhujau bhujA lamvakA viSame // 110 // atroddezakaH / he dvitribIjakasya kSetrabhUjArthena cAnyamutthApya / tasmAdviSamatribhuje bhujabhUmyavalambakaM brUhi // 111 // iti janyavyavahAraH samAptaH // paizAcika vyavahAraH. itaH paraM paizAcikavyavahAramudAhariSyAmaH / samacaturazrakSetre vA AyatacaturazrakSetre vA kSetraphale rajjusaGkhyA same sati, kSetraphale bAhusaGkhyA same sati kSetraphale karNasaGkhacayA same sati, kSetraphale rajjvardhasaGkhacayA same sati, kSetraphale vAhostRtIyAMzasaGkhyA same sati, kSetraphale karNasaGkhyAyAzcaturthAMzasaGkhyA same sati, dviguNita karNasya triguNitavAhozca caturguNitakoTezva rajjo saMyogasaGkhyAM dviguNIkRtya taddddiguNitasaGkhyA kSetraphale samAne sati, ityeva - mAdInA kSetrANAM koTibhujA karNakSetraphalarajjuSu iSTarAzidvayasAmyasya ceSTarAzidvayasyAnyamiSTaguNakAraguNita phalavat kSetrasya bhujAkoTisaha khyAnayanasya sUtram - svaguNeSTena vibhaktAsveSTAnAM gaNaka gaNitaguNitena / guNitA munA bhujAH syuH samacaturazrAdijanyAnAm // / 1123 // atroddezakaH / 125 rajjurgaNitena samA samacaturazrasya kA tu mujasaGkhyA / aparasya bAhusadRzaM gaNitaM tasyApi me kathaya // 113 //

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126 gaNitasArasaGgrahaH. karNo gaNitena samaH samacaturazrasya ko bhavedvAhuH / rajjurdviguNo'nyasya kSetrasya dhanAcca me kathaya // 114 // Ayatacaturazrasya kSetrasya ca rajjutulyamiha gaNitam / gaNitaM karNena samaM kSetrasyAnyasya ko bAhuH || 1993 // kasyApi kSetrasya triguNo bAhurdhanAcca ko bAhu: / karNazvaturguNo'nyaH samacaturazrasya gaNitaphalAt // 116 // Ayatacaturazrasya zravaNaM dviguNaM trisaGguNo bAhuH / koTizcaturguNA tai rajjuyutairdviguNitaM gaNitam // 117 // Ayatacaturazrasya kSetrasya ca rajjuratra rUpasamaH / kAMTiH ko bAhurvA zIghraM vigaNayya me kathaya // 118 // kaNoM higuNo bAhutriguNa. koTicaturgaNA mizraH / rajvA saha tatkSetrasyAvatacaturazrakasya rUpasamaH || 1993 // punarapi janvAyatacaturazrakSetrasya bIja saGkhyAnayane karaNasUtram -- koTyUna karNadalatatkarNAntaramubhayayozdha pade / Ayatacaturazrasya kSetrasyeyaM kriyA janye // 120 // atroddezakaH / Ayatacaturazrasya ca koTiH paJcAzadadhikapava bhujA / sASTAcatvAriMzatrisaptatiH zrutirathAtra ke bIje // 121 // iSTakalpitasaGkhyApramANavatkarNasahitakSetrAnayanasUtram - padyatkSetraM jAtaM bIjaissaMsthApya tasya karNena / iSTaM karNaM vibhajellAbhaguNAH koThidoH karNAH // 1223 // atroddezakaH / ekadvikAhika trikacatuSkasaptaikasASTakAnAM ca / gaNaka caturNAM zIghraM bIjairutthApya koThibhujAH // 123 //

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kSetragaNitavyavahAraH. AyatacaturazrANAM kSetrANAM viSamabAhukAnAM ca / karNo'tra paJcaSaSTiH kSetrANyAcakSva kAni syuH // 124 // iSTajanyAyatacaturazrakSetrasya rajjusAM ca karNasaGkhyAM ca jJAtvA tajjamyAyatacaturazrakSetrasya bhujakoTisaGakhyAnayanasUtra karNakRtau dviguNAyAM rajvardhakRti vizodhya tanmUlam / rajvarthe saGkramaNIkRte bhujA koTirapi bhavati // 121 // atroddezakaH / paridhiH sa catustriMzat karNazcAtra trayodazo dRSTaH / janyakSetrasyAsya pragaNayyAcakSva koTi bhujau || 126 // kSetraphalaM karNasakhyAM ca jJAtvA bhujakoTisaGakhyAnayanasUtram-karNakRtau dviguNIkRtagaNitaM hInAdhikaM kRtvA / mUlaM koTibhRjau hi jyeSTha ikhena maGkramaNe // 127 // atroddezakaH / Ayatacaturazrasya hi gaNitaM paSTistrayodazAsyApi / karNastu koTibhujayoH parimANaM zrotumicchAmi // 128 // kSetraphalasakhyAM rajjusakhyAM ca jJAtvA Ayatacaturazrasya bhujakoTisakhyAnayanasUtram- rajjvardhavargarAzergaNitaM canurAhataM vizodhyAya / mUlana hi rajjvarthe saGkramaNe sati bhujAkoTI / / 129 // atroddezakaH / saptatizataM tu rajjuH pazvazatottarasahastra miSTadhanam / nanyAyatacaturazre koTibhujau meM samAcakSva // 130 // 127

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198 gaNitasArasAhaH. AyatacaturazrakSetradvaye rajjusaGkhyAyAM sadRkSAyAM satyAM dvitIyakSetraphalAt prathamakSetraphale dviguNite sati, athavA kSetradvaye'pi kSetraphale sadaze sati prathamakSetraspa rajjusaGkhyAyA api dvitIyakSetrarajjusaGkhyAyAM dviguNAyAM satyam, athavA kSetradvaye prathamakSetrarajjusaGkhyAyA api dvitIyakSetrasya rajjusaGkhyAyAM dviguNAyAM satyAM dvitIyakSetraphalAdapi prayamakSetraphale dviguNe sati, tattatkSetradvayasthAnayanasUtram-- skhAlpahatarajjudhanahatakRtiriSTa traiva koTissyAt / vyekA dostulyaphale'nyatrAvikagaNitaguNiteSTam // 131 / / vyekaM tadUnakoTiH triguNA doH pAdathAnyasya / rajjvardhavargarAzeriti pUrvoktena sUtreNa / tadgaNitara jjamititaH samAnayettagujAkoTI // 133 // ___atroddezakaH / asamavyAsAyAmakSetre dve dvAvatheSTaguNakAraH / prathamaM gaNitaM dviguNaM rajjU tulye kimatra koTibhuje // 134 // Ayatacatura zre he kSetre dvayameva guNakAraH / gaNitaM sadRzaM rajaDhigaNA prathamAt dvitIyasya // 135 // Ayatacaturazre dve kSetre prathamasya dhanamiha dviguNam / dviguNA dvitIyarajjustayorbhujAM koThimapi kathaya // 13 // dvisamatribhujakSetrayoH paraspararajjadhanasamAnasaGkhyayoriSTaguNakaguNitarajuSanavato; dvisamatribhujakSetradvayAnayanasUtram rajjukRtinAnyonyadhanAlpAptaM Sadvighamalpamekonam / taccheSaM dviguNAspaM bIje tajjanyayorbhujAdayaH prAgvat // 137 // atroddezakaH / dvisamatribhunakSetrahayaM tayoH kSetrayossamaM gaNitam / rajjU same tapossyAt ko bAhuH kA bhavedrUmiH // 13 //

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kSetragaNitavyavahAraH. 129 dvisamatribhujakSetre prathamasya dhanaM dvisaGgaNitam / . rajjuH samA dvayorapi ko bAhuH kA bhavedbhamiH // 139 // dvisamatribhujakSetre he rajjadigaNitA dvitIyasya / * gaNite dvayossamAne ko bAhaH kA bhavedbhUmiH / / 140 // dvisamatribhujakSetre prathamasya dhanaM hisaGgaNitam / dviguNA dvitIyarajjuH ko bAhuH kA bhavemiH / / 141 / / ekaTyAdigaNanAtItasaGkhyAsu iSTasaNyAmiSTavastuno bhAga - sayAM parikalpya tadiSTavastubhAgasaGkhyAyAH sakAzAt samacatura zrakSetrAnayanasya ca samavRttakSetrAnayanasya ca samatribhujakSetrAnAnasya cAyata. catura kSetrAnayanasya ca sUtrama vasamIkRtAvadhRtihatavanaM catughnaM hi vRttasamacatura zrayAsaH / SaDguNitaM tribhujAyatacatura zrabhujArdhamapi koTiH // 112 // atrauddezakaH / svAntaHpura narendraH prAsAdatale nijAnanAmadh / divyaM sa ranakambalamapIpatattacca samavRttam // 143 / / tAbhirdevIbhidhRtamabhirmujayozca muSTibhirlabdham / paJcadazaikasyAH syuH kati vanitAH ko'tra viSkambhaH / / 14 4 // samacatura zrabhujAH ke samatribAhI bhujAdhAtra / Ayatacaturazrasya hi tatkoTibhujo sarakha kathaya / / 145 // kSetraphalasaGkhyA jJAtvA samacatura kSetrAnayanasya cAyatacaturazrakSetrAnayanasya ca sUtram sUkSmagaNitasya mUlaM samacaturazrasya bAhuriSTahatam / dhanamiSTaphale syAtAmAyatacaturazrakoTimujau // 14 //

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130 gaNitasArasabahaH. atroddezakaH / kasya hi samacaturazrakSetrasma phalaM catuSSaSTiH / phalamAyatasya sUkSmaM SaSTiH ke vAtra koTibhuje // 147 / / iSTadvisamacatura kSetrasya sUkSmaphalasaGkhyAM jJAtvA, iSTasakhyAM guNakaM parikalpya, iSTasakhyAGabIjAbhyAM janyAcatacatura zrakSetraM parikalpya, tadiSTadisamacaturazrakSetraphalavAdiSTadvisamacatura zrAnayanasUtram -- taDanaguNitaSTakatirjanyadhanonA bhujAhatA maravaM koTiH / / dviguNA samuravA mudalimvaH karNo bhane AdiSTahanAH // 148 // atroddeshkH| sUkSmadhanaM sapteSTaM trikaM hi bIje dvi ke trika dRSTe / hisamacaturazrabAhU muravabhUmyavalambakAn hi / / 149 / / iSTasUkSmagaNitaphalavatrisamacatura kSatrAna nasUtram ... iSTaghanabhaktadhanakRtiriSTayutAdhaM bhujA dviguNitaSTama / vibhujaM muravAmiSTAptaM gaNitaM hyavalambakaM trisamajanya / / 15 // atroddezakaH / kasyApi kSetrasya trisamacaturbAhukasya sUkSmadhanam / SaNNavatiriSTamaSTau bhUvAhamuravAvalambakAni vara // 151 // sUkSmaphalasaGkhyAM jJAtvA caturbhiriSTacchedaizca viSamacatura kSetrasya muravabhUmujApramANasaGkhyAnayanasUtram dhanakRtiriSTacchedai zratubhirAptaiva labdhAnAm / yutidalacatuSTayaM tairUnA viSamAkhyacatura zrabhujasaGkhyA // 152 // atroddezakaH / navatirhi sUkSmagaNitaM chedaH paJcaiva navaguNaH / dazadhRtiviMzatiSatihataH krmaadvissmcturshre|| muravabhUmimunAsaGkhyA vigaNayya mamAzu saGkathaya / / 153 //

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181 kSetragaNitavyavahAraH. 131 sUkSmagaNitaphalaM jJAtvA tatsUkSmagaNitaphalavatsamatribAhukSetrasya bAhusayAnayanasUtram gaNitaM tu caturguNitaM vargIkRtvA' bhajet tribhirlabdham / . tribhujasya kSetrasya ca samasya bAhAH ranavargaH // 154 // atroddezakaH / kasyApi samadhya kSetrasya va gaNitamuddiSTam / rUpANi trINyeva brUhi pragaNayya me bAhum // 155 // sUkSmagaNitaphalasakhyAM jJAtvA tatsUkSmagaNitaphalavAhasamatribAhu. kSetrasya mujabhUmyavalambakasalA nayanasatama icchAptavanecchAkRti(timUlaM doH kssitidignntNcchaa| icchAptadhanaM lambaH kSatre hisamavivAha janye syAt // 156 // atroddezakaH / kasyApi kSetrasya hirAmatribhujamya sUkSmagaNitaminAH / trINIcchA kathaya sarava bhujammavalambakAnAzu // 157 // sUkSmagaNitaphalasaLyAM jJAtvA tatsUkSmagaNitaphalavadvipamatribhujAnayanasya sUtram bhaSTaguNiteSTakRtiyutadhanapadaghanamiSTapada hadiSTArtham / bhUH syAnaM dvipadAhaneSTavarga bhuje ca saGkamaNam / / 158 // atroddezakaH / . kasyApi viSamavAhAmya akSatrasya sUkSmagaNitamidam / he rUpe nirdiSTe trINISTaM mibAhavaH ke syuH // 119 // punarapi sUkSmagaNitaphalasakhyA jJAtvA tatphalavahiSamatrimujAnayanasUtram 'vIkRtvA ought to be vargIkRtya : but this form will not suit the require. ments of the metre.

Page #170
182 gANatasArasagrahaH. khASTahatAtseSTakateH kRtimUlaM ceSTamitaraditarahatam / jyeSThaM svalpA!naM svalpArdha tatpadena ceSTena / / 160 // kramazo hatvA ca tayoH saGkamaNe bhUbhujau bhavataH / iSTArthamitaradoH syAdviSamatrakoNake kSetre // 161 // ' __ atroddezakaH / he rUpe sUkSmaphalaM viSamatribhujasya rUpANi / trINISTaM bhUdoSo kathaya sarave gaNitatattvajJa // 162 / / sUkSmagaNitaphalaM jJAtvA tatsUkSmagaNitaphalavatsamavRttakSetrAnayanatram gaNitaM caturabhyastaM dazapadabhaktaM pade bhavedyAsaH / sUkSmaM samavRttasya kSetrasya ca pUrvavatphalaM paridhiH // 163 // __ matroddezakaH / samavRttakSetrasya ca sUkSmaphalaM paJca nirdiSTam / viSkambhaH ko vAsya pragaNayya mamAzu taM kathaya // 1646 / / vyAvahArikagaNitaphalaM ca sUkSmaphalaM ca jJAtvA tayAvahArikaphalavatatsUkSmagaNitaphalavahisamacaturazrakSetrAnayanasya trisamacaturazrakSetrAnayanasya ca sUtrama dhanavargAntarapadayutiviyutISTaM bhamurave bhuje sthUlam / hisame sapadasthUlAtpadayutiviyutISTapadahataM trisame // 1653 // atroddezakaH / gaNitaM sUkSma paJca trayodaza vyAvahArikaM gaNitam / dvisamacaturazramamuravadoSaH ke SoDazecchA ca // 119 //

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188 kSetragaNitavyavahAraH. trisamacatura shrsyodaahrnnm| gaNitaM sUkSmaM pazca trayodaza vyAvahArikaM gaNitam / trisamacaturazrabAhUna sazcintya sarave mamAcakSva / / 167 // vyAvahArikasthUlaphalaM sUkSmaphalaM ca jJAtvA tadyAvahArikasthUlaphalavatsUkSmagaNitaphalavatsamatribhujAnayanasya ca samavRttakSetravyAsAnayanasya ca sUtram dhanavargAntaramUlaM yattanmUlATvisaGgaNitam / bAhustrisamatribhune samasya vRttasya viSkambhaH // 168 // atroddezakaH / sthUlaM dhanamaSTAdaza sUkSmaM trighano navAhataH karaNiH / . vigaNayya samve kathaya trisamatribhujapramANaM me // 119 // pazcakRtervoM dazagaNitaH karaNirbhavadidaM sUkSmama / sthUlamapi pazcasaptatiretako vRttavikambhaH // 170 // vyAvahArikasthUlaphalaM ca sUkSmagaNitaphalaM ca jJAtvA tadyAvahArikaphalavattatsUdamaphalavaDisamatribhujakSetrasya bhabhajApramANasaGkhyayorAnayanasya sUtram phalavargAntaramUlaM dviguNaM bhAvahArikaM bAhaH / mamyardhamUlabhakte dvisamatribhujasya karaNamidam / / 171 // atrodezakaH / mUlmadhanaM SaSTiriha sthUladhanaM pacapaSTiruddiSTam / gaNayitvA hi sarave hisamatribhujasya pujasaGkhyAma / / 172 //

Page #172
134 gaNitasArasAhaH. iSTasaGkhyAvAvisamacaturazrakSetraM jJAtvA taTvisamacatura kSetrasya sU. kSmagaNitaphalasamAnasUkSmaphalavadanyadvisamacaturazrakSetrasya bhabhujamuravasA - khyAnayanasUtramlambakatAviSTenAsamasaGkamaNIkRte bhujA jyeSThA / hasvayutiviyuti murava bhayutidalitaM talamurave dvismcturshre|| 173 // bhatroddezakaH / bhUrindrA dorvizve vakraM gatayo'valambako ravayaH / iSTaM dik sUkSmaM tatphalavaDisamacaturazramanyat kim // 174 // dvisamacatra kSetravyAvahArikasthUlaphalasakhyAM jJAtvA tadyAvahArikasthUlaphale iSTasaGkhyAvibhAge kRte sati tadvisamacatura kSetramadhye tattadAgasya mUmisaGkhyAnayane'pi tattatsthAnAvalambakasahakhyAnayane'pi sU. tram ---- ravaNDayutibhaktatalamuvi katyantaraguNitaravaNDamuravavargayutam / mUlamadhastalamukhayutadalahatalabdhaM ca lambakaH kramazaH // 175 // atroddezakaH / vadanaM saptoktamadhaH kSitistrayoviMzatiH punastriMzat / bAhU dvAbhyAM bhaktaM caikaikaM labdhamatra kA bhUmiH // 176 // mamirdiSaSThizatamatha cASThAdaza varanamatra sandRSTam / lambazcatuzzatIdaM kSetraM bhaktaM narezcaturbhizca // 177 // ekadvikatrikacatuHravaNDAnyakaikapuruSalabdhAni / prakSepatayA gaNitaM talamapyavalambakaM brUhi // 178 // bhUmirazItirvadanaM catvAriMzaJcatargaNA SaSTiH / avalambakapramANaM trINyaSTau paJca ravaNDAni // 179 / /

Page #173
kSetragaNitavyavahAraH. 135 stambhahayapramANatayAM jJAtvA tatstambhavayAgre sUtradvayaM baddhA tatsU. tradvayaM karNAkAreNa itaretarastambhamUlaM vA tatstambhamUlamatikramya vA saMspR. zya tatkarNAkArasUtradvayasparzanasthAnAdArabhya adhaHsthitabhUmiparyantaM tanmadhye eka sUtraM prasArya tatsUtrapramANasaGkhyaiva antarAvalambakasaMjJA bhavati / antarAvalambakasparzanasthAnAdArabhya tasyAM bhUmyAmbhavapArzvayoH karNAkArasUtradvayasparzanaparyantamAvAdhAptaMjJA syAt / tadantararAvalambakasaGkhyAnaya. nasya mAbAdhAsakhyAnayanasya ca sutrama ----- stambhau rajjvantara bhUhatI skhayogAhanau ca bhaguNitau / AvAdhe ne vAmaprakSepaguNo'ntaravalambaH // 180 // ___ atroddezakaH / SoDazahanocchAyo stambhava panizca pADazoddiSTau / AbAdhAntarasaharU pAmatrApyavalambakaM vArDa // 181 // stambhakasyocchrApaH SaTtriMzadizA madinIyasya / midiza hastAH kAcAdhA ko'yamavalamba // 182 // dvAdaza ca paNaza ca sampAnna bhUmigapi ca catvAraH / dvAdazakastambhApAdajaH pani ganyato malA / / 183 / / Akramya caturha nApAsya mUlaM tathaika :mnAJca / patinAgrAtkAbAdhA ko minnavalambako bhavati // 184 // bAhupratibAha ho trayodazAniyiM catuza ca / vadane'pi caturhastAH kAvAdhA kA larAvalambazca / / 185 // kSetramidaM murava bhUmyorekakonaM pAspagagrAJca / rajjuH patitA mUlAtvaM brahmavalambakAbAdhaM // 18 //

Page #174
136 gaNitasArasAhaH. bAhupayodazaikaH paJcadaza pratibhujA muravaM sapta / bhUmiriyamekaviMzatirasminnavalambakAbAdhe // 18 // samacatura prakSetraM viMzatihastAyataM tasya / koNebhyo'za catubhyo vinirgatA rajjavastatra // 18 // mujamadhyaM dviyugabhuje' rajjuH kA syAtsusaMvItA / ko vAvalambakaH syAdAbAdhe ke'ntare tasmin // 1896 // stambhasyonnatapramANasaGkhyA jJAtvA tasmin stambhe yenakenacitkAra. Nena bhagne patite sati tatstambhAramUlayormadhye sthitau bhUsaGkhyAM jJAtvA tatstambhamUlAdArabhya sthitaparimANasaGkhyAnayanasya sUtram- . nirgamavargAntaramitivargavizeSasya yadbhavedardham / nirgamanena vibhaktaM tAvasthitvAya bhamaH syAt // 1909 // ___atroddezakaH / stambhasya paJcaviMzatirucchrAyaH kazridantare bhagnaH / stambhAramUlamadhye paca sa gatvA kiyAn bhanaH // 191 // vegRcchrAye hastAH saptakRtiH kazcidantare bhagnaH / bhUmizca saikaviMzatirasya sa gatvA kiyAn manaH // 192 // vRkSocchrAyo viMzatiragrasthaH ko'pi tatphalaM puruSaH / karNALatyA vyakSipadatha tarUmUlasthitaH puruSaH / / 193 // tasya phalasvAbhimuravaM pratibhunarUpeNa gatvA ca / phalamagrahIJca tatphalanarayorgatiyogataGkhyaiva // 1943 // paJcAzadabhUttatphalagatirUpA karNasaGkhyA kA / tahasamUlagatanaragatirUpA pratibhujApi kiyatI syAt // 195 // 'bhujacaturSa ca is the reading found in the Ms.., but it is not correot. * The Hundbi in 7 in Klammatically indurrect; but the author rooms to baru intend the phonetio fusion for the sake of the met; vide sans 2047 of this o bapter.

Page #175
137 kSetragaNitavyavahAraH. jyeSThastambhasaGkhyAM ca alpastambhasaGkhyAM ca jJAtvA ubhayasta. mbhAntarabhUmisaGkhyAM jJAtvA tajjyeSThasaGkhye bhagne sati jyeSThastambhAgre aspastambhAgraM sTazati sati jyeSThastambhasya bhanasaGkhyAnayanasya sthita. zeSasaGkhyAnayanasya ca sUtram jyeSThastambhasya kRterhasvAvanivargayutimapohyArdham / stambhavizeSeNa hRtaM labdhaM bhagnonnatirbhavati // 196 // atroddezakaH / stambhaH paJcocchrAyaH parastrayoviMzatistathA jyeSThaH / madhyaM dvAdaza bhagnajyeSThAgraM patitamitarAgre // 197 // Ayatacatura zrakSetrakoTisaGkhyAyAstRtIyAMzadvayaM parvatotsedhaM pari. kalpya tatparvatotsedhasaGkhyAyAH sakAzAt tadAyatacaturazrakSetrasya bhuja. saGkhyAnayanasya karNasaGkhyAnayanasya ca sUtram -.. giryutsaMdho dviguNo giripurama yakSitigirarardham / gaganaM tatrotpatitaM giryardhavyAsasaMyutiH karNaH // 198 // atroddezakaH / Sar3ayojano ziravariNi yatIzvarI tiSThatastatra / eko'nicaryayAgAttatrApyAkAzacAryaparaH / / 199 // zrutivazamutpatya puraM girizigvarAnmUlamavaruhyAnyaH / samagatiko sAtau nagaravyAsaH kimutpatitam / 200 // DolAkArakSetre sambhadvayasya vA giridvayasya vA utsedhaparimANa. sakhyAmeva AyatacaturazrakSetrahaye bhujadvayaM parikalpya tadvighiyAntara. mumyAM vA nastambhadyAnnara pUmyAM vA AbAdhAhagaM parikalpya tadAbAvA. 12

Page #176
138 gaNitasArasagrahaH. iyaM vyutkrameNa nikSipya tadyutkramaM nyastAbAdhAdvayameva AyatacaturazrakSetra haye koTidvayaM parikalpya tatkarNadvayasya samAnasaGkhyAnayanasUtram DolAkArakSetrastambhadvitayordhvasaGkhye vaa| zikharidvayordhvasaGkhye parikalpya bhujadvayaM trikoNasya / / 2013 // tahotiyAntaragatabhUsaGkhyAyAstadAbAdhe / AnIya prAgvatte vyutkramataH sthApya te koTI // 202 / / syAtAM tasminnAyatacaturazrakSetrayozca taddobhyAm / koThibhyAM karNau dvau prAgvatsyAtA samAnasaGkhyau tau // 203 // atroddezakaH / stambhastrayodazaikaH pazcadazAnyazcaturdazAntaritaH / rajjurbadA ziravare bhUmIpatitA ka' AbAdhe / / te rajjU samasaye syAtAM tadrajumAnamapi kathaya // 20 // dvAviMzatirutsedho girestathASTAdazAnyazailasya / viMzatirubhayormadhye tayozca ziravayossthitau sAdhU // 20 // AkAzacAriNau tau samAgatau nagaramatra bhikSAyai / samagatiko saJjAtau tatrAbAdhe kiyatsaGkhye // samagatisaGkhyA kiyatI DolAkAre'tra gaNitajJa // 2074 / / viMzatirekasyonnatiradrezca jinAstathAnyasya / tanmadhyaM dvAviMzatiranayorayozca zRGgayoH sthitvA // 208 // AkAzacAriNau dvau tanmadhyapuraM smaayaatii| bhikSAyai samagatiko syAtAM tanmadhyaziravarimadhyaM kim // 209 // viSamatrikoNakSetrarUpeNa hInAdhikagatimatornarayoH samAgamadina. sayAnayanasUtrama 1. 31 is grammatioally incorrect since there onn be no candhi hotwroon in the dual number and a ; vide footnoto on page 136.

Page #177
kSatra gANatavyavahAraH. dinagatikRtisaMyogaM dinagatikRtyantareNa hatvAtha / hatvodaggatidivasaistallabdhadine samAgamaH syAnnoH // 210 // * atroddezakaH / yojane prayAti hi pUrvagatistrINi yojanAnyaparaH / uttarato gacchati yo gatvAsau taddinAni paJcAtha // 211 // gacchan karNAkRtyA katibhirdivasairnaraM samApnoti / ubhayoryugapadgamanaM prasthAnadinAni sadRzAni // 212 // pazcavidhacaturazrakSetrANAM ca trividhatrikoNakSetrANAM netyaSTavidhabAhya 103 vaktavyAsa sakhyAnayanasUtram ---- zrutiravalambakabhaktA pArzvazujannA caturbhuje tribhuje / bhujaghAto lambahRto bhavedbahirvRttaviSkambhaH // 2133 // atroddezakaH / samacaturazrasya trikabAhupratibAhukasya cAnyasya / koTiH paca dvAdaza bhujAsya kiM vA bahirvRttam || 214 // bAhU trayodaza mukhaM catvAriM dharA caturdaza proktA / dvisamacaturazra bAhiraviSkambhaH ko bhaveda / / 299 / / pacakRtirvadanamunAzratvAriMzam bhUmirekonA / trisamacaturazrabAhiravRttavyAsaM mamAcakSva // 216 // vyekA catvAriMzadvAhuH pratibAhuko dvipacAzat / SaSTirbhUmirvadanaM pazJcakRtiH ko'tra viSkambhaH // 217 // trisamasya ca SaT bAhustrayodaza dvisamabAhukasyApi / bhUmirdaza viSkambhAvanayoH kau bAhyattayoH kathaya // 298 // 18

Page #178
140 gaNitasArasaGgahaH. bAhu paJcavyuttaradazako bhUmizcaturdazo viSame / tribhujakSetre bAhiravRttavyAsaM mamAcakSva // 219 // dvikabAhuSaDazrasya kSetrasya bhavedvicintya kathaya tvam / bAhiraviSkambhaM me paizAcikamatra yadi vetsi / / 220 // iSTasaGakhyAvyAsavatsamavRttakSetramadhye samacaturazrAdyaSTakSetrANAM murava. mamujasakhyAnayanasUtram -- labdhavyAseneSTavyAso vRttasya tasya bhaktazca / labdhena bhujA guNayedbhavecca jAtasya bhujasakhyA // 221 // atroddezakaH / vRttakSetravyAsastrayodazAbhyantare'tra saJcintya / samacatura zrAdyaSTakSetrANi sarave mamAcakSva // 222 // bhAyatacaturazraM vinA pUrvakalpitacaturazrAdikSetrANAM sUkSmagaNitaM pa rajjusaGkhyAM ca jJAtvA tattatkSetrAbhyantarAvasthitavRttakSetraviSkammAnayana - sUtram---- paridheH pAdena ajedanAyatakSetrasUkSmagaNitaM tat / kSetrAbhyantaravRtte viSkambho'yaM vinirdiSTaH / / 223 / / atroddeshkH| samacatura zrAdInAM kSetrANAM pUrvakalpitAnAM ca / kRtvAbhyantaravRttaM brahmadhunA gaNitatattvajJa // 224 // samavRttavyAsasaGkhyAyAmiSTa saGkhyA bANaM parikalpya tadvANaparimANasya jyAsakhyAnayanasUtrama---

Page #179
141 kSetragaNitavyavahAraH. vyAsAdhigamonassa ca caturguNitAdhigamena saGgaNitaH / yattasya vargamUlaM jyArUpaM nirdizetprAjJaH // 225 // atroddezakaH / * vyAso daza vRttasya hAmyAM chinno hi rUpAbhyAm / chinnasya jyA kA syAtpragaNathyAcakSva tAM gaNaka // 226 // samavRttakSetravyAsasya ca mauAzca saGkhyA jJAtvA bANasakhyA. nayanasUtram vyAsajyArUpakayorvargavizeSasya bhavati yanmUlam / tadviSkambhAcchodhyaM zeSAmiSaM vijAnIyAt // 227 // ___ atroddeshkH| daza vRttasya viSkambhaH ziainyabhyantare sarave / uSTASTo hi punastasyAH kaH syAdadhigamo vada // 228 // jyAsaGkhyAM ca bANasaGkhyAM ca jJAtvA samavRttakSetrasya madhyavyAta. sakhyAnayanasUtram bhaktazcaturguNena ca zareNa guNavargarAziriSusahitaH / samavRttamadhyamasthitaviSkambho'yaM vinirdiSTaH // 229 // atroddezakaH / . kasyApi ca samavRttakSetrasyAbhyantarAdhigamanaM he / jyA dRSTASTo daNDA madhyavyAso bhavetko'tra // 230 // samavRttadvayasaMyoge ekA matsyAkatirbhavati / tanmatsyasya murakhapuccha. vinirgatareravA kartavyA / tayA revayA anyonyAmimuravadhanurbayAlatirma

Page #180
gaNitasArasAhaH. vati / tanmukhapucchavinirgatareravaiva tahanu yasyApi jyAkatirmavati / taddhanuIyasya zaradvayameva vRttaparasparasampAtazarau jJeyo / samavRttadvayasaMyoge tayoH sampAtazarayorAnayanasya sUtram grAsonavyAsAbhyAM grAse prakSepakaH prakartavyaH / vRtte ca parasparataH sampAtazarau vinirdiSTI // 231 // atroddezakaH / samavRttayoIyorhi dvAtriMzadazItihastavistRtayoH / grAse'STau ko bANAvanyonyabhavau samAcakSva // 232 // iti paizAcikavyavahAraH smaaptH|| iti sArasahe gaNitazAstre mahAvIrAcAryasya kRtau kSetragaNitaM nAma SaSThavyavahAraH samAptaH //

Page #181
saptamaH khAtavyavahAraH, sarvAmarendramakaTArcitapAdapIThaM sarvajJamavyayamacintyamanantarUpam / bhavyaprajAsarasijAkarabAlabhAnu bhaktyA namAmi zirasA jinavardhamAnam // 1 // kSetrANi yAni vividhAni pugeditAni teSAM phalAni guNitAnyavagAhanAni (nena) / karmAntikauNDraphalasUkSmavikalpitAni vakSyAmi saptamamidaM vyavahAraravAtam // 2 // sUkSmagaNitam. atra paribhASAzlokaH-- hastaghane pAMsUnAM dvAtriMzatpalazatAni pUryANi / utkIryante tasmAt SaTviMzatpalaza nAnIha // 3 // khAtagaNitaphalAnayanasUtram --- kSetraphalaM vedhaguNaM samaravAne vyAvahArikaM gaNitam / muravatalayutidalamaya satsaGkhyAptaM syAtsamIkaraNam // 4 // atrAzakaH. samacatura zrasyASTI bAhuH pratibAhukazca vedhazca / kSetrasya vAtagaNitaM samavAte kiM bhavedatra // 5 // tribhujasya kSetrasya dvAtriMzadvAhakasya vedhe tu / SaTtriMzadRSTAste SaDalAnyasya kiM gAMNatam // 6 // sASTazatavyAsasya kSetrasya hi patraSaSTisahitazatam / vedho uttasya tvaM samaravAte kiM phalaM kathaya // 7 // Ayatacaturazrasya vyAsaH paJcAyaviMzatirbAhuH / SaSTivaiSo'STazataM kathayAzu samasya vAtasya // 8 //

Page #182
144 gaNitasArasahaH. ____ asmin vAtagaNite karmAntikasaMjJaphalaM ca auNDUsaMjJaphalaM ca jJAtvA tAbhyAM karmAntikoNDUsaMjJaphalAmyAM sUkSmakhAtaphalAnayanasUtram - bAhyAbhyantarasaMsthitatattatkSetrasthabAhukoTabhuvaH / khapratibAhusametA bhaktAstatkSetragaNanayAnyonyam // 9 // guNitAzca vedhaguNitAH kAntikasaMjJagaNitaM syAt / tadbAhyAntarasaMsthitatattatkSetre phalaM samAnIya // 10 // saMyojya saGkhyayAptaM kSetrANAM vedhagaNitaM ca / auNDraphalaM tatphalayorvizeSakasya tribhAgena / / saMyuktaM karmAntikaphalameva hi bhavati sUkSmaphalam // 11 // atroddezakaH / samacaturazrA vApI viMzatiruparIha SoDazava tle| vedho nava kiM gaNitaM gaNitavidAcakSva me zIghram / / 12 / / vApI samatribAhuviMzatiruparIha SoDazaiva tale / vedho nava kiM gaNitaM karmAntikamauNDumapi ca sUkSmaphalam / / 13 // samavRttAsI vApI viMzatiruparIha SoDazaiva tale / veSo dvAdaza daNDAH kiM syAtkarmAntikauNDUsUkSmaphalam // 14 // Ayatacaturazrasya tvAyAmappaSTireva vistAraH / hAdaza murave tale'dhaM vedho'STau kiM phalaM bhavati // 15 // navatirazItiH saptatirAyAmazcordhvamadhyamUleSu / vistAro dvAtriMzat SoDaza daza sapta vedho'yam / / 15 / / vyAsaH SaSTirvadane madhye triMzattale tu paJcadaza / samavRttasya ca vedhaH SoDaza kiM tasya gaNitaphalam // 17 //

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khAtavyavahAraH. 145 tribhujasya murave'zItiH SaSTirmadhye tale ca pazcAzat / bAhutraye'pi vedho nava kiM tasyApi bhavati gaNitaphalam // 18 // khAtikAyAH svAtagaNitaphalAnayanasya ca svAtikAyA madhye sUcI muravAkAravat utsedhe sati khAtagaNitaphalAnayanasya ca sUtram parirakhAmuravena sahito viSkambhastribhujavRttayostriguNAt / AyAmazcaturazre caturguNo vyAsasaGguNinaH // 19 // sUcImuravavadvedhe pariravA madhye tu pariravArdham / muravasahitamatho karaNaM prAgvattalasacivedhe ca / / 20 // atroddezakaH. tribhujacaturbhujavRttaM puroditaM pariravayA parikSiptam / ' daNDAzItyA vyAsaH parigvAzcatururvikAstrivedhAH syuH // 21 // AyatacaturAyAmo viMzatyuttarazataM punAsaH / catvAriMzat pariravA catururvIkA trivadhA snAt // 22 // utsedhe bahuprakAravati sati vAtaphalAnayanasya ca, gasya kasyacit svAtaphalaM jJAtvA tattvAtaphalAt anyakSatrasya khAtaphalAnayanasya ca sUtram--- vedhayutiH sthAnahanA vedho mugvaphalaguNaH sva vAtaphalam / tricaturbhujavRttAnAM phalamanyakSetraphalahataM vedhaH // 23 // atroddezakaH / samacatura kSetre bhUmicaturhastamAtravistAre / tatraikadvitricaturhastanikhAte kiyAn hi samabaMdhaH // 24 // 14-A

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146 gaNitasArasahaH. samacaturazrASTAdazahastabhujA vApikA caturvedhA / vApI tajjalapUrNAnyA navabAhAtra ko vedhaH / / 25 // yasya kasyacitkhAtasya urdhvasthitabhujAsaGkhyAM ca avassthita . bhujAsaGkhyAM ca utsedhapramANaM ca jJAtvA, tatkhAte iSTotsedhasaGkhyAyA bhujAsaGkhyAnayanasya, adhassUcivedhasya ca saGkhyAnayanasya sUtram mukhaguNavedho muravatalazeSahato'traiva sUcivedhaH syAt / viparItavedhaguNamuravatalayutyavalambahayAsaH // 26 // atroddezakaH / samacaturazrA vApI viMzatirUve caturdazAdhazca / vedho mukhe navAdhastrayo bhujAH ke'tra sUcivedhaH kaH // 27 // golakAkArakSetrasya phalAnayanasUtram - vyAsArdhadhanArdhaguNA nava golavyAvahArikaM gaNitam / tadazamAMzaM navaguNamazeSasUkSmaM phalaM bhavati // 28 // bhatroddezakaH / SoDazaviSkambhasya ca golakavRttasya vigaNayya / kiM vyAvahArikaphalaM sUkSmaphalaM cApi me kathaya // 29 // zRGgATakakSetrasya rakhAtavyAvahArikaphalasya khAtasUkSmaphalasya ca sUtram --- bhujakatidalaghanaguNadazapadanavahadyAvahArikaM gaNitam / triguNaM dazapadabhaktaM zRGgATakasUkSmaghanagaNitam // 30 // atroddezakaH / vyazrasya ca zRGgAThakaSaDbAhughanasya gaNapitvA / kiM vyAvahArikaphalaM gaNitaM sUkSmaM bhavetkathaya // 31 //

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khAtavyavahAraH. 147 vApapriNAlikAnAM 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 / dvitricatuHpaJcAMgairdinasya katibhirdinAMzaistAH // 34 // trairAzikAkhyacaturthagaNitavyavahAre sUcanAmAtrodAharaNameva ; atra samyagvistArya pravakSyate - samacaturazrA vApI navahastaghanA nagasya tale / tacchiravarAjaladhArA caturazrAGgulamamAnaviSkammA // 35 // patitAgre vicchinnA tayA ghanA sAnnarAlajalaparNA / zailotsedhaM vApyAM jalapramANaM ca me bRhi // 31 // vApI samacaturazrA navahastaghanA nagasya tale / / alasamavRttaghanA jaladhArA nipatitA ca tacchikharAt / / 37 // agre vicchinnAbhattasyA vApyA mukhaM praviSTA hi / sA pUrNAntaragatajaladhArotsedhena shailsy| utsedhaM kathaya sAve jalapramANaM ca vigaNayya // 38 // samacatara zrA vApI navahassaghanA nagamya tale / tacilakhagajaladhArA patitAGga rAyanatrikoNA sA // 30 // vApImuravapraviSThA sAgraM chinnAnta galajalapurNA / kathaya sarave vigaNayya ca giryutsedhaM jalapramANaM ca // 40 //

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148 gaNitasArasaGgrahaH. samacaturazrA vApA navahastaghanA nagasya tale / aGgulavistArAGgularavAtAGgulayugaladIrghajaladhArA // 41 // patitAne vicchinnA vApImuravasaMsthitAntarAlajalaiH / sampUrNA syAdvApI giryutsedho jalapramANaM kim // 42 // iti vAtavyavahAre sUkSmagaNitaM sampUrNam / citigaNitam. itaH paraM vAtavyavahAre citigaNitamudAhariSyAmaH / atra pari bhASA-- hasto dIrgho vyAptastadardhamaGgalacatuSkamutsedhaH / dRSTastatheSTakAyAstAbhiH karmANi kAryANi / / 43 // iSTakSetrasya vAtaphalAnayane ca tasya khAtaphalasya iSTakAnayane ca sUtram-- muravaphalamudayena guNaM tadiSTakAgaNitabhaktalabdhaM yat / citigaNitaM tadvidyAttadeva bhavatISTakAsaGkhyA // 44 // atroddeshkH| vediH samacaturazrA sASTabhujA hastanavakamutsedhaH / ghaTitA tadiSTakAbhiH katISTakAH kathaya gaNitajJa // 45 // aSTakarasamatrikoNanavahastotsedhavedikA rcitaa| pUrveSTakAbhirasyAM katISTakAH kathaya vigaNayya // 46 // samavRttAkRtivedirnavahaso; kraassttkvyaasaa| ghaTiteSTakAbhirasyAM katISTakAH kathaya gaNitajJa / / 47 / / Ayatacatuzrasya tvAyAmaH SaSTireva vistAraH / paJcakatiH Sar3a vedhastadiSTakAcitimihAcakSva // 48 //

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khAtavyavahAraH. 149 prAkArasya vyAsaH sapta caturvizatistadAyAmaH / ghaTiteSTakAH kati syuzcocchrAyo viMzatistasya // 49 // vyAsaH prAkArasyorce paDagho'thASTa tIrthakA dIrghaH / ghaTiteSTakAH kati syunocchrAyo viMzatistasya / / 50 // dvAdaza SoDaza viMzatirutsaMdhAH sapta SaT ca paJcAdhaH / vyAsA murave catustrihikAzcaturviMzatirdIrghaH // 11 // iSTavedikAyAM patitAyAM satyAM sthitasthAne iSTakAsaGkhyAnayanasya ca patitasthAne iSTakAsaGkhyAnayanasya ca satram ---- muravatalazeSaH patitotsedhaguNaH sakalavedhahatsamuravaH / mukhabhUmyobhUmimukhe pUrvoktaM karaNamavaziSTam // 52 // atroddeshkH| dvAdaza deyaM vyAsaH pazcAdhazcordhvamekamutsedhaH / daza tasmin paJca karA bhagnAstatrepTakAH kati syustAH // 53 // prAkAre karNAkAreNa bhagne sati sthitaSTakAnayanasya ca patitaSTakAnayanasya ca sUtram bhUmimurave dviguNe mugvabhUmiyate manmabhUdagayatone / daiyodayapaSThAMzane sthitapatitepTakAH krameNa syuH // 54 // atroddezakaH / prAkAro'yaM mUlAnmadhyAvartana vAgunA vidvaH / karNAkRtyA manastatsthitapatitapTakAH kityaH syuH // 55 // prAkAro'yaM mUlAnmadhyAvartana caikahamtaM ganvA / karNAkRtyA bhagnaH katIpTakAH syuH sthitAzca patitAH kAH // 16 //

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gaNita sArasaGgrahaH. prAkAramadhyapradezotsedhe taravRddhayAnayanasya prAkArasya ubhayapArzvayoH tarahAnerAnayanasya ca sUtram - 150 iSTeSTako hRto vedhazca tara pramAName konam / mukhatalazeSeNa hRtaM phalameva hi bhavani tarahAniH // 57 atroddezakaH / prAkArasya vyAsaH sapta tale viMzatistadutsedhaH / ekenAyeM ghaTitastarabRddhayUne karodayeSTakathA // 983 // samavRttAyAM vApyAM vvAsaca tupke 'rdhayuktakara bhUmiH / ghaTiteSTakAbhirabhitastasyAM vedhastrayaH kAH syuH / ghaTiteSTakAH sarakhe me vigaNayya brUhi yadi vetsi // 60 // iSTakAghaTitasthale adhastalavyAsai sati UrdhvatalavyAse sati ca gaNitanyAyasUtram-- dviguNaniveza vyAsAyAmayuto dviguNitastadAyAmaH / Ayatacaturazre syAdutsedhavyAsasaGgaNitaH // 61 // 11 atroddezakaH / vidyAdharanagarasya vyAso'STau dvAdazaiva cAyAmaH / paJca prAkAratale mukhe tadekaM dazotsedhaH // 62 // iti svAtavyavahAre citigaNitaM samAptam / * krakacikAvyavahAraH. itaH paraM krakacikAvyavahAramudAhariSyAmaH / tatra paribhASA - hastadvayaM SaDaGgulahInaM kiSkvAyaM bhavati / iSTAdyantacchedanasaGkhyaiva hi mArgasaMjJA syAt // 63 // atha zAkAkhyadyAdidrumasamudAyeSu vakSyamANeSu / vyAsoddayamArgANAmaGgulasaMkhyA parasparannAptA // 64 //

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151 ravAtavyavahAraH. hastAGgulabargeNa krAkacike paTikApramANa syAt / zAkAhayadrumAdidrumeSu pariNAhadairghyahastAnAm // 15 // saMkhyA parasparamnA mArgANAM saMkhyayA guNitA / tatpaTTikAsamAptA krakacakanA karmasaMkhyA syAt // 66 // zAkArjunAmlavetasataralAsitasarjaDuNDukAkhyeSu / zrIparNIulAkhyadrumezvamIcekamArgasya / / SaNNavatiraGgulAnAmAyAmaH ki kureva vistAraH // 17 // atroddezakaH / zAkAyatarI dIrghaH SoDaza hastAzca vistAraH / sArdhatrayazca mArgAzcASTau kAnyatra karmANi // 68 // iti vAnavyavahAre krAMcakAvyavahAraH samAptaH // iti sArasaGgAhe gaNitazAstre mahAvIrAcAsya kRtI saptamaH vAtavyavahAraH samAptaH //

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aSTamaH. chAyAvyavahAraH, zAntirjinaH zAntikaraH prajAnAM jagatprabhutisamastabhAvaH' / . yaH prAtihAryASTavivardhamAno namAmi taM nirjitazatrusaGgam // 1 // AdI prAcyAdyaSTadiksAdhanaM pravakSyAmaH saliloparitalavasthitasamabhUmitale liravedattam / bimbaM svecchAzadviguNitapariNAhasUtreNa // 2 // tavRttamadhyasthatadiSTazakozchAyA dinAdau ca dinAntakAle / tahattareravAM spRzati krameNa pazcAtpurastAca kakup pradiSTA // 3 // tagriyAntargatatantunA liravenmatsyAkarti yAmyakuberadisthAm / tatkoNamadhye vidizaH prasAdhyAzchAyaiva yAmyottaradigdizArdhajAH // 4 // ajadhaTharavisaGkamaNadhudalajabhaikyArdhameva viSuvadA // 4 // laGkAyAM yavakovyAM siddhapurIromakApuryoH / viSuvagA nAstyeva triMzadvaThikaM dinaM bhavettasmAt // 5 // dezeSvitareSu dinaM triMzannAjyAdhikonaM syAt / meSadhaTAyanadinayostriMzadaThikaM dinaM hi sarvatra // 6 // dinamAnaM dinadalabhAM jyotizzAstroktamArgeNa / jJAtvA chAyAgaNitaM vidyAdiha vakSyamANasUtrIdhaiH // 7 // IM reads tattva:.

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chAyAnyavahAraH. 153 viSuvacchAyA yatrayatra deze nAsti tatratatra deze iSTazaGkoriSTakAla. cchAyAM jJAtvA tatkAlAnayanasUtrama chAyA saikA dviguNA tayA hRtaM dinamitaM ca pUrvAddhe / aparAhne taccheSaM vijJeyaM sArasaGgAhe gaNite // 8 // atroddezakaH / pUrvAddhe pauruSI chAyA triguNA vada kiM gatam / aparAle'vazeSaM ca dinasyAMzaM vada priya // 9 // dinAMze jAte sati ghaTikAnayanasUtrama aMzahataM dinamAnaM chedavibhaktaM dinAMzake jAte / pUrvA gatanAjyastvaparAddhe zaMpanAjyastu // 18 // atroddezakaH / viSuvacchAyAvirahitadeze'STAMzo dinasya gAH / zeSazcASTAMzaH kA ghaTikAH syuH gvAgninADyohaH // 11 // mallayuddhakAlAnayanasUtram -- kAlAnayanAddinagatazeSasamAsonitaH kAlaH / stambhacchAyA stambhapramANabhaktaiva pIrupI chAyA // 12 // atroddezakaH / pUrvAdde zaGkasamacchAyAyAM mallayuddhamArabdham / aparAhne dviguNAyAM samAptigatIcca guDhakAlaH kaH // 13 // apagardhasyo hAharaNam / dvAdazahastastambhacchAyA caturuttareva viMzatikA / tatkAle pauruSikacchAyA kiyatI bhavedgaNaka // 14 //

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154 gaNitasArasaGgrahaH. viSuvacchAyAyukte deze iSTacchAyAM jJAtvA kAlAnayanasya sUtram' zakyuteSTacchAyA madhyacchAyonitA dviguNA / tadavAptA zaGkamitiH pUrvAparayodinAMzaH syAt // 15 // atroddezakaH / dvAdazAGgulazaGkovudalacchAyAGguladvayI / iSTacchAyASTAGgulikA dinAMzaH ko gataH sthitaH / vyaMzo dinAMzo ghaThikAH kAstriMzannADikaM dinam // 17 // iSTanADikAnAM chAyAnayanasUtramdviguNitadinabhAgahRtA zaGkamitiH zaGkamAnonA / dhudalacchAyAyuktA chAyA tatveSTakAlikA bhavati // 18 // achoddezakaH / dvaadshaanggulshngkoryudlcchaayaangguldvyii| dazAnAM ghaThikAnAM mA kA chiMzannADikaM dinam // 19 // pAdacchAyAlakSaNe puruSasya pAdapramANasya paribhASAsUtram puruSonnatisaptAMzastatpuruSAGgestu dairdhya syAt / yadyevaM cetpuruSaH sa bhAgyavAnAGgibhA spaSTA // 20 // ArUDhacchAyAyAH saGkhyAnayanasUtram - nRcchAyAhatazaGkabhittistambhAntaronito bhaktaH / nRcchAyayaiva labdhaM zaGkobhittyAzritacchAyA // 21 // ____ achoddezakaH / viMzatihasaH stambho bhittistambhAntaraM karA aSTau / puruSacchAyA hinA bhittigatA sambhabhA kiM syAt // 22 // Not found in any of the M88.

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chAyAvyavahAraH. stambhapramANaM ca bhicyArUDhastambhacchAyAsayAM ca jJAtvA bhitti stambhAntarasaGkhyAnayanasUtram -- puruSacchAyAnimnaM stambhArUDhAntaraM tayormadhyam / stambhArUDhAntarahRtatadantaraM pauruSI trAyA // 23 // atroddezakaH / viMzatihastaH stambha: SoDaza mittyAzritacchAyA / dviguNA puruSacchAyA bhittistambhAntaraM kiM syAt // 24 // aparArdhasyodAharaNam / vizanihataH stambhaH SoDaza bhicyAzritacchAyA / kitI puruSacchAyA bhittistambhAntaraM cASTau // 25 // ArUDhacchAyAyAH sakhyAM ca bhittistambhAntara bhUmisakhyAMca puruSacchAyAyAH saGkhyAM ca jJAtvA stambhapramANasaGgakhyAnayanasUtrama- nRcchAyAnArUDhA bhittistambhAntareNa saMyuktA / pauruSabhAhRtalabdhaM viduH pramANaM budhAH stambhaM // 26 // atroddezakaH / SoDaza bhicyArUDhacchAyA dviguNaiva pauruSI chAyA / stambhotsedhaH kaH syAdbhittistambhAntaraM cASTau || 27 || 155 zaGkapramANazaGkacchAyAmizravibhaktasUtram - zaGkupramANazaGkacchAyAmizraM tu saikapIruNyA | maktaM zaGkumitiH syAcchaGkucchAyA tadUnamizraM hi // 28 // atroddezakaH / zaGkupramANazaGkucchAyAmizraM tu paJcAzat / zaGkatsedhaH kaH syAccaturguNA pauruSI chAyA || 29 //

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156 gaNitasArasagrahaH. zaGkacchAyApuruSacchAyAmizravibhaktasUtram-- zaGkanaracchAyayatirvibhAjitA zaGkusaikamAnena / labdhaM puruSacchAyA zaGkucchAyA tadUnamizraM syAt // 30 // ___ atroddeshkH| zaGkorutsedho daza nRcchAyAzaGkabhAmizram / pazcottarapaJcAzanRcchAyA bhavati kiyatI ca // 31 // stambhasya avanatisaGkhyAnayanasUtram- --- chAyAvargAcchodhyA narabhAkatiguNitazaGkakRtiH / saikanaracchAyAkRtiguNitA chAyAkRteH zodhyA // 32 // tanmUlaM chAyA zodhyaM nrbhaanvrgruupenn'| bhAgaM hRtvA labdhaM stambhasyAvanatireva syAt // 33 // atroddezakaH / dviguNA puruSacchAyA vyuttaradazahastazaGkorbhA / ekonatriMzatsA stambhAvanatizca kA tatra / / 34 // kazcidrAjakumAraH prAsAdAbhyantarasthassan / pUrvAhne jijJAsurdinagatakAlaM naracchAyAm // 35 // dvAtriMzaddhasto jAle prAbhittimadhya AyAtA / ravibhA pazcAdittI vyakatriMzatkarordhvadezasthA // 36 / / tadbhittidvayamadhyaM caturuttarAviMzatiH karAstasmin / kAle dinaganakAlaM nRcchAyAM gaNaka vigaNayya / kathayacchAyAgaNite yadyasti parizramasava cet // 37 // samacatura zrAyAM dazahaslaghanAyAM naacchaayaa| puruSotsedhadviguNA pUrvAdde prAktaTacyA // 3 // is the rending given in the 21SS, for AT HTT; but it is metrically innorradt.

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chAyAvyavahAraH. tasmin kAle pazcAttAzritA kA bhavedgaNaka / ArUDhacchAyAyA AnayanaM vetsi cetkathaya // 393 // zaGkordIpacchAyAnayanasUtram * zaGkanitadIponnatirAptA zaGkapramANena / tallabdhahRtaM zaGkoH pradIpazaGkuntaraM chAyA // 40 // atroddezakaH. zaGkapradIpayormadhye SaNNavasyaGgalAni hi / dvAdazAGgalazakostu dIpacchAyAM bAzu me / SaSTirdIpazikhotsedho gaNitArNavapAraga / / 42 / / dIpazakuntarAnayanasUtram - zaGkanitadIponnatirAptA zaGkaprAmANena / talabdhahatA zaGkucchAyA zaGkupradIpamadhyaM syAt // 43 // aAdezakaH / zaGkucchAyAGgulAnyaSTau SaSTirdIpazikhodayaH / zaGkudIpAntaraM brUhi gaNitArNavapAraMga // 44 // dIponnatisaGkhyAnayanamUtram - zaGkucchAyAmaktaM pradIpazaGkantaraM saikam / zaGku pramANaguNitaM labdhaM dIponnatirbhavati / / 41 / / atroddezakaH / zatrucchAyAdvinibhaiva dvizataM zaGkudIpayoH / mantaraM hyaGgulAnyatra kA dIpasya samunnatiH // 46 // 157

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168 gaNitasArasamahaH. zApramANamatrApi hAdazAGgulakaM gate / jJAtvodAharaNe samyagvidyAtsUtrArthapaddhatim / / 47 // puruSasya pAdacchAyAM ca tatpAdapramANena vRkSacchAyAM ca jJAtvA vRkSo. nateH saGkhyAnayanasya ca, vRkSonnatisaGkhyAM ca puruSasya pAdacchAyAyAH saGkhyA ca jJAtvA tatpAdapramANenaiva vRkSacchAyAyAH saGkhyAnayanasya ca sUtramsvacchAyayA bhaktanijeSTavRkSacchAyA punassaptabhirAhatA sA / vRkSonnatiH sAdrihatA svapAdacchAyAhatA syAmabhaiva nanam // 48 // atroddezakaH / AtmacchAyA catuHpAdA vRkSacchAyA zataM padAm / vRkSocchrAyaH ko bhavetsvapAdamAnena taM vada // 49 // vRkSacchAyAyAH saGkhyAnayanodAharaNam / AtmacchAyA catuHpAdA paJcasaptatipiryaMtam / zataM vRkSonnativRkSacchAyA syAtkiyatI tadA // 50 // . purato pojanAnyaSTau gatvA zailo dazodayaH / sthitaH pure ca gatvAnyo yojanAzItitastataH // 51 // tadagrasthAH pradRzyante dIpA rAtrau pure sthitaiH / puramadhyasthazailasyacchAyA pUrvAgamUlayuk / asya zailasya vedhaH ko gaNakAzu prakathyatAm // 52 / / iti sArasaGgahe gaNitazAstre mahAvIrAcAryasya kRtau chAyAvyavahAro nAma aSTamaH samAptaH / samApto'yaM sArasaGgahaH //

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ENGLISH TRANSLATION AND NOTES.

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CONTENTS. CHAPTER 1. PAGE 1o. do. TERMINOLOGY Holutation and Bonediction ... ... ... An appreciation of the science of oalculation Terminology relating to the measurement of Njiace time Do. do, grain ... Do. du. old Do. do. D. other metals ... .. Names of the operations in Arithmetis ... General rules in regard to zero and positive and negative quantities. Words denoting numbers ... The names of national placun ... ... ... ... ... - Qualities of an arithmeticion silver do. CHAPTER 11. ARITHMETIIN. OPERATION Multiplication Division Squaring Square root Cubin Cube root ... Summation ... Vyutkahta (subtraction CHALTER III FRACTIONS Multiplication of fractions .. ... Division of friction .. ... .. ... Squaring, Syu:10-rot, C'u link and Cube-root of huction Summation of fractional acrius in progression . Vyutkalita of fractions in series ... ... Six varieties of fractions ... ... Simplo fructions (weldition and subtractiun) Compound and complex fractione Bhd yanubandha fractions ... Bha ga pura ha fractions Bhd yandts fractions

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CONTENT $. CHAPTER IV. PAGE 70 MISCELLANEOUS PROBLEMS (on fractions) ... Bhd ga and Sera varieties Mala variety .. ... Seramula variety ... Sapamala variety involving two known quantities Ambamdla variety ... ... ... " Bhd gasa marga variety ... Avitavarga variety characterized by the subtraction or addition of known quantities .. Malamiera variety ... .. Bhinnadraya variety CHAPTER V. RULE OF THREE Inverse double and treblo rule of three ... ... Invorso quadruple rule of threc .. . Probloins bearing on forward and backward movement Doublo, treblo and quadruple rule of three CHAPTER VI. MIXED PROBLEMS Sankramana and Visamanaikramana Double rule of three Problems bearing on interest Proportionate division Vallika-Kutrikdre ... Virama-Kutikdra ... Sakala-kutikdra ... Swarya-Ktikara ... Visitra-Kuttikdra ... Summation of sories 117 125 126 188 140 188 CHAPTER VII. CALCULATIONS RELATING TO THE MEABUREMENT OF AREAS ... ... Calculation relating to approximate measurement of areas ... The minutoly socurato calonlation of the meuure of areas ... Subject of treatment known as the Janya operation ... ... Do. do. Paitdcika or devillohly difficult problems ... ... ... ... . .. .. . ... 184 187 197 209 890

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CONTENTS. CHAPTER VIII. PAGE 26R CALICLATIONS REGARDING EXCAVATION ... .. Calculations relating to piles of bricks ... ... .. . Operations relating th work done with sewn in a wir woond ... 2 78 CHAPTER IX. CALCULATION RELATING TO SHADOW 278

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GANITA-SARA-SANGRANA. ENGLISIL TRANSLATION, CHLAPTER 1. ON TERMINOLOGY. Salutation and Benediction, 1. Salutation to Mahavira, the Lord of the lines, the protector (of the faithful), whose four infinite attributes, worthy to love esteemed in (all) the three worlds, arrudistirpasable (in excellence). 2. low to that highly glorious Lord of the Jinan, 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 Amoghavarna (i.e., one who showorn down truly useful rain), who (over) wishes to do good to those whom he loves, and by whom the whole body of animals and vegetabler, having been freed from the effects of) p and drought, has been made to feel delighted : 4. Hle, in whose mental operatious, conceived nn firo, the enemies in the form of sins have all been turned into the condition of nehen, 'and who in consequence has become one Woche anger is not futilo: 5. Hle, who, having brought all the world under his control and being himself independent, has not been overcome by (nny) opponents, and is therefore au absoluto lord (like) Inow God of love: 11. lle, to whom the work of service) is rendered by a cirolo of kings, who have beon overpowered by the progress of (his) heroism, and who, being Cubrikubhanjanin by name, in in reality & cnkrikabhanjana (1.p., the destroyer of the cycle of recurring re-births) : * There foar attributes of Jina Mahavira are said to be hin faith, understand. ing, blissfulness and power.

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GANITABARA SANGRAHA. 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 (bis rulo)--the rule of that sovereign lord who has destroyed in philosophical controversy the position of single oonclusions 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, Vedic or (othor) similarly religione 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 pootry, 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 estoem. 12. In relation to the movements of the sun and other heavenly bodies, in connection with colipses and the conjunctions of planets, and in connection with the triprasna + and the courso 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 tho The nyddydila is a process of reasoning adopted by the Jainas in relation to the question of the reality or otherwisu of the totality of the perceptible objects found in the phenomenal aniverse. The word is translatable at the may-lo-argument; and this may-bo-argament declares that the phenomenal nnivorne (1) may be real, (2) may not be real, (3) may and may not be real, (4) may be indescribablo, (6) may bo real and indescribable, (6) may bu nareal and indoscribable, and (7) may be real and unreal and indescribable. The position represented by this argument is not, therefore, one of a single conolusion. The tripraina is the name of a chapter in Sanskrit astronomical worko; And the fact that it deals with three questions in responsible for that nam.. The question dealt with sro Dik (diroption), Dita (position) and Kila (time) apportaining to the planets and other heavenly bodied,

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3 (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. CHAPTER 1-TERMINOLOGY. 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 manner 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 orocodile 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 excavations; and wherein (finally) shines forth the advancing tide represented by the chapter on

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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 renson that it is not possible to know without (proper) terminology the import of anything, at the (very) commencement of this science the requirol torminology is mentioned. Terminology relating to the measurement of) Space. 25-27. That infinitely minuto (quantity of) matter, which is not destroyed by water, by fire and by other such things, is oalled a paramiinu. An endless number of them makes an amu, which is the first (measure) hore. The trasarenu which is derived therofrom, the rathurenu, thence (derivod), tho bair-measure, tho louso-measuro, the sesanium-measure, which (last) in the sanie as the mustard-mcasure, 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 worlls of work, which ure (all) differentiated as superior, middling and inferior- eight-fold (as measured in relation to what immediately prcerles cach of them), in the order (in which they are mentioned). This angula is known as ryaraharangula. 28. Those, who are acquainted with the processes of measuromont, say that five-hundred of this (rynvaharangula) coustitutos (another magula known as) pramaao. The finger measure of men now existing forms their own anyulu. 29. Thoy hold that in the established usage of the world the angulo is of threo kinils, wyarahara and pramana constituting two (of them), and then thoro boing) one's own angula ; and six angulas make the foot-measure as measured across. 30. Two (nuch) feet mako a viiasti; and twice that is a hasta, Four hastas make a denda, and two thousands of that make a krdxa. 31. Those who are well vorsed in the measurement of spaoc (or surface-aroa) say that four krosas form a yojana. After this, I montion in duo order the terminology relating to the monsuroment of) tine.

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CHAPTER 1-TERMINOLOGY. Terminology relating to the measurement of) Time. 32. The time in which an atom (moving goes beyond another atom (immedliately next to it) is it sumaya; innumerablo xamayas make an avali. 33. A mensured number of mulis inakes an urhreisil; soven uccharsex make one stokie ; soven stokrx make one lura, 'aud with thirty-eight and a half of this the ghuli in formed. 34. Two ghatis make oue muhurlu; tbirty muharlux make ono day ; fiftoen days make ono pokre; and two parkos ir tukou to be a month. 35. Two months make one rtru; three of these are understood to make one nyinu ; two of theso form one year. Next, I give the grain-moasine. Terminology relating to (the measurement of) Grain. 36. Know that four solaikas form here we kwuhu ; four kulahux ono prasth.; and four posthus one indhaha. 37. Four colhukus make one drown, ami four times one drona make one mant; four waniy make one khuri; live khurin muke one pravurtiku. 35. Four times that ramo (prarartikit) in it maha ; five pravartikis make (3kumbht. After this the terminology relating to the measurement of gold is described. . Terminology relating to the measurement of) Gold, 39. Four yanulakas make one yunja; five yunga maku oje pana, and cight of this (puni) make one dhuruwa: two charanma mako one kars, and four karsus make one pala. Terminology relating to the measurement of) Silver. 40. Two grains make one yunji; two yunjiix make one masa ; sixteon minax are sail loro to make one tharum. 41. Two and u half of that (dharana) make one kursa; four puriinux (or karxis) make one pla--80 day personin well worror in calculation in respoct of the m isurement of silver wecording to the

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GANITABARASANGRAHA. Terminology relating to the measurement of) Other Metals. 12. What is known as a kala consists of four padas; six and a quarter kalis make one yava; four yavas make one amsa; four ambas mako one bhaya. 43. Six bhagax 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 palax make ono tula; ten tulas make one bhara. Thus say thogo 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 oporations) is gunakara (multiplication), and it is also called) pratyutpanna ; the second is what is known as bhagahara (division); and krti (squuring) is said to be the third 47. The fourth, as a matter of courso, is varga-mula (square root), and the fifth is said to be ghana (cubing); then ghanamula (cube root) is the sixth, and the seventh is kuown as citi (summation). 48. This is also spoken of as sankalita. Then the eighth is oyutkulita (the subtraction of a part of a series, taken from the beginning, from the whole sories), and this is also spoken of as sega. All these eight (operations) appertain to fractions also. General rules in regard to sero and positive and negative quantities. 49. A number multiplied by zero is zero, and that (number) remnius unchanged when it is divided by," combined with (or) * It can be easily seen here that a number when divided by zero door not really remain unchanged. Bhakara calls the quotient of anch xero-divisions khahara and rightly atrigna to it the value of infinity. MahaviraoArys obvionsly think that a division by zero io no division at all.

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CHAPTER I-TERMINOLOGY. 7 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 positivo (quantity). But it is a nogative 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 adition 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 a (given) number becomes negative, and a negative (quantity) which has to be (so) subtracted becomes positive. 52. The square of a positive as well as of a negative (quantity) 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. [The stanzas give cortain 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 daka-sahasra (ten-thousand) and the sixth is no other than laksa (lakh). The seventh is dasa-lakya (ten-lakh) and the eighth is said to be koti (crore).

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GANITASARASANGRAHA. 65. The ninth is lasa-koti (ten-croro) and the tenth is katakoli (hundred-orore). The (place) characterised by cleven is arbuda and the twelfth (place) is nyarbuda. 66. The thirteenth place is kharva and the fourteenth is maha. Kharva. Similarly the fifteonth is padma and the sixteenth maha. padma, 67. Agnin the seventeenth is knona, the eighteenth maha-ksona. The nineteenth place is kmkha anul the twentieth is maha-sankha. 68. The twenty-first place is hsitya, the twonty-second mahaksitya. Then the twenty-tbird is ksobhu and the twenty-fourth maha-kxobha. 69. By means of tho (following) eight qualities, viz., quick method in working, forethonght as to whether a desirable result may be arrival 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 (wknown) quantities known-(by means of these qualities) an arithmetician is to bo known as such. 70. Great sages have briefly stated the terminology thus. What has to be further sail (abont it) in detail must be learnt from a study of the science (itself). Thus ends the chapter on Terminology in Sara. sangraha, which is a work on arithmetic by Maha. viracarya.

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CRAFT OE 11-ARITHMETICAL OPBRATIONS f. CHAPTER II. ARITHMETICAL OPERATIONS. The First Subject of Treatment. Hereafter we shall expound the first subjoot of treatment, which is named Parikarman. Multiplication. The rule of work in relation to the operation of multiplication, which is the first (among tho parikarman operations), is as follows: 1. After placing (tbo 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, iu Avoordance with (oither of) the two methods of normat (or) rosvorse working, by adopting the process of (i) dividing the multiplicand and multiplying the multiplior by a factor of the multiplicand, (ii) of dividing the multiplier and multiplying the multiplicand 1. Symbolically expressed, this role works out thun: In multiplying al by ed, the product in (1) 90 * (a xed), or (ii) (ab xe)x d; or (lii) ab * cd. Obvionsly the object of the first two devices here is to facilitate working through the choiou of suitable factors. Tbe anuloma or normal method of working in the ope that is generally followed. The viloma or the reverse method of working is as follows: To multiply 19:18 by 37: 1998 27 a - x 9 8x8 7x1 = 7x9 = 7 9 * 7*8 = 8 9 4 6

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10 GANTTABARASANGRAHA. by a factor of the multiplier, or (iii) of saling them multiplication) as they are (in themselves). in the Examples in illustration thereof. 2. Lotuses were given away in offering)-eight of them to sach Jina temple. How many (were given away) to 144 temples ? %. Nine parlmaraga 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 ? 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) con. sisting of 1, 8, 6, 4, 8, 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 P 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 (produot) has been declared by the holy prooeptor Mahavira to constitute the necklace of Narapala. 4. Here, 199 is mentioned in the original - 40 + 100 -1. 6. Here, 1998 is mentioned in the original ar 1098 + 900. 10. Here a well as in the following stansas, oortain nombers are said to constitute different kinds of neoklaors on sooount of the symmetrical arrango. ment of dimilar figures whioh is readily noticeable in relation to them,

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CHAPTER II-ARITHYSTTOAL OPERATIONE. 11. Six 3's, five o's, and (one) 7, which is at the end, aro pat down (in the descending order down to the units' placo); and this (number) multiplied by 33 has (alsu) been deolared to be a (kind of) neoklace. 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 gems. 13. Write down (the number) 142867148, and multiply it hy 7; and then say that it is the royal neoklace. 14. Similarly 37037137 is multiplied by 3. Find out the result) obtained by multiplying (this product) again to get such multiples (toreof) as bavo one is the firat and nine as the last (of the multipliers in order). 15. The (figures) 7,0, 2, 2, 5 and I are put down (in order from the units' place upwards); and then this (number) whiob is to be multiplied by 73, should also be called a neoklaco (when 80 multiplied). 16. Write down (the number reprosented by) the group (of figures) consisting of 4, 4, 1, 2, 6 and 2 (in order from the unita' place upwards); and when this is) multiplied by 64, you, who know arithmetio, tell me what the (rosulting) number is. . 17. In this (problem) put down in order (from the unita' place upwarda) 1,1,0,1,1,0, 1 and 1, which (figures so placed) give the measure of a (partioular) number; and (thon) if this (number) is multiplied by 91, there results that necklace which is worthy of prinoe. Thus ends multiplication, the first of the operations known as Parikarman. 11. The multiplicand here is 328333886667. 14 This problem rodnoos itaelf to this: multiply 37037037 * 3 by 1, 2, %, 4, 5, 6, 7, 8, and 9 in order

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12 GANITASABABANGBABA. Division. The rule 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.., 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 porson when 2701 picces 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 (givon) to each (temple) P 24. Jambu fruits amounting to) 31317 bave been divided between 39 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. 28. 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 unita' place upwards are) such as 20. Here, 8192 is mentioned in the original as 8000 +92 + 100. 98. In the original, 10349 is given w 10000+800+7'. 28. Here, 10141 is given as 10000 + (40+4000 + 1 + 100). 24. Here, 31817 is given me 17+ 300 + 81000. 26. Here, 31313 is given as 18 +800+81000. %. Here, 36261 is given as 80000 +1+(60 + 200 + 6000). 87. Lore, the given dividend is obviously 18846654321.

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OTAPTER I-ARITHMETICAL OPARATIONS. CO begin with 1 and end with 6, and then become gradually diminished, are divided between 441 persone. What is the share of each? 28. Gems (amounting to) 28483 (in number) are given in offering) to 13 Jina teinpley. Givo out the share of each (temple). Thus ends division, the second of the oporations kuown as Parikarman. Squaring. The rulo of work in relation to the operation of squaring, which is the third (among tho parikaman operations), is as follows: 29. The multiplication of two equal quantities: or the multi. plication of the two quantities obtained (from tho givon quantity) hy tho subtraction (therefroin), and tho addition (therounto), of any chosen quantity, together with tho addition of the square of that chosen quantity (to that product): or the sum of a series in arithmetical progrossion, of which 1 is the first torm, 2 is the common difference, and the number of terms whorein is that (of which tho square is) required : givos rise to the (roquired) square. 30. Tho square of pubers consisting of two or more places ** is equal to) the sum of the squaros of all the numbors (in all the placos) combined with twice the product of thono (vumbers) taken (two at a time) in order. 28. Here, 28483 is given a 83 + 400 + (4000 x 7). 29. Tho rule given hereiu, exproewed algebryically, comow out. thus : (i) axa - q'; (ii) (u + a)(a -2) + x = c; (iii) 1 + 8 + 6+ 7+ ... op to a terms sal 80. The word translatod by place hero in P T; it ubviously means a pluce in notation. Here, as a commentary interpruts it, it may also denote the cons. ponent parts of rum, 4 ench kuch part han place in the sam. According to both these interpretations the rule workout correctly. For instance, (1284)' = (100* + 200EUR + 80EUR + 4*) + 2 x 1000 200 + 2 x 1000 x 80 +2 x 1000 X 4+2 x 200 x 30+ 2 * 200 X 4+ 2 * 30 * . Similarly (1+2+3+4=(1+2deg +81 +4*) + 2(1 x 2+1 3+1x 4+2 3+%** +8 x 4).

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GANITASARASANGRAHA. 31. Get the square of the last figure 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 syuaring. 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, 4601 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 squarod. O clever arithmetician, tell me, after multiplying well, the value of those threo (squares). Thus ends aquaring, the third of the operations known as Partkarman, 31. Tho pushing on to the right mentioned herein will become clear from the following worked out oxamples: To smuaru 131. To sqgoro 182 To square 555. 1 ili 5* 11 2 x 132x1x1 = 2 x1 x3 12x1x2 2 x 5 x 2 x 5 x 5 = 1 2 x 3 x 2 = 1121 2x 5x5= 5? = 2x9x1 = 1 |(2) (1) (5) (8) (6) (2 11716 17 4 2 4 30 8 0 % 5 33. Here, 4661 is given am 4000 +61 +600. 35. Here, 7135 is given as 135+ (1000 x 7).

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CHAPTER 11--ARITHMETICAL OPERATIONS. 16 Square Root. The rule of work in relation to the operation of (oxtracting) 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 tho right), subtract the (highest possible) square number; then multiply the root (of this number) by two, and divide with this (product the number represonted 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) old place. (If it is so continued till the end), the half of the (last) doublert quantity (conues to be the rosulting syuare root. Examplex in illustratiom thereof. 37. O friend, tell mo quickly the roots of the squares of the numbers from 1 to 9, and of 256 and 570. 38. Find out the square root of 6501 and of 08536. 39. What are the square roots of 4294967296 and 622521 P 40. What are the square roots of 63664441 and 1771501 P 41. Tell me, friend, after considering well, tho rquare roots of 1296 and 625. 36. To illustrate the rule, the following example in worked out below:To extract the quare root of 65738 0155 36 2 x 2-4)5( 20 5 - 65 25 25 2 - 50)8036 30/0) 86 6* - 88 266 * 2 - 612) 0 (0 612 Square root required - " 00 - 256,

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GANITASARASANGRAHA. 42. Tell me, O leading arithmetician, the square roots of 110889, 12321, and 844561. Thus ends aquare 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 eqnal quantities: or the product obtained by the multiplication of any (given) quantity by that (given quantity) as diminished hy a chosen quantity and (then again) by that (given quantity) as increased by the (samo) chosen quantity, when combined with the square of the chosen quantity as multiplied by the least of the above three qunntities) 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 giveu quantity). Or, the square of the quantity (the cube whoreof is required), when combined with the product (obtained by the multiplication) of this giveu quantity diminished by one by the sum of a serios in arithmetical progrossion in which the first term is one, the common difforence is two and the number of torms is (equal to) the given quantity, (gives rise to the oube of the given quantity). 49. Synbolically expressed, this rule works ont thus: (i) a xaxa=al: (ii) a (a + b)(a-b) + be (a - b)+ b al 44. Algebraically, this rule means (i) ad == a + 3a + 5a + 7 + .........to a terme. (ii) a=a'+ (a-1) (1+3+5+7+ ........to a terms).

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CHAPTER II-ARITA VETICAL OPERATIONS. 17 45. In an arithmetically progressive series, wherein gun is the first term ns well as the common liflorone, and the number of terms is (egnal to the riven number, multiply tho preoeding 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 serios in arithmetical progression), bocomes the cubo (of the given quantity). 46. (In a given quantity, the squares of (the number reproBeutel by the figures in the last place as also (ly those in the other (remaining placos) are taken ; and cach of these (squarcs) is multiplied by the number of the other place and also by three; thu sum of the two (quantities resulting thus), when combined again with the cubes of the numbers oorrosponding to all tho (optional) places, (gives riso to) the cube (of the given quantity). 17. Or, the cubo of the last figuro (in the number counted from right to left is to be obtained); and thrior the square (of that last figure is to be pushed on to the right by ono notutional plaor) and multiplied by the number represented by the figures found in the remaining (placos); theu the Hybro of this number represented by the figures found in the remaining (placos) is to be pushed on (us above) and multiplied by thrico the Inrt figuru (above-mentioned). These three quantities are thon to bu placod in position and then summed up). Such is the rule to be carried out) here. Framplex in illustration thereof. 48. Give out tho cubes of the numbers from 1 to 9 and of 13, 25, 35, 77 and 96. 49. Give out the cubes of 101, 172,516,717 and 1341. ........ +: 3 1x2 + 2x3+3 **+ **5+ ... +a- + -a! 40. 3ab+34) + a + ( + ) To make the rule general and applicable to numbers having more than two place, it is clearly insplied here that 3 1 + + 3u (b + c)* + a+(+ c) (a + b + c)', and it is obvious that any number my be represented as the sum of two other suitably chosen number 47. 'llo pnahing on of a figure here referred to in vinilar to what in exhibited in the note under stants 31 in this chapter.

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18 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 enbes. Give out those cubes 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 cubing, 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; thon 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) Odhya place (after it is taken into position) the square of the (abovo) quotient as multiplied by three and by the already mentioned (root of the highest possible cube); and then (subtract) from 53 and 54. The figures in any given number, the cube-root whereof is required, are conceived in these rules to be divided into groups, each of which consists as far as possible of three figures, named, in the order from right to left, na ghana or that which is cubic, that is, from which the cube is to be subtracted, as sodhya cor that which is to be subtracted from, and as bhajya or that which is to be divided. The bhajya and sidhya are also known as aghana or non-cubie. The last group on the left need not always consist of all these three figures; it may

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CHAPTER II-ARITHMETICAL OPERATIONS. the (number represented by the figure in the next) ghana placo (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-cubie. Divide (the non-cubie figure) by three times the square of the cube root. From the (next) non-enbie (figure) subtract the square of the quotient (obtained as above and) multiplied by three times the previously mentioned (cube-root of the highest enbe that can be subtracted from the previous cubie 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-rt-figures (so obtained (and taken into position, the procedure is) as before. Examples in illustration thereol. 55. What is the cube root of the numbers beginning with 1 and ending with 9, all cubed; and of 4013; and of 1860867? 56. Extract the cube root of 13821. 36926087 and 618170208. consist of one Ao or three figures, as the case may be. will be clear from the following worked out example. To extract the cube root of 77308776: gh. bi gh..... bi..... T .. 2 x 3 x 1 * 4. W. 7 73 15 + x - 1)133(2 IN 24 1 ... 6 x 12 370 48 322N 8 ... 42 x 5292)32207(6 31752 4536 216 216 * " X 19 The rule mentioned Litt 7 7 6 Cube root 121. but it is meant The rule does not state what figures constitute the cube rent that the cube root is the number made up of the figures which are enbed in this operation, written down in the order from above from left to right

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GANITASARASANGRAHA. 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 cubic quantity Thus ends cube 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 (aniong tho parikurman operations), is as follows: 61. The number of terms in the sorice is (first) diminished by one and is thon) halvoi and multiplied by the common difference; this when combined with the first trrm in the series and (then) multiplied by the number of terms (therein) becomes the sun of all (thre terms in the series in arithmetical progression). Tho rule for obtaining the sum of the series in another manner: 62. The number of torms (in the series) as diminished by one and (then) multiplicd by the common difforence is combined with twice the first torm in the sories; and when this combined sum) is multiplied by the nunber of terms (in the series) and is (then) divided by two, it becomes the sum of tho scries in all cases. / 01. This rule cones ont tous wben expressed algebraically : 17-0+ a ju == s, where a is the tirat term, t the common difference, n Abe number of terms, and the sum of the whologeries. 12. Similarly,

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CHAPTER II-ARITHMETICAL OPERATIONS. 21 The rule for finding out the vididhana, the uttaradhana and tbe sarvadhana : 63. The ididhawa is the first term multiplient by the number of terms (in the series. The uttarakhand is the product of) the number of terms multiplied by the common difference (and again) multiplied by the half of the number of terme less box one. The sum of these two gives) the surrulhand, ie, the sum of all the terms in the scries; and this sum will be the name as that of a series wbich is characterised by a negative common difference, when the order of the terms in the series in reversed so that the last term is made to be the first term, The rule for finding the antyethana, the wurdhyadhana and the sarradhana :-- 61. The number of terms in the arriva lossche boy www and multiplied by the cominion difference and then combined with the first torm (given the antycellunt. - llall of the sum of 03-64. In the rule of the terms in all withoutically g i ve WHICH I S wbtained cling to the litora tre multiple of tho comun lift wonen, the nuts of thin iple bring the teamured by the Rition which thy Kefim term bol in the Nor d ing to this concepti. wolve to find in to the main the first teemalong with u multiple of the common difference. The wurms of all uchun IN NO found in what in her called the adidhane; the sum of H much multiples of the common difference Corintitute the tawhanu; and the marvaillant which is obtained liv uilding these two mums is of course the sum of the whole mriew. The expronion tyadhana (lenotch the value of the last t ill en arithmetically pro v o Borice. And marthyadha wa means the value of the middle form which value bowever, coitoond to the arithmetical ban of the first and the lat terminin the series, that when there are 2n + 1 teria in the pink, the volor of the (* + 1)th town in the wadhyadhana, but win there are 2.1 termin in the merica the arithmeticul menn of value of the l and of that of the (n + 1) torm becomes tho madhyadhana. Accordingly weber (1) Adidhani xa. (2) Uttarad . l 6) Aniyda (n-1) + (4) Jodhyadhan su cuilhan (1) + (%) (*)

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22 GANITASARASANGRAHA. this (antyadhuna) and the first term (gives) tho madhyadhuna. The product of this (madhyarthana) and the Dumber of terms (in the series gives) the desired suin of all the terms therein. E.camples in illustration thereof. 65. (Euch of) ten merchants gives away money in an arithmetically progressive series) as a religions offering, the first terms of the (ton) series being from 1 to 10, the common difference in cach 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 soring). Calculate the sims of those scries). 66. A certain excellent sricaha gave gems in offering to 5 temples (one after another) commencing the offering) with 2 gems), and then increasing (it successively, hy 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 $; and the number of terms is 13. All these three quantities) are (gradually) increascd lay 1, antil (there are) 7 (series). O arithmetician, givo o:t the sums 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 iu relation to 1000 cities, conmcucing the offering) with 4 ani iucreasing 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 quito obvious that an arithnetically progressive series having a negative common difference becomes cluanged into one with a positive common difference when the order of the terms is rerersed throughout so an to make the last of them become the first. 60. A fraraka is a lay follower of the Juina religion, who merely hears, i.e.. listens to and learns the hernias or dutice, as opposed to the datetice who art entitled to teach thoro religiour lutios. 69, Alyobrnically this rulu works oat thus V'(2a - 2)2 + 80S + b

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CHAPTER I-ARITHMETICAL OPERATIONS term and the common difference to 8 times the common diffor. once multiplied by the sum of the series, the common differenco is added, and the resulting quautity is bulved ; and when (again) this is diminished by the first term and then dividel loy the common difference, we get the number of terms in the series. Th. rule for finding out the number of terms intated) in another manner :-- 70. When, from the square ront of the quantity obtained by) the addition of the square of the difference between twice the first term and the common difference to ti differenor multiplied by the sum of the comes the comman side the common subtracted, and the resulting quantity is! pries, the lixeippada in this is divided by the common differs the " Sulsel; and when again of ternis in the series. we'm loco, (wo get the number em in Eramples in /wer of w 71. The first term is 2, rutrulion thereus are increased necessively 1.48 ori, the common difference 8; these two The aims of the three serie der DN 1 till three (sories are so made up). is the number of tormt ter Usare 0,276 and 1110, in order. What 72. The firsts of termy. of tho serieparat.jewermix 5. the common differentie 8, and the sum The led by 533 What is the number of terms? 8, and 08 18 rst term of another series) is 6, the common difference be sum +20. What is the number of terms? . 81. T' divided be rule for finding out the common difference 13 well as the (prodt term : 73. The sum of the series) (liminished by the adren, and (then) divided by half (the quantity reprehented by tho square .18 in each sering of the serieperst jewel 70. Kwjapaila ishnlf of the difference between lwire the hotter and the difference.. a-0. It is obvious that this the varies the role mentioned in the previous stanzu ply to the font r ent by the introduction of thin kripapuda therein. 73. For adidhand and utturadhana, me noto unor stanza 03 und 14 in this chapter. Symbolically expressed this stanza work out thus .. ( - 11

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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 d by the half of the number of terms less by one, gives when) the sum, divic series. The common difference is (obtained, uished by the first ternt by the number of terms and then dimiterms less by one. is divided by the half of the number of Two rules for finding out, i ence and the first term: nother way, the common differ 75. Understand that the common a li the sum of the series, multiplied by ifference is (obtained, when) number of terms (therein), is diminished by and is (then) divided by the number of terms lessened by one. fo and divided by the twice the first term, 76. Twice the sum of the series divided by the enumber of terms therein, and (then) diminished by the number of gs terms as lessened by one and multiplied by the common differeno 4 divided by two, (gives) the first term of the series. when 0, Examples in illustration thereof. in 77. The first term is 9; the number of terms is 7; and the sun. of the series is 105. Of what value is the common difference? 74. Algebraically, a 75. Symbolically, b 76. Algebraically, a 8 21 2S " # 28 21 21 1 2 b; and b = - 2 a - (n-1) b 2 S " -a N-1 2

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OHAPTER 11 - ARITHMETICAL OPERATIONS. The common difference (in respect of another series) is 5, the number of terms is 8, and the sum is 15EUR. Tell me the first term. The rule for finding out how (wben 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 boomes the first term (in the porics); and the remainder (got after this subtraction) wheu divided by the half of the number of terme lessoned by one becomes the common difference. Example in illustration therrul. 79. The sun given in this problem in 540. () crest-jowel of arithmeticians, tell me the number of terms, the common difforence, 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 torm, or witb the common differences, or with the number of terms, or with all thehe. 80 () errost-jewel of calculators, understand that the misraithana diminished by the uttarudita, and (then) divided by the number of terms to which one has been added, gives rive to be first torm. . 81. The wirudhawl, criminished by the w hana, and (then) ilivided by the quantity obtained by the) aldition of me to be product of the number of terms multiplied by the half of tho number of turmus lossened by www. (gives rise to the common 78. Symbolically, the problem herein is to find out when Min kiven, and a and are allowed to be chosen loption. Naturally, there may be in solution to any given value of Sony values of b, which went on the chosen value of a and . When the values of and re delinitely chon, the rule heroin kiven for finding onto turn out to be the ramuri lat niven in Mtunan 74 v. 40-82. The expression inadhara mak #mixed num. Itin used here to depote the quantity which may be obtained by adding the first t o the common difference or the number of termin or a thire of these to the sum of

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GANITASARASANGRAHA. difference. (In splitting up the namber of terms from the mieradhana), 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 (80 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 those) 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 differenco (required here). This is the mathod of work in (splitting np) the all-combined (mirulhana). Scamples in illustration thereof. 83. Forty exceeded hy 2, 3, 5 and 10, represonts (in order) the adi-mikrauthanul and the other (mirathanax). Toll me what (respeotively) happens in these cases to be the first term, the common difference, the number of terms and all (these three). serios in urithmetical progression. There are accordingly four different kinds of mifradhava mentioned hero; and they are respectively adi-migrudhana, wtura. mitrachana, yaccha-nisradhana and sarva-nisradhana. For adidhana and uttara. Ihana noo noto under stanzas 63 and 6 in this chapter. Sa- (n-1) 2 + whore Algebruicully, stanza 80 works out thus : a = _So is the adi-misradhana, i.e., 8 + a. 80 - ma where So is the uttura-mibradhaga, And stanza 81 giver b = 32 (n-1)+1 1.6., 8 +0: and further points out that the value of may be foun out, when the value of $n, which, being the gaccha-miuradhanu, it equal to $ + , in given, from the fact that, when S a + (a + b) + (a + 2b) + ...up tan terms, Sn = (a + 1) + (a + 1 + b) (a + 1 + 21) + ..... up to the same n terms. Simco, in stanza 82, the choice of u and n aro left to our option, the problem of finding out a, m, and l from the given value of Sand. which, being the marva. misradhana, is oyual to 8 + 0 + + + b, resolves itself canily to the finding out of 5 from any giveu value of St in the manner above explained. 83. The problom expressed in plainer terms is :-(1) Find out a when &a = 42, b=3 and 5. (2) Vind out b, when 80 = 43, q=2 and 5. (3) Find out when S + - 3, 4 * 2 and b 3. And (4) find out a, b, and when 3 + 4 + 0 + = 50,

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CHAPTER II-ARITHMETICAL OPERATIONS. 27 The rule for finding out, from the known rum, first term, and common difference of a given is in arithmetical progression), the first term and the commons terence (of auotlr sories), the optionally chosen sum (whereof) is wice, throe Ames, half, onethird, or sone such (multiple or fraction of the known sum of the given series):- . 84. Put down in two places for facility of working) the choson sum as divided by the known vie., the given) sum; this (quotient) whou multiplied by the (known) corumou difference gives the required) common difference; and that (name) quotient when multiplied by the (known) first torm gives the required) first term of the series of which the sum is cither a multiple or a fraction of the known sum of the given series) Eramples in illustration the rol. 85. Sixty is the (known first term, and the known) common difference is twice that, and the number of forms in the sume, 1.m., 4 (in the giveu scries is 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 (number) beginning with 2. The rule for finding ont, in relation to two (ories, the number of terms wherein are optionally whore'n, their mutually interchangod first term and common difference, it also their name whicb . may be cqual, or (one of which may be) twice, thrice, hall, or onethird, or any such (multiple or fraction of the other) : 86. The number of terins in one horios), multiplied by itself as lessened by me, and then multiplied by the chosen (ratio between the sums of the two Horics), and then diminished by Rek. Symbolically, as = sa, l's 1,w.rw. ;. ", und long nor the am, thin Arst torm and the common difference, in order of the mories whomum in chromon. Given the of two vericu, the ratio between the two firnt terms and that between the dommon differences need not alwaye bwa The wolution here given in hence plicablo only to curtain particular casen. 86. Algebraio ly, a - *(n-1) - 2n, and I, () - -2pn, whoro a, b and are the first term, the common difference and the number of

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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 (serics), 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). Examples in illustration thereof. 87. In relation to two men, (whose wealth is measurert 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 aud the interchangeable first term and common difference after calculating (them all) well. *8. In relation to two serios (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 interchangeable first term and common difference). The rule for finding out the first terms in relation to such (series in arithmetical progression) as are cbaracterised by varying oommon 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 euins : G and b being thus found ont, the first term and the oommon differonce of the second series are 8 and a respectively in value. 89. The solution herein given is only partioular case of the general rule *, ** (, --- ) + a, where a and an are the first terms of two verios, and

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OHAPTER JI-ARITHMETICAL OPERATIONS. largest common difference and any other remaining common difference is multiplied by the half of the number of terms lessened by one ; and when thie (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 serieu 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 scuond sets) have respectively common differences beginning with I and ending with 6 (in one onso) and have 1, 3, 5 and 7 as the common differences in the other case). The role for finding out the common difference in rolatiou to such (series in arithmetical progression) as are charactorised by varying first torms, equal numbers of terms and cqual sume :-- 91. Of that (series) wbich has the largest first term, one is taken to be the common difference. The difforonce between this largest first term and (each of the) remaining (smaller) first toring is divided by the half of the number of terms lessened by ine; and when this (quotient in each (ANC) in combined with ono, (we get the conuinon differences of the various scries having) the remaining (smaller) first terms. An (xample in illustration thereof. 9.2. O arithmetician, who bave seen the other shore of caloulation, give out the common differences of (all) those (series) which are characterized by equal gums and havd 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, und, are given, a, in determined by chooning any valon for a; and one in chosen as the value of a in the rule here. 91. The general formula in this charis b = " + b, wherein alno the value of bis taken to be one in the rule 1-1 given above.

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GANITASARASANGRAHA. The rule for finding out the gunadhana and the sum of a serios 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 (produot the frequency of the courrenge 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 eum 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 sero 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 sero 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 zero bappens to be the denoting item). When the result S = arxa 93. The guradhana of u series of terms in goometrical progression corresponde in value to tho (n + 1)th term thereof, when the series is continued. The valuo of this gunadhana algebrnioally stated is r xrx..... up to n such fnotors x a, i.e., ara. Compare this with the uttaradhana. This role for finding out the sum may be algebraically expressod thus : , whero a is the first torm, the common ratio, and n tho number of tormg. 94. This rule differs from the previous one in so far as it gives a new method for finding out by using the processon of squaring and ordinary multiplication; and this niethod will become clear from the following cxample: Letnin r bo equal to 12. 12 is even; it has therefore to bo divided by 2, and to be denoted 0: f = 3 is odd; 1 is it subtraoted from it, and it is , 1: 3-12 is even; it has divided by 2, and to be =lis odd;l is subtracted from it, and it is 1 1-1-20, which concludes this part of the operation.

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CHAPTER II --ARITHMETICAL OPERATIONS. 31 of) this operation) is diminishod by one, and (is thon) multiplied by the first term, and is then) divided by the common ratio lessened by one, it becomes the snm (of the series). The rule for finding out the last term in a goonetrically progressivo series as also the sum of that (sories) : 95. The antyadhana or the last term of a series in geometrical progression is the yumuidhiana (of another sories) wheroin the number of terms is less by me. This (antyodhana), whon multiplied by the common ratio, aud (then) diminishod by the first term, and (thon) divided by the common ratio lessened by one, gives rise to the sum of the sorios). An crample in illustration thereof. 96. Having (first) ohtnined 2 golden coins (in some wity), a man goes on from oity to city, carning (overywhere three timos (of what he carned immediately before. Say how much he will make in the eighth city. Now, in the representative column of figuras derived and given in tho margin o the lowest 1 in multiplied by r, which giver since this lowest I haw 0 above it, thu robinet before in ured, which gives wince this 0) i ham ) above it, the now btained is multiplied by r, which gives mi 0 wince this I har 0 above it, this in nuured, which gives': And Ninco I again this has another above it, this in nurod, wluch gives Thus the value of why be arrived at ly uning n bow time is powmible the proceMON of qualing and imple multiplication. The object of the method in to facilitate the ditermination of the value of dit in early on that tho method hold true for all positive and integral values of .. ar ixra 95. Exprowned algebraically, $ : ". " The antyathunu In the value of the last termin # mereu in krometrien progression for tho meuning and value of yunadhana, za %3 above in this chapter. The antyadhana of geometrically progressive series of n terminal, while the annadhana of the same nerica in ar". Similarly the antywland of geometrically progressive series of n - 1 terus in ar" , while the adhama thurwol in arr-, Here it in ovident that the antyalhana of the worin of terms in the mamo the qunadhana of the sering of -1 terme.

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32 GANITABARASANGRAHA. 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 <Page #235
CHAPTER ARITHMETICAL OPERATIONS. term, the common ratio and the number of torms thereof being 3, 5 and 15 (respectively)? The rule for finding out the common ratio and the first term in relation to the (given) sum of a series in geometrical progression: 10. 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 5, 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? 101. The first par of the rule will become clear from the following example:-- The sum of the series is 4095, the first team 3, and the number of terms 6. Here, dividing 40:55 by 3 we get 1365. Now, 1365-1=1364. Choosing by trial 4, we have 341; 341 - 1 1364 1 * 310 3-44; 85, 85 - 1 81: 8.4 -21; 21 - 1 20; 20 + 5; 5-1 4; J. Hence 4 is the common 4 ratio. The principle on which this method is band will be clear from the following:-- " (TN-1) by r. - /4 33 "-1 r-1 ; and 11-1 7-1 - 1 TH-T 1-1 The second part expressed algebraically is a which is obviously divisible (-1) 7-1 7-1 -1 6

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GANITASARASANGRAHA. Tho rule for finding out the number of terms in a geometrically progressive series : 103. Multiply the sum of the given series in geometrical progression) by the common ratio lessoned by one ; (then) 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 commou ratio- that gives the measure of the number of terms in the series). Examples in illustration thereof. 104. O my excellently able 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) bave 5 for the first term, 2 for the common ratio, 1275 for the sum : 7 for the first term, 3 for the cominon ratio, 68887 for the sum : and 3 for the first term, 5 for the common ratio and 22888183593 for the sum ? Thus enda summation, the seventh of the operations known as Parikarman. Vyutkalita. The rule of work in relation to the operation of Vyutkalita, which is the cighth (of tho Purikarman operations), is as follows : 106. (Tuko) the chosen-off number of terms as combined with the total number of time (in the series), and (take) also your own chosen-off number of terms (simply); diminish (each of) * In a given series, any portion chosen of from the beginning is called i la or the chosen-olf part; and the rest of the periog is called fra, and it oontains the remaining terms and forms the remainder-series. It is the onm of these a terms which is called vyutkalita. 104. Algehrnically, vyutkalita or Sp = {*+ -?o+a} (n - A), and the aum of the ista or Si whered in the number of terms in the chosen-off part of the series. d

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CHAPTER II --ARITHMETICAL OPERATIONS. these (quantities) by one and (then) halve it and multiply it by the common difforonow; and (then) add the first term to (cach of) these (resulting products). Anil these resulting quantities), when multiplied by the remaining number of terms and tho chosen-off number of terme (reupeotively), give riso 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 remninder-acrice and also the sum of the chosen-off part of the given series :-- 107. (Tako) the chosen-off number of terminas combined with the total number of terms in the series), and (tako) algo tho choseu-off number of terms (simply); diminish (cach of) thum by one, and (thon) multiply by the common differences, and (then) add to (cach of these resulting products) twice the first term. These (resulting quantities), when inultiplied by the half of tho remaining number of terins and by the half of the chosen-off number of torms (respectively), give rise to the sum of the romainder-sories and to the sum of the clowon-off' part of the orien (in onler). The rulo for finding out the sum of the remainder-acrion in respect of an arithmetically progressive in well as a geometrically progressive series, as also for finding out the remaining nuubor of terms (belonging to the remainder-nerics) : 108. The sum of the given morics) diminished by the sum of the chosen-off part (of the series) gives rine to the moun of the remainder-serien in respect of the arithmetically progressive as well as the goomctrionlly progressive series; and when the difference between the total oumber of teronth and the obosco-off number of terms in the series) is obtained, it becomes the romain. ing number of terms belonging to that (remainder-serior). 197. Again, S = {(u + d = 1) 6+20) "z", ond S. -- {(- 1)0*24

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36 GANITASARASANGRAHA. Tho rule for finding out the first term in relation to the remaining number of terms (bolonging to the remainder-series) : 109. The chosen-off number of terms multiplied by the common difference and thon) combined with the first terin (of the given series) gives rise to the first term in relation to the remaining terms (belonging to the remainder-series) " he already mentioned common differonco is the common difference in relation to these (remaining torms also); And in relation to the choseu-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 goometrically progressive series : 110. Even in respect of a geometrically progressive series, the cominon rati) and the first term ar exactly alike in the given serios and in the chosen-off part thereof). There is (however) this difference bere in respect of the first term in relation to the remaining number of torms (in the remainder-serios), viz., that the first term of the 'givon) 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 teims, gives rise to the first term of the remainder-series) . Examples in illustration thereuf. 111. Calculate what the sums of the remainder-sorios are in rospect 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 conuection with a series in arithmetical progression) here (given), the first term is 6, the common difference is 8, the number of torms is 36, and the chosen-off numbers of turms are 10, 109. The first term of the remainder series = db + a. The series dealt with in this rulo is obviously in arithmetical progression . 110. The first term of the remainder series is art.

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CHAPTER II-ARITHMETICAL OPERATIONS. 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-series. 37 113. The number of terms (in a series in arithmetical progression) is 216; the common difference is ; 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 remainderscries 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 have 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 ryutkalita, the eighth of the operations known as Parikarman. Thus ends the first subject of treatment known as Parikarman in Sarasangraha, which is a work on arithmetic by Mahaviracarya. 115. In this problem, there are given 7 different fruit trees, each of w: ich has 16 bunches of fruits. The lowest bunch on each tree has 4 fruits, the fruits in the higher bunches are geometrically progressive in number, the common ratio being 2; and 10, 9, 8, 7, 6,5 and 1 represent the numbers of the bunche We have to find out the total removed from below in order from the 7 trees. Maltehhavi. of the remaining fruits on the top of the various good tree kridita, 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.

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38 GANITASARASANGRAHA. CHAPTER III. FRACTIONS. The Second Subject of Treatment. 1. Unto that excellent Lord of the Jinus, by whom the tree of karman has been completely uprooted, and whnae lotus-like feet are ouveloped in the halo of splendour proceeding from tho tops of the crowns belonging to the chief sovereigns in all the three worlds (unto that Lord of the Jinas), I bow in devotion. Herenster, we shall expound the second subject of treatment known as Kalisuvurna* (i.e., fractions). Multiplication of Fractions: The rule of work hero, in relation to tho multipliention of fractions, is as follows: 2. In tho multiplication of fractions, the numerators are to be multiplied by the numerators and the denominators by the denominators, after carrying out the process of croes-roduction, if that be possible in relation to thein. Examples in illustration thereof. 3. Tell me, friend, what a person will got for of a pala of dricd gingor, if he gots of a pana for 1 pala of such ginger. 4. Where the price of 1 pala of pepper is of a puna, there, say, what the price will be of of a pala. 5. A person gets i of a pali of long peppor for 1 pana. O arithmetician, mention, after multiplying, what (he gets) for v rapav. i. Where a merchant buy's of a pala of cumin seeds for 1 pana, there, you who possess complete knowledge, mention what (ho buys) for panus. 7. The numerators of the given fractious begin with 2 and go on increasing gradually by 2; again their denominators begin Kaldmarna literally means parts resembling io, since kald donoter the sixteenth part. Honce the term Kulisavara has come to signify fraotions in general. 2. Whon fx is reduced as the process of cross-redaction is applied. 7. The fractions hereiu mentioned are: %, $,, da.

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CHAPTER PRACTIONS. 39 with 3 and go on increasing by 2; thoso (numerators and denominators) are, in both (the cases), 10 in number. Mention, of what valno the products here will be, when those (fra-tions) are multiplied, they being takeu two by two. 'Thus cuds multiplication of fractions. Division of Fractions. The rule of work, in relation to the division of fractions, is an follows: 8. After making the denominator of the divisor its numcritor (and vice versa), the operation to be comincted then in an in the multiplication (of fractions). Or, when the fractions consiituting) the divisor and the dividend are multiplied by the denominutora of cach other and these two produrts) are (thus reduced sons to be without denominators, (the operation to be conducted in as in the division of whole numbers. Ecumples in illustration thereof. 9. When the cost of half n pah of infection is of a p a, what does a person get it be wolle 1 pole at that (OMC) rate? 10. In case a person gels?" of : pou for of pulu of rol sandalwood, what will be get for 1 perlu (of the manur woul)? 11. When ? palas of the perfume sukha in obtainable for 1 of & pann, wlnt (will be obtainable) for 1 ju at that (naine mnto)? 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 throughout) lor liy one (ihan the corresponding pumciators). Tell me what the mult is when the encoreding (fmctions hero) aro divided in order by the proceding ones). Thus ends the division of fractions. 8. 6) + ) * = ud + bc.

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GANITABARASANGRAHA. The Squaring, Square-Root, Cubing, and Cube Root of Fractions. In connection with the squaring, the square root, the oubing, and the cube root of fractions, the rule of operation is as follows: 13. If, after getting the square. the square root, tho cube (or) the cube root of the (simplificd) 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 ns the c180 may be) iu relation to fractions. Eramples in illustration thereof. 14. O arithmetician, tell me the synares of , 1, 1, 19, 20, 1.54 and 299 15. The numorators (of the given fractions) begin with 3 and (gradualls) riso by 2; tho denominators begin with 2 and (gradunlly) riso by 1; the number of these (fractions) is known to be 12. Tell me quickly their snares, you who are foremost among arithmetioins. 16. Tell me quickly, O arithmetician, the square roots of 1, }, I'm z's and 17. O clever man, tell me what the squaro roots are of the squared quantities which are found in the (examples bearing on the) squaring of fractions and also of 75% 18. The following quantities, uamely', , , , , , , }and , Are given. Toll me their cubes separately. 19. The numerators (of the given fractions) bogin with 3, and (gradually) rise by 4; the denominators begin with 2 and (gradually) riso by 2; the number of such (fractional) terms is 10. Tell me their cnbes quickly, friend who are possessed of koen intelligonce in caloulation. 17. Ilere 078 is given in the original ax 700-3* 8. 25

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CHAPTER III-YRACTIONS, 20. Give the oube roots of 1 and 24. 21. O friend of prominent intelligonoo, give the oube roots of the cubed quantities found in (the examplog on) the obing of fractions and (givo also the cube root) of 2172. 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 umber of terms making up the fractional series in arithmetical progression) is multiplied by the common difference, and then it) in combined with twice the first term and diminished by the common difference. And when this resulting quantity) is multiplied by the half of the number of turmy, it gives rise to the sum in relation to a fractional serinn sin arithmetical progression). Examples in illustration thereuf. 23. Tell me what the sun is (in reIntion to a series) of which 3. . and are the first term, the common difference and the number of torms (in order); is also in relation to another of which }, { and ] (constitnto these elements) 21. The first torm, the common difference and the number of terms are , and (in order in relation to a given series in arithmetical progression). The numerators and donominators of all (these fractional quantities) are (noncessively) increasod by 2 and 3 (respectively) until seven (series are so made up). What is the sum (of each of thes)? 22 Algebrnicnly $ = in6 + 2a-6)". Ci nota under o2, Chap. 11 23. Whenever the number of termin a win in viven ** fruction, a hero, it in evident that such nerics cannot generally be formed actually number of terms. But the intention peil to be to show that the rule holda good ovon in such coses.

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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 (so) chosen is the number of terms, and one is the first term. The number of terms diminished 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. 42 Examples in illustration thereof. 26. The optional number of terms (in a given series) is (taken to be); 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 arithmetical progression) which (sum) happens 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 n 25. It is obvious that, in the formula S= 2 (2a+1.6), the value of S (n-a)2 In the multiplication of becomes equivalent to n whon a 1, and b 2-1 this sum by ", there is necessarily involved the multiplication of a as well as of b by n, so that, when an and b=! (na) 2n, Sn. A little consideration will show n-1 makes it possible to arrive at 2 as the value of S 2(a) how the value of b as 1-1 whatever may be the value of a, whether fractional or integral. 27. This rule gives only a particular caso of what may be generally applied. 3. 52 The rule as given here works out thus: 4 + ++ +up to 2s terms 4

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CHAPTER 11-PRACTIONS. common difference. The common difference multipliod by four is the number of terms (in the roquired sories). "The sum as related to these is the cube (of the chosen quantity). Eramples in illustration thereof. 28. The numerators begin with 2 and aro muiocoksi voly increased by 1; the denominators begin with 3 and are also) NUCrosnively increased by l; and both these kinds of terms (namely, the numerators and the denominaturn aro (severally ) five in mumlor). In relation to these (chon fractional quantities), vive out, 0) friend, the cubic sum and the corresponding) first term, common difference, and number of terms. The rule for finding out, from the known aum, first term and common difforence (of a given series in arithmetical progression), the first term and the common difference (of a serice), the optionally chosen gumi (whoreof) is twice, three times, half, one-third, or somo such (multiple or fraction of the known some of the givon 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 (known) common difference, gives the required) common difference and that name quotiont), when multiplied by the (known) first term, gives the (required) first tern--of (the series of which) the sun is either a multiple or a fraction of the known sum of the giveu series). Examples in illustration thereof. 30. The first term (of a series) is ], the coinmon difference in 1, and the number of terms common (to the given as well as the = (26)* =* The general applicability of thin pronunn can be utonou mado out from the equality, *(px) = ', so that in all such canot the number of terte in the series is obtained by multiplying by po the first term, which is refrowentable and the common difference is of course taken to be twice this first torm in every case. 20. See note ander 84, Chap. II.

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GANITASARASANGRAHA. 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) it. The sum of the required series is 7. Find out the first term and the common difference. 32. The first term is 1, the common difference is and the number of torms 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 role for finding out the number of torms in a series in arithmetioal 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 oommon 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 (thun) divided by the common difference, (we get) the number of terms in the series. He (the author) states in another way (the role 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 twioe the common differenoe multiplied by the sum of the series, the ksepapada is subtracted, and whon (this resulting quantity is) divided by the common difference, (we get) the number of terms in the series. - 2 det ) 88. Symbolically expressed, - 60, in Chap. II. 84. Por kipapada, see note under 70 in Chap. II. Cf. not under

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CHAPTER HI-FRACTIONS. 45 Examples in illustration thereof. 35. In relation to this (given) sories, the first torm is $, tho common differonce is , and the sun given is sh; again in relation to another series), the common difference is the value of the first term is, and the sum is In respect of those two (scries), o friend, give out the number of terms quickly. The rule for finding out the first tenu as well as the common difference : 36. The sum of the series) divided by the number of torms (therein), when diminished by (tho product of) the common difference multiplied by the half of the number of terms loss by one, (gives) the first term (of the series). The common difforonoo is (obtained when) the sum, divided by the number of terms and (then) diminished by the first term, is divided by the half of tho number of torms less by une. Examples in illustration thereuf. 37. Give out the first term and the common differenco (rospectively) in relation to the two sories characterised by), as the sum, and having in one case) an tho common differenco and as the number of terms, and in the othor oase) | as the first term and 4 as the number of terms. The rule for finding out in relation to two (sories), the number of terms wherein is optioually chosen, thoir mutually intorobanged first term and the common difference, as also their sums which may be equal, or (one of which may be) twice, thrioo, half or onethird of the other) : 38. The number of terms in one scrics), multiplied by itwulf as lessoned by one, and then multiplied by the chosou (ratio between the sums of the two serios), 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. 8ee note uader 74 Chap. U. 88. 800 note under 86 Chap. II.

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46 GANITABARABANGRAHA. (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). Examples in illustration thereof. 39. In relation to two series, baving 101 and 0} 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) suins, 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 produot 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 lessoned 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 gumalhana 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. Soo note under 98, Chap. II. 41. Bee note under 95, Chap. II.

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CHAPTER III-PRACTIONS. 47 divided ty the common ratio lessened by one, gives rise to the sum (of the series) ottes). An example in illustration thereof. 42. In relation to a serios in geometrical progression, the first term is t, the common ratio is and the number of terins is hero 5. Tell me quickly the sum and the last term of that (sories). 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, abould also be found out by means of the rules utatod already in the last chapter).* The rule for finding out the common) first term of two serios having the saine sum, ono of them being in arithmetical progression and the other in geometrical progression, their optionally oboson number of terms being equal and the similarly chosen common difference and common ratio also being oqual in value. 43. One is taken as the first term, the numbor of torms and the cominon ratio as well as the common difference (which is equal to it) are optionally choson. The uttaradhana (bero), divided by the sum of this geomotrically progressive series Hy diminished by the adidhana (thercof), and (then) multiplied by whatever is taken as the first term, gives rise to tho (required common) first term in relation to the two sories, (one of which is in goometrical progrosion and the other in arithmetical progression, and both of) wbich are characterised by sums of the same value, . For those ralen, nce 87, 94, 101 and 103, Chap. II. 43. For ddidhana and uttardhana, noc note onder 03 and 64, Ch. II. This rule, ayrubolically expressed, works ont thus : a = - wlore = ( -1) Por facility of working, 1 is chosen as the provisional first term, but it is obvious that any quantity may be no provisionally choren. The one of the provisional Brat term is seen in facilitating the statement of the rule by means of the expressions ddidhana und wftaradhawi. The formula bere given in obtained by equating the formulw kiving the sume of the krometrical and the arithmetical series. It is worth noting that the word caya ig und bere to denoto both the common difference in an arithmetical spd the common ratio in a geometrical weries,

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48 GANITASA LA SANGRAHA. Examples in illustration thereof. 44. The number of terms are 5, 4 and 3 (respeotively) and the common ratios as well as the equal) common differences are 4, 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 ternis (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 tiro; and these (resulting quantities), whon multiplied by the half of the remaining number of terms and by the balf of the chosen-off number of terms (respectively), give rise to the sum of the remainder-sories and to the sum of the chosen-off part of the (given) serios (in order). The rule for finding ont 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 ooinmon differenog, and combinerl with the chosen-off number of terms as multiplied by the common difference, as also with the balf 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 ander 106, Chap. II. 48. Cf. noto under 109 Chap. II.

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CHAPTER III-PRACTIONS 47. Even in respect of a geometrically progressivo series, the common ratio and the first term are exactly alike in the given series and in the chosen-off part thereof). Thero is (howevor) this difference here in respect of the first term among) the remaining number of torms (constituting the remainder-series), viz., that the first term of the (given) series multiplied by that self-multiplied product of the common ratio, in which (prodlnot) the freqnonoy of occurrence of the common ratio is measured by the chosen-off numibor of terms, gives rise to the first torm (of the remainder-series). Examplis in illustration thereof: 48. Culoulate what tho sum of the remainder-Ecries is in relation to that (series) of which is the common differenco, the first term, and is (taken to bo) the number of terms, when the chosen-off number of terms (to be remored) is (takon as) . 49. In relation to a series in arithmetical progression, the first term is , the common difference is }, and the number of terms is (taken to be). When the choson-off number of forms to be removed) is taken as) , give out, you who kuow calculation, the sum of the remainder-series. 50. What is the value of the sum of the remainder-serica in relation to a series of which the first term in 1, the common difference is l, and the number of terms is taken to hu) , when the chosen-off number of terms is '? 51. The first term is , the common difference is $, and the number of teins is taken as), and tho choyan-off number of terug is taken to lie , or l. () you, who, being the abode of kaliy, are the moon shining with the moonlight of winilon, tell me tho sum of the remaining number of torn. 52. Calculate the sum of the remaining number of terms in relution to a series of which the number of terny in 12, the common difference is minus , and the first term is 41, the chosenoff number of terms being 3. 1,5 or & 47. Sme note under 110, Chap. II. * Kals is here bred in the double benim of noun Journing' and the divit of the

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GANITASARASANG RAHA. Example in illustration of vyatkalita in relation to a series in geometrical progression. 53. The first term is 74, the common ratio is y, 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 tering in relation to the (rospective) remainder-series ? Thus ends the vyutkalita of fractions, The six varieties of fractions. Lorcaftor we shall expom 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 iu dissociation), together with Bhayamatr (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 sinple 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 alteration by the quotients obtained 55. 'l he ineth of reducing fractions to common denominntors described in this rule applics only to pains of fractions. The rule will be clear from the following worked out example: To simplify " + " Here, a and Jy are to be multiplied by a which is tho quotiont obtained by dividing yz, the denoninator of the other fraction, by y which is the common faotor of the douaminntors. Thus we got . TUZ Similarly in the second fraction, by multiplying band ya by : which is the qaotient obtained by dividing the first denominator zy by y the common factor, weget b Now + bx a3 + bx 2 yo w y sys sya

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CHAPTER 11-FRACTIONS. 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 au reduced us to have equal denominators. (Then) removing ono 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 numerator 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) common 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 multiplying those (denominators and numerators by means of the quotients derived from the division of the niruddha by the (respective) denominators, the denominators become equal (in value). Examples in illustration thereof. 57 and 58. Aravaku purchased, for the worship of Jina, jambu fruits, limes, cranges, cocoanuts, plantains, mangoes and pomegranates for 4, 4, 12, 20, 30, 2, and of the golden coin in order; tell (me, what the result is when these fructions) are added together. 59. Add together 15, 20, 36, 4 and 60. (There are 3 sets 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 GO. The resulting problems are to find the values of 1 ) (i) + 1X2 2 (ii) .in ru 2 X 3+ the given h is the numerato,+ 2x2+5x1 + 3 2 3 X 4 3 4X5 + UI + 2 4 X 5 3 XB 1 8X9 2 79 x 10 3 16 10 3 16 x 18+ iu 1 H

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52 GANITASARASANGRAHA. 16 (in order in the respectiva dets); the numerators (of these sets of fractions are of the same value as the first number in theso sets of denominators), and overy one of these (above-mentioned denominators in oach sot) is multiplied by the next one (the last denominator, however, remaining in each case unchanged for want of a further multiplying denominator). What is the sum of (each of) theso (finally resulting sets of fractions)? 61 and 02. (There are 4 sets of fractions), the denominators *wbercof begin with 1, 2, 3 and + (respectively) and rise suocegsively in value by 1 until 20, 42, 25 and 36 become the last (denominatora in the several sets) in order; the numerators of theso (sets of fractions) are of the saine 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 addivg these (finally resulting sets of fractions) ? 63. A man purchasod ou account of a Jina-festival sandalwood, camphor, agaru and saffron for,., and of a goldon coin. What is tho remainder (loft thereof)? 64. A worthy sravaka gave me two golden coins and told me that I should bring, for the purpose of worshipping in the tumplo of Jina, blossomed wbite lotuses, thick curds, ghee, milk and sandal-wood for }, }, }, and is of a golden coin, (respectively, out of the given amount). Now tell me, O arithmetician, what remains after subtracting the (various) parts (80 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 theso (sets of fractions) are of the same value as the first term in cach (of these sets of denominators). And every one of tho denominators (in each sot) is multiplied by the next one, the last (denominator) being (in each oase) multiplied by 4. After subtracting from 1, (each of) those two (sums obtained by the addition of the sets of fractions finally resulting as above), tell me, O friend, who have gone the other shore of the ocean of simple fractious, what iya by which in 'n8. hy u the oommon fac

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CHAPTER III-FRACTIONS. 53 67 to 71. The denomiuntors of certain given fractions aro stated to he 19, 23, 62, 29, 123, 35, 188, 37, 9, 17, 140, 141, 116. 31, 92, 57, 73, 55, 110, 19, 74, 219, in orler); and the numorators begin with 1 and rise accessively in value by in order. Add (all) these (fractions) and give the result, () you who have reached the other shore of the ocean of simple fractions. Here, the rule for arriving at the numerators, when the deno. inators and the sum of a muinber of fraction are given, in an Make one the numerater in relation to all the given deno. 6); then, multiply by means of such numbers us are Ally chosen those numeriutors which are derived from these 18 go as to have a common denominator, llere, those ors) turn out to be the reynirl nuncrators, the son of the hereof, obtained by multiplying them with tho numeraved its above, is equal to f lumerator of the given I the fractions concerned. The rule for arriving at the numerater's, the denominators and ar sum being given as before in relation to such fractional) quantities an have their inmerator'L ively rising in value by one, when, in the given: oan if these fractions, the denominator is higher in value than the morator: 73. The quotient obtained by dividing tho (given him (of tho fractions conoernod) by the sum of those (tentative fructions) 73. This rule will become clear from the king of the example in stansu No. 74, wherein weilsaume 1 to be the provisional meritor in ration to each of the given denominator ; thun we get to and ;', which, being record so as to haron common denoininator, become , and .. When the numerators are multiplied by 3, 3 und 1 in order, the sum of the products thus obtained becomes equal to the numarrator of the riven uni, tinmols, 477. Hence, 2, 3, and 4 are the required namerators. Here it may be pointed out that this givon sum aluo muat be understood to bave the same denominator in the common denominator of the fractions. 78. To work out the sum given under 74 below, according to thin rule: .. Rodaoing to the sam, denominator the fractions formed by umowning 1 to be the apmerator in relation to each of the given denominators, we get 19. apdoh. Dividing the giren kam #1 by the nom of thoso fractions 18%, we get the quotient 1, which is the namerutnr in relation to the first denominator. The remaindet 279

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54 GANITABARASANGRAUA. which, (while having the given denominators), have one for the numorators and are then reducel so as to have a common denominator, becomes the first required) numerator among thoso which (successively) rise in value hy one (and are to be found out). On the remainder (obtained in this division) being divided by the aim of the other numerators (having the common denominator as above), it, (i.e., the resulting quotient), becomes another (viz., tho second required) numerator (if added to the first one already obtained). In this manner the problem has to be worked out) to the end. Un example in illustration thereof. 74. The sum of certain numbers which are divided (respectively) hy 9, 10 and 11 is 871 as divided hy 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 AH, beginning with one, are in order multiplied (nuccessively) by obtained in this division in the divided by the son of the remaining provisional numerators, i... 189, giving the ypotient 1, which, combined with the numerator of the tirat fraction, namely %, becomca the numerator in relation to the scoond denominator. The remainder in this second division), viz., 90, is divided by the provisional numerator 90 of the Inst fraction, and the quotient I, when combined with the numerator 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 to and fr. It is notiovable here that thu honeruto's successively found out thus become the required numeratore in relation to the given denominators in the order in which they are given. Algebruically #lo, given the denominators a, c, in respect of 3 fructions lica + ( + 1) ac + (x + 2) ab whose sumn is the numerators + 1 and abc # + ? are easily found out by the method as given abovy. 75. In working on an example according to the method stated berein, it will be found that when there are fractions, thero nre, after leaving out the first and the last fructions - 2 terni in geometrical progressicy with an the first term and to the conimon l'utio. The bune of these # - 2 terms is . -) , which when reduced becomies ! - Ry" which is the same which is the

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CHAPTER III--PRACTIONS 66 three, the first and the last donowinators we obtained Wing (however multipliert (again, by 2 and respectively. Examples in illustration thereof. 76. The sum of five or six or seven (ilifferent fractioual) quantities, having l fot (ouch of) their umemtors, is 1 (in each case).. () you, who know arithmetic, say what the required) donominators are. The rule for finding out the denominators in the case of an mnoven number of fractione) : 77. When the sum of the different fruitional, quantities, having one for oach of their nuncrators, is one, the required) denominatong are such as, beginning with two, go on a successively rising in value by one, each (such denominator) being further) multipliol by that AK! From thin it is clear that, wlun the firme fraction ou the last fraction are added to this lust rosult, the um boumon 1. In this connection it may be noted that, in a series in geometrical progression consisting of n terms, having us the first toru and the common ratlo, the vum in, for all positive integral values of a, loan thun (n + 1)th term in the series. Therefore, if we udel to the sum of the series in geometrical progressivu , x the (+ 11th torm, which in the lunt fraction necording to tbe rule stated in this stanza, weet, have to adi - in order to get I am tlir om. Things in oneineet in this rulo as the first fraction, and wo 3 in the valne chun for a, nince the numerator of all the fractions has to be l. 17. Bere noteszi mitrinovavett on var? = z{ckstatoest...+ aco', +]

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50 GANITARARASANGRAHA. (number) which is immediately) next to it in value) and then halved. The rule for arriving at the required) denominatorg (in the case of certain intended fractions), when their numerators are (each) one or other than one, and when tlfe (fraction coustituting their) sum has one for its numerator : 78. When the sum of certain intended fractions) bas one for its numerator, thon (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) combinod 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) heing (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) aud (thirdly) . Say what the denominators of those fractional quantities) are. The rule for arriving at the denominators (of certain iutendod fractions) having one for their numerators, when the gum (of those fractions) has one or any quantity) other than one for its numerator: - 78. Algebraically, if the sum is -, and a, b, c, nld are the given numeratore, the fractions bummed up aro an below: -Hin + a) * (* + a) (n + a + b) (+ a + b) (x + a + 0 + c)" d ( + a + b + c) a(n + a + b) + b (+ ) ( + a + b) c + " + a + b (n + a + (n + a + b + c) * N * + a)( + b + # + a + b 1 + a + 6

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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 leave 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. CHAPTER III-FRACTIONS. 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). 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). of the remaining fractions in mentioned in the role to be a a 80. Algebraically, if is the sum, the first fraction is optionally chosen quantity. This " 1 + P P n + P a # 57 1 n + p a P n + P 26 & ; and the sum 'where p is the is obtained obviously by simplifying a We must here give such a value top that n+p becomes exactly divisible by a. 8

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GANITASABASANGRAHA. Examples in illustration thereof. 83. Say what the denominators are of three (different fractional) quantitien each of which has 1 for its numerator, when the sum (of those quantities) is it. 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 78. 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 (ohtained 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 to The first rule for arriving at the denominators of two (intended fractions) which have cither one or (any numbor) other 86. Alyobraioally, whon in the bim of two intended fractions, the fractions according to this rulo ure in and out, where p is any chosen quantits. It will pn pri P-1 Le soon at onoo that the sam of these two fraotions is - Or, whon the sam is an the fractions may be taken to be alat b) and Na+)

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11 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, thon 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 donominator of the other (fraction), however, is this (denominator) multiplied by the (above) ohosen (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; us also of two (other fractional quantities) which have 7 and 9 (rospectively) for (their) numerators. The second rule (is as follows):-- 29. The numerator (of one of the intended fractions) as mplied 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 59 as their numerators, then the fractions are mo up+b th and a ap+b n m is any number so chosen that ap + b in divisible by m. it will be found, is 118 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 in itself substituted in this rule for the quantity to be chosen in the previous rule. x-xp 1 where p m P The sum of these fractions,

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60 . GANITABIBASANGKAHA. 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 (rospectively) for (their) numerators, 91. The sum of , , and 13 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 }, }, } and as 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 samo 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 (soverally) as the sums of the pairs; (and from them) the (roquired) denominators are to be found out in accordance with the rule relating to two (such component fraotional quantities). Examples in illustration thereof. 94. What are tho 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 aro 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 aro arrived at according to any other of those 93. The rules renting to two fractional quantities hate been given in stanzas 86, 87 and 89.

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CHAPTER 11-PRACTIONS. 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 donominators derived in accordanoe with another rule, become the (required) denominators. The sum (of all the fractions), diminishod by the sum of the specified part (thereof), gives the measure of the optionally leftout part. Eramples in illustration thercol. 96. The number of fractions (obtained by rulo 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 tho (component) fractions? 97. The number of fractions (obtained) by rule No.78 is 7, and 3 (is obtained) by rulo No. 77. When the sum of the fraotions arrived at) with the help of these (rulen) is 1, how many pro tho (component) fractions ? 98. Certain fruotions 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 uni of all (these five) being I, what is that (fraotional) quantity (which is left out) Hero ond Simple Fractions. Compound and Complex Fractions. l'ho 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 forning) tho denominator (becomes) tho multiplior of the number forining the numerator (of the given fraction). 4). The complex fraction here dealt with in of the soft which how all integer for the numerator and a fruction for the denominator

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02 GANITASIBASANGBAHA. Examples in compound fractions. 100 to 102. To offer in worship at the feet of Jina, lotuses, jasamines, ketakis and lilies were prrchased in return for the payment of f of 1, of }, of of }, } of fof, of } of s, of of }, of } of of }, { of of of of }, } off of of }, and } of }, of a pana. Sum up these (paid quantities) and give out the result. 103 and 101. A cortain person gave (to a vendor) of 1, of of }, of, of }, and of b, (of a pana) out of the 2 panas (in his posscasion), and brought fresh ghee for (lighting) the lamps in a Jina temple. O friend, give out what the remaining balance is 106 and 106. If you have taken pains in connection with com pound fracticus, give out (the rosulting sum) aftor adding these (following fractions) :-} of }, 1 of , l' of 'n o of, is 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 whatover happens to be the sum arrived at in accordance with the rulo (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. 108. The sum of $, of 1, $ of , t of of , of a certain quantity is 1. What is this unknown (quantity) P The rulo for finding out more than one unknown (clement, one such occurring in cach of a set of compound fractions whose sum is given) 109. Make the unknown (values of the various partially known compound fractions) to he (equivalent to) such optionally chosen 100. This rule will be clear from the following working of the problem giren in stanta No. 110: Splitting up the sum of the intended fractions, into 3 fractions according to rule No. 78, we get, undt. Making these the values of the three

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CHAPTER III-FRACTIONS. quantities, as, (being equal in number to the given compound fractions), have their sum 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. 111. (Given) An example in illustration thereof. 110. (The following partially known compound fractions, viz.,) of a certain quantity, of of another (quantity), and of of (yet) another (quantity give rise to) as (their) sum. What are the unknown (elements here in respect of these compound fractions)? 68 Examples in complex fractions. and # ; say what the sum is when these are added. 112. After subtracting, and also and, from 9, give out the remainder. 4 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 Bhaganubandha class (of fractions), add the numerator to the partially known compound fractions, we divide them in order by t,t of i, and of respectively. The fractious thus obtained, viz. 1, 1, and, are the quantities to be found out. 118. Bhdganubandha 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 of this associated quantity. The expression"] associated with" means + of. The meaning of the other example here is + of 1+1 of (+1 of 3). This kind of relationship is what is denoted by association in additive consecution.

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64 GANITABARASANGRAHA. (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 numeraton, and by the denominator (itself, of this other fraction). A 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 24 (niskas). purchased for 14, camphor for 10, and What is (their total) value when added ? 116. O friend, subtract from 20 (the following):-8, 61, 21 and 34. 117. A person, after paying 74, 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 associated quantity); and also (is similarly associated); ciated with its own and with (of this associated quantity). What is the value when these are (all) added ? (of this is asso 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, t, and (thereof); and incense (purchased) for (niska) associated (similarly) with and (thereof): what is the sum when these (niskas) are added ?

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CHAPTER III-PRACTIONS 65 120. O friend, subtract (the following) from 3: associated with $ of itself and with of this (associated quantity), 3 Assvoiated with, and of itself (in additive consoontion), similarly) Associated with (fractions thereof) commencing with and onding with , and associated with of itself. 121. O friend, you, who have a thorough knowledge of Bhaga. nubandha, give ont (the result) after adding associated with of itself, to associated with of itself, l associated with of itself, associated with of itself, and associated with of itself. Now the rule for finding out the ono unknown (clement) at the beginning (in cach of a mumber of associated fractions, their sum being given): 122. The optionally split up parts of the given) sum, which are equal (in mumber) to the intended) component clements (thercol), when divided in order by the resulting quantities arrived at by taking one to be the associate quantity (in relation to those component elements, give rise to the value of the (required) unknown (quantities in association). E.complex in illustration t ool. 123. A certain fraction is associated with lud of itself (in additive consecution); another (in similarly associata) with 1, 2, and, of itself; and another again is similarly storiated) with }, . and 1 of itself; the sum of those (threr fructions 40 MHHO) ciated) is 1 : what are these fractions ? 124. A certain fraction, when associated (analove) with }, }, 1 and 1 of itself. lecomes. Toll me, friend, quickly the measuro of this unknown (fraction). 122. This rule will be clear from the working of Naple No. 123 * *xplained below:-- There are three oth of fraction given and splitting up the mum, 1into three fractions recording to rule No. 75. Wek a nd. By dividing them fructions by thu quantities obtained by simplifying the three kiyn Nets of fractions wherein 1 is usilmed as the unknown quantity, webtuin ! and it, which are the required quantitire.

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66 GANITASARASANGRAHA. The rulo 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 Bhaganubanitha class (of fractions). Bhagapavaha Fractions. Thon (comes) the rule for the (simplification of) Bhagapaviha (or the dissociated) variety (in fractions) : 124. 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 donominator. (When, however, the dissociated quantity is not integral, but is fractional, mulitiply (respectively) the numerator and the donominator of the first (fraction to which the other fraction is negatively attached) by tho denominator diminished by the numerator, and by the denominator (itself, of this other fraction). Examples on Bhigipavaha fractions containing dissociated integera. 127. Karsus 3, 8, 4 and 10, diminished by 1, 1, !, and of a karsa, are offered by certain men for the worship of tirthaikaras. What is (the sum) when they are added ? vrally weapouleu. Wheneracted from 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 cas, ench diminished by one, 126. Bhriga pariha literally means fractionnl dissociation. As in Bhaganu bandha, there are two varieties here also. When an integer and a fraction are in Bhaiyapavviha relation, the fraction is simply subtracted from the integer. Two or more fructions may also be in such relation, as for example, disguci. ated from of itself or dissociated from , and, of itself. It is meant bere " that of is to be subtracted from in the first esample; and tbe second example comes to 9-of 9 - 60f (9-4 of 9)-6 or.{l-/of B-1 of (9 - of 9)}.

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CHAPTER III-PRACTIONS. 87 128. Tell me, friend, quickly the amount of the money romaining after subtracting from 6 x 1 of it, (the quantities) 9,7 and 9 as diminished in order by ,, and Examples on Bhagapava ha fractions containiny dissociated fractions. Alld }, , , & and; which aro (respectively) diminisbod by ,,, and of themselves in order; and (then) givo out (the result). 130. (Given) of a para diminished by A and of itself (in consecution); (similarly) diminished by, and of itself; & (similarly) diminished lay 3, 3 anil of itself; and another (quantity), viz., diminished by of itself-when these aro (all) added, what is the result? 131. If you huve taken paine, O friend, in relation to Bhigia pavila fractions, give out the remainder after subtracting from 1! (the following quantities): diminished (in rousaution) by , and) of itself; alsof (similarly) diminished by A, Land of itself; and (also) (similarly) diminished by A and f of itself. Here, the role for finding out tho (ono) unknown olemment at the beginning in cach of a uumber of dissociated fractions, their num being given): 132. The optionally split up parts of th: (givell mum which aro equal (in number) to the intended) component cloments (thereof), when divided in order by the resulting quantitich arrived at by taking one to be the dissociated quantity (innlation to these component elements), give rise to the valac of the required) unknown (quautities in dissociation). Examples in illustration thereof. 133. A certain fraction is diminished (in consecution) by 1, $ and l of itself; another fraction is similarly) diminished by t. * and of itself; and (yet) another is (similarly diminished by 2, 132. The working in similar to what has bech explained undur stupis No. 122.

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68 GANITASARASANGRAHA. and of itself. The sum of these (quantities so diminished) is 1. What are the unknown fractions here? 134. A certain fraction, diminished (in conscention) by,,, 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 Bhayapavaha 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 theso (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. 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 fraction 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 stansas 132 and 135.

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CHAPTER 11-FRACTIONS. Bhagamat! Fractions. The rule for the simplification of) that class of fructions which contains all the foregoing varieties of fractions : 138, In tho case of the Bhiyamatr class of fractions (or that class of fractions which contains all the foregoing varieties), the respectivo rules pertaining to the different) variction beginning with simple fractions (hold wood). It, .., Bhigamali, is of twenty-six kinds. One is taken to be the denominntor in the case of a quantity which has no denominator. Eramples in illustration thereof. 139 and 140. (Given)!; ; ! of }; ofiki !;!; 11; sociated with of itself; then associated with of itself; 1 diminished by ); 1 diminished by 'o ; A diminished by 8 of itself; and diminished by of itself: after adding these acconling to the rules which are strong together in the manner of a parland of blue lotuses made up of fractions, give out, ( friend, what the result is). Thus ends the Bhiyamit, variety of fractions. Thus ends, second subject of treatment known as Fractions jarasangraha which is a work on arithmetic by Mahaviracarya. 138. The twenty-six varieties here mirationed arr Ishan, Praban, Bhaga. bhaga, Bhaganufaniha. and Bhayapa vald, in combination of two), Ilirer, four or five of theme at a timu; Huch ar, the variety in which Bling and Prabhagnare mixed. or Bhagaand Bhagabagat are mix, no on. The number of variction obtained by mixing two of them at a time is 10, Ly mixing three of them ha time is 10, and by mixing fuor if them at a time in and by mixing all of them at a time in l; ko there are 2 varieties. The example visen in mtanza 139 belong to this last-mentioned variety of Bharumats in which all the five simple varietium are found. 139. The word uprilamalki, which occurs in this mtanza, mean gurlund of blue lot unen, at the Mame time that it happens to be the frame of the metro in which the stanga is composed.

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GANITASARASANGRAHA. 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 praise worthy, 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 syaulvvida, and is the joy of learning, and is the great disputant and the best of sages, bo (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, Mali, Sesamula, the two varieties 3. The Bhaiya variety consists of problemy 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 tre each of thom called u bhiya, and the nomorical value of the known remainder is termed driyat. The a variety consists of problems wherein the numerical value is given of the portion remaining after removing a known fructional part of the total quantity to be found out is also after removing curtain known fractional parts of the successive di as or remainders. The Mala variety consists of problems whervin the numerical value is given of the portion remaining after subtracting from the total quantity certain fractional parts thereof as I it multiple of the Nuare root of that total quantity. The Sumulle variety is the name as the maila variety with this difference, vis., the Biulare root here is of the remain-ler after subtracting the given fructional parts, instead of being of the whole. The Dviraya.pamul variety consists of problems wherein a known number of things in the removed, then some fractional parts of the successive remainders and then some multiple of the square foot of the further remainder are removed, and lastly the numerical value of the remaining portion is given. 'The known number first removed is called firingra. In the d iamella variety, a multiple of the muare root of a fractional part of the total number in fupponed to be first removed, and then the numerical value of the remaining portion is given.

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CHAPTER IV-MISCELLANEOUS PROBLEMS (ON PRACTIONS). 71 Dvirayraseramila and Arukamula, and then Bhagabhyasa, then Ankararga, Mulamiira and Bhinnad, sya. Thu rule relating to the Bhoga and the Seru varieties theroin, (.e., in miscellaneous problemas on fractions). 4. In the operation relating to the Bliga variety, the required) rosult is obtained by dividing the given quantity log one as diminished by the knowl) fractions. In the operation reluting to the st varioty, (the required result) is the given quantity divided by the product of (the quantities obtained rrspectively by) sub). tracting the (known) fractions from one. Examples in the Bhiga variety 5. Of a pillar, part was seen by me to be (lauriel) under the ground, in water, in moys, and 7 hastas thereof was free) in the air. What is the length of the) pillar? In the B rithya or B r arit varity, the numerical value in given of the portion remaining after removing from the while the produtor products of certain fractional part of the whole fathen tulny w The murary variety Coint of problems wherein the meal value in Kivin of the remainder after removing from the whithe r of fractional part berof, thin fractional punt being at the same time t o read by a given number The Mulawira variety consists of problems wherein in viven the munericul value of the kunt of the quarrant of the whole when to the wore root of the wholu un increase or diminished by a given h er of thing. in the B ade variety: Oructional pent of the whole in multiplied by another fractional post o in cm from it, the remaining portion in Xpred as a fraction of the whole. He it will l. men that unlike in the other varieties the numeri uc of the lont remaining portion is not actually gisen, but in expr ...ction of the whole 4. Algebraically, the rule relating to the Bhaye variety in Sai, whez is the unknown collective quantity to hu found out, in the d r ama, und bis the bhd ja or the fractional part or the sum of the fractional part kiven. It is obvir , that this is virivable from the cquutions-los. The rulotting to the Briv variety, w. lgebraically p rocl, com to (1-b where , lz, I., &c., arr fructionul parts of the -0) 1 - 1)*&. mucoensivi enunindra. This formula also is derivable from the contion 3-148-13 (3 - 0,7) -{ 2=bs-bs (e1921} &..=a.

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GANITASARASANGRAHA. 6. Out of a collection of excellent bees, * took delight in patali trees, in kadamba tree, ; in mango trees, in a campaka tree with blossoms fully opened ; sh in a collection of full-blown lotuses, oponed 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 sruvaka, having gathered lotusce, and loudly uttoring hundreds of prayers, offered those (lotuses) in worship, of those lotuses and }, and of this (1) respectively to four tirthankaras commcucing with the excellent Jina Vrsabha; then to Sumati as well as is of this (same of the lotuses); (thereafter he offcrod in worship to the remaining (19) tirthankuras 2 lotuscs 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 sonses under control, who had driven away the poison-like sin of karma, who were adorned with rightcous conduct and virtuous qualities and whose bodies had been cm. braced by the Lady Mercy. Of that (collection), 1, was made up of logicians; this (2) diminished hy of itself was made up of the teachers of the true religion; the difference between these two (, namely, it and 11- of 13 ) was made up of those that know the Vedas; this (last proportional quantity) multiplied by 6 was made up of the preachers of the rules of conduct, and this very, samo (quantity) diminished by of itself was mado up of astrologers; the difference between these two (last mentioned quantities) was made up of controversialists; this (quantity) multiplied by 0 was made up of penitent asceties; and 9x 8 leading ascetics were (further) seen by me near the top of a mountain with their shining bodies highly heated by the rays of the sun. Tell me quickly (the monsure of this collection of prominent sages. 12 to 16. (A number of) parrots descended on a paddy-field beautiful with the crops) bent dowu through the weight of the ripe corn. Being scured away by men, all of them suddenly flew up. One-half of them went to the east, and I went to the south-cast; the difference between those that went

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CHAPTER IV-MISCELLANEOUS PROBLEMS (ON PRACTIONS). 73 to the cast and those that went to the south-enst, diminished by half of itself and (again) diminished by the half of this (resulting difference), went to the south; the differenco between those that went to the south and those that went to the south-onst, 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 (alors. How many were the parrots (in all)? 17 to 2:. One night, in a mouth of the spring nongon, il certain young lady ... was lovingly happy along with her husband on ... the floor of a big mangion, white like tho moon, and situated in a pleasure-karlen with noch beut down with the load of the bunches of howers and fruits, and resonant with the sweet sounds of parrots, Cuckooh and we which were nll intoxicated with the honey obtained from the flowers tburoin. Then on a love.guarrolarining between the husband and the wife, that lady's necklace made up of pearls locum numered and foll on the floor. Oue-third of that nevklare of pearls reached the maill-scrvant ther; ! full on the bed; then of what romained (and one-half of what remained thereafter and again. of what remained thereafter) and so on, counting wix times in all), fell all of them everywhere; and there worr. found to remain (unicattored) 1,161 pearls; and if you know how to work) miscellanoous probleme (on fractions), vive out the numerical) mengure of the pearls (in that necklace). 23 to 27. A collection of bech characterized by the blue color of the shining indranilo rem was seen in a flowering pleiuurn 17. Certain epithet her: huve not be considered it for transition. 10

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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 axoka trecs, respectively, multiplied by 6, hecame hidden in a crowd of big patali trees. The difference between those that hid themselves in the patali trees and the axoka trees, diminished by of itself becanze hidden in an extensive forcst of sila 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, sarala 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. Givo out, () you arithmetician, the numerical) moasure of that collection of bees. 28. Of a herd of cattle, is on a mountain; of that is at the lase of the mountain ; and 6 more parts, cach being in value hall of whnt preories it, are found together in an extensive forest, and there are the remaining 82 cows seen in the neighbourhood of a city. Tell me you my friend, the numerical) measure of that hord of cattle. Here end the examples in the Bhaga variety. Examples in the Sera variety. 29-30. Of a collection of manyo fruits, the king (took) ; the queen (took) of the remainder, and three chief princes took and 1 (of that same remainder); and the youngest child took the reinaining 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 heri. measuring from in order to }. in the end, of every successive remainder, wander about in a forest; and the remaining 6 (of them) are seen neur a lake. How many aro those olephants ?

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CHAPTER IV-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 purinas were seen by me (to remain). What is the numerical measure of the purunas contained in the treasury? Here and examples in the Sexu variet. The rule relating to the Mila variety of miscellaneous problend ou fractions!: 33. lalf of the coetticient of the sylare root of the unknown quantity and (the) the known remainder would loc cach divided by one as diminished by the fractional cocflicient of the unknown) quantity. The squaro root of be 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 18 a love, and then associated with the similarly treated coellicient of the square root of the unknown quantity), and thereafter nun redan a whole, gives rise to the required unknown yuantity in this mula varicts (of miscellaneous problem on fraction). Eroompies 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 (wers, however, found) to remain ou the bank of a river. What is the (mmerical measure of that herd of camels? 35. After listening to the listinct 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 .gladilcued with joy, kept on dancing on 33. Algebraically expr , this rule come to < 3 i this in easily obtained from the LA equation - bs + Na+a)= 0. l'hin tuation in the allorical expromjon of problems of this variety. Iferec mtand. for the coefficient of the wuare root of the unknown quantity in t found ont.

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1 76 GANITASAKASANGRAHA. 1 the big theatre of the mountain top; and 5 times the square root (of that collection) stayed in an excellent forest of cakulu trees; and (the remaining) 25 wore seen on a punnaga tree. O arithmetician, give out 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 are 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 lilaka tree. No (multiple of the) square root (of their number was to be found anywhere). Give out (the numerical value 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 monsuro of that (collection). Here ends the Mula variety (of miscellaneous problems on fractions). The rule relating to the Sesamule 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 2 40. Algebraically, br {S + (2)2 + a} 2 From this the value of r is to be found out according to rule 4 given in this chapter. This value of abe is obtained easily from the equation x-br+ (ex-bx + a ) = 0.

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CHALTER IV-MISCELLANEOUS PROBLEMS (ON FRACTIONS). 77 combine it with the known number remaining, and (then cxtract) the sqnare root of this sum, and make that square root becomo) combined with half of the previously mentioned (coefficient of the) syuare root of the remaining part of the unknown collective quantity). The square of this last sum) will hore be the required result, when the remaining part of the unknown collective quantity) is taken as the original collective quintity itself. But when that remaining part of the nknown collective quantity is treated merely as a part, the rule relating to the Whiye variety (of miscellaneous problemy on fractions) is to be applied. Examples in illustration thereof. 11. Onc-third of a herd of elephants and three times the nuure root of the remaining part of the herl) were Moon on a mountainslope; and in a Inke was seen a male elephant along with three femalo clephants (constituting the ultimate remainder. llow many wore the elephants here! 12 to 45. In al garden beantified by groves of various kinds of trees, in a place free from all living animals, many asoctics were scated. Of them the number (quivalent to the square root of the whole collection were practising yoyu at the foot of the tren. One-tenth of the remainder, the square root (of the remnjuder after deducting this), (of the remainder after deducting this, theu the square root (of the remainder after deducting this), (of the remainder after deducting this, the Nqunre root (of the remainder after deducting this), ! (of the remainder after deducting this, the square root of the remainder ofter deducting this, (of the remainder after leducting this, the square root of the remainder after deducting this), ! (of the remainder after deducting this), the square root (of the remainder after desueting this)- thene parts consisted of those who were learned in the teaching of literature, in religious law. in logic, and in politicr. as alwo of those who were verscil in controversy. prosody, astronomy. magic, rhetoric and grammar and of those who poschued the power derived from the 12 kinds of austerities, as well as of those who pokecsked an intallimont knowledge of the twelve varictics of the anga-natra; and

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GANITASARASANGRAHA.. at last 12 ascetics were seen (to remain without being included among those mentioned before) 0 (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 olephants 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 clephants here? Here ends the Sesamila variety of miscellaneous problems on fractions). The rule relating to the Sesamula variety involving two known (quantities constituting tho) remainders : 47. T'ho (cocfficient of the) square root (of the unknown collective quantity), and the (final) quantity known (to remain), should (hoth) be divided by the product of the fractional (proportional) quantities, is subtracted from one (in cach case); then the first known quantity should be added to the (other) known quantity (troated as above). Thereafter the operation relating to the Sesamula variety (of miscellaneous problems on fractions is to be ndopted). 47. Algebraically, this rule enables 18 to arrive at the exprcasions (1 -- ;) (1 --ba) * &c and (1 - 0) (1 6.) * &c. + a which are required to be substitntot for c and respoctively in the formalt for sisamulu, which is fre 17c2"2 +aj. In applying this formula the value of o becomes zero, an thumla or wyttare root involved in the dvirajra. Amikla is that of the total collective quantity and not of fractional part of that quantity. Solstitating as desired, we get 7 == b) (1 -b) x &c. * 2 12 ( 1 01) (1 - box do.) + (1 - 1) (1 - 0) < &c. + 1). This result may onsily be obtained from the equation - - (- ) -bi ....-CVI - 2, 0, where O, ba, c., are, the various Cractional parts of the successive remainders; and a, and , are the first known quantity and the final known quantity respectively.

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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 bees 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 scen) 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 Sexamula variety involving two known quantities. 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) haying been 30. The word harini occurring in this stanza not only means a female deer' but is also the name of the metre in whiel: the stanza is composed. bg= 51. Algebraically stated, this rule helps us to arrive at cb and ab, which are required to be substituted for and respectively in the formula amala variety. As pointed out in the note { 2 + (2)2 + ~ } , as in the

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80 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 fractious). | Another rule relating to the Amsamula variety. 52. The known quantity given as the (ultimate) romainder 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 coofficient 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? under stanza 47, xbx becomes z here also. After substituting as desired, and dividing the result by b, we get x= = { cb + (cb)2 + ab } 2 / b. This value of 2 may be easily arrived at from the equation x-cbx-u = 0. 4a 2 *={c + 2++)2. bxb. This is obvious from 2 the equation given in the note under the previous stanza. 52. Algebraically stated, <

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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 plar; 8 times the squiaro root of of the remaimler (of the colleotion) went to a mountain ; and 9 times the square root of of the (further) remainder (left thereafter) went to the bank of a river; and boars equivalent in (numerical) measure to bi were seen ultimately, to remain (where they were in the forest. live out the numerical) moasure of (all) those boarn. Thus ends the saunila variety The rule relating to the Bhagas cirurya variety of mincollaneous problems on fractions): 57. From the winplifier) denominator of the procities compound fractional part of the unknown collective quantity, divided br its own related numerator also wimplitied, subtract four tince the given known part of the quantity, then multiply this (resulting difference) by that same simplified) denominntor (dealt with as abovel. The wquare root of this product is to be aideil to as well as subtracted from that hilme denominator (NO dealt with; (then) the half of wither) of those two quantities resulting as sim or ifference is the known collective qu:tity (required to be found out). Eramples in illustration thereof. 58. A cultivator obtained first of a hoap of paddy ily mula tiplied by (of that name hoap); and (then) he had 24 urihan (left in addition). Give out what the measure of the heap is. 59. One-sixteenth purt of a collection of pearocks as multiplied by itself, (ie., by the same 'part of the collection), was found 56. The word saribulovokrilita in this lanz means 'tiyern ut play.' und at the same time happens to be the one of the metre in which the mannen in composed. " //ng ng 37. Algebraically nealed - men (me to ) map and the value of a TIP and thin vulue of : may camily be obtained from the equation - the fractions contemplatus in the rule.

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GANITASARASANGRAHA. on a mango tree; of thu remainder as multiplied by that same ( part of that same remainder), as also (the remaining) fourteen (peacocks) were found in a grove of tamala trece. How many are they (in all)? 60. One-twelfth part of a pillar, as multiplied by o part thereof, was to be found under water ; b of the remainder, an 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. Hore onds the Bhayasamvarya variety. The rule relating to the Aviso carga variety of miscollaneous problems on fractions), characterised by the subtraction or addition (of known quantities) :-- 61. ('Take) the ball of the denominator (of the specified fractional part of the unknown collective quantity), as divided by its own (rolated) muncrator, 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 squaro root of the square of this (resulting quantity), an diminished by the square of (the abovo known) quantity to be subtracted or to be added nud (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 thon divided by the (spocified) fractional part (of the unknown collectivo quantity), gives tho (required) valno (of the unknown collective quantity). Examples of the minus rariety. 62. (A number) of buffaloes (equivalent to) the square of (of tho whole hord) minus 1 is sporting in the forcst. The 61. Algebraioally, -={+N ( + )" -.-a + ( + ) } = ".. This vulua is obtained from the equation :-( Fa) - << = , where d in thu given known quantity.

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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? An example of the plus variety. 64. (A number of peacocks equivalent to) is of their whole collection plus 2, multiplied by that same (s of the collection plus 2), are playing on a jamb treo. The other (remaining) proud peacocks (of the collection), numbering 2x 5, are playing on a mango tree. O friend, give out the numerical measure of (all) these (peacocks in the collection). Here ends the Asararga variety characterised by plus or minus quantities. The rule relating to the Malamisra 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 valne (of the unknown collection). In relation to the working ont of the Malamisra variety of problems, this is the rule of operation. 64. The word maltamaytira orcarring in the stanga means a proud peacock' and is also the name of the metre in which the stanza is composed. m2 d - {~2~ d}" This is easily derived from the equa. 65. Algebraically = tion ++-d = m. The quantity is here the known combined sum mentioned in the rule. >

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GANITASARASANGRAHA. Examples of the minus variety. 66. On adding together (a number of pigeons equivalent to) the square root of tho (whole) collection of pigeons and another number cquivalent to the square root of the whole) collection as diminished hy 12, (exactly) 6 pigeons are seen (to be the result). What is the numerical value of) that collection (of pigeong, ? 67. The sum of two (quantities, which are respectively oquivalent 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 swand there are in that collection. Here ends the Malamisra variety. The rule relating to the Bhinnudraya variety of miscellaneous problems ou fractions): 69. When we, diminished by the given) fractional remainder (related to the nuknown quantity), is divided by the product of the (specified fractional parts (related thereto), the result which is (thus) arrived at hecomos the (required) answer in working out the Bhinneyu variety of problems on fractions). Examples in illustration thereof. 70. Que-eighth part of a pillar, as multiplied by the part (of that same pillar), wus found to be buried) in the sands; } of the pillar was visible (above). Say how much the (vertically measured) longth of the pillar is. 09. Alyebraically stated, 03. algebraically atutoit, e = (1-1) = equation * - - = "P. This is obvious from the ng

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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 berd) as divided by 2, are in a happy condition on a plain. The romaining ones forming (of the herd), resembling exceedingly dark masses of clouds in form, are playing on a mountain. O friond, you tell me now the numerical measure of the hord of elephauts. 72. (Ascetics equivalent in number to a part of a collection of ascetics, us divided by 3 and as multiplied by that same Ga part divided by 3), are living in the interior of a forest ; (the remaining ones forming) i part of that collection) are living on a mountain. 1) you, who have crossed over to the other shore of the ocean-like miscellaneous problems ou fractions, tell me quickly the numerical) value of that collection of ancotics). Here ends the Bhinnulya variety Thus ends the third subject of treatment known as Prakirnuku in Sarasangraha which is a work on arithmetic by Mahaviracarya. 71. The word prthvi occurring in this stanza wenn 'threarth, and in its the name of the metre in which the tunza in compared

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GANITASARASANGRAHA. 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) ike the sun in the matter of absolute knowledge, and has cast off the taint of) all tho karmas. Next we shall expound the fourth subject of treatmont, viz., ule-of-three. The rule of operation in respect thereof is as follows: 2. Here, in the rule-of-three, Phala multiplied by Iccha and livided by Promina, becomes the (required) answer, when the Techa and the Promana are similar, (i.., in direct proportion); and n the case of this (proportion) being inverse, this operation involving multiplication and division) is reversed, (so as to have livision in the place of multiplication and multiplication in the place of division). Eramples relating to the former half of the above rule, i.e., on the direct rule-of-three. 3. The man who in 31 days goes over 51 yojanas--give out what (distance) he (goos over) in a year and a day. 4. A lame man walks over of a kvoxer together with $ (thereof) in 7 days. Say what (distance) he (goes over) in 3 years at this rate). 5. A worin goes in & of a day over of an angulu. In how many lays will it reach the top of the Meru mountain from its bottom? 6. The man who in 3 days usos up 11 karse panas--- in what rime (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 samo kind is the required answer and Pramana in of the same kind as Iccha. l'hin Iccha is the quantity about which something is required to he found out at ho givon rate. For instance in the problem in stansa 3 here, 3 days in the Pramdna, 67 yojunas is the Phala, and 1 year and 1 day is the Iccha. 3. The height of the Miru mountain a supposed to be 99,000 yojanas or 78,032,000,000 angulus,

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CHAPTER V-RULK-OF-THREE. 87 7. A good piece of krsnagaru, 12 hastes 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 raha of very superior black gram, along with 1 drone, 1 adhaka and 1 kudura (thereof), has been purchased by means of 10 svarnas, what measure (may we purchase of it) by means of 100 svarnas! 9. Where 1 pala of kunkuma is obtainable by means of 31 puranas, what measure (of it) may (we obtain) there by means of 100 puranas? 10. By means of 73 palas of ginger, 13 say, O friend, what (may be obtained) in of ginger? panas were obtained: return for 32 palas 11. By means of 44 karsapanas, a man obtains 16 palus of silver; what (weight does he obtain thereof) by means of 10,000 kargus? 12. By means of 7 palas of camphor, a man obtains 5 dinarus along with bhaga, 1 ansa and 1 kala. What does he obtain) here by means of 1,000 palux (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 1001 karsas! 14. By means of 5 puranas, 163 pairs of cloths were obtained. O friend, say what number of them may be obtained) by means of 61 kargus 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 ont, from the given diameter, the aron of the Cross-Rection of a cylinder is supposed to be known. This is given in the sixth Iyavahara, in the 19th stanza, where the area of a circle is said to be approxi mately equal to the diameter squared and then divided by 4 and multiplied by 3. Kranagar is a kind of fragrant wood burnt in fire as incense. 15-16. In this problem, the stream of water in 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 Vahar, etc., the relation between cubical measure and liquid measure should have been given. The Sanskrit commentary in l' and the Kanarese fika in B state that 1 cabic angula of water is equal to J'karga thereof in liquid measure.

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88 GANITASABASANGRAHA. thereof flows down, (to the bottom) a crystal-clear stream of water having 1 angula for the diameter of its circular section, and the well becomes quite filled with water within. What is the height of the hill, 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-beau, 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. Ilere ends the (direct) rule-of-three. * Examples on inverse rule-of-three as explained in the fourth pada (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 gunja thereof made up of gold of 10 varnus! 19. There are 300 pieces of China silk of 6 hastas in breadth as well as in length; give out, O you who know the method of inverse proportion, how many pieces (of that same silk) there are (in them, each) measuring 5 by hastas. Here onds 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 longth, are to be found in 70 (piecos) 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 Tirthankaras, (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 in the passage of the sun from one zodiacal sign to another. 18. Pure gold is here taken to be of 16 rarnas. The reference here is to the fourth quarter of the second stanza in this chapter.

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CHAPTER V-RULE-OP-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 templo, (may be obtained, each) moasuring 2 by 8 by 1 (hastas), and being 5 in worth? Thus ends the invorso double, treble and quadruplo rule-ofthree. Tho rule in regard to (problems bearing on uspooiatel) forward and backward movement. 23. Write down the net daily movement, as derived from the difforence of the given rates of) forward and backward movemouts, each (of these rates) being (first) divided by its own (epooified) 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 courso of of a day, a whip goch over of a krosa in the ocean; being upposed by the wind who goes back (during the same time) e of a krosul. Give ont, U you who havo powerful arms in crossing over the ocean of numbers woll, in what time that (ship) will have gone over 993 yojanux. 26. A man omrning (at the rate of) 11 of it gold coin in 31 daye, spends in 41 days of the golil coin is also of that (1) itsolf; by what time will be own 70 (of those gold coins as his net earnings)? 27. That excellent elephant, wbieb, with temples that are attacked by the feet of bees greedy of the flowing) ichor, goes over as well as of a yojana in 5 days, and moves back in 31 days over of a kronut : say in wbat time he will have gone over (a net distance of) 100 yojumax lens by i kro a. 28-30. A well completely filled with wntor in 10 dumtur in depth ; a lotus sptting up therein grows from the bottom 28-30. The 'dopth' of the will is mentioned in the original u 'height' * mrvured from the bottom of it. 12

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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; 1} angulas of water in depth) are lost in a day by evaporation owing to the heating) rays of the sun; a tortoise below pulls down 57 angulas of the stalk of the lotus plant in 34 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) 74 angulas in y of a day; and in the course of of a day its tail grows by 23 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-throo. 32. Transpose the Phala from its own place to the other place (wherein a similar concrete quantity would occur); (theu, for the purpose of arriving at the roquired 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 82. The transference of the Phala and the other operations herein mentioned will be clonr from the following worked out example. The data in the problem in stanzn No. 36 aro to be first represented thus:9 Manis. 1 Viha + 1 Kumbha. 3 Yojanax. 10 Yijanas. 00 Panas. When the Phala hero, viz., 60 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. Thon we have 1 x 10 x 60 9 X 3 The result hero gives the nomber of panas to We found out.

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CHAPTER 1-RULE-OF-THREE. smaller number (of different quantities, after these aro also similarly thrown together and multipliod); but in the matter of the buying and selling of living animals (the transposition is to take place only in relation to the nunbers representing) thom. 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 ly a person desiring profit. How much interest docs ho obtain in ten months ? 34. The interest on 80. gold coins for of a month is 14. How much (will it be) on 907 gold coins for months? 35. He who obtajus 20 gems in roturn for 100 gold pieces of 16 varnas--what (will he obtain) in return for 288 gold pieces of 10 ydrnas "36. A inan, by carrying 9 minix of wheat over 3 yojunus, obtained 60 panas. How much (would lo obtain) w currying one kumbha along with one ruka (thereof) over 10 yojunux P Examples on barter, 37. A man obtaing 3 karras of musk for 1) gold coins and 2 karxas of camphor for o gold coins. Ilow many (karray of camphor does he obtain) in return for 300 kursus of munk? 38. In return for 8 (misax in weight of silver), a mau i obtains 60 jack fruits; and in return for 10 miinus (in weight of silver be obtains) 80 pomegranates. How muy pomogranates (does he obtain) in return for 900 jack fruits ? Examples of (problema liraring on the buying and selling of animals. 39. Twenty horses, (cach) of 16 years of age), aro worth 100,000 gold coins. O leading aritlin.etician, say how much 70 horses, (each) of 10 years of age), will be (worth) at this (rute). 40. Three hundred gold coins form the price of O damsula, (each) of 10 years of age). What is the price of 30 damgols, (cach), of 16 years (of age) ?

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92 GANITASARASANGRABA. 41. What is the interest for 10 months on 90, invested at the rate of 6 per 100 (per month)? (you, who are a mirror to the face of arithmeticians, way, 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-nf-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 tho 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 ; 0 you, who are learned, give out how much (wator) I wells, (each being) 7 hastax in broadth, 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 inlini ocurring in this stanze indicates the name of the metre in which the stanz is com poned, at the same time that it means 'belonging to # bouro.'

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CHAPTER VI-MIXBD PROBLEMB. CHAPTER VI. MIXED I'ROBLEMS. The Fifth Subject of Treatment, 1. For attaining the supreme good, we worshipfully saluto the holy Jinas, who are in possession of the fourfold infinito attributes, who are the makers of tirthas, who have attaineil self-conquoat, aro puro, are honoured in all the threr worlds and are also excellent preceptors--the Jinas who have gone over to the (other) shore of the oocan 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 oxpound the fifth subjest of troatmont known as mixed problems. It is as follows: Statement of the meaning of the technical trrms kaikramana and visama-sankramane : 2. Those who have gone to the end of the orcan of calculation say that the halving of the sum and of the difference (of any two quantities) is (known ay) suikramma, and that the sakramona of two quantities which are respectively) the divinor and the quotient is that which is visuma (c., winnuna sakramana). Examples in illustration thereof. 3. What is the sakramau where the number 12 (is associatod) with 2; and what is the divisional visama-sankramana of that (same) number (12 in relation to 2) 1 Tirtha in interpreted tu nan ford intended to cross the river of mandare existence which is subject to karma und reincarnation, 'l be Jinul are conceived to be capable of enabling the souls of men to get out of the stream of madra or the recurring cycle of embolied existence. The Jinin are therefore called tirthaikarax. 2. Algebraically the sankramana of any two quantities a und bin firding out and after thuir virumu-nunkrumura in arriving at "** - and

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GANITASARASANGRAFA. Double Rule-of-three. The rule for arriving at the value of) the interest which (Oporation) is of the nature of double rule-of-three :-- 4. The number representing the Iccha, i.e., the amount the interest wheroon 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 arithmetio, the interest of the desired amount. Examples in illustration thereof. 5. Puranas, 50, 60, and 70 (in amount) wero lent out on interest at the rate of 3, 5 and 6 per cent (per menfem respectively); what is the interest for 6 months ? 6. (A sum of) 30 karsiponas and & panas were lent out on intorest at the rate of 74 per cent (per month); what is the interest produced in exactly 7 vonths? 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 is 15. What would be the interest obtained at this rate on 300 for 10 months ? 9. A mercbant lent out 63 karsripanas at the rate of 8 for 108 (per month). What is the interest) for 7 months ? The rulo for finding out the capital lent out: 10. The capital quantity (involved in the rate of interest) is multiplied by the time connected with itself and is then divided 4. Symbolically i where T, C and 1 aro respeotively the time, capital and interest of the proming or the late, and t, cand i are respectively the time, capital aud interest of the iccha. For un explanation of pramana, icchd, &c., eee note under Ch. V. 2. 5. Unless otherwise mentioned, the rate of interest is for 1 month. CxTxi 10. Syn bolionlly a Ixc.

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CHAPTER VI-MIXED PROBLEMS. 98 by the interest connected with itself. (Then) this (quotient) has to be divided by the time connected with the capital lont out; (this last) quotient when multiplied by the interost (that has acorued) becomes the capital giving rise to thnt (interest). Examples in illustration therrof. 11. In lendiug out at the rato of 2} per cent (per monsom), a month and a half (is the time for which interest has acernod), and a certain person thus obtains 5 purinas as the interest. Tell me what the capital is in relation to that (interest). 12. The interest on 70 for 11 months is exactly 2). When the interest is ?} for 7 months what is the cupital lont out! 13. In lending out at the rates of 3, 5 and 6 per cont (per mensem), the interest has so accrued in 6 months as to be 9, 18 and 25+ (respectively); what are the capital amounts lent ont? The role for finding out the time (during which interest has accrued) : 14. Take the cupital amount involved in the given rute of interest) as multiplied by the time (connected therowith); 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 (prodluct) is the timo (for which the interest has accruel). Examples in illustratiqu thereof. 16. O friend, mention, after calculating the time, by what timo 28 will be obtained as interest on 80, lent out at the rato of 34 per cent (per menser). 16. The capital amount lont out at the rate of 20 per 600 (por mensom) is 4:20. The intercet also is 84. O friond, you tell me quickly the time (for which the interest has accrued). 14. Symbolically,

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96 GANITASARASANGRAHA. 17. It is 96 that is lent out at the rate of 6 per cent (per mensem); the interest thereon is seen to be 57. What is the time (for which interest has accrued)? The rule regarding barter or exchange of commodities : 18. I ho quantity of the commodity taken in exchango 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 oxchanged. 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 it panas and palas 5 of long pepper for 8 panas. Think out and tell mo quickly, O you who know arithmetic, how many palas of long pepper have been purchased by one (at the above rate) by means of 80 polas of dried ginger. Thus end the problems on double rule-of-three in this chapter on mixed problemy. Problems bearing on interest. Next, in the chapter on mixed problems, we shall expound problons bearing on interest. The rule for the separation of the capital and interest from their mixed sum : 21. The result arrived at by carrying out the operation of division in relation to the given mixed sum of capital and interest 21. Symbolically, c x x1 , where m c + i; bende i=m - c. 1+ TxC

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CHAPTER VI-MIXED PROBLEMS. 07 by means of one, to which the interest tbcreon for the (given) time is added, (happens to be the required) capital; and the interest required is the combined sum minus this oapital. An example in illustration thereof. 22. If one lends out money at the rate of 5 por cont (per month), the coinbined 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 rato-intorost, 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 uuderstood as the (required) capital. An example in illustration thercofa 24. Having given out on interest some money at tho rate of 24 per cent (per monsem), ono obtains 33 in 4 months 18 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 rato-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 become 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. Hynibolically c=m +1}. It is evident that this is very much the same as the formula given under 21. 25. Symbolically i=m+ 1 +1= i, where mei + l. Cx7 1xc 13

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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 13 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 37 por 80 for 21 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 interost from their mixed gum :-- 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 23. The interest here in the given instance) is 24, and m- OxT x 4im 29. Symbolically, ... = cort as the case may be, where m e + t. The value of the quantity under the root, as given in the rule, in (c-t)* and the square root of this and the mijra have the operation of savikramana performed in relation to them. For the explanation of sankranana see Ch. VI. 2.

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CHAPTER VI-MIXED PROBLEMS. 90 60 is (the value of) the time combined with the capital lont out. (What is the time and what the capital ?) The rule 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-timo, by the given interest and by four, and is thon 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 80 as to carry out the process of swikramana. An rcample in illustration thereof. 34. The mixed sum of tho rate-interest and of the timo (for which interest has acerned) at the rate of the quantity to be found out per 100 per month and a half is 12, the capital lont out being 30 and the interest accrming thereon beliny 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 tho interest from their mixeil eum : 35. Any (optionally chosen) quantity subtracted from the given mixed eum may happen to be the time reunired. By mennis of the interest on one for that same time, to which interest one is a ldod, (the quantity remaining after the optionally chosen time is subtractal, from the given mixed suni) is to be divided. (The resulting quotient) is the required capital. Tho mixod 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 cont (per monsem), the quantities representing the time, the capital and the interest 33. Symbolically ?...*T*1in used with m in carrying out the required sankramara, m being equal to 1 +t, 35. Here, of the three unknown quantities, the value of the time is to be optionally chosen, and the other two quautities aro arrivod at in socordance with rule in Ch. VI. 21.

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100 GANITASARASANGRAHA. connected with the loan) are not known. Their sum however is 32. 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 correspozrding) time and multiplied (also) by the (given) total (of the various imounts) 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) dcolured. An example in illustration thereof. 38. In this (problem), the (given) capitals are 40, 30, 20 and 50; and tho months arc 5, 4, 3 and 6 (respoctively). The sum of he amounts of interest is 34. (Find out cach of these amounts.) The rulc for soparating the various capital amounts from their mixed sum : 39. Let the quantity representing the mixed sum of the various apitals lent out be divided by the sum of those (quotients) which tre obtained by dividing the various amounts of interest by their sorresponding periods of time, and let the resulting) quotient be nultiplied (respcotively) by (tho various) quotients obtained by and_ 0, tig m 87. Symbolically, tynn ot,+ og ty + Cots+ ... ----- =ig: where m- ,+ i+is+. . City + 0,13 + Cats +. . . * and C1, C, Cs, etc., are the various caepitals, and t1, ts, ts, etc., are the various periods of tino. 39. Symbolically, and x = C: tots . eto. whero de + 03 +09+. = .

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CHAPTER VI-MIXED PROBLEMS. 101 dividing the various amounts of interest by their corresponding periods of time. Thus the varioue capital amounts happen to bo found ont. Examples in illustration thereof. * 40. (Sums represented by) 10, 6, 3 and 15 are the (various given) amounts of interest, and 5, 1, 3 and 6 are the corresponding) months (for which those amounts of interest have accrued); the mixed sum of the corresponding) cupital amounts is soen to be 140. (Find out these capital amounts.) 41. The (various) amounts of interest aro 2, 6, 10, 16 and 30; (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 those amounts rospectively ?) Tho rule for arriving separately at the various periods of timo from their given mixed sum : 42. Let the quantity representing tho mixod sum of tho (various) periods of time be diviiled by the sum of those (various quotients) obtained hy dividing the various amounts of interest by their corresponding capital amounts; and (then) let the (rcaulting) quotient bomultiplied (separately by cach of the abovomentioned quotients). (Thus) tho (various) periods of time happon to be found out. An erample in illustration thereof. 43. Here, (in this problem,) the (gives) capital amounts aro 40, 30, 20 and 50 ; and 10, 6, 3 and 15 are tho (corresponding) amounts of interest; 18 is the quantity representing the mixed sum of tho (respoctive) periods of time (for wbich intorest has accrued. Find out these periods of timno separately). 42. Symbolically,--. " = 1., where m=t,+ +,+to+&o. C1 C CH 3imilarly tz, ts, etc., may be found out.

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102 GANITASA BASANGRAHA. 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 out: 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). Examples 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 casos, and the interest (thercon 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 = I which is equal to c. 44. 8ymbolically, N 6?xm +()2 CT N e **+ ( 21 ) - 47. Symbolically, mit ungt * By "the poriod of interest "here is meant the time for which interest has acorued in connection with any of the given mixed sums of capital and interest.

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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). CHAPTER VI-MIXED PROBLEMS. Examples in illustration thereof. 48. The mixed sums aro 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, 334 and 35, (these) being 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 value 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, or mixed sums. my is my i!-- c, where m,, m,, etc., are the various mitras 4154

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104 GANITASARASANGRAHA. The rule for arriving at the capital dealt out at two differen rates of interest : 54. Let the balance quantity (i.e., the difference between th two amounts of interest,) be divided by the difference between those (two quantities) which form the interest on one for the giver periods of time; (this quotient) becomes the capital thought of b ono's self before. Examples in illustration thereof. 55. Borrowing at the rate of 6 per cent, and then lending ou at the rate of 9 per cont, ono obtains in the way of the difforentia gain 81 duly at the end of 3 months.. What is the capita (utilized here)? 56. Borrowed at the rate of 3 per cent per mensem, a certain oa pital amount is put out to interest at the rate of 8 per cent po mensom. The differential gain is 80 at the end of 2 months How much is the capital (80 usod) ? The rulo for arriving at the time whon both capital and interes will become paid up (by instalmonts) : 57. The capital lent out is multiplied by its time (of instal mont) 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 instal mont) and the timo (of discharge of the debt are to bo made out * as before from (this) interest. Eramplis in illustration thereof, 58. The rate of interest is 5 for 70 per nensem; the amount o: the) instalment to be paid is 18 in (every) 2 months; the capita lent out is 84. What is the time of discharge ? C. 54. Symbolically, "1x, xl, 1; * C skhrkh I 67. Symbolically, skh t time of each instalment. 1x tex I 7, RC, = interest in the instalment, where p is the

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CHAPTER VI-MIXED PROBLEMS. 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 ara arrived at as 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. Ilero 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 leut out at the rates of 3 per cent, 5 per 70, and 34 per 60 (per mensem); the interest (accrue) 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 ont capital in relation to the known time of discharge by instalments: 63. Let the amount of the instalinent s divided by the time thereof and as multiplied by the time of discharge be divided by 112 t CTG x T I J. are found out by the rule in Ch. VI. 10, t 60. Symbolically. 63. Symbolically. + 1 + 105 + &c. P 1 x tx 1 Tx C p the time of an instalment, and the time of discharge. , where a from this, the capitals amount of instalment. 14

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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). Kaamples in illustration thereof. 64. In accordance with the rate of 5 per cent (per mensem), 2 months is the time for cach instalment; and paying the instalment of 8 (on each occasion), a man here became free (from debt) in 60 months. What is the capital (borrowed by him)? 05. A certain person gives once in 12 days an instalment of 2%, the rate of interest boing 3 per cent (per mensem). What is the capital amount of the debt discharged in 10 months ? The rule for arriving separately at the varions capital-amounts which, when combined with or diminished by their respective interests, are equal to one another, from their mixed sum, (the interests being either added to the capital amounts in all the given cases or subtracted from them similarly in all the given cuses) : 66. One is to be either combined with or diminished by the interest (accruing) thereon for the given period of time (in cach caso in accordance with the respectively given rate of interest ; then again in each case, one is divided respectively by these (combined or diminished yuantities arrived at as hefore). Thereafter the (given) mixed suun (of the various capital amounts lont out) is divided by the sum of these (resulting quotients), and in relation to the mixed num (so treated) the process of multiplication is to be conducted (separately in each case by multiplying it) by (the corresponding) proportionate part of the ahove. inentioned sum of the quotients). This gives rise to the cupital Hi. Symbolically, Ixtxl, +80 1. It T.C Tix C, if T, XC Similarly. do. do. xxl Itt, Cs And so on for Crato,

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CHAPTER VI-MIXED PROBLEME. 101 amounts lurt out, which on being combined with or diminished by thoir rospective amounts of interest are equal (in valne) Kramples in illustration thereuf. 67. The total capital represented by ,20 is invested in parts) at the respective) rates of , j ind per cent (per month). Then, in this investment, in 0 months the capital amounts lent out are, on being diminished by the respective) aniounts of interest, (secn to he) equal in value. What are the respective amounts invested thus?) 68. The total capital represented by 1,250 in invented in parts) at the (respective rates of . 6 au for 60 for 2 months; then, in this investment, in 8 months the capital amounts ont out arr, on being diminished by the respective amounts of interest, en tu be) equal in value. What are the respective amounts investod thus ) 69. The total capital represented by 13,7l0 is invested in parts) it the respective) rates of 2, 5 and ! per cent (risonth); then, in thie investment, in f months the capital amount lont out are, on being combined with the respective) amounts of interest, (soon to be) cqual in value. (What are the respective amounts investcil thus?) 70. The total capital represented by 5,016 in invested in parts) at the respective) rates of 11, , and, for 0 (per month); then, in this investnunt, in 8 months (the capital amount lent out are, ou being combined with the respertive amounts of interest, nech to lo equal in valne. What are the rispective 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 debiti in that capital amount (which results) by adding the product of the 71. The rule in very elliptical and will become wear from the following working of the cample contained in Alanzan 7 77 Hero the milla or the maximum availnble amount of an inntalment in this when divided by 7, the amonnt of the first instalment. Kivonor M, of which

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108. GANITASARASANGRABA. optionally chosen (maximum available amount of an instalment) hy (whatever happens to he) the outstanding (fractional part of the number of terms in the scrics), to the amount of the (first) instalment as multiplicd 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 torms the integral value of) the quotient obtained by dividing the above optionally chosen maximum) amount of debt (dischargod at an instalment) by the (above amount of the first) instalment. The interest thereon is that which accrues for the period of an instalmont. 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 dobt). Examples in illustration thereof. 72 and 731. A certain man ntilised, (for the discharge of a debt) bearing interest at 5 por cent (per month), 60 (as the available maximum amount) with 7 as the first instalment amount, increasing it by 7 in Bucoceive instalments due every of a month. ile thus gavo in discharge of the debt the sum of a series in arithmetical progrossion consisting of forms, and gave also the interest accruing on those multiples of 7. What is the debt anivunt corresponding to the sum of the series, whut is that interont (which he paid), and (what is the time of discharge of that debt ? 734 to 76. A certain man utilised for the discharge of u delt, bearing interest at 5 per cent (per munscm), 80 (as the available maximum amount) with 8 as the first instalment amount, increasing it by 8 in successivo instalments due every of a month. He thus 8 ropresents the number of terus of the series in arithimetical progression. which has 1 for the first term und 1 for the common difference; and in the agru or the outstanding fractional part. The sum of the above-mentioneel serios, viz., 30, 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 = 4, which is the required capital amount in the due debt. The interest on 844 fort of a month at the rate of 5 per cent per mengem will be the interest paid on the whole. The time of discharge will be ( 7) X 20 = 4 months.

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gave in discharge of the debt the sum of a series in arithmetical progression consisting of terms and gavo 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. 3 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. An example in illustration thereof. 7e. 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? CHAPTER VI-MIXED PROBLEMS. Thus end the problems bearing on interest in this chapter on mixed problems. and Bre the various amounts of 77 and 774. The various accruing interests interests acerning on the several amounts at the varions rates for their respective periods. Symbolically. Tx & ta x r, x 1, x 1, Ty C [r, x 1, x 1, 1 T C x 1 x 1. Tx C C2 x 1 x 1. Tx C r. x t. x 1. 7x C + +.. 109 +. }: [ x 1 x 1, Tx C i on average interest. + to or average time; (e1je2+. ..)

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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 operatiou of proportionate division is that wherein the (giveu) collective quantity to be divided) is first divided by the sum of the numerators of the cominon-denominator-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 cach case) by (these) proportional numerators. This is called kuttikora by the Icarned. Examples in illustration thereof. 80%. Here, (in this problem, 120 gold pieces are divided anong 4 servants in the respective) proportional parts of , and . O arithmetician, tell me quickly what they obtained. 813. (The sum of) 363 dinaras was divided among live, the first ono (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 82; to 85). A certain faithful xravaka took a number of lotus Aowers, and going into thc Jina temple conducted (theroin) with devotion the worship of the chief Jinas that were worthy of worship. He offered part to Vrsabha, , to worthy Parava, and th to Jinapati, and to sage Suvrata ; he dovotedly gave to Aristanemi who destroyed all the eight kinds of karmas and who was beloved by the world, and I of 1 to Jinasanti: 480 lotuscs were brought (for this purpose.) By adopting the operation known 794. In working the example in stanos 804 according to this rule we get: t =a.1. After removing the denominators here, we have 6, 4, 3 and 3. These are also called praksepas or proportional numerators. The sum of these io 15, by which the amount to be distributed, vis., 120, is divided ; and the resulting quotient 8 in separately multiplied by the proportional namerators 6, 4, 3 und 3. Then the mounts thus obtained are 6 x 8 or 48, + x 8 or 32, 3 x 8 or 94,X 8 or 16. It is worthy of note that praksapre moans both the operation of proportionate division and proportional numerator.

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CHAPTER VI-KIXED PROBLEMS. 111 as praksepaka, give out the proportionate distribution of the flowers. 861. (A sum of) 480 was divided among five mon in the proportion of 2, 3, 4, 5 and 6 ; 0 friend, give ont (the sbare of each). Che rule for arriving at certain) results in required proportions : 87!. The number representing tho) rate-price is divided by (the number representing) the thing purchasable thorewith ; (it) is (then) multiplied by the (given) proportional number; by means of this, (we got at) the sum of the proportionate parts, (through) the procese of addition. Then the given amount multiplied by tho (respective) proportionate parts and then divided by this om of) the proportionate parts gives rime to the value (of the various things in the required proportion). Another rule for this (name) purpose :- 881. Multipls the numbers representing the rute-priocs (respectively) by the numbers representing the (given) proportions of the (various) things to be purchased); theu divide (the result) by the respective numbers measuring the things purchasable for the rate-price ; the resulting quantities bappeu to be tho (reynisite) multipliers in the operation of pralinipaka. The intelligent man may (then) give out the required Answer by adopting the rule-of-three. Agaiu a rule for this (same) purpose : 891. The numbers representing the various) ralr-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 should be carriol ont) in before. $7to 89. Tu working the example in tanzA 90 and 1 ceording to those rules 3, 3 and 5 are divided by 8.5 and 7 respectively and are similarly multi. plied hy 6,3 and 1. Thus we have 6, 3, 14, . These arv the proportional part. The roles in tanzas y nod ng; require thereafter the operation of praki pu to be applied in relation to tbree proportional parts ;) but the rulo in tanza 871 expressly describes this operation.

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112 GANITASARASANG RAJA. The required result is well arrived at by going through the process of the rule-of-thrcc. Examples in illustration thereof. 90 and 91. Pomegranates, mangoes and woodapples are obtainable at the respective) rates of 3 for 2, 5 for 3, and 7 for 5 panax. () you friend, wbo kuow 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 inuch. 92 to 941. A follower of Jina had the image of Jina bathed in potfuls of cards, ghee and milk. Three pots becamo filled with 72 palar (of these); 32 palas were fomd in the first pot and 24 in the second pot and 16 in the third pot. From these (potfuls of mixedup) curls, ghee and milk, find out each of those ingredients) separately and give them ont, there being altogether 24 palox of ghuo, 16 pulas of milk and 32 polus of curls. 95! and 96}. Three puruna, formed the pay of one man who is a mounted soldier; and at that rate there were 65 men in all. Somo (among them) broke down, and the amount of their pay was given to those that remained in the field. Of this, cach man obtained 10 puranas. You tell me, after thinking weli, low muny 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 ynantities : 97). The given total quantity is diminished by the integral quantition that aro to be adoled, or is combined with the positivo integral quantities that are to be subtractel; thou with the help of this reenlting 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 thom), or they are diminished (respectively) by those integral quantities that are to be subtracted). 971. The operation of proportionato dirision to le condected here is according to any of the rules in stanzas 875 to 89).

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CHAPTER VI-MIXED PROBLEMB. 113 Examples in illustration thereof. 981. Four men obtained their shares in successively doubled proportions and with successively doubled differences in addition, the first mon obtaining one share : 67 is the quantity so to be distributed) here. What is the share of cach? 991. (A sum of) 78 is divided in these four (nmong themselves) in proportions which are successively from the frst 11 times (what preceden) and with differences (in addition, which,) commencing with 1, (go out increasing three-fold. Give ont the (valne of tho) parts obtained by sach.) 1001. The shares of fire (persony) are (nuccessively) from the first 11 times (what goes before), and the differences in audition sro quantities which are (necessive)?times the preceding tifference) 514 is (the total quantity) to be divided. Find out the values of * the portions obtained by cach.) . 101. A sum of) 400 winux 15 is divided by four men (among themselves) in proportions which from the first are 24 timos (what preceden), and which besides) aro luns by differences which are (MucocoHively) 4 times (the preceding difference). (Pind out the values of the various portions obtained.) The rule for arriving at the value of the pricos produing equal salo-pro-code and at the value of the highest capital (invented in the transactions concerned): 1023. The largost capital investoid) combined with one bocomes the vending rate of the commodity to be sold). Tont (same vending rate), multiplied by the (riven) price at which the remuant is to be sold, and diminished by one, gives rise to the 98. The difference untity to be added to the hun brin I in the case of the second man, and twire the green ling cliff-riner in the case of schrif the remaining two men; In this difference in the care of the MUCond men in act ""preonly mentioned 11 l in this complete well as in the tuinplo in tanzu 10). 1027. The examples bearing on this rule contemplate the purchase of . commodity at a certain common rate for various capital Amount; then the coinmoility so purchased is to be sold # certain other common rute. That quantity of the couumolity which is left wver, owing to its nt beingrayli to bo sold for a unit of the kind of muney employed in the traductiou, io horo 16

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114 GANITASABASANGRAHA. purchasing rate. By reversing the processes, one may arrive at tho valuation of the highest capital (invested in the transaction). Eramples in illustration thereof. 1033. Tho capital amounts invested by (three) men are (respectively) 2, 8 and 36; 6 is tho price at which the remnants of the cominodity are to be sold. llaving purchased and sold at the name rates, they became possessors of equal wealth. (Find ont the buying and selling prices.) 104. Those three persons took up 13, 1 and 2} (as their rospective capital amounts) and conducted the operations of buying nad selling in relation to the same commodity at the same rates of price); by selling the remnant (in the end) at a price represented by 6, they became possessors of equal wealth. Find out their buying and selling prices.) 105). The quantity measuring the qual wealth is 41, and the price at which the remnants of the commodity are sold is 6. ( arithinetician, tell me quickly what the liighest capital invested) is, and what the (various) capitals are. 106). In the case where 35 linaras give the numerical measuro of the cynal wealth, and 4 is the price at which the remuant is to bo gold, you tell me, O arithmetician, what the highest capital (investerl) is. spoken of in the remnant, and the price at which this remnant is sold is the remnant-price'. Symbolionlly, leta, a + band a + b + c be the capitals, where the last is the Fay or the largest capital, and let p be the TTATE or tho remnantprice; then, focording to the rule, a + b + c + 1 = the viniling rate; nnd ( + b + C + !) P-= the purchasing rate. From those, it can be easily shown that the sum of the amonnte realised by welling tho com modity at the vending rate and the rennent at the remnantprior turns out to be the same in ench Case. It may be noted that the purchasing rate happens in problems bearing on this rule to be the same in value as the 47 or the equal sa le proceeds. 1054. It may be noted here that, according to the rule, it is only the largest pitnl thnt is found out; while the other apitals required in the problem are optionally chosen, 90 as to be lous than the largest capital.

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CHAPTER VI-MIXED PROBLEMS. 115 The rule for arriving at the value of the prices producing equal sale-proceeds when the price of the remnant in fractional in character:- . 107). When the romant 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 (thou nie spectively) to be multiplied by this denominator in the synaro of (this) denominator (for arriving at the required selling and buy. ing rates). The value of the cyual sale-proceeded in the obtained) by means of the rule-of-threo. An ermmple in illustration thereof. 104. (In a transaction) !. are the capital amount (investei respectively by three porod); the romant-price is 3. By purhaning and selling at the same prices, they becamo possessed of cynnal sale-proceeds. What is the buying price what the selling price, and what the equal malo-mount!) Again, another rule for arriving at the value of the equal wale. proceerle, when the remnant-price is frational: - 1094. l'ho continued product of the highest numerator, of turo, and of (all) the denominator (to be found in the value of the capital amounts invested, when combined with the last) denominator belonging to the value of the romant-price, gives rise to the selling rate. This multiplied by the remnant-price, and then diminished by one, and then multiplied (HUICively) lay two and all the denominators, becomes the purchasing rate. Then the rule-of-thrre (is to be used for arriving at the common value of the sale-amounts). An crampla in illuxtration thereof. 1103. Having invested l, , $ (respectively), and having bought and sold (the same commodity), and with as the remnant price, three merchants became pos8olgors of equal sale-firoccods

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116 GANITASARASANGRAHA. (in the end. What is the buying price, what is the selling price, and what the oqual sale-amount ?) Tho rule for arriving at the solution of a problem wherein) optionally chosen quantities (are) bestowed in optionally chosen multiplos for an optionally chosen number of timos: 111.3. Let the penultimate quantity bo addol to tho ustimate quantity as divided by its own corresponding multiple number, and lot tho rosult of this operntion be divided by thut (multiple number which is associated with this) penultimate quantity (given in the problem). What results (from carrying ont this operation throughout in relation to all the various quantities bestowed) happens to be tho (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 takon (with him) fragrant flowers offered them (thus) in worship with devotion :At the four gate-ways, they became doubled, then trellod, then quadrupled and then qnintuplod (respectively in order.) Tho number of flowers offered by him was five at every (gato-way). llow many were the lotuses (originally taken by him)? 1114. Flowors were obtained and offered in worship by devoteos with devotion, the flowers (so offc rod) being (succossively) 3, 6, 9 and 8; (their corresponding) multiple quantities being }, }, 1 and (in order. Find ont the original number of flowers). * This onds proportionate division in this chapter on mixed problems.

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CHAPTBR TI-MIXED PROBLEMS. 117 Prillika-kuttikara. Hereafter we shall explain the process of calculation known as Vallika-kuttikara":-- The rulo underlying the process of calculation known as Vallika, in relation to Kuttikor (which is a special kind of division or distribution) : 1154. Divide the (given) group-number by the (givon) divisor; discard the first quotient; then put down one below tho other the (various) quotient obtained by the successivo 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 hulrikara explained in the rule in Larod upon a creeper.like chain of figuren. 115+. The rule will become oleur froin the following working of the problem in stanzi No. 1175. Here it is stated that 63 honps of plantaina together with 7 mert fruita are exactly divisible among 23 per'noni; it is required to find out the number of fruita in a henp. Here the 83 is called the group number, the numerical value of the fruits contained in each leap in called the group-value'; and it in this latter which lie to be found out. Now, according to the rule, we divide first the rani, or group-lumber 63, ly 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)6362 48 Iter, the first quotient 2 in dieserded; the other 17)2301 quotients at written down in a linn one below the other in in the markin; then we have to choon Auch number air, when multiplied by the last 6)1713 12 reminder I, and then combined with 7, (the number of parte fruits given in the problem, 68.1 will be divinible by the laat divinar 1. We accordingly cho 1, which is written dowi below the last figure in the chain; and below this chonendum er, again, is written down thu quotient ubtained in the above division with tl:e help of thu choroni number. Here we stop the divinion with the fifth remainder as it in the least remainder in the odd position of order in the series of divisione carried out hero. 1)6(4

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118 GANITASARASANGRAHA. with which the least remainder in the odd position of order (in the above-mentioned 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 lowermost 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 column of figures in the margin. Then we multiply the penultimate figure below in the olaiu, viz., 1, by 4, which is above it, and add 8, the last number in the chain; the resulting 12 is written down so as to be in the place corresponding to 4; then multiplying this. 12 by 1 which is the figure above it in the creeper chain, ind adding 1, the figure similarly below it, we get 13 in the place of 1; proceeding in the same manner 38 and 51 are obtained in the places of 2 and 1 respectively. This 51 is divided by 23, the divisor in the problem; and the remainder 5 is seen to be the least nambor of fruits in a bunch. The rationale of the rule will be clear from the following algebraical representation:--- 151 2-38 1--13 4-12 1 8 B+ b A = y (an integer) P1 = p1, where p = ,(whero r, B Ag the first remainder) Pb TI dp b 21 P+P, where p = ra the second remainder. Hence, P1 = Tip: + b Tip:+P3 where p1 = "xp: + b ". third quotient and rs the third remainder. T Similarly, Pa Ta Pa-b = 13 P4+b 20 Thus we have, 72 P1+ Pri 98 P+Pa P1 Pa 20 18 P+Ps, where Ps Ps =94 P3 + P1, where p1 = " P + b "1 (BA)x + b A and 9 is the second quotient and Pa + Pa: 96 P1+ Pe. "P-b 21 -i and is the

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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 above in relation to either of two specified distributions), is to be divided (as above) by the divisor (related to By choosing a value for 7, such that "s P1 + b 74 which is, as shown above, the value of ps, becomes an integer, and by arranging in a chain a 4, P4 and ps we get at the value of r hy 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 a so obtained is divided by 1, and the remainder represents the lost Br+ b value of ; for the values of which satisfy the equation, an integer, A are all in an arithmeticul progression wherein the common difference is 4. This same rule contemplates problems where two or more conditions are given, such as the problems given in stanzas 1214 to 1294. The problem in 1215 may he thus worked out according to the rale: It is given that a heap of fruits when diminished by 7 is exactly divisible among 8 men, and the same hemp when diminished by 3 is exactly divisible among 18 men. 8)13(1 8 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 the group-values 16 and 16 respectively. 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 rallika chain. Thus dividing 13 by 8 and continuing the division, we have 15551 CHAPTER VI-MIXED PROBLEMS. 6)8(1 3)5(1 119 2)3(1 1)2(1 From this the Vallika chain comes out thus: Chooning 1 as the mati, and adding the difference between the two group-values already arrived at, that in, 16-15, or 1, to the product of the mati and the 1 1 1 1 last divisor, and dividing this sum by the Inst divisor, we have 2, which is to be written down below the mati in the Fallika chain. Then pro1 cording as before with the rallikd, we got 11, 2 which, when divided by the first divisor 8, leaves the remainder 3. This in multiplied by the divisor related to the larger group-value, vis., 13, and then is combined with the larger group-value. Thus 65 is the num' or of fruits in the heap. J

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120 GANITASARASANGRAHA. the smaller group-valuo, so that a creeper-like chain of suocessive 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 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 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 : B. + O, is an intever: (i) B:1 + is an integer; and By x + og in an integer. In (i) Lot the lowest value of x=8 In (ii) = 89 In (iii) . . . (iv) When both (1) and (ii) : re to be satisfied, dA, + & has to be equal tw kA, + Ag, No that sy - Ny=kA, - A. That is 1.2 + (1 -- 8) From (iv), which is an indeterminate cguntion with the values of dand k unknown, we arrivo, according to what has been already proved, at the lowest positive integral value of d. "This valpe of d multiplied by A, and then increased by , gives the valne of which will satisfy (i) and (ii). Let this be t,; and let the next higher value of r which will satisfy both these equations be tz. (v) Now, t, + 14 =ta; (vi) and ti + mA, = tz. * Thus A, = mp, nnd Ag = np. where p in the highest common factor between A, and Az. cm = 4, and n=4 pen Substitating in (v) or (vi), we have Tp From this it is obvious that the next higher value of satisfying the two equations is obtained by adding the least common multiple of .4, and A, to the lower value. Now again, let v be the value of 2 which satisfios all the three equations. + 4,4 x . (where is a positive integer) = (say), + Ir; " P and v=ny + CA,= t; + lr.

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CHAPTER VI - MIXED PROBLEMS. 181 above division chain. Thus the croeper-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 bofore by the first divisor in this last division chunin. The reinniudor obtained in this operation is then) to be multiplied by the divisor (rolated to tho 'larger grunp-value, and to the rusuliing produot, this) larger group-value is to be addel. (Thus the value of the requireil multiplier of the riven group-lumber is obtained ; aud this will satisfy both the specified distributions taken together into consideration). Exemples in illustration thereof. 116. Into the bright and refreshing outskirts of a forrat, which were full of merous trece with their branches bent down with the weight of flowers and fruits, trees such ins jambit troos, limo troes, plantairs, aroca pulms, jack trors, dute-palme, hintala trees, palmyras, punniyu trees and mango troos (into the outskirts, the varions quarters wherruf were filled with the many sounds of crown of parrots and cuckoos found nonr aprings containing lotuses with bees ronming about them into anoh forest outskirts) a number of wears travellers entored with joy. 117. (There were) 6:3 (numerically equal) leaps of plantain fruits put together and combined with 7 (more) of those same fruits; and these were (qually) distributed among 23 travollers Ho as to leave no rimainder. You tell me now) the (numorical) measur, of a heap of plantains.) 1184. Agein, in relation to 12 (numerically oqunl) heaps of pomegranates, which, after having len put together and Hy applying the principle of vullikd.hullikdra in the lot equation, the valor of c in obtainet, and thence the value of v can be mwily arrived at It is seen from this that, when, in order to find out, we deal with l, and , in nccordance with the kufikira method, the choda or the divinor to on taken in relation to this A r the least common naltiple of the diviso in the first, two ypatione. 16

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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 honps less by 3 (fruits); and when the remainder (of those 31 heaps) is equally) divided among 73 men, there is no remainder. Give out the numerical value of one (of these heaps). 1201. In the forest 37 heaps of wood-apples were seen by the travellers. After 17 fruits were removerl (therefrom, the remainder) was (oqually) divided among 79 persons (80 ns 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) (liviiled among 13 of them; it left 110 remainder (in both cases). O arithmetician, tell me (tho numerical measure of this) single heap. 122}. A single heap of wood-apples divided among 2, 3, 4, or 5 (persons) leaves 1 as romainder (in ench case). O you who know arithmetic, tell me the (mumerical) measure of that (heap). 123]. When (divided) hy 2, the remainder is 1; when hy 3, it 18 ?; whou by 4, it is 3; when by 3, it is 4. Tell me, O friend, what this heap is. 1241. Whon (divided) by 2, the remainder is 1; when by 3, there is no remainder; when by 4, it is 3 ; when by 5, it in 4. Toll me, O friend, what the heap is (in numerical value). 125. When divided by 2, there is no remaindor; when by 8, there is 1 as remainder; when by 4, there is no romainder; and when by 5, there is one as remuiudor. What is this quantity ? 1267. When divided by 2 (the remainder is) l; when by 3, there is no renainder; when by 4, (the remainder is) 3; and whon divided by 5, there is 110 remainder. Tell me now what (this) quantity is. 1271. The travellers saw on the way cortain (equal) heaps of jambu fruits. Of them, 2 (heape) were equally divided among 9

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CHAPTER VIMIXED PROBLEMA. 128 ascetics and left 3 (fruits) as remainder. Again 3 (heaps) were (similarly) divided among 11 persone, and the remainder war 5 fruits; then again 5 of those heups were similarly divided among 7. and there were + more fruits (left ont) of them. () you arith. metician who know the meaning of the kuttikira process of distribution, tell me after thinking out well the numerical measure of a heap (hero) 128. In the interior of the forest, 3 heaps (equal in value) of pomegranates were divided (equally) among 7 travellers, louving .1 (fruit) as remainder; 7 (of such lon) were divided (similarly) among 9, leaving I remainder of 3 (fruits; again) 5 (of such henpo) were (similurly) 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 cqual in numerical value), which after being combined with 2 (fruits of the same kind) wero (equally) divided amony 9 travellers and no remainder); 6 (heape) combined with 4 (fruits) were (nimilarly) divided among 8 of thein; and 4 (heaps) combined with 1 (fruit) were also similarly) divided among 7 of them. Give out the numerical mcasure (of a heap here). The rule for arriving at the original quantity distributed (An desired), aftor obtaining the remainder due to the removal of certain specified) known quantities : 1304. (Obtain) the produrt of the (given, known quantity (to bo removed), as multiplied lip the fractional proportion of what is poft (after a Apocified fructional part of wlint romains on the removal of the given known quantityling ber giver athy). The next quantity is (ohtained by means of this produrt), to which 180. Lore the known quantity to be removed in called the agri, What remains ufter the removal of the yra is the manner that faction of this remainder which is given or taking in the cast and what is left of the remainder after the Ogranica is given nr taken away in the word of the romaining fraotional proportion of the remainder. For example, w e in the quantity to be found out, and a is the ayra in relation to the first dintrilotin with sthe fractional proportion distributed, " heppino to be thirdjed vita, end (*-u) -*" to be the sepanba.

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124 GANITABABASANGRAHA. 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 beon 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 ol their denominators; and these denominator-less quantities and the ancocesive products of the above-mentioned remaining fractional proportions of the remainder) aro (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. 131}. On a cortain man bringing mango fruits (honie, his elder son took one fruit first and then hulf of what remained. (On the elder son going away after doing this, the younger (son) did similarly (with wbat was left there. He further took half The rulo will be clear from the following working of the problem in 132) -- 193: Here 1 in the first agra, and in the first nyra visa; thereforu 1- or in the teminda. Now, obtain the product of agra and sera visa or 1 x 3 or 3. Writo it down in 2 plucos. O Repeat the qui add the second ayra 1 (to one of the quantitice) Thon we hav ultiply both by the next sexa msal - or $, so that you get Tako then lignres and add the third agru 1 as before ; and you have i multiply by the next sopa msu 1 - 3 or $ ond by the lantan sa or); and you have {8} 111 The denominators of the first fractions in these three sets of fractions Marked I, II, III, are dropped, and the numerator's represent negative ayras in problem on Mallika-kuffikara, where in the numerator and the denominator of oach of the second fractions in those sote represent respeotively the dividend ooeficients and the divisor. Thus we have 22 - 2 in an integer; 3 The value of satisfying these three oonditions gives the number of towers, 4% - 10 is an integer; and 81 - 38 in an integer. 81

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CHAPTER VI- MIXBI) PROBI.EMS. 125 of what was theriafter left); and the other (son) took the other half. (Find the number of fruits brought by the father.) 132} and 133;. A certain person weut (with flowers) into a Jina temple which was (in height) three times the height of a man. At first he offered ono (out of those flowers) in worship at the fout of the Jina and then offered in worship) one-third of the remaining number of flowers to the first height-meanitro of the Jina). Out of the remaining two-thirds of the number of flowers, ho conduce ted worship) in the filme manner in relation to the secondi leightmo'sure; and then loodud the same thing in relation to the third height-mensure also. The twi-thirds which remained at last wero also made into 3 equal parto by bim); and having worshipped the 24 tirthanikurax (with these parts at the rate of ciglit tir thunkurus for each part), he went away with 110 (flower), on hand. (Find out the number of lower's taken by him.) Thus ends simple Kuttikar in this chapter on mixed problems. *l'isema-kuttikini'n. Hereafter we shall expound complex kullikura The rule relating to complex kuttikira : 134). The (given) divisor, (written down) in two placow, in to be multiplied (in each place; by an optionally chosen number; and tho (known) quantity given in the problem for the purposo of hoing alded is to be subtrarte (from the product in one of these places); and tho quantity given in the problem) for the purpose of boing subtracted is to be added to the product noted down in the othor place. The two quantities thus obtained aro) to bo divided by the known (coetficient) multiplier of the unknown * The words Vipama and Bhinna lvre used in relation to Kuffikura hara obviourly the name meaning and refer to the fractional character of the dividend, quantities occurring in the problema contemplated by the rulo.

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126 GANITASARASANGRAHA. quantities to be distributed in accordance with the problem). Each (of the quotients so obtained) happons 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, ( certain other quantity multiplied by 6, then) diminished by 10 (and then divideil by 9 leaves no remaindor). Tell me quickly what those two quantitios are (which are thus multiplied by the given multiplier bere). Sakala-kuttikara. The rulo in relation to sakala-kuttikara. 1361. The quotient in the first among the divisions, carried on by means of the dividend-coofficient of the unknown quantity to be distributed), as well as by means of the divisor and the (sucoessively) rosulting remainders, is to be discarded. The other quotients obtained by means of this mutual divisiou (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 givon kuown quantity in the problem by the divisor tberein), 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 vumber) the product of the two 136. This rulo will become clear from the following working of the problem given in 1375:-- The problem is, when 177240 is an integer, to find out the valaos of e. 201 Removing the oonimon factors, we have is an integer,

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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 problem), which (values)are related oither to the known given quantity that is to be added or to the known given quantity that is to be subtracted, according as the number 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 67)59(0 ( 59)67(1 59 Below this are next written down 1 and 1, the last equal divisor and remainder. Here also, as in Fallika-huftikara, it is worthy of note that in the last division there can be really no remainder, as 2 is fully divisible by 1. But since the last remainder is 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 hore, add 13, which is the remainder obtained by dividing 80 y 67; the 14 so obtained in also written down at the bottom of the chain, which now becomes complote. 1 Now, by the continued process of multiplying and adding the figures in this explained in the note under stanza No. 115, chain, as already we arrive at 592. This is then divided by 67; and the remainder 57 is one of the values of r, when 80 is taken as negativo owing to the number of figures in the chain being odd. When NO is taken as positive, the value of e is 6757 or 10. If the number of figures in the chain happen to be even, then the value of a firet arrived at is in relation to the positive agra, if this value betracted from the divisor, the value of in relation to a negative agra is arrived at, 8)59(7 56 1 14 3)8(2 C 2)3(1 2 1--382 7-345 2-47 1--16 1-15 After discarding the first quotient, the others are written down in a chain thus: 1)2(1 1 1 7 2 1 1 1 *

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128 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 ralo is the same as that. explained in the rule regarding Vallika-kuttikara--but with this difference, namely, that the last two figares in the chain here are obtained in a different way. Again, from the rationale given in the footnote to rule 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. Ilence, when the continued division is carriel up to a romainder in the odd position of order, the value of arrived at therefrom is in relation to ach an agra as has its sign unchanged; on the other hand, when the continuel division is carried up to a remainder in the even position of order, the value of r arrive 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 rulo always taken to be positive, the value of z 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 ayra, 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. The value of a in relation to the positive or the negative agra being thus found out, the other value is arrived at by snbtracting this value from the divisor in the problem. How this turns out will be clear from the following representation:--- Az + b an integer. Here let ac; then Ac + b B know that AB B is also an integer. Hence AB B an integer. We B Ac + b A(B c) b B B or is an integer. It has to be noted here that the common fuctor, if any, of the three given numerical quantities is be removed before the operation of continued division is begun. The last divisor and the ast 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-kuttikdra and required to be writton 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 l'allika-kuffinira. It will be seen further that the last figure of the chain obtained according to this rule, i...+agra, is the same as the last figure in the chain obtained in the Fallikd-katikara by dividing by the last divisor the sum of the aura 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 reforrei to, to the product of the least romainder in the odd position of order multiplied by the abovo optionally choseu multiplier thereof, and then by dividing this resulting sum by the last divisor in the above divisiou chain. Thus the creeper-like chain of figures required for the solution of this latter kind of problem is obtainel. This chain is to bo dealt with as before from below warls, and the resulting number is to be divided is before by the first divinor in this last division chain. The remainder obtained in this operation is thon to le) multiplied by the divisor (related to the larger group-value); and to the resulting product this larger group-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 specifici distributions taken into consideration) Examples in illustration thereof. 1371. One hundred and seventy-seven in the dividond-cn. efficient of the unknown factor), 2.40 is the known yuantity associated (with the product soon to be added to or subtractod from it); the whole is divided lov 201 (and leaves no remainder). What is the unknown factor hore (with which the given dividend. coefficient is to be multiplied? 1381. Thirty-five and other quantities, 16 in number, riving (thenco successively in value) ly , fure the given dividond-00efficients). The given divisors are : 2 (1 others) as successively increased br 2. And l successively increased by 3 gives rise to the associated known (positive and negative) quantities. What are the values of the unknown factors of the known dividond. coefficients), according as they are additively associated with positive or negative known) numbers? 17

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130 . GANITASARASANGRARA. The role 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 : 1391. From the higher price-sum, as multiplied by the corresponding larger number of one of the two kinds of things, subtract the lower prioe-number as multiplied by the smaller pumber 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. 1404 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 tho price of a citron and of a wood-apple hero, having distinctly separated those prices well. The rule for separating the prices and the numbers of different mixed quantities of different kinds of things from their given mixed price and given mixed values : 143. The (difforont) 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 139). Algebraically, if az + by=m and bx + ay = 1, then * + aby = um and b's + aby = bn. .. (-") =an-bn. sum - bu 143. The rule will become clear by the following working of the problem in stancas 144, and 145:-- The total pumber of fruits in the first heap is 21. do. second do. 29. Do. third do. 23. Do. do.

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CHAPTER VI- MIXED PROBLEMS. 131 are to be divided (ouo 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 (suocessively) diminished by the corresponding quotients in the aboven procoss. (In this manner the numerical values of tho 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 is also that multiplier constitute (respectively) the prices of a singlo thing in oach of the varieties of the given different things). Choose any optional number, way y, and multiply with it there total timbers: we get 42, 44, 46. Subtract those from 72, the price of the rewportive brain. The remainders are 31, 29, and 27. These are to be divided by another optionally ohusen number, way 8. The quotients aro 3,3,, and the remainders are 7, 6 and 3. These renuinders are ngain divided by third optionally chonon number way 2. The quotients are 3, 2, 1, and the remaindern aro 1, 1, 1. Then laat remainders are in their turn divided by fourth optionally chowa nuinber which is 1 hore. The quotiontie 1, 1, 1 with no remaindern. The quotionta derived in relation to the first total number ure to wubtracted from it. Thus we ket 21-(3 + 3 + 1) = 11; thin number and the quotinta 3, 3, 1 represent the number of fruits of the different morts in the tirnt heap. Similarly wu ket in the sound group 16, 3, 2, 1, and in the third group 1, 8, 1, 1, as the number of the different kotin of fruita. The prices are the firmt chosen multiplier, viz., 2, and its hin with the other optionally chonon multipliers. Thus we get 2, ? + Mor 10,2 + 2 or 4, and 2 + 1 ur 3, 140 the prio of who the four different kinds of fruit in orler. The principlo underlying this rethod will be clear from the following algebraical representation : - az + by + ca + du = p ... ... .. . ... ... 1 Let w =w. Stultiplying II by, we have . (a + b + + d in Subtracting 111 from I, wu get a (+-) + b (y - ) + c(-) ... ... IV Dividing IV by I-, W got in the quotient, and (y - ) + c(-) AR the remainder, where -Dinn muitable integer. Similarly we proceed till the end. Thus it will bo acen that the KUCCA Dively choron divinor , y , and --, when combined with X, give the valon of the various prices, by itself being the price of the first thing; and in the successive quotinen a, b, c, along with - (a + b + c) are the nurobers measuring the various kinds of think. It may be noted that, in this role, the number of divisions to be carried out in ono less than the number of the kinds of things given, and that there should be no remainder left in the last division.

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182 GANITASARASANGRAHA. An example in illustration thereof. 1447 and 145]. There are here fragrant citrons, plantains, wood-apples and pomogranates mixed up (in threo 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 (heaps) is 73. What is the number of the (various) fruits (in cach of the kea ps), 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):-- 1464. Divide (the rate-quantities of the given things) by their rate-prices. Diminish (these resulting quantities geparately) 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 romainder optionally) into as many bits as thore aro remainders of tho above quotientquantities left after subtraction); and then divido (thieso bits by thoso 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 parchase. Exemples in illustration thereof. 1474 to 149." In accordance with the ratos of 3 peacocks for 2 panas, 4 pigoons for 3 panas, 5 swans for 4 panas, and 6 sirasa 148). The rulo will be clear from the following working of the problem givon in 147--149:-- Divide the rate quantities 3, 4, 5, 6 by the respective rate-priors 2, 3, 4, 5; thus wo lave ., ... Subtract the least of these from onoh of the other three. Wo get is, in By multiplying tho given mixed price, 56, by the obrve. ruentioned leant quantity,%, we have 10x. Subtract this from the total number of birds, 72. Split up the remainder into any three parts, say 1, 5, 1. Dividing these respectively by it is to wo get the prices of the first three kinds of birdig 1, 12, 36. The price of the fourth variety of birds can be found out by subtracting all these three prices from the total 6.

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CHAPTER VI-MIXED PROBLEMS. 133 birds for 5 panas, purchase, O friend, for 56 panas 72 birds and bring thein (to me)". So saying a man gavo over the purobasemoney (to his friend). Calculato quickly and find out how many birds (of cach variety he bouglt) for how many punas. 150. For. 3 panas, 5 pilis of ginger aro oltained ; for 4 panax, 11 palas of long pepper; and for 8 panis, I pula of pepper is obtained By means of the purchase-money of 60 parnus, 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 sume of monoy as their total price : 151. The rato-values of the varions things purchased aro cach separately) multiplied by the total value (of the purchase-money), and the varions values of the rate-monoy are (aliko scparately) 151. The following working of the problem given in 152-153 will illustrato the rule :5 7 93 Write down the rate-things and the rate-prices in 500 500 900 30) two rows, on below other. Multiply by the total 300 500 700 M) price and by the total number of things in pectively. Thon mubtruct. Remove the common furtur 100. Multi0 0 0 0 200 2001 2000 ply by tho chen humb 3, 4, 7, 1. Add the numbers in ach horisontal row and remove the commun factor (: 0 0 0 0 Change the position of these figuren, ud write down in 2 2 2 0 two rows owch figuro non many times in there are compo. nent elements in the corresponding sun change in pomition. 0 0 0 36 Multiply i bo two rows litho rate prices and the ruto6 X 100 things respectively. Then remove the common factor 6. Multiply by the already chon number 3, 4, 5, 8. The numbers in the two POW represent the proportion according to which the total price and the Wotal number of thingn become distributed 6 6 6 18 30 30 42 42 64 This rule relate ton problemi in indeterminato 4 equations, and an mich, thoro may be many wel of answers, thene Answers obviously depending upon the quantition 36 chosen optionally an multipliers. 12 It can be camily mean that, only when certain not of numbers are chosen an uption multipliern, integral 7 answers are obtained ; in other CAKCH, fractional anwera are obtained, which are of course not wantod. For an 36 explanation of the rationale of the proces, weo thu note 13 giron at the oud of the chapter. 9 18 20 28 36 48

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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 choson quantity ; (then thogo 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 lowor row of numbers being written above and the sum of the upper row being written below. Theso 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 writton down twioo, (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 rato-prices and rate-values of things, (the rate-prico multiplication being conducted with one row of figuros and the rato-number multiplication being in relation to the other row of figures. The products so obtained aro ngain reduced to their lowest terms by the removal of such factors as are common to all of them. The resulting figures in oach Vertical row are separately) multiplied (cach) by (moans of its corresponding originally chosen) optional multiplier. (And the products should be written down as before in two horizontal rows. The numbers in the uppor 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 parohased are distributed. Therefore) what remains thereafter is only the operation of praksepaka-karana (proportionato distribution in scoordance with rule-of-three). An example in illustration thereof. 152 and 153. Pigeons are sold at the rate of 5 for 3 (panas), Farasa birds at the rate of 7 for 5 (panas), swans at the rate of 9 for 7 pan.), and poacooks at the rate of 3 for 9 (panas). A certain man

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CHAPTER VI-MIXED PROBLEMS. 135 was told to bring at these rates 100 birds for 100 panas for the amusement of the king's son, and was sont to do so. What (amount) does ho give for each (of the various kinds of birds that he buys)? The rule for arriving at the incastro of two given commoditios whose prices are interchanged: 154. Let (tho numerical value of) the sum of the (total selling and buying) nonoy-prices of the two givon commodities) bo divided by the numerical measure of) the sum of the commodities put togother); then lot the differonce between thonbove-mentioned buying and selling prices) bo divided by the (numerical measuro of any such) difference as may bo obtained by subtracting any optionally chosen commodity-quantity from the given moasure of the sum of the given coinmodlities. If the oporation 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 bo oltained in the second operation), the rates at wbich those commodities are purchased is obtained. Then if tho samo operation of sankramana is 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). Tho alternation (of these above-mentional purchago-ratos) gives rise to the yale-rates. This is the solution of (this kind of) problems an propounded by the learned ; and the rulo (itself) has bocu doclared by the great Jina. 154. The algebraical representation of the method describwd in the rulo may he given thus in relation to the problem proposed in stunzi 155 and 160. Let az + by == 101... ... ... ... ... ... 1 ay + lz=116 .. ... dding I and II, we have (a + b) (x + y) = 220 1, 2 + y=11 ... ..... .... ) Again subtracting from 11, we get (a-6) (-) = 12 Now 2b in optionally chosen to be equal to 6. a+b - 20 or a-b=20 -or 14 ... ... ... .. VI ..y-I=11 ... ... ... ... ... ... VII Carry out the operation of sankramana with reference to VII and V, aud Vi sod Ill; and thu values of z, y, a and bare all made out.

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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 panns; when after a time they were sold with their prices mutually alterod, 116 panas were obtained. You give out their buying and selling rates and the numerical measure of the commodities, taking 0 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. Tho number representing the total yojanas, divided by the total number of horses, gives the yojanas (which each has at a stage to travol) in turn Theso yojanas, as multiplied by the optionally chosen uumber of horses to be yoked, give the measure of the distance to be travelled over by each horse. An example in illustration thereof. 158. It is well-known that the horses belonging to tho sun's chariot are 7. Four horson (have to drag it along, being harnessed to the soko. They have to do a journey of 70 yojanas. How many times are thoy 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 oach (of a body of joint proprietors), from the Bonjoint remainder left after subtracting whatever is desired from he total value of all the commodities : 159. Let the sum of the values of the conjoint remainders) of ho commodities be divided by the number of men lessened by one; ihe quotient will be the total value of all the commodities (owned n common). This total valuo as diminished by the specified values rives 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 ommon were asked oach separately by the customs officer what he value of the commodity (they were dealing in) was ; and one

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CHAPTER VI-XIXID PROBLEMI. 187 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 tho fourth said that it was 27; (in saying so) each of them deduotoil bis own investod amount (from the total value of the commoility for sale, O friend, tell mo separately the value of the (share in the commodity owned by each! The rule for arriving at qual amounts of wealth, (ng owned in precions gems, after mutnally exchanging any desired number of gome: 163. The number of gems to be given away is multipliod by tho totul mumber of men (taking part in the exchange). This product is (neparately) subtracted from the umber (of the gems) for sale (owned by each); the continued product of the romainders (no obtained) gives riso to the value of the gein (in each onko), provided the remainder relating to it is given up Tin obtaining such a procluct). Eramples in illustration thereuf. 104. The first man hal 6 azuro.blue gom (of equal value), tho second man had 7 (similar) omoralds, and the other--tbe third man--had 8 (similar) (diamonds. Each (of thom), on giving to oach of the others) the value of a single gem (owned by himself), became oyunl (in wealth-value to the others. What is the value of a gem of ench variety ?) 165 and 166. The first man has 16 azure-blue yems, the Bocond has 10 emeralds, and the third man has 8 diamonds. Each among them gives to each of the others 2 goins of the kind owned by hiinself; and then all three inen come to be possessed of equal 163. Let m, n, p, be rispectively the numbers of the three kinds of soms owned by three different persons, and a the number of goms inutanlly erohangod and let z, y, , be the value in oriler of mingle gain in tho three varieties concerned. Then it may be ensily found out as required that < < (n - 8a) (p - lei; im-8a) (p - Bay, 18

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188 JANITASARASANGRAHA. wealth. Of hat nature are the prices of those azure-blue gems, emeralds, and diamonds ? The role 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-measuros of the com. modity are each multiplied alternately by the rate-prices ; (the product obtained with the help of) the buying rate-measure is divided by the other product obtained with the aid of) the selling rate-measure. The profit, divided by the rosulting 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 & profit of 72 panus. What is the capital employed in this transaction P Thus ends Sakali-kuttikaru in the chapter on mixed problems. Suvarna-kuttikara. Hereafter wg ahall explain that kuttikara which consists of oaloulations relating to gold. The rulo for arriving at the rarna of the resulting mixed gold obtained by putting together (different component varieties of) gold of (various) desired varnas : 109. It has to be known that the sum of the various) products of (the various component quantities of) gold as multiplied by (their respective) marnas, whon divided by (the total quantity of) 107. If the buying rate is a things for b and the selling rate is c things for d, and it m in the gain by the trannction, then the capital invested in

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CHAPTER VI-MIXBD PROBLIMS. 189 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 samo component part thereof). an example in illustration thereof. 170 to 171). There are 1 part (of gold) of 1 varna, 1 part of 2 varras, 1 part of 3 varnis, 2 parts of 4 varnar, parts of 5 varnas, 7 parts of 14 varnas, and 8 parts of 15 varus. Throwing thoso into the fire, make them all into one (mans), and then (say what the varna of the mixoil gold is. This mixed gold is distribuwd among the owners of the foregoing parts. What does caoh of thom get? The rule for arriving nt the reynired weight of gold (of any desired varna equivalent in value to given quantities of gold) of given wirnar: 172. The given quantities of gold are all (soparatoly) multiplied by their respective varniis, and the products are addod. The resulting sum is divided by tho total weight of the mixed gold ; the quotiont is to be understood as the resulting avorago virna, This (above-mentioned sum of the products) is separately divided by the desired virnus (to nrrive at the required oquivalent weight of this gold). Examples in illustration tharenf. 173. Twenty panas (in weight of gold) of 16 varnas bave boon oxchanged for (gold of; 10 rarnas in quality; you give out how many purunas (in weight) they become now. 1744. One hundred and eight (in weight of) gold of 112 durnas is exchangod for (gold of) 14 varnas. What is the oquivalent quantity of this new) gold? The rule for finding out the unknown warna : 1759. Froin the product obtained by multiplying the total quantity of gold by tho resulting varna of the mixture, the sum of

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140 GANITABARABANGRAHA. tho 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 vurna (of an unknown component quantity of gold) gives rise to the (required weight of that) gold. Another rule in relation to the unknown varna : 1761. The sum of the products of the various component quantities of) gold as multiplied by their respective vurnas 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 respeotively hy 13, 8 and 6 as their rarnas, 5 in weight of gold of an unknown varna is mixed. The resulting vurna of the mixed gold is 11. O you, friend, who know the socrets of calculation, tell me the numerical value of this unknown varna. 179. Seven in woight (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 rosulting varna is 10. Give out the anknown varna (of this second specimen of gold). T'ho rule for arriving at the unknown weight of gold : 180. Subtract the sum, obtained hy adding together the products of the (various component quantities of) gold as multiplied by their respectivo varnas, 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) yarna of the uuknown quantity of gold and the resulting durable marwa (of the mixed gold) gives rise to the weight of) gold.

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OHAPTER VI-MIIRD PROBLEYS. 141 An example in illustration thereof. 181. Threo pioccs of gold, of 3 onoh in weight, and of 2, 3, and 4 varrus (rospectivoly), are aldod to an unkuown woight of) gold of 13 varnas. 'The rosulting varna comes to be 10. Tell mo, O friend, the moasuro (of the unknown weight) of gold. Tho rule for arriving at the weights of) gold (corresponding to two given varnas) from (the known weight and rarnu of) the mixture of two (given spocimens of) gold of (givon) rarnas : 182. Obtain the differences between the resulting varna (of the mixture on the oue hand) and the known higher and lower eyrnas (of the unknown component quantities of gold on tho other band); divide one by these differences (in order); the carry out as boforo the operation of praksipukul (or proportionate distribution with the aid of these various quotionts). In this mannor it is possiblo to arrive oveu at the value of many coin poucnt quantities of gold also. Again, the rule for arriving at (the weights of) gold (corrosponding to two given rarnax) from (the known weight and varna of) the mixture of two (givon specimens ot) gold of (given) varnas : 183. Write down in inverse order the difference lictwoon thio resulting varna and the higher (of the two given varnar of the two oomponont quantities of gold), and also the difforonco butween the rosulting varna and the lower of the two given rarnas). The result arrived at by means of the operation of proportionato distri. bution carried out with the aid of these inversely arranged differencos),--that (result) gives the required (weights of the component quantities of) gold. An example in illustration thereof. 184. If gold of 10 rarnas, on being combined with gold of 16 varnas, produces as result 100 in weight of gold of 12 varnas, give out separately (tbe measuros in weight of the two different varieties of gold.

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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. Thon 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 bo 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. Divido one (soparately) by the two (given weights of) gold; multiply (neparately cach of the quotients thus obtained) by (the weight of) the corresponding quantity of) gold and (also) by tho (resulting) varna ; write down (both the products so obtained) in two different) places; (each of those in onch of the two sets,) if diminished and increased alternately by one os divided by (the 186. The rule reforred to hore is found in stanza 180 above. 187. The rule will become clear by the following working of the problom in stanna 188* 18 1) and in * 10 x 11 uro writton down in two places 11 11 aro added and subtracted alternately in each of the two Thon io ando sote thus : 11+ al fuciom 16 Jand od 11 - 16 m m . '. Tbeo give the two sets of answers. Lil + 10

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CHAPTER VI-XIXED PROBLEMS. 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) carnax not being known, tho resulting parna ohtained by mrcans of two different kinds of) gold weighing 16 and 10 (respectively) happens to be 11, what would lio tho (respective) varnas of those two (different kinds of) gold P Again, the rule for arriving at the unknown rurnax of two (known quantities of gold, when the resulting varna of thu mixture is known) - 189. Choose an optional rarna in relation to one (of the two given quantities of gold); what remains to be found out) mny then be arrived at ns before. In relation to the knowll quantities of all) the numerous varieties of gold oxcepting one, the rarnas nro optional; then (proceed) us beforo. An example in illustration thereof. 190. On fusing together (two different kinds of) gold which are 12 and 14 (respectively in weight), the resulting uw na is made out to be 10. Think out and way (what) tho varnux 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 respectivoly of four different kinds of gold, the resulting mixture turns out to be (gold of) 12 varnas. Give out the rurrus (of the various component kiuds of gold) separately. The rulo regarding how to arrive at (an estimate of the valuo 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 no obtained) are (all) to be added together. The resulting sum gives (tho measure of) the required quantity of (pure) gold. From the summed up

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144 GANITABIRABANGRAHA. (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 stioks. 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 virnas. The (various specimens of the test sticks of) gold in the case of the first (merchant) wero (regularly) less by 1 (in varno); those of the second were less by and ; and those of the third were (in regular order) less by . The specimens of test gold) possessoil by the first (morchant) hogan with that of (his) maximum varna and ended with that of 1 varna ; (similarly, those of the second hegan 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 vurnas. Every test stick is 1 masu in weight. O mathematician, if you indeed know gold calculation, tell me separatoly and soon what the measure of pure gold here in, and what that of the baser metal mixed. Tho rule for arriving at the difforent weights of) gold obtained in oxchange and characterised by two given) varnas : 1977. 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 speoified) varnas (of the oxohanged gold)-(and, secondly, between the Brst product above-mentioned and tho product of the weight of 1975. This role will be clear from the following working of the problem given in stanga 1985: 700 x 16-1008 x 10, and 1009 12-700 x 16 are altered in position and written down - 896 and 1120 ; and these, whon divided by 12-0 or , give rive to the answers, namely, 448 and 560 in weight of gold of 10 and 18 varias roupectively,

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CHAPTER VI-MIXED PROBLEMS. 145 gold obtained in exchange ay multiplied by the second of the specified urnas of the exchanged gold - these two differences) havo to be written down. If theu, they are altorod in position and divided by the difference between the two sperilicl) murras (of the two varieties) of the exchanger gold, the result happons to be the (two required) quantities of the two kinds of gold (obtained in exchange). Au couple in illustration thereus. 1987. Seven hundroil in weight of gold characterise! by 16 virus produccs, on being exchanged, 1,008 (in weight) of two kinds of gold characterised (respectively) by 12 and 10 virnas. Now, what is the woight (of cach of these two varieties of gold! The rule for finding out the various woights of) gold obtamod as the result of many (nprocitied) kinds of exchange : 1997. If the (vivon) weight of gold (to be exchanged) as multiplied by the varna (theroof is divided by tho quantity of) the desirod gold (obtained in oxchange), there arises the uniform average vurna. On carrying ont further) operations is mentioned hefore, the result arrived at given tho required weights of th:0 various kinds of gold obtaincel in oxchange. An example in illustration thereof. 2001-201. In the case of a man exchanging 300 in weight of gold ch: racteriscil by 1.1 varnwx, the gold (obtained in oxchange) is seen to be altogether 500 in weight, (the various parts whoroof aro respectively characterized by 12, 10, and 7 varnis. What in the woight of gold soparutely corrospopiling to each of thoso (different) ournax ? The rule for arriving at the various weights of goli wbtained in exchange which are characterisou ly known rurnax and ure (elefinite) multiples in proportion : 202-203. The sum of the (given) proportional multiple mumbors is to be divided by the sum of the proucts (obtained) by . 1994. The operation which is stated here as having been mentioned before is what is given in stanza 185 aburu. 19

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146 GANITASARASANGRAHA. multiplying the (given proportional quantities of the various kinds of the exchanged) gold by (their respectivo specified) varnas. (The resulting quotient) is to be multiplied by the original varna (of the gold to be exchanged). If by this product as diminished hy one, the increase in the weight of gold due to exchango) 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 unoxchanged gold) is then to be subtracted from the sum of the weight) of the original gold and the incrcase (in weight due to exchange). Then if the result ing remainder (here) is divided by the sum of the proportional multiple numbers connected with the exchango, and is then multiplied by each of those) proportional numbors (soparately), the (various woights of) gold obtained in exchange and characterised by the specific varnas and the specified proportions are arrived at. An example in illustration thereof. 201--205. Thero is a certain merchant desirous of obtaining profit; and the gold in his possession) is of 16 carnus 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 varnits, (so that those varicties 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? Toll me also the weights of gold obtainoil in exchange corresponding to thoso (above-mentioned varnas). The rule for arriving at (the weight of the original (qnantity of) gold with the aid of the gold exchange (in part), and with the aid (of the weight) of gold seen to be in cxcogs (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

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CHAPTER VI-XIXED PROBLEMS. 147 optionally chosen varna (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 bolonging to a merchant is taken; and 4, 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 tho original gold. Then 1,000 is observed to be in excess on removing from the account the weight of the original gold. What thon is (the weight of this) original gold? The rule for arriving at the desired earna 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 he noted down in reverse order; aud (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 213-815: Dividing 1 by and, we get respectively 2,; altering their position and multiplying them by any optionally chosen number, say 1, we get 2, 2. These two numbers represent the quantities of gold owned respectively by the two merchants. Choosing 9 as the varna of the gold owned by the first merchant, we can easily arrive, from the exchange proposed by him, at 13 as the vary 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 varpa as given in the sum has to be 12 or . If 8 is chosen instead of Therefore the varnas 9 and 13 have to be altered. 9, 13 has to he increased to 18 in the first exchange. Using these two varnas, 8 and 16, in the second exchange, we obtain as the average varna, instead of 14.

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148 GANITABARASAKURAHA. optionally choson quantity, (it) gives 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 optiou become the verified varnas montioned in relation to the two (given) little halls of gold. If, (howevor, those answers do) not (tally), the carnas belonging to the first set (of answers) have to be made (as the case may be) a little less or a littlo moro; (then the averago 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 requirod proportionate quantities) are (therefrom) derived by means of the operation of the Rule of Three: and the varnas (arrived at according to the option chogen before, when respectively diminished by one of these two quantities and increased by the other, turu out to be evidently tho reyuired varnas (here). An example in illustration thereof. 213--215. Two morchants will versed in cstimating tho 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 (tho whole become gold ot) 10 rurnas." Then this other Baid--"If I only obtain your rold by ouc-third (thereof), I shall likowiso make the whole (gold in my possession become Thus, in the second exobungo, we see an increase of 40-35 or 5 in the som of the products of weight and varna, while tho (leoreude and the inoronse in relation to the originally choson rarnas are 9-8 or 1 and 18-13 or 3. But the required increase in the sum of the products of weight and varra in the second exchange is 36-35 or 1. Applying the Rule of Three, we get the corresponding deorease and increase in the varpas to be and Therefore, the varras are 9- or 8 and 18+ or 187.

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CHAPTER VI-MIXED PROBLEMS. 149 gold of) 12 rarnas 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 Sucarna-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 these (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 bare liked. The number of utterances is a, and each statement refers to a persons. Hence the total number of statements is a xa or a3. * Now, of these a person, bare liked, and a-b are not liked. When each of the number of persons is told "You alone are liked," the number of antruthful statements in each case is b 1. Therefore, the total number of untruthful statements in b statements is b (b. 1)..... I. When, again, the same statement is made to each of the ab persons, the number of untruthful statements in each case is b+1. Therefore, the total number of untruthfal statements in a- butterances is (a) (b + 1) .11. Adding I and 11, we get b (11) + (a b) (b + 1) = u(b + 1) 20. This represents the total of untruthful statements; and on subtracting it from a, which is the measure of all the statements, truthful and untruthful, wo arrive obviously at the measure of the truthful statements.

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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)". low inany (of her statemonts, explicit n3 well as implicit) are true ones? 10. The rulo rogarding the (possible) varieties of combinations (among given things) : 218. Beginning with one and increasing by one, let the numbors going up to the given number of things be written down in regular order and in the inverse order (rospectively) in an upper and a lower (horizontal) row. (If) the product of one, two, threo, or more of the uumbers in the upper ruw) taken froin right to left (be) diviled 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 roquired in oach such case of combination) is (obtained as the result. Examples in illustration thereof. 219. Tell me now, O mathematician, the combination varietion is also the combination quantities of the tastes, viz., the astringent, the bitter, the sour; the pungent, and the saline, together with the sweet taste (as the sixth). 220. O friond, you (tell mo quickly how many varieties there may be, owing to variation in combination, of a (single string) uceklace made up of diamonds, sapphiros, oneralds, corals, and poarls. 221. () (my) friend, who know the priociples of calculation, tell (mo) how many varieties there may be, owing to variation in combination, of a garland made up of tho (following) flowersketaki, asoka, campaka, and nilotpala. 218. This rulo relates to a problem in combination. The formula given baru ien. (n-1). (* -- 2) .... ( r ). and this is obviously equal 1. 2. 3. . . . .

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CHAPTER VI-MIXED PRORLEUR 151 The rule to arrivo 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) profiis 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 hs that (samo common factor which has been lisod in the process of proportionate distribution above). An exemple in it/ustration thereof. 223-22:). According to agreement some three merchants carried out (the operation of) buying and selling. The capital of tric first of them) consisted of six purinas, that of the second of cight purrins, but that of the third was not known. The profit obtained by all those (three) men was 96 puramus. lu fnet the profit obtained by him (this third person) on the imknown capital happened to be 10 purinas. What is the amount thrown by him (into the transaction), and what is the prolit (of each) of the other two merchants? O friond, if you know the operation of proportionate distribution, tell me this) after making the (110centary) calouiation. The rule for arriving at the wagos (due ini kinl for having carried certain given things over a part of the stipulated distance according to a given rate):-- 226. 1'rom the square of the product (of the numerical valun) of the weight to be carried and hail of the stipuuted distalloe 220. Algebraically, the formulas given in the rule in -N OW ) - abd (D-R) were worn to found out, D-d a= the total weight to be carrive, D = the total dimance, the distance KON over, and the total Waris promined. It may be noted here that the rate of the wagen for the two waves of the journey in the name, althongh the amount paid for each mage of the jonrney is not in accordance with the promised rate for the whole journey. The formula is rasily derived from the following equation containing the duta in the problem : adla --) (D-d).

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152 GANITASARASANGRAHA. measured in) yojana, subtract the continued) product of (the numerical value of) the weight to be carriod, (that of the stipulated) wages, the distance already gone over, and the distanoe still to be gone over. Then, if the fraction (viz., half) of the weight to be carried over, as multiplied by the wholo of the stipulated) distanco, and then as diminished by the square root of this (difference above mentioned), be divided by the distance still to be gone ovor, the required answer is arrived at. An example in illustration thereof. 227. Here is a man who is to receive, by carrying 2 juckfruits over 1 yojuna, 7% of them as wages. le broaks down at half the distanco. What amount within the stipulated wagos) is (then) due to him ? Tho rule for arriving at the distances in yojanas (to be tr:velled over) by the second or the third weight-carrier (aftor tho first or the second of them breaks down) : 228. From tbe product of the wholo) weight to be carried as multiplied by the value of the stipulated) wages, subtract the square of the wagos givon to the first carrior. This difference bas to ho userl as the) divisor in relation to tho (continued) product of the difference between thu (stipulated) wages (and the wages alroady givon away), the (wholo) weight to be oarried, and tho (wholo) distance (over which the weight has to be carriol. The resulting quotient gives rise to the distance to ho travelled over by the scoond (person). An example in illustration thereof. 229. A man by carrying 21 jack-fruits over (a distance of) five yojanas has to obtaia 9 (of them) as wagos therofor. When 6 of these bave been givon away as wagos (to the first carrier), what is the distance the second carrio, bas to travol over to obtain the remainder of the stipulated wagos)? 228. Algebraically D-d= the equation in the last note. -*) ab , which can beeasily found out from

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CHAPTER VI-MIIED PROBLEMS. 103 The role for arriving at (the valuo 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 dividol by the nunbers of the men that are doing the work of carrying) thoic. The quotionts (80 obtained) have to bo combined so that the first (of them is taken at first separately and then) hus (1, 2, and 3, ote., of) the followiug (quotients) added to it. Thesr quantities no resulting are to be respectively) multiplied by the numbers of the men thut turn away (from the journoy at the various stage. Theu) by adopting (in relation to these resulting products) the process of proportionate distribution (aksepaka), the wige due to the mon leaving at the different stiges) may be forind out. Anecommple in illustration thereof. 231-232 Twenty men have to carry A palanguin over ( distance of) 2 yojunas, and 720 dinaras form their wagen. "I'wo men stop away after going over two krosus ; after going ows two (more) kroxas, three others (ntop away); after going over half of the remaiving distance, five men stop away. What wagen do thoy (the various bearers) obtain The rule for arriving at (the value of the money contents of) & purgo which (when added to what is on hand with cach of certain persons, becomes a specified multiple (of tho nun of what is on hand with the others) :--- 233-235. The quantities obtained by adding one to (cach of the specified) multiple numbers (in the problem, and then) 238-238. In the problem kiven in 236--237, let z, y, z represent the munoy. on hand with the threw Inwrchants, and the money in the porse. Then * += a (y + 2) where a, b, c represent the multiples * + y = b (z +I) viven in the publere *+ = (2+ y)) Now ++y+= (a + 1) (y + 1) = (b + 1) (2+2) = (c+1) (*+y).

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154 GANITASIBASANGRAHA. 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 money, 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 Then, (n+1) (6+1) (c+1) (y + 8) = (6+1) (c+1). .....I. where T u +*+ y +. Similarly, (+12 +12 (0+1)x(+) * (*+ x) = (c+1) (a +1) ... II. and (a+1)(6+1) (C+1) ! (*+y) = (a + 1) (6 + 1)... III. Adding I, II and III, (x + 1) (6+1) (C+1) "x 2 (a + y + z) = (+ 1) (EUR +1) +(c + 1) (a + 1) + (a +1) (0+1) * 8(way) ... ... ... ... IV Sabtracting separately I, II, III, each multiplied by 2, from IV, we have (a + 1)(b + 1) (+1) X 2 = S-3(6+1) (c+1), (a + 1)(8 +1 (+1) x 2y = 8 - % (c + 1) (a + 1), ST (a +1) (6+1) (c+1) x 2:=S-3 (a +1) 86 + 1). -T .. :y::::8-2 (6+1) (6+1): 8-2 (c + 1)(a + 1): 8-2 (a + 1) (6+1). By removing the common factors, if any, it the right-hand side of the pro. portion, wo get at the smallest integral values of , y, #. This proportion is given in the rule as the formuls. It may be noted that the square root mentioned in the rule has reference only to the problem given in the stanzes 836-237. Correctly speaking, instead of "equare root", we must have '3'. It can be seen easily that this problem is possible only when the sum of any two of t he seti is greater than the third.

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CHAPTER VIMIXED PROBLEMS. 186 to be respectively) multiplied by the specified) multiple quantities (mentioned above); from the several products so obtained the (already found out) values of the moneys on hand aro (to bo separately subtracted). Then the sume) value of the money in the purse is obtained separately in relation to onch of the several moneys on hand). * An example in illustration thereof. 236-237. Three merchants suw (dropped) on the way a purso (uontaining money). Ono (of theni) said (to the others), "If I secure this purse, I shall become twice as rich as both of you with your monoys on hand." Then the second (of them) said, "I shall become throo times as rich." Then the 'othor, (the third), said, "I shall become five timos as rich." What is the valuo of tho monoy in the purse, as also tho money on hand (with cach of the three merchants)? Tho rule to arrive at the value of the moneys on hand am also the money in the pursu (when particular spocified fractions of this latter, added respectively to the moneys on hand with each of a given number of persons, make their wealth becomo in each case) the saniy multiple of the sum of what is on hand) with all (the others) : 238. The sum of (all the specifiod) fractions in the problom) ---the denominator being ignored--is multiplied by tho (speci. tied common) multiple number. From this product, the products obtained by multiplying each of the ubove-mentioned) fractional parts as reduced to a common denominator, which is then ignored), by the product of the number of cases of persons minus one and the specified multiple number, this last product being diminishod 288. The formula given in the rule is =m (a + b + c)-a (2m-1), where x, y, are the money on hand, m yom (+ b + c)- (2 -1), the common multiple, and a, b, c, the and =m(a + b + c)-c (2m-1), specified fractional purts given. These values can be easily found out from the following equations : Pa +2=m (y + 2), 1 Pb + y =m (8 + x), whero P is the money in the purse. and Pc+s="(* + y))

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156 GANITASABASABARAHA. 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 to respectively) of the contents of the purse, they would cach becomo with what he had on hand three times as wealthy ay all the remaining others with what they had on band 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 measuro of the money contents of & purse, when specified fractional parts (thereof added to what may be on hand with one among a number of persons) makes him & 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 viow) are (reduced to a common denominator, which is ignored for practical purposes. These are severally) multiplied by the specified multiple number (rolating to the person in view). To these products, the fractional part (relating to the person) in view (and treated like otber 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 (80 obtained in relation 241. The formula given in the role is la + mb a + med + md -1*+1+2+1 +*+T+ ...--- 2)} + (m + 1) 10+ na + nuc 6+nd Im +19+1++ I ...-(-2) -(+ I) and so on; where #, y,... are moneys on hand; a, b, c, d,, fractional parts; m, , 4 ,.. various multiplo numbers; and the number of persons concerned in the transaction.

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CHAPTER VI-MIXED PROBLEMF. 167 to the several casos) are diminished by the product of the partioular specified fractional part as multiplied by the number of cuses loss by two. The difference is divided by the particular specified multiple quantity as increased by one. The result is tho monoy on hand (in the particular ouse). Examples in illustratim thereof. 242-243. Two travellers saw & purse containing money (dropped) on the way. One of the said to the other), "By scour. ing half of this money in the purse), I shall become twice as rich (as you)." The othor said, " By scouring two-thirils (of the money in the purso), I shall, with the money I have on hand, have threo times as much monoy ng wbat you have on hand." What are the moneys on hand, and what the money in the purso? 241-2441. Two travellers saw on the way I purse containing monoy; and the first of tbem took it up and said, that) that money along with the money that he had on hand horamo twice tbo money of the other (traveller. This) other said that that nionay in the purse with the aid of what he had on hand would be) throo times (the money in the hand of the first traveller). What is the money on hand in the case of cuch of them), and what the money in the purse? 2454-247. Four mon anw on the way a purse containing money. The first among them said, "If I seeure this purxc, I shall with the money already on hand with me become (poemessed of money which will be) eight timno' (the money on hand with the remaining travellera)." Anothor said, that the money in the purso with what he had on hand) would be nine times tho monoy on hand with the rest (among them). Another (said that similarly ho) would he possessed of ton times the mouny, 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 band of each of them was. 248. Four men saw on the way & Purs containing money. (Then), with what each of them had on hand, the 3, 4, ), and parts (respectively) of this (monoy in the purse) became twice,

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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 (bocome possessed of) thrice (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 you)." Tell me quickly, O mathematician, what the money on hand with each of them was, and what was the money in the purso. The rule for arriving at the money on hand, which, with the moneys begged (of othors), becomes a specified multiple (of the money on hand with the others) : 2513-252;. The sums of the moneys beggod are multiplied each by its own corresponding multiple quintity 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 quautities as increased by one. (The rosulting quotients themselves should be understood to be the moneys on band with the various persons). 2511-2523. Algebraically, (a+b) (m+1) +mc+d) (x + 1). (a+b) (m + 1) +m (f+ 9) (p+1) *+1 p+1 etc.- (0-2) (a+b) (on +1) } + (m+1)]+ Similarly for y, , etc. llere a, b, c, d, f, g, are sums of money beggod of each other.

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CHAPTER VI-MIXED PROBLEMS. 169 Examples in illustration thereof. 253-255). Three merobants begged money from the hands of each other. The first begged 4 from the scoond and 6 from the third man, and became possessed of twice the money (then on hand with both the others). The second (merobant) 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 tho others togethor). The third man beggod 5 from the first and 6 from the second, and (thus) became 5 times (as rich as the other two). O mathematician, if you know the mathematical proces known as citrakuttikara-mikra, tell mo quickly whnt may be the moneys they respectively bad on hand. 256-2584. There were threo very clever porsone. They begged money of each other. The first of them begged 12 from the second and 1: from the third. and became thus 3 times no rich as these two were then. The second of them bruged 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 socond and 10 from the first, and becamo (similarly) 7 times as rich. Their intentions wero 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 tbo penultimate man an anonnt equal to his own, (and again on this mau doing the namo in relation to the man who comes behind him, and so on): 2591. One divided by the optionally chosen multiple quantity (in respect of tbe amount of money to be given by the one to the other) becomes the multiple in relation to the ponultimato man's amount. This (multiplier) increased by one becomes the multiplier of the amounts (in the hands of the othors. The 2694. The rule will be clear from tho following working of the problem given in st. 263 : 1+ or 3 is the multiple with regard to the penultimate man' amount; this I combined with 1, s.e., 3 becomes the multiple in relation to the amounts of the other.

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GANITASARASANGRAHA. amount of the last person (so arrived at) is to have one added to it. This is the process to be adopted. 160 Examples in illustration thereof. This middle son 2604-261. 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. 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 what 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. Five 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 Now Multiplying the penultimate 1 by 2 and the other by 3, we get Adding 1 to the lust Write down Multiply the penultimate 4 by 2, and the others by 3, and add 1 to the last... Again Repeating the same operations as above we *** ... get *** ... ... ... ... 1,1. 2, 3. 2, 4. 2, 4, 4. 6, 8, 13. 6, 8, 13, 13. 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. Algebraically a + b = fd fo <=c+a=/ =1; where, a, b, c, d, fare the amounts on hand with the 5 merchants.

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CHAPTER VI-VIXBD PROBLEMB. 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 bad on hand, to thoso who were next younger to them exactly two-thirds (of what they respectively had on hand). Afterwards, they all became possosaed of equal amounts of money. What were the amounts of money thoy severally) had on hand to start with)? The rule for arriving at cqual amounts of money on hand, aftor a number of persons give each to vio others among thoin as much as they (respectively) huve (then) ou hand :-- 2654. One is divided by the optionally chosen multiple quantity (in the problem). (To this, the mumber corresponding to the men (taking part in the transaction) is added. The first (man's) amount (on hand to start with is tbus arriveil nt). This (and the results thereafter arrived at) aro written down (in order), and each of thom is multiplied by the optional multiple number as increased by one ; and the result is then diminished by one. (Thus the money on band with each of the others to start with is arrived at). Erumplex in illustration thereof. 266}. Each of three merchants gavo to the othors what cach of these had on hand at the time). Then they all became pokroskod of equal amounts of money. What are the amounts of money wbich they (respectively) lad on hard (to start with)! 265+. 1 he rule will be clear from the following working of the problem given in st, 266) : 1, divided by the optionally chosen inultiple l, and increased by the nomilier of perdona, 3, vivant; this is the money in the hand of the first man. This 4, moltiplied by the optionally chomin multiplo, I, am increased by 1, becomes 8; when 1 in sabtracted from this, we get 7, which is the money on hand with the second person. Tbin 7, ngain, treated an above, 1.c., multiplied by 2 and thon diminimbed by 1, gives 13, the money on hand with the third man. This solution can be easily arrived at from the following equation -- +(a - b -c)= 2 26 - (a - b-c) - 2 = 40 - 2 (a - -c) -{26 - (4 - 5 - c) - 3: 31

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162 GANITASARASANGRAHA. 2677. 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 Auccess and failure (in a gambling operation) :-- 2684-2694. 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. Tho (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 sct) 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 thoso arrived at in relation to the second set are written down) in the form of numerators: (and the difference between tho denominator and numerator in cach set is noted down). Then by means of these differenoes 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 inverso order the measure of the moneys on hand (with the gamblers to stake). . An example in illustration thereof. 270-2721. A great man possessing powers of magical charm and medicine saw a cock-fight going on, and spoko separately in (a + b) a 2681-2695. Algebraioally, (c + d) b (c + d) 6 - (a + b) * * * * T. * p, and y = and y are the moneys on hand with the gamblers G + b) d - (+ d) * P, where fractional parts taken from them, and y the gain. This follows from

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OHAPTER VI-XIXRD PROBLBX6. 188 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-lonoy on similar conditions). From both of them the gain to him could live only 12 (gold-pieces in cach case). You tell me, 0 ornament on the forehead of mathematicians, the (values of the stako-inonoy which each of) the cock-owncra had on band. The rule for separating tho (unknown) dividend number, tho quotient, and the divisor from their combinod sum : 2734. Any (snitable optionally chosen) number (which has to be) subtracted from the (given combinod rum happous to be tho divisor (in question). On dividing, by this (divisor) as increased by one, the remainder left after subtracting the optionally choson number from the given combined sum), tho (required) quotient is arrived at. Tho vory samo remainder (nhove mentionell), as diminished by this quotient becomes the required dividend) number. An example in illustration thereof. 2745. A certain unknown quantity in divided by a certain (other) unknown quantity. The quotient bere as combined with the divisor and the dividend number is 53. What is that divinor, and what (that) quotient ? The rulo for arriving at that number, which becomes a squaro 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 assoointed with whatever may happen to be the excess above the even number (nonrest to 2751. Algebrsically, 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)2 -1 +1+a-ti 2

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164 GANITASARASANGRAHA. that sum). The resulting product 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) 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 than 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. 276. 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. 277. 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. 2784. A certain quantity diminished by 3, or again that same (quantity) increased by 4, yields the square root (exactly). Tell me that quantity quickly, O arithmetician, after thinking out what it may be. The rationals of this may be made out thus: (n + 1) n3 2 n + 1, an odd number; and (n + 2)2 -- n2 = 4 n + 4, an even number; where n is any integer. From 2+1, and 4 n + 4, the rule shows how we may arrive at n2 + a when we know 2n+1, or 4n+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 it and respectively then these quantities are multiplied by (21), so as to yield 294 and 189, which are assumed to be the b and the a in the problem. The result arrived at with these assumed values of b and a is divided by (81), and the quotient is taken to be the answer of the problem.

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CHAPTER VI-XIIBD PROBLEMS. 186 The role for arriving at the square root of an unknown) number as inoroased or diminished by a known munber : 2797. The known quantity which is given is first halved and (then) Squared and then one is added (to it). The reulting quantity wither when incroased by the desired given qnantity or when diminished by the (same) quantity yiolds the square root (exactly). . . An example in illustration thereof. 2804. Here is a number which, whou increased by 10 or diminished by the same 10, yields an exact square root. Think out and tell me that number, O mathematician. l'he 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 (tho rame known number as forming the value of the squaro root of the difference (between these products): - 281}. The given number is increased by one; and the given number is also diminished by one. The resulting quantitice when halved and then squared give rise to thu two (required) quantities. Then if these be (separately) multiplied by tho given quantity, the syuro root of the difference betwoon those (productos) becomes the given quantity. An example in illustration thercof. 282-283. Two unknown squared quantities aro 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 calculation known as citra-kutsikara, calculate and tell me what (those two ankuown) quantities are. 2797. This is merely a particuler cane of the rule given in ntaosa 2767 wherein a is taken to be equal to b. 281). Algebraically, wben the given number ind are the required quero quantities.

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186 GANITABARASANGRAHA. The rule for arriving at the required increase or decrease in relation to a given multiplicand and a given multiplier (80 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 desirod) is (severally) divided in the inverse order by the gums (resulting thus). These give rise to the quantities that are to be added (respectively to the givon multiplicand and the multiplier) or (to the quantities that are to be respectively) subtracted (from thom). E.camples 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 (rospectively to the multiplicand and the multiplier) here, or what to be subtracted from them)? The rule for arriving at the required result hy) the procese of working baokwards: 286. To divide where there has been a multiplioation, to multiply where there has beon a division, to subtract where thero 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 quantition !o be added or subtracted sre-- d~ad and -46. d+ b a +i For (a + b ) (0+ 4+ )=d, where a sad b aro the given factors, where a and b are the given factors, and a the required multiple,

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CHAPTER VI-MIXED PROBLEM. 167 CHAPTER VIMIXED PROBLEMS. 167 divided by , (then) halvod, and then reduced to its square root, happens to be the number 5 ? The rule for arriving at the number of arrow 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 (rosulting sum) and add again three to this square quantity). If this be further divided by 12, the quotiont becomes the number of arrows to be found in the bundle. An crample in illustration thereof. 289. The circumferential orrows are 18 in number. llow many in all) are the arrows to be found in the bundlo) within the quivur ? () mathematician, give this out if you have taken paine in relation to the process of calculation knowit ar vicitrakuttikara. Thus ends vicitra-kuttikara in the chapter on mixed problems. 288. The formula here given to find out the total number of arrOWN IN (n+3)' + 3 where in the numbor of circumferentini arrow. This formulit can 12 be arrived at from the following considerations. It can be proved krometrically tbat only six circles can be doncribed ronnd another circle, 11 of them being equal and each of them touching its two neighbouring circle HR well All the central circle ; that, round theme circles again, only twelve circle of the same dimotinion can be describedl sirnilarly; and that round the ongoin, only 18 euch circles are possible, and no on. Thus, the first round him ii circles, the second 12, the third 18, and so on. 8o that the twmber of circle in any round, 18y p, is equal to p. Now, the total number of circles in the given number of rou , calculated from the central cirsle, in 1 +1 6+ 2 + 3 H + .. . + y * B 1 + 6(1 + 2 + 3 + ... + p) = 1 + 2?*: 1 + 3 p (p+1). If the 2 . + 1). value of Op is given, say, *, the total number of circlon in 1 + 8x which is easily reducible to the formula given at the beginning of this note.

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168 GANITASARASANGRAHA. Summation of Series. Hereafter we shall expound in (this) chapter on mixed problems the summation of quantities in progressive series. 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 h alving 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 negativo 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 difforence 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 produot obtained by multiplying the common differenco by the hali of the number of terms in the sories 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 (theroin). 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 differenoo. 390. Algebraioally #, where w is the number of terme, a the first term, the common difference, and the spin of the series. 992. Algebraically, a = : - 1 0; and 16-)?

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CHAPTER VI-XIXRD PROBLEMS. 169 Examples in illustration thereof. 293. The sum of the sories is 40; the number of terms is 5; and the common difference is 3; the first torm is not known now. (Find it out.) When the first term is 2, find out the cominon difference. The rule for arriving at the aum and the number of terms in a series in arithinetical progression (with the aid of tho known labha, which is the same as the quotiont obtainod by dividing the sum by the unknown number of terms therein) : 294. The libhre in diminished by the first term, and (thon) divided by the half of the common differonco; and on adding one to this samo resulting quantity), the number of terms in the serios (is obtained). The number of torins in the series multi plied by the labha becomos the sum of the series. An example in illustration thereof. 295. (There were a nuinber of utpali flowers, representable a: the sum of a serios in arithmotical progression, whoroof) is the first term, and 3 the common difforence. A number of womon divided (these) utprla flowers (equally among them). Each woman had 8 for her share. How many were the women, and bow. 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) qnarod ; (this squared quantity is multiplied by two, and (then) diminished by the given quantity as inorcased by one. (The remainder thus 284. Algebraically, + 1, where l= , which is the libha. 295. The number of women in this problem is conceivod to be equal to the number of terms in the series. { 2 (n+1)!-(n+1) 990. Algebraioally, 38, which is the sum of the squares of the natural numbers up to th.

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170 GANITASARASANGRAHA. 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 nuinber). 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 mo 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. (Toll 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 aritbmotical progression, whereof the first term, the oommon difforonco, and the number of terms are giveu : 298. Twice the number of terms is diminished by one, and (then) multiplied by the square of the common difforence, 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 Bum multiplied by the number of terms becomes the sum of the squares of the torms 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 differecoe, and (also) by the number of torms as diminished by one. This 22 298. [(*.7730* + ab }(n-1)+ a* fn = sum of tho squares of the termo in Rories in arithmetical progression.

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CHAPTER VI-MIXED PROBLEMS. 171 product is divided by sir. (To this resulting quotiont), the square of the first term and tho (continued product) of the number of terms as diminished by one, the first term, and the common differenoe, are added. The whole of this multiplied by the number of torms becomes the required result. . Examples in illustration thereuf. 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 squaros (of tho torms) in the series. (Similarly, in another series), 5 is the first torm, 3 the common difference, and 7 the number of terms. What is tho sun of the squaros (of the torms) in this series? The rule for arriving at the sum of the uubos (of a givo number of natural numbers) :: - 301. The quantity represented by the square of half tho (given) number of torms is multiplied by the wuare of the sun of one and the number of terms. Tu this (nience of) arithmetic, this result is said to be the sum of the cubos (of the given number of natural numbers) by thoso who know the socrot of oalculation. Erumples in illustration thereof. 302. Give out (in ench case the sum of the cubes of (tho natural numbers up to) 6, 8, 7, 25 and 250. The rule for arriving at the sum of the cubes (of the torme in a series in arithmetical progression), the first torm, the commou difference, and the number of terms wkeroof aro optionally chosen : 303. The sum of the simplo terms in the given sorios), as multiplied by the first torm (therein), is (further) multiplied by the 301. Algebraically =), which is the sum of the cubes of the natural numbers up to n. 303. Algebruioally, tva (a b) + b = the sum of the cubox of the terms in a series in arithmetical progression, where I = the sum of the simple tornis of the series. The sign of the first tertu in the formula is + or - according as .> or Page #374
172 GANITASIRASANGRAHA. difference between the first term and the common diffurence 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 oommon 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 oubes when the first term is 3, the common difference 2, and the number of terins 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 sumns of the natural numbers (from 1 up to a specified limit, these limiting numbers being the terms in the given sories in arithmetioal progression) : 305-305}. 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 differeuce. The sum (so obtained) is multiplied by the number of terms as diminishod by one and then increased by the product obtained by multiplying the first term as inoreased by one by the first torm itself. The quantity (80 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-R057. Algebraically: [ {{Sm = 2%B9+ + a0 }(m- 1) +0 (4+1)]; is the . bum of the series in arithmetical progression, wherein esob term represents the om of a series of natural numbers up to a limiting number, whiob is itself a member in a series in arithmetioal progression. P4, s .

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CHAPTER VI-MIXED PROBLEMS. 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. 178 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 (thon) by seven. From the (resulting) product, the given number is subtracted; and the (rosulting) 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 quantitios, 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) :-- 309. The number of terms is combined with three; it is (then) multiplied by the fourth part of the number of terms; (then) one *x (n + 1) x 7 2 307. Algebraically, x (n + 1) is the sum of the four quantities specified in the rule. 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 n; and (iv) the cube of n. 809. Algebraically, {(n +3)+1(+) in the collective sum of the sums, namely, of the sums of the different series dealt with in rules 20, 801, 805 to 805 above, and also of the sum of the series of natural numbers up to n. *

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174 GANITASA RASANG RATA. 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. 3101. What would be the required) collective sum in relation to the (various) serios represented by each of) 49, 66, 13, 14, and 25 ? Tho rule for arriving at the sum of a series of fractions in geometrical progression :-- 311). The number of terms in the sories) is caused to be marked in a separate column) by sero and by one (respectivoly), corresponding to the oven (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 inultiplied so as to obtain the square (whor. ever zero happens to be tho denoting item). Tho result (of this operation) is written down iu two positions. (In one of thom, 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 (80 obtained).when divided by one as diminished by the common ratio, gives rise to the required sum of the series. Examples in illustration thereof. 3127-313. 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 tern is , the common ratio is - - - - -- --- -- .. . 311. In this rolo, the numerator of the fractional common ratio ia taken to be always 1. See stansa 94, Ch. Il and the note thereunder,

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CHAPTER VI-MIXED PROBLEMS. 176 7, 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 progrossion here is. The rule for arriving at the sum of a series in geometrical progression wherein the terms are either inoronsod or decreased (in a specified manper 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 Bums so written down) is divided by the given) first torm. From the resulting) quotient, the given) number of torme is subtracted. The (rosulting) 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 eum 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 adiled, then the sum of the series in geometrical progression written down in the other position) bas to be increased by tho resulting quotient (already referred to. The result in either caru gives the required gum of the specified series). Examplex in illustration thereof. 315. The common ratio is 5, the first term is 2, and the quantity to be mulded (to the various termy) iw 3, and the naniber of terms is 4. () you who know the secret of caleulation, think out and tell me quickly the sum of the seriod in geometrical progression, wherein the terms are increased by the specified quantity in the specified manner). 814. Algebraically, +(5-) m + (1 - 1) + o in the sum of the series of the following form : a, arm,(arn) m).+m and so on.

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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). 176 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 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) : 817. See stanses 80---82 in Ch. II and the note relating to them.

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CHAPTER VI--MIXED PROBLEMS. 177 1 "B) 319. The unchanging velocity is diminished by the first torni (of the velocities in sories in arithmetical progression), and is (then) divided by the half of the commou difference. On adding one (to the resulting quantity), the (required) time (of moeting) is arrived at. (Where two porsons travel in opposite directious, cach with a detinite velocity), twice (thio Avoruge distanco to bo covered by cjther of thom) is the whole) way to he travelled). This when divided by the sum of their volocitios gives rise to the time of (their) meeting. An example in illustration thereof. 320. A certain person yoes with a velocity of 3 in the beginning increased (rogularly) by 8 as the (nuccosive) conmon differonoo. The stond y luchanging velocity (of another person) is 21. What may he the time of their meeting (again, if they start from tho simo place, at the same time, and move in the same direction) An example in illustration of the latter half (or the rule yiten in the stanza alove). 321-321. One man travels at the rate of Oyonnux and another at the rate of 3 yojiinas. The average) distance to be covered by either of them moving in opposite directions in 108 yojanas. O arithmetician, toll me quickly what the time of their meeting toyether is. The rule for arriving at the time and distance of merting together, (when two persous start from the sume place at the same time And travel with varying) velocities in arithmetical progression. 322). The difference between the two first terms divided by the difference between the two common differences, when multiplied by furu and increased by one, gives rise to the line of coming together on the way by the two person travelling simultaneously (with two series of velocities varying in arithmetical progression). + 1 = 1, whero in the unchanging volucity, 319. Algebraically(-) + and the time. 3924. Algeyraioally, n=% %* 2+ ). 23

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178 GANITASARASANGRAHA, An example in illustration thereof. 3239. A person travels with velocitios beginning with t, and Increasing (successively) by the common difference of 8. Again, a second person travels with velocitios 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 caso) being positive, and in the other) negative: 3244. The difference between the two first terms is divided by half of the sum of the numbers representing the two (given) common differenoes, and (then) one is added (to the resulting quantity). This becomes the time of meeting on the way by tho two persons (starting at the same time and) truvelling simultaneously (with velooities in arithmetical progression, the common difference in the one casc being positive and in the other negative). An example in illustration thereof. 325. The first man travels with velocities beginning with 5, and increased (sucocesively) 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 personis, (starting at different times and) travelling (respeotively) with a quioker and a less quick velocity (in the same direction) : 3264. He who travels less quickly and he who travols more quickly-both move in the same direction. Wbat happens to be the distance to be overtaken here is divided by the differenco between those two velocities. Iu the course of the number of days reprosented by the quotiont (here), the more quickly moving person goes to the less quickly moving one. 324). Compare this with the rule given in 322above.

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CHAPTER VI-MIXED PROBLEMS. An example in illustration thereof. 327. A certain person travels at the rate of 9 yojanas (a day); and 100 yojanas 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 (mossenger) meet him? 179 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): 328. The number of the circumferential arrows is increased by three and (then) halved. This is squared and (then) divided by three. On adding one (to the resulting quantity), the num ber 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). case. Examples in illustration thereof. 329. 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 arith metician, the number of the circumferential arrows also in this The rule for arriving at the number of bricks to be found in structures made up of layers (of bricks one over another) :--- 330. 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. n--1 xax 830. Algebraically, 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. * (n + 1) (" 2 is the total number of

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180 GANITASARASANGRAHA. Examples in illustration thercof. 331}. 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 caloulation of mixed problems, tell me how many bricks there are (hero in all). 332. Thore is a structure built up of successive layers of bricks, which is in the form of the nanlyavarta. There are 4 layers built symmetrically with 60 (as the numerical measure of the top-brioks in single row). Tell me how many are all the bricks (here). Rules regarding the six things to be known in the scienco of prosody : 3333-336. The number of syllables in a given syllabic inetre or chandas is causod to be marked in a soparate column) by zero and 332. The nandyavarta figure referred to in the stansa is 333-330. As each ayllable found in a lino forming a quarter of a stanza may be aliort or long, there arises a number of varieties corresponding to the different arrangements of long and short Ayllables. In arranging those varieties, a certain order is followed. The rules given here enable us to find out (1) the number of variotics 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 syllablos, (6) the number of varieties containing & specified number of long or short syllables, and (8) the nmoont of vertical Apace required for exhibiting the varieties of a particular metre. The rules will become clear from the following working of the problems given in stanza 337) :-. (1) l'hore are 3 ayllables in a metre; now, we proceed thus: Now, multiplying by 8 the figures in the 3-1 1 2 % 0 right-hand chain, we obtnin 0. By the procons of 1-1 1 2 multipliontion and squaring, an explained in the note to stanza 94, Ch. II, we get 8; and this is the number of varieties. (2) 'The manner of arrangement of the syllables in each variety is arrived at ttus lat variety : 1, being odd, donotes a long syllable; so the first syllablo is long. Add 1 to this 1, and divide the num by 3; the quotient is old, and denotes another long syllable. Again, 1 in m od to this quotient 1, and divided by 2; the result, 2

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CHAPTER VI--XIXED PROBIENS. 181 by one (respectively), corresponding to the even (value) which is balved, and the uneven (value from which one is subtractod, till by continuing these processes zero is ultimately roached. The numbers in the chain of figures Bo obtained are all doublel, (and then in the process of continued multiplication from the bottom to the top of the chain, those figures which come to live a to above them)" are. squared. The resulting product (of this continue multiplication gives the number of the varieties of stanzas possible in that syllabic metre or chandas). The arrangement of short and long Avllables in all the varic. ties of stanza: 80 obtained) is shown to be arrived at thus :- . (Tho natural numbers commencing with me and cuding with the measure of the maximum number of porrible tauzas in the given metro being noted down), every odd number (therviu) has onr added to it, and is (then) halved. Whenevor this process is gone through), a long syllable is decidedly indicated. Where ognin odd, lenoto's third long wlable. Thus the first variets copaints of three long asllablom in indicated thos II. 2nd variety : 2, being aveu, indicates a whort wylo bilo; whon this in divided by the quotient in I, which bring udd indicates long syllable. And I to thin 1, and divide the sun by 2, the quotient being odd imicates a long wylable: thun Similarly the other six varieties are to be found out. (3) The fifth variety, for instance, may be found out an above. (4) To find out, for instance, the ordinal powition of the variety, wo proceed thus:-. Below these syllables, write down the ternit of a sories in geomotricul poro Krenrion, liaving 1 as the first form and 2 on the common ratio. Add the I l figures 4 and 1 under the the short syllable and incrone the sum ly 1, 1 9 4 we get 0: and we, therefore, way that thim in the sixth variety in the tri-syllabic metre. 16) Suppose the problem in: How many varieties onntnin? short myllables ! Write down the nntoral numbers in the regular and in the informe order, OH below the other thur: 1. Taking two terme from right to left, both from 1. Taking two terma from above and from below, we divide the product of the former by the prolurt of the Inttor. And the qoutient 3 is the Annwer requirol. (C) It is prescribed that the symbole reprcmenting the long and short Myllables of any variety of metre should occupy an argula of vertical space, and that the intervening space between any two varieties should also be an angwla. 'The amount, therefore, of vertical jare required for the varieties of thm metre is 2 x 8-1 or liangulas.

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GANITASARASANGRAHA. the number is even, it is (immediately) halved and this indicates a short syllable. In this manner, the process (of halving with or without the addition 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 syllables 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). 182 Where (a stauza of a particular varioty is given, and) its ordinal position (among the variotics of stanzas possible in the metre) is to be found out, the terms (of a series in geometrical progression) commencing with one and having to 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) bolow the position of short syllables are all added; the sum (so obtained) is increased by one. (This gives the required ordinal number.) Natural numbers commencing with one, and going up 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 yarieties of stanzas in the given metre, with 1, 2, 3 or more) short or long syllables (in the verse).

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CHAPTER VI-NIXED PROBLENS. 183 "The possible number (of the varieties of stanzas in the given metre) is multiplied by two and (then) diminishod by ono. This rosult gives (the measure of what is onlled) zdhvan, (wheroin an interval oquivalent to a stanza is conocived to oxist hetwoon every two guccousive varieties in the metre). Examples in illustration thereof. 337. lu 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 mtanzas in the metre). (2) the manner of arrangement (of the syllables in those wtanzan), (3) the arrangement of the syllables (in a given variety of the stunza, the ordinal position whereof among the possible varieties in the metre is known), (4) the ordinal position (of a given tunza), (6) the number (of stanzas in the given motro containing any given number) of short or long syllables, and (6) tho (quantity known as) adhran. Thus ends the process of summation of crics in the chapter on mixod problems. 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. CALOULATION RELATING TO THE MEASUREMENT OF AREAS. 1. For the accomplishment of the object held in view, I bow again and again with true earnostness to the most excellent Siddhas who have realized the knowledge of all things. Hereafter we shall expound the sixth variety of calculation forining the subject known by the name of the Measurement of Areas. And that is as follows:-- 2. (The measurement of) area has been takou to be of two kinds by Jiun in avvordance with the nature of) the result, namely, that which is (approximate) for practical purposes and that wbicb is minutely accurate. Taking this into consideration, I shall clearly explain this subject. 3. (Mathematical) teachers, who have reached the other shore of the coug of calculation, have given out well (the various kinds of) areas as cousisting of those that are trilatorul, quadrilateral aud curvi-linear, being differentiated into their respective varieties. 4. A trilateral arou is differentiated in three ways; a quadrilatoral one in five ways; and a curvilincar one in eigh ways. All the romaining (kinds of) areas are indood variations of tho varieties of these (ditferent kinds of areas). 5. Learned men sily that the trilateral arou may be equilatoral, isosceles or scalone, and that the quadrilateral aren also may be 3 and 8. "The various kinds of unclosed areas gentioned in these lansas are illustrated below:--- Swmatribhuja = Equilaterul trilnteral figure. Dvissmatribhaja= laundeles trilatural Agaro Visamatrivhuja = Boulene trilloral figaro.

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OHAPTER VII--MBASUREMENT OF ARBAS. .186 equi-lateral, equi-diohastic, equi-bilateral, equi-trilateral and in. equi-lateral. Samncaturasra = Equilateral quadrilateral. Dvidvinamacatures - Equi-dichaatio quadrilatorni. Dvinamacaturaara Eybi bilateral Trinamacaturuara Equi-trilateral quadrilateral quarilateral. Vifamacaturuara = Inequi-lateral quadrilateral. Nomavrttr=Circle. Ardhavrtta = Bemloirole. Ayatartta - Elupu. 24

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186. GANITASARABANGBAHA. 6. (The ourvi-linear area may be) a circle, a semicirole, an ellipse, a conchiform area, & concave circular area, a convex oironlar area, an out-lying annulus or an in-reaching annulns. Ooi Kambukartta = conchiform Nimnavrtta = QUOAVO area. cironlar area, Unnatavrtta = convex circular area. Bahiboakravalavrtta = Out-lyiug annulae. Antabcakravalavrtta =Ine reaching sanulos. From a consideration of the rules given for the measurement of the dimensions and areas of quadrilateral Agures, it has to be concluded that all the quadrilatorul figures mentioned in this ohapter are oyolio. Hence an equilateral quadrilateral is. qnare, an equidichastio quadrihteral is an oblong; and equ. bilateral and eqal-trilateral quadrilateralo have their topaide parallel to the bene

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CHAPTER VII-MEASUREMENT OF AREAS. 187 * Calculation relating to approximate measurement (of areas). The rule 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 annulus like) the rim of a wheel, half of the sum of the (inner and outer) oircumferences 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 orescent 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 calculating, 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 arca.) 7. A trilateral figure is here conceived to be formed by making the topside, i.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 opposite 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 rale cannot be acon. rate 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. Nomi is the area enclosed between the circumferences of two concentric oircles; and the rule here stated for finding out the approximate measure of the area of a Namikitra happens to give the accurate measure thereof, In the case of a figure resembling the crescent moon, it is evident that the result arrived at socording to the rule gives only an approximate measure of. the area.

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188 GANITASARASANGRAHA. 10. In the case of a scalone 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 (figare) P 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) thd 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 quiokly, 0 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 oqual 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 figuro ?) 15. In the case of a quadrilateral figure with three equal sides, (each of these) three sides measures 108 dandas, the (romaining side here called) mukha or top-side measures 8 dundas and 3 hastas. Accordingly, tell me, 0 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 tase measures 38 dandas, the side forming the top is 32 dandas : one of the lateral sides is 50 dindas and the other is 60 dandas. What is the area of this (figure) P 17. In an annulus, the inner circular boundary ineasures 30 dandow; the outer cironlar boundary is seen to be 800. The breadth 11. The shape of the figure mentioned in this stansa sooms to be what is given here in the margin: it is intended that this should be treated as a trilateral fgare, and that the area thereof should be found out in acoordano with the rule given in relation to trilateral figuros.

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CHAPTER VIL MEASUREMENT OF ARBAS. 189 of the annulus is 45. What is the caloulated measure of the area of (this) annulus ? 18. In the case of a figure resembling the crescent moon, the broadth is seen to be 2 hastas, the outer curvo 68 hastas, and the inner onrve 82 hastas. Say what the resulting) area is. The rule for arriving at the (pructioally approximato value of the) area of the cirole : 19. The (measure of the) diamotor multiplicd by three is the measure of the circumference; and the number representing the squaro of half the diameter, if multiplied by thrre, gives the (resulting) area in the case of a completo virolo. Tenchers say that, in the case of a semicircle, half (of those) givo (respootively) the measure (of the ciroumference and of the arca). Examples in illustration thereof. 20. In the case of a cirolo, the diameter is 18. What is the circumference, and what the (rcsulting) area (thereof) In the case of a semicircle, the diameter is 18: toll me quickly what the calculated measuro is (of the area as well as of tho circumference). The rule for arriving at (tho valuo of) the area of an elliptical figure : 21. The longer diameter, increasod by half of the shorter) diameter and multiplied by two, gives the measuro of the cir-( cumference of the elliptical figuro. Onc-fourth of the (shortor) diameter, multiplied by the circumference, gives riso to the (measure of tho) area (thereof). 19. The approximate character of the measure of the circamforenco aw woll w of the ares as given bere io dao to the value of being taken 6 3. 21. The formula given for the circumference of an ellipms is evidently an approximation of different kind. The area of an ollip ir r. a.b, where 4 and I are the semieres. Ils is taken to be equal to 8, tbon r. a.b, = 8 a. But the formula given in the stansa makes the area equal to 2ab + 69.

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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) arca (thereof) ? The rule for arriving at the resulting) area of a conchiform curvilinear figuro: 23. In the case of a conobiform curvilinear figure, 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. 21. In the case of a conchi-form figure the breadth is 18 hastax, and the moasure of the month thereof is 4 (hastax). You tell me what the perimeter is and what the caloulated 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. Thonoo, in the case of ooncave and convex areas like that of a 23. If a in the diametor and m in the measure of the month, then 3 (a - m) is the measure of the circumference; and ( 41) and * 3* *13) is tho moscuro of the aroa. The exact shape of the figore is not clear from the description given; but from the values given for the circumference and the Arda, it may bo conceived to consist of 3 unoquel semicircles placed so that their diameters ooincido in position as shown in figure 12, givon in the foot-note to abansa 6, in this chapter. 25. The area here specified soems to be that of the surface of the segment of a apliere ; and the messare of the area is stated to be, when symbolically represented, equal tox d, where c is the circumference of the sectional cirole, and d is the diameter thereof. But the area of the surface of a spherical megment of this kind is equal to 2 r., where is the radius of the sectional oirolo aud * is the height of the spherical segment.

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CHAPTER VII-XEABUREMENT OF ARRAS. 191 ssorificial fire-pit and like that of (the back of) the tortoise, (the required result is to be arrived at). A8 example in illustration thereof. 26. In the caso of the area of a sacrificial firo-pit the measure of the diameter is 27, and the monsure of the circuni feronco is seea to be.56. What is the caloulatat measuro of the area of that same (pit) ? An example about a conver circular surface resembling (the back) of a tortoise. 27. The diameter is 15, and the circum furonon is soon to be 36. In the case of this area roscmbling the back of a) tortoiso, what is the practically approximate mousure as calculated ? The rulo for arriving at the practically approximato value of the area of an in-lging annular figure as well as of an out-ronching acnular figure : 28. The (inner) diameter inoroased by the breadth (of the annular aren) when multiplied by three and by the breadth (of tho annular area) gives the calculatod moasuro of the area of the ontreaching annular figure. (Similarly the measure of the calculated area) of the in-lying annular figure is to be obtainod) from the dia.neter as diminished by the breadth of the annular area). Eramples in illustration thereuf. 29. The diameter is 18 hastas, and tho'breath of the outreaching annular area is 3 in this case : the diamctor is 18 hastas and again the breadth of the in-lying annular area is 3 hastas. What may be (the area of the annular figuru in each case) ? 38. The shape of the F ACTA us well as of thu Ena is identical with the shape of the TTATGT mentioned in the note to stansa 7 in this chapter. Hence the rule givon for arriving at the area of all these figures works ont to be the same praotically.

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192 GANITABARASANGRAHA. The rule for arriving separately at the numerical measures of the ciroumference, 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 squaro 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 cirolo) is 1116. Tell me what the measure of the) circumference is, what (that of) the caloulatod area and what (of) the diameter is. The rule for arriving at the practically approximate value of surface-arons resembling (the longitudinal sections of the yava grain, (of) the mardula, (of) the panuva, and (of) the vajra : 32. In tho case of areas shaped in the form of the yava grain, of the muraja, of the parava and of the vajra, the 80. This rule will be olear from the following algebraioal representation - Let c bo the ciroumference of the circlo. As is taken to be oqual to 8, in the diameter and in 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 stanga to the effect that c= N12 m + 64 - 84 may be easily arrivod at from the qualratio equation containing the data in the problem :-- 6+ s + 3 8 =m. 33. Muraja means the same thing as marda la and mrdanja. The shape of the various figures montioned in this alansa is as follows: Yavakirakaotra. Morajakarakpotra. Papavilarak etn. Vajribirakpetra.

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CRAPTER VII-XBASURBMENT OF A REAB. 193 (require dmeasurement of) area is that which results by multiplying half the sum of the end measure and the middle moasure by the length Examples in illustration thereof. 33. In the case of an arcn resembling the coufiguration of a yava grain, the length is 80 and the breadth in the middle is 40. Tell me, what may be the oaloulated measure of that arca P 34. Tell (me what may be the calculated measure of the aron) in relation to a field which has the outline configuration of tho mrdanga, and of which the length is 80 dandax, the end measuro is 20 and the middle measure is 40 dandas. 35. In the case of a field having the outline of tho panava, the length is 77 dandas, the measure of each of the two ends in 8 dandas, and the measure in the middle is 4 dandas. (What is tho measure of the area P) 36. Similarly in the case of a field having the outline of the vajra, the lengtb is 96 dandas, in the middle there is the middle point ; and at the ends tho mensure is 134 dandns. (What is the meaeure of the aroa ) 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 longth into hall the breadth from the produot of the length into the breadth, you The measures of the area arrived at according to the rule given in this stany. are approximately correct in the case of all the Aguros, as the rulo is based on the wsumption that onch of the bounding curved lines may be taken to be equal to the rim of two straight lines formed by joining the ends of the curven with tho middle point thereof. 87. The figare mentioned in this stanse are those given below: These are looked apon w being derived from & quadrilateral Agaro which is divided into four triangles by moone of ita diagonale ronging each other. The .26

Page #396
194 GANITASAHASANGRAHA. deolare 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. 18-1 * 1/2 This " 39. The rule stated in this stanza gives the area of figures made up of any number of sides. If 8 is half the sum of the measures of the sides, and n the number of sides, the formula is found to give the approximato value of the area in the case of a triangle, a quadrilateral, a hexagon and a cirole conceived as a figure of infinite number of sides. The other part of the rale deals with the interspace bounded by parts of circles in oontact, 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. area is said to be equal to. 88 .

Page #397
CHAPTBR VI BABURKYBNT OF ARRA8. 196 inoluded between ciroles (in contact), one-fourth (of the result thus arrived at gives the required measure). Examples in illustration thereof. 40. In the case of a six-sided figure the measure of a sido is 6, and in the case of another figure of 16 sides the measure of a side i, 3. Give put (the measure of the area in each care). 41. In the case of a trilateral figure one of the sides is 5, the opposite (ie., the otbor) side is 7, and the base is 6. In the case of another bexalateral figure the sides are in measure from 1 to 8 in order. (Find out the value of the aroa in each case). 12. (Give out) the value of the interspaco inoluded inside foar (equal) circles (in contact) having a diameter which is 9 in measure ; and (givo out) the value of the area of the interspace inolnded inside three circles having diameters measuring 6, 6 And 4 (respeotively). The rule for arriving at the practically npproximate aroa of & field resembling a bow in outlino : 43. In the case of a bow-shaped field the calculatod measure (of the area) is obtained by adding together (the measure of the arrow and (that of the string and multiplying the sum by balf (the measure) of the arrow. The squaro root of the square of the (measure of the) arrow as multiplied by 5 and (then) as onmbined with the square of the (measure of tho) string gives the measure of the bent) stick" (of the bow). 43. The field resembling a bow in outlinu in in fact the segment of u oirolo, the bow forming the aro, the bow-string forming the chord, and the arrow mensuring the groatest porpondicolor distunov Lotween the aro and tho obord. It a, c, and represnnt the lengths of these throu liues, then, according to the rules given in stanga 43 and 45 - Area = c + p) - Length of bow =15' + c of arrow = 8 of bow-string van Bp* For mourate value sa stangas 781 and 741 in this chapter.

Page #398
196 GANITASARASANGRAHA. An example in illustration thereof. 44. A bow-shaped field is seen whereof the string-measure is 28, 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 aro) 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 calonlation, give out both these measures. The rule for arriving at the practically approximate value of the area of the circle which is oiroumsoribed about or inscribed within & four-sided figuro: 47. Half of three times (the measure of the area of the inscribed quadrilateral figure) gives the measure of the area of the oirole in the case in which it is circumscribed outside. In the case where it is inscribed within and the quadrilateral is the other way (1.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 quare, but only approximate in the one of other quadrilaterale, if 8 bo takon to be the correot valao otr.

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CHAPTER VII-XBASUREMENT OP ARKAS. 197 An example in illustration thereof. 48. In relation to a quadrilateral figure, each of whose sidos is 15 (in measure), tell me the practically approximato value of the inscribed and the escribed ciroles. Thus ends the calculation of practically approximato value in relation to areas. The Minutely Accurate Calculation of the Measure of Areas. Hereafter in the caloulation rogarding the mowurement of areas we shall expound the subject of treatment known as minutely acourato caloulation. And that is as follows: The rule for arriving at the measure of the perpondicular (from the vertex to the base of a given trianglo) and (also) of tho sogmonts into which the base is thoroby divided): 49. The process of sankramana carried out between the baro 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 (oither of these (segments) and of the (corresponding adjacent) eide gives rise to tbo mousure of the perpendicular. 40. Algebraically represonted-- 5= (+4*50*)**: Va' - or Vbi - C. Here a, b, c, represent the meannres of the sides of trianglo, G, C, the measures of the segments o! the browbone total length iso, and y represents the length of the perpendioular.

Page #400
198 GANITASARASANGRAHA. The rule for arriving at the minutely socurate measurement of the area (of trilateral and quadrilateral figures) - 50. Four quantities represented (respectively). by balf the sum of the sides as diminished by (each of) the sides (taken in order) are multiplied together; and the square root (of the produot 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 sam of the top measure and the base measure. (The lattor 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 is also of each of the two sides. You, who know caloulation, 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 (ouch of the) two (equal) sides measures 13 dandax, and the base measures 10. (What is) the accurate measure of the area thereof, and of the perpendi 50. Algebraioally represented : Aron of a trilateral figure = V (-a) (A-6) (a-c); where # is half the sum of the sides, a, b, c, the respective measures of the sides of the trilatorul figaro; P, whero p is the perpendioclar distance of the vertex from the baso. Aroa of a quudrilateral figure = V TR-a) (8-6) (0-c) (-d) where is half the sum of the vides, and a, b, c, d the measures of the respootive sides of the quadrilateral figure ; * p (except in the 0.40 of an inequilateral quadrilateral) where is the measure of either of the porpendicolars drawn to the base from the extremi. ties of the top aide. The formulas here given for trilateral figures are oorroot; but those given for quadrilatral figures hold good only in the case of oy olio quadrilaterals, as in these formulas vight is loat of the faot that for the same messore of the sides the value of the area as well as of the perpendioular may vary. or = 0+d

Page #401
CHAPTER VII-MEASUREMENT OP ARRAN. 199 onlar (to the base) as also of the segments (of the baso caused thereby)? 53. In the case of a soalene triangle one of the sides is 13 (in moagure), the opposite side is 15, and the base is 14. What indeod is the caloulated measure of the area of this figure), and what of the perpendicular (to the base) and of the basal segments Hereafter (wo give the rule for arriving at the valuo of the Jiagonal of the five varieties of quadrilateral figures. 54. The two quantities obtained by multiplying the basal sido by the larger and the smaller of the right and the loft) sides are (respectively) combined with the two (other) quantitics obtained by multiplying the top side by the smaller and the larger of the right and the left sides. T'ho (rosulting) two suns constitute tbe 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 thoroquired measures of the diagonals. Examples in illustration thereof. 55. In the case of a cqnilateral quadrilateral which has all around a sido measure of 5, tell me quickly, () friend who know the secret of caloulation, the value of the diagonal and also the acourate value of the rea. 64. Algebraically representod the meaRDM of the diagonal of quadrilateral figure as given here ik -- lac+bd) (ab + cd) /(ac + bd) (ad + bc) N ad + bc of N abi od These formula alan are oorrect only for oyclic quadrilaterals. Bhankara. cirys is aware of the futility of attempting to give the monsure of the wes of 4 quadrilateral without previously knowing the values of the perpendicular or of the diagonal.. Vide the following stanza from his Lild vati : lambayoH karNayorvekamanirdizyAparAn katham / pRcchatyaniyatatve'pi niyataM cApi tatphalam !! sa pRcchakaH pizAco vA vaktA vA nitarAM tataH / yo na veti caturbAhukSetrasyAniyatAM sthitim / /

Page #402
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 quiokly what the measure of the diagonal is and what the acourate 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 same. The top side is 14. What is the diagonal and wbat the socurate measufe of the area P 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 arca ? 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 5%, 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 minately acourate values relating to curvilinear figures. Among them tho rule for arriving at the minutely accurate values relating to a circular figure is as follows: 80. The diamotor of the circular figure multiplied by the square root of 10 becomes the circumference (in measure). The cimumference 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 oircle). Examples in illustration thereof. 61. In the case of one (cironlar) 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 ciroumferences and the areas ? 60. The value of a given in this stansa is vlo, which is equal to 8-16........ Compare this with the more approximate value in (= 8:1416) given by Aryabhata. Bhankaricdrys also given to it the mme valde, and represents it in reduced tormi sa

Page #403
CHAPTER VII-KRASUREMENT OF AREAS. 201 62. In the case of a semiciroular field of a diameter measuring 12. and of another) field having a diameter of 36 in measure what is the oiroumference and what the area ! The role for arriving at the minutely accurate valuos 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 tho circumforence 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. 84. In the case of an elliptical figure, the length (as moasured by the longer diameter) is 36, and tho broudth (as measured by the shorter diameter) is 12. Tell me, after caloulation, what the measure of the circumference is, and what the minutoly acourato measure of the area. The rule for arriving at the minutoly acourato valuos in relation to a conchiform figuro :-- 05. The maximum moasure of the) breadth (of the figure), diminished by half (the measure of the broadth) of the mouth, and (then) multiplied by the square root of 10, gives rise to the measure of the perimeter. The square of half tho (maximum) 69. Il a represents the measure of the longer diameter and b that of the shorter diameter of an ollipse, then, k'cording to the rule given here, the oir. onmference is V 66 + 4a, and the area in bx V 60' + tal. It may be noted that this stanse, u found in the M88., omits to mention that the square root of the quantity is to be taken for arriving at the value of the cironmferonoo. The formula for the area given here is only an approximation, at noon to be based on the analogy of the srea of a circle us represented by ad *-. where d is the diameter and rd in the circumforenco. 864. Algebraically, circumference = ( a m ) * 10; 2 arow = [ {6-4m) **}'+4)*]x vio; whero a ia tho monsivo of the

Page #404
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 conobiform figure. An example in illustration thereof. 66,. In the case of a conchiform ourvilinear 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 acourate measure of the area as caloulated P The rule for arriving at the minutely acourate measures in relation to outreaching and inlying andular 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. . 881. Eighteon dandas measure the inner or the onter) dia meter of the annulus (as the case may be); the breadth of the annulus is, however, 3 (ilandas). You give out the minutely accurate value of the area of the outreaching as well as the inlying annular figure. 694. 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 mesuure of the month of a conchiform figare. As observed in the note relating to stanna 23 of this chapter, the figure intepded in obviously made ap of two unequal semigiroles.

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80EUR CHAPTER VIMBABUREMENT OF AREAS. The rule for arriving at the minutely accurate values relating to & figore resembling (the longitudinal section of) the yara grain, and also to a figure having the outline of a bow : 701. It should be known that the measure of the string (chord) multiplied by one-fourth of the moasure of the arrow, and then multiplied by the square root of 10, gives rise to the (800urate) 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. 711. In the case of & figure resembling (the longitudinal) section of the yara grain, the (maximum) length is 12 landas; the two ends are noodle points, and the broadth in the middle is 4 dandas. What is the area ? 721. 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-moasure. What may be the minutely accurate value of the aroa P 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 : 734. The square of the arrow moasure is multiplied by 6. To this is added the square of the string measure. The square 704. The figare reseinbling a bow is obviously tho segment of a cirole. The area of the segment as given here = c * ?X v10. This formula is not accurate. It seems to be bred on the analogy of the role for obtaining the aron of semi-circle, which area is ovidently equal to the pro. duct of the diameter and one-fourth of the radius, 1.6., * 3r . The figure resembling the longitudinal soction of yava grain may be evil: men to be made up of two similar and equal segments of a circle applied to osol other so to have a common chord. It is evident that in this case the value o the arrow-line becomes doubled. Thus the samo formula is made to hold goo here sloo. 784 & 74. Algebraically, arc=v Ray perpendicular - chord Wap.

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204 GANITASABASANGRAHA. 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): -: 744. 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. 75. 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 76. To the rosulting 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 ohord and the largest perpendicular distance betweeen the arc and the chord, the aro forming a semicirole is taken as the basis, and the formula obtained for it is utilised for arriving at the value of the arc of any segment. The semicircular aro = x 10 10r 16r+: based on this is the formula for any aro; where p the largest perpendicular distance between the arc and the chord, and c the chord. 76. The rationale of the rule here given will be clear from the figures given in the note under stanza 39 above.

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CHAPTER VII- EASUREMENT OF ARIAS. 208 of those two (other) figures which resemble (the longitudinal section of the Panavil, and (of) the Vajra, that (same resulting area, which is obtained by multiplying the maximum length with the measure of the breadth of the month), is diminished by the measure of the areas of the associated bow-shaped figures. (The remainder gives the requird measuro of the area concorned.) Examples in illustration thereof. 774. In the case of a figure having the outline configuration of a Urdanga, the (maximum) length is 24; the breadth of (each of) the two mouths is 8; and the maximum) broadth in the middle is 16. What is the aroa ? 787. In the case of a figure having the outline of a Panava, the (maximum.) length is 24; similarly the measuro (of the breadth of either) of the two mouths is 8; and the central breadth is 4. What is the area ? 797. In the case of a figure having the outline of a Vajra, the (maximum) length is 21; 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 acourato valne of the areas of figures resembling the annulus making up) the rim of a whoel, (resembling) the crescent moon and the (longitudinal) section of the tusk of an elephant : 801. In the case of (a circular annulus rosembling) the rim of & wheel, the sum of the measures of the inner and the outer ourvos is divided by 6, multiplied by the measure of the breadth 80%. The role hero given for the area of an annulus, if expressed algebraically, comes to be *" * p* v 10, where a, and a, are the mouwdren of the two circumferences, and p is the measure of the breadth of the annulae. On comparison of this value of the area of the annulos with the approximate value of the same as given in stanza 7 above (vida note thereunder), it will be evident that the formala here does not give the mourato valge, the valde mentioned in the rule in stanza 7 being itsell the accurate value. The mistake seems to have arisen from wrong notion that in the determination of the value of this won, is involved even otherwise than in the value of a, and a

Page #408
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. 81. In the case of a field resembling (the circalar 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 orescent 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) oiroles (touching each other) : 82. 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 included within four equal circles (touching each other). An example in illustration thereof. 83. 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 figure below: 88

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CHAPTER VII- BASUREMENT OF ABRAS. 207 The rule for arriving at the minutely accurate value of the figure formed in the interspace caused by three (oqual) oiroular figures touching each other : 84). The minutely accurate measure of the area of an equilateral triangle, each side of which is oqual in measure to the diameter (of the ciroles) is diminished by half the area of any of the three equal) ciroles. The remainder happens to be the measure of the interspace are caused by three (mutually touching equal oiroles). An example in illustration thereof. 851. What is the minutely accurate oalculated value of a figure forming the interspace enclosed by three mutually touching (equal) ciroles the diameter (of each) of wbich is 4 in measure P 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 : 864. In the case of a (regular) six-sided figure, the measuro of the side, the square of the side, the square of the square of the , side multiplied respectively by 2, 3 and 3 give riso, in that same order, to the values of the diagonal, of the squaro of the perpendi. cular, and of the square of the measure of the area. 84). Bimilarly the figure here eluci. dates at once the reason of the rule : 861. The role seems to contemplato a regular hexagon. Tho formula given for the value of the area of the hexagon is 3a, where a is the length of sida The oorrect formula, however, is

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208 GANITAS RASANGRAHA. An example in illustration thereof. 877. In the case of a (regular) six-sided figure each side is 3 dundas in measure. In relation to it, what are the squares of the measures of the diagonal, of the perpendioular and of the minutely accurate area of the figure ? The rule for arriving at the numerical measure of the sum of a number of squnre 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 (ouch) a (common) faotor (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 caloulation in regard to (all kinds of) square root quantities. An example in illustration thereof. 891. O my friend who know the result of caloulations, 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 oalonlation of the measure of areas). 887. The word karans ocourring here denotes any quantity the square root of which is to be found out, the root itself being rational or irrational as the onse may be. The rule will be clear from the following working of the problem given in stanza 89: To find the value of 718+ 36+ 100, and V100-(186- 18). There ure to bo ropresented as va(v +19+w25), vi{w25- (Vo-VA)}. =V 4 (2+8+8); =v7{6 - 18 - 2). EN 4 (10), -Vix 100; =v7x18. . = 100;

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CHAPTER VII-XBASUREMENT OF AREAR. 209 Subject of treatment known as the Janya operation. Hereafter we shall give out the junya operation in caloulations relating to measurement of areas. The rule for arriving at a longish quadrilateral figure with optionally chosen numbers as bijas :-- 902. In the case of the optionally dorived longish quadrilateral figure the difference between the squares (of the bija numbers) constitutes the measure of the perpendicular-side, the product (of the bija numbers) multiplied by two becomos tho (other) side, and the sum of the squares (of the bija numbers) becomes the hypotenuse. Examples in illustration thereof. 911. In relation to the geometrical figure to bo derived optionally, 1 and 2 are the bijas to be noted down. Toll (me) quiokly after calculation the moasurements of the perpendicular-side, tho othor side and the hypotenuso. 927. Having noted down, 0 friend, 2 and 3 as the bijax in rolation to a figure to be optionally derived, give out quickly, after caloulating, the measurements of the perpendicular-wido, tho 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: 937. The product of the sum and the difference of the bijas forms the measure of the perpon lioular-side. The sankramana of 201. Janya literally means "arining from" or "ant to be derived", hence it rofors here to trilateral and qundrilateral figuren that may be dorivo out of certain given data. The operation knowo 48 junya relates to the finding out of the length of the sides of trilateral and quadrilateral figuram to be no derived. Bija, a given bere, generally happens to be a puitive integer. Two such are invariably given for the derivation of trilateral and quadrilateral tigures dependent on them. The rationale of the role will be clear from the following algebraicei representation : If a and bare the bija numbers, then a win the measure of tho perpendi. cular, 2 ab that of the other side, and a + b that of the hypotenone, of o oblong. From this it is evident that the bijas are norbers with the aid of the prodnot and the squares wheroof, w forming the measures of the eldes, right. angled triangle may be oonstructed. 27

Page #412
210 GANITASARASANGRAHA. the squares of that (sum and the difference of the bijas) gives rise (respeotively) 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 bijas). 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 bijas, and then think out and mention quickly the numbers measuring the perpendioular-side, the other side and the hypotenuse (thereof). The role for arriving at the bija nombers relating to a given figuro capablo 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) bijas. (An optionally chosen exact) divisor of half the measure of the other side and the resulting quotient (also) form the hijas (required). Those (bijas) are, (respectively), the square roots of half the sum and of half the difference of the moasure of thu hypotenuse and the square of a (suitably) chosen optional number. An example in illustration thereof. 964. In relation to a certain geometrical figure, the perpendionlar is 16: what are the bijas ? Or the other side is 30 : what are the bijas P The hypotenuse is 34 : what are they (the bijas) P The rule for arriving at the numerical measures of the other side and of the hypotenuse, when the numerical measure of the perpendionlar-side is known; for arriving at the nomerical measures of the perpendioular-side and of the hypotenuse, when the numerical measure of the other side is known; and for arriving 984. In the role given here, a'-0", 8 ab, and a + b are represented an (a + b) (a - b) (a + b) + (a - b)'. (a + b)(a - b). 96). The processes mentioned in this role may be seen to be converse to the myperations mentiored in stansa 90). 9

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OHAPTER VII-MBABUREMENT OF AREAS. 201 at the numerioa) measure of the perpendionlar-ride and of the other side, when the numerical measure of the hypoten use is known : 971. The operation of sankramana, conduoted between (an optionally choson exact) divisor of the square of the measure of the perpendicular-side and the resnlting 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 mcasures of the perpendicular-side and of the hypotonuse). Or, tho squaro root of the difference between the squares of the hypoten uso and of a (suitably chosen) optional number forms, along with that chosen number, the perpendicular-side and the other side respectively, An example in illustration theref. 981. In the case of a certain (geometrical) figure, the porpondioular-side is 11 in measure ; in the case of another figure, the (other) side is 60 ; and in the cas of (still) another figure the hypotonuso is 61. Tell me in these cases the measures of the uumentioned elements. The rule regarding the manner of arriving at a quadrilateral figure having two equal sidos (with the aid of the given bijas : 991. The perpendicular-side of the primary figure derived (with the aid of the given hajius), on being adiled to the perpendicularside (in another figure) derived with the aid of tho (two optionally chosen) factors of half the base of (this original) dorivod II. - 200 971. This rule depends on thu following identities: (a - b) I. { 1 +(2-6)* -- 2 = a' + 1" 06 2ab us tho cane may bo S (2 ab)* Low 26" } - =' + b or vid -6deg III. Via' + bay' - (2 ab) = a'-. 90). The problem solved in the ralo stated in this stans is to construct with the aid of two given bijas & quadrilateral having two equal widon. The lengtha of the sides, of the diagonalo, of the perpendicular from the oud-points of the top. uide to the base, and of the segments thereof canged by the perpendicular are all derived from two rectangles constructed with the aid of the given Wijas, The Art of these rootangles is formed according to the rale given in stants 904 above. The second reotangle is formed socording to the same ralo from two optionally chosen factors of half the length of the base of the first reolangio,.

Page #414
212 GANITASARASANGRAHA. figure (taken as the bijas), 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 diagonals (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 second rectangle. The rationale of the rule will be clear from the following diagrams illustrating the problem given for solution in stanza 100. Here 5 and 6 are the bijas given; and the first rectangle or the primary figure derived from the bljas is ABCD :-- D A B Half the length of the base in this figure 11 is 30; and two factors of this, namely, 3 and 10 may be chosen. The rectangle constructed with the aid of these numbers. as bijas is EFGH: E 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 rectangle 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 HA'FC'. F 60 60 61 109 C H G

Page #415
CHAPTER VII-NBASUREMENT OF ARBA8. 218 figure ; and the measure of the base (of either of the derived figures of reference) happens to be the measure of the perpendi. cular (dropped to the base from other of the ond-points of the topside in the required figure). An example in illustration thereof. 1007. In relation to a quadrilateral with two oqual sides construeted. with the aid of 5 and 6 as bijas givo o'rt tho measures of the top side, of the base, of (either of the two equal) sides, of the perpendioular (from the top to the base), of the diagonal), of the (lesser) segmeat (of the base), and of the arva. The rule for arriving at the measures of the top-side, of the base, of (any one of) the equal) sides, of the perpendioular (from the top to the base), of the diagonal, of the (losser) segment of the base) and of the area, in relation to a quadrilatorul having throu equal sides (with the aid of given bijax) : The process will be clear from a comparison of the diagrams: JI Area of the required quadrilateral, HAFC 61 = area of the second rootangle, EFGH. to 109 60 A' E 102 Bano A'F = perpendioulnr-side of the first rectanglu plus perpendicular. wide of the nocund rectangle, ..., AB + EF. Top side HC = porpendicular-side of the wicond rectanglo minta per. pendicular-side of the first rectangles, 6., GH - CD. Diagonal HF = diagonal of tno necond rectangle. Smaller negment of the base, .. , A'E = jerpendicular-side of the first rectangle, .e., AB. Perpendicular HE = base of the first or of the mocond rectangle, e., BC or FG. Each of the lateral equal siden A'll and YC' = diagonal of the Art rec. tangle, i.e., AC

Page #416
214 GANITASIRASANGBAHA. 1014. The difference between the (given) bijzs 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 80 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 meusures of the sides of the immediately dorived quadrilateral are : Perpondicular-ride =a- b? Buse = 2ab Diagonal= 2% + 1 Aros = 2ab x (a - b) As in the case of the construction of the quadrilateral with two equal sides (vide stanza 99 ante), this rulo proceeds to construct the roquired quadrilateral with threo equal sides with the aid of two derived roctangles. The bijas in relation to the first of theso rectangles are :2ab a- bo): 1.A., V 2ab (a + b), and V2ab (a - b). V Zab * (a - b) Applying the rulo given in stanza 90 above, we have for the first roc. tangle: Porpendicular-side = (a + b) x 2ab - (a - b)? 2ab or 8 a?bo. Bnso = 2 x V 2ab (a + b) x V 2ab (a - b) or tab (a - b). Diagonal = (a + b)2ab + (a - b) * 2ab or 4ad (a+ b*). The dijas in tho case of the second rectangle are: 4-and 2ab. The various elements of this rectangle are : Porpendicular-side = 4a6% - (a - b) ; Bage=4ab (az - 6'); Diagonal = 4a b2 + (a - bay or (a + b*) With the holp of theso two rectangles, tho moasuros of the siden, diugonals, eto., of the roquirod quadrilateral aro ascertained as iu tho rule given in stansa 991 above. Thoy aro: Base = sum of the perpendioular sides = 8ab' + 4a98-a'-6")". Top-side = greater perpendioular-side nuinua smaller perpendioular-side = Sab* - 4a?b!-(a'-6")* = (a'+)'. Either of the lateral sidus = smaller diagonal=(a+b). Losgor segment of the base = smaller perpendionlar-sido = 4a"6" - (a' --0')', Perpondioular = base of either rectangle = fab (a'-'). Diagonal tho grenter of the two diagonale =4ab (a+b). Area = area of the larger reotangle 80'' x 4ab (a'-6"). It may be noted here that the measure of either o! the two lateral sides is equal to the moosure of the top-side. Thus is obtained the required quadrilateral with threo equal sides.

Page #417
CHAPTER VII-MEASUREMENT OF AREAS. * roference 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. 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 perpendicular (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: 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 elements are as follow: Lateral sides 2ab (c+d')(a+b) and (ab) (c2 + d1) (a* +b2). Base2cd (a*b*Xa3 +b"). Top-side (cd) (a+b) (a+b2). Diagonals = {(a - b) x 2ed + (c1-d1)2ab} x (u* + b2) ; and {(a"_b "-b") (c(r) -d") + babed} x (n*+b2) Perpendiculars= {(a* -- b*) x 2cd + (c2 --da) 2ab } >> x 2ah; and {(a* - b )(c" --d") + 4abcd} x (a*b*) + 4abcd Segmenta { (a*--b(r)) x 2cd + (ca --d1) x 2ab | (a --b"); and {(a*--b) (c*--*) cd} x 2ab. *

Page #418
216 GANITASARASANGRABA. of given bijas), the base and the perpendicular-side (of the smaller and the larger derived figures of reference) are respectively multiplied. The products (80 obtained) are (separately) multiplied (again) by the shorter diagonal. The resulting products give the meagures 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 roference), 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 baso and the perpendioular-side (respectively) of the smaller figure (of referenoe), give rise to the measures of tho perpendiculars (dropped from the ends of the diagonals); and when multiplied (respectively) by the perpendioular-side and the base of the same figure of referenco), give rise to the measures of the segments of the base (caused by the perpendichlars). The measures of these segments, when subtracted from the mcasure of the base, give the values of the (other) segments (thereof). Half of the product of the diagonals (of the required figuro arrived at as above) gives the measure of the area (of the required figure). An example in illustration thereof. 104). After forming two derived figures (of reference) with 1 and 2. and 2 and 3 as the requisite bijas 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 perpendioulars, of the diagonals, of the segments of the base), and of the area. Again another rule for arriving at (the measures of the sides, eto., in relation to a quadrilateral, the sides of which are all nneanal:

Page #419
CHAPTER VII-XEASUREMENT OF ARBAS. 217 1051-1077. 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 perpendioular-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 referenoe, each) multiplied successively by the two diagonals (one of each of the oblongs of referenco), give rise to the measures (res peotively) of the two (lateral) sides (of the required quadrilateral). The difference between the bago and the perpendioular-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 porpondioular-side of the smaller (oblong of reference) with the perpendiculor-side of the larger (oblong of referenco). The two sums (80 obtained), when multiplied by the diagonal of the smaller (oblong of reference), give rise to the values of the two diagonale (of the roquired quadrilateral). The diagonals (of the required qualrilatoral) are (separutely) divided by tho diagonal of the smaller (oblong of 106-107). The name values as are mentioned in the footroto to mtanza 103 above are given here for the measure of the wides, etc.; only they are stated in a slightly different way. Adopting the same symbols as in the bote to stanza 103}, we hayo: Diagonals- [ { 2cd - (c-d") } 21b + {2a1 +(24-40)] co-)] *(* +"); pa[ {2cd=(0) -d")}(20-0deg)+ {2ab + (a1 +00)}(co-4"}]x (2"+W). [{2cd=(09-")} * 2ab + { 2ab +(29-60, } (c- d") ]ca+61) . [{2cd=(co-d") }(2deg -6deg +) 2ab +(a2-1", }(- ) Top+b). Perpendiculars = *2 ab. 'The above four expressions can bo reduced to the form in which tho mbasures of the diagonalo and the perpendicalors are given in stanza No. 108). The mesures of the segments of the balle are here derived by extracting the square root of the difference betweon the squares of the side and of the perpend oular corresponding to the segment. 98

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218 GANITAS BASARGRAHA. reference). The quotients (80 obtained) are multiplied respectively by the perpendicular-side and the base of the smaller (oblong of reference. The resulting) produots give rise to the measures of the perpendioulars (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 irosceles triangle with the aid of single derived oblong (of reference). 1081. The two diagonals (of the oblong of reference constructed with the aid of the given bejas) 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 referenoe). 108. The rationale of the rule may be made out thus :- Let ABCD be an oblong and let AD be produoed to E so that AD =DE. Join EC. It will be teen that ACE is an isoscoles triangle whose oqual sides are equal to the diagonale of the oblong and whose area is equal to that of the vblong. 0

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CHAPTER VIIMEASUREMENT OY AREAS. 819 'An example in illustration thereof. 1097. 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 isoroeles triangle derived with the aid of 3 and 5 as bijas. The rule regarding the manner of constructing a trilateral figure of unequat sides : 1101. Half of the base of the (oblong of reference) derived (with the aid of the given bijas) is divided by an optionally chosen factor. With the aid of the divisor and the quotiont (in this operation as bijas), another (oblong of reference) is derived. The sum of the perpendicnlar-sides belonging to these two (oblongs of reference) gives the moasure of the buso of the (roquired) 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 eithor of the two oblonge of reference) gives the measure of the perpendicular (in the case of the l'equirod triangle). An eaample in illustration thereof. 1113. 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 lijas, 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 ng the Janya operation, 1104. The rule will be clear from the following construction :-Lot ABCD and EFGH be the two BK derived oblongi, such that the base AD = the base EL. Produce BA to K 10 that AK= EF. It can be sually shown that DK = A DE EG and that the triangle BDK has its base BK = BA + Er, called the perpendiculars of the oblongn, and has its sides equal to the diagonals of the same oblongu. .

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220 GANITASIRABAAGRARA. Subject of treatment known as Paisacika or devilishly difficult problems. Hereafter we shall expound the subject of treatment known as Paikacika. The rule for arriving, in relation to the equilateral quadri. lateral or longish quadrilateral figures, at the numerical measure of the base and the perpendicular-side, when, out of the perpendicular sido, 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 choson 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 lougish quadrilateral figure.) when the area of the figure is (numerically, equal to the measure of the porimeter (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 numerioally 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 mesaare 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 : '1121. 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 role will be clear from the following working of the first example given in stanga 1181 :--Here tho problem is to find out the measure of the wide of an oquilateral quadrilateral, the numerical value of the area where. of is equal to the numerical value of the perimeter. Taking an equilateral upadrilateral of any dimension, say, with 6 as the measure of its side, we have the portreter equal to 10, and the area gal to 85. The factor with which

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OHAPTER VII-MBASUREMENT OF AREAS. 221 of the said optionally chosen figure happons to be arrived at); or the base of such an optionally chosen figure of the requisito 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 basos of the (required) equilateral quadrilateral and other kinds of dorivod figures. . . Examples in illustration thereof. 1134. In the case of an equilateral quadrilateral figure, the (numerical measure of the) perimeter is equal to that of) the aroa. What then is the numerical measure of its) base? In the case of another similar figure), the numerical measure of tho) area is equal to (thut of) the base. Toll me in relation to that (figure) algo (the numerical measure of the base). 1141. 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 (ite) 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 tho measure of its base). 1157. Here in the case of a longish quadrilateral figure, the (numerical) measuro of the area is equal to that of the perimeter; and in the case of another (similar)i 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) P 1167. 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) measiure of the diagonal is four times that of the area. What is the measure of the base in each of those cases) tho measure of the perimeter, viz. 20, has to be multiplied in order to make it equal to the monstro of the area, vis., 25, ist. If 6, the monstro of side of the optionally choron quadrilateral is divided by this factor 4, the mosnre of the wide of the required quadrilateral is arrived at. The rule gives also in another manner what is practically the same procon thus: * The factor with which the measure of the area, vis. 26 bus to be maltplied in order to make it equal to the measure of the perimeter, vis. 20, in f. 116, the moncare of side of the optionally chosen figure is multiplied by thin factor 1, the montre of the side of the required figuro is arrived at.

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222 GANITABARASANGRAHA. RO. 117). 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.) 1187. In the case of a longish quadrilateral figure, the (numerical) measure of the perimeter is 1. Tell me quiokly, after calculating, what the measure of its perpendicular side is, and what that of the base. 1194. In the case of a longish quadrilateral figure, the (numerical measures of twice the diagonal, three times the baso, and four times the perpendioular, on being added to tho (numerical) moasure 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 bejax in relation to the derived longish quadrilateral figure : 1207. The operation to arrive at the generating (bijas) in relation to a longish quadrilateral figure consists in getting at the square roots of the two qnantities 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. 121*. 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 bijas here ! 1204. The role in starea 951 of this chapter relates to the method of arriv. ing at the bljar from the base or the perpendicular or the diagonal of a longish quadrilateral. But the role in this stansa gives a method for finding out the bijgs from tho perpendionlar and the diagonal of longish quadrilateral. The procons denoribed is based on the following identities :Ja + b-(03-08) = 0; and Notty_a+b-a? 2 whers o p in the measure of the diagonal, and al-is the momenre of the perpendioular-side of longiah quadrilateral, and d being the required bijas. Ng

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CHAPTER VII-MEASUREMENT OP AKRAS. 228 The rule for arriving at the longish quadrilateral) figura associated with a diagonal having a numerioal value optionally determined : 1221. Each of the various figures that are derived with the aid of the given (bijas) is written down; and by means of the measure) of its diagonal the measure of the) givon diagonal is divided. The perpendioular-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. 1231-1241.0 mathematician, quickly bring out with the aid of the given (bijas) the (value of the) perpendicular-sides and the bados of the four longish quadrilateral figures that havo respectivoly 1 and 2, 2 and 3, 4 and 7, and 1 and 8, for their bijas, and are also oharacterised by different basee. And, (in the problemn) boro, the diagonal is (in value) 65. Give out (the measures of) what may be the (required) geometrical figures in that caso). 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 :-- 1251. Multiply the square of the diagonal by two; (from the resulting product), subtract the squaro of half the perimotor; (then) get at the square root of the resulting difference). If (this square root be thereafter) utilized in the performanco of the 1291. The rule is based on the principle that the sides of a right anglod triangle vary as the hypotonune, although for tho samo neanuro of the hypo. tangse there may be different sets of values for the sides. 1864. If a and 6 represont the sides of rectangle, then a+b in thu meagare of the dingonal, and 2a + 2b is the measuro of the perimotor. It can be been esily that These two formales represent algebrsically the method donoribed in the rule hero,

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224 GANITASARABANGRABA. operation of sarkramana along with half the perimeter, the (required) base and also the perpendicular-side are arrived at. An example in illustration thereof. 1261. The perimeter in this case is 34; and the diagonal is been to be 13. Give out, after calculating, the measures of the perpendicular-side and the base in relation to this derived figare. The rnle for arriving at the numerioal values of the base and the perpendionlar-side when the area of the figure and the value of the diagonal are known : 127). 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 (roquired) perpendicularside and the base, if the larger (of the square roots) is made to undergo the prooCB8 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 perpendioular-side and the base. The rulo 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 : 1297. 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 symbole as in the note to stanss 125$, we have the following formula to represent the rule here given :-- { w[ver+o) lva + o) - Zante +soor dy as the chae tay be. 1994. Here we have Su + 90 17 2a + 36 -4b += or b, u the case may be.

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OBAPTER VII-XEASUREMENT OF ARBAS. 225 Bubtracted. Then, on carrying out the process of sankramina with the square root (of this resulting differenco) in relation to half the measure of the perimeter, the values of the required) base and the perpendioular-side are indeed obtained. An example in illustration thereof. 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 tho numerioal measures of the perimeter are equal, and the area of the first figure is double that of the socond ; or, (2) when the areas of both the figures aro equal, and the numerical measure of the perimeter of the second figure is twice the numerical measuro 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 figuro, and tho area of the first figure is twice the area of the second tiguro : 1317--133. (The larger numbers in the givoa ratios of) the perimeters as also (of) the areas (relating to the two required longish quadrilateral figures,) are divided by the smaller (numbero) corresponding to them. The resulting quotients) are multiplied (between themselves) and (then) squared. This same quantity.) 1814 to 1.33. If and y roprenent the two adjaount sides of the Arst rectangle, and a and & the two adjacent video of the second rootangla, the conditions mentioned in the three kinds of problems proposed to be solved by this role may be represented thus : (1) +y=a+: xy = 2ab. (2) 2(2+ y) = a +b: by = ab. (8) 2 (8+ y) =a +b: zy = 2ab. The solation given in the role seems to be correot only for the particular 01909 given in the problems in stan 186 to 186. 29

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226 GANITASARASANGRAHA. on being multiplied by the given optional multiplior, 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 respeot of arriving at the other of the two required quadrilateral figures), its base and perpendioular are to be brought out with the aid of the (now knowable) measure of its area and perimeter in acoordance with the rule already given in stanza 1297). Examples in illustration thereof. 134. Thore are two (quadrilateral) figures, each of which is charaoterised 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 porimeters are equal. What are the perpendioular-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 (figure) is twice that of the first. Find out their perpendioular-sides and bases.) 136. There are two longish quadrilateral figures. The area of the firt (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 being equal to each other, or by the values of their perimeters and of their areas forming multiples of each other :

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CHAPTER VII-MBASUREMENT OF AREAS. 227 137. 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 produots 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 bijine in relation to the longish quardrilateral figure) from which one (of the required triangles) is to be obtained. The differonoe hetween those (two bijas above noted) and twico the analler ono (of those bijax) constitute the tijas (in relation to the longish quadrilateral tigure) from which the other (required triangle) is to be obtainod. (From the two longish quadrilateral figures forined with tho aid of their respective bijas), tho sides and the other things (rolating to the required triangles) are to be arrivod at as (explainod) before. 187. When a : b in the ratio of the perimeters of the two inosoelos triunglon, and c: d the ratio of their arean, thon, according to tho rule, and 10 1 48 C and a a + 1 and a de - 2 ure the two sets of bijas, with the help of which the values of the various required elements of the two isosoolen trianglou may be arrived at. The measures of the sides and the altitudin, calculated from those ljen nooording to stanea 108 in this chapter, when multiplied ronpootively by a and b. (the quantities occurring in the ratio of iho periunetors), kive the required measures of the sides and the ultitude of the two iDonooles triangles. They are a follow : Equal sido = a* {(VI) + ( 2 )"} Bmw = x 2 x 2 x - ?) Altitudo = ax { ) - -)} 11 Equal oido = 6 * { leta 1) -)"} Antado =bx { lett +1)-( -2)} Now it may be easily proved from these values that the ratio of the perime. tors is a: 6, and that of the areas inc: d, as taken for granted at the beginning.

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228 GANITASABASANGRAHA. Examples in illustration thereof. 138. There are two isosceles triangles. The perimeters are (also) equal in value. their sides, and what of their bases P Their area is the same. What are the values of 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 (their) sides, and what of (their) bases ? The perimeter of the 140. There are two isosceles triangles. 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 equila. teral 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), when 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 (so obtained), if multiplied by four, gives rise to the measure of the breadth of the circle and 142. In problems 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

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OHAPTER VII- MEASUREMENT OF ARBAS. 229 (also) of the square. (That same) quotient, it multiplied by s6, gives rise to the required ineasure of the base of thu (equilatoral) triangle as also of the longisb quadrilateral figuro. Half (of this) is the measure of the perpendicular-side (in the case of the longisb quadrilateral figure). An example in illustration thereof. 143-145. A king caused to be dropped an excellent carpot on the floor of (his) palace in the inner apartments of his zenana amidst the ladies of his harem. That (cnrpet) was in shape) # regular circle. It was held in band) by those ladies. The fistfuls of both their arms made each of them) acquire 15 (dandas ont of the total area of the carpot). How many are the ladies, and what is the diameter of the circle) here? What are the sides of the square (if that samo carpot be square ip shupo) ? 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. Ilm represents the arou of ouch part and n the length of a part of the total porimeter, the formulas given in the rule aro x 4 = diamotor of the circle, or side of the square ; and "x 6 = side of the equilateral triangle or of the oblongi and half of x 6 = the longth of the perpendicular-wide in the came of the oblong. The rationals will bo cleur from the following aquations, where represents the nnulor of parts into which ench figure is divided, a in the length of the radius in the case of the circle, or the length of a side in the case of tho other figures ; and 6 is the vertical side of the oblouk : x mwa In the case of the Cirolo ... = X 2 ! Xm In the case of the Square = X 1 In the case of the Equilateral Triangle = 2 1 x 1 3 In the cro of the Oblong tkh takh' = ; horu b is taken to be equal * Xn2(a + b) nor to half of a It has to be noted that only the approximate value of the ares of the equilater triangle, w given in stanza 7 of this chapter, is adopted bero. Otherwise the formula given in the role will not hold good. 148-146. Wat is called Astful in this problem is cquivalent to four angulas in measure.

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* 230 GANITASARABANGRAHA. sides of the equilateral trianglo (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 rule for arriving at an equilaterally quadrilateral figure or at a longish quadrilateral figure when the numerical value of the area of the figure is known : 146. The square root of the acourate 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 valne of the given area, this) optionally chosen quantity and the resulting quotient constituto the valocs 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 P The accurate value of the area of the longish (quadrilateral) figure is 60. What are the values of the perpendioular-side and the base here? The rule for arriving at a quadrilateral figure with two equal sides baving the given area of such a quadrilateral figure with two equal sides, after gotting at a derived longish quadrilatoral figure with the aid of the givon numerical bijax and also after 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 oqual sides is known : 148. The square of the given (multiplier) is multiplied by the that (given) area. The (resulting) produot is diminished by the value of the area (of the longish quadrilateral figure) derived (from the given bijas). The romainder, when divided by the base 148. The problom here is to construot . quadrilateral figare of golvo area and vith two equal siden. For this purpose an optionally chosen number and set of two bijas are given. The process deuoribed in the role will become olour by wpplying it to the problem given in the next'tanse. The hijau mentioned therein are 8 and 8; and the given area is 7, the given optional number badag 9.

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CHAPTER VII-MEASUREMENT OF AREAS. 231 of this derived longish quadrilateral figure), gives rise to the measure of the top-side. The value of the perpendicular-side of he derived longish quadrilateral figure), on being multiplied by mo and increased by the value of the top-side (already arrivod it), gives rise to the value of the base. The value of the baso (of he derived longish quadrilateral figuro) is (the same as that 1 % The first thing we have to do is to constrnot a rectangle with the aid of ho given bijas in accordance with he rule laid down in Atanza 904 in his chapter. That rectangle oomos o have 6 for the mensure of its imaller side, 12 for the measure of ts larger side, and 13 for the neagre of its diagonal; and its irea is 60 in value. Now the aren given in the problem is to be multiplied by the square of the given yptional nunbor in the problein, so that we obtain 7 x3 = 63. Frourth is 03, We have to subtract 60, which is he measure of the area of the rectangle oonstructed on the bnais of the given bijn: and this gives 3 12 18 the remainder. Then the thing to be done is to construct a rectangle, the iron whereof is equal to this 8, and one of the sides is equal to the longer side of the rectangle derived from the same bijas. Binoe this longer vide is equal to 12 in valile, the smaller side of the required reotangle 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 un each sido to this last rectangle, so that the sides measuring 12 in the case of these triangles coincido with the sides of the rectangle having 12 as their measure. The figure here exbibits the operation. Thus in the end we get the quadrilateral figare having two equal wider, each of which measures 18, the value of the other two sides being + and 10% respectively. From this the values of the sides of the quadrilateral required in the problem may be obtained by dividing hy the given Optional namaber damely 3, the values of its sides reprosettod by 13, , 18 and 101

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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 bajas 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 rule for arriving at a quadrilateral figure with three equal sides, having an accurately measured given area, (with the aid of a given multiplior) : 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 moasure (of one of the equal) sides. The given 150. It is stated in the rulo 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 prodact of the perpendicular 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-sido. If ABOD bo A quadrilateral with three equal sides, and CE the perpendicular from C on AD, then AE is half the Bam of AD and BC, and is equal to the given optional anmber. It oan be easily shown that 9 AD. AE =CE+ AE! +42 AE +AE AE CE' x AE (CEAE)' AE. CE+AESCE AE. .. AD= 2AE A E 9= Here CEAE 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 vides of the quadrilateral contemplated in the problem. pilateral. This Month

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CHAPTER VIIMEASUREMENT OF ARFAS. 238 (multiplier) as multiplied by two and (thon) diminished by the value of the side (just arrived at) givos riso 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 (dropprd 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 sideg, 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) sites 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 (those) quotients is (noted down) in four positions, and is (in order) diminished (respectively) by those (quotients noted down abovo). The remainders (80 obtained) give rise to the numerical values of the sides of a quadrilateral figure (having unequal wides and 00180quently) named uneqnal.' 162. The area of a quadrilateral with unequal side. bas already been men. Hobed to be -a) (-ba-o le-d), whero e = ball tho perimetor, and a, b, c, and d are the measures of the siden (tide potato tante 30 in this chapter). The rule here given requires that the numerical valoe of the area should be squared and then divided separately by the four optioually chosen divlegru. 11 (-a) (1-0)(1-c) (-d) in divided by four suitably choron divinors so an to give u quotient -9,-6, .-C, and Id, then on adding these quotionta and halving their sam, the result is seen to be .. 11 , is diminiebed in order by i-,.-,.-C, and ,-d, the remaindera refinent respectively the valdes of the sides of the quadrilateral with unequal sidou. 80

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284 GANITASIRABANGRAEA An example in illustration thereof. 153-1537. In the the case of a quadrilateral figure with unequal sides, the (given) sonurate measure of the area is 90. And the produot of 5 multiplied by 9, as multiplied by 10, 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 numerioal value of the sider of an equilateral triangular figure possessing a given acourately measured area, when the value of (that) accurately measured area is known : 1547. Four times the (given) area is squared. The resulting quantity) is divided by 3. The quotient (80) obtained happeni to be the square of the square of the value of the side of ar equilateral triangular figure. An example in illustration thereof. 156). In the case of a certain equilateral triangular figure the given area is only 3. ('alculate 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 figur having that same accurately measured area (as its own) : 1567. In the case of the isosceles triangle (to be so) construo ted, the square root of the sum of the squares of the quotien obtainod by dividing the (given) area by an optionally chosen quantity, as also of (that) optionally chosen quantity, gives risi to the value of the side : twice the optionally chnsen quan tity gives the measure of the base; and the area divided by 1644. The role here given may be seen to be derived from the formula to the area of an equilateral triangle, vis., area = a'* 7 where a is th monaro of side. 166). In problems of the kind contemplaind in this rolo, the monare of the Area of an inomoeles triangle is given, And tho ralae of hall the base chouen option is also given. The mourires of the perpendicnlar and the side are the Soily derived from these kaown quantitie.

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CHAPTER VI-MEASUREMENT OF AREAS. 236 the optionally ollosen quantity gives rise to the measure of the perpendicular, An example in illustration thereof. 157}. 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, aftor knowing the exact numorical measure of a (given) area, at a triangular figure with upoqual sides, having that same accurately measured area (as its own) :-- 1581. The giveu area is multiplied by eight, and to the rosulting produot the square of the optionally chosen quantity is added. Then the square root (of the sum so resulting is obtainod). The cube (of this square root) is (thereafter) divided by the optionally chosen number and (also) by the square root (obtained as abovo). Half of the optionally chosen number gives the measure of the base (of the required triangle). The quotiont (obtained in the previous operation) is lessened (in valne) by tho (measuro of this) base. The resulting quantity) is to be used in carrying out the sankramana process in relation to the egnaro of the optionally chosen quantity as divided by two as woll as the square root (mentioned above). (Thuy) the values of the sides arc arrived at. 1584. If A represents the area of & triangle, and in the optionally chosen number, then according to the rule the required values are obtainod than : 2 -- bago; (W8A+dad & N 8A+H 2 3 N 8A + d' = siden. and 2 When the area and the base of a triangle arc kiven, tbo locos of the vertor is a lino parallel to the bano, and the sides can have any met of valoon. In onder to arrive at spuoifc set of values for the siden, it is evidently msumod here that the sum of the two sides is equal to the sum of the base and twise the altitudo, 6..., equal to +24. With this serumption, the formula nbove "2d24 given for the measure of the sides can be derived from the general formula for the area of the triangle, Nele-a) ( b) ( c) given in apzs 60 of this chapter.

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286 GANITASI KASANGRAHA. An example in illustration thereof. 1694. In the case of a certain triangular figure with unequa sides, it has been pointed out that 2 constitutes the accurate measure of its area and 3 is the optionally choson 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 (acourately measured) area (as ite own) : 1601-1615. The square root of the measure of the given area as multiplied by eight and as increased by the square of an option. ally 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 respeotively 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 aoourate measure of the area is 2, and the optionally chosen quantity is 3. O friend who know the secret of caloulation, 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 acourately measured area (as its own) : 1634. The accurate measure of the area is multiplied by four and is divided by the square root of ten. On getting at the square 10. 1684. The rule in this stans is derived from the formule, aros = Where d in the diameter of the oirole.

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CHAPTER VIIMBABURRNENT OF ARRAS. 887 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. Caloulate quickly and tell me what the diameter of this circlo) may bo. 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 figuro with throo equal sider, baving those samo approximate and accurate mcasures (as such measures of their aroas) : 165). In the case of the quadrilateral with) two equal siden, the square root of the difference between the squares of the (approximate and accurate) mensures of the aron is to ho ohtainod. On adding this square root) to the optionally chosen quantity and on subtraoting (the samo square root from the same optionally obosen quantity), the base and the top-sido are so obtained as to have to be divided by the square root of the optional quantity, The approximato measure of the area gives rise to the valuo of the sides so as to bave to be divided by the square root of the optional quantity. 165. If R represents the approximate area of a qurulrilateral with two oqual sides, and the accurate valao thoroof, and p in the optionally choron number, then NR - bage = P P-NR- ! op-side and cach of the WP equal sides - If a, b, c and d bo tho moasures of the wider of NP the quadrilateral with two equal sides, then it may be sewn that

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238 GANITAPARASANGRAHA. In the oase of the quadrilateral figure witb) three equal sides, the square root (of the difference between the two area-equares 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 80 88 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 (tbis) optional quantity, givos rise to the moasure of the other sidos. An example in illustration thereof. 1661. The accurate moannre 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 eaample relating to a quadrilateral figure with three equal sides. 167). The accurate measure of the aroa is 5; and the approximate monsure of the area is 13. Think out and tell me, O friend, the values of the sides of the quadrilateral figure with three equa) sides. l'ho rule for arriving, when the approximate and the acourate ineasures of an area are known, at the equilateral triangle and also at the diameter of the circle, having those samo approximate and accurate moasures (for their area):-- 1687. That which happens to be the square root of the square root of the difference between the squares of the (approximate meagure and of the acourate measure of the given) area is to be R = 4(6.4 ), n= + 9) *; and r = 0 4 dx No - 10 mayo. The formulas given above for the base and the top-side can be easily verified by substitating these values of Rr and p thereia. Similarly the rule may be been to hold good in the case also of a quadrilateral Agare with throe equal idem. 1881. For the approximate and accurate values of an oquilateral triangla ne pulos a stangas 7 and 60 of this chapter.

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CHAPTER VII-MEASUREMENT OF AREAS. 289 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 oircle. Examples in illustration thereof. 169. 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. 170. 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 caloulated measures of an area are known, the rule for arriving at the numerical values of the baso and the side of an isosceles triangle baving the same approximate and accurate measures for its arca :-- 171. Twice the square root of the difference between the squares of the (approximato 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. 1724. 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 quadri lateral figure (with two equal sides) which has an accurate measure

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240 GANITASABASANGRAHA. @ 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 visamasankraman 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 b and d be the top side and the base thereof respectively; and let the value of the perpendicular distance be p. If a, b, o, di, bo 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, d2 + b1 = d + b (I): and and q2 - (di = b1) * - = p2 (II): that is, (4, +52 d1 = b1 = 2; 2 2 Let a db N; then a, + - di ; bi = (~ + d) (~,~-~~-~1) - N 11. +N. b + (1) (a, - d2 = 1) = 3. 2 pl N-N 2 + (III); - - |( ~ ( d1 + b1 + 2 1 a, + 1-01 ) ( ) ( d, or b1; (IV) Here N is what is called or the optionally given number in the rule, and formulas III and IV are those that are given in the rule for the solution of the problem.

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CHAPTER VII-MEASUREMENT OF AREAS. 4n example in illustration thereof. 1744. The base of the given quadrilateral figuro) 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. What is that other quadrilateral figure with two equal sides, the acourato measure (of the area of which is the same as (the acourate measure of) the area of (this given quadrilateral) ? When an area with a given practically approximate measure is divided into any required number of parts, tho rule for arriving at the numerical measure of the bases of those various parts of the quadrilateral figure with two equal siden, as also at the numerical measure of the sides as measured from the various divisioni-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 (80 obtained), Nn 1767. If ABCD bo e quadrilateral with two equal aidos, and if RF, all and KL divide the quadrilateral so that the divided portions A b D are in the proportion of m, #, pand q in respect of area, then according to tho rule, EF = 2 - 0 - xm + 1; GREN di - D2 + + P + 9 K Nm + + D + 9 - * (m +n + p) +0?; and so on. + + + P +9 G KL EN Similarly AB= + + PRG DA RP + AD 1 min + P +91 ON + RP (a*** GK=**7 * ***+p+e, and so on. XL + GU It can be eile shown that AB BC-AD 81

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242 GANITASARASANGBAHA. 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. * 1764. The measure of the top-side is given to be 7 ; tbat of the base below is 23, and that of each of the remaining) sides is 30. The area (inoladed within such a figure) is divided het wonn two so that each obtains one (share). What is the value of the base (to be found out here) 1774-1781. The measure of the base (of a quadrilateral with two equnl sides) is 162, and that of the top-ride (thereof) is seen to be 18. The value of each of the two equal) sides is 400. The area of this (figure so enolosed) 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 acoordance with this propor. tionate distribution, the values of the area, of the base, and (of either) of the two equal) sides in each case). 1797. The measure of the base (of the given quadrilateral figure) is 80, that of the top-side is 40; the moasure (of either) of EPS (BC-A 02 AB (BC + AD) - BC-AD ..AE (BP + AD) - EF-AD AB (B C + AD).m + + + P + 9. But a BC-AD + + + + WAE (LY+AD) m 'EPCADI--- m (BC-A D') + AD' = de-16 +* + P 11 +* + P + 4 dt - J And Fav. V + + + xm + 0. Similarly the other formula, 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 one. That is, in the figure on the previous page, to arrive at GH, for instance, has to be multiplied by m + * and not by merely.

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CHAPTER VII-MEASUREMENT OF AREAS. 848 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 hoight, two strings are tied, one to the top of each. Each of these two strings is strotched in the form of a hypotenuso 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 by potonuso strings meet, another string is suspended (perpendioularly) till (it touohos) the ground. The measure of this (lust) string goes by the name antaravalambaka or the inner porpendicular. 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: 1807. The measurement of each of tho pillars is divided by thu muasurement of the baso 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 (80 obtained) is 1804. If a and represent the height of the pillars in the diagram, o the distance between the two pilla r, and m and the respoctive distanoos of the pillars from the point where the string stretched from the top of the other pillor meets the earth, then, scoording to the rule, * (c + m) + 0(c + )) (c + m + *); whoro e, and C, sto segments of ( m) ( + ) the base us whole; and p a *. , or cg * , whero p in the mentare of the inner perpendioular. From consideration of the similar triangles in the diagram it may be won that "+and ". . C + m C +

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244 GANITASABARANGRAHA. (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. 1814. (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. 182). 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! 1837-1841. (The two pillars are) 12 and 15 hastas (respeotively); tho 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 horo, and what of the inner perpendicular ? 1854. (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 From these ration wo get = (+) ; C + C a(c+ m) + Blatni.cz = *(c+m) (c+ m + n) a (c+m) + b(c+) Similarly C, =- 0(c+n) (c+m+ >>). k; and p=gX , or a(c + m) + b (C++)' X C + m. 185. Here a quadrilateral with two equal siden is given; in the next stensa quadrilateral with three equal sides, and in the one next to it . quadrilateral rith unequal sides are given. In all these cases the diagonals of the quadri. lateral have to be first found out in aocordance with tht rule given in stansa

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CHAPTER VII--MEASUREMENT OF AREAS. 246 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-sido and the base are each to be taken to be less by 1 hasta. From the top of each of the two perpendiculars, & string is stretched so as to reach the foot of the other perpendioular). You givo out the measures of the innor perpendi. oular and of the basal segments caused thereby). 187}. (In the case of a quadrilateral with uncqual sides), one side is 13 hastas in measure ; the opposite side is 15 haxtas; the top-side is 7 hastas; and the base here is 21 hastax What aro the values of the inner perpendicular and of the basal segmenta (caused thereby)? 1887-1894. There is an equilateral quadrilateral figure, measuring 20 hastas at the side. From the four angles of that VII-54, and then the measuron of the perpendiculors from the end of the topside to the base as also the measures of the boxn onts of the long rauwed by those perpendiculars have to be arrived at by thu application of the rule given in stanga VII. 46. Thon taking these MCDMUCH of the perpendiculars to be those of the pillars, the rulo given in stanza 180 abrivo in applied to rrive at the measures of the inner perpendioular and the banul Beginnt rutined thereby. The problom given in stanza 187, in however worked in it lightly different way in the Kanarono commentary. The top-Hide is supposed to be parallel to the lane, and the measures of the perpendicular und of the bunul seguents caused thoroby are arrived at by construoting & trianglo who widos are the two willen of the quadrilateral, and whose baso is equal to the differendu between the base ond thu top-side of the quadrilateral. v 1881-1891. The figure contemplated in this problem seeds to bo this: The inner perpendiculars referred to herein are GA and KL. To find oat thene, FE in first determined. FE, AC cording to the commentary, is said to be equal to CN DY+DE+(ID)) Then with PE and BC or AD taken as pillars, the rule under referenco may be applied. HEL

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246 GANITASABASANGRAHA. (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 so stretched out P 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 measuro of the remaining part of the pillar measured from its foot:-- 1907. The half of the difference between the squaro of the total height and the square of the (known) measure of the basal distance, whep 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. 1913. The hoight of a pillar is 25 hastas. It is broken somewhere between the top and the foot). The distance between the (fullen) top (on the floor) and the foot of the pillar is 8 hastas. How far away (from the foot) is it (viz., the pillar) broken? 1904. If A B C is a right-angled triangle, ond if the mossuros of AC and of the sum of AB and BC are given, then AB and BC on be found out from the fact that BC = AB + AC?. The formula given in the rule is AB = (AB+ RC)' - AC 9 (AB+BO) and this 04n be easily proved to be true from the above equality.

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CHAPTER VII-MEASUREMENT OF AREAS. 247 192). 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? 1934-1954. 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 patkr fornting a hypotenuse. Then another man standing at the foot of the tree went towards that fruit taking a path reprosenting the other side (i.e., the base of the triangle in the situation ) and received that fruit. The sum of the distancos travelled by that fruit and this man turned out to be 50 hastas. What is the numerical value of the hypotonuso 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 The numerical value (of the height) of a taller pillar as glotho numerical valao (of the height) of a shorter pillar is known. Tho numerical value of the length) of the interveuing space between the two pillars is also kuowl. The taller (of the two pillars) gets broken and falls so that the top thereof rosta ou the top of tho 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 tho length) of the broken part of the taller pillar as also at the numerioal value of the height) of the remaining part of the same tailor pillar) : 1961. From the square of the numorical measure of) the taller (pillar), the sum of the square of the measure of the shorter 1981. Il a represents the height of the taller pillar and b that of the aborter pillar,c the length of the intervening proe between them, and a, the height of the standing portion of the broken pillar, then, according to the rule, 452(0-6)

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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) interveuing 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 oity from the foot of the mountain). An example in illustration thereof. 1991-2003. On a mountain having a height of 6 yojanas there wore 2 ascetics. One of them went walking on foot. The other 1994-2004. If in the marginal figure, a reprobents the height of the mountain, o the distance of the city from the foot of the mountain, and c the length of the hypotenuse course, then a in, according a to the supposition made in the preamble to the role in 1981, of the side AB, Therefore the height of the flight upwards 1..., EB., is $a ... ... ... I AL As the courses of the two ascetica are oqual, c+ ta= a +b; ::c=ta + ... II = a + b + ab. But of = fal + ,.. ab = 3 ..b= la ... ... ... ... 11 The three formulas marked I, II and III abovo are those given in the rule. (

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OHAPTER VII-MEASUREMENT OF AREAS. 249 was capable of moving in the sky. This ascetio flew up and then came down to the oity taking the hypotenuse course. The other Ascetio descended from the suminit (vertically) to the foot of the mountain and walked alung) to the city. (It was found that) both of them had travelled ovor 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 rosting 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 quadrilatoral figuros. Then, (with the aid of these kuown horizontal sides and) in relation to the base lino either betweon the two bills or between the two villars, (as the oase may be), the values of the two sogments (caused by the meeting point of the perpendicular) aro arrived at. These two segments are written down in the inverse ordor. The values of the two segments so written down in the inverso order are taken to be the values of the two perpendicular sides of tho two longish quadrilateral figures. And, now, the rulo for arriving at the equal numerical value of the diagonala of thoso (two longish quadrilateral figures) : 2011-2031. In relation to a figuro representablo by a (918ponded) swing (and its vertical supports resting on the ground), the measares of the heights of either two pillars or two hills are taken to be the mensures of the two sides of a triungle. Then, in relation to the value of the base (lino) enclosod hetweon thoso two 2011-2031. In the two quadrilaterals of the kind contemplated in this rule, let the vertical sides be reprownted by a, , d let the base bec; and let C, CA, its weg. * ment and I the leugth of each of the equal portions of the ropa.

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250 GANITABARASANGRAHA, sides (which has to be the same in value as the base line between the given pillars or hills), 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 acoordance 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 namorioal value. Ecargples in illustration thereof. 2043-205. One pillar is 13 (hastas in height). The other is 15 (hastas in height). The intervening distance (hetween 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, (80 cansed, 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. 206-207. 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 mendioante, (one) on the top of each, who can move along the sky. For the purpose of begging (their food), they came down Now, a + G9 = 18+ (cy + c) ( - Ca)= -1; and C + 0,=c; 00cc ...C= - woda Those values are obviously those of the segments of the base of a triangle having the sides a and b, the segments having been caused by the perpendicular from the verter. This is what is stated in this role. Vide rule given in stansa 49 above,

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CHAPTER VII-MEASUREMENT OF AREAS. 261 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 distaudes. (Under these ciroumstancos), of what numerical value were the segments (of the basal line between the two hills) P Of what value, O you who know oalculation, is the numerical measure of the equal distance travelled in this (arca) representable by a (suspended) swing. 2087-209}. The height of one hill is 20 yojanas; and similarly, that of another (bill) 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 cach), and were able to move through the sky, came down, for the purpoee of begging their food, to the city situated betwoon thoso (two hills), and were fonnd 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 reprosentable by the boundary of) a triangle oonsisting of (three) unequal sides : 2104. The sum of the squares (of the numerical valuos) 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 (80 obtainct) 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 mai). The meeting together of these two men takes place at the end of the number of days measured by this product. 3107. The coure contemplated here is that along the side of a right anglod triangle. The formula given in the role, if algebraically represented, in b' +'xd. .ba where in the number of daye taken to go through tbe hypotonuso course, a and b the rates of journey of the two men, and d the uumber of days taken in going Dorthwards. This follows from the under mentioned equation whiohin bused on the data given in the problem:

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252 GANITASARASANGRAHA. An example in illustration thereof. 211-212. The man who travels to the east moves at the rate of 2 yojanas (a day); and the other man who travels north. wards 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 day spent (by both of them) in journeying out is the same. The rule for arriving at the numerical value of the diameter of circles described about the eight kinds of figures consisting of the five kinds of quadrilateral figures and the three kinds o: triangular figures (already mentioned) : : 213. 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 trilatera 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. 214. In the case of an equilateral quadrilateral figure having 8 as the measure of each of (its) sides, and also in the case o another (quadrilateral figure) of which the vertical side measures! 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 similar. .. AB: AD-BE: BC .. AD ABX BC BE This is the formula given in the rule A for the diameter of a circle circumscribed about a quadrilateral or a triangle. 6 B D C

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OHAPTER VIIMBASUREMENT OF AREAS. 253 2154. The two lateral sides are (each) 13 in measure; the topside is 4; and the hase is said to be 14 in measure. In this case, what may be the diameter of the circle describod about (Buch) & quadrilateral figure with two equal sides ? 216%. The top-side and the (two) lateral sides are oach 25 in measure. The base is 39 in moasure. Tell me (here) the measuro 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-widdo is 25. In relation to this (quadrilateral figure), what is tho value of the diameter of the circumscribod circle) ? 2184. The measure of the side of an equilateral triangle is 6; and that of an isosceles triangle is 13, the baso in this caso) being 10 in measure. Givo out what tiie values are of tho diameters of the oiroles described about these triangles. 219]. In the case of a triangle with unequal sides the two sides are 15 and 13 in monsure; the base is 14. Tell mo the value of the diameter of the circle described about it. 2204. If you know the paikacika (processes of ouloulation), tell me after thinking well what may be the value of the dinmeter of the circle described about a (regular) six-sided figuro having 2 as the measure of cash of (its) sides. The rule for arriving at the numerical values of the base, of the top-side and the othor) sides of the eight different) kinds of Agures beginning with the aquare, which aro inscribed in a regular ciroular 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 (bypothetically) arrived diameter of the circle (described about an optionally choson figure belonging to ""6408. 220. The Kadarese commentary on this stansa works out this problem by pointing out that the diagonal of regular hexagon is equal to the diameter of the circumscribed oircle. 9914. The rule follows a matter of coure from the similarity of the required and the optionally chosen Algure.

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254 . GANITASABASANGRAHA. the specified variety). The values of the sides (of this optionally 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. 222}. 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 rule 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 figare, when the accurate measure of the area and the numerioal value of the perimeter are known in relation to those same) quadrilateral and other figuros :-- 2231. The (known) accurate moasure of the area of any of the figures other than thr longish quadrilateral figure should be divided by a quarter of the numerical value of the perimeter (of tbat figuro). The result is pointed out to be the diameter of the cirolo inscribed within that figure. Eramples in illustration thereof. 2247. Having drawn the inscribed circle in relation to the already specified figures beginning with the squaro, O you who know the secret of calculation, give out now (the value of the diameter of each such insoribed circle). 2981. If o represents the sum of the sides, and a the dian eter of the inscribed oirole, and A the area of the quadrilateral or the triangle in which the circle is insoribed, then Monoe the formula given in the rule ind

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CHAPTER VII-MEASUREMENT OP AREAS. ' 235 Within the (known) numerical measuru of the diameter of a regular circle, any known number being taken as the measure of an arrow, the rule for arriving at the numerioal value of the string (of the bow) having an arrow of that same measure : -- 2257. The difference between the giveu value of) the diameter and (the known value of) the arrow is multiplied by four times the value of the arrow. Whatever is the squaro root of tho resulting product), that the wise man should point out to be tho (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 tho punerion value of the arrowline, when the qumerical value of the diameter of a (given) regular cirole and the value of x bow-string line in relation to that circle) are (both) known : 2271. That which happens to be the squaro root of the difference between the squares of the (known) values of the diameter and the bow-string line (rolating to the given circlo)-- that has to bo 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 thereos. 2284. The diameter of the (givon) circle is 10 in moasure. Moreover, the bow-string line inside is known to be 8 in measure. Give out, o friend, wbat the value of the arrow-line may bo in relation to that (bow-string). 9254. The rules given in stansas 2254, 227), 2291 and 23|| are all based on the foot that in circle the rectangles contained by the segments of two intersecting ahords are equal.

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256 GANITASARASANGRAHA. The rule for arriving at the numerical value of the diameter of a (given) cirole when the namerical values of the related) bow-string line and the arrow line are known : 2294. 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 cirole measured through the centre. An example in illustration thereof. 2304. In the case of a regular circular figure, it is known that the arrow line is 2 dundas 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 bowe 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 linos connected with the overlapping portion when two regular ciroles out each other : 2317. With the aid of the values of the two diameters (of the two outting ciroles) as diminished by the value of the greatest breadth of) the overlapped portion (of the ciroles), 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 281). The problem here contemplated may be seen to bave been alao solved by Aryabhata, and the rule given by bit ooinoides with the one andor redaredoe hore.

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CHAPTER VIIMBABUREMENT OF ARBAS. 267 of (such) oiroles pointed out to be the values, each of the other, measnring the two arrow lines related to the overlapping (oircles). An exampk: in illustration thereof. 2321. In relation to two oiroles 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 respeotivoly to those two (circles), are (here). Thus ends the section treating of devilishly difficult problema. Thus ends the sixth subject of treatment, known as Caloulations regarding Areas, in Sarasangraha, a work on arithmetic by Mahaviracarya.

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258 GANITABARASANGRAHA, CHAPTER VIII. CALOULATIONS 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 vorious 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 ou bio hasta is 3,200 palas. From that (same oubio volume of excavation) 3,600 palas (of earth) may be taken out. The rule for arriving at the oubical contents of excavations : 4. Ares multiplied by depth gives rise to the approximate measure of the cubical contents in a regular excavation. The gums 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. 8. The term Andra in Aundraphala is rather strange Banokrit and in perhape related to the Hindi word sfing neaning deep.' 3. The idea in this stanza evidently is that one oubio hasta of compressed earth weighs 8,600 palas, whilo 3,200 palas of orth are sufficient to al loosely the pace of 1 cubio hasta. 4. The latter half of this stansa evidently gives the process by which we Day arrive at the dimensions of a ropular excavation fairly eqaivalent to say piron Irregular excavation.

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OILAPTER VIII.-CALCULATIONS REGARDING EXCAVATIONS. 259 Examples in illustration thereof. 5. In relation to (an equilateral) quadrilateral area (represent. ing 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 ench, and in the depth there are found 36 hastar and 6 angulas. What is the calculation (of the contents) here? 7. In relation to a (regular) circular arca reprosenting (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 cubioal contents are. 8. In relation to a longish quadrilateral area (forming the section of a regular ozonvation, the breadth is 25 hastas, the side (measuring the length) is 60 hastas and the depth (of the exoavation) is 108 hastas. Quickly give out (the cubical contents of this regular excavation). The rule for arriving at the accurato valuo of the oubical contents in the calculation relating to excavations, after knowing the result designatod karmantika as woll as the rosult dosignated aundra and with the aid of these results : 9-11. The values of the base and the other sides of the figure representing the top sectional aroa are added rospectively to the values of the base and the corresponding sides of the figure representing the bottom sectional area. The (Beveral) sums (80 arrived at) are divided by the number of the sectional areas taken into consideration (in the problem). The (rosulting) quantitios are 9-117. The figures dealt with in this rule aro tranoated pyramide with rectangular or triangular bases, or troncated cones all of which have to bo oodopived w turned upside down. The rule deals with three different kinds of meware of the obical contents of exostations. Of these, two, vis.. the Karmantika and Andra measures give only the approximate values of the contenta. lhe acourte measure is chloulated with the help of those valore. It K representa the Karmantika-phuls and A representa the Aundraphala then the socurate measure is mid to be equal to + K, ..., * + + 4.

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260 GANITASARABANGRAHA. 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 Oase 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 (aeparately) arrived at. The area values (80 obtained) are added together and thon divided by the number of (seotional) areas (taken into consideration). The quotient (80 obtained) is multiplied by the value of the depth. This gives rise to the oubical measure designated) the aundra result. If one-third of the difference between these two results is added to the karmantika result, it indeed becomos the accurate value (of the required cubical contents). Examples in illustration thereof. 124. There is a well whose (sectional) arca happens to be an equilateral quadrilatoral. 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 oalculation, tell me quickly what the oubical measure here is.. 131. There is a well wboso (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 be the measures of a side of the top and bottom surfaops respectively of truncated pyramid with a squaro baso, it can be ossily shown that the acourate measure of the cubivel ountonte is equal to sha' + b + ab), where is the buight of the truncated pyramid. The formula given in the rule for the scourate measure of the cubical oontents may be verified to be the same as this with the help of the following values for tho Karmantika and Aundra results given in the rule: K= (*+b)xhi A=4'11" xh. Bimilar vorifications may be arrived at in the case of tranoated pyramida aaving an equilateral triangle or rectangle for the base, and also in the one of tranoated cones..

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CHAPTER VOI-CALOULATIONS REGARDING EXCAVATIONS. .261 aundra cubical measure, and of the acourate oubical measure here? 141. There is a well whoso (scctional) area happons to bo regularly circular. The (diameter of the) top (sectional arou) is 20 dandas, and that of the bottom (sectional area) is ouly 16 dandas. The depth is 12 dandas. What may be the karmantika, the coundra, and the acourate cubical measures here? 154. In relation to (an excavation whose sectional arca happens to be) a longish quadrilateral figure (i.e., oblong), the length at the top is 60 (hastas), the brendth is 12 (hastax); at the buttom, these are (respectively) half (of what they measuro at the top). The depth is 8 (hastas). What is the cabical measure here? 164. (Here is another well of the samo kind), the longthe (of whose sectional areas) at the top, at the middle, and at the bottom are (respectively) 90, 80, and 70 (hastus), and the breadths are (respectively) 32, 16, and 10 hastas. This is 7 (hastus) in depth. (Find out the required cubical measuro.) 174. 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 oubical measure ? 187. In relation to (an excavation whoso sectional Arca happens to bo) a triangle, each of the three sides measures 80 hastas at the top, 60 hastas in the middle, and 50 hastas at tho bottom. The depth is 9 hastas. What is the calculated result giving its cubical contents ? The rule for arriving at the value of the cubical contents of a ditch, as also for arriving at the value of the cubical contents of an excavation having in the middlo (of it) a tapering projection (of solid earth) : 191201. The breadth (of the central mass) increased by the top-breadth of the surroupding ditch, and (then) multiplied by 197-20. Those stanga deal with the messurement of the cabio oontents of . ditoh dug round contral mass of earth of any shape. The coptoel mens may be in section square, a rootangle, an aquilateral trianglo, or oirolo

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+362 GANITASIBASANGBAHA. 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 rosults) 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 Karmantskaphala) is to add the value of) half the breadth of the excavation to that of the breadth of) the central nass, 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. 214. The already mentioned trilateral, quadrilateral, and circular (areas) have ditohes 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 ont the cubioal contents.) 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 role enables us to find out the total longth of the ditoh in all these cases. I. When the width of the ditch is uniform, the length of ditch = (a + b)x 3 in the orso of nn equilateral triangular or ciroular ditch, where d is the measure of a side or of the diameter of the oentral mass and b is the width of the ditch: bat this length = (d + b) x 4 in the case of a square oxcavation with a cuntral mass, square in section. II. When the ditch is ta pering to a point at the bottom or the top, the length of the ditoh for finding out the Karmantika-phala = * 4, sooording as the oentral masa (1) is in section trilateral or circular, or (3) square. Length of ditoh for finding out Aundra-phala = (a + b) x 8 and (a + b) x 4 respectively, Those expressions have to be multiplied by half of the width of the ditob and by ito dopth for finding out the respective oubioal phalas. The formulas given above in relation to triangular and oiroular excavations give only approximate results. With the aid of the total length of the ditoh to obtained, the oubica! contents are found out in the one of ditches with Wloping side by applying the ralo piven in start to 11} above.

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OHAPTER VIN-QALOULATIONS REGARDING EXCAVATIONS. 263, 224. The length of a longish quadrilateral is 120 (dandas) and the breadth is 40. The ditch around is as big as 4 danlas in breadth and 3 in depth. (Find out the cubical contents.) The rule for arriving at the value of the oubionl contents of an excavation, when the depth of the excavation varios (at various points), and also for arriving, when the cubioal contents of an excavation are known, at the depth of digging noocesory in the case of another (known) area (so that the oubical contents may be the same) :-- 23}. The sum of the depths (measured in different places) is divided by the number of places; this gives rise to the (avorago) depth. This multiplied by the top area (of the oxcavation) gives rise to the (required) cubical contents of the excavation in the case where that area is trilateral, quadrilateral or circular. The cubical contente (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. 241. In an equilateral quadrilateral field, the ground covered by which has an extent measured by 4 hartas (in longth 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)? 254. There is a well with an equilateral quadrilateral section, the sides whereof aro 18 hastas in measure; its depth is 1 hantas. With the water of this (well), another woll measuring 9 hastas at each of tho sides of the section) is filled. What is the depth (of this other well)? When the measnres of the sides of the top (sectional area) and also of the bottom (sectional aren) are known, and when the 297. Yor finding out the total fength of the surrounding ditoh whon the contral mon of earth is rectangular in section, the measures of the sidor M iporowsed by the width or hall the width of the ditch are added together, according tbe Ka mantika or tho Andra result is required.

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264 GANITASABASANGRAHA. measure of the depth also is known, in relation to a cortain 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 reduoed to a mere point : 264. The product resulting from multiplying the (given) depth with (the given moasure of a sido 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 mcasured (from the pointed bottom) upwards (to the position required) multiplied by the moasure 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) moasure at the top is 20 and at the bottom 14. The depth given in the heginning is 9. (This depth has to be) further (carried) down by 3. What will be the side value (of the bottom here) P What is the measure of the depth, (if the bottom is) made to end in a point ? 26+. The problems contemplated in this stnnes are (a) to find out the full latitudo of an inverted wyramid or cone and (b) to ficd out the dimensiona of the cross section thereof at a desired level, when the altitude and the dimensious of the top and bottom surfaces of a truncated pyramid or oone are given.. Jl, in a truncated pyramid with square base, a is the monore of a side of the base and b that of a side of the top surface and the height, then 80001 ding to the rulo given here, I taken 48 the height of the whole pyramid , and the treasure of a side of the cross section of the pyramid at any given height represented by h al--11). H These formulas are applicable in the case of a cono as well. In the rule the monstre of the side of section forming the pointed , art of the pyramid is required to be added to H, the denominator in the second formula, for the reuson that in some cases the pyramid may not actually end in a point. Where, however, it dote ond in a point, the value of this side has to be soro na matter of ourno.

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CHAPTER VIII-CALCULATIONS REGARDING EXCAVATIONS. 485 The rule for arriving at the value of the cubical contents of . spherically boundod space : 287. The half of the cube of half the diameter, multiplied by nine, gives the approximate value of the oubical contents of a sphere. This (approximate value), multiplied by nine and divided by ten on noglecting the remainder, gives rise to the aourate value of the cabical measure. An example in illustration thereof. . 291. In the case of a sphere measuring 16 in diameter, oalou. late and tell me what the approximato value of its) cubioal measure is, and also the acourate measure (thereof). The rule for arriving at the approximate value as well as tho accurata 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) : 304. The oube of half the square of the side of the basal equilateral triangle) is multiplied by ten; and the equare 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 281. The rolame of a sphere an given bero in (1) approximately= * and (3) acourately= 14 The correct formula for the cubi. calomatonta of sphere is on; and this becomes comparable with the above valne, it is taken to be v Tu. Both the M88. read T4412 Tay, making it appear that the socurato value is of the approximau value, but the text adopted in 1977 a u which iskon the accurate valge of the approximate one. It is cany to me that this gives more socurata result in regard to the measure of tho cubical contents of a sphere than the other reading. 304. Algebraically represented the approximato valdo of the cabloul contonte of a triangular pyramid socording to the rule oomes to a l.com No, and the courate valae becomes equal to . * V *; whorse .

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266 : GANITA IRABANGBAHA. ton, gives rise to the accurately caloulated oubical contents of the pyramidal excavation. An example in illustration thereof. 317. Calculate and say what the approximate value and the acourate value of the cu bical 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 80 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 whioh 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. 8. 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) P In the Fourth Subjeot of Treatment named Rule of Three, an example (like this) has (alroady) been given merely as a hint; the e gives the measure of the altitude of the pyramid m also of a side of the bacal equilateral triangle. It may be easily seen that both these valuo yo soncewhat wide of the mark, and that the given approximate value is Tourer the correot valde that the no-onlled aponrato value,

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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 chaunel, 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 hill and also the measure of the water in the well. 37-884. 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 the measure of the water. 39-40. 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 obannel, (the section whereof is throughout) triangular, each side measuring 1 angula. 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. O friend, calculate and tell (me) the height of the mountain and the measure of the water. 88 to 42. The reference here is to the example given in stanzas 16-16 of chapter V-ide also the footnote thereunder. The volume of the water is probably intended to be expressed in dhas. (Vide the table relating to this kind of volume measure in slansas 36 to 38, chapter 1.) It is stated in the Kanarese commentary that 1' onbio angula of water is equal to 1 karpa. Then according to the table given in stanza 41 of chapter I, 4 karpas make one pala; according to stanza 44 in the same chapter, 12 palas make one prastha; and stansas 36 to 87 therein give the mlation of the prasthe to the vha. *

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288 QANITASIRASANGBAHA. 413-42}. 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 obannel, (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 of 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 exoavations. Calculations Relating to Piles (of Brioks). Hereafter, in (this) chapter treating of operations relating to excavations, we will expound calonlations relating to (briok) piles. Here there is this convention (regarding the unit brick). 43%. The (unit) briok 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 oubioal contents of a given excavation in a field and also at the number of brioks corresponding to the above oubioal oontents. 44. The area at the mouth of the excavation) is multiplied by the depth; this (resulting product) is divided by the oubio measure of the (unit) briok. The quotient so obtained is to be understood as the (oubioal) measure of a (brick) pile; that same (quotient) also happens to be the measure of the number of the bricks. Examples in illustration thereof. 464. There is a raised platform equilaterally quadrilateral (in sootion) having a side measure of 8 hastas and a height of 9 14. The oubloul mossure of the briok pile bore is evidently in terms of the unit brick

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OH APPER VIII-CALCULATIONS REGARDING EXCAVATIONS. 209 hastao. That (platform) is built up of brioks. O you who know oalculation, say how many bricks there are (in it). 404. A raised platforin, equilaterally triangular (in section), having 8 hastas (as its side measure), and 9 hartas as height, has been construoted with the aforesaid bricks. Calculate and say how many brioks there are in this (structure). 47. A raised platform, circular in section, baving a diameter of 8 hastas and a height of 9 haslas is built up with the same aforesaid) brioke. O you who know caloulation, say how many brioks there are in it. 481. In the case of (& 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 briok pile. 491. A boundary wall is 7 hastas in thickness, 24 hastas in length and 20 hastas in height. How many are the bricks used in building it P 50$. The thiokness of a boundary wall is 6 hastas at the top and 8 hastas at the bottom; its length is 24 hustas and height 20 hastas. How many are the bricks used in building it P 514. (In the case of a raised sloping platform), the heights are (respectively) 12, 16 and 20 hastas (at three different points); 101-811. In finding out the oubical contents of the wall, the average breadth calculated sooor ding to the rule, given in the lat. ter half of the 4 sboro, in uned Ao the Karman. tika valuo taken into 000 aideration here. 61}. This sloping platform la bounded at ito two ends by two Vertical plane, the top and the uide fortuos slany belos sloptag. The top forms na faolipod + -

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270 GANITAS BASANGRAHA. the measures of the breadth at the bottom are respectively) 7, 6 and 5, (the same) at the top being 4, 3 and 2 hastas ; the length is 24 hastas. (Find out the number of brioks 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 (intaot) in the nnfallen (part) and also at the number of bricks found in the fallen (part) :-- 524. 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 desoribed. 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 all through is 10 hastas. (A measure of) 5 hastas (in height) of that (platform) gets broken down and falls. How many are those (unit) brioks 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 brioks that have fallen down : plane, the breadth of which is 8 hastas at the raisod end and hastas at the other end. Vide diagram in the margin of page 369. 62. 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 platformelgebraically 2 + 0; where a in the bottom-broadth, in the topbreadth, the total height and d the height of the fallen part of the rained platform. This formula can be easily shown to be correos by applying the properties of similar triangles. The operation referred to in the role as having been already doucribed in what in given in stann 4 sbove.

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CHAPTER VIII-CALCULATIONS BEGARDING EXCAVATIONS. 271 544. 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 aro (rospootively) inoreased and decreased by the height (above the ground) of the unbroken (part of the wall); and (then the quantities so obtained) are multiplied by the length and also by the sixth part of the (totalt height. . (Thus) the number of bricks intact and the number of bricks fallen off may be ubtained in order. Examples in illustration thereof. 551. This high fort-wall (of measurements already given, struck by 8 oy olonic wind) bas been (obliquely) from the bottom, broken down along the diagonal section. In relation thereto, how mans are the bricks intact and the bricks fullen down 561. The same high fort-wall has been broken down by the cyclone obliquely after leaving over 1 hasta from the bottom. How many are the bricks that remain intact and how many the bricks that have fallen down? The rule for arriving at the growing number of layers (of brioks) 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 1 height and the length of the wall, and a the height above the ground of the unbroken part of the wall then 12a + 0 + d), and (20 + a-d) represent the vumber of bricks intact and the number of brick. fallen 12 oft. The Bgare in the purio shows the wall mentioned in Atanen 68), and ABCD adlante the plane along which the wall fractured when it broker

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272 GANITASABASANGRAHA. (happening to be the diminution in breadth) on both the aides (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 bricka). 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. 58. 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). 59-60. In a regularly circular well, 4 hastas in diameter, a wall of 1 hastas in thickness is built all round by means of (the already mentioned typical) bricks. The depth of that (well) is 3 hastas. If you know, calculate and tell me, () friend, how many are the bricks used in the building. In relation to a structure built of brioks (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) in (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 48 above. This problem does not illustrate the rule given above in stanza 57. but it has to be worked according to the rales giran in stanzas 191-20) and 444 of Chis chapter.

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CHAPTER VIII-CALOULATIONS 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 I at tho top. Its height is 10. (What is the cubic measure of this wall ?) Thus ends (the seotion on) the measurement of (brick) piles in the operations relating to excavatione. Hereafter, we shall expound the operations relating to the work done with saws (in sawing wood). The definitions of torms in relation thereto are as follow : 63. Two hastas less by six angulax is what is onlled 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 angulus measuring the breadth, and those measuring the length, and the number of margas are all three) multiplicd together. The resulting product is divided by the square of the number of wigulus found in a hasta. In operations relating to Haw-work, this gives rise to a valuation (of the work as measured) in what is known 48) patrikas. In relation to logs (of wood) consisting of teak logs and other such logs, the number of hastus measuring the breadth and of those measuring the length are multiplied with each other, and (then) multiplied by the number of margar, 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. 68 to 87. Kipku = hasta. Marga is the name given to any desired course or line of us wing in log of woud. The extent of the out surface in log of wood measures ordinarily the work done in sawing it provided that the wood is of definite hard nye sastmed to be of unit valge. This extent of the cat marface is menanred by means of special unit area which is called pati and is 96 angular in length and one kipku or 49 angul asin breadth. It io. owy to Ree that paffim is thus equal to seven square hastan. 86

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274 GANITASABASANGRANA. 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 riparni 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 3 hastas and the margas (or saw-courses) are 8 in number. How many are (the units of saw-work) done here ? U 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.

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CHAPTER IX-CALCULATIONS RELATING TO SHADOWS. 275 CHAPTER IX. VALOULATIONS RELATING TO SHADOWS. 1. That Jiga, Santi, who bestows peaoo 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 bis enemies, I bow in salutation. In the beginning, we shall give out the means of determining the eight directions commencing with the oast. 2. On an even ground-surface which is (a horizontal plane) like the apper 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 (fixod in the centre). 8. The shadow of that optionally chosen style fixed in the centre of that circle touches the circumferential) line of that oirole 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 diroo. tion are pointed out in order. 4. By means of the string running in the line of these two (Avoertained) directions, a fish-shaped figure (or luno) should be R 4. The string with the aid of which the fish-shaped figure is drawn should be longer than the radius of the ofrolo drawn sooording to starsza 3 sboro. It OE and Ow in the annosed diagram represent the enstern and the westera directions, W NPSR will be the lace drawn by dosaribing two circles with contros redpootively at E and W and with XR and WP M equal radii. The line N8 cutting the angles of the lune marks the northern and the southern directions. E

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276 GANITASARASANGBAKA. drawn which will extend from north to south. The straight line running through the middle of the angles of this (fish-shaped figure) represents of itself the northern and the southern direo. tions. The intermediate directions have to be ascertained as being derivable from half the (interspace between these) direotions. 41. 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. 54. In Lalka, Yavakoti, Siddhapuri, and Romakapuri, there is no (such) eqninootial shadow at all; and, therefore, the day-time is of 30 ghatis. 04. 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) 74. 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 & given time in the forenoon or afternoon) in relation to a place where there is no equinootial shadow : 81. One is added to the measure of the shadow (expressed in terms of the height of an objeot), 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 olapse is, according to the rule given hero, equal to 1 ,6- 1 , where A is the angle repro 26+1) 2 (oot. A + 1)' senting the altitude of the sup at the time. It may be seen that this formula rives only the approximate value of the time of the day in all cores erompt when the altitude in 46', and that the approximation is very rough only in the ouse of largo altitudes, nearing 90deg. The forman la seems to be based on the fact that for a ball values the angles in right-Apgled triangle are approximately proportional to the opposite sides.

Page #479
CHAPTER IX-CALCULATIONS RELATING TO SHADOWS, 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) Sarasangraha, 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. . 9. The (length of the) shadow of a man is 3 times (his 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) ghatis, when the portion of the day (elapsed or to elapse) has been arrived at (already). 10. The (known) measure of (the duration of) the day multiplied 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. 11. 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 clapse) is also t. What are the ghatie (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. 12. 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 terms of his own height). *

Page #480
278 ::.. GANITASARASANGBAHA. ..-. 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. (Ita) conclusion 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). 141. 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 equinoctiul shadow, when the measure of the shadow at any time is known : 151. 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 (80 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 (elapoed or to elapse) happens to be , what are the ghatis (ourresponding to it), the duration of the day being 80 ghatis ! 167. Algebraically the formula given here for the measure of the time of the whore in the length of the equinoctial shadow of the stylen Phan formula is obviously based on the formula giren in the note to the nuta sanaa 8 sbor. " day i got a

Page #481
CHAPTER X-CALCULATIONS RELATING TO SHADOW8. 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 quotiont) 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 angular, the (equinoctial) shadow is 2 angulas, what is the measure of the shodow (of the style) at a time when i ghatis have clapeod or have to elapso, the duration of the day-time being 30 gharis? The definition of the ineasure of a man's foot in rolation to measurements carried out by means of the foot-measure as involved in the shadow. 20. One seventh of the beight of a person happens to be the length of that person's foot. If this be so, that person shall bo 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 asoended up (& perpendicular wall). 21. (The height of) the style is multiplied by the measuro of tho human shadow (in terms of the mau's height). Thu i resulting) 18. Algebrsically 12 .a + , where y in the measure of the time of the day in ghafis. Thin formula may be seen tu follow frm thnt given in the bute to stanza 161 above. 91. Aigebraically, h hosil *. * **, where a in the ultitado of the shadow. casting stylo, h the height of the shadow on the wall, o the measure of the human shadow in terms of the man's height, and a the distance be tween the pillar and the wall. The diagram here given clacl. dates the rule. It has to be noted here that the distance between the pillar and the wall has to be measured alongelbe time of the shadow which is cast in sunlight,

Page #482
280 GANYALBASANG RANA, product is diminished by the measure of the interval between the wall and the style. The difference (80 obtained) is divided by the very measure of the human sbadow (referred to above). The quotient so obtained happens to be the measure of (tbat 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 (wall), multiplied by the moasure 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 tho height of the pillar and that of the portion of) the shadow thereof cast on (the wall), gives rise to the measure of the buman shadow (in terms of the man's height). An esample in illustration thereof. 24. A pillar is 20 hastas (in height); and the (portion of its) sbadow on a wall is 10 (hastas in height). The buman shadow (at the time) is twioe (the human height). What may be the measure of the inters pave between the pillar and the wall ? 14. This rule and the one in atann 98 following fire the converse Onsen of ther the instans$1 above,

Page #483
CHAPTER 1X.CALCULATIONS RELATING TO SHADOWS. 281 An exampld in illustration of the rule in the) latter half (of the stanga). 25. 'A pillar ia 20 hastas (in height) and the (portion of its) sbadow on a wall is 16 (hastas in height). The moasure of the interspace between the wall and the pillar is 8 (hastus)What is the measure of the human shadow (in torms of the man's height) P The rule for arriving at the numerical value of the boight of & pillar, when the numerical ineasure of the portion of its) shadow oast on (a wall and the measure of the interspace botween (that) pillar and (that) wall, and also the human shadow (in terms of the human height) are knowi. 26. The measure of the (pillar-shadow) cast on (the wall) is multiplied by the measure of tbe buman shadow (in terms of the human height); and to this product the measure of the interspoo between the pillar and the wall is aided The quotient obtained by dividing the sum so resulting) by the measure of the human shrlow in terms of the human height) is mado out by the wise to be the moasuro (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 (hastax). 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 pillor 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 gum of the measure of the stylo and the measure of the shadow (thereof), when diviiled by the measuro of the buman shadow (in terms of the hunian height) as increaecd by one, gives rise to the measure of the beight of the style. The measure of the sbadow of the style is of course the (given) combined sum diminished by this (measure of the style). 26. Vide note ander stansa 23 above. 88 and 80. The rales here given are based on the rule stato in the latter half of the range 134 aboro. 86

Page #484
282 GANITASARASANGRAHA, An example in illustration thereof. 29. The combined som 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 stylo and the measure of the human shadow in terms of the human height) from (their) combined sum : 30. The comhined sum of the moasures 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 (80 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-measure 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 produot of the square of the human shadow and the square (of the height) of the style is to be subtracted from the 32-83. Let AB represent the position of a slanting pillar, and AC its shadow : and let AD be the same pillar in the vertical D AE position and AE its shadow. Then Shadou. " AD is equal to the ratio of the shadow of a van to his height at the timo; and let this ratio be. BC, the perpendiovlar from Bon AD, representa the amuant of slanting of the pillar, AB. It can be easily 1 shown that N AB' - BG|_ AD AU - BGA From this it can be seen that BG= AO NAC'-(AO - AB'xr) (+1). +1 > The ralo here rire this samo formala.

Page #485
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 romainder) is to be subtracted from the (given) measure of the shadow; and, whon (the quantity thus obtained is) divided by the sum of) one and the square of the human shadow, there results oxactly the measure of the inclination of the pillar. An exumple in illustration thereuf. 34. The human shadow (at the time) is twioe (the human height). Tho shadow of a pillar, 13 hastas in height, in 29 (hastas). What is the measure of the slanting of the pillur horo? (General Examples). 35-37. A certain prince, staying in the interior of x polaco, was, (at a certain moment) in the course of a forenvon, 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 throngh a window at a height of 32 hastas in the middle of the eustern 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 mathon aticiun, if you havo taken paine (to acqnaint yourself) with shadow-problems, calculate and give out the measure of the time clapsed then, on that day, and also the measure of the human shadow (at that time in terms of the human height). 381-391. 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 (sectiou) measuring 10 hantar in cach dimension, will be the height of the shadow on the westorn wall caused by the eastern wall (thereof) ? O matbomatician, give out, if you know, how you may arrive at the value of the shadow that has ascended up (a perpendicular wall). 86-874. This example beurs on the roles given in stanza 84 and 28 abovo. 381-391. This exau ple has to be worked out according to the tale giron in tann 91 above.

Page #486
284 QANITABARABANGBAKA. The rule for erriving at the shadow of a style due to (the light of) a lamp : 401. The height of the lamp as diminished by the height of the style is divided by the heigbt 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. 411-42. The (horizontal) distance between a style and a lamp is in fact 98 angulas. The height of the flanie of the lamp (above the floor) is 60 (angulas). () 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). The rule for arriving at the (horizontal) distance between the lamp and the style :- . 43. The height of the lamp (above the floor) is diminisbed 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 qnotiont 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). Tho height of the flame of the lamp (above the floor) is 60 (angulas). 401. Algebraically stated the rule 8=c 0-4; where s is the length of the shadow of the style whose 6 height is represented by a, b is the height of the lamp above the ground, and the horisontal distance between the lamp and the style. The formula may be soon to be correct by means of the diagram here given. 43. Using the same symbolo, c = 1 x 4. The given measure of the height of the style is 18 angulae, vide stanga 46-47 below.

Page #487
CHAPTEB 1X.-CALCULATIONS RELATING TO SHADOWS. 286 O yon who have gone to the other shore of the ocean of caloulation, 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 no 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 tho) sbadow of the stylo is oxactly twice (its height). The measure of the intervening (horizontal) distance between the style and the lamp is 200 angular. What is the measure of the height of tho 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 rulo works out is to be learnt well. The rule for arriving at the numerical mousuro 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 kuown; as also for arriving at the numerical measure (of the length) of the shadow of the tree in terms of that same foot-mtasure, when the anmerical 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. The measure (of the length) of the shadow of the tree chosen by a person is divided by (the foot-measure of the length 46. Similarly, o=( + 1) ... 48. This deals with a converse came of the rule given in the latter hall of sans 12 above. The relation between the height of a man and his foot- ARITO I alised in the statomopt of the rule w given here.

Page #488
286 GANITASARABANGRAHA. 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 inoasure (of the length) of the shadow of the tree exaofly. An example in illustration thereof. 49. The foot-measury (of the length) of one's own shadow is t. I'he (length of the shadow of a tree is 100 in terms of the (samo) 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 tree then P 51-52. After going over (& distance of) 8 yojanas (to the east) of a city, there is a hill of 10 yojanas in height. In the city glso there is a bill of 10 yojanas in height. After going over (a distanou of) 80 3ojanas (from the eastern bill to the west, there is another hill. Lights on the top of this (last mentioned hill) are seen at nights by the inhabitants of the city. The shadow of the hill lying at the centre of the city touches the base of the eastern bill. Give out quickly, O mathematician, what the height of this (western) bill is. Thus ends the eighth subject of treatment, known as Calculations relating to shadows, in Sarasangraha, which is a work on arithmetic by Mahaviracarya. 80 ENDS TAIS 8 ARASANGRA RA. 61-685. This example is intended to illustrato the rulo given in stanna 46 above.

Page #489
287 APPENDIX I. SANSKRIT WORDS DENO PING NUMBERS WITH THEIR ORDINARY AND NUMERICAL SIGNIFICATIONS. appy ... The eye ... 2 Men have two eyes. affra ... Fire ... ... 3 The number of sacrificial fires in threr, viz., , rafty, and for 372 ... Number ... There are only nine numerical figuren oxcluding the zero. IF .. An auxiliary divi 6 There are wix auxilinry department of sion or depart study in relation to the Valus, viz., ment of science. 797, 4. ETT TOT, PET, 37, ura. 37 .. A mountain ... 7 Bewen principal mountains called Kuld. calas are recognised in the geography of the Porunas, viz., 6-4,471 AU, 3 , , fq7, MITTE. apr ... A mountain ... 7 VidH 3TEC. 27 ... The sky ... 0 The way in considered to be void. 37773 ... Fire ... ... 3 Vide 3fa. ariis ... An atmy ... 8 "Thero nre right kind of army mentioned in Sanakrit, viz.,, ofer, aarga. TA To, arteaft, 7777, 7a, staffiaft Batu ... The sky ... Vide 37777. afry .. The ocean ... It io hold that there are four venna, vir., Artern, southern, western nod northern. apr ... The age ... ... 3 Vide af. PTT . The oky ... ... Vide 3777. Hits ... Tho ocean ... 4 Vide affou. autis ... The ocean ... Vide itu.

Page #490
288 azva azvin AkAza ina indu indra indriya ibha iSu ikSaNa udadhi upendra Rtu ... ... ... ... P ** A horse.... ... Consisting of horse. The sky The sun ... ... An .. The moon The god Indra ... GANITASARASANGRAHA. organ songe. An elephant An arrow The eye The ocean God Visnu ... A season ... ... ... of ... ... ... ... kara The hand karaNIya That which has to be done : an aot of devotion or austerity. 7 7 0 12 1 14 Fourteen Indras are usually reckoned at the rate of one Indra for each of the fourteen manvantaras. There are five organs of sense, vis, nose. tongue, eye, skin and ear. Eight elephants are said to guard the eight cardinal points of the world. They are airAvata, puNDarIka, vAmana, kumuda, aJjana, puSpadanta, sArvabhauma, and supratIka. The arrows of Manmatha or the Indian Cupid are declared to be five, viz. aravinda, azoka, cUta, navamallikA, Bud nIlotpala. 8 Vide akSi. 4 Vide abdhi. 9 5 8 The horses of the sun's chariot are supposed to be seven. Vide azva. Vide ananta. The number of suns is reckoned to be 12 corresponding to the 12 months of & mitra, aryamanU, rudra, year, viz., dhAtR, varuNa, sUrya, bhaga, vivasvat, pUSan, savitR, tvaSTR and viSNu. They are called the twelve adityan. We have only one moon. 5 6 2 5 There are said to be nine Vispus corresponding to the nine past incarnations of Vippu. There are, scoording to Sanskrit literature, six seasons in year, viz. & vasanta, prISma, varSa, zarada, hemanta, and zizira. Human beings have two hands. There are 5 vratas or austerities to be observed according to the Jaina religion, vis., ahiMsA, sUnRta asteya, brahma, and aparigraha.

Page #491
APPENDIX I. 289 karin ... An elephant ... 8 Tide ima. karman ... Aotion : the effect_8 Acoording to Jainaa there are eight of action as its kinds of karma, vir. jJAnAvaraNIya, karma. darzanAvaraNIya, mohanIya, antarAya, vedanIya, nAmika, gotrika and Ayaka. kalAdhara, ... The moon ... 1 Vide indu. kaSAya ... Attachment to 4 According to the Jains religion there are wordly objects. four causes for such attachment, viz., krodha, mAna, mAyA, lobha. kumAravadana The faces of Ru. G This War-god ia gupposed to have aix mara or the faces. C. SaNmukha. Hindu war-god. kezava ... A name of Vian. vide upendra. . kSapAkara ... The moon ... | Vide indu. kha ... Bky ... ... o ride ananta. khara .... ....... 6 gagana ... sky ... ... 0 Vide ananta. gaja Elephant ... 8 Vide ibha. gati ... Passago : passage 4 According to the Jaina roligion nouin may into re-birth. have four kinds of embodimen", vis., ___4 deva, tiryaka, manuSya, naraka. giri ... Monntain ... 7 Vide acala. guNa ... Quality ... 3 Primordial matter is maid to have three __ qualition', vir., satva, rajas, tamas. ha ... A planet ... 9 Nine planets are recognised in llinda Astronomy, viz., Mars, Moroury, Jupi. ter, Venus, Hatorn, Rahul, Kotu the 8nn and the Moon. cakSusa ... The eye ... ! Vide akSi. cana ... The moon ... 1 Hide indu. candramasa ... The moon ... 1 Vide indu. jalaparapatha 8y ... .. o Vide ananta. jalAdhi ... Ocean ... ... . Vide adhi . jalaniSi.. Ooosn ... ... - Vids Andha.

Page #492
290 GANITASARAMANGRALA. jina 11 TEE ... Name of a Jaina 24 According to the Jainas there are 84 saint. tirthankaras or saints. 3897 ... Fire ... ... 3 Vide fora. T ... Elementary prin. 7 The Jainas reoognise soven suoh prinoi. ciples. ples, viz., sta, arsta, 314, TT, ples, viz., 114,8 Aar, fast, ATT. 77 . Body ... 8 siva is considered to have his body nade ap of eight things, viz., gerai, 371, Ant, aty, 3977, art, T yajamAna. at ... Evidence ... 6 The six kinds of evidence are 97 39417, JUHTA, 375, suffer, and bhanupalabdhi. arealet. Viepu ... ..., 9 Vide Sta. a ... Tirthankars or 34 Vide 157 Jina. Port ... An elephant ... 8 Vide . TITT ... Worldly aotion ... 8 Video Name of a mani. 9 Nine separate manifestations of Durga are festation of Par. recognised vati or Durga. f ... Quarter or 8 There are eight cardinal points of the cardinal point of universe. the universe. to ... Do. 10 Ten directions are recognised, namely the eight cardinal points of the universe, the upward and the downward direotions. ... Sky ... .. 0 Vide 3PTAT. 6 ... The eye ... Pido f. gte ... The eye ... ... % Fide 377. O ... Elementary s Elementary sah. Awoording to the Jains there are six stanoe. varieties of elementary mabstance, vis., fta, a ph, park, C, T, a 1. EEEE

Page #493
dvipa dvirada dvIpa dhAtu dhRti waa, naga manda nabhas naya nayana* nAga fadha netra padArtha pacana ... ... ... ... ... ... 100 ... 000 *** An elephant An elephant A Puranio insular division of the terrestrial world. Constituent principles of the body. Name of a kind 18 of metre. Mountain Name of a dynasty of kings. Sky Method of oom The eye, An elephant Treasure payodhi Ocean payonidhi... Ocean pAvaka Fire prehending things from particular stand-points. The eye Category of things. The serpent 10. ... *** ... ... ... *** APPENDIX I. ... 000 ... 7 7 9 8 8 0 2 7 2 8 9 3 0 7 Vide ibha. Vide ibha. There are seven such divisions, vis. jambU, prakSa, zAlmalI, kuza, krIzca, zAka, pauSkara 291 These are said to be seven, viz., chylo blood, flesh, fat, hone, marrow, semen. Each line of a stanza in this metre contains 18 syllables. Vide acala. Nine Nanda kings are said to have reigned in Magadha. Vide ananta. According to Jains there are two Nayas: dravyArthikanaya and paryAyArthikanaya Vide akSi. Vide ibha. Nine famous treasures are maid to belong to Kabers, the god of wealth, viz., padma, mahApadma, zaGkha, makara, kacchapa, mukunda, kunda, nIla, kharva. Vide akSi. The Jainas recognize nine categories of things. Sometimes eight and sometimes seven principal serpents are reckoned in Hindu mythology. $ Vide Agdha. 4 Vide abdhi. 8 Vide agni.

Page #494
292 GANITAS RASANGBABA. go ... City ... ... 3 Three oitios representing three Asuras are said in the Paramas to have caused great bevoo to the gods, and Siva is said to have destroyed them. Of. goatia ... Elephant 9.9TEI ... The moon qayt ... Bondage ... ... 8 vide 5a. 1 Vido 75. 4 The Jainas recognize four kinds of spiritual bondage, vie., fa, ferra, arata and 41. T 7 ... Arrow ... ... ... A constellation 6 27 Vido . The Hindu astronomers count 27 chief stelllar constellations or luuar man. sions around the ecliptio. 774 ... Elemento ... 5 TFT ... The sun 497 ... The world ... 12 ... 3 Five elements are recognized, vis., feeft, 379, iste, aty, 3174137. Vide 7. The number of worlds ordinarily counted are three, vis., the upper, the lower and the middle worlds. Vido T. Pide BTT. TOT TET H HEINT ... Element ... Mountair ... Passion ... Mountain . A guddess ... ... ... ... ... 6 7 8 7 7 8 grat ... Sage ... ... Tide acala. Generally seven of these goddesues are enumerated. soven chief sages are usually mentioned they are, t, f, TETIT, famant, of , fiefon, afhs. Pide Fe. It is held that there are oloven Rudru. Tout ... The moon ... 1 ... A name of Sirs or 11 Rodra para ... Sago.. . y ... Tho mean . 1 Fido tanto ride F.

Page #495
rakha rava randhra rasa rudra rUpa laggha labdhi lekhya loka locana varNa vasu ... * BBBB ... ... ... vaDi vAraNa varSa vidhu viSAdha viSanidhi.. viSaya BARE:: ... Excellent thing.... A precious gem... Opening Tasto Name of a deity Form or shape ... Attainment; attainment of the nine powers. Attainment World The eye A class of Vedic deities. Fire Elephant Ocean ... The moon viyat Bky Ocean Osevd Object of sense... APPENDIX I. ... ... 3 9 9 8 11 1 9 9 6 3 2 G 8 8 8 4 1 " 4 5 0 There are three excellent things for Jainas, vis, samyagdarzana, samyagjJAna, samyakcArita. Nine gems are usually recokned, viz., vajra, vaidUrya, gomeda, puSparAga, padmarAga, 93 marataka, nola, muktA, pravAla. There are nine chief openings in the human body. The six principal tastes are madhura, amla, lavaNa, kaTuka, tikta, kaSAya. Vide mRDa. Everything has its one only shape. The nine powers to be attained are anantadarzana, anantajJAna kSAyakasamya, kstvakSAyakacAritra, anantadAna, anantakAbha, amantabhoga, anantopabhoga and anantavIrya. Vide labdha. Vide bhuvana. Vido akSi. Those deities are considered to be eight in number. Vido agni. Vide ibha. Vide abdhi. Vide indu. Vide abdhi. Vide effort. The objects oognisable by the five organs of sense are five, viz. gandha, rasa, rUpa, sparza, zabda. Vide ananta.

Page #496
294 GANITASIBASANGRAHA. ar ... A group of Vedic 18 deities. rasyara... Sky .....0 at ... The Vedas ... 4 This gruap of deities is said to consist of 18 members. Vida 3777. There are four Vedas, **, JH, FTA, apere. a TTT ... Fire H 8 Vide 37 a. vyasana An anwholesome 7 Beven each addictions are prohibited in uddiction. the case of kings. OTA ... Bky ... ... 0 Vide 377777. T ... Act of devotion 6 Vido mirofit. or austurity. TFT ... Name of Rudra... 11 Vide 98. 37 ... Arrow ... ... 5 Vide . 73797 ... The inooc ... 1 Vide Frs. 77615339 The moon ... Vide Fe. Trg ... The moon i Vide fr. 22 ... The moon ... 1 Vide 575. TH ... Arrow ... ... 6. Vido T. f eat .. Fire ... ... 3 Vido syfta. fretaanThe lege of a 8 The logs of a bee are bold to be six. bee. te ... Mountain ... ... Vide 3 . with ... ..... salilAkara Ooen ... ... ATT ... Dovan ... Vide apoy. arga ... Arrow ... ... Vide . Peraye .... Elephant ... 8 Vido 4. prof . The sun ... ... 18 vide 97. ATH ... The moon . Viders. Frota ... Elephant ... 8 Tide 74. RR ... A note of the 7 dovon notes are rooognised in the Hindu musical voule. musical soalo, vis., H, A, T, #, 1,

Page #497
APPENDLX I. 296 80 ... Horne .... ... 7 Pide app. ... Name of Radra ... 11 Vido He. S ET ... Siva's eyon ... 3 Siva is said to have one extra eye in the middle of the forehead making up three in all. gan ... l'iro ... ... 3 Vide 3 GOTT ... Fire .... ... 9 Vide affia. ... 'The moon 1 Vido FS fond ... The mood ... 1 Vido 75. fchig ... The moon ... i Vide Fs. himakara

Page #498
296 APPENDIX II. SANSKRIT WORDS USED IN THE TRANSLATION AND THEIR EXPLANATION. Abadhd ... Adhaka Adhvan ... Adidhana ... ... ... Begment of a struight line forming the base of triangle or a quadrilateral. A measure of grain. Vide Table 3, Appendix IV. The vertical space required for prosenting the long and the short syllables of all the possible varieties of mutre with any given number of syllables, the pace 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-833 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. Se note to II-80 to 82. ... A kind of fragrant wood; Amyris agallooha. ... A kind of sorrel; Rumex vesicariwe. ... Name of a king; lit: one who showers down truly useful rain. A messure of weight in relation to metals. 86 Table 6, Appendix IV. Square root of a fractional part. 8o9 note to IV-. Bquare of a fractional part. So note to IV-8. An auxiliary science. ... A measure of length; finger meneru. 80 1-95 to 29 also Table 1, Appendix IV. ... Inner perpendioular; the mossure of . string suspended from the point of intersection of two stringa stretohed from the top of two pillars tos point in the lino pusing through the bottom of both the pillars, Agar ... Amla-pitam : Amoghavarra ... ... Ana .. Amiamala ... Ambavarga ... Anga-Sastra Adgula ... ... Anidriva lambako ...

Page #499
Antyadhana Am Aristanemi Arbula Arjuna Asita Aicka Avali Ayana Bija Bhaga ... ... Aundra-Aundraphala Bhayabhdga Bhd gabhyd sa Bhajya *** Bhd gandra... Bhagamatr BMra Bhaganubandha Bhd gd pava ha Bhd gasa mearga ***B Bhinnadriya ... *** ... ... ... ... APPENDIX II. 297 The last term of a series in arithmetical or geometrical progression. Atom or particle. See stanzas 25 to 27, Chaptor 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; Grialea Tomentosa, Name of a tree; Jonesia Asoka Rorb. A kind of approximate measure of the cubical contents of an oxoavation or of a solid. See note to VIII-2. This kind of approximate measure is called Auttra by orahmsgupta. A measure of time. Vide Table 2, Appendix IV. Do. do. Literally seed; here it is used to donote a set of two positive integers with the aid of the product and the squares wheroof, as forming the measure of the sides, a right angled trianglo may be cons tructed. Vide note to VII--201. A measure of baser metals. Vide Table 6, Appendix, IV. A simple fraction. A variety of miscellaneous problems on fractions. See note to 1V-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, Prabldga, Bhagabhaga, Bhagdnubandha, and Bhagipavaha fractions. See note to III-138. Fractions in association. Vide note to III-118. 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 cabe root group; that which has to be divided. See note to II--53 and 54. A measure of baser metals. Vide Table 6, Appendix IV. A variety of miscellaneous problems on fractions. See note to IV-8. * 38

Page #500
298 GANITARSRASANGRAHA. Bhinnakufftkdra ... Cakrikabhanijana ... Campaka ... Chandas ... Citi Citra-kuffikdra Citra-kuffikdramira Danda .. Data Dala-koti ... Dasa-lakpa Dabaraharra Dharana ... Dindra ... ... Proportionate distribution involving fractional quantities. See footnoto in page 136. ... The destroyer of the oyole of recurring rebirtha, also the name of a king of the Raptrak ata dynasty. Name of 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 & curious and interesting nature involving the application of the operation of proportionate division. ... A measure of distance. Vide Table 1 of Appendix IV. 'Tenth place. Ten crore. 'Ten lakhs or one million. Ten thousand. A weight measure of gold or silver; Vide Tables 4 and 5 of Appendix IV. .. A weight measure of baser metalo. Vide Table 6 of Appendix 1V. Also gred as the name of a coin. ... A weight measure of baner metalo. Vide table 6 of Appendix IV. ... A measure of capacity in relation to grain. Vide Table 8 of Appendix IV. .. Name of a tree. ... A. variety of miscellaneous probleme un fractions. ... Unit place. ... A weight measure of gold. Vido Table 4, Appendix IV. Cubing; the first figure on the right, among the three digits forming & group of Agares into which a numerical quantity whose cube root to to be found out bas to be divided. 860 note to 11-53, 54. Cube root. ... A measure of tinie. Vide table 2 of Appendix IV. ... Multiplication. ... The produot of the common ratio taken many times w the number of terma in goometri. cally progressive series multiplied by the first term. So note to 11--98, Drakpuna ... ... Drona ... ... Dunduka ... ..... Dviragrafipa mula Eka ... ... Gampaka ... ... Ghana ... ... Ghanamula Ghaf ... Gupakdra ... Gumadhana ...

Page #501
APPENDIX ir. 299 Gunja . Basta Bintilla Iccha ... ... ... ... .... Indranila Jambu Janya ... ... Jinas Jinapati ... ... Jina odnti ... ... Jina-Vardhamana ... Kadamba ... Kali ... . ... Kald sa vara Karmas ... ... ... ... A weight measure of gold or silver. Vide Tables and 5 of Appendix IV. ... A measure of length. Mide l'ablo 1 of Appendix IV. Name of a trav, Phones or Elate Paludona. That quantity in a problem on Rule-of-Threo in relation to which something is required to bo found out according to the givon rate. ... Sapphire. Name of a tree; Eugenia Jambalona. ... Trilatoral and quadrilntoral figures that may be derived out of certain giren data called bijax, The great teachers of the Jaina religion; the Jains Tirthankaras, ... The Chief of the Jinon, Vardhamana. Name of Jaina saint; a Tirtharkara. ... Vardhamana, the great propagator of thu lwind religion and the last of tho Tirthankarui. ... Name of a trvo; Naurlet Cadambu. ... A weight measure of baner motala. Pide Tablu 0, Appendix IV. ... Fration. See footnote ch page 38 ... Consequence of acts done in provious birth. Ac. cording to Juinus the Karman are of eight kind. See under Hin Appendix 1. A kind of approximate measure of the cubioul contents of an excavation or of a solid, 8ro noto to Chapter VIII A. ... A weight mr.aspre of gold or milvor. Yule Tablon Hand 6, Appendix IV. A Karpa. ... Name of a tree, Pandanus Odory tiamimua. A nealore of capacity in relation to grain. The 13th place in notation. A moscore of longth in relation to the sewing of wood. Crore, the 8th place in notation. ... A numericul mossure of clotha, jewols and ance. Vide Table 7, Appendix IV. A casure of lengtb. Vide Tablo 1 of Appendix IV. ... A kind of fragrant wood, a Llauk variety of Agallo. chvn. Syaaring. Half of the difference between twice the first torm and the common difference in a wrio in arithmeti. oal progrynaion. Karmantika Karpa ... .. Karad pana XMA Iharda KA Katika Kniha Arrigaru... ... pipapada

Page #502
300 GANITASARASANGRAHA. Kritya Koobha .. Koori Kudaha or Kudaba ... Kwmbha ... Kunkuma ... Kuravaka ... Kufaja Kutfikara ... Labha Lakra Lanka Lana .. Madhuka ... Madhyadhana ... Mahakharga Malakoitud Mahaksobha Mahaksoni Mahapadma Maha sankha Mahavira ... Mani ... ... The 21st place in notation. The 23rd place in notation. The 17th place in notation, A measure of capacity in relation to grain. Vide Table 3 of Appendix IV. ... DS. do. ... The pollen and filaments of the flowers of saffron, Croens sativue. Name of a tree; the Amaranth or the Barlaria, Name of a tree ; Wrightia Antidysenterica. Proportionate division. See VI-797. Quntii nt or share. Lakh, the Cth place in notation. ... The place where the meridian passing through Ujjain meets the equator. A measure of tine. Vide Table 2 of Appendix IV. ... Name of a troe, Bassia Latifolia. ... The middle term of a series in arithmetical progrce sion. See poto to II-63. The 14th place in notation. The 22nd place in notation. The 24th place in notation. The 18th place in notation. The 10th place in notation. The 20th place in notation. ... A name of Vardhamana. ... A measure of capacity in rolation to graiu. Vid. Table 3 of Appondix IV. ... A kind of drum; for a longitudinal section, see noto to VII -32. ... Seotion; the line along which a piece of wood in out by & saw. A weight moasure of silver. See Tables 6, Appendix IV. ... Name of a fabulous mountain forming the oentre of Jambda vipa, all planets revolving round it. Mixed oum. See note to 11-80 to 82. ... A kind of dram; for a longitudinal section, see noto to VIII--32. A measure of time. Vide Table 8, Appendix IV. ... The topaide of quadrilateral. Square root; - variety of miscellaneous problems on fractions. Vide note to IV-3. ... Involving square root; * variety of miscellaneous * problems on fractions. Vide note to IV . ... Mardala Marga khara khn Mifradhana Urdanga ... Muharta Mukha Nola Xolamitra...

Page #503
APPENDIX 11. 801 Muraja .. Nandyd varla Narapala ... Nilotpala Niruddha ... Nipka Nyerbuda ... Pada Padma ... Padmard ga. Paisdcika ... ... .. . Pakya Paal Papa '. , . Panava ... Paramanu.. Parikarman Pariva Pafali ... A kind of drom; same as Mrdanga ... Name of a palace built in particular form. 8. note to V1-3325 King; probably name of a kiny. Blue water-lily. Lcast common multiple. A golden coin. The 12th place in notation. A measure of length. Vide Table 1, Appendix IV. The 15th place in potation. A kind of gem or precious stono .. Relating to the devil; hence vory cliffoult or com plex ... A moasure of time. Vide 'Table 3 of Appendix IV. .. A weight measure of gold, silver and other metals. Vide Tables 4, 5, 6 of Appendix IV. . A weight mensure of gold; vide table 4 of Appendix IV; algo a golden coin. ... A kind of drum; for lungitudinal nection weg toto to VII-32. Smallest particle. Vide Table 1, Appendix IV. ... Arithmetical operation. .. A Jainn saint ; one of the T'irlhunkaran. A treo with swect-scouted blussronia; Bignonia Suuveolens. .. A measure of saw-work. Vid. Tablo 10, Appendix IV; also note to VIII 68 to 077. ... A given qnantity corresponding to what he to be found out in a problem on the Hulo-of-Throu. See note to V-2. Name of a tree; the waved-leaf ny true, b'ious In fectoria or Religiosa. Fraction of a fraction. Miscellaneous problema. Proportionate distribution. An operation of proportionate distribution. . A measure of length. Vide Table 1 of Appendix IV. The given quantity correoponding to lechd, in problem on Rule.of.Three. See note to V-2. ... Literally, that which completes or fille, here, bewer metalo mixed with gold, drom. ... A measure of capacity in relation to grain. Wie Tables 8 and 6, Appendix IV, ... Kaltiplication. ... Paffikd ... ... Phela Plakps Prabhdga ... ... Protirpaka... ... Prakpipa ks ... Prakpopaka-karana Prawda ... ... Prepargid ... Prastha Tabaco Pratyuipenna ...

Page #504
Bo . GANITABIRASLAGBAHA. Prawartika Punndga ... Purdna NAM Pupyard ga Ratharonu Rimaka puri Rtu ... ... Sahasra ... Saka ... Sakala kuifikira .. ... Sala Sallaki Sama ya ... Sankalita .. Sankha ... Sunkramana Sankranti ... A measure of capacity in relation to grain. Name of a tree; Rottieria Tinctoria. A weight measure of silver. Vide Table 6, Appendix IV; probably also & onin. ... A kind of gem o: precious stone. . ... A particle. Pide Table 1, Appendix IV, A place 90deg to the west of Lankt. . Season, here used * & measure of time. Vid. Table 2, Appendix IV. Thousand, The teak tree. .. Proprrtionate distribution, in which fractious are not involved. The sal tree; Shorea Robusta or Valeria Robusta. Name of a tree; Boswellia Thurifera. ... A measure of time. Pide Tablo 2, Appendix IV. Summation of series. The 19th place in notation. An operation involving the halves of the sum ond tho difference of any two quantities. See note to VI-2. The passage of the son from one zodiacal sign to another. Name of a Jaina saint. Seo Jina-Santi. ... Nunie of a tree; Pinus Longifolia, A kind of bird. ... Literally, a brief exposition of the essentials or principles of a sabjoot ; here, the name of this work on arithmetic. Name of a trec; same as the sal trec. The sumn of a series in arithmetical progression. 866 note to II-68 and 64. ... A hundred. ... A handred crores. ... A weight measure of baser metale. Vide Table 6, Appendix Iv. The terms that remain in a serien after a portion of it from the beginning is taken away. Buo nota on page 84. A variety of miscellaneous problems on frobione. See note to IV 3. ... A variety of mipoeldous problems on fractions. See note to IV-3. .. . The antipodes of Lettled. #duti Sarala ... Sarasa ... Sarasangraha Sarja ... Sarvadhana .. Sata Batakofi datdra " .. Sino . Siamila . Soldhapurt .

Page #505
APPENDIX II. 808 Biddhat ... " Sodalikd ... ... Sodhya Brdraka ... Griparni ... Btoka ... 8.kymaphala ... ... Suvarna-kuffikdra... ... Ewurata ... Svarna Syddudda ... Tamd la Tilaka Tirtha Tirtharkaras Tramaranu ... Triprama. ... Those who have attained to the highest position in "" TO regard to spiritoal knowlodge. ... A rendore of capacity in relation to praia. Vid. Table 3 of Appendix IV. One of the thro figures of a oubio root group. See noto to II-53 and 54. A lay follower of Jainism. ... Name of a tree; Preonna Spinosa, ... A measure of time. Vide Table 2, Appendix IV. ... Acontrate measure vf the aron or of the cubical contents. ... Proportionate distribution an applied to probletus relating to gold. ... Name of a Jaina saint; one of the Tirtharikaras. A gold coin, ... "The argument of may be.' See footnoto on page 2 Name of a tree; Xanthochymna Pictrix, Name of a tree with buautiful Howers. Ford. See noto to VI - 1. The 24 fainous Jaina saints and truchers. See note to VI-1. .... A partiolu. Vide Table 1, Appendix IV. ... Name of a chapter iu Banskrit ostro..omioul works. She footuote on luge 2. A weight moupure of barer mot al. A di-dubicient quadrilateral. Ser not to VII 37. A Monsuru of time. Vide Table 2, Appendix IV. The water-lily lower. ... Tho sum of all the multiples of the common difer. onoe found in a series in arithmetioul progression. See note to 1183 and 04. A mixed sum obtained by wling together the common difference of # merico in arithmetical progression and the sam thereof. See note to !! -80 to 82. ... Deasure of oupacity in relation to grain. ... A weapon of Indra; for longitudinal wootion we BOLO to Chapter VII-38. ... Cross reduction in multiplication of fractions. So note to 111 -2. ... Name of a tree; Mimusope Elengi. ... Proportionate distribution baand on creeper Hike ... } chain of figures. Be noto to V1--1164 ... ... Ubhayenipodha Ucohod na Utpala .. Uttaradhana ... ... Utaramitradhana ... ... tha Vafra Vajrd parartona Vakula ... . VIH Voluwmf filed na ...

Page #506
804 GANITASI RASANGRALA Vardhamana ... Vargamela Varna ... * Vicitra-kutfikara ... Vidyddhara-nagara Vivamakuifikara ... Pipamasankramana ... Name of the chief of the Jinas; vide Jina-Pundiline ** mana, ... Square root. ... Literally colour; here denotes the proportion of pure gold in any given piece of gold, pare gold being taken to be of 16 sarnas. ... Curious and interesting problems involving propor. tionate division. ... A rectangular town is what 19oms to be intended bere. ... ProportionatA 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 poto to VI-3. A measure of length. 'ide Table 1 of Appendix IV. ... Name of & Jaina saint ; one of the Tirthankaras. A measure of length. Vide Table 1 of Appendix IV, Subtraction of part of a series from the whole series in arithmetical progression. Se note on rago 84. A kind of grain ; a measure of length. Vide Table 1, Appendix IV. Longitudinal section of a grain ; for diagram - note to VII-32. A place 90deg to the East of Lanka. ... Penance; practice of meditation and mental oan. centration. A mesure of length. Vide Table 1, Appendix II. Vitasti ... Vrracha ... ... Vyavahard ngula ... Vyutkalita Kana Yavakoti Yoga ... ... ... Tojana . ..

Page #507
806 APPENDIX III. ANSWERS TO PROBLEMS. CHAPTER II, () 1169 lotunes. (8) 2592 gems. (4) 16151 goms. (6) 68948 lotades. (6) 9255827948 lotuses. (7) 12845854321, (8 43048721, (9) 1419147. (10) 111111111. (11) 11000011000011. (12) 200010001. (13) 1000000001. (14) 111111111;222228222; 333333333; 144444444; 656555566; 880000006 77777777; 888888888, 999999999. (16) 111111l. (16) 16777216. (17) 1002002001. (90) 128 Dindras. (21) 78 pieces of gold. (a) 181 Dindras. (88) 179 pieces of gold. (84) 808 fruite. (28) 178 fruits. (88) $029 gems (97) 27994681 gold pieces. (88) 191 gems. (82) 1; 4; 9; 16; 25; 38; 12; 14; 81, 825, 266, 036; 1298 ; 5626. (83) 114244; 91724921; 65636. (84) 4294967296, 162399026, 11108889. (85) 40799769; 50906226; 104484. (87) 1, 2, 3, 4,5:8; 7; 8; 9 16, 24. (88) 81, 866. (29) 86636, 789. (40) 7079, 1881.

Page #508
806 GANITA BASAKGRAHA. (41) 88; 26; (42) 833; 111; 919. (48) 1; 8; 27; 84; 125; 216 ; 348 ; 619;729 ; 9875; 16625; 46666 ; 486588 884786. (40) 1030301 ; 5088448; 137388096; 848601818; 2437716684. (50) 9663597; 77308778; 260917119; 618470208; 1207949625. (51) 4741632 ; 37933056; 128024064; 303464448 ; 698704000, 1024103512; 1626879778; 2427716584. (52) 869011889946948864. (65) 1; 2; 3; 4; 6; 6; 7; 8; 9; 17; 123. (56) 24 ; 333; 852. (57) 8484 ; 4242. (58) 426; 689. (59) 1344; 1176. (80) 950804. (66) 55; 110; 165; 220; 375; 380; 885; 440; 406; 650. (86) 40. (67) 664; 754; 980; 1245; 1652; 1904; 2804. (68) 4000000. (71) 6; 8; 16. (72) 9: 10. (177) 3; 3. (79) 2,620; 10; when the chosen numbers are 8 and 10. (83) 2; 3; 6: 2; 8; 6. (85) 120 ; 24; when the sam of the required series is twioe the known am: 80; 60; when the sum of the required series is balt of the known lam. (87) 48; 4; when the sums are equal : 96; 84 ; when one of the runs in twice the other: 44; 26; when one of the soms is thriothe other. (28) 100; 216; when the sums are equal : 283; 193; when one of the roma in twice the other: 84, 238 ; when one of the sums is half of the other. (90) 21; 17; 13; 9; 6; 1: 25; 17; 9; 1. (92) 8; 6; 4; 8; 8; 1. (96) 4874 coins. (99) 1276 dindras. (100) 68887, 22888188608. (102) * ; 3. (104) (106) 8; 9; 15. (111) 994; 201 ; 176; 944; 261. (118) 4886; 4866; 4900; 76960. (118). 189988; 6846. (114) 180 ; 118, 60, 60. (116) 1099, 2044; 1020; 608 ; 988; 184, 00.

Page #509
APPENDIX III. 807 CHAPTER III. (4) papras. (5) 2 bo papas. (6) 24 palas. (7) 8.18 120 W 16 7 148 (9) Dopas. (10) 177 papas. (11) 14. palce. 224 120 265 * 133 * (14) 25 40 81 366 400 10000 40000 asi 26 40 81 121 189 226 289 861 441 629 (40) T oi 18deg 26deg 38 49: *84 81: 100121 14 626 160 (10) (17) Bec exemplos 14 and 16 in this chapter ; 26. (18) 1 1 1 1 1 1 1 1 77: 84 128 216 343 : 512: 72 848 1881 8275 6850 12107 19083 29791 42876 84 216 612 : 1000: 1728 3744 ww 6 889 89819 8000 (80) 191) 1 1 1 1 1 1 1 1 8 7 11 16 10 28 87 (31) ji z T5 Bi 7 gigi 2: 7 Big, To' 19' iti 86 987 8877 1891 8686 Ti 88 2388 8704 : 201022284 (30) In och of the series the first term is 1, and the common difference in 2. 4 9 16 26 8840 84 M The squares of the some are to 26.38 39 100 10015. The onbow of the some are in 1950 m m

Page #510
808 GANITASIRAKANGRADA. 10 In (86) Tho cubio mums aro de gare not eno Art terme aro ben the common differences troba , the numbers of terms aroma, m los con la olac bojm lo ola 4108 (30) 76 76 are the are the interchangeable first term and common difference when the sums are equal; and -- od 1651488 in the equal sum. When the sims are 226 7853 in the ratio of 1 to 2, 2558 0 are the first term and the common ditter 75 ence; and the double com ir 3102876. When tbe vamo are in the proportion of LORA 76 and the 1 to , the first term and the common difference are halved eum in 7661488 341 1 2048 2048 226 (60) 100 (62) 90, 31, 30. 87186 the same we 8661 176 852 704 (08) The first termo are 81 248 729 298FO 18976. then : the pumbers of terms are 6;6,4 8681' 6661 (87 4 68) 1. (80) 1. (60) 1;1;1.

Page #511
APPENDIX tit. 300 (01 & 82) (68) O. 1;X; 1; 1. do do (86 & 88) (87 to 71) 4, (74) %; 8; 4. (76) a) 3; 8; 9, 27; 54. (6) 2, 3; 9; 27;, 81; 162. (c) 2; 8; 9; 47.81 848, 486. (78) (1) 8; 186; 340; 260. (2) 44; 220; 480 389. (3) 78 288, 560; 895. (81) (1) 6; 21; 420; when the optionally chosen quantity in 1 throughont; (2) 3; 11 ; 232 ; 53602 ; when the optionally chonon quantition nro 2, 1, 1. (88) *; - when the chosen quantites are 6, 8, 9. (84) 8; 13; 18; when the chosen quantites are 6, 4, 3. (88) (1) 18; 9; when the chosen number is 8. (2) 80, 16; when again the chosen number is 3. (84) (1) 6; 18; the chosen number being 2. (2) 8; 16 (8) 46; 92 (1) 22 ; 110 (90) (1) 4; 28; (3) 26, 175. (91) 18, 20. (99) 151, 8020. (94) (1) 32, 44, 83, 66, 68, 116; wbeu the sum in split up into and and the chosen number in 2. (9) 11; 32; 19; 236; 191; 38; 20; wbon the sum is uplit up into 9' ' (86) 8. (07) 1. (98) (100 to 108) 1. (108 & 104) L. (106 & 108) 1. (108) are the optionally chosen quantitio. ws 11 (111) ? (129) (114) O.

Page #512
810 GANITASARAAANGRAHA. (116) 14 - niykas. (116) 0. (117) 2 droras and 8 mifu. (118) 15 (119) % * nipkas. (120) 1. (10) 1 (128) i to i a re the optionally split up parta (124) (187) 24 karpas. (128) (129) 1. (180) 1. (181) 1. (198) , 1. g; wben and are the optionally uplit up parte. (184) (187) , when it comes to o are the optionally obonen fractions in places other than the beginning; 77 Tsimilar fraction. (189 & 140) 8 76 OHAPTER IV. (6) 4 hastas. (6) 80 boom (7) 108 lotare. (8 to 11) 988 sapor. (19 to 16) 2590 parrots. (17 to 29) 8468 pensla. (98 to 87) 7600 boos, (28) 099 vow.. (19 and 80) 18 madou. (01) 48 alophants. (12) 208 por pas. (W) * malo. () 16 poucooke.

Page #513
811 ATTENDIX III. (36) 676 birda. (87) 84 monkoyo. (88) 88 onokoos. (89) 100 swans. (4) 44 elephante. (12 to 45) 100 ascetios. (48) 14 elephants. (48) 46 bees. (19) 196 lions. (50) 894 deer. (68) 48 angulas (64 & 56) 160 elephants. (58) 200 boars. (58) 96 or 82 od has. (69) 144 or 11peacocks. (80) 240 or 120 hastas. (82) 64 or 16 battaloes. (68) 100 or 40 elephants. (64) 120 or 45 peacooko. (68) 16 pigeons. (67) 100 pigeons. (88) 258 sw&ns. (70) 73. (71) 834 elephants. (79) 1798 soation. CHAPTER T. (3) 688 to yojamas. (1) 6 yojanas. (5) 105800000 (0) 10day.. (7) 10 years. (9) settem whes. () ost palaa RO) 7 sapae. (11) 1964 bhores. (LS) 686 . Condres.

Page #514
318 GANITASIRASANGRAHA. (18) 3880 palas. (14) 168 pairs. (16 & 18) 18 in yojanas ; 87 7vihas. (17) 112 dronas of kidney bean ; 604 kudabas of gheo; 888 dromas of rloe 448 pairs of cloth : 836 cows ; 188 svarnas. (18) 160 ; 112 100 dharanas. (18) 720 piecor. (20) 626 pieces. (91) 24 Tirthankaras. (29) 216 blooka. (24 & 25) 6 years and 117 days, (28) 218 7 daya. (27) 10 years and 346 18 days. (28 to 80) 861 daya. (81) 76 days. (88) 10 purd pas; 18 purdnas ; 28 purdnas. (84) 29 1120 gold coins. (86) 86 gemi. (88) 4000 panas. (87) 250 karpas (88) 960 pomegranater. (89) 660000 gold coins. (40) 750 gold coins. (41) 64. (42) 962 gold coins (49) 945 wd has. CHAPTER TL (8) 7, 6:4, 6. (5) 9; 18; and 26. pwrdnas. (6) 17 hod podpamas. (7) 61 pure pas and 14 popull (8) 200. (8) 88 7 8 karripapas, (11) 188 surfpas.

Page #515
APPENDIX 111. 318 (18) 14. (18) ; 60; 70. (15) months. (16) 8 months. (17) 10 months (10 & 20) 35 palas. (22) 30 ; 18. (24) 80. (26) 's months. (87) 5 months ; 75. (98) 4 months; 314. (30) 311. (81) 60; 6 months (32) 24 months ; 30. (34) 10; 2 months. (88) 48 ; 10 months; 24. (38) 10; 6; 3; 15. (40) 40; 30; 20; 50. (41) 6; 10; 15; 20; 3.), (43) 5 months ; 4 months ; 8 months; fi months. (45) 8. (46) 6; 18. (48) 20; 28; 36. (49 & 50) 25. (52) 18. (63) 30. (65) 900. (56) 800. (58) 28 months. (69) 18 months. (61) 2400; 800; 1200; 96. (82) 1000, 420, 480; 2. (64) 60. (86) 60. (67) 9400, 2720; 84011. * (68) 1060, 1400 ; 1800. (89) 6100; 4590; 4060. (70) 1800 ; 110EUR; 1160. (72 and 731) 2004; 8 H1; 5 montha. (784 to 76) 440 ; 11 ; 5 inontbe. (781) montbond (801) 48; 89 ; 24; 16.

Page #516
314 GANITASARASANGRABA.. (811) 3; 9; 27; 81 ; 243. (82 to 857) 120; 80 ; 40; 180; 60; 20. (M8) 48 ; 72; 96; 120 ; 144. (901 and 9:) 70 pomegranatos; 36 mangoen; (92% to 941). Curd. Ghee. Milk. 35 wood appler I pot 128 32 64 pou 98 II pot 32 8 18 64 18 32 Ill pot 9 (961 sud 96}) 15 men ; 50 mon. (981) 4; 9; 18; 36. (901) 8; 13; 21; 36. . (1004) 2; 4; 7; 13; 25. (101) 16; 38; 4; 234. (1084) 220; 37. (1041) 20; 5. (105) 6; 4 ; 3. (the intter two having been optionally choosen). (1061) 8. (1084) 8081600; 1860); 2231. (1101) 148; 8532; 184. (1124 und 1137) Howoro. 10+ (114) Howers. (117) 6. (118) 17. (1191) 26. (1201) . (1211) 55. (122) 61. (138+) 69. (1241) 30. (1267) 16. (1261) 15. (1373) 637. (1987) 188. (1201) 194. (1811) 11. (1824 and 133.) 26. (1871) 10,57. (1887) In the onse of positive associated numbers : , 91;18; 18; 11 ; 21 ; 10; 37; 7; 87; 6: #1; 18 ; 6; 12; 1 ; .

Page #517
In the case of negative associated numbers: 11; 18; 23, 27; 19; 23; 7; 39; 11; 44; ;41; 51; 46; 59; 37. 83 '65 (140 to 1424) 8; 5. (1444 and 1454) I heap II" III Price (147 to 149). 29 Number Price in panas (150)-- Peacocks. Pigeons. Swann. Sarasa-birds. 7 16 45 4 14 10 3 3 Citrons. 14 (160 to 182) 10; 9; 8; 5. (164) 20; 15 and 12. (165 and 166) 8; 20; 40. (168) 243 panas. 16 18 2 (1784) 82. (1741) 874. (177) and 178) 14, (179) 8. (181) 21. 200 100 Ginger. Long pepper. Pepper. 20 4 32 Quantity Price in panas 12 (152 and 153) Panas 9; 20; 35; 36. (155 and 156) When the optional number is 6; 3 (184) 8 (186) 20; 4; 4; 4; 4; 24. (188) (180) ; 18. APPENDIX III. (191) 8; 13; 10; 2 4 2 16 40 28 80 (170 to 171) 10; 21; 217 21 H 21 3 7 Plantains. Wood-apples. Pomegranates. 3 3 1 3 2 1 3 1 10 4 (1964) 560; 448. When the optional number is 8; 5; 6; 18; 4. (158) Length of a stage 10 Yijanas; each horse has to travel 40 Yojanas. 12 117 109 175 111 16 10 or 10 18 29 (198 to 1964) (a) 17 44 16 (200 to 1) 200, 100; 7 18,100 1007 36 3 *; (b) 32 1800 800 83 14 (904 and 905) 47; 17; 84; 68; 186. (307 and 208) 2400. ; 1 11 75 889 27 82 8 71 14; 3;7. 41" 315

Page #518
816 GANITASIBASANGRAHA. (N18 to 816) 8; 2. ; 68 (917) 11. (319) 6; 15; 20; 16; 6; 1: 88 (980) 6; 10; 10; 6; 1 : 31. (281) 4; 6; 4; 1; 15. (228 to 326) 10; 24; 32. (227) 4 jaok fruits. (299) 9 Yojanas. (281 and 289) Dindras 18; 67; 166; 400. (936 and 237) 16;1; 3; 6. (289 and 240) 261; 921; 1416; 1801; 2009; 110880. (942 and 243) 11 ; 18; 30. (944 and 2441) 3; 4; 5. (946) and 247) 6177 ; 103 ; 189; 223 ; 138. (248) 14760, 356; 585 ; 445 ; 624. (249 to 250) 55 ; 71; 66; 876, (2687 to 2567) 7; 8; 9. (2681 to 268) 11; 17; 90. (960} and 2811) 7; 3 ; 2. (2621) 8; 12; 14 , 16; 31. (2831) 64 ; 72; 78; 80 ; 121. (2644) 1876; 2625; 2925; 3045 ; 3093 ;-6187. (9661) 4; 7; 13. (987) 12; 16; 32; 31. (970 to 272) 427; 40. (2741) 6; 8. (276) 188. (8771) 161. (2781) 58875 441 (2904) 26. (282Lo 283) 1996 ; 1926. (2986) (a), (o) (187) sa (289) 87, (291) 40; 184. (298) 8; 3. (996) 6 women, 40 flowers. (207) 204 ; 9109; 2870; 78810 ; 180441 ; 26908. (800) 1095; 1824. (809) 441 : 1996 ; 784, 106626 ; 1087 46816. (804) 2566; 128296. (8061) 87668. (8081) 604; 788 ; 1020 ; 1875; 5804 , 160876 ; 872804

Page #519
APPENDIX III. 317 (810) 1663100, 5038869 ; 9646:12705; 114400. (8127-813) 121 * 5461 162' 18288 (815) 426. (316) 416348873. (818) 2; 3; 6 ; 40. (329) 7. (821 to 321) 24 daye. (3287) , (325) 6. (327) 26 dnys. (3294) 18:18. (381) 55. (332) 620.(337) For answer see footnote in the translation, CHAPTER VII. (8) 32 sq. dandas, (9) 866 84. dandas and 4 mq.hastus, (10) 98 sq. dardas. (11) 1200 sq. donlas. (12) 3600 sq. dandas. (13) 1852 sq. danda. (14) 23781 sq. dandas. (15) 6304) 84. dandas. (16) 1925 sq. dandas. (17) 7425 sq. dandan. (18) 60 Rq. hastan. (20) (1) 64 ; 243. (ii) 27'; 121. (22) 84; 252. (24) 48 hastas; 195 4. hastas. (26) 378 (37) 135. (29) 189 sq. hantas ; 135 sq. hastas, (81) 108; 072; 26. (88) 1600. (84) 2,400 sq. dandar. (86) 469 9. dando (86) 640 sq. dandas. (88) 34 nq. damdarl; 486.4. dander. (40) ; 180. (41) 18, 804.

Page #520
818 GANITAJIRASANGRAHA. . (42) 201; 84. (44) 2583, 89. (48) 18; 26. (48) 675 676 (51) ~ 768 sq. dandaa; dandas 48; 4; 4. (E2) 80 sq. dandas; dandas 12; 5; 5. (68) 84; 12; 5; 8. (56) V50; 25. (56) 18; 60. (57) 65; 1500. (58) 312; 288; 119; 120; 34660. (69) 318, 280; 48; 252, 132; 188; 224; 189; 44100. (61) V3210; V65610; v36000; 18100000 ; 4840; v146410. (62) V380; V3240; V3240 ; V 262440. (6+6V6048; V54432. (661) 2640 dandas; V 42250 mq. dandan, (88}) V 39880 sq. dundas; V20260 sq. dandas. (891) 31380 sq. dandas. (713) V 1440 sq. dandas. (724) 5780. (751) V380 ; 12; 6. (77}) 192+ V 23040. (785) 102-V5780 (794) 193-V23040. (814) V 19380, 4840: 1 4840 0 ;" ; (88+) 18-V180. (861) V48-740 (87) 16, 18; 48. (891) 90 ; 8. (911) 8; 4, 6. (991) 8; 12; 18. (941) 16; 80; 34 (981) 5; 8; for the three onnes. 198+) (i) 60; 81. (ii) 11 ; 01. (iii) 11; 60. (1004) 80 ; 102, 61; 60; 109; 11 ; 6480. (109+) 189; 407; 169, 120; 812 ; 119; 84680. (1044) 136 ; 800 ; 980, 196, 944; 180; 48 1 26%, 108189, 44100 (1004) 3; 60, 16.

Page #521
APPENDIX III. 819 (111) 18; 16; 14; 12. (1181) 4; 1. (1144) v2: .. (1164) B; 8. (1181) I NA (1171) 83; (perpendicular 34). (1181) . (1194) zi (perpendionlar 16). (1917) 8; 8. (1337 and 1247) 89; 52; 26; 60; 88, 66; 68; 18. (1961) 6, 12. (1284) 3; 13. (1801) 35; 60. (184) 8; 16; 8; 20. (185) 8; 7; 2; 38. (186) 82; 87, 6; 982. (188) 87; 24; 29; 40. (189) 17; 16; 18; 24. (140) 626; 672; 970; 1904. (141) 281 ; 920; 442 ; 890. (148 to 145) Circle : 25920 ladies; 720 dandan square : 84660 ladion, 720 dandas. Equilateral triangle: 88880 ladies ; 1080 damda.. Longish quadrilateral: 38880 ladies : 1080 dundas; 540 dandas. (147) (i) Side 8 (il) Base 18; perpendicular 6. 18 18 1. 41 (149) aici iz is: 4. (181) 18; 18; 18;; 8; 13. (183 to 1684) 8; 16; 11; 12. (1661) 8 (167) 5; 6; 4 * 89 116 (1601) sc 30 30 (1094) 198, 89, 48 2001 (1044) VTO. 1601) 7,1, (1677) (1601) 6. (1704) 10.

Page #522
820 . GANITASIBMANGRAHA. (1724) 10; 18. (174) Biden if ; top-side base (178) 17. (1771 to 1781) (a) 8800 ; 7200 ; 10800 ; 14400 ; (6) 64; 90, 126 , 169 ; (c) 100 100; 100; 100. (1791) (a) 2700 ; 7200 ; 4500 ; (6) 60 : 70; 80; (c) 60; 120 ; 60. (1815) 8 hastas; 8 hastas. (1827) * hastas ; *, hastas ; hastas. (1834 and 184) 3 hastas ; 6 hastas; 9 hastas. (2863) 7 hastao ; 7 hastao ; hastas. 80 128; 2 hastas; hastas. (187}) hastas; 12 hastas ; 9 hastas, (1887 and 1894) 8 hastas; 2 hastas; 4 hastas. (1915) 13 hastas. (1921) 20 hastas. (1981 to 195}) 29 hastas ; 21 hastas. (197}) 10 hastas. (1997 to 2005) 12 yonnas; 8 yojanas. (204to 205) 9 hastas; 6 hastas; V 250 hastas. (206 to 207) yojanas ; 14 yojanas; V 520 yojanus. (208) to 209_) 15 yojanas ; 7 yojanas. (3111 to 212 ) 18 days. (2141) 18; 13. (9165) (316+) 125. (8177) 65. (2184) V (2104) $50 (2204) 4. (9221) Square: N1. Oblong : 8; 19. Quadrilateral with two oqualoidov: oides is top-side ; baro . Quadrilateral yith three comid istine nidos en un baro 1983 Inoquilateral qandrilateral: aideo s top-sido By base;12. Equilateral triangle : 1 bo loosoolos triangles aidoo 136 6. bos: 190 Boulono trianglo: idee, 13, , ben

Page #523
APPENDIX III. 321 (2241) Square, 3. Quadrilateral with two equal sidos: Quadrilateral with three equal nider : 51.2. Inequilateral quadrilatoral: * Equilateral triangle : V12. Isosceles triangle: Scalene triangle : 8. Hexagon : 116 N , if the area of the same in taken as v 48 in acourdance with the rulo given in itanA 864 of this chapter. (2261) 8. (3281) 2. (2804) 10. (282) 6; 2. CHAPTER VIII. *(5) 612 cubio hastas. (6) 18580 cubic hastas. (7) 144320 cubic hastas, (8) 162000 cubio hastas, (121) 8928 oubio hastas. (18+) 1438 oubio hastas; 1478 cobio haxtas; 1464 oubio hastus. (141) 2016 oubio hastas; 2052 cubio hastus; 2928 cnbio hastus. (151) 8880 oubio hastas. (161) 98980 cabio hastas. (171) 18100 oubic hastas. (18+) 18288t cabic hastas. (311) (1) 8024 cabio dandas ; 8024 oubio dangas; 4032 cubic dandan; (ul) Central man in tapering; 1488; 1488; 1984 onbio dandan (891) 4032; 1984 oubio dandas. (941). 40 onbio hastas. (961). 16 hastas. (871). 12, 80. (804). 8804; 9078. (814). 790; 648. 93 4 6 (84)., Tof a day. i' i' T' of the well. (85 sad 86). 18 ydjanss, and 976 dandas ; 89 14 whos. (87 to 881). 17 yojanas, 1 krada and 1968 danda. (89 and 404). 26 yojance and 1952 dandas. (47 and 44). yojanas, 2 lbhas and 488 dandas (464). 1913 upit brioke. (161). 8466 unit brioks.

Page #524
822 GANITABIRASANGRAHA. (47+). 5184 unit bricks. (484). 108000 unit bricks. (491). 40820 unitebricks. (60%). 40820 anit brioks. (514). 20738 unit bricks. (587). 1440 unit bricks; 2880 unit bricks. (50). 2640 anit bricko; 1680 unit bricks. (567). 2880 upit bricks ; 1440 unit bricks. (584). 20; (59-60). 891 anit bricks. (62). 18,720 unit brioks. (884). 64 pattikas. CHAPTER IX. (91). 5 of a day. (114). Of ghafis (18+). i of a day. (141). 2. (167 to 17). A of a day; 10 ghafis. (19). 8 angulas. (22). 18 hastas. (24). 8 hastas. (86). 2. (87). 90 hastas. (99). 10. (81). 6; 60. (84). 6 hastas. (86 to 871). is of a day; 8. (884 sud 80+). 6 hastas, (414 to 48). 84 angulas. (4%). 88 angulas. (46 and 47). 118 angulas. (49). 175 foot-mosaures. (50). 100 foot-menores. (51 to 58). 100 yojanas.

Page #525
323 APPENDIX IV. TABLES OF MEASURES. 1, LINEAR MEASURE. Infinity of Paramdrus 8 Apus 8 Trasaronne 'Ratharius 8 hair-measures 8 louse-measures A sesamum-measures 8 barley-measures 600 Vyavahdrangulaa 8 Angulas (finger-measure) feet 3 Vitastis 4 Hostas 2000 Dandas 4 Kritas. = 1 Am. 1 Trasarim. = 1 Ratharonu. = 1 hnir-messure. = 1 louse-measure. = 1 sesamum-measure or mustard. measuro. = 1 barley.messare. = 1 angula or VyavaMdrangula. = 1 Pramd pa or Pramd nd Agula. = 1 foot-mouraro (mourred acros). == 1 Vitasti. = 1 Ranta. = 1 Danda. = 1 Kroka. El Yojana. 3. TIME MEASURE. Infnity of Samayas A number of palis 7 Uchado 7 Salas .884 Land 3 Ghafte * Wwhertos 16 dayo 3 Pakopos 3 months = 1 Aval. = 1 Uechnd na. = 1 Suka. - 1 Lana. El Chapt. 1 wirta. = 1 day. 1 Pakpe. - 1 month. # 1 Ayane. 1 year yance

Page #526
824 GANITASABAKANGBANA. 3. MEASURES OF CAPACITY (GRAIN MEASUREMENT). 4 fodaftada 4 Xudahas 4 Prasthas 4 Adhakas 4 Dronas 4 Manis 6 KMris 4 Pravartikdo 6 Pravartikas = 1 Kudahan = 1 Prastha. = 1 Adhaka. = 1 Drona. - 1 Mdni. = 1 Khari. = 1 Pravartika, - 1 Vaha. = 1 Xwmbha. 4. MEABURES OF WEIGHT-GOLD 4 Gandakas 5 Guide 8 Panas 9 Dharapas Karjas = 1 Gunja. = 1 Papa. = 1 Dharana = 1 Karja. = 1 Pala. 8. MEASURES OF WEIGHT-SILVER. Graine 3 Gwritje 16 Mapas #Dharanas 4 Karpas or Purdpas = 1 Gunjd. = 1 Mdge. = 1 Dharama = 1 Karya or Pwrdna, -- 1 Pala. 6. MEASURES OF WEIGHT-OTHER METAL8. 4 Padas of Kalls 4 Yavas 4. Anas 6 BM gas 2 Drakpapas Dindras 18 Palas 200 Palas 10 Tulls = 1 Kali. = 1 Yava. = 1 Am . = 1 BM ga. = 1 Drakpia. = 1 Dindra. 1 Satora. = 1 Prastha. = 1 Tuu. * 1 BMrch . 7. MEASUREMENT OF CLOTHES, JEWELS AND CASH . 20 paint = 1 Kl.

Page #527
APPENDIX IV. 8. BARTH MEASUREMENT. 1 cubic Hasta of compressed earth 1 cubic Hasis of loose earth * Brick of 1 hasta x Harta x 4 Angulas 9. BRICK MEASUREMENT. 3000 Palas. 8900 Palas. Unit brick. 10. WOOD MEASUREMENT. 1 Kipku. 1 Haste and 18 angulas Work done in cutting along by means of a saw a piece of wood 96 Angu las long and I Kipku broad of a man's height = 1 Paffib. 11. SHADOW MEASUREMENT. his foot measure, 825

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12499 ROYAL ASIATIC SOCIETY OF BENGAL LIBRARY Author Lahavira carya. . Title sanita-sara-3ansraha. Call No. 510.254 M 214 3.. Date of issue Issued to Date of Return Library of the BOYAL ASIATIC SOCIETY OF BENGAL Call No.510-254.M214.9 Res. * Ancension: No...12499..