Ganitasara Sangraha of Mahavira

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

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THE
GANITA-SĀRA-SANGRAHA
OF
MAHAVIRÁCARYA
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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महावीराचार्यप्रणीतः
गणित सा र स द य ह :
9919
म. रगाचार्येण
गरिशचन:
आगलभाषाननाद टीकाभ्यां म.गजका याज्ञानमारण प्रकाशितध।
अन्नपूर्वा राजकीय मुद्राक्षरवालायो (म्परिन्टेडण्याफयम) अनिवारण महितः।
२२१२.

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TABLE OF TRANSLITERATION.
Conremunten
Vowelve
Diphthongs.
Cutturala
..k, kb,g.gb, ii, h, h. ..
क, स्व . ग, घ, ङ, ह.: ..अ
.. ) आ
(6) ai.
Palatale
.
..
ए
.
c, ch, j, jh, ii, .. च, छ. ज. स. न. य. श
.... । .. ई
Linguals
...
..tth, a, dh, II, I,
ट. ठ, ड, ढ, ण, र. प
...,! .. .. क, क
Dentals
!
t.th, d,dh,n, I, .. त, थ, द, घ.न, ल, स ..
. Labials
.. pph, b, bh, m, v
प. फ., घ, म. म. व
.. u, opan. ..उ उ .. ओ. श्री.

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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
GAŅITASĀRASANGRAHA.
Linc
Page X
(Preface)
XIV
XX
do.
For
Read wiksu
mokat. Nakalya-rrimihitu
Sakalya-8ahitd. Māli pirácāry
Mahaviricirya. Do. owit much of' before küsirakuta Aryabhula
Arynlibutan TIPS
गशि वालका
a . मङलित
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 amüla. 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
अम्बीध wordly
289
292
मनि
298
नलि क्षायकसम्य, क्त्व
अम्बुधि worldly.
TA. ata. क्षायकसम्यक्त्व.
A. FAHETA. Pala. cube, chose'll. Srokar.
294
301
स्तम्ब्रेम 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 Mahāvīrācārya's Ganita-sära-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
GAŅITABİRABANGRARA. - 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 -sära. sangraha that the time when its author Mahavirácărya 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 Amõghavarsa Nņpatnnga; 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-sāra-sangraha was in some way attached to the court of this Rāstrakūta king Amõghavarşa 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 Bhāskarācārya in the twelfth century of the Christian era; and chronologically, therefore, Muhāvisācārya comes between Brahmagupta and Bhāskarācārya. This in itself is a point of historical noteworthiness; and the further fact that the author of the Ganita-sāra-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. virācārya 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 anuyôgas or auxiliary sciences indirectly serviceable for the attainment of the salvation of soul-liberation known as moksa.
A comparison of the Ganita-sāra. sangraha with the corresponding portions in the Brahmasphuta-siddhānta 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, Mabăvirãoárya was familiar with the work of Brahmagupta and endeavoured to improve upon it to the extent to which the scope of his Gunita-sāra. sangraha permitted such improvement. Mahăvirácărya's classification of arithmetical operations is simpler, his rales are fuller and he gives a large number of examples for illustration and exercise. Prthūdnkasváinin, the wellknown commentator on the Brahmasphuta-siddhānta, could not have been chronologically far romoved from Mahăvirācārya, 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 Mabávirácārya wrote his Ganita-sära-sangraha, Brahmagupta must have been widely recognized as a writer of authority in the field of Hindu astronomy and mathematicy. Whether Bhaskarācārya was at all acquainted with the Ganitasāra-sangraha of Mahavirácārya, it is not quite easy to gay. Since neither Bhaskarācārya nor any of his known commentators seem to quote from him or mention him by name, the natural conclusion appears to be that Bhag. karácărya's Siddhānta-firõmani, including his Lilavati and Bijagaạita, was intended to be an improvement in the main upon the Brahmasphutamsiddhanta of Brahmagupta. The fact that Mahavirácārya was a Jaina might have prevented Bhaskarácărya 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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GANITARĂRABAKGRAHA.
ander the stimulating influence of the enlightened rule of Rājarājanarēndra of Rajahmundry, it was translated into Telugu in verse by Pāvulāri 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 Mahāvirācārya's Ganita-sāra-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-sārasangraha 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-särasangraha, 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-bāra-sangraha twenty-four notational places are mentioned, commencing with the units place and ending with the place called mahāksõbhain

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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 Mahavirácārya 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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GANITARİRASANGRAHA. are related to the Brahma-siddhānta; and in a work of this name, which is said to form a part of what is called Sakalya-sāmihitā 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 Ārayabhata 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 Sünya, 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 sūnya, 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. Mahävirācārya gives, in the very first chapter of his Ganita-sära-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 Bhaskarācārya, that the quotient in such a case is infinity, still the very mention of operations in relation to zero is enough to show that Mahaviracårya must have been aware of some symbolic representation of the zero quantity. Since Brahmagupta, who must have lived at least 150 years beforo Mahāvīrācārya, 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, . Mahavirácarya 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-sära-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-sāra-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-sära-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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GANTTASĪBASANGRAHA.
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. Rangåcårya of Madras, somo years ago became interested in the work of Mahaviråcårsa, 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.
Mabăvirácărya appears to have lived in the court of an old much of Râştrakūta monarch, who ruled probably over much of what is now the kingdom of Mysoro and other Kanarese tracts, and whose name is given as Amõghavarşa 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 Bīja-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 Bbåskara's Lilavati are included in the Ganita-sära-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 Rangiicàrya 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.

Page #28

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. Rangācārya of Madras, somo years ago became interested in the work of Mahavirăcārya, 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.
Mahäviräcārya appears to have lived in the court of an old much of Rastrakūta monarch, who ruled probably over much of what is now the kingdom of Mysore and other Kanarese tracts, and whose name is given as Amõghavarşa 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 Bhāskara’s Lilavati aro included in the Ganita-sära-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

Page #29

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.

Page #30

xxü
.
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 Bháskura 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 Mahavirăcărya ? 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, Mahüvirácarya, and Bhāskara 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 Mahăviripärya 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 Mahavirkoårya are much better than the one to be found in either Brahmagupta or Bhaskara, and no questions are duplicated.

Page #31

INTRODUCTION.
xxü
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 thatés 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

Page #32

Xxiv
GAŅITASĀBASANGRAHA.
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 Bláskara, who was practionlly his contemporary, shows of the Ocoidontal.
Professor Rangācārya 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.

Page #33

गणित सारस दु य ह :. .

Page #34


Page #35

·
संज्ञाधिकार :
मङ्गलाचरणम्
गणितशास्त्रप्रशंसा
संज्ञा
क्षेत्रपरिभाषा..
कालपरिभाषा
धान्यपरिभाषा
सुवर्णपरिभाषा
रजत परिभाषा
लोहपरिभाषा परिमानि
वनणेशत्यविषयकमामा-यनियमाः
मडुगामंज्ञा
स्थाननामानि
गणकगणानरूपणम
प्रथमः परिकर्मव्यवहारः
प्रत्युत्पन्न
भागहारः
वर्ग:
वर्गमूलम
घनश्
...
CONTENTS.
घनमूलम्
मङ्कलितम्
च्युत्कलितम् ... द्वितीयः कालास वर्णव्यवहारः
भित्रप्रत्युत्पन्न: भिन्नभागहार:
:::::::
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Page
I
2
3
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6
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12
13
16
17
45
28

Page #36

CONTENTS.
Page
330
भिनवर्गवर्गमूलघनघनमूलानि भिन्नसङ्कलितम् भिन्नव्युत्कालतम् कलासवर्णघडजाति: भागजाति: ... प्रभागभागभागजाती भागानुबन्धजातिः मागापवाहजातिः
भागमातृजाति: तृतीयः प्रकीर्णकव्यवहारः -
भागजातिशेषजाती मल जातिः शेषमलजातिः द्विरग्रशेषमूलजाति: अंशमलजातिः भागसंवर्गजाति: उनाधिकांशवर्गजाति: मलमिश्रजाति:
भिन्नदृश्य जातिः चतुर्थः ।राशिकव्यवहारः
त्रैराशिकः ... गतिनिवृत्तिः
पश्चसप्तनवराशिकाः ... पश्चमः मिश्रकव्यवहारः--
सक्रमणसूत्रम् पश्चगशिकविधिः वृद्धिविधानम् प्रक्षेपकुटीकारः वल्लिकाकुटीकारः विषमकुकिारः

Page #37

OONTENTS.
Page
101
110
मकलकुटीकारः सुवर्णकुटीकारः . ... विचित्रकुट्टीकार:
श्रेाबद्धमडालितम् पष्टः क्षेत्रगणितव्यवहारः --..
व्यावहारिक गणितम् सूक्ष्मगणितम् जन्यव्यवहार:
पैशाचिकव्यवहारः सप्तमः खातव्यवहार :---
खातगणितम नितिगाणितम्
क्रकचिकाव्यवहार: अष्टमः छायाव्यवहारः
110 122 126
148
160
168

Page #38


Page #39

गणित सारसह : महावीराचार्यप्रणीत ।
संज्ञाधिकारः ।
मङ्गलाचरणम् ।
अलक्ष्यं त्रिजगत्सारं यस्यानन्तचतुष्टयम् । नमस्तस्मै जिनेन्द्राय महावीराय तायिने ॥ १ ॥ सङ्ख्याज्ञानप्रदीपेन जैनेन्द्रेण महा त्विषा । प्रकाशितं जगत्सर्वं येन तं प्रणमाम्यहम् ॥ २ ॥ प्रीणितः प्राणिसस्योत्रो निरीतिर्निरवग्रहः । श्रीमतामोघवर्षेण येन वेष्टहितैषिणा ॥ ३ ॥ पापरूपाः परा यस्य चित्तवृत्तिहविर्भुज । . भस्मसा'द्भावमीयुस्तेऽवन्ध्यकोपोऽभवत्ततः ॥ ४ ॥ वशीकुर्वेन जगत्सर्वं स्वयं नानुवशः परैः I नाभिभूतः प्रभुस्तस्मादपूर्व मकरध्वजः ॥ ५ ॥ यो विक्रमक्रमाक्रान्तवकि चककृतक्रियः । चक्रिकाभञ्जन) नाम्ना चक्रिकाभञ्जनोऽअसा ॥ ६ ॥ यो विद्याद्यधिष्ठान मर्यादावन्त्रवेदिकः । रत्नगर्भो यथाख्यात चारित्रनलधिर्महान् ॥ ७ ॥ विध्वस्तैकान्तपक्षस्य स्याद्वादन्यायवादिनः । देवस्य नृपतुङ्गस्य वर्धतां तस्य शासनम् ॥ ८ ॥
1 M and B 4°.
*M and K HI.
की.
• अ प्रणीतः.
• K, P and B भवेत्.
• M and B T.
स
• B यांऽयं.
• P वेदिन:.

Page #40

गणितसारसङ्ग्रहः
गणितशास्त्रप्रशंसा। लौकिके वैदिके वापि तथा सामायिकेऽपि यः। व्यापारतत्र सर्वत्र सङ्ख्यानमुपयुज्यते ॥ ९॥ कामतन्त्रेऽर्थशास्त्रे च गान्धर्वे नाटकेऽपि वा। . सूपशास्त्रे तथा वैद्ये वास्तुविद्यादिवस्तुषु ॥ १० ॥ छन्दोऽलङ्कारकाव्येषु तर्कव्याकरणादिषु। . कलागुणेषु सर्वेषु प्रस्तुतं गणितं परम् ॥ ११ ॥ सूर्यादिग्रहचारेषु ग्रहणे ग्रहसंयुतौ। त्रिप्रश्ने चन्द्रवत्तौ च सर्वत्राङ्गीकृतं हि तत् ॥ १२ ॥ द्वीपसागरशैलानां सङ्ख्याव्यासपरिक्षिपः। भवनव्यन्तरज्योतिर्लोककल्पाधिवासिनाम ॥ १३ ॥ नारकाणां च सर्वेषां श्रेणीबन्धेन्द्रकोत्कराः । प्रकीर्णकप्रमाणाद्या बुध्यन्ते गणितेन ते ॥ १४ ॥ प्राणिनां तत्र संस्थानमायुरष्टगुणादयः । यात्राद्यास्संहिताद्याश्च सर्वे ते गणिताश्रयाः ॥ १५ ॥ बहुभिर्विप्रलापैः किं त्रैलोकये सचराचरे । यत्किश्चिद्वस्तु तत्सर्वं गणितेन विना न हि ॥ १६ ॥ तीर्थकद्भयः कृतार्थेभ्यः पूज्येभ्यो जगदीश्वरैः । तेषां शिष्यप्रशिष्येभ्यः प्रसिद्धाद्गुरुपर्वतः ॥ १७ ॥ जलधेरिव रत्नानि पाषाणादिव काथनन् । शुक्र्मुक्ताफलानीव सङ्ख्याज्ञान महोदधे : ॥ १८ ॥
14 स्यात् ; B चापि. • Mand B दण्डा". 1K,M and B से
'B च. IM and R पुरा.
Kand x महा . R.HT क्षिपा. K and PT for .
•M वसु.

Page #41

संज्ञाधिकारः किश्चिदुइत्य तत्सारं वक्ष्येऽहं मतिशक्तितः । अल्प ग्रन्थमनल्पार्थ गणितं सारसङ्गहम ॥ १९ ॥ संज्ञाम्भोभिरयों पूर्णे परिकर्मोस वेदिके। कलासवर्णसंरूढलुठत्पाठीनसङ्कलें ॥ २० ॥ प्रकीर्णकमहामाहे वैराशिकतरङ्गिाणि । मिश्रकव्यवहारोद्यत्सूक्तिरनाशपिअरे ॥ २१ ॥ क्षेत्रविस्तीर्णपाताले खातारूप सिकताकुले । करणस्कन्धसम्बन्धच्छायावेलाविराजिते ॥ २२॥ गणकैर्गुणसम्पूर्णस्तदर्थमण योऽमलाः । गृह्यन्ते करणोपायैस्सारसङ्गहवारिधी ॥ २३ ॥
अथ संज्ञा । न शक्यतेऽयों बोडे यत्सर्वस्मिन संशया विना । आदावतोऽस्य शास्त्रस्य परिभाषाभिधारयते ।। २४ ॥
तत्र तावत् क्षेत्रपरिभाषा। जलानलादिभिनाश यो न यानि स पुद्गलः । परमाणुरनन्तस्तैरणुस्सोऽत्रादिरुच्यते ॥ २५ ॥ त्रसरेणुरतस्तस्माद्रयरेणु: शिरोरुहः । परमध्यजघन्याख्या भोगभूकर्मभूभुवाम ॥ २६ ॥ लीक्षा तिलस्स एवेह सर्षपोऽथ' यवोऽङ्गलम् । क्रमेणाष्टगुणान्येतद्व्यवहाराङ्गलं मतम् || २७ ।।
Hand B अल्प'. K संज्ञातोयसमा". ME (Probably a scrild's mistake for tथ) ad, मटे. P. Kaod l' .
'Mud Ba". .P and B-स्य.
P.N.

Page #42

गणितसारसङ्ग्रहः
तत्पश्चशतं प्रोक्तं प्रमाणं मानवेदिभिः । वर्तमाननराणामङ्गुलमात्माङ्गुलं भवेत् ॥ २८ ॥
व्यवहारप्रमाणे द्वे' राद्धान्ते लौकिके विदुः ।
आत्माङ्गुलमिति त्रेधा तिर्यक्पादः षडङ्गुलैः ॥ २९ ॥ .
9
पादद्वयं वितस्तिस्स्यात्ततो हस्तो द्विसङ्गणः । दण्डो हस्तचतुषेण क्रोशस्तद्विसहस्रकम् ॥ ३० ॥
योजनं चतुरः क्रोशान्प्राहुः क्षेत्रविचक्षणाः । वक्ष्यतेऽतः परं कालपरिभाषा यथाक्रमम् ।। ३१ ॥ अथ कालपरिभाषा |
अरण्वन्तरकाले व्यतिक्रामति यावति । स कालस्तमयोऽसङ्ख्यैस्समयैरावलिर्भवेत् ॥ ३२ ॥ सङ्ख्यातावलिरुच्छ्रासः स्तोकस्तूच्छाससप्तकः । स्तोकास्तप्त लवस्नेषां साधीष्टात्रिंशता घटी ॥ ३३ ॥ घटी मुहूर्तोऽत्र मुहूर्तेस्त्रिंशता दिनम् । पश्वनौस्त्रादिनैः नैः पक्षः पक्षौ द्वौ मास इष्यते ॥ ३४ ॥ ऋतुर्मासद्वयेन स्यात्रिभिस्तैरयनं मतम् । तद्दुयं वत्सरो वक्ष्ये धान्यमानमतः परम् ॥ ३५ ॥ अथ धान्यपरिभाषा |
विद्धि षोडशिकास्तत्र चतस्रः कुडहो' भवेत् । कुset चतुरः प्रस्थश्वतुः प्रस्थानथादकम् ॥ ३६ ॥ चतुर्भिदकैोणो मानी द्रोणेश्वतुर्गुणैः ।
खारी मानीचतुषेण वार्यः पञ्च प्रवर्तिका ॥ ३७ ॥
• K and B वां.
'Kai.
M Sम्ये.

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संज्ञाधिकारः . सेयं चतुर्गुणा वाहः कुम्भः पञ्च प्रवर्तिकाः । . .. .. इत : परं सुवर्णस्य परिभाषा विभाष्यते ' ॥ ३८ ॥
अथ सुवर्णपरिभाषा । चतुर्मिर्गण्डकैर्गुञ्जा गुमाः पञ्च पणोऽष्ट ते । धरणं धरणे कर्षः पलं कर्षचतुष्टयम् ॥ ३९ ॥
अथ रजतपरिभाषा। धान्यद्वयेन गुजैका गुञ्जायुग्मेन मापकः । माषषोडशकेनात्र धरणं परिभाप्यते ॥ ४० ॥ तद्दयं सार्धकं कर्षः पुराणांचतुरः पलम् । रूप्ये मागधमानेन प्राहुस्सख्यानकोविदाः ॥ ४१ ॥
अथ लोहपरिभाषा। कला नाम चतुष्पादाः सपादाप्षटुला यवः । यवैश्चतुर्मिरंशस्स्यावागोऽशानां चतुष्टयम् ॥ ४२ ॥ द्राणो मागषट्रेन दीनारोऽस्मादिसङ्गणः । द्वौ दीनारी सनेरं स्यात्माहुलौहेऽत्र सूरयः ।। ४३ ॥
I For the whole of धान्यपरिभाषा, 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.
आद्या पोशिका तत्र कुर(दु)व: प्रस्थ आठकः । प्रोणो मानी ततः खारी क्रमेण (मश: * ) चतुगहताः ।। (सहस्त्रैश्च त्रिभिष्परिश्शतैश्च श्रीहिभिस्समम् । यस्सम्पूर्णोऽभवत्सोऽयं कुलः परिभाष्यते ॥)
प्रतिकात्र ताः पत्र वाहस्तस्यावतुर्गुणः । .. कुम्भस्सपादवाहस्स्यात् (पञ्च प्रवतिका: कुम्भः) स्वर्णसंज्ञाय वर्यते ॥ .yसतेराव्यम.
•In Balno.

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गणितसारसङ्ग्रहः पलैादशभिस्साधैः प्रस्थः पलशतद्वयम । तुला दश तुला मार': सङ्ख्यादक्षाः प्रचक्षते ॥ ४४ ॥ वस्त्राभरणवेत्राणां युगळान्यत्र विंशतिः । कोर्ट कानन्तरं भाष्ये* परिकर्माणि नामत : ॥ ४५ ॥
____ अथ परिकर्मनामानि। आदिमं गुणकारोऽत्र प्रत्युत्पन्नोऽपि तद्भवेत्। द्वितीयं मागहाराख्यं तृतीयं कृतिरुच्यते ॥ १६ ॥ चतुर्थ वर्गमूलं हि भाप्यते पश्चमं घनः । घनमूलं ततष्षष्ठं सप्तमं च चितिस्स्मृतम् ॥ ४७॥ . तत्सङ्कलितमप्युक्तं व्युत्कलितमतोऽष्टमम् । तञ्च शेषमिति प्रोक्तं भिन्नान्यष्टावमून्यपि ॥ ४८ ॥ __ अथ धनर्णशून्यविषयकसामान्यनियमाः । ताडितः वेन राशिः व सोऽविकारी हतो युतः । हीनोऽपि ववधादिः खं योगे खं योज्यरूपकम् ॥ ४९ ॥ अणयोर्धनयो_ते भजने च फलं धनम । ऋणं धनर्णयोस्तु स्यात्स्वर्णयोर्विवरं युतौ ॥ ५० ॥ ऋणयोधनयोोगो यथासङ्ख्यमणं धनम् । शोध्यं धनमृणं राशेः ऋणं शोध्यं धनं भवेत् ॥ ५१ ॥ धनं धनर्णयोर्वर्गो मूले स्वर्णे तयोः क्रमात् । ऋणं स्वरूपतोऽवर्गों यतस्तस्मान्न तत्पदय ॥ ५९ ॥
अथ सङ्ख्यासंज्ञाः। 'शशी सोमश्च चन्द्रेन्दू भालेयांश रजनीकरः । श्वेतं हिमगु रूपञ्च मृगाङ्कश्च कलाधरः । १३ ॥
विद्याकलासवर्णस्य.
IN डि. • 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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संज्ञाधिकारः हि वे डावुभौ युगलयुग्मं च लोचनं द्वयम् । रष्टिनेत्राम्बकं द्वन्द्वमक्षिचक्षुर्नयं दशौ ॥ ५४॥ हरनेत्रं पुरं लोकं त्रै(त्रि)रलं भुवनत्रयम् । गुणो वह्निः शिरवी ज्वलनः पावकश्च हुताशनः ॥ ५५ ॥ अम्बुधिषिधिर्वाधिः पयोधिस्सागरो गतिः । नलधिर्बन्ध चतुर्वेदः कषायस्सलिलाकरः ॥ ५६ ॥ इषुर्बाणं शरं शस्त्र भूतमिन्द्रियसायकम् । पच व्रतानि विषयः करणीयस्तन्तुसायकः ॥ १७ ॥ अतुजीवो रसो लेख्या द्रव्या षटुकं ग्वरन । कुमारवदनं वर्ण शिलीमुरवपदानि च ॥ १८ ॥ शैलमद्रियं मधो नगाचलमनिर्गिरि : । . अश्वाश्विपन्नगा द्वीप धातुर्यसनमातृकम् ॥ ५९ ।। अष्टी तनुर्गजः कर्म वसु वारणपुषरम् । द्विरदं दन्ती दिग्दुरितं नागानीकं करी यथा ॥१०॥ नव नन्दं च रन्ध्रश्च पदार्थ लब्धकेशवी। निघिरनं ग्रहाणां च दुर्गनाम च सङ्ख्यया ॥ ११ ॥ आकाश गगनं शून्यमम्बरं वं नभो वियत् । अनन्तमन्तरिक्षं च विष्णपादं दिवि. स्मरेत् ॥ ६२ ॥
मथ स्थाननामानि । एकं तु प्रथमस्थानं द्वितीयं दशसंक्षिकम् । तृतीयं शतमित्याहुः चतुर्थ तु सहस्रकम् ॥ १३ ॥ पचमं दशसाहस्रं षष्ठं स्यालक्षमेव च। सप्तमं दशलक्षं तु मष्टमं कोठिरुच्यते ॥ १५ ॥

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नवमं दशकोत्यस्तु दशमं शतकोठयः । अर्बुदं रुद्रसंयुक्तं न्यर्बुदं द्वादशं भवेत् ।। ६५ । रवर्व त्रयोदशस्थानं महारवर्वं चतुर्दशम् । . पद्मं पञ्चदशं चैव महापमं तु षोडशम् ॥ १६ ॥ ' क्षोणी सप्तदशं चैव महाक्षोणी दशाष्टकम् । शङ्ख नवदशं स्थानं महाशझं तु विशकम् ॥ ६ ॥ क्षित्यैकविंशतिस्थानं महाक्षित्या द्विविंशकम् । त्रिविंशकमथ क्षोभं महाक्षोभं चतुर्नयम् ॥ ८॥
मथ गणकगुणनिरूपणम् । लघकरणोहापोहानालस्यग्रहणधारणोपायैः । व्यक्तिकराङ्कविशिष्टेणकोऽष्टाभिर्गुणै यः ॥ ६९ ॥ इति संज्ञा समासेन भाषिता मुनिपुङ्गवैः । विस्तरेणागमाद्वेद्यं वक्तव्यं यदितः परम् ॥ ७० ॥
इति सारसहे गणितशास्त्र महावीराचार्यस्य कृती संज्ञाधिकारसमाप्तः ॥

Page #47

प्रथमः परिकर्मव्यवहारः.
इतः परं परिकर्माभिधानं प्रथमव्यवहारमुदाहरिष्यामः ।
प्रत्युत्पन्नः । तत्र' प्रथमे प्रत्युत्पन्नपरिकर्मणि करणसूत्रं यथा
'गुणयद्गुणेन गुण्यं कवाटसन्धिक्रमेण संस्थाप्य । राश्यर्घखण्डतत्स्थैरनुलोमविलोममार्गाभ्याम् ॥ १ ॥
अत्रोद्देशकः । दत्तान्येकैकस्मै जिनभवना याम्बुजानि तान्यष्टौ । वसतीनां चतुरुत्तरचत्वारिंशच्छता य कति ॥ २ ॥ नव पद्मरागमणयस्समर्चिता एकजिनगृहे दृष्टाः । साष्टाशीतिद्विशतीमितवसतिषु ते कियन्तस्स्युः ॥ ३ ॥ चत्वारिंशश्चैकोनशताधिकपुप्यरागमणया ाः । एकस्मिन जिनभवने सनवशने ब्रुहि कति मणयः ॥ ४ ॥ पद्मानि सप्तविंशतिरे कस्मिन जिनगृहे प्रदत्तानि । माष्टानवतिसहस्त्रे "सनवशते नानि कति कथय ॥ ५ ॥ "एकैकस्यां वसतावष्टोत्तरशतसुवर्णपद्मानि । एकाष्टचतुस्सप्तकनवषटुवाष्टकानां किम् ॥ ६ ॥
KIच. Kand B विन्यस्योभी गशी. Kand Bसाणयेत. •B यहि. नस्या.
'शतस्य कति भवनानाम् . Nand B चत्वारिंशश्पका शताधिका. M च्छा.. • ४ ते कियन्तस्स्यु:.
M एकैकजिनालयाय इतानि. प्रयतनवशतगृहाणां किम् । Thinatana is found only in M and B.

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गणितसारसङ्ग्रहः शशिवसुरवरजलनिधिनवपदार्थभयनयसमूहमास्थाप्य । हिमकरविषनिधिगतिभिर्गुणिते किं' राशिपरिमाणम् ॥ ७ ॥ हिमगुपयोनिधिगतिशशिवह्निव्रतनिचयमत्र संस्थाप्य । सैकाशीत्या त्वं मे गुणयित्वाचक्ष्व तत्सङ्ख्याम् ॥ ८ ॥ अनिवसुरवरभयेन्द्रियशशलाञ्छनराशिमत्र संस्थाप्य' । रन्धैर्गुणयित्वा मे कथय मरवे राशिपरिमाणम् ॥ ९ ॥ नन्दायूतुशरचतुस्त्रिद्वन्द्वैकं स्थाप्य मत्र नवगुणितम् । आचार्यमहावीरैः कथितं नरपालकण्ठिकाभरणम् ॥ १० ॥ षट्विकं पश्चषटुश्च सप्त चादौ प्रतिष्ठितम् । त्रयस्त्रिंशत्सङ्गणितं कण्ठाभरणमादि शत्॥ ११ ॥ हुतवहगतिशशिमुनिभिर्वसुनयगतिचन्द्रमत्र संस्थाप्य । शैलेन तु गुणयित्वा कथयेदं रत्नकण्ठिकाभरणम् ॥ १२ ॥ अनलाब्धिहिमगुमुनिशरदुरिताक्षिपयोधिसोममास्थाप्य । शैलेन तु गुणयित्वा कथय त्वं राजकण्ठिकाभरणम् ॥ १३ ॥ गिरिगुणदिविगिरिगुणदिविगिरिगुणनिकरं तथैव गुणगुणितम् । पुनरेवं गुणगुणितम एकादिनवोत्तरं विद्धि ॥ १४ ॥ सप्त शून्यं 'द्वयं इन्हें पचैकश्च प्रतिष्ठितम् । वयःसप्ततिसङ्गण्यं "कण्ठाभरणमादिशेत् ॥ १५ ॥ जलनिधिपयोधिशशधरनयनद्रव्याक्षिनिकरमास्थाप्य । गुणिते तु चतुष्षष्टया का सङ्ख्या गणितविहि ॥ १६॥
In and B किन्तत्य. • ॥ प्यम्. अहो. • मे शीघ्रम्. .B PURI. Stanwas from 10 to 15 are found only in N and B. All the Mes. road स्थाप्य तत्र. 'B शे.
Bनयं. lo All the Msg. give the metrimls erroneous reading कण्ठाभरणं विनिर्दिशेत ।

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परिकर्मव्यवहारः शशावेन्दुवैकेन्दुशून्यैकरूपं निधाय क्रमेणात्र राशिप्रमाणम् । हिमांश्वप्ररन्धैः प्रसन्ताडितेऽस्मिन्
भवेत्कण्ठिका राजपुत्रस्य योग्या ॥ १७ ॥ • इति परिकर्मविधौ प्रथमः प्रत्युत्पन्नः समाप्तः ॥
भागहारः। द्वितीये भागहारपरिकर्मणि करणसूत्रं यथा
'विन्यस्य भाज्यमानं तस्याधस्थेन भागहारेण ।
सहशापवतविधिना भागं कृत्वा फलं प्रवदेत् ॥ १८॥ अथ वा
प्रतिलोमपथेन मजेदाज्यमधस्स्थन भागहारेण । सहशापवर्तनविधिर्यद्यस्ति विधाय तमपि तयोः ॥ १९॥
अत्रोदेशकः । दीनाराष्टसहस्रं वानवतियुतं शतेन संयुक्तम् । चतुरुत्तरषष्टिनरैर्भक्तं को शो नुरेकस्य ॥ २० ॥ रूपाग्रसप्तविंशतिशतानि कनकानि यत्र भाज्यन्ते । सप्तत्रिंशन्पुरुषैरेकस्यांशं ममाचक्ष्व ॥ २१ ॥ दीनारदशसहस्रं त्रिशतयुतं सप्तधर्गसम्मिश्रम् । नवसप्तत्या पुरुभक्तं किं लब्धमेकस्य ॥ २२ ॥ 'अयुतं चत्वारिंशश्चतुस्सहस्त्रैकशतयुतं हेनाम् । नवसप्ततिवसतीना दत्तं वित्तं किमेकस्याः ॥ २३ ॥
K
कोशी नंगकरण.
| This stanon is not found in
P This stanza is not found in P.
. Band K हेमम्.

Page #50

गणितसारसाहः 'सप्तदशत्रिशतयुतान्येकात्रिंशत्सहस्रजम्यूनि । भक्तानि नवत्रिंशन्नरैर्वदैकस्य भागं त्वम् ॥ २४ ॥ 'ध्यधिकदशत्रिशतयुतान्येकत्रिंशत्सहस्रजम्बूनि । सैकाशीतिशतेन प्रहृतानि नरैदैकांशम् ॥ २५ ॥ • त्रिदशसहस्री सैका षष्टिद्विशतीसहस्रषट्युता। रबानां नवपुंसां दत्तैकनरोऽत्र किं लभते ॥ २६ ॥
एकादिषडन्तानि क्रमेण हीनानि हाटकानि सरवे । विधुजलधिबन्धसङ्ख्यैर्नरैर्हतान्येकभागः कः ॥ २७ ॥
व्यशीतिमिश्राणि चतुश्शतानि चतुस्सहस्रननगान्वितानि । रत्नानि दत्तानि जिनालयानां
त्रयोदशानां कथयैकभागम् ॥ २८ ॥ इति परिकर्मविधौ द्वितीयो भागहारः समाप्तः ॥
वर्गः ।
तृतीये वर्गपरिकर्मणि करणसूत्रं यथा
द्विसमवधो घातो वा खेष्टोनयुतद्वयस्य सेष्टकृतिः। एकादिहिचयेच्छागच्छयुतिर्वा भवेद्वर्गः ॥ २९ ॥
.
M rends the problem contained in this stanzu thus:--
त्रिशतयुतैकत्रिंशत्सहस्रयुक्ता दशाधिका: सक्ष ।
भक्ताश्चत्वारिंशत्पुरुषरेकोनैस्तत्र दीनारम् ॥ • This stanza is found only in M. " एकद्वित्रिचतुःपञ्चषटूहीना: क्रमेण सम्भक्ताः ।
सैकचतुःशतसंयुतचत्वारिंशग्जिनालयाना किम् ॥

Page #51

परिकर्मव्यवहारः
18 द्विस्थानप्रभृतीनां राशीनां सर्ववर्गसंयोगः । तेषां क्रमघातेन द्विगुणेन विमिश्रितो वर्गः ॥ ३० ॥ कृत्वान्त्यकति हन्याच्छेषपदैद्विगुणमन्त्यमुत्सार्य । शेषानुत्सायैवं करणीयो विधिरयं वर्गे ॥ ३१ ॥
अत्रोद्देशकः । एकादिनवान्तानां पञ्चदशानां द्विसङ्गणाष्टानाम । व्रतयुगयोश्च रसाग्न्योश्शरनगयोर्वर्गमाचक्ष्व ॥ ३२ ॥ साष्टाविंशत्रिशती चतुस्सहस्कषष्टिषटछतिका । द्विशती षट्पश्वाशन्मिश्रा वर्गीकृता किं स्यात् ॥ ३३ ॥ लेख्यागुणेषुबाणद्रव्याणां शरगतित्रिसूर्याणाम । गुणरत्नामिपुराणां वर्ग भण गणक यदि वेत्सि-॥ १५ ॥
सप्ताशीतित्रिशतसहित पदसहस्रं पुनश्च पश्चत्रिंशच्छतसमधिकं सप्तनिघ्नं सहस्रम । हाविंशत्या युतदशशतं 'वर्गितं तत्रयाणां
ब्रूहि त्वं में गणक गुणवन्सङ्गणय्य प्रमाणम् ॥ ३५ ॥ इति परिकर्मविधौ तृतीयो वर्गस्समाप्तः ।।
वर्गमूलम् । चतुर्थे वर्गमूलपरिकर्मणि करणसूत्रं यथा
अन्त्यौजादपहृतकतिमूलेन द्विगुणितेन युग्महतो । लब्धकृतिस्त्याज्योने द्विगुणदलं वर्गमूलफलम् ॥ ३ ॥
IP, Kand B राशिरेनस्कृतीनाम.

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गणितसारसाहः
अत्रोद्देशकः । 'एकादिनवान्तानां वर्गगतानां वदाशु मे मूलम् । अतुविषयलोचनानां द्रव्यमहीधेन्द्रियाणाञ्च ।। ३७ ।। एकाग्रषष्टिसमधिकपञ्चशतोपेतषट्सहस्राणाम् । षडर्गपञ्चपञ्चकषण्णामपि मुलमाकलय ॥ ३८ ॥ ' द्रव्यपदार्थनयाचललेख्यालबध्यब्धि निधिनयाब्धीनाम् ।
शशिनेत्रेन्द्रिययुगनयजीवानाश्चापि किं मूलम् ॥ ३९ ॥ चन्द्राब्धिगतिकषायद्रव्यर्तहुताशनतराशीनाम् । विधुलेख्येन्द्रियहिमकरमुनिगिरिशशिनां च मूलं किम् ॥ ४० ॥ द्वादशशतस्य मूलं षण्णवतियतस्य कथय सश्चिन्त्य । शतषट्कस्यापि सरवे पश्चकवर्गण युक्तस्य ।। ४१ ॥
भाभकर्माम्बरशङ्कराणां सोमाक्षिवैश्वानरभास्कराणाम् । चन्द्र बाणाब्धिगतिहिपानामाचक्ष्व
मूलं गणकाग्रणीस्त्वम् ॥ ४२॥ इति परिकर्मविधौ चतुर्थ वर्गमूलं समाप्तम् ।।
घन: । पश्चमे घनपरिकर्मणि करणसूत्रं यथा --- त्रिसमाहतिर्घनस्स्यादिष्टोनयुतान्यराशिघातो वा । अल्पगुणितष्टकृत्या कलितो बृन्देन चेष्टस्य ॥ ४३ ॥ इष्टादिहिगणेष्टप्रचयेष्टपदान्वयोऽथ वेष्टकृतिः। ठयेकेष्टहतैकादिद्विचयेष्टपदै ययुक्ता वा ॥ ४४ ॥ 'एकादिचयेष्टपदे पूर्व राशि परेण सङ्गणयेत् । गुणितसमासस्त्रिगुणश्चरमेण युतो घनो भवति ॥ ४५ ॥ Pand M वर्गगतानो शीघ्रं रूपादिनवावसानराशीनाम् । मूलं कथय सखे त्वं.
'M नब.
This stanza is not found in P.

Page #53

परिकर्मव्यवहारः
अन्त्यान्यस्थानकृतिः परस्परस्थानसङ्गणा त्रिहता । पुनरेवं तद्योगस्सर्वपदधनान्वितो बृन्दम ।। ४६ ।। 'अन्त्यस्य घनः कृतिरपि सा त्रिहतोत्सार्य शेषगुणिता वा । शेषकृतित्र्यन्त्यहता स्थाप्यत्सायैवमत्र विधिः || ४७ अत्रोद्देशकः ।
'एकादिनवान्तानां पञ्चदशानां शरेक्षणस्यापि । रसवह्नयोगिरिनगयोः कथय घनं द्रव्यध्योश्च ॥ ४८ ॥ पिकरगगनेन्दूनां नयगिरिशशिनां वरेन्दुवाणानाम् । वद मुनिचन्द्र वृन्दं चतुरुदविगुणशशिनान् ॥ ४९ ॥ राशिनीकृतोऽयं शयं मिश्रित त्रयोदशभिः । तद्दिगुणोऽस्मात्रिगुणश्चतुर्गुणः पञ्चगुणितश्च ॥ ५० ॥ शतमष्टषष्टयुक्तं दृष्टमभीष्टं घने विशिष्टतमैः । एकादिभिरष्टान्यैर्गुणितं वद तद्धनं शीघ्र ॥ ११ ॥
बन्याम्बर्तृगगनेन्द्रिय केशवानां सङ्ख्याः क्रमेण विनिधाय घन गृहीत्वा । आचक्ष्व लब्धमधुना करणानुयोगगम्भरिसारतरमागरपारदश्वन ।। ५२ ।।
इति परिकर्मविध पश्चमो घनस्समाप्तः ||
PM रपि.
• M गां वा.
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
स्तद्वर्गत्रिगुणितां हतशेषः । उत्सार्य शेषकृतिरथ निष्टा त्रिगुणा घनस्तथा वा ॥
15
Instead of stanzas 48 and 49, M reads
एकादिनवान्तानां रुद्राणां हिमकांन्दूनाम् 1 वद मुनिचन्द्रयतीनां वृन्दं चतुरुदधिगुणशशिनाम् ॥

Page #54

16
गणितसारसङ्ग्रहः
घनमूलम् ।
षष्ठे घनमूलपरिकर्माणि करणसूत्रं यथा
अन्त्यघनाद पहृतघनमूलकृतित्रिहतिभाजिते भाज्ये । प्राहिताप्तस्य कृतिश्शोध्या शोध्ये घनेऽथ धनम् ॥ ५१ ॥
'घनमेकं द्वे अघने घनपदकृत्या भजेत्रिगुणयाघनतः । पूर्वत्रिगुणाप्त कृतिस्त्याज्याप्तघनश्च पूर्ववल्लब्धपदैः ॥ ५४ ॥ अत्रोद्देशः ।
एकादिनवान्तानां घनात्मनां रत्नशशिनवाब्धीनाम् । 'नगरसव सुवर्त गजक्षपाकराणाश्च मूलं किम् ॥ ९५ ॥ गतिनयमदशिविशशिनां मुनिगुणरवत्वीक्षनव' खरानीनाम् । 'वसु 'वयुगवाद्रिगतिकरिचन्द्रर्तनां गृहाण पदम ॥ ५६ ॥ चतुः पयोध्यशिराक्षिदृष्टि ये भव्योमभयेक्षणस्य ।
दाष्टकर्माब्धिरघातिभाव द्विवह्निरत्नर्तुनगस्य मूलम् ॥ ५७ ॥ द्रव्याश्वशैल दुरितखवह्नयद्रिभयस्य वदत घनमूलम् । नवचन्द्र हिमगु मुनिशशिलब्ध्यम्बरखरयुगस्यापि ॥ ९८ ॥ 'गतिगज विषयेषुविधुत्वराद्रिकरगतियुगस्य मण मूलम् । लेख्याश्वनगनवाचलपुरखरनयजीवचन्द्रमसाम् ॥ ५९ ॥
गतिरखरदुरितेभाम्भोधितार्क्ष्यध्वजाक्षद्विकृति नवपदार्थद्र व्यवहीन्दुचन्द्र । जलधरपथरन्ध्रवष्टकानां घनानां गणक गणितदक्षाचक्ष्व मूलं परीक्ष्य ॥ ६० ॥
इति परिकर्मविधौ षष्ठं घनमूलं समाप्तम् ॥
M गिरि.
M रसा.
This stana is not found in M
This stanza is not found in M.
M विधुपुरखरस्वरर्तुज्वलनधरार्णा.

Page #55

17
परिकर्मव्यवहारः
सङ्कलितम् । सप्तमे सङ्कलितपरिकर्मणि करणसूत्रं यथा -- रूपेणोनो गच्छो दलीकृतः प्रचयताडितो मिश्रः ।
प्रभवेण पदाभ्यस्तस्सङ्कक्रितं भवति सर्वेषाम् ॥ १ ॥ प्रकारान्तरेण धनानयनसूत्रम्एकविहीनो गच्छः प्रचयगुणो द्विगुणितादिसंयुक्तः । गच्छाभ्यस्तो द्विहतः प्रभवेत्सर्वत्र सङ्कलितम् ॥ १२ ॥ आधुत्तरसर्वधनानयनसूत्रम्--- पदहतमुवमादिधनं व्येकपदार्धनचयगुणो गच्छः । उत्तरधनं 'तयोोगो धनमूनोत्तरं मुखेऽन्त्यधने ॥ १३ ॥ अन्त्यधनमध्यधनसर्वधनानयनसूत्रम्-..
चयगुणितैकोनपदं साद्यन्त्यधनं तदादियोगार्थम् । मध्यधनं तत्पदवधमुद्दिष्टं सर्वसङ्कलितम् ॥ ६४ ॥
____ अत्रोद्देशकः । एकादिदशान्ताद्यास्तावत्प्रचवास्समर्चयन्ति धनम् । वणिजो दश दश गच्छास्तेषां सङ्कलितमाकलय ॥ १५ ॥ द्विमुरवत्रिचयैर्मणिमिः प्रानर्च श्रावकोत्तमः कश्चित् । पञ्चवसतीरमीषां का सङ्ख्या धूहि गणितज्ञ ॥ ६ ॥ आदिस्त्रयश्चयोऽष्टौ द्वादश गच्छस्त्रयोऽपि रूपेण । आ सप्तकात्मवृद्धास्सर्वेषां गणक भण गणितम् ॥ १७ ॥ विकृतिर्मुरवं चयोऽष्टी नगरसहस्रे सर्चितं गणितम् । गणिताब्धिसमुत्तरणे बाहुबलिन् त्वं समाचक्ष्व ॥ ६ ॥
। तद्ना सैक(व !)पदाप्ता युतिः प्रभवः । This stanu is omitted in K.
..
ली.
12499.

Page #56

गणितसारसङ्ग्रहः
गच्छानयनसूत्रम्अष्टोत्तरगुणराशेर्द्विगुणाद्युत्तरविशेषकृतिसहितात् । मूलं चययुतमर्धितमाघूनं चयहतं गच्छः ॥ १९ ॥ प्रकारान्तरेण गच्छानयनसूत्रम्
अष्टोत्तरगुणराशेर्टिगुणाद्युत्तरविशेषकृतिसहितात् । मूलं क्षेपपदोनं दलितं चयभाजितं गच्छः ॥ ७० ॥
अत्रोद्देशकः ॥ आदिौं प्रचयोऽष्टौ हौ रूपेणा यात्क्रमाबदौ । वाको रसाद्रिनेत्रं रवेन्दुहरा वित्तमत्र को गच्छः ।। ७१ ॥ आदिः पत्र योऽष्टौ गुणलानिधनमत्र को गच्छः । षट् प्रभवश्व चयोऽष्टौ स्वद्विचनुस्त्वं पदं किं स्यात् ॥ ७२ ॥
सोर
उत्तराद्यानयनसूत्रम्--
आदिधनोनं गणितं पदोनपदकृतिदलेन सम्भजितम् । प्रचयस्तदनहीनं गणितं पदभाजितं प्रभवः ॥ ७३ ॥ आधुत्तरानयनसूत्रम्प्रभवो गच्छाप.(धनं विगतैकपदार्धगुणितचयहीनम् । पदहाधनमायनं निरकपददलहत प्रचयः ।। ७४ ॥ प्रकारान्तरेणोत्तराद्यानयनसूत्रद्वयम् -- द्विहतं सङ्कलितधनं गच्छहतं द्विगुणितादिना रहितम् । विगतैकपदविभक्तं प्रचयस्स्यादिति विजानीहि ।। ७५ ॥ द्विगुणितसङ्कलितधनं गच्छहतं रूपरहितगच्छेन । ताडितचयेन रहितं. द्वयेन सम्भाजितं प्रभवः ॥ ७६ ॥

Page #57

परिकर्मव्यवहारः
19
अत्रोद्देशकः । नव वदनं तत्त्वपदं भावाधिकशनधनं कियान्प्रचयः । पश्च चयोऽष्ट पदं षट्पनाशच्छतधनं मुरवं कथय ॥ ७७ ॥ खेष्टायुत्तरगच्छानयनसूत्रम् - सङ्कलिते स्वेष्टहते हारो गच्छोऽत्र लब्ध इष्टोने । ऊनितमादिश्शंषे व्येकपदार्थोद्धृते प्रचयः ।। ७८ ॥
अत्रोद्देशकः । चत्वारिंशन्सहिना पशती गणितमत्र सन्दृष्टम । गच्छप्रचयप्रभवान गणितज्ञशिरोमणे कथय ॥ ७९ ॥
आयुत्तरगच्छसर्वमिश्रधनविलेपणे सूत्रत्रयम ---. उत्तरधनेन रहितं गच्छेनैकन संयतेन हतम् । मिश्रधनं प्रभवरस्यादिति गणकशिरोमणे विन्हि ।। ८० ॥ आदिधनोनं मिश्रं रूपोनपदार्थगणितगच्छेन । सैकेन हंत प्रचयो गच्छविधानात्पदं मुरवे सैके ॥ ८१ ॥ मिश्रादपनीतेप्टी मुरवगच्छौ प्रचमिश्रविधिलब्धः । यो राशिस्स चयस्यात्करणमिदं सर्वसयोगे ॥ ८२ ॥
भत्रोद्देशकः । द्वित्रिकपणदशाग्रा चत्वारिंशन्मुग्वादिमिश्रधनम् । तत्र प्रमवं प्रचयं गच्छं सर्व च मे बेहि ॥ ८३ ॥
11 विगणय्य सखे ममाचक्ष्न.
पदोनपदकृतिदलन सकेन । म. प्रत्योऽत्र पदं गच्छविधानान्मुखे सके ।

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गणितसारसङ्ग्रहः ___ दृष्ठधनायुत्तरतो द्विगुणत्रिगुणविभागविभागादीष्टधनाद्युत्तरानयन सूत्रम्
इष्टविभक्तेष्टधनं द्विष्ठं तत्प्रचयताडितं प्रचयः । तत्प्रभवगुणं प्रभवो 'गुणभागस्येष्टवित्तस्य ॥ ८४ ॥
अत्रोदेशकः। समगच्छश्चत्वारःषष्टिर्मुखमुत्तरं ततो द्विगुणम् । तयादिहतविभक्तस्वेष्टस्यायुत्तरे ब्रूहि ॥ ८५ ॥
इष्टगच्छयोर्व्यस्ताद्युत्तरसमधनद्विगुणत्रिगुणद्विभागत्रिभागादिधनानयनसूत्रम् -
व्येकात्महतो गच्छस्वेष्टनो द्विगुणितान्यपदहीनः । मुरवमात्मोनान्यकृतिदिकेष्टपदघातवर्जिता प्रचयः ॥ ८६ ॥
___ अत्रोद्देशकः। पश्चाष्टगच्छपुंसो यस्तप्रभवोत्तरे समानधनम् । द्वित्रिगुणादिधनं वा ब्रूहि त्वं गणक विगणय्य ॥ ८७ ॥ द्वादशषोडशपदयोर्व्यस्तप्रभवोत्तरे समानधनम् । यादिगुणभागधनमपि कथय त्वं गणितशास्त्रज्ञ ॥ ८ ॥ असमानोत्तरसमगच्छसमधनस्याद्युत्तरानयनसूत्रम्-- अधिकचयस्यैकादिश्चाधिकचयशेषचयविशेषो गुणितः । विगतैकपदार्धेन सरूपश्च मुरवानि मित्र शेषचयानाम् ॥ ८९ ॥
अत्रोदेशकः । एकादिषडन्तचयानामेकत्रितयपश्चसप्तचयानाम् । नवनवगच्छानां समवित्तानां चाशु वद मुखानि सखे ॥९०॥ 1 x गुणभागायुत्तरेच्छायाः. ' गुण. ' गणकमुखतिलक ।

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परिकर्मव्यवहारः
श विसदृशादिसडशगच्छसमधनानामुत्तरानयनमूत्रम्
अधिकमुखस्यैकचयश्चाधिकमखशेषमुखविशेषो भक्तः । विगतैकपदार्धन सरूपश्च चा भवन्ति शेषमुग्वानाम ॥ ११ ॥
अत्रोद्देशकः। एकत्रिपश्चसप्तनवैकादशवदनपत्रपत्रिपदानाम् । समवित्तानां कथयोत्तराणि गणिताब्धिपारदश्वन गणक ॥९२ ।। अथ गुणधनगुणसङ्कलितधनयोस्तुत्रम् ---- पदमितगुणहतिगुणितप्रभवस्म्याद्गुणधनं नाद्यनम् । एकोनगुणविभक्तं गुणसङ्कलितं विजानीयात् ॥ ९३ ॥ गुणसङ्कलिते अन्यदपि सुत्रम् -- समदलविषमखरूपो गुणगणितो वर्गनाडिनो गच्छ । रूपोनः प्रभवनो व्येकोत्तरभाजितस्मा म ॥ ९४ ।। गुणसङ्कलितान्त्यधनानयने मत्सलिाधनानाने व सूत्रम् -- गुणसङ्कलितान्त्यधनं विगतकपदग्य गणधनं भवति । तद्गुणगुणं मुखोनं व्यकोत्तरभाजित माम ॥ ९५ ॥
गुणधनस्योदाहरणम् । वर्णद्वयं गृहीत्वा त्रिगुणधनं प्रतिपुर समा यति । यः पुरुषोऽप्टनगर्या तस्य किर्याहत्तमांचक्ष्व ॥ ९६ ॥ गुणधनस्यायुत्तरानयनसूत्रम् - गुणधनमादिविभक्तं यत्पदमितवघसमं स एव चयः । गच्छप्रमगुणघाताहतं गुणितं भवेत्प्रभवः ॥ ९७ ॥
IN समयति.

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9.2
गणितसारसङ्ग्रहः गुणधनस्थ गच्छानयनसूत्रम् -- मुखभक्ते गुणावते यथा निरग्रं तथा गुणेन हते। यावयोऽत्र शलाकास्तावान गच्छो गुणधनस्य ॥ ९ ॥
गुणसङ्कलितोदाहरणम् । । दीनापनकादिद्विगुणं धनमर्जयन्नरः कश्चित् । प्राविक्ष टनगरीः कति जातास्तस्य दीनाराः ॥ ९९ ॥ सप्तमुखात्रिगुण वयत्रिवर्गगच्छस्य किं धनं वणिजः। , त्रिपक्षकप-शप्रभवगुणोत्तर परस्यापि ॥ १० ॥ गुणसङ्कालिनोत्तराधानयनसूत्रम् -
असकृयेकं मुखहतवित्तं येनोद्भुतं भवेत्स चयः । व्येकगुणगुणिनगणितं नोकपद-मात्रगुणवयातं प्रभवः ।। १०१ ॥
अत्रोदेशकः । त्रिमुरव (गच्छ यासाङ्काम्बर गलनिधियने कि प्रचयः । षङ्गणवाप पावरशशि हिमगुत्रित्तभत्र मुखं किम् ।। १०२ ॥ • गुणसङ्कलिनगच्छानवनसूत्रम - एकोनगुणाभ्यरत प्रभवहनं रूपसंयु वित्तम् । यावत्कृत्वो अक्तं गुणेन तद्वारसम्मितिर्गच्छः ॥ १०३ ॥
अत्रांद्देशकः। त्रिप्रभवं पटुगुणं सारं सप्तत्युपेततप्तशती। सप्तामा ब्रूहि सरवे कियत्पदं गणक गुणनिपुण ॥ १० ॥
पचादिद्विगुणोत्तरे शरगिरिध्येकप्रमाणे धनेसप्तादि त्रिगुणे नगेअदुरितस्तम्बेरमर्तुप्रमे ।

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परिकर्मव्यवहारः त्र्यास्ये पश्चगुणाधिके हुतवहोपेन्द्राक्षवहिद्विप
श्वेताशुद्विरदेभकर्मकरदृड्पानेऽपि गच्छ: किवान ॥ १०५ ॥ इति परिकर्मविधौ सप्तमं सङ्कलितं समाप्तम ।। .
व्युत्कलितम्। अष्टमे व्युत्कलितपरिकणि कगसूत्रं यथा ... सपदेष्टं वेष्टमपि व्येकं दलिनं चाहतं समत्वम् । शेषेष्टगच्छगुणितं व्युत्कलितं खंट वित्तं च ॥ १०६ ॥ प्रकारान्तरेण व्युत्कलितधनस्वेष्टधनानयन सूत्रः ... गच्छसहितेष्टमिष्ट चेकोनं चयहत विहादियुतम् । ।
शेषेष्टपदार्धगुणं व्यत्काल स्वष्टवित्तापि ॥ १०७ ॥ चगगुणभवव्युत्कलिनधनानयने व्युत्कलितधनस्य शेषेष्टगच्छान यने च सूत्रम
इष्टधनोनं गणित व्यवकलितं च यभवं गणोत्थं च । सर्वेष्टगच्छशेषे शेषपदं जायते ।म्य ॥ १०८ ॥ शेषगच्छस्याद्यानयनसूत्रम् प्रचयगुणिनेष्टगच्छस्मादिः प्रभवः पदस्य, शेषस्य । प्राक्तन एव चयस्स्याद्गच्छस्वष्टस्य मावेव :: १०९ ॥ गुणव्युत्कलितशेषगच्छस्थाद्यानयनसूत्रम्---
गुणगुणितेऽपि चयादी तथैव भदोऽयमत्रशेपपदे । इष्टपदमितिगुणाहतिगुणितप्रभवो भवेदकम् ।। १११ ॥
IM गणितं.

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गणितसारसग्रहः
अत्रोद्देशकः । द्विमुग्यस्त्रिचयो गच्छश्चतुर्दश खेप्सितं पदं सप्त । अष्टनवषट्रपन च किंव्युत्कालतं समाकलय ॥ १११ ॥
षडादिरष्टौ प्रचयोऽत्र पटुतिः पदं दश द्वादश षोडशेप्सितम्।। मुखादिरन्यस्य तु पञ्चपञ्चकं
शतद्वयं ब्रूहि शतं व्ययः कियान् ॥ ११२ ॥ षड्डनमानो गच्छः प्रचयोऽष्टौ द्विगुणसप्तकं वक्रम् । सप्तत्रिंशत्खेष्टं पदं समावि फलमुभयम ।। ११३ ।। अष्टकृतिरादिरुत्तरमूनं चत्वारि षोडशात्र पदम् । इष्टानि तत्त्वकेशवरुद्रार्कपदानि कि शेषम् ॥ ११४ ॥ गुणव्युत्कलिनस्योदाहरणम्चतुरादिद्विगुणात्मकोत्तरयुतो गच्छ चतुर्णा कृतिदश वाञ्छापदमसिन्धुरगिरिद्रव्येन्द्रियाम्भोधयः । कथय व्यत्कलितं फलं सकलसन्जाग्रिम 'व्याप्तवान करणस्कन्धवनान्तरं गणितविन्मत्तेभविक्रीडितम् ॥ ११५ ॥ इति परिकर्मविधावष्टमं व्युत्कलितं समाप्तम् ।।
इति सारसनाहे गणितशास्त्रे महावीराचार्यस्य कृतौ परिकर्मनामा प्रथमो व्यवहारः समाप्तः ॥
18.

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अथ द्वितीयः कलासवर्णव्यवहारः ।
'त्रिलोकराजेन्द्रकिरीठकोटिप्रभाभिरालीदपदारविन्दम् ।
निर्मूलमुन्मूलितकर्मवृक्षं जिनेन्द्रचन्द्रं प्रणमामि भक्त्या ॥ १ ॥ इतः परं कलासवर्णं द्वितीयव्यवहारमृदाहरिष्यामः ।।
मिनप्रत्युत्पन्नः । तत्र भिन्न प्रत्युत्पन्ने करणसूत्रं यथा -- गुणयेदंशानशैहारान हारैर्घटेत यदि तेषाम । व चापवर्तनविधिविधाय तं भिन्नगणकारे ॥ २ ॥
अत्रोद्देशकः । शुण्ठ्याः पलेन लभते चतुर्नशिं पणस्य यः पुरुषः । किमसौ ब्रूहि सवे त्वं त्रिगणेन पलाष्टभागेन ॥ ३ ॥ मरिचस्य पलस्यार्घः पणस्य सप्ताष्टमांशको यत्र । तत्र मवल्कि मूल्यं पलपटपञ्चांशकस्य वद ॥ ४ ॥ कश्चित्पणेन लभते त्रिपञ्चभागं पलस्य पिप्पल्याः । नवभिः पढिभक्तैः किं गणकाचक्ष्व गुणयित्वा ॥ ५ ॥ क्रीणाति पणेन वणिग्रजीरकपलनवदशांशकं यत्र । तत्र पणैः पश्चाः कथय त्वं कि समग्रमते ॥ ६ ॥ यादयो द्वितयवृद्धयोऽशकारूयादयो द्वयचया हगः पुनः । ते द्वये दशपदाः कियत्फलं बृहि तत्र. गुणने द्वयोईयोः ॥ ७ ॥ इति भिन्नगुणकारः।
. मी.
This stanza inomitted in P.

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26
गणितसारसाहः
भिन्नभागहारः। भिन्न भागहारे करणसूत्रं यथा - अंशीलत्यच्छेदं प्रमाणराशेस्ततः क्रिया गुणवत् । प्रमितफलेऽन्यहरने विच्छिदि वा सकलवच्च भागहतौ ॥ ।
अत्रोद्देशकः। हिङ्गोः पलार्धमौल्यं पणत्रिपादांशको अवेद्यत्र । तत्रार्धे विक्रीणन पलमेकं किं नरो लभते ॥ ९ ॥ अगरोः पलाष्टमेन त्रिगुणेन पणस्य विंशतिव्यंशान् । उपलभते यत्र पुमानेकेन पलेन किं तत्र ।। १० ।। पणपञ्चमैश्चतुभिर्नरवस्य पलसप्तमो यशीतिगणः । संप्राप्यो यत्र स्यादेकेन पणेन किं तत्र ।। ११ ॥ ध्यादिरूपपरिवद्धियुजोडशा वावदष्टपदमेकविहीनाः । हारकासत इह द्वितयाद्यैः किं फलं वद परेषु हतेषु ॥ १२ ॥
इति भिन्नभागहारः ।
भिन्नवर्गवर्गमूलघनघनमूलानि ॥ 'भिन्नवर्गवर्गमूलघनघनमूलेषु करणसूत्रं यथा
कृत्वाच्छेदांशकयोः कृतिकृतिमूले घनं च घनमूलम् । तच्छेदैरंशहती वर्गादिफलं भवेशिने ॥ १३ ॥
1 4 भिन्नवर्गभिमवर्गमूलभिमघनतन्मूलेषु.

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कलासवर्णव्यवहारः.
अत्रोद्देशकः । पञ्चकसप्तनवानां दलितानां कथय गणक वर्ग त्वम्। षोडशविंशतिशतकद्विशतानां च त्रिभक्तानाम् ॥ १४ ॥ त्रिकादिरूपद्वयवृद्धोऽशा द्विकादिरूपोत्तरका हराश्च । पदं मतं द्वादश वर्गमेषां वदाश मे त्वं गणकाग्रगण्य ॥ १५ ॥ पादनवांशकपोडशभागानां पञ्चविंशतितमस्य । षत्रिंशदागस्य च कृतिमूलं गणक भण शीघ्रम् ।। १६ ॥ भिन्ने वर्ग राशयो वर्गिना ये. तेषां मूलं सप्तशत्याश्च किं स्यात् । त्र्यष्टोनायाः पञ्चवर्णोद्धताया बेहि वं मे वर्गमूलं प्रवीण ॥ १७ ॥ अर्धत्रिभागपादाः पञ्चांशकषष्ठसप्तमाष्टांशाः । दृष्टा नवमश्चैषां पृथक पृथगहि गणक घनम् ॥ १८ ॥ त्रितयादिचतश्चयको शगणो हिमुखद्विचयोऽत्र हरप्रचयः । .. दशकं पदमाशु तदीयघनं कथय प्रिय सूक्ष्ममते गणिते ॥ १९ ॥ शतकस्य पञ्चविंशस्याष्टविभक्तस्य कथय घनमूलम् । 'नवयुतसप्तशतानां विशानामष्टभक्तानाम् ॥ २० ॥
14 सप्तशतस्यापि सर्व व्येकानशिकाष्टकाप्तस्य ॥

Page #66

गणितसारसहः
भिन्नघने परिदृष्टघनानां मूलमुदग्रमते वद मित्र । त्र्यूनशतद्वययुग्द्विसहस्याश्रापि नवप्रहतत्रिहतायाः ॥ २१ ॥
इति भिन्नवर्गवर्गमूलघनघनमूलानि ।। - भिन्नसङ्कलितम् । भिन्नसङ्कलिते करणसूत्रं यथा ----- पदमिष्टं प्रचयहतं द्विगुणप्रभवान्वितं चयेनोनम् । गच्छार्थेनाभ्यस्तं भवनि फलं भिन्नसङ्कलिते ॥ २२ ॥
__ अत्रोद्देशकः । द्विध्यशष्षहभागस्त्रिचरणभागो मुरवं चयो गच्छः । द्वौ पचमौ त्रिपादो द्विव्यंशोऽन्यस्य कथय कि वित्तम् ॥ २३ ॥ आदिः प्रचयो गच्छस्त्रिपञ्चमः पञ्चमस्त्रिपादांशः । सर्वांशहरौ वृद्धौ द्वित्रिभिरा सप्तकाञ्च का चितिः ॥ २४ ।। इष्टगच्छस्याद्युत्तरवर्गरूपघनरूपधनानयनसूत्रम् -- पदमिष्टमेकमादिव्य॒केष्टदलोद्भुतं मुरवानपदम् । प्रचयो वित्त तेषां वर्गो गच्छाहतं वृन्दम् ॥ २५ ॥
अत्रोद्देशकः। 'पदमिष्टं द्विव्यंशो रूपेणांशो हरश्च संवृद्धः । यावद्दशपदमेषां वद मुवचयवर्गवृन्दानि ॥ २६ ॥
This stanza is not fourd in N.

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20
कलासवर्णव्यवहारः. इष्टघनधनाद्युत्तरगच्छानयनसूत्रम् ---- इष्टचतुर्थः प्रभवः प्रभवात्प्रचयो भवेद्दिसङ्गणितः । प्रचयश्चतुरभ्यस्ता मच्छस्तेषां युतिवृन्दम् ॥ २७ ॥
अत्रोद्देशकः । द्विमुग्वैकचया अंशास्त्रिप्रभवकोत्तग हग उभये । पचपदा वद तेषां घनधनमुरवचयपदानि सवं ॥ २८ ॥ 'दृष्टयनाद्युत्तरतो द्विगणत्रिगर्णाद्रभागत्रिभागादीष्टधनाद्यत्तरानयन सूत्रम -
दृष्टविभक्तेष्टधनं द्विष्ठं तत्प्रचयताडितं प्रचयः । त्प्रभवगुणं प्रभवो गुणभागस्येष्टवित्तस्य ॥ २९ ॥
_अत्रोद्देशकः । प्रभवरूपी रूपं प्रवयः पञ्चाष्टमसमानपदम् । इच्छाधनमपि नावत्कथय सब को मुखप्रचयां ॥ ३० ॥ प्रचयादादिगिणस्त्रयांदशाप्टादशं पदं वेष्टम।। वित्तं तु सप्तपष्टिः पचनभक्ता वदादिचयौ ॥ ३१ ॥ 'मुखमकं द्विव्यंशः प्रचना गच्छरसमश्चतुर्नवमः ।। धनमिष्टं द्वाविंशतिरकाशीत्या वदादिचयौ ॥ ३२ ॥ 'गच्छानयनसूत्रम् ---- द्विगुणचयगुणितवित्तादुत्तरदलमखावशेषकतिर्माहतात् ।
मलं प्रचयार्धयुतं प्रभवानं चयहतं गच्छः ॥ ३३ ॥ '५ गुणभागाद्युनरानयनसूत्रम् ।
५ ॥ प्रचगन. : M गुणभागागुत्तरच्छाया:. This stanza takes the place of stanza No.31 in Maul is viitted in B.
Insterd of the following two stanzar Mod. अात्तग्गुणगशान्यादिना इ. धनगच्छ आनंतव्यः and tepent stanza So. 70 give m... परिकर्मव्यवहार,

Page #68

गणितसारसङ्ग्रहः प्रकारान्तरेण तदेवाह - द्विगुणचयगुणितवित्तादुत्तरदलमुखविशेषकृतिसहितात् । मूलं क्षेपपदोनं प्रचयेन हृतं च गच्छस्स्यात् ॥ ३४ ॥
अत्रोद्देशकः । द्विपञ्चांशो वकं त्रिगुणचरणस्स्यादिह चयः पडंशस्तप्तन्नस्त्रिकतिविहतो वित्समुदितम् । • चयः पश्चाष्टांशः पुनरपि मुखं व्यष्टममिति त्रिचत्वारिंशास्स्वं प्रिय वद पदं शीघ्रमनयोः ॥ ३५ ॥
आधुत्तरानयनसूत्रम् -- 'गच्छाप्तगणितमादिविगतैकपदार्धगुणितचयहीनम् । पदहनधनमान निरेक पददलहतं प्रचयः ॥ ३६ ॥
अत्रोद्देशकः । त्रिचतुर्थचतुःपञ्चमचयगच्छे खेषुशशिहतैकत्रिंशद्- । वित्ते व्यंशचतुःपञ्चममुखगच्छे च वद मुखं प्रचयं च ॥ ३७ ॥
इष्टगच्छयोर्व्यस्ताद्युत्तरसमधनद्विगुणत्रिगुणविभागत्रिभागधनानयनसूत्रम् -
व्येकात्महतो गच्छस्स्वेष्टनो द्विगुणितान्यपदहीनः । मुखमात्मोनान्यतिर्दिकेष्टपदघातवर्जिता प्रचयः ॥ ३८ ॥
अत्रोद्देशकः । एकादिगुणविभागस्त्वं व्यस्ताद्यत्तरे हि वद मित्र । द्विव्यशोनैकादशपश्चांशकमिश्रनवपदयोः ॥ ३९ ॥
-
-
IK and B प्रभवो गच्छगप्तश्नम,

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81
कलासवर्णव्यवहारः. गुणधनगुणसङ्कलित धनयोः सूत्रम्पदमितगुणहतिगुणितप्रभवः स्याद्गुणधनं तदाधूनम् ।
एकोनगुणविभक्तं गुणसङ्कलित विजानीयात् ॥ ४० ॥ • गुणसङ्कलितान्त्यधनानयने तत्सकलितानयने च सूत्रम्--
गुणसझलितान्त्यधनं विगतैकपदस्य गुणधनं भवति । तद्गुणगुणं मुरवोनं व्यकोत्तरभाजितं सारम् ॥ ४१ ।।
अत्रोदेशकः । प्रभवोऽष्टमश्चतुर्थः प्रचषः पञ्च पदमत्र गुणगुणितम् । गुणसङ्कलितं नस्यान्यधनं चाचक्ष्व में शीघ्रम् ॥ ४२ ॥ 'गुणधनसकलितधनयोरागृत्तरपदान्यपि पूर्वोक्तसूत्रैरानयेत् । समानेष्टोत्तरगच्छ सङ्कलिनगुणसङ्कलितममधनस्याद्यानयनसूत्रम्-- मुखमेकं चयगच्छाविष्टौ मुखवित्तरहितगुणचित्या । हतचयधनमादिगुणं मुखं भवेदिचितिधनसाम्ये ॥ ४३ ॥
___ अत्रोद्देशकः । भाववाधिभुवनानि पदान्यम्भोधिपश्चमुनयस्त्रिहतास्त । उत्तराणि वदनानि कति स्यु. युग्मसङ्कलितवित्तसमेषु ॥ ४४ ॥ इति भिन्नसङ्कलितं समाप्तम् ॥
भिन्नव्युत्कलितम् । भिन्नव्युत्कलिने करणसूत्रं यथा - गच्छाधिकष्टमिष्टं चयहतमुनोत्तरं बिहादियुतम् । शेषेष्टपदार्धगुणं व्युत्कलितं खेष्टवित्तं च ॥ १५ ॥
Pound only in B,

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32
गणितसारसग्रहः
शेषगच्छस्याद्यानयनसूत्रम् - 'प्रचया?नः प्रभवो युतश्चयनेष्टपदचयार्धाभ्याम् । शेषस्य पदस्यादिश्चयस्तु पूर्वोक्त एव भवेत् ॥ ४६ ॥ गुणगुणितेऽपि चयादी तथैव भेदोऽयमत्र शेषपदे। . इष्टपदमितगुणाहतिगुणितप्रभवो भवेद्वक्रम् ।। ४७ ।।
अत्रोदेशकः । . पादोत्तरं दलास्य पदं त्रिपादांशकस्समुद्दिष्टः ।
खेष्टं चतुर्थभागः किं व्युत्कलितं समाकलय ॥ ५८ । प्रभवोऽधं पञ्चांशः प्रचयो द्विव्यंशको भवेद्गच्छः । पश्चाष्टांशस्वेष्टं पदमणमाचक्ष्व गणितज्ञ ।। ४९ ॥ आदिश्चतुर्थभागः प्रचयः पञ्चांशकास्त्रिपञ्चाशः । गच्छो वाञ्छागच्छो दशमो व्यवकलितमानं किम् ॥ ५० त्रिभागौ हौ वकं पञ्चमांशश्चयस्स्यात् पदं त्रिनः पादः पञ्चमस्वेष्टगच्छः । षडशस्सप्तांशो वा व्ययः को वद त्वम् कलावास प्रज्ञाचन्द्रिकाभास्वदिन्दो ॥ ५१ ॥ द्वादशपदं चतुर्थर्णोत्तरमधौनपञ्चकं वदनम् । त्रिचतुःपञ्चाष्टष्टपदानि व्युत्कालतमाकलय ॥ ५२ ॥
1 प्रचयगुणितेष्टगच्छत्सादिः प्रभवः पदस्य शेषस्य ।
पूर्वोक्तः प्रचयस्स्यादिष्टस्य प्राक्तनादेव ॥ • ' च चतुर्भागः.
'M किं व्युत्कलितं समाकलय.

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38
कलासवर्णव्यवहारः. गुणसङ्कलितव्युत्कलितोदाहरणम् । द्वित्रिभागरहिताष्टमुखं द्वित्र्यंशको गुणचयोऽष्ट पदं भोः। मित्र रत्नगतिपश्चपदानीष्टानि शेषमुखवित्तपदं किम् ॥ ५३ ॥
इति भिन्नव्युत्कालितं समाप्तम् ॥
कलासवर्णषड़जातिः ।।
इतः परं कलासवणे पड़जातिमुदाहरिष्यामः ---
भागप्रभागावथ भागभागो भागानुबन्धः परिकीर्तितोऽतः । भागापवाहस्सह भागमात्रा षड़जातयो मुत्र कलासवणे ॥ ५४ ॥
भागजातिः । तत्र भागजातौ करणसूत्रं यथासदृशहतच्छेदहतो मियोंऽशहारौ समच्छिदावंशी ।
लुप्तैकहरौ योज्यौ त्याज्यौ वा भागजातिविधौ ॥ ५५ ॥ 'प्रकारान्तरेण समानच्छेदमुद्रावयितुमुत्तरसूत्रम -
छेदापवर्तकानां लब्धानां चाहती निरुद्धः स्यात् । हरहतनिरुद्धगुणिते हारांशगुणे समो हारः ॥ ५६ ॥ Kand M add after this इति सारमङ्गडे महावीराचार्गस्य कृती द्वितीयच्या.
रस्सामत:. This, howerer, seems to be a nistake.
* This and the stanze following are not found in M.

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84
गणितसारसङ्ग्रहः
अत्रोद्देशकः । 'जम्बूजम्बीरनारङ्गचोचमोचाम्रदाडिमम् । अझैषीद्दळषड्भागद्वादशांशकविंशकैः ॥ ५७ ॥ । हेम्रस्त्रिंशचतुर्विशेनाष्टमेन यथाक्रमम् । श्रावको जिनपूजाये तद्योगे किं फलं वद ।। ५८ ॥ अष्टपञ्चदशं विशं सप्तषट्त्रिंशदंशकम् । एकादशत्रिषष्टयंशमेकविंशं च सक्षिप ॥ ५९ ॥ 'एकतिकत्रिकायेकोत्तरनवदशकषोडशान्त्यहराः । निजनिजमुखप्रमांशास्त्वपराभ्यस्ताश्च किं फलं तेषाम् ॥ ६ ॥ एकदिकत्रिकाद्याश्रतुराद्याश्चैकवृद्धिका हाराः । निजनिजमुखप्रमांशाः स्वासन्नपराहताः क्रमशः ॥ ६१ ॥ विंशत्यन्ताः षड्गुणसप्तान्ताः पश्चवर्गपश्चिमकाः । षटत्रिंशत्पाश्चात्याः सङ्क्षेपे किं फलं तेषाम् ॥ ६२ ॥ 'चन्दनघनसारागरुकुङ्कममकेष्ट जिनमहाय नरः। चरणदळविंशपभमभागैः कनकस्य किं शेषम् ॥ ६३ ॥ पादं पश्वांशमधे त्रिगुणितदशमं सप्तविंशांशकश्च स्वर्णद्वन्द्वं प्रदाय स्मितसितकमलं स्त्यानदध्याज्यदुग्धम् ।
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

कलासवर्णव्यवहारः.
श्रीखण्डं त्वं गृहीत्वानय जिनसदन प्रार्थनायाब्रवीन्मा
मित्यद्य श्रावकार्यो भणगणक कियच्छेषमंशान्विशोध्य ॥ ६४ ॥
'अष्टपञ्चमुखौ हारावुभयेऽप्येकवृद्धिकाः ।
'त्रिंशदन्ताः पराभ्यस्ताश्चतुर्गुणित पश्चिमाः ॥ ६५ ॥
"
'स्वस्ववक्र प्रमाणांशा रूपात्संशोध्य तद्द्वयम् । शेषं सरखे समाचक्ष्व प्रोत्तीर्णगणितार्णव || ६६ || एकोनविंशतिरथ क्रमात् त्रयोविंशतिद्विषष्टिश्च । रूपविहीना त्रिंशत्ततस्त्रयोविंशनिशतं स्यात् ॥ ६७ ॥ पश्चत्रिंशत्तस्मादष्टाशीतिकशतं विनिर्दिष्टम् । सप्तत्रिंशदमुष्मादष्टानवतित्रिकोनपञ्चाशत् ॥ ६८ ॥
चत्वारिंशच्छतिका सैका च पुनश्शतं सषांडशकम् । एकत्रिंशदतरस्यादद्वानवतिः सप्तपञ्चाशत् ॥ ६९ ॥
त्र्यधिका सप्ततिरस्मात्सप पचाशदपि च सा द्विगुणा । सप्तकृतिः सचतुषा सप्ततिरेकोनविंशनिद्विशतम् ॥ ७० ॥
हारा निरूपिता अंशा एकाद्येकोत्तरा अमून | प्रक्षिप्य फलमाचक्ष्व भागनात्यब्धिपारग || ७१ ॥
अशोत्पत्तौ सूत्रम् -
एकं परिकल्प्यांशं तैरिष्टैस्समहरांशकान् हन्यात् । यद्गुणितांशसमासः फलसढर्शोऽशास्त एवेष्टाः || ७२ ||
This stanza is omitted in M.
This stanza is not found in M.
B प्रीतीर्णगणितार्णव.
• B विंशत्य.
• K and B भागजात्यन्धिपारन.
36

Page #74

गणितसारसङ्गहः 'एकांशद्धीनां राशीनां युतावंशाद्वारस्याधिक्ये सत्यंशोत्पादक सूत्रम्--
समहारकांशकयुतिहतयुत्यंशोऽश एकवडीनाम् । शेषमितरांशयुतिहतमन्यांशोऽस्त्येवमा चरमात् ।। ७३ ॥ ..
अत्रोद्देशकः । नवकदशैकादशहतराशीनां नवतिनवशतीभक्ता ।
यूनाशीत्यष्टशती संयोगः केंऽशकाः कथय ॥ ७४ ॥ छेदोत्पत्तौ सूत्रम्-- रूपांशकराशीनां रूपाद्यास्त्रिगुणिता हराः क्रमशः । द्विदिव्यंशाभ्यस्तावादिमचरमौ फले रूपं ॥ ७५ ॥
अत्रोद्देशकः। पश्चानां राशीनां रूपांशानां युतिर्भवेद्रूपम् । षण्णां सप्तानां वा के हाराः कथय गणितज्ञ ॥ ७६ ॥ विषमस्थानानां छेदोत्पत्ता सूत्रम्--- एकांशकराशीनां व्याद्या रूपोत्तरा भवन्ति हराः । स्वासनपराभ्यस्तास्सर्वे दलिताः फले रूपे ॥ ७७ ॥ एकाशानामनेकांशानां चैकांशे फले छेदोत्पत्तौ सूत्रलब्धहरः प्रथमस्यच्छेदः सस्वांशकोऽयमपरस्य । प्राक् स्वपरेण हतोऽन्त्यः स्वांशेनैकांशक योगे ॥ ७८ ॥
अत्रोद्देशकः। सप्तकनवकत्रितयत्रयोदशांशप्रयुक्तराशीनाम् । । रूपं पादः षष्ठः संयोगाः के हराः कथय ॥ ७९ ॥
1 B सशसंशराशीना अंशोत्पादकसूत्रम् ।

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37
कलासवर्णव्यवहारः. एकांशकानामेकांशेऽनेकांशे च फले छेदोत्पत्ती सूत्रम्सेष्टो हारो भक्तः स्वांशेन निरग्रमादिमांशहरः । तद्युतिहाराप्तेष्टः शेषोऽस्मादित्यमितरेषाम् ॥ ८ ॥
अत्रोद्देशकः । त्रयाणां रूपकांशानां राशीनां के हरा वद । फलं चतुर्थभागस्स्याञ्चतुर्णा च त्रिसप्तमम् ।। ८१ ॥ ऐकांशानामनेकांशानां चानेकांशे फले छेदोत्पत्ता सूत्रम्इष्टहता दृष्टांशाः फलांशसहशा यथा हि तद्योगः । निजगुणहतफलहारस्तद्धारो भवति निर्दिष्टः ।। ८२ !!
अत्रोदेशकः । 'एककांशेन राशीनां त्रयाणां के हरा वद । हादशाप्ता त्रयोविंशत्यशंका च युतिर्भवेत् ।। ८३ ॥ त्रिसप्तकनवांशानां त्रयाणां के हरा वद । य्यूनपश्चाशदाता त्रिसप्तत्यंशा युतिर्भवेत् ॥ ८४ ।। एकांशकयो राश्योरेकांशे फले छेदोत्पत्ती सूत्रम्वाम्छाहतयुतिहारश्छेदः स व्यंकवाञ्छ्याप्तोऽन्यः । फलहारहारलब्ध स्खयोगगुणिते हरौ वा स्तः ॥ ८५ !!
अत्रांद्देशकः । . राश्योरेकांशयोश्छेदी को मवंता. तयोयुतिः । पशो दशमागो वा ब्रूहि त्वं गणितार्थवित् ।। ८६ ॥
Stadzus 88 and 84 are omitted in B.

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38
गणितसारसङ्घः
एकांशकयोरनेकांशयोश्च एकांशेऽनेकांशेऽपि फले छेदोत्पत्तौ प्रथम
सूत्रम् —
'इष्टगुणांशोऽन्यांशप्रयुतः शुद्धं हृतः फलांशेन । इष्टतयुतिहरनो हरः परस्य तु तदिष्टहतिः ॥ ८७ ॥ अत्रोद्देशः । रूपांशको राश्योः कौ स्यातां हारकौ युतिः पादः । शो वा द्विहतस्तप्तकनवकांशयोश्च वद ॥ ८८ ॥
द्वितीयसूत्रम् -
फलहारताडितांशः परांशसहितः फलांशकेन हृतः । स्यादेकस्यच्छेदः फलहर गुणितोऽयमन्यस्य ॥ ८९ ॥ अत्रोद्देशकः ।
राशिद्वयस्य को हाराकांशस्यास्य संयुतिः । द्विसप्तांशो भवेद्ब्रूहि षडष्टांशस्य च प्रिय ॥ ९० ॥ अर्धश्यंशदशांशकपचदशांशकयुतिर्भवेपम् ।
T
त्यक्ते पश्चदशांश रूपांशावत्र कौ योज्यौ ॥ ९१ ॥
दलपादपश्वमांशक विशानां भवति संयुती रूपम् । सप्तैकादशकांशी कौ योज्याविह विना विंशम् ॥ ९२ ॥
युग्मान्याश्रित्यच्छेदोत्पत्तौ सूत्रम् --
युग्मप्रमितान् भागानेकैकांशान् प्रकल्प्य फलराशेः । तेभ्यः फलात्मकेभ्यो द्विराशिविधिना हरास्साध्याः ॥ ९३ ॥
1P and B add as another reading.
शुद्धं फलांशभक्तः स्वान्यशयुतो निजेष्टगुणितांशः ।

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कलासवर्णव्यवहारः.
अत्रोद्देशकः । त्रिकप चकत्रयोदशसप्तनवैकादशांशराशीनाम् ।
के हाराः फलमेकं पञ्चाशो वा चतुर्गुणितः ॥ ९४ ।। एकसूत्रोत्पन्नरूपांशहारैस्सूत्रान्तरोत्पन्नरूपांशहारैश्च फले रूपे छेदोपत्तौ नष्टभागानयने च सूत्रम--
वाञ्छितसूत्रजहारा हरा भवन्त्यन्यसूत्रजहरमाः । दृष्टांशैक्योनं फलमभीष्टनष्टांशमानं स्यात् ।। ९५ ॥
अत्रोद्देशकः । परहतिदलनविधानाश्रयोदश स्वपरसङ्गणविधानात् । . भागाश्चत्वारोऽतः कति भागास्युः फले रूपे ॥ ९६ ॥ प्राकस्वपरहतविधानात्सप्तस्वासनपरगुणार्धविधानात् । भागास्त्रितयश्चातः कति भागास्स्युः फले रूप ।। ९७ ॥ रूपांशका द्विषटकहादशवितिहरा विनष्टोऽत्र । पञ्चमराशी रूपं मर्वसमासस्स राशिः कः ॥ ९८ ।।
- इति भागजातिः। प्रभागभागभागजात्योस्सूत्रम
अंशानां सङ्गणनं हाराणां च प्रभागजातौ स्यात् । गुणकारोंऽशकराशेहरिहरो भागभागजातिविधी ॥ ९९ ॥
प्रभागजातावुद्देशकः । रूपार्ध त्र्यंशार्घ त्र्यंशार्षि दलार्धपांशम् । पाशार्धत्र्यंशं तृतीयभागार्थसप्तशिम् ॥ १० ॥

Page #78

गणितसारसग्रहः 'दलदलदलसप्तांशं त्र्यंशव्यंशकदलार्धदलभागम् । अर्धध्यंशव्यंशकपश्चाशं पश्मशिदलम् ॥ १०१ ॥ क्रीतं पणस्य दवा कोकनदं कुन्दकेतकीकुमुदम् । जिनचरणं प्रार्चयितुं प्रक्षिप्यैतान् फलं ब्रूहि ॥ १०२ ॥" । रूपार्ध त्र्यंशका पादसप्तनवांशकम् । द्वित्रिभागद्विसप्ताशं द्विसप्तांशनवांशकम् ॥ १०३ ॥ दया पणद्वयं कश्चिदानैषीन्नूतनं घृतम् । जिनालयस्य दीपार्थ शेषं किं कथय प्रिय ॥ १०४ ॥ त्र्यंशाद्विपश्चमांशस्तृतीयभागात् त्रयादेशषडशः । पश्चाष्टादशभागात् त्रयोदशांशोऽष्टमानवमः ॥ १०५ ॥ नवमाञ्चतुस्त्रयोदशभागः पञ्चांशकात् त्रिपादार्धम् । सक्षिप्याचक्ष्वैतान प्रभागजातौ श्रमोऽस्ति यदि ॥ १०६ ॥ अत्रैकाव्यक्तानयनसूत्रम्-- रूपं न्यस्याव्यक्ते प्राविधिना यत्फलं भवत्तेन । भक्तं परिदृष्टफलं प्रभागजातौ नदज्ञातम् ॥ १०७ ॥
__ अत्रोद्देशकः । राशेः कुतश्चिदष्टांशस्यशपादोऽर्धपञ्चमः । षष्ठत्रिपादपश्चाशः किमव्यक्तं फलं दलम् ॥ १० ॥ अनेकाव्यक्तानयनसूत्रम्कृत्वाज्ञातानिष्टान् फलसहशी तबुतिर्यया भवति । विभजेत प्रथग्व्यक्तैरविदितराशिप्रमाणानि ॥ १०९ ॥

Page #79

कलासवर्णव्यवहारः.
अत्रोद्देशकः। राशेः कुतश्चिदर्ध कुतश्चिदष्टांशक त्रिपश्चाशः । कस्माद्दियंशाधू फलमधू के स्युरज्ञाताः ॥ ११० ॥
भागभागजातावुद्देशकः । षट्सप्तभागभागस्यष्टांशांशश्चतुर्नवांशांशः । त्रिचतुर्थभागभागः किं फलमेतद्युती ब्रूहि ॥ १११ ॥ दिव्यंशाप्तं रूपं त्रिपादभक्तं द्विकं वयं चापि । द्विव्यंशोदृतमेकं नवकात्संशोध्य वद शेषम् ॥ ११२ ॥
इति प्रभागभागभागजाती । भागानुबन्धजातौ सूत्रम् ---
हरहतरूपेष्वंशान् सक्षिप भागानुबन्धजातिविधौ । 'गुणयानांशच्छेदावंशयुतच्छेदहाराभ्याम् ॥ ११३ ॥
रूपभागानुवन्ध उद्देशकः । "द्वित्रिषट्राष्टनिष्काणि द्वादशाष्टषडंशकैः । पश्चाष्टमैस्समेतानि विंशतेश्शोधय प्रिय ॥ १११ ॥ सार्धेनकेन पके साष्टांशैर्दशमिहिमम । . सार्धाभ्यां कुकर्म द्वाभ्यां क्रीत योगे कियद्भवेत् ॥ ११५ ॥ 'साष्टमाष्टौ षडंशान षडद्वादशांशयुतं द्वयम् । त्रयं पचाष्टमोपेतं विशनशोधय प्रिय ॥ ११६ ॥
B roads गुणयेदप्रांशहरौ महितांशमंद. Thisrtansa is not found in P.
दत्. This stanga i found only in P,

Page #80

गणितसारसङ्ग्रहः.
सप्ताष्टौ नवदशमाषकान् सपादान् दत्त्वा ना जिननिलये चकार पूजाम् । उन्मीलत्कुरवककुन्दजातिमल्लीमालाभिर्गणक वदाशु तान् समस्य ॥ ११७ ॥ . .
__ भागभागानुबन्ध उद्देशकः । स्वयंशपादसंयुक्तं दलं पञ्चांशकोऽपि च । व्यंशस्वकीयषष्ठार्धसहितसातौ कियत् ॥ ११८ ॥ व्यंशाधेशकसप्तमांशचरमैस्वैरन्वितादर्धतः पुष्पाण्यर्धतरीयपश्चनवमस्वीयैर्युतात्सप्तमात् । गन्धं पश्चमभागनोऽर्धचरणव्यंशांशकैमिश्रिताद्धूपं चाचयितुं नरो जिनवरानानेष्ट किं तद्युतौ ॥ ११९ ॥ स्वदलसहितं पादं स्वयंशकेन समन्वित द्विगुणनवमं स्वाष्टांशव्यंशकाविमिश्रितम् । नवममपि च स्वाष्टांशाद्यपश्चिमसंयुतं निजदलयुतं व्यंशं संशोधय त्रितयात्प्रिय ॥ १२० ॥ स्वदलसहितपादं सस्वपादं दशांशं निजदलयतषष्ठं सस्वकव्यंशमर्धम् । चरणमपि समेतस्वत्रिभागं समस्य प्रिय कथय समग्रप्रज्ञ भागानुबन्धे ॥ १२१ ॥
अत्राग्राव्यक्तानयनसूत्रम्लब्धात्कल्पितभागा रूपानीतानुबन्धफलभक्ताः । क्रमशः रवण्डममानास्तेऽज्ञातांशप्रमाणानि ॥ १२२ ॥
'B स्वचरणावर्धान्तिमैः.

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कलासवर्णव्यवहारः.
49
अत्रोद्देशकः । काश्रित्वकैरर्धतृतीयपादैरंशोऽपरः पञ्चचतुर्नवांशैः। अन्पत्रिपञ्चाशनवांशकायुतो युती रूपमिहाशकाः के ॥ १२३ ॥ कोऽप्यंशस्वार्धपञ्चांशत्रिपादनवमैर्यतः । अर्ध प्रजायते शीघ्रं वदाव्यक्तप्रमा प्रिय ॥ १२४ ॥ शेषेष्टस्थानाव्यक्तभागानयनसूत्रम -- लब्धात्कल्पितभागास्सवर्णितैर्व्यक्तराशिभिभक्ताः । 'क्रमशो रूपविहीनास्स्वेष्टपदेप्वविदितांशास्स्युः ॥ १२५ ॥
इति भागानुबन्धजानिः । अथ भागापवाहनातौ सूत्रम् हरहतरूपेप्वंशानपनय भागापवाहजानिविधौ । 'गुणयाग्रांशच्छेदावंशोनच्छेदहागभ्याम् ॥ १२६ ॥
रूपभागापवाह उद्देशकः । म्यष्टचतुर्दशकर्षाः पादाद्विादशांशषष्ठोनाः । सवनाय नरैर्दत्तासीर्थकतां नाती किं स्यात् ॥ १२७ ॥ त्रिगुणपाददलत्रिहताष्टमैविरहिता नव सप्त नव क्रमात् ।
गुणयद प्रांशहरी हितांशदहागभ्याम् ।
6-A

Page #82

गणितसारसङ्ग्रहः. प्रिय विशोध्य चतुर्गुणषटुतः . कथय शेषधनप्रमिति द्रुतम् ॥ १२८ ॥
भागमागापवाह उद्देशकः । द्विगुणितपञ्चमनवमत्र्यंशाष्टांशद्विसप्तमान क्रमशः । खषडंशपादचरणव्यंशाष्टमवर्जितान् समस्य वद ॥ १२९ ॥ षट्सप्तांशस्वषष्ठाष्टमनवमदशांशैर्वियुक्तः पणस्य स्यात्पश्चद्वादशांशस्वकचरणतृतीयांशपश्चांशकोनः । खद्विव्यंशद्विपञ्चांशकदलवियुतः पञ्चषड्भागराशिदिव्यंशोऽन्यस्वपश्चाष्टमपरिरहितस्तत्समासे फलं किम् ॥ १३० । अर्ध व्यष्टमभागपादनवमैस्वीयविहीनं पुनः . खैरष्टांशकसप्तमांशचरणैरूनं तृतीयांशकम् । अध्यर्धात्परिशोध्य सप्तममपि स्वाष्टांशषष्ठोनितं शेषं ब्रूहि परिश्रमोऽस्ति यदि ते भागापवाहे सखे ॥ १३१ ॥
अत्राग्राव्यक्तभागानयनसूत्रम्लब्धात्कल्पितभागा रूपानीतापवाहफलभक्ताः । क्रमशः खण्डसमानास्तेऽज्ञातांशप्रमाणानि ॥ १३२ ॥
अत्रोद्देशकः । कश्चित्स्वकै चरणपञ्चमभागषष्ठैः कोऽप्यंशको दलषडंशकपञ्चमांशैः । हीनोऽपरो हिगुणपश्चमपादषष्ठैः तत्संयुतिर्दलमिहाविदितांशकाः के ॥ १३३ ॥

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कलासवर्णव्यवहारः. कोऽप्यशस्वार्धषड्भागपञ्चमाष्टमसप्तमैः । विहीनो 'जायते षष्ठस्स कोऽशो गणितार्थवित् ॥ १३४ ।। शेषेष्टस्थानाव्यक्तभागानयनसूत्रम्• लब्धात्कल्पितभागास्तवर्णितैफक्तराशिभिर्भक्ताः । रूपात्प्टथगपनीतारखेष्टपदेप्वविदितांशास्स्युः ॥ १३५ ॥
इति भागापवाहजातिः । भागानुबन्धभागापवाहनात्योस्सर्वाव्यक्तभागानयनसूत्रम्-- त्यक्त्वैकं स्वेष्टांशान् प्रकल्पयंदविदितंषु सर्वेषु । ऐतैस्तं पुनशं प्रागुक्तैरानयेत्सूत्रैः ।। १३६ ॥
अत्रांद्देशकः । कश्चिदंशोंशकैः कैश्चित्पश्चभिम्स्वैर्यतो दलम् । वियुक्ती वा भवेत्पादस्तानंशान् कथय प्रिय ॥ १३७ ॥ मागमातृजाती सूत्रम् - भागादिमजातीनां स्वस्वविधिर्भागमातृजाती स्यात् । सा षड्रिंशतिभेदा रूपं छदो-च्छिदा राशः ॥ १३८ ।
अत्रोद्देशकः । यंशः पादोऽर्धा पञ्चमषष्ठस्त्रिपादह मेकम् । पश्चार्धहतं रूपं सषष्ठमेकं सपञ्चमं रूपम ॥ १३९ ॥ खीयतृतीययुग्दलमतो निजषष्ठयुतो द्विसप्तमो हीननवशिमेकमपनीतदशांशकरूपमष्टमः ।
'P, K And B नयुति: for जायते.

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गणितसारसङ्ग्रहः. खेन नवांशकेन रहितश्चरणस्स्वकपश्चमोज्झितो ब्रूहि समस्य तान् प्रिय कलासमकोत्पलमालिकाविधौ ॥ १४० ।
. इति भागमातृजातिः । इति सारसङ्घहे गणितशास्त्रे महावीराचार्यस्य कृतौ कलासवर्णों नाम द्वितीयव्यवहारस्समाप्तः ॥

Page #85

तृतीयः प्रकीर्णकव्यवहारः.
• प्रणुतानन्तगणौघं प्रणिपत्य जिनेश्वरं महावीरम् ।
प्रणतजगत्रयवरदं प्रकीर्णक गणितमभिधास्ये ॥ १ ॥ 'विध्वस्तदुर्नयध्वान्तः सिद्धः स्याहादशासनः । विद्यानन्दो जिनो जीयाद्वादीन्द्रो मुनिपुङ्गवः ॥ २ ॥ इतः परं प्रकीर्णकं तृतीयव्यवहारमृदाहरिप्यामः---
भागश्शेषो मूलकं शामलं स्यातां जानी दे द्विरप्रांशमूले । भागाभ्यासोऽनाऽशवर्गीय मूल
मिश्रं तस्मादिन्नदृश्यं दशामूः ॥ ३ ॥ तत्र भागनानिशेषजात्यो मूत्रम् --- भागोनरूपभक्तं दृश्यं फलमत्र भागजातिविधौ । अंशोनितरूपाहतिहतमग्रं शेषजातिविधो ॥ ४ ॥
___ भागजातावुद्देशकः । दृष्टाऽष्टमं पृथिव्यां मम्भम्य ऽयंशको पया तोये। पादांशः शैवालं कः स्तम्भः मप्त हस्ताः ग्वं ॥ ५ ॥ षड्डागः पाटलीषु भ्रमरवरततस्तत्रिभागः कदम्ब पादश्रूतद्रुमंषु प्रदलितकुसुमे चम्पके पश्चमांशः । प्रोत्फुष्ठाम्भोजषण्डे रविकरदलिते त्रिंशदंशो-मिरेम तत्रैको मत्तभृङ्गो भ्रमति नभसि का तस्य बृन्दस्य सळ्या ॥६॥
Rund Monit this planza

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॥१०॥
गणितसारसङ्ग्रहः. आदायाम्भोरुहाणि स्तुतिशतमुखरः श्रावकस्तीर्थकद्भयः पूजां चक्रे चतुभ्यो दृषभजिनवरात यंशमेषाममुष्य । त्र्यंशं तुर्य षडशं तदनु सुमतये तन्नवद्वादशांशी शेषेभ्यो द्विद्विपद्मं प्रमुदितमनसादत्त किं तत्प्रमाणम् ॥ ७ ॥ स्ववशीकृतेन्द्रियाणां दूरीकृतविषकषायदोषाणाम् । शीलगुणाभरणानां दयाङ्गनालिङ्गिताङ्गानाम् ॥ ८ ॥ साधूनां सद्वन्दं सन्दृष्टं द्वादशोऽस्य तर्कज्ञः । खत्र्यंशवर्जितोऽयं सैद्धान्तश्चान्दसस्तयोश्शेषः ॥ ९ ॥ षडनोऽयं धर्मकथी स एव नैमित्तिकः स्वपादोनः । वादी तयोविशेषः षगुणितोऽयं तपस्वी स्यात् ।। १० ॥
गिरिशिरवरतठे मयोपदृष्टा यतिपतयो नवसङ्गुणाष्ट सङ्ख्याः । रविकरपरितापितोज्ज्वलाङ्गाः
कथय मुनीन्द्रसमूहमाशु मे त्वम् ॥ ११ ॥ फलभारनम्रको शालिक्षेत्रे शुकास्समुपविष्टाः । सहसोत्थिता मनुष्यैः सर्वे सन्त्रासितास्सन्तः ॥ १२ ॥ तेषामधं प्राचीमामेयी प्रति जगाम षड्भागः । पूर्वानेयीशेषः स्वदलोनः स्वार्धवर्जितो यामीम् ॥ १३ ॥ याम्यानेयीशेषः स नैर्ऋतिं स्वद्विपञ्चभागोनः । यामीनैर्मत्यशकपरिशेषो वारुणीमाशाम् ॥ १४ ॥ नैर्मत्यपरविशेषो वायव्यां सस्वकत्रिसप्तशिः । वायव्यपरविशेषो युतस्वसप्ताष्टमः सौमीम् ॥ ११ ॥

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प्रकीर्णकव्यवहारः. वायव्युत्तरयोयुतिरेशानी खत्रिभागयुगहीना । दशगुणिताष्टाविंशतिरवशिष्टा व्योनि कति कीराः ॥ १६ ॥ काचिद्वसन्तमासे प्रसूनफलगुच्छभारनम्रोद्याने । कुसुमासवरसरञ्जितशककोकिलमधुपमधुरानिस्वननिचिते ॥ १७ ॥ हिमकरधवले पृथुले सौधनले सान्द्ररुन्द्र मृदुतल्पे । फणिफणनितम्बबिम्बा कनदमलामरणशोभाङ्गी ॥ १८ ॥ पाठीनजठरनयना कठिनस्तनहारनम्रतनुमध्या। सह निजपतिना युवती रात्रौ प्रीत्यानरममाणा ॥ १९ ॥ प्रणयकलहे समत्थं मुक्तामयकण्ठिका तदबलायाः । छिन्नावनौ निपतिता तयशश्चेटिकां प्रापत् ॥ २० ॥ षड्भागः शय्यायामनन्तगनन्तरार्धमितिभागाः। ' षटमङ्ख्यानास्तस्याः सर्वे सर्वत्र सम्पतिताः ॥ २१ ॥ एकाग्रषष्टिशतयुतसहवमुक्ताफलानि दृष्टानि । तन्मौक्तिकप्रमाणं प्रकीर्णकं वेसि चन् कथय ॥ २२ ॥ 'स्फुरदिन्द्रनीलवर्ण षट्पदबृन्दं प्रफुल्छिनोद्याने । दृष्टं तस्याष्टांशोऽशोके कुटजे षडंशको लीनः ॥ २३ ॥ कुटजाशोकविशेषः षड्गुणिता विपुल पाटलीषण्डं । पाटल्यशोकशेषः वनवांशोनो विशालसालबने ॥ २४ ॥ पाठल्यशोकशेषा युतस्वसप्तांशकेन मधुकवने । पश्चशिस्सन्दृष्टो वकुलेषुत्फुल्लमुकुलेषु ॥ २५ ॥ तिलकेषु कुरवकंषु च सरलेषामेषु पद्मषण्डंषु । वनकरिकपोलमूलपि सन्तस्थे स एवांशः ।। २६ ॥
'M reads स्फु तिंन्द्र'.

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50
गणितसारसङ्ग्रहः.
किञ्जल्क पञ्जरकञ्जव मधुकरास्त्रयस्त्रिंशत् ।
दृष्टा भ्रमरकुलस्य प्रमाणमाचक्ष्व गणक त्वम् ॥ २७ ॥ गोयूथस्य क्षितिभूति दलं तद्दलं शैलमूले पद तस्यांशा विपुलविपिने पूर्व पूर्वार्धमानाः ।
सन्तिष्ठन्ते नगरनिकटं धेनवो दृश्यमाना द्वात्रिंशत् त्वं वद मम सखे गोकुलस्य प्रमाणन् || २८ ॥
इति भागजात्युद्देशकः ॥
शेषजाता वुद्देशकः ।
बड़भागमा म्रराशं राजा शेषस्य पञ्चमं राज्ञी | तुर्यत्र्यंशदलानि त्रयोऽग्रहीषुः कुमारवराः ॥ २९ ॥
शेषाणि त्रीणि चूतानि कनिष्ठां दारको ग्रहीत् । तस्य प्रमाणमाचक्ष्व प्रकीर्णकविशारद ॥ ३० ॥
चरति गिरौ सप्तांशः करिणां षष्ठादिमार्धपाश्चात्याः । प्रतिशेषांशा विपिने षदृष्टास्सरसि काने ते स्युः ॥ ३१ ॥ कोष्ठस्य लेभे नवमांशमेकः परेऽष्टभागादिदलान्तिमांशान । शेषस्य शेषस्य पुनः पुराणा दृष्टा मया द्वादश तत्प्रमा का ॥ ३२ ॥ इति शेषजात्युद्देशकः ॥
अथ मूलजातो सूत्रम—
मूलार्था छिन्द्यादंशांनैकेन युक्तमूलकृतेः ।
दृश्यस्य पदं सपदं वर्गितामह मूलजातौ स्वम् ॥ ३३ ॥

Page #89

प्रकीर्णकव्यवहारः.
अत्रोद्देशकः ।
दृष्टया यूथस्य पादो मूले च द्वे शैलसानौ निविष्टे ।
'उष्ट्रास्त्रिन्नाः पञ्च नद्यास्तु तीरे किं तस्य स्यादुष्ट्रकस्य प्रमाणम् ।। ३४ ।।
श्रुत्वा वर्षाभ्रमाला पटपटुखं शैलशृङ्गारुरङ्ग नाट्यं चक्रे प्रमोदप्रमुदितशिखिनां षोडशांशोऽष्टमश्च ।
त्र्यंशः शेषस्य षष्ठो वखकुलवने पन मलानि तस्थुः पुन्नाग पख दृष्टा भण गणक गणं बर्हिणां सङ्गणय्य ।। ३५ ।।
चरति कमलषण्डे सारसानां चतुर्थो नवमरणभाग सप्त मूलानि चाद्री । विकचवकलमध्ये सप्तनिप्राष्टमानाः
कति कथय स त्वं पक्षिणो दक्ष साक्षात् ॥ ३६ ॥ न भागः कपिवृन्दस्य त्रीणि मूलानि पर्वत । चत्वारिंशद्वने दृष्टा वानरास्तद्रणः कियान ॥ ३७ ॥
५
कलकण्ठानामधं सहकारतरोः प्रफुल्लशाखायाम् । तिलकेऽष्टादश तस्थुनी मुलं कथय पिकनिकरेंम ॥ ३८ ॥
हंसकुलस्य दलं वकुले स्थान पश्च पदानि तमालकुजाग्रे ।
Brends हस्ति. * Breads नागाः
51
* Bds किं स्याने कृगणां प्रमाणम् ।

Page #90

गणितसारसङ्ग्रहः. अत्र न किञ्चिदपि प्रतिदृष्टं तत्प्रमितिं कथय प्रिय शीघ्रम् ।। ३९ ॥
इति मूलजातिः ॥
अथ शेषमूलजाती सूत्रम्पददलवर्गयुताग्रान्मूलं सप्राक्पदार्धमस्य कृतिः । दृश्ये मूलं प्राप्ते फलमिह भागं तु भागजातिविधिः ॥ ४० ॥
अत्रोद्देशकः ॥ गजयूथस्य व्यंश'श्शेषपदं च त्रिसङ्गणं सानौ । सरसि त्रिहस्तिनीभिर्नागो दृष्टः कतीह गजाः ॥ ४१ ॥ निर्जन्तुकप्रदेशे नानाद्रुम षण्डमण्डितोद्याने । आसीनानां यमिनां मूलं तरुमूलयोगयुतम् ॥ ४२ ॥ शेषस्य दशमभागो मूलं नवमोऽथ मूलमष्टांशः । मूलं सप्तममूलं षष्ठो मूलं च पञ्चमो मूलं ॥ ४३ ॥ एते भागाः काव्यप्रवचनधर्मप्रमाणनयविद्याः । वादच्छन्दोज्यौतिषमन्त्रालङ्कारशब्दज्ञाः ॥ ४४ ॥ द्वादशतपःप्रभावा द्वादशभेदाङ्गशास्त्रकुशलधियः । द्वादश मुनया दृष्टाः कियती मनिचन्द्र यतिस मितिः ॥ ४५ ।
मूलानि पञ्च चरणेन युतानि सानी शेषस्य पश्चनवमः करिणां नगाग्रे । मूलानि पञ्च सरसीजवने रमन्ते नद्यास्तठे षडिह ते द्विरदाः कियन्तः ॥ १६ ॥
इति शेषमूलजातिः ॥
| B reads शेषस्य पदं विसंगणं.

Page #91

58
प्रकीर्णकव्यवहारः. अथ द्विरमशेषमूलजातौ सूत्रम्मूलं दृश्यं च भजेदंशकपरिहाणरूपघानेन । पूर्वाग्रमग्रराशौ क्षिपेदतश्शेषमूलविधिः ।। ४७ ॥
अत्रोद्देशकः ।
मधुकर एको दृष्टः वे पद्मे शेषपश्चमचतुर्थी । शेषव्यंशो मूलं द्वावाने ते कियन्तः स्युः ॥ १८ ॥ सिंहाश्चत्वारोऽद्रौ प्रतिशेषषडंशकादिमार्धान्ताः । मूले चत्वारोऽपि च विपिने दृष्टाः कियन्तस्ते ॥ ४९ ॥
तरुणहरिणीयुग्मं दृष्टं द्विसङ्गणित वने कुधरनिकटे शेषाः पनांशकादिदलान्तिमाः। . विपुलकलमक्षेत्रे तासां पदं त्रिभिराहतं कमलसरसीतीरे नस्युर्दशैव गणः कि यान् ॥ ५० ॥
इति द्विरग्रशेषमूलजातिः ॥
अथांशमूलजातौ सूत्रम्भागगुण मूलाग्रे न्यस्य पदप्राप्तदृश्यकरणंन । यल्लब्धं भागहतं धनं भवेदंशमूलविधौ ॥ ५१.॥
अन्यदपि मूत्रम - दृश्यादेशकभक्ताश्चतुर्गुणान्मूलकृतियुतान्मूलम् । सपदं दलितं वर्गितमंशाभ्यस्तं भवेत् सारम् ॥ ५२ ॥
| B reads द्वौ चाम्र.

Page #92

गणितसारसङ्ग्रहः.
अत्रोद्देशकः । पद्मनालत्रिभागस्य जले मूलाष्टकं स्थितम् । षोडशाङ्गलमाकाशे जलनालोदयं वद ॥ ५३ ॥ द्वित्रिभागस्य यन्मूलं नवप्नं हस्तिनां पुनः । शेषत्रिपश्चमांशस्य मूलं पहिस्समाहतम् ॥ ५४ ॥ विगलद्दान'धारार्द्रगण्डमण्डलदन्तिनः । चतुर्विशतिरादृष्टा मयाटव्यां कति द्विपाः ।। ५५ ॥ क्रोडौघार्धचतुःपदानि विपिनं शार्दूलविक्रीडितं प्रापुश्शेषदशांशमूलयुगलं शैलं चतुस्ताडितम् । शषार्धस्य पदं त्रिवर्गगुणितं व वराहा वने दृष्टास्सप्तगुणाष्टकप्रमित यसैषां प्रमाणं वद ।। ५६ ।।
इत्यंशमलजातिः ॥
अथ भागसंवर्गजातौ सूत्रम्स्वांशाप्तहरादनाच्चतुर्गणाग्रेण तहरेण हतात् । मूलं योज्यं त्या तच्छेदे तद्दलं वित्तम् ॥ ५७ ।।
अत्रोद्देशकः । अष्टमं षोडशांशघ्नं शालिराशेः कृषीवलः । चतुर्विशतिवाहांश्च लेभे राशिः कियान् वद ॥ ५८ ॥
1 B rends वारा.
After this stanzn all the MSS. have the following stanza; but it is simply aparaphrase of stanza No. 57:
अन्यच्च
चतुर्हतदृष्टेनोनाद्भागाहन्यंशहतहारात्।। तडेदन हतान्मूलं योज्य न्याज्यं तच्छेदे तदर्थं वित्तम् ॥

Page #93

प्रकीर्णकव्यवहारः.
शिखिनां षोडशभागः स्वगुणभूते तमालषण्डेऽस्थात् । शेषनवांशः स्वहतश्चतुरग्रदशापि कति ते स्युः ॥ ५९ ॥ जले त्रिंशदशाहतो द्वादशांशः स्थितश्शेषविशेो हतः षोडशेन ।
त्रिनिन प करा विंशतिः खे सरखे स्तम्भदैर्घ्यस्य मानं वद त्वम् ॥ ६० ॥ इति भागसंवर्ग जातिः ||
अथोनाधिकांशवर्गजानौ सूत्रम्-स्वांशकभक्तहरार्धं न्यूनयुगंधिकोनितं च तद्वर्गात् । व्यूनाधिकवर्गाग्रान्मूलं स्वर्ण फलं परें शहतम् ।। ६१ ।।
'हीनालाप उदाहरणम् ।
महिषीणामष्टांशी व्येक वर्गीकृत वनं रमते । पश्चदशाद्रौ दृष्टास्तृणं चरन्त्यः कियन्त्यस्ताः ।। ६२ ।। अनेकपानां दशमी द्विवर्जितः
स्वङ्गणः क्रीडति सल्लकीवने ।
ु
चरन्ति षडुर्गमिता गजा गिंगै
कियन्त एते भवन्ति दन्तिनः ।। ६३ ।।
'अधिकालाप उदाहरणम् ।
जम्बूवृक्षे पदशांश कियुक्तः स्वेनाभ्यस्तः केकिकलस्य द्विकृतिघ्राः ।
Momite हीन,
55
[+ Momats this as well as the following stanz

Page #94

56
繁
गणितसारसङ्ग्रहः.
पश्चाप्यन्ये मत्तमयूरास्सहकारे रंरम्यन्ते मित्र वदेषां परिमाणम् ॥ ६४ ॥
इत्यूनाधिकांशवर्गजातिः ॥
अथ मूलमिश्रजातौ सूत्रम् -
मिश्रकृतिरूनयुक्ता व्यधिका च द्विगुणमिश्रसम्मक्ता । वर्गीकृता फलं स्यात्करणमिदं मूलमिश्रविधौ ॥ ६५ ॥ हीनाला उद्देशकः ।
मूलं कपोतवृन्दर द्वादशोनस्य चापि यत् । तयोर्योगं कपोता षड् दृष्टास्तन्निकरः कियान् ॥ ६६ ॥ पारावतसङ्घ चतुर्धनोऽपतत्र यन्मूलम् ।
योगः षोडश तद्वृन्दे कति विहङ्गाः स्युः ।। ६७ ।।
अधिकालाप उद्देशकः ।
राजहंसनिकरस्य यत्पदं साष्टषष्टिसहितस्य चैतयोः ।
संयुकविहीनषट्कृति
कति मरालका वद || ६८ ||
इति मूलमिश्रजातिः ॥
अथ भिन्नदृश्यजातौ सूत्रम्-
दृश्यांशने रूपे भागाभ्यासेन भाजिते तत्र ।
लब्धं तत्सारं प्रजायते भिन्नदृश्यविधौ ॥ ६९ ॥
1 B reacts योग:.

Page #95

प्रकीर्णक व्यवहारः
अत्रोद्देशकः ।
सिकतायामष्टशस्तन्दृष्टोऽष्टादशांशसङ्गणितः ।
1
स्तम्भस्यार्ध ' दृष्टं स्तम्भायामः कियान् कथय ॥ ७० ॥
द्विभक्तनवमांशकप्रहतसप्तविंशांशकः प्रमोदमवतिष्ठते करिकुलस्य पृथ्वीतले । विनीलजलदाकृतिर्विहरति त्रिभागो नंगे वद त्वमधुना सरवे करिकुलप्रमाणं मम ।। ७१ ।। साधूत्कृतोर्निवसति षोडशांशक
विभाजितः स्वकगुणितो वनान्तरे ।
पादो गिरौ मम कथयाशु तन्मित प्रोत्तीर्णवान् जलधिसमं प्रकीर्णकम् ।। ०२ ।।
57
इति भिन्नदृश्यजातिः ॥
इति सारसङ्घहे गणितशास्त्रे महावीराचार्यस्य कृते प्रकीर्णको नाम
तृतीयव्यवहारः समाप्तः ॥
B, M and K read गगने.

Page #96

त्रिलोकबन्धवे तस्मै केवलज्ञानभानवे । नमः श्रीवर्धमानाय निर्धूताखिलकर्मणे ॥ १ ॥
चतुर्थः त्रैराशिक व्यवहारः ।
इतः परं त्रैराशिकं चतुर्थव्यवहारमुदाहरिष्यामः ।
तत्र करणसूत्रं यथा -
त्रैराशिकेऽत्र सारं फलमिच्छासङ्गणं प्रमाणाप्तम् ।
ु
इच्छाप्रमयोस्साम्ये विपरीतेयं क्रिया व्यस्ते ॥ २ ॥
पूर्वार्धोद्देशकः ।
दिवसे स्त्रिभिस्तपादैर्योजन षटुं चतुर्थभागोनम । गच्छति यः पुरुषोऽसौ दिनयुतवर्षेण किं कथय ।। ३ ।।
G
व्यर्धाष्टाभिरहोभिः क्रोशाष्टांशं स्वपञ्चमं याति । पङ्गसपश्च मागैर्वर्षैस्त्रिभिरत्र किं ब्रूहि ॥ ४ ॥ अङ्गलचतुर्थभागं प्रयाति कीटो दिनाष्टभागेन । मेरोर्मूलाच्छिखरं कतिभिरहोभिस्तमाप्नोति ॥ ५ ॥ कार्षापणं सपाद निर्विशति त्रिभिरहोभिरर्धयुतैः । यो ना पुराणशतकं सपणं कालेन केनासौ ॥ ६ ॥ "कृष्णागरु सत्खण्डं द्वादशहस्तायतं त्रिविस्तारम् । क्षयमेत्यङ्गलमह्नः क्षयकालः कोऽस्य वृत्तस्य || ७ || स्वर्णैर्दशभिस्सार्धेर्द्रोणाढककुडब मिश्रितः क्रीतः । वरराजमाषवाहः किं हेमशतेन सार्धेन ॥ ८ ॥
JP, Kand M road स for स्व.
G
*B reade
सत्कृष्णागरु खण्डं.

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त्रैराशिकव्यवहारः सास्त्रिभिः पुराणैः कुङ्कुमपलमष्टभागसंयुक्तम् । संप्राप्यं यत्र स्थात् पुराणशतकेन किं तत्र ॥ ९ ॥ सार्धाकसप्तपलैश्चतुर्दशाोनिताः पणा ' लब्धाः । द्वात्रिंशदाकपलैस्सपञ्चमैः किं सरवे ब्रूहि ॥ १० ॥ कार्षापणैश्चतुभिः पञ्चांशयुनः पलानि रजतस्य । षोडश सार्धानि नरो लभते किं कर्षनियुतेन ॥ ११ ॥ कर्पूरस्याप्टपलैख्यंशोनैर्नात्र पभ दीनारान् । भागांशकलायुक्तान् लभते किं पलसहस्रेण ।। १२ ।। साथूस्त्रिभिः पणैरिह घृतस्य पलपञ्चकं सपनांशम् । क्रीणाति यो नरोऽयं किं साष्टमकर्षशतकेन ॥ १३ ॥ . सार्धेः पञ्चपुराणैः षोडश सदलानि वस्त्रयुगलानि । लब्धानि सैकषष्टया कर्षाणां किं सरवे कथय ।। १४ ।। वापी समचतुरश्रा सलिलवियुक्ताप्टहस्तघनमाना । शैलस्तस्यास्तीरे समुत्थिताश्शखरतस्तस्य ।। १५ ।। वृत्ताङ्गलविषम्भा जलधारा स्फटिकनिर्मला पतिता । वाप्यन्तरजलपूर्णा नगोच्छृितिः का च जलसङ्ख्या ॥ १६ ॥ 'मुन्द्रोणयुगं नवाज्यकुडवान् षट् तण्डुलद्रोणकानष्टौ वस्त्रयुगानि वत्ससहिता गाप्पट सुवर्णत्रयम् ।
'M and B read लभ्याः .
Breads समुत्थिता शि'. • Band K read the following for this stanza :
दुरवद्रोणयुगं नवाज्यकुडवान् षट् शर्कंगद्रोणकानष्टी चोचफलानि मान्द्रदधिखार्यषट् पुराणत्रयम् । श्रीखण्ड ददता नृपेण सवनाथै पजिनागारक पत्रिंशतिशतेषु मित्र वर मे तत्तदुग्धादिकम् ॥ 7-A

Page #98

60
गणितसारसनाहः सस्क्रान्तौ ददता नराधिपतिना षड्भ्यो द्विजेभ्यस्सखे षट्त्रिंशत्रिशतेभ्य आशु वद किं तद्दत्तमुद्गादिकम् ॥ १७ ॥
इति त्रैराशिकः ॥ व्यस्तत्रैराशिके तुरीयपादस्योद्देशकः ।। कल्याणकनकनवतेः कियन्ति नववर्णकानि कनकानि । साष्टांशकदशवर्णकसगुञ्जहेम्नां शतस्यापि ॥ १८ ॥
व्यासेन दैर्येण च षट्कराणां चीनाम्बराणां त्रिशतानि तानि । त्रिपञ्चहस्तानि कियन्ति सन्ति व्यस्तानुपातक्रमाविद्वद त्वम् ॥ १९ ॥
इति व्यस्तत्रैराशिकः ॥
व्यस्तपश्चराशिक उद्देशकः । पश्चनवहस्तविस्तृतदायां चीनवस्त्रसप्तत्याम् । द्वित्रिकरव्यासायति तच्छुतवस्त्राणि कति कथय ॥ २० ॥
व्यस्तसप्तराशिक उद्देशकः । व्यासायामोदयतो बहुमाणिक्ये चतुर्नवाष्टकरे। द्विषडेकहस्तमितयः प्रतिमाः कति कथय तीर्थकृताम् ॥ २१ ॥
व्यस्तनवराशिक उद्देशकः । विस्तारदेयोदयतः करस्य षट्त्रिंशदष्टप्रमिता नवार्ण । शिला तया तु द्विषडेकमानास्ताः पञ्चकार्याः कति चैत्ययोग्याः ॥ २१ ॥
इति व्यस्तपञ्चसप्तनवराशिकाः ॥

Page #99

त्रैराशिकव्यवहारः गतिनिवृत्तौ सूत्रम्-- निजनिजकालोद्धृत योर्गमननिवृत्त्योर्विशेषणाजाताम् । दिनशुद्धगतिं न्यस्य त्रैराशिकविधिमतः कुर्यात् ॥ २३ ॥
अत्रोद्देशकः । क्रोशस्य पञ्चभागं नौर्याति दिनत्रिसप्तभागेन । 'वार्धी वाताविद्या प्रत्येति क्रोशनवमांशम् ॥ २४ ॥ कालेन केन गच्छेत् त्रिपञ्चभागोनयोजनशतं सा ।
सङ्ख्याब्धिसमुत्तरणे बाहुबलिंस्त्वं समाचक्ष्व ।। २५ ।। सपादहेम त्रिदिनैस्सपञ्चमैनरोऽर्जयन् व्येति सुवर्णतुर्यकम् । . निनाष्टमं पञ्चदिनैदलोनितैः स केन कालेन लभेत सप्ततिम् ॥ २६ ॥
गन्धेभो मदलुब्धषट्पदपदप्रोद्भिन्नगण्डस्थलः । सार्ध योजनपञ्चमं व्रजति यष्षभिर्दलोनैदिनैः । प्रत्यायाति दिनैस्त्रिभिश्च सदलैः क्रोशद्विपञ्चांशकं
ब्रूहि क्रोशदलोनयोजनशतं कालेन केनामुयात् ॥ २७ ।। वापी पयःप्रपूर्णा दशदण्डसमुच्छ्रिताब्जमिह जातम् । अगलयुगलं सदलं प्रवर्धते सार्धदिवसेन ॥ २८ ॥ निस्सरति यन्त्रतोऽम्भः सार्धेनाहारले सविंशे हे। शुप्यति दिनेन सलिलं सपञ्चमाङ्गुलकमिनकिग्णैः ॥ २९ ॥ क्र्मों नालमधस्तात् सपादपञ्चाङ्गलानि चाकृषति । साखिदिनैः पद्मं तोयसमं केन कालेन ॥ ३० ॥
IBand K read तस्मिन्काले वाधी.

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गणितसारसग्रहः द्वात्रिंशद्धस्तदीर्घः प्रविशति विवरे पश्चभिस्सप्तमाधैः कृष्णाहन्द्रिो दिनस्यासुरवपुरजितः सार्धसप्ताङ्गुलानि । पादेनाहोऽङ्गले द्वे त्रिचरणसहिते वर्धते तस्य पुच्छं रन्ध्र कालेन केन प्रविशति गणकोत्तंस मे ब्रूहि सोऽयम् ॥ ३१ ॥
इति गतिनिवृत्तिः ॥ पञ्चसप्तनवराशिकेषु करणसूत्रम्'लाभं नीत्वान्योन्यं विभजेत् पृथुपतिमल्पया पलया। गुणयित्वा जीवानां क्रयविक्रययोस्तु तानेव ॥ ३२ ॥
___अत्रोद्देशकः । द्वित्रिचतुश्शनयोगे पञ्चाशत्षष्टिसप्ततिपुराणाः । लाभार्थिना प्रयुक्ता दशमासेष्वस्य का वृद्धिः ।। ३३ ।। हेम्रां सार्धाशीतेसिव्यंशेन वृद्धिरध्यर्धा । सत्रिचतुर्थनवत्याः कियती पादोनषण्मासैः ॥ ३४ ।। षोडशवर्णककाञ्चनशतेन यो रलविंशतिं लभते । दशवर्णस्वर्णानामष्टाशीतिद्विशत्या किम् ॥ ३५ ॥
preads as variations the following: प्रकारान्तरेण सत्रम्
सड़क्रम्य फलं छिन्द्यालघपङक्त्यानेकराशिका पडाक्तम। स्वगुणामश्वादीनां क्रयविक्रययोस्तु तानेव ॥ अन्यदपि सूत्रम्सङक्रम्य फलं छिन्द्यात् पृथुपयत्यभ्यासमल्पया पङ्कत्या।
अश्वादीनां क्रयविक्रययोरश्वादिकांश्च सङ्क्रम्य ॥ B gives only the latter of these stanzas with the following variation in the second uarter:
पथपत्यभ्यासमल्पपङ्क्त्याहत्या.

Page #101

त्रैराशिकव्यवहारः गोधूमानां मानीव नयता योजनत्रयं लब्धाः । षष्टिः पणाः सघाहं कुम्भं दशयोजनानि कति ॥ ३६ ॥
भाण्डप्रतिभाण्डस्योद्देशकः । कस्तूरीकर्षत्रयमुपलभते दशभिरष्टभिः कनकैः । कर्षद्वयकर्पूरं मृगनाभित्रिशतकर्षकैः कति ना ॥ ३७ ॥ पनसानि षष्टिमष्टभिरुपलभतेऽशीतिमातुलुङ्गानि । दशभिर्माषैर्नवशतपनमैः कति मातुलुङ्गानि ॥ ३८ ॥
जीवक्रयविक्रययोरुद्देशकः । षोडशवर्षास्तुरगा विंशतिरर्हन्ति नियुतकनकानि । ' दशवर्षसप्तिसप्ततिरिह कति गणकाग्रणीः कथय ॥ ३९ ॥ स्वर्णत्रिशती मूल्यं दशवर्षाणां नवागनानां स्यात् । षट्त्रिंशन्नारीणां षोडशसंवत्सराणां किम् ॥ ४० ॥ षट्कशतयुक्तनवतेर्दशमासैवृद्धिरत्र का तस्याः । - कः कालः किं वित्तं विदिताभ्यां भण गणकमुखमुकुर ।। ४१।।
सप्तराशिक उद्देशकः । त्रिचतुळसायामौ श्रीखण्डावहतोऽष्टहेमानि । षण्णवविस्तृतिदैर्ध्या हस्तेन चतुर्दशात्र कति ॥ ४२ ॥
इति सप्तराशिकः ||
Badde 71 at the end
K, M and B read हमको :
ना.

Page #102

गणितसारसङ्ग्रहः
नवराशिक उद्देशकः । 'पञ्चाष्टत्रिव्यासदैर्योदयाम्भो धत्ते वापी शालिनी वाहषट्कम् । सप्तव्यासा हस्ततः षष्टिदैर्ध्याः
पासधोः किं नवाचक्ष्व विद्वन् ॥ ४३ ॥ इति सारसङ्घहे गणितशास्त्रे महावीराचार्यस्य कृतौ त्रैराशिको नाम चतुर्थव्यवहारः.
The following stanza is found in Kanit B in addition to stanza No. 48.
द्वथष्टाशीतिव्यासदैयोनताम्भो पत्ते वापी शालिनी सार्धवाहौ। हस्तादष्टायामकाः षोडशांच्छाः पटकव्यासा: कि चतना वर लम् ।

Page #103

पचमः
मिश्रकव्यवहारः.
प्राप्तानन्तचतुष्टयान् भगवतस्तीर्थस्य कर्तृन् जिनान् सिद्धान् शुद्धगुणांस्त्रिलोकमहितानाचार्यवर्यानपि । सिद्धान्तार्णवपारगान् भवभृतां नेतनुपाध्यायकान् साघून् सर्वगुणाकरान् हितकरान् वन्दामहे श्रेयसे ॥ १॥
इतः परं मिश्रगणितं नाम पञ्चमव्यवहारमदाहरिष्यामः । तद्यथासस्क्रमणसंज्ञाया विषमसङ्क्रमणसंज्ञायाश्च सुत्रम्
युतिवितिदलनकरणं सक्रमणं छेदलब्धयो राश्योः । सक्रमणं विषममिदं प्राहुगणितार्णवान्तगताः ॥ २ ॥
अत्रोद्देशकः । द्वादशसङ्ख्याराशेाभ्यां सक्रमणमत्र किं भवति । तस्माद्राशेर्मक्तं विषमं वा किं तु सहक्रमणम् ॥ ३ ॥
पश्चराशिकविधिः ॥ पश्चराशिकवरूपवृद्ध्यानयनसूत्रम्इच्छाराशिः स्वस्य हि कालेन गुणः प्रपाणफलगुणितः । कालप्रमाणभक्तो भवति तदिच्छाफलं गणिते ॥ ४ ॥
अत्रोद्देशक । त्रिकपश्चकपदृशते पचाशत्वष्टिसप्ततिपुराणाः । लाभार्थतः प्रयुक्ताः का दिर्मासषट्स्य ॥ ५ ॥

Page #104

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गणितसारसङ्ग्रहः व्यर्धाष्टकशतयुक्तास्त्रिंशत्कार्षापणाः पणाश्चाष्टौ ॥ . मासाष्टकेन जाता दलहीनेनैव का वृद्धिः ॥ ६ ॥ षष्ट्या वाद्विदृष्टा पञ्च पुराणाः पणत्रयविमिश्राः । मासद्वयेन लब्धा शतवृद्धिः का तु वर्षस्य ॥ ७ ॥ सार्धशतकप्रयोगे सार्धकमासेन पञ्चदश लामः । मासदशकेन लब्धा शतत्रयस्यात्र का वृद्धिः ॥ ८ ॥ साष्टशतकाष्टयोगे त्रिषष्टिकार्षापणा विशा दत्ताः । सप्तानां मासानां पञ्चमभागान्वितानां किम् ॥ ९ ॥
मूलानयनसूत्रम्--
मूलं खकालगुणितं स्वफलेन विभाजितं तदिच्छायाः। • कालेन भजेल्लब्धं फलेन गुणितं तदिच्छा स्यात् ॥ १० ॥
अत्रोद्देशकः । पश्चार्धकशतयोगे पञ्च पुराणान्दलोनमासौ द्वौ । वृद्धिं लभते कश्चित् किं मूलं तस्य मे कथय ॥ ११ ॥ सप्तत्याः सार्धमासेन फलं पश्चार्धमेव च । व्यर्धाष्टमासे मूलं किं फलयोस्सार्धयोईयोः ॥ १२ ॥ त्रिकपश्वकषटूशते यथा नवाष्टादशाथ पञ्चकृतिः। पश्चांशकेन मिश्रा षट्सु हि मासेषु कानि मूलानि ॥ १३ ॥
कालानयनसूत्रम्-- कालगुणितप्रमाणं स्वफलेच्छाभ्यां हृतं ततः कृत्वा । तदिहेच्छाफलगुणितं लब्धं कालं बुधाः प्राहुः ॥ १५ ॥

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मिश्रकव्यवहारः
67
अत्रोद्देशकः । सप्तार्धशतकयोगे वृद्धिस्त्वष्टायविंशतिरशीत्या । कालेन केन लब्धा कालं विगणय्य कथय सरवे ॥ १५ ॥ विंशतिषट्शतकस्य प्रयोगतः सप्तगणषष्टिः । वृद्धिरपि चतुरशीतिः कथय सरवे कालमाशु त्वम् ॥ १६ ॥ षटूशतेन हि युक्ताः षण्णवतिर्दृद्धिरत्र सन्दृष्टा । सप्तोत्तरपश्चाशत् त्रिपञ्चभागश्च कः कालः ॥ १७ ॥ भाण्डप्रतिभाण्डसूत्रम् - भाण्डस्वमूल्यभक्तं प्रतिभाण्डं भाण्डमूल्यसङ्गणितम् । स्वेच्छामाण्डाभ्यस्तं माण्डप्रतिभाण्डमूल्यफलमेतत् ॥ १८ ॥
__ अत्रोद्देशकः । कीतान्यष्टौ शुण्ठ्याः पलानि षभिः पणैः सपादांशैः । पिप्पल्याः पलपञ्चकमथ पादोनैः पणैवभिः ॥ १९ ॥ शुण्ठ्याः पलैश्च केनचिदशीतिभिः कति पलानि पिप्पल्याः । क्रीतानि विचिन्त्य त्वं गणिनविदाचक्ष्व मे शीघ्रम् ॥ २० ॥ इति मिश्रकव्यवहारे पश्चराशिकविधिः समाप्तः ॥
दृद्धिविधानम् ॥ इतः परं मिश्रकव्यवहारे डिविधानं व्याख्यास्यामः । मूलद्धिमिश्रविभागानयनसूत्रम्-- रूपेण कालवृद्ध्या युतेन मिश्रस्य मागहारविधिम् । कृत्वा लब्धं मूल्यं वृद्धिभ्लोनमिश्रधनम् ॥ २१ ॥
Both Mand Bhave the rroneoun reading कथित स्वशातिभिः सर पलानि पिप्पत्या:.

Page #106

गणितसारसहिः
अत्रोद्देशकः। पश्चकशतप्रयोगे द्वादशमासैर्धनं प्रयुक्त चेत् । साष्टा चत्वारिंशन्मिश्रं तन्मूलवृद्धी के ॥ २२ ॥ पुनरपि मूलादिमिश्राविभागसूत्रम्इच्छाकालफलनं स्वकालमूलेन भाजितं सैकम् । सम्मिश्रस्य विमक्तं लब्धं मूलं विजानीयात् ॥ २१ ॥
अत्रोद्देशकः । सार्धद्विशतकयोगे मासचतुषण किमपि धनमेकः । दत्वा मिश्रं लभते कि मूल्यं स्यात् त्रयस्त्रिंशत् ॥ २४ ॥ कालद्धिमिश्रविभागानयनसूत्रम्मूलं स्वकालगुणितं स्वफलेच्छाभ्यां हृतं ततः कृत्वा । सकं तेनाप्तस्य च मिश्रस्य फलं हि वृद्धिः स्यात् ॥१५॥
अत्रोद्देशकः । पञ्चकशतप्रयोगे फलार्थिना योजितैव धनषष्टिः । कालः खवृद्धिसहितो विंशतिरत्रापि कः कालः ॥ २६ ॥ अर्धत्रिकसप्तत्याः सार्धाया योगयोजितं मूलम् । पश्चोत्तरसप्तशतं मिश्रमशीतिः स्वकालवृद्ध्योहि ॥ २७ ॥ व्यर्धचतुषाशीत्या युक्ता मासद्वयेन साधैन ।
मूलं चतुश्शतं षटत्रिंशन्मिश्रं हि कालवृद्ध्योर्हि ॥ २८ ॥ मूलकालमिअविभागानयनसूत्रम्
वफलोद्धृतप्रमाणं कालचतुर्वृद्धिताडितं शोभ्यम् । मिश्रकृतस्तन्मूलं मिश्रे क्रियते तु सङ्क्रमणम् ॥२९॥

Page #107

मिभकन्यवहारः
89
मत्रोद्देशकः ।
सप्तत्या वृद्धिरियं चतुःपुराणाः फलं च पञ्चकतिः । मिश्र नव पञ्चगुणाः पादेन युतास्तु किं मूलम् ॥ ३० ॥ त्रिकषश्या दत्वैकः किं मूलं केन कालेन । प्राप्तोऽष्टादशवृद्धि षट्षष्टिः कालमूलमिश्रं हि ॥ ३१ ॥ अध्यर्धमासिकफलं षष्ट्याः पश्चार्धमेव सन्दृष्टम् । वृद्धिस्तु चतुर्विंशतिरथ षष्टिर्मूलयुक्तकालश्च ॥ ३२ ॥ प्रमाणफलेच्छाकालमिश्रविभागानयनसूत्रम्
मूलं स्वकालवाद्धिहिकतिगुणं छिन्नमितरमूलेन । मिश्रकृतिशेषमूलं मिश्रे क्रियते तु सङ्क्रमणम् ॥ १५ ॥
अत्रोद्देशकः । मध्यर्धमासकस्य च शतस्य फलकालयोन मिश्रधनम् । द्वादश दलसंमिश्रं मूलं त्रिंशत्फलं पश्च ।। ३४ ॥ . मूलकालवृद्धिमिश्रविभागानयनसूत्रममिश्रादुनितराशिः कालस्तस्यैव रूपलाभेन । सैकेन भजेन्मूलं स्वकालमूलानितं फलं मिश्रम ॥ ३५ ॥
मत्रोदेशकः । पचकशतप्रयोगे न ज्ञातः कालमूलफलराशिः। तन्मिश्र 'हाशीतिर्मूलं किं कालवृद्धी के ॥ ३१ ॥
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

गणितसारसहः बहुमूलकालादिमिश्रावमागानयनसूत्रम्विमजेत्स्वकालताडितमूलसमासेन फलसमासहतम् । कालाभ्यस्तं मूलं पृथक् पृथक् चादिशेदृद्धिम् ।। ३७ ॥
अत्रोद्देशकः । चत्वारिंशत्रिंशविंशतिपञ्चाशदत्र मूलानि । मासाः पश्चचतुस्त्रिकषट् फलपिण्डश्चतुस्त्रिंशत् ॥ ३८ ॥ बहुमूलमिश्रविभागानयनसूत्रम्स्वफलैस्स्वकालभक्तस्तद्युत्या मूलमिश्रधनराशिम् । 'छिन्द्यादशं गुणयेत् समागमो भवति मूलानाम् ।। ३९ ॥
अत्रोद्देशकः। दशषत्रिपञ्चदशका रद्धय इषवश्चतुस्त्रिषण्मासाः । मूलसमासो दृष्टश्चत्वारिंशच्छतेन संमिश्रा ॥ ४० ॥ पश्चार्धषड्दशापि च सार्धाः षोडश फलानि च त्रिंशत् । मासास्तु पश्च षट् खलु सप्ताष्ट दशाप्यशीतिरथ पिण्डः ॥ ४१ बहुकालामश्राविभागानयनसूत्रम्--- स्वफलैः स्वमूलभक्तैस्तद्युत्या कालमिश्रधनराशिम् । 'छिन्द्यादशं गुणयेत् समागमो भवति कालानाम् ॥ ४२ ॥
अत्रोद्देशकः । चत्वारिंशत्रिंशदिशतिपञ्चाशदत्र मूलानि । ' दशट्विपञ्चदश फलमष्टादश कालमिश्रधनराशिः ॥ ४३ ॥ प्रमाणराशी फलेन तुल्यमिच्छाराशिमूलं च तदिच्छाराशौ वृद्धि
The M99. road fpoort
wbiob does not seem to be corroot.

Page #109

मिभकव्यवहारः च संपीज्य तन्मिश्रराशौ प्रमाणराशेः वृद्धिविमागानयनसूत्रम् - कालगुणितप्रमाणं परकालहृतं तदेकगुणमिश्रधनात् । इतरार्धकृतियुतात् पदमितरार्धानं प्रमाणफलम् ॥ ४ ४ ॥
अत्रोद्देशकः । मासचतुष्कशतस्य प्रनष्टद्धिः प्रयोगमूलं तत् । स्वफलेन युतं द्वादश पञ्चकृतिस्तस्य कालोऽपि ।। ४५ ॥ मासत्रितयाशीत्याः प्रनष्टद्धिः स्वमूलफलराशेः । पश्चमभागेनोनाथाष्टी वर्षेण मूलब द्धी के ॥ ४६ ॥ समानमूलदृद्धिमिश्रविभागसूत्रम्अन्योन्यकालविनिहतमिश्रविशेषस्य तस्य भागाख्यम् । कालविशेषेण हृते तेषां मूलं विजानीयात् ॥ ४७ ॥
अत्रोद्देशकः । पश्चाशदष्टपश्चाशन्मिश्रं षट्षष्टिरेव च । पश्च सप्तेव नव हि मासाः किं फलमानय ॥ १८ ॥ त्रिशचकत्रिंशहित्र्यंशाः स्युः पुनस्त्रयस्त्रिंशत् । सत्र्यंशा मिश्रधनं पश्चत्रिंशश्च गणकादात् ॥ १९ ॥ कश्चिन्नरश्चतुर्णा त्रिभिश्चतुर्मिश्च पथभिः षभिः । मासैर्लब्धं किंस्यान्मूलं शीघ्रं ममाचक्ष्व ॥ ५० ॥ समानमूलकालमिश्रविभागसूत्रम्अन्योन्यद्धिसङ्गणमिश्रविशेषस्य तस्य भागाख्यम् । इदिविशेषेण इते लब्धं मूलं बुधाः प्राहुः ॥ ५१ ।।

Page #110

गणितसारसग्रहः
मत्रोद्देशकः । एकत्रिपञ्चमिश्रितविंशतिरिह कालमूलयोर्मिश्रम् । षड् दश चतुर्दश स्युर्लाभाः किं मूलमत्र साम्यं स्यात् ॥ ५२ ॥ पश्चत्रिंशन्मिश्रं सप्तत्रिंशच नवयुतत्रिंशत् । विंशतिरष्टाविंशतिरथ षट्त्रिंशच्च वृद्धिधनम् ॥ ५३ ॥ उभयप्रयोगमूलानयनसूत्रम् -- रूपस्येच्छाकालादुभयफले ये तयोविशेषेण । लब्धं विभजेन्मूलं स्वपूर्वसङ्कल्पितं भवति ॥ ५४ ॥
__ अत्रोद्देशकः । उदृत्त्या षटूशते प्रयोजितोऽसौ पुनश्च नवकशते । मासैस्त्रिभिश्च लभते सैकाशीति क्रमेण मूलं किम् ॥ ५५ ॥ त्रिवद्धचैव शते मासे प्रयुक्त श्वाष्टभिश्शते । लाभोऽशीतिः कियन्मूलं भवेत्तन्मासयोईयोः ॥ ५ ॥ इद्धिमूलविमोचनकालानयनसूत्रम्मूलं स्वकालगणितं फलगुणितं तत्प्रमाणकालाभ्याम् । भक्तं स्कन्धस्य फलं मूलं कालं फलात्प्राग्वत् ॥ ५७ ॥
भत्रोद्देशकः । मासे हि पश्चैव च सप्ततीनां मासदयेऽष्टादशकं प्रदेयम् ।
This carno rule is Binewhat delectively stated again with modification in ronding than :
पुनरप्युभयप्रयोगमलानयनसूत्रम् --
इच्छाकालादुभयप्रयोगवृद्धिं समानीय । तपन्तरभक्तं लभ मूलं विजानीयात् ॥

Page #111

·
मिश्रकव्यवहारः
स्कन्धं चतुर्भिस्सहिता त्वशीतिः
मूलं भवेत्को नु विमुक्तिकालः ॥ ५८ ॥
षष्ट्या मासिकवृद्धिः पश्चैव हि मूलमपि च पञ्चत्रिंशत् ।
मासत्रितये स्कन्धं त्रिपञ्चकं तस्य कः कालः ।। ५९ ।।
समान वृद्धिमूल मिश्र विभाग सूत्रम् --
मूलैः स्वकालगुणितैईद्धिविभक्तैरसमासकैर्विभजेत् ।
मिश्रं स्वकालनिघ्नं वृद्धिर्मूलानि च प्राग्वत् ॥ ६० ॥
अत्रोद्देशकः
ः ।
द्विकषट्टु चतुश्शतके चतुरसहस्त्रं चतुश्शतं मिश्रम् । मासद्वयेन वृद्धया समानि कान्यत्र मूलानि ॥ ६१ ॥
त्रिकशतपञ्चकसप्ततिपादोनचनुष्कषष्टियांगंषु । नवशतसहस्रसङ्ख्या मासत्रितये समायुक्ता ॥ ६२ ॥
विमुक्तकालस्य मूलानयनसूत्रम -
स्कन्धं स्वकालभक्तं विमुक्तकालेन ताडितं विभजेत् । निर्मुक्तकालवृद्धया रूपम्य हि सकया मूलम् || ६३ ||
8
अत्रोद्देशकः ।
पश्वकशतप्रयोगे मासौ द्वौ स्कन्धमष्टकं दत्वा । मासैष्षष्टिभिरिह वै निर्मुक्तः किं भवेन्मूलम् || ६४ ||
सत्रिभागी स्कन्धं द्वादशदिनैर्ददात्येकः ।
78

Page #112

गणितसारसाहः त्रिकशतयोगे दशभिर्मासैर्मुक्तं हि मूलं किम् ।। ६५ ॥ हडियुक्तहीनसमानमूलमिश्रविभागसूत्रम्कालस्वफलोनाधिकरूपोद्धृतरूपयोगहतमिश्रे । प्रक्षेपो गुणकारः स्वफलोनाधिकसमानमूलानि ॥ ६६ ॥
भत्रोद्देशकः । त्रिकपश्चकाष्टकशतैः प्रयोगतोऽष्टासहस्रपञ्चशतम् । विशतिसहितं वृद्धिभिरुद्धृत्य समानि पञ्चभिर्मासैः ॥ ६ ॥ त्रिकषट्राष्टकषष्ट्या मासद्वितये चतुस्सहस्राणि । पधाशद्विशतयुतान्यतोऽष्टमासकफलाढते सहशानि ॥ ६८ ॥ द्विकपञ्चकनवकशते मासचतुष्के त्रयोदशसहस्रम् । सप्तशतेन च मिश्रा चत्वारिंशत्साहिसममूलानि ॥ ६९ ॥ सैकार्धकपश्चार्धकषडर्घकाशीतियोगयुक्तास्तु । मासाष्टके षडधिका चत्वारिंशच्च षटुतिशतानि ॥ ७० ॥ सङ्कलितस्कन्धमूलस्य मूलवृद्धिविमुक्तिकालानयनसूत्रम्स्कन्धाप्तमूलचितिगुणितस्कन्धेच्छाग्रघातियुतमूलं स्यात् । स्कन्धे कालेन फलं स्कन्धोइतकालमूलहतकालः ॥ १ ॥
अत्रोद्देशकः। केनापि संप्रयुक्ता षष्टिः पश्चकशतप्रयोगेण । माप्तत्रिपञ्चभागात् सप्तोत्तरतश्च सप्तादिः ॥ ७२ ॥ तत्षष्टिसप्तमांशकपदमितिसङ्कलितधनमेव । दत्वा तत्सप्तांशकवृद्धि प्रादाच चितिमूलम् ।।
PN: in the reading found in the MS.; is adopted as being more mtalsstory from matioally

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मिश्रकव्यवहारः
____78
किं तदृद्धिः का स्यात् कालस्तदृणस्य मौक्षिको भवति । ७३, । केनापि संप्रयुक्ताशीतिः पञ्चकशतप्रयोगेण ॥ अष्टाद्यष्टोत्तरतसदशीत्यष्टांशगच्छेन । मूलधनं दत्वाष्टाद्यष्टोत्तरतो धनस्य मासार्धात् ।। ७५ ॥ वृद्धि प्रादान्मूलं सद्धिश्च विमुक्तिकालश्च । एषां परिमाणं किं विगणय्य सरखे ममाचक्ष्व ॥ ७ ॥ एकीकरणसूत्रम्-- वृद्धिसमास विभजेन्मासफलैक्येन लब्धमिष्ठः कालः । कालप्रमाणगुणितस्तादष्टकालेन सम्भक्तः ॥ दृद्धिसमासेन हतो मूलसमासेन भाजितो वृद्धिः । ७७।
अत्रादेशकः । युक्ता चतुश्शतीह द्विकत्रिकपनकचतुषशतेन । मासाः पञ्च चतुर्दित्रयः प्रयोगैककालः कः ॥ ७० ॥
इति मिश्रकव्यवहारे वृद्धिविधानं समाप्तम् ।।
प्रक्षेपककुटीकारः ॥ ___ इतः परं मिश्रकव्यवहार प्रक्षेपककुट्टीकारगणितं व्याख्या. यामः ।
प्रक्षेपककरणमि: सवर्गविच्छेदनांशगुतिहृतमिश्रः। . प्रक्षेपकगुणकारः कुट्टीकारो वृधैम्समुद्दिष्टम् ॥ ७९ ॥
अत्रांद्देशकः । द्वित्रिचतुष्पदभागैविभाज्यते द्विगुणषष्टिग्हि हेम्राम । भृत्येभ्यो हि चतुभ्यों गणकाचक्ष्वाशु मं भागान ॥ ८॥
8-A

Page #114

गणितसारसङ्ग्रहः प्रथमस्यांशत्रितयं त्रिगुणोत्तरतश्च पञ्चभिर्भक्तम्।। दीनाराणां त्रिशतं त्रिषष्टिसहितं क एकांशः ॥ ८१६ ॥ आदाय चाम्बुजानि प्रविश्य सच्छ्रावकोऽथ जिननिलयम् । पूजां चकार भक्त्या पूजाहेभ्यो जिनेन्द्रेभ्यः ॥ ८२३ ॥ वृषभाय चतुर्थाशं षष्ठांशं शिष्टपार्थाय । द्वादशमथ जिनपतये व्यंशं मुनिसुव्रताय ददौ ॥ ८३ ।। नष्टाष्टकर्मणे जगदिष्टायारिष्टनेमयेऽष्टांशम् । षष्ठघ्नचतुर्भागं भक्त्या जिनशान्तये प्रददौ ॥ ८४ ॥ कमलान्यशीतिमिश्राण्यायातान्यथ शतानि चत्वारि । कुसुमानां भागाख्यं कथय प्रक्षेपकाख्यकरणेन ॥ ८५ ॥ चत्वारि शतानि सरवे युतान्यशीत्या नरैविभक्तानि । पञ्चभिराचक्ष्व त्वं द्वित्रिचतुःप चषडणितैः ॥ ८६ ॥ इष्टगुणफलानयनसूत्रम्भक्तं शेषैर्मूलं गुणगुणितं तेन योजितं प्रक्षेपम् । तद्रव्यं मूलपघ्नं क्षेपविभक्तं हि मूल्यं स्यात् ।। ८७ ॥ अस्मिन्नर्थे पुनरपि सूत्रम्-- फलगुणकारैर्हत्वा पणान् फलैरेव भागमादाय । प्रक्षेपके गुणास्स्युस्त्रैराशिकतः फलं वदेन्मतिमान् ॥ ८॥ अस्मिन्नर्थे पुनरपि सूत्रम्स्वफलहताः स्वगुणनाः पणास्तु तैर्भवति पूर्ववच्छेषः । इष्टफलं निर्दिष्टं त्रैराशिकसाधितं सम्यक् ॥ ८९ ॥

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मिश्रकव्यवहारः
अत्रोद्देशकः । द्वाभ्यां त्रीणि त्रिभिः पञ्च पञ्चभिस्सप्त मानकैः । दाडिमाम्रकपित्थानां फलानि गणितार्थवित् ॥ ९० ॥ कपित्थात् त्रिगुणं ह्यानं दाडिमं षड्गुणं भवेत् । क्रीत्वानय सरवे शीघ्रं त्वं षट्सप्ततिभिः पणैः ॥ ९१ ॥ दध्याज्यक्षीरघटैर्जिनबिम्बस्याभिषेचनं कृतवान् । जिनपुरुषो द्वासप्ततिपलैस्त्रयः पूरिताः कलशाः ॥ ९२ ॥ द्वात्रिंशत्प्रथमघटे पुनश्चतुर्विंशतिढितीपघटे । षोडश तृतीयकलशे पृथक पृथक कथय म कृत्वा ॥ ९३ ॥ तेषां दधिघृतपयसां ततश्चतुर्विंशतिघृतस्य पलानि । षोडश पयःपलानि द्वात्रिंशद दधिपलानीह ॥ ९४ ॥ वृत्तिस्त्रयः पुराणाः पुंसवारोहकस्य नत्रापि । सर्वेऽपि पञ्चषष्टिः कचिद्गना धनं तेषाम् ॥ ९५ ॥ सन्निहितानां दत्तं लब्धं पंसा दशैव चैकम्य । के सन्निहिता भनाः के मम सचिन्त्य कथय त्वम् ।। ९६ ॥
इष्टरूपाधिकहीनप्रक्षेपककरणमूत्रम्-- . पिण्डोऽधिकरू पोनो हीनोत्तररूपसंगुनः शेषात् । प्रक्षेपककरणमनः कर्तव्यं तयुता हीनाः ॥ ९७ ।।
अत्रोद्देशकः ।
प्रथमस्यैकांशोऽतो द्विगुणद्विगुणोत्तगरनन्ति नगः । चत्वारा शः कस्स्यादेकस्य हि मतष्टिरिह ॥ ९८॥

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7R
गणितसारसाहः प्रथमादध्यर्धगुणात् त्रिगुणाद्रूपोत्तराद्विभाज्यन्ते । साष्टा सप्ततिरेभिश्चतुर्भिराप्तांशकान् ब्रूहि ॥ ९९ ॥ प्रथमादध्यर्धगुणाः पश्चार्धगुणोत्तराणि रूपाणि । पञ्चानां पञ्चाशत्सैका चरणत्रयाभ्यषिका ॥ १० ॥ प्रथमात्पश्चार्धगुणाश्चतुर्गुणोत्तरविहीनभागेन ।
भक्तं नरैश्चतुर्भिः पञ्चदशोनं शतचतुष्कम् ॥ १०१६ ॥ समधनार्यानयनतज्ज्येष्ठधनसङ्ख्यानयनसूत्रम्---
ज्येष्ठधनं सैकं स्यात् स्वविक्रयेऽन्त्यार्घगुणमपैकं तत् । क्रयणे ज्येष्ठानयनं समानयेत् करणविपरीतात् ॥ १०२ ॥
अत्रोद्देशकः । हावष्टौ षट्रिशन्मूलं नृणां षडेव चरमार्घः । एकाधैंण क्रीत्वा विक्रीय च समधना जाताः ॥ १०३६ ।। साधैकमर्धमर्धद्वयं च सगृह्य ते त्रयः पुरुषाः । क्रयविक्रयौ च कृत्वा षड्डिः पश्नार्धात्समधना जाताः ॥ १० चत्वारिंशत् सैका समधनसङ्ख्या षडेव चरमार्घः। . आचक्ष्व गणक शीघं ज्येष्ठधनं किं च कानि मूलानि ॥ १०५ ॥ समधनसङ्ख्या पश्चत्रिंशद्भवन्ति यत्र दीनाराः । चत्वारश्चरमा? ज्येष्ठधनं किं च गणक कथय त्वम् ॥ १० ॥ चरमार्घभिन्नजाती समधना_नयनसूत्रम्तुख्यापच्छेदधनान्त्याीभ्यां विक्रयक्रया! प्राग्वत् । छेदच्छेदकृतिघावनुपातात् समधनानि भिन्नेऽन्त्यापें ॥ १०॥

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मिश्रकव्यवहारः
अर्धत्रिपादभागा धनानि षट्पञ्चमांशकाश्चरमार्घः । एकार्घेण क्रीत्वा विक्रीय व समधना जाताः ॥ १०८ ॥
पुनराप अन्त्यार्धे भिन्ने सति समधनानयनसूत्रम्ज्येष्ठांशद्विहरहतिः सान्त्यहरा विक्रयो ऽन्त्यमूल्यन्नः । नैको द्व्यखिलहरघ्नः स्यात्क्रयसङ्ख्यानुपातोऽथ ॥ १०९ ॥ अत्रोद्देशकः ।
अर्थ हौ ध्यंशौ च त्रीन् पादांशांश्च सङ्गृह्य | विक्रीय क्रीत्वान्ते पचभिरक्ङ्ग्यंशकैस्समानधनाः ॥ ११० ॥
इष्टगुणेष्टसङ्ख्यायामिष्टसङ्ख्यासमर्पणानयनसूत्रम्-
अन्त्यपदे स्वगुणहते क्षिपेद्पान्त्यं च तस्यान्तम् । तेनोपान्त्येन भजेद्यलब्धं तद्भवेन्मूलम् ॥ १११ ॥ अत्रोद्देशकः ।
कचिच्छ्रावक पुरुषश्चतुर्मुखं जिनगृहं समासाद्य । पूजां चकार भक्त्या सुरभीण्यादाय कुसुमानि ॥ ११२ ॥
द्विगुणमभूदाद्य मुखे त्रिगुणं च चतुर्गुणं च पवगुणन् । सर्वत्र पश्ञ्च पञ्च च तत्सङ्ख्याम्भोरुहाणि कानि स्युः ।। ११३ ।।
चतुर्भागगुण पथार्धगुणास्त्रि सप्ताष्टौ ।' भक्तैर्भक्त्याभ्यो दत्तान्यादाय कुसुमानि ॥। १९४३ ।।
इति मिश्रकव्यवहारे प्रक्षेपक कुकारः समाप्तः ॥
79
The following stanza is added in M after stanza No. 119g, but it is not found
B:
resureभागा धनानि पपश्वमांशकान्त्यार्थः ।
एकार्घेण क्रीत्वा विक्रीय न ममधना जाताः ॥

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80
गणितसारसङ्ग्रहः
काकुकारः ॥
इतः परं वल्लिकाकुट्टीकारगणितं व्याख्यास्यामः । कुट्टीकारे बल्छिकागणितन्यायसूत्रम्-
छित्वा छेदेन राशि प्रथमफलम पोह्याप्तमन्योन्यभक्तं स्थाप्योर्ध्वाधर्यतोऽधो मतिगुणमयुजालपेऽवशिष्टे धनर्णम् । छित्वाधः स्वोपरिनो परियुतहर भागोऽधिकाग्रस्य हारं छित्वा छेदेन साग्रान्तरफलमधिकाग्रान्वितं हारघातम् ॥ ११९ ॥ अत्रोद्देशकः ।
5:
जम्बूजम्बीररम्भाकमुकपनसरखर्जूर हिन्तालतालीपुनागाम्राद्यनेकद्रमकुसुमफलैर्ननशास्वाधिरूढम् । भ्राम्यङ्गानवापीशुकपिककुलनानाध्वनिव्याप्तदिकं
पान्थाः श्रान्ता वनान्तं श्रमनुदममलं ते प्रविष्टाः प्रहृष्टाः ॥ ११६ 11
राशित्रिषष्टिः कदलीफलानां सम्पीड्य संक्षिप्य च सप्तभिस्तैः ।
पान्थैस्त्रयोविंशतिभिर्विशुद्धा
राशेस्त्वमेकस्य वद प्रमाणम् ॥ ११७ ॥
राशीन् पुनर्द्वादश दाडिमानां समस्य संक्षिप्य च पञ्चभिस्तैः ।
पान्यैर्नरैविंशतिभिर्निरेकै -
भक्तांस्तथैकस्य वद प्रमाणम् | ११८३ ॥
वाशी पथिको यक
त्रिंशत्समूहं कुरुते त्रिहीनम् ।

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मिश्रकव्यवहारः
81 शेषे हृते सप्ततिभिस्त्रिमित्रै
नरैर्विशुद्धं कथयैकसङ्ख्याम् ॥ ११९ ॥ दृष्टास्सप्तत्रिंशत्कपित्थफलराशयो वने पथिकैः । सप्तदशापोह्य हते व्येकाशीत्यांशकप्रमाणं किम् ॥ १२०३ ॥ दृष्ट्वाम्रराशिमपहाय च सप्त पश्चादक्तेऽष्टभिः पुनरपि प्रविहाय तस्मात् । त्रीणि त्रयोदशभिरुद्दलिते विशद्धः पान्थैर्वने गणक मे कथथैकराशिम् ॥ १२१ ॥ द्वाभ्यां त्रिभिश्चतुर्भिः पनाभिरेकः कपित्थफलराशिः । भक्तो रूपाग्रस्तत्प्रमाणमाचक्ष्व गणितज्ञ ॥ १२२ ॥ द्वाभ्यामेकस्त्रिभिट्ठौं च चतुर्भिर्भाजिने त्रयः । चत्वारि पञ्चभिश्शेषः को राशिर्वद में प्रिय ॥ १२३ ।। द्वाभ्यामेकस्त्रिभिश्शुद्धश्चतुभिजिते त्रयः । चत्वारि पञ्चभिश्शेषः को राशिर्वद में प्रिय ॥ १२४ ॥ द्वाभ्यां निरग्र एकाग्रस्त्रिभिर्नाग्रो विभाजितः । चतुर्भिः पञ्चभिर्भको रूपायो राशिरेष कः ॥ १२५ ॥ द्वाभ्यामेकस्त्रिभिश्शद्धश्चतुभिर्भाजिते त्रयः ।। निरग्रः पञ्चभिर्भक्तः को राशिः कथयाधुना || १२६ ॥ दृष्टा जम्बूफलानां पथि पथिकजनै राशयस्तत्र गशी द्वौ व्यग्रौ नौ नवानां त्रय इति पुनकादशानां विभक्ताः । पश्चाग्रास्ते यतीनां चतरधिकतराः पन ते सप्तकानां कुट्टीकारार्थविन्मे कथय गणक सञ्चिन्त्य गशिप्रमाणम् ॥ १२७॥

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गणितसारसग्रहः बनान्तरे दाडिमराशयस्ते पान्यैस्त्रयस्सप्तभिरेकशेषाः । सप्त त्रिशेषा नवभिर्विभक्ताः पञ्चाष्टभिः के गणक हिरग्राः ।। १२८॥
भक्ता द्वियुक्ता नवभिस्तु पश्च युक्ताश्चतुर्भिश्च षडष्टभिस्तैः । पान्थैर्जनैस्सप्तभिरेकयुक्ता
चत्वार एते कथय प्रमाणम् ।। १२९ ॥ अग्रशेषविभागमूलानयनसूत्रम्शेषांशाग्रवधो युक् स्वाग्रेणान्यस्तदंशकेन गुणः । पावदागास्तावद्विच्छेदाः स्युस्तदग्रगुणाः ॥ १३० ॥
अत्रोद्देशकः । . मानीतवत्याम्रफलानि पुंसि प्रागेकमादाय पुनस्तदर्धम् । गतेऽग्रपुत्रे च तथा जघन्यस्तत्रावशेषार्धमथो तमन्यः ॥ ११ ॥ प्रविश्य जैनं भवनं त्रिपूरुषं प्रागेकमभ्यर्च्य जिनस्य पादे'। शेषत्रिभागं प्रथमेऽनुमाने तथा द्वितीये च तृतीयके तथा ॥ १२ ॥ शेषत्रिभागह यतश्च शेषऽयंशद्वयं चापि ततस्विभागान् । कृत्वा चतुर्विशतितीर्थनाथान् .
समर्चयित्वा गतवान् विशुद्धः ॥ १३३ ॥ इति मिश्रकव्यवहारे साधारणकुटीकारः समाप्तः ॥ The M88. giro on, which does not seem to be correct bero. B roade sur
पारे.

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मिश्रकव्यवहारः
8 विषमकुटीकारः ॥ इतः परं विषमकुट्टीकारं व्याख्यास्यामः । विषमकुट्टीकारस्य
सूत्रम्--
• मतिसङ्गाणिती छेदी योज्योनत्याज्यसंयुतौ राशिहतौ । भिन्ने कुट्टीकारे गुणकारोऽयं समुद्दिष्टः ॥ १३४३ ॥
अत्रोद्देशकः । राशिः षट्रेन हतो दशान्वितो नवहतो निरवशेषः । दशभित्नश्च तथा तद्गुणको' को ममाशु सङ्कथय ।। १३५ ॥
मकलकुट्टीकारः॥ सकलकुट्टीकारस्य सूत्रम्भाज्यच्छेदाग्रशेषैः प्रथमहतिफलं त्याज्यमन्यांन्यभक्तं न्यस्यान्ते साग्रमूवँरुपरिगुणयुतं तैस्समानासमाने । वर्णनं व्याप्तहारौ गुणधनमृणयोश्चाधिकाग्रस्य हार हत्या हत्वा तु सामान्तरधनमाधिकाग्रान्वितं हारघातम ।। १३१॥
भत्रोद्देशकः । सप्तोत्तरसप्तत्या युतं शतं योज्यमानमष्टत्रिंशत् । सैकशतद्वयभक्तं को गुणकारो भवेदत्र || १३७ ॥ पश्चत्रिंशत् व्युत्तरषोडशपदान्येव हाराश्च । । द्वात्रिंशयाधिकका व्युत्तरतोऽग्राणि के धनर्णगुणाः ॥ १३॥ अधिकाल्पराश्योर्मूलमिश्रविभागसूत्रम्-- ज्येष्ठनमहाराशेर्जघन्यफलनाडितानमपनीय । फलवर्गशेषभागो ज्येष्ठार्थोऽन्यो गुणस्य विपरीतम् ॥ १३९ ॥
'B गुणकारी.

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गणितसारसङ्गहः
अत्रोद्देशकः । नवानां मातुलुङ्गानां कपित्थानां सुगन्धिनाम् । सप्तानां मूल्यसम्मिश्रं सप्तोत्तरशतं पुनः ॥ १४०३॥ सप्तानां मातुलुङ्गानां कपित्थानां सुगन्धिनाम् । नवानां मूल्यसम्मिश्रमेकोत्तरशतं पुनः ॥ १४ १ ॥ मूल्ये ते वद मे शीघ्रं मातुलुङ्गकपित्थयोः । अनयोर्गणक त्वं मे कृत्वा सम्यक् पृथक् पृथक् ॥ १४२ ॥ बहुराशिमिश्रतन्मूल्यमिश्रविभागसूत्रम्इष्टप्रफलैरूनितलाभादिष्टाप्तफलमसकृत् । तैरूनितफळपिण्डसच्छेदा गुणयुतास्तदर्घास्स्युः ॥ १ ४ ३ ॥
अत्रोद्देशकः । अथ मातुलुङ्गकदलीकपित्थदाडिमफलानि मिश्राणि । प्रथमस्य सैकविंशतिरथ द्विरग्रा द्वितीयस्य ।। १४४३ ॥ विंशतिरथ सुरभीणि च पुनस्त्रयोविंशतिस्तृतीयस्य । तेषां मूल्यसमासस्त्रिसप्ततिः किं फलं कोऽर्घः ॥ १४५३ ॥ जघन्योनमिलितराश्यानयनसूत्रम्-- पण्यहृताल्पफलोनैश्छिन्द्यादल्पनमूल्यहीनेष्टम् । कृत्वा तावत्खण्डं तदूनमूल्यं जघन्यपण्यं स्यात् ॥ १४६ ॥
अत्रोद्देशकः । द्वाभ्यां त्रयो मयूरास्त्रिभिश्च पारावताश्च चत्वारः । हंसाः पञ्च चतुर्भिः पञ्चभिरथ सारसाषट च ॥ १४७ ॥

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मिश्रकव्यवहारः यत्रास्तत्र सरवे षट्पञ्चाशत्पणैः रवगान् क्रीत्वा। द्वासप्ततिमानयतामित्युक्त्वा मूलमेवादात् । कतिभिः पणैस्तु विहगाः कति विगणय्याशु जानीयाः॥ १४९ ॥ त्रिभिः पणैः शुण्ठिपलानि पञ्च चतुर्भिरेकादश पिप्पलानाम् । अष्टाभिरेकं मरिचस्य मूल्यं षष्ट्यानयाष्टोत्तरषष्टिमाश ॥ १५० ॥ इष्टांर्घरिष्टमूल्यौरष्टवस्तुप्रमाणानयनसूत्रम्मूल्यनफलेच्छागुणपणान्तरेष्टनयुतिविपर्यासः । द्विष्ठः स्वधनेष्टगुणः प्रक्षेपककरणमवशिष्टम् ।। १५१ ॥
अत्रोद्देशकः । त्रिभिः पारावताः पञ्च पञ्चभिस्सप्त सारसाः । सप्तभिनव हंसान नवनिश्शिरिवनस्त्रयः ॥ १५२ ।। क्रीडार्थ नृपपुत्रस्य शतेन शतमानय । इत्युक्तः प्रहितः कश्चित् तेन किं कस्य दीयते ॥ १५ ॥ व्यस्तार्थपण्यप्रमाणानयनसूत्रम्'पण्यैक्येन पणैक्यमन्तरमतः पण्येष्टपण्यानरेश्छिन्द्यात्सङ्कमणे कृते तदुभयोरभवतां पुनः । पण्ये ते खलु प रयोगविवरे व्यस्तं तयोर्घयोः । प्रश्नानां विदुषां प्रसादनमिदं सूत्रं जिनेन्द्रादितम् ।। १५४ ।।
अत्रोद्देशकः । आद्यमूल्यं यदेकस्यं चन्दनस्यागरोस्तथा । पलानि विंशतिमिश्रं चतुरग्रशतं पणाः ॥ १५५ ॥
INot found in any of the Mgs. consuited.

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गणितसारसहः कालेन व्यत्ययार्घस्स्यात्सषोडशशतं पणाः । तयोरर्घफले धूहि त्वं षडष्ट पृथक् पृथक् ॥ १५ ॥ सूर्परथाश्वेष्टयोगयोजनानयनसूत्रम्
अरिवलाप्तारिवलयाजनसङ्ख्यापर्याययोजनानि स्युः । तानीष्टयोगसङ्ख्यानिनान्येकैकगमनमानानि ॥ १५७ ॥
मनोद्देशकः। रविरथतुरगास्सप्त हि चत्वारोऽश्वा वहन्ति धूर्युक्ताः । योजनसप्ततिगतयः के व्यूढाः के चतुर्योगाः ॥ १५८ ॥ सर्वधनेष्टहीनशेषपिण्डात् स्वस्वहस्तग धनानयनसूत्रम्रूपोननरैविभजेत् पिण्डीकृतभाण्डसारमुपलब्धम् । सर्वधनं स्यात्तस्मादुक्तविहीनं तु हस्तगतम् ॥ १५९ ॥
अत्रोद्देशकः । वणिजस्ते चत्वारः पृथक् पृथक शौल्किकेन परिपृष्टाः । किं भाण्डसारमिति खलु तत्राहैको वणिकच्छेठः ॥ १६० ॥ आत्मधनं विनिगृह्य द्वाविंशतिरिति ततः परोऽवोचत् । त्रिभिरुत्तरा तु विंशतिरथ चतुरधिकैव विंशतिस्तुर्यः !! १६१ ।। सप्तोत्तरविंशतिरिति समानसारा निगृह्य सर्वेऽपि । ऊचुः किं ब्रूहि सरखे पृथक पृथग्भाण्डसारं मे ॥ १६२ ॥ भन्योऽन्यमिष्टरत्नसङ्खयां दत्त्वा समधनानयनसूत्रम्-- पुरुषसमासेन गुणं दातव्यं तद्विशोध्य पण्येभ्यः । शेषपरस्परगुणितं खं खं हित्वा मणेर्मूल्यम् ॥ १६३ ।।

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मिभकव्यवहारः
अत्रोद्देशकः।
प्रथमस्य शक्रनीलाः षट् सप्त च मरकता द्वितीयस्य। वजाण्यपरस्याष्टावेकैकार्घ प्रदाय समाः ॥ १६४ ॥ प्रथमस्य शक्रनीलाः षोडश दश मरकता द्वितीयस्य । वजास्तृतीयपुरुषस्याष्टौ हौ तत्र दत्वैव ॥ १६५॥ तेषेकैकोऽन्याभ्यां समधनतां यान्ति ते त्रयः पुरुषाः । तच्छकनीलमरकतवज्राणां किंविधा अर्घाः ॥ १६६ ॥ कयविक्रयलाभैः मूलानयनसूत्रम्---
अन्योऽन्यमूल्यगुणिते विक्रयभक्ते कयं यदप कब्धम् ।। तेनैकोनेन हतो लाभः पूर्वाहृतं मूल्यम् ॥ १६७ ।।
अत्रोद्देशकः । त्रिभिः क्रीणाति सप्तव विक्रीणाति च पश्मभिः । नव प्रस्थान् वणिक् किं स्याल्लाभो हासप्ततिर्धनम् ॥ १६॥
इति मिश्रकव्यवहारे सकलकुटीका': समाप्तः ॥
सुवर्णकुटीकारः ॥ इतः परं सुवर्णगणितरूपकुटीकारं स्याम्यामः । समस्तेष्टवगैरेकीकरणेन सङ्करवर्णानयनसूत्रम्---
कनकक्षयसंवर्गों मिश्रस्वर्णाहतः क्षयो ज्ञेयः । परवर्णभविमकं सुवर्णगुणितं फलं हेम्नः ॥ १६९ ॥

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गणितसारसङ्ग्रहः
अत्रोद्देशकः । एकक्षयमेकं च द्विक्षयमेकं त्रिवर्णमेकं च । वर्णचतुष्के च हे पञ्चक्षयिकाश्च चत्वारः ॥ १७० ॥ सप्त चतुर्दशवर्णास्त्रिगुणितपञ्चक्षयाश्चाष्टौ । एतानेकीकृत्य ज्वलने क्षिप्त्वैव मिश्रवर्ण किम् । एतमिश्रसुवर्ण पूर्वेभक्तं च किं किमेकस्य ॥ १७१ ॥ इष्टवर्णानामिष्टस्ववर्णानयनसूत्रम्स्वैस्स्वैर्वर्णहमिश्रं स्वर्णमिश्रेण भाजितम् । लब्धं वर्ण विजानीयात्तदिष्टाप्तं पृथक् पृथक् ॥ १७२३ ॥
अत्रोद्देशकः। विंशतिपणास्तु षोडश वर्णा दशवर्णपरिमाणैः । परिवर्तिता वद त्वं कति हि पुराणा भवन्त्यधुना ॥ १७३ ॥ अष्टोत्तरशतकनकं वर्णाष्टांशत्रयेण संयुक्तम् ।। एकादशवर्णं चतुरुत्तरदशवर्णकैः कृतं च किं हेम ।। १७४ ॥ अज्ञातवर्णानगनसूत्रम्कनकक्षयसंवर्ग मिश्रं वर्णनमिश्रतः शोध्यम् । स्वर्णेन हृत वर्ण वर्णविशेषेण कनकं स्यात् ॥ १७५ ॥ अज्ञातवर्णस्य पुनरपि सूत्रम्स्वस्वर्णवर्णविनिहतयोगं स्वर्णैक्यदृढहताच्छोध्यम् । अज्ञातवर्णहेना भक्तं वर्णं बुधाः प्राहुः ॥ १७६३ ।।
अत्राद्देशकः। 'षड्जलधिवहिकनकैस्त्रयोदशाष्टर्तुवर्णकैः क्रमशः। Hore वहि is ruletituted for रनल, and तुवर्णकैः for टावृतुक्षयः, us thereby
___
bhe reading will be better grainmatically,

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मिश्रकव्यवहारः अज्ञातवाहन : पञ्च विमिश्रक्षयं च सैकदश । अज्ञातवर्णसङ्खयां ब्रूहि सरवे गणिततच्वज्ञ ।। १७८ ।। चतुर्दशैव वर्णानि सप्त स्वर्णानि तत्क्षये'। चतुरवणे दशोत्पन्नमज्ञातक्षयकं वद ॥ १७९ ॥ अज्ञातवर्णानयनमूत्रम्खस्वर्णवर्णविनिहतयोगं स्वर्णेक्यगुणितदृढवर्णात् । त्यक्त्वाज्ञातस्वर्णक्षयदृढवर्णान्तराहृतं कनकम् ।। १८० ।।
अत्रोद्देशकः । द्वित्रिचतुःक्षयमानास्त्रिस्त्रिः कनकास्त्रयोदशक्षयिकः !
वर्णयुतिर्दश जाता बहि सम्वे कनकपरिमाणम् ॥ १८१ ॥ गुग्मवर्णमिश्रसुवर्णानगनसूत्रम्- .
ज्येष्ठाल्पक्षयशोधितपक्वविशेषाप्तरूपकैः प्राग्वत् ।
प्रक्षेपमतः कुर्यादेवं बहुशोऽपि वा साध्यम ॥ १८२ ॥ पनरपि युग्मवर्णमिश्रस्वर्णानयनसूत्रम्
इष्टाधिकान्तरं चैव हीनेष्टान्तरमेव च। उभे ते स्थापगड्यस्तं स्वर्ण प्रशंपतः फलम् ।। १८३ ॥
अत्राद्देशकः। दशवर्णसुवर्णं गत षोडशवर्णन संयुतं पक्वम् ।
द्वादश चेत्कनकशतं द्विमंदकनकं पृथक पृथग्यहि ।। १८४ ॥ बहुसुवर्णानगनमूत्रम्
व्येकपदानां क्रमशः स्वर्णानाष्टानि कल्पगच्छपम् । अव्यक्तकनकविधिना प्रसाधयंत प्राक्तनायव ।। १८५ ॥
The reading in the M88 is 1979, which is obviously run ou.

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गणितसारसङ्ग्रहः
अत्रोद्देशकः। वर्णाश्शरतुनगवसुमृडविश्वे नव च पक्ववर्णं हि।
कनकानां षष्टिश्चेत् पृथक पृथक कनकमा किं स्यात् ।।१८६॥ द्वयनष्टवर्णानयनसूत्रम्--
स्वर्णाभ्यां हतरूपे सुवर्णवर्णाहते द्विष्ठे । स्वस्वर्णहृतैकेन च हीनयुते व्यस्ततो हि वर्णफलम् ॥ १८७॥
अत्रोद्देशकः।
षोडशदशकनकाभ्यां वर्णं न ज्ञायते' पक्वम् ।
वर्ण चैकादश चेद्वौँ तत्कनकयोर्भवेतां को ॥ १८ ॥ पुनरपि द्वयनष्टवर्णानयनसूत्रम् --
एकस्य क्षयमिष्टं प्रकल्प्य शेषं प्रसाधयेत् प्राग्वत् । बहुकनकानामिष्टं व्येकपदानां ततः प्राग्वत् ॥ १८९ ॥
__ अत्रोद्देशकः। द्वादशचतुर्दशानां स्वर्णाना समरसीकते जातम् । वर्णानां दशकं स्यात् तद्वौँ ब्रूहि सश्चिन्त्य॥ १९० ॥
अपरार्धस्योदाहरणम् । सप्तनवशिविदशानां कनकानां संयुते पक्वम् ।
द्वादशवर्ण जातं किं ब्रूहि पृथक् पृथग्वर्णम् ॥ १९१ ॥ परीक्षणशलाकानयनसूत्रम्
परमक्षयाप्तवर्णाः सर्वशलाकाः पृथक् पृथग्योज्याः । वर्णफलं तच्छोध्यं शलाकपिण्डात् प्रपूराणका ॥ १९२॥
Budda hers यते।

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मिश्रकव्यवहारः
अत्रोद्देशकः। वैश्याः स्वर्णशलाकानिकीर्षवः स्वर्णवर्णज्ञाः । चक्रुः स्वर्णशलाका द्वादशवर्णं त दाद्यस्य ॥ १९३॥ चतुरुत्तरदशवर्ण षोडशवर्णं तृतीयस्य । कनकं चास्ति प्रथमस्यैकोनं च द्वितीयस्य ।। १९४ ॥ अर्धार्धन्यूनमथ तृतीयपुरुषस्य पादोनम् । परवर्णादारभ्य प्रथमस्यैकान्त्यमेव च द्यन्त्यम् ॥ १९५ ॥ उयन्त्यं तृतीयवणिजः सर्वशलाकास्तु मापमिताः । शुद्धं कनकं किं स्यात् प्रपूरणी का पृथक् पृथक् त्वं मे।
आचक्ष्व गणक शीघ्रं सुवर्णगणितं हि यदि वेत्सि ॥१९६ ॥ विनिमयवर्णसुवर्णानयनसूत्रम् -
क्रयगुणस्वर्णविनिमयवर्णेष्टनान्तरं पुनः स्थाप्यम। व्यस्तं भवति हि विनिमयवर्णान्तरहृत्फलं कनक :। १९७६॥
___ अत्रोद्देशकः। पोडशवर्ण कनकं सप्तशतं विनिमयं कृतं लभते ।
द्वादशदशवर्णाभ्यां साष्टसहवं तु कनकं किम् ॥ १९८॥ बहुपदविनिमयमुवर्णकरणसूत्रम् ---
वर्णनकनकमिष्टस्वर्णनाप्तं दृढक्षयां भवति । प्राग्वत्प्रसाध लब्धं विनिमयबहुपदसुवर्णानाम ॥ १९९, ॥
अत्रोद्देशकः। वर्णचतुर्दशकनकं शतत्रयं विनिमयं प्रकुर्वन्तः । वर्गादशदशवसुनगश्च शतपचकं स्वर्णम् ।
.
9-A

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गणितसारसङ्ग्रहः एतेषां वर्णानां पृथक् पृथक वर्णमानं किम् ॥ २०१ ॥ विनिमयगुणवर्णकनकलाभानयनसूत्रम्
वर्णनवर्णयुतिहृतगुणयतिमूलक्षयनरूपोनेन। आप्तं लब्धं शोध्यं मूलधनाच्छेषवित्तं स्यात् ।। २०२ ।। तल्लब्धमूलयोगाद्विनिमयगुणयोगभाजितं लब्धम्। प्रक्षेपकेण गुणितं विनिमयगुणवर्णकनकं स्यात् ।। २०३ ।।
अत्रोद्देशकः। कश्चिद्वणिक् फलार्थी षोडशवर्णं शतद्वयं कनकम् । यत्किचिद्विनिमयकृतमेकाद्यं द्विगुणितं यथा क्रमशः।। २०४॥ द्वादशवसुनवदशकक्षयकं लाभो द्विरग्रशतम्।
शेषं किं स्याद्विनिमयकांस्तेषां चापि मे कथय । २०५॥ दृश्यसुवर्णविनिमयसुवर्णेमूलानयनसूत्रम् ---
विनिमयवर्णेनाप्तं स्वांशं वेष्टक्षयनसंमिश्रात् । अंशैक्योनेनाप्तं दृश्यं फलमत्र भवति मूलधनम् ।। २०६ ॥
अत्रोद्देशकः । वणिजः कंचित् षोडशवर्णकसौवर्णगुलकमाहृत्य । त्रिचतुःपञ्चमभागान् क्रमेण तस्यैव विनिमयं कृत्वा ।। २०७॥ द्वादशदशनववर्णैः संयुज्य च पूर्वशेषेण ।
मूलेन विना दृष्टं स्वर्णसहस्रं तु किं मूलम् ॥ २०८ ।। इष्टांशदानेन इष्टवर्णानयनस्य तदिष्टांशकयोः सुवर्णानयनस्य च सूत्रम
अंशाप्तकं व्यस्तं क्षिप्त्वेष्टन्नं भवेत् सुवर्णमयी । सा गुलिका तस्या अपि परस्परांशाप्तकनकस्य ॥२०९ ॥

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02
मिश्रकव्यवहारः खदृढक्षयेण वर्णी प्रकल्पयेत्प्राग्वदेव यथा। एवं तवययोरप्युभयं साम्यं फलं भवेद्यदि चेत् ।। २१०।। प्राक्कल्पनेष्टवो गुलिकाभ्यां निश्चयौ भवतः । नो चेत्प्रथमस्य तदा किञ्चिन्न्यूनाधिको क्षयौ कृत्वा ॥ २११ ।। तत्क्षयपूर्वक्षययोरन्तरिते शेषमत्र संस्थाप्य । त्रैराशिकविधिलब्धं वर्णी तेनोनिताधिको स्पष्टौ॥ २ १२ ॥
अत्रोद्देशकः । वर्णपरीक्षकवणिजौ परस्परं याचिती ततः प्रथमः । अर्धं प्रादात् तामपि गुलिकां स्वसुवर्ण आयोज्य ॥ २१३ ।' वर्णदशकं करोमीत्यपरोऽवादीत् त्रिभागमात्रतया। लब्धे तथैव पूर्ण द्वादशवर्ण करोमि गुलिकाभ्याम् ।। २१४ ।। उभयोः सुवर्णमाने वर्णी सविन्त्य गणिततत्त्वज्ञ ।
सौवर्णगणितकुशलं यदि तेऽसि निगद्यतामाशु ॥ २१५।। इति मिश्रकव्यवहारे सुवर्णकुट्टीकारः समाप्तः ॥
विचित्रकुट्टीकारः। इतः परं मिश्रकव्यवहार विचित्रकुटीकारं व्याख्यास्यामः । सत्यानृतसूत्रम्
पुरुषाः सैकेष्टगुणा द्विगुणेष्टोना भवन्त्यमत्यानि । पुरुषकृतिस्तैरूना सत्यानि भवन्ति वचनानि ।। २१६ ॥
अत्रोद्देशकः। कामुकपुरुषाः पञ्च हि वेश्यायाश्च प्रियास्त्रयस्तत्र । प्रत्येकं सा बृते त्वमिष्ट इति कानि सत्यानि ॥ २१७ ॥

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गणितसारसङ्ग्रहः
प्रस्तारयोगभेदस्य सूत्रम
एकाद्यकोत्तरतः पदमूर्ध्वाधर्यतः क्रमोत्क्रमशः । स्थाप्य प्रतिलोमन्नं प्रतिलोमन्नेन भाजितं सारम् ॥ २१८॥
अत्रोद्देशकः । वर्णाश्चापि रसानां कषायतिक्ताम्लकटु कलवणानाम् । मधुररसेन युतानां भेदान् कथयाधुना गणक ॥ २१९॥ वजेन्द्रनीलमरकतविद्रुममुक्ताफलस्त रचितमालायाः । कति भेदा युतिभेदात् कथय सरवे सम्यगाशु त्वम् ।। २२०॥ केतक्यशोकचम्पकनीलोत्पलकुसुमरचितमालायाः । कति भेदा युतिभेदात्कथय सखे गणिततत्त्वज्ञ ॥ २२१ ।। ज्ञाताज्ञातलाभैर्मूलानयनसूत्रम्---
लाभोनामश्रराशेः प्रक्षेपकतः फलानि संसाध्य । तेन हृतं तल्लब्धं मूल्यं त्वज्ञातपुरुषस्य ॥ २२२ ।।
अत्रोद्देशकः । समये केचिद्वणिजस्त्रयः क्रयं विक्रयं च कुरिन् । प्रथमस्य षट् पुराणा अष्टौ मूल्यं द्वितीयस्य ॥ २२३ ।। न ज्ञायते तृतीयस्य व्याप्तिस्तै रैस्तु षण्णवतिः । अज्ञातस्यैव फलं चत्वारिंशद्धि तेनाप्तम् ॥ २२४ ॥ कस्तस्य प्रक्षेपो वणिजोरुभयोर्भवेच्च को लाभः ।
प्रगणय्याचक्ष्व सरवे प्रक्षेपं यदि विजानासि ॥ २२५ ॥ भाटकानयनसूत्रम्
भरभृतिगतगम्यहतिं त्यक्त्वा योजनदलघमारकृतेः । तन्मूलोनं गम्यच्छिन्नं गन्तव्यभाजितं सारम् ।। २२६ ।
Mund Budd त here; metrically it is faulty.

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मिश्रकव्यवहारः
अत्रोद्देशकः । पनसानि द्वात्रिंशन्नीत्वा योजनमसौ दलोनाष्टौ ।
गृहात्यन्तर्भाटकमधै भनोऽस्य किं देयम् ॥ २२७ ॥ • द्वितीयतृतीययोजनानयनस्य सूत्रम्
भरभाठकसंवर्गोऽद्वितीयभतिकृतिविवर्जितश्छेदः । तत्यन्तरभरगतिहतेर्गतिः स्याद द्वितीयस्य ॥ २२८ ॥
अत्रोद्देशकः । पनसानि चतुर्विंशतिमा नीत्वा पञ्चयोजनानि नरः । लमते ततिमिह नव षड्भृतिवियुते द्वितीयनृगतिः का।। २२९॥ बहुपद भाटकानयनस्य सूत्रम्
सन्निहितनरहतेषु प्रागुत्तरमिश्रितेषु मार्गेषु । व्यावृत्तनरगुणेषु प्रक्षेपकसाधित मूल्यम ।। २३० ॥
अत्रोद्देशकः। शिबिका नयान्त पुरुषा विंशतिरथ योजनद्वयं नषाम् । वृत्ति-नाराणां विंशत्यधिकं च सप्तशतम् ॥ २३१ ॥ क्रोशद्वये निवृत्ती वावुभयोः क्रोशयोस्त्रयश्चान्ये ।
पश्च नरः शेषार्धाद्यावृत्ताः का भृतिस्तपाम ॥ २३२ ॥ इष्टगुणितपोट्टलकानयनसूत्रम्----
सैकगुणा स्वस्वेष्टं हित्वान्यान्यवशेषमितिः । अपवर्त्य योज्य मूलं(विष्णोः) कृत्वा ढंगकेन मूलेन ।।२३३ ।। पूर्वापवर्तराशीन् हत्वा पूर्वापवर्तगशियुतेः । पृथगेव पृथक् त्यक्त्वा हस्तगताः स्वधनसङ्ख्याः स्युः ।।२३ ४ ।। ताः स्वस्वं हित्वैव त्वशेषयोगं पृथक पृथक स्थाप्य । स्वगणनाः स्वकरगतैरूनाः पोटलकसमयाः स्युः ।। २३५।।
'Bonits पद here.

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अत्रोद्देशकः ।
मार्गे त्रिभिर्वणिग्भिः पोदृलकं दृष्टमाह तत्रैकः । पालकमिदं प्राप्य द्विगुणधनोऽहं भविष्यामि ॥ २३६ ॥ हस्तगताभ्यां युवया स्त्रिगुणधनोऽहं द्वितीय आहेति । ' पगुणोऽहं त्वपरः पोलह स्तस्थमानं किम् ॥ २३७॥ सर्वतुल्यगुणकपोट्टल कानयनहस्तगतानयनसूत्रम्-
96
व्येकपदव्येक गुणेष्टांशवघोनितांशयुतिगुणघातः ! हस्तगताः स्युर्भवति हि पूर्ववदिष्टांशभाजितं पोट्टलकम् ॥ २३८ ॥ अत्रोद्देशकः ।
वैश्यैः पञ्चभिरेकं पोट्टलकं दृष्टमाह चैकैकः । पोट्टलकषष्ठसप्तमनवमाष्टमदशमभागमाप्त्वैव ॥ २३९ ॥ स्वस्वकरस्थेन सह त्रिगुणं त्रिगुणं च शेषाणाम् । गणक त्वं मे शीघ्रं वद हस्तगतं च पोट्टलकम् ॥ २४० ॥ इष्टशिष्टगुणपोट्टलकानयनसूत्रम -
इष्टगुणन्नान्यांशाः सेष्टांशाः सैकनिजगुणहता युक्ताः । पदनेष्टशिन्यूनाः सैकेष्ट गुणहता हस्तगताः ॥ २४१ ॥ अत्रोद्देशकः ।
द्वाभ्यां पथि पथिकाभ्यां पोट्टलकं दृष्टमाह तत्रैकः । अस्यार्ध सम्प्राप्य द्विगुणचनोऽहं भविष्यामि ॥ २४२ ॥
अपरस्त्र्यंशद्वितयं त्रिगुणधनस्त्वत्करस्थधनात् । मत्करधनेन सहितं हस्तगतं किं च पोट्टलकम् ॥ २४३ ॥
दृष्टं पथि पथिकाभ्यां पोट्टलकं तद्गृहीत्वा च । द्विगुणमभूदाद्यस्तु स्वकरस्थधनेन चान्यस्य ॥

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मिश्रकव्यवहारः
97 हस्तस्थधनादन्यस्त्रिगुणं किं करगतं च पोहलकम् । २४४ ॥ मार्गे नरैश्चतुर्भिः पोट्टलकं दृष्टमाह तत्राद्यः ।
पोट्टलकामद लब्ध्वा ह्यष्टगुणोऽहं भविष्यामि ॥ २४५ ॥ • स्वकरस्थधनेनान्यो नवसङ्गणितं च शेषधनात् ।
दशगुणधनवानपरस्त्वेकादशगुणितधनवान स्यात् । पोट्टलकं किं करगतधनं कियहि गणकाशु ॥ २४७ ।। मार्गे नरैः पोट्टलकं चतुभिर्दष्टं हि तस्यैव तदा बभूवुः । पञ्चांशपादार्धतृतीयभागास्तद्वित्रिपननचतुर्गुणा'श्च ।। २४८ ॥ मार्गे त्रिभिर्वणिग्भिः पोट्टलकं दृष्टमाह तत्राद्यः । यद्यस्य चतुर्भागं लभेऽहमित्याह स युवयोर्डिगुणः ॥ २४९ ॥ आह त्रिभागमपर: स्वहस्तधनसहितमेव च त्रिगणः । अस्याधं प्राप्याहं तृतीयपुरुषश्चतुधनवान् स्याम् । आचक्ष्व गणक शीघ्रं किं हस्तगतं च पोट्टलकम् ॥ २५० ।। याचितरूपैरिष्टगुणकहस्तगतानयनस्य सूत्रम्याचितरूपैक्यानि स्वसैकगुणवर्धितानि तैः प्राग्वत् । हस्तगताना नीत्वा चेष्टगुणग्नेति सूत्रेण ॥ २५१ ॥ सदृशच्छेदं कृत्वा सैकेष्टगुणाहनष्टगुणयुत्या । रूमेनितया भक्तान तानेव करस्थितान् विजानीयात् ॥ २५२ ॥
भत्रोद्देशकः । वैश्यस्त्रिभिः परस्परहस्तगतं याचितं धनं प्रथमः । चत्वार्यथ द्वितीयं पश्च तृतीयं नरं प्रार्थ्य ॥ २५३ ॥
'M and Bread
: ; and it is obviously inappropriu in.

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गणितसारसङ्ग्रहः द्विगुणोऽभववितीयः प्रथमं चत्वारि षट् तृतीयमगात् । त्रिगुणं नृतीथपुरुषः प्रथमं पञ्च द्वितीयं च ॥ २५४३ ॥ षट् प्रार्थ्याभूत्पञ्चकगुणः स्वहस्तस्थितानि कानि स्युः । कथयाशु चित्रकुट्टीमिश्रं जानासि यदि गणक ॥ २५५ ॥ पुरुषास्त्रयोऽतिकुशलाश्चान्योन्यं याचितं धनं प्रथमः । स द्वादश द्वितीयं त्रयोदश प्रार्थ्य तत्रिगुणः ॥२५६ ।। प्रथमं दश त्रयोदश तृतीयमभ्यर्थ्य च द्वितीयोऽभूत् । पश्चगुणितो द्वितीयं द्वादश दश याचयित्वाद्यम् ॥ २५७३ सप्तगुणितस्तृतीयोऽभवन्नरो वाञ्छितानि लब्धानि । कथय सखे विगणय्य च तेषां हस्तस्थितानि कानि स्युः ।।२५८ अन्त्यस्योपान्त्यतुल्यधनं दवा समधनानयनसूत्रम्वाञ्छाभक्तं रूपं स उपान्त्यगुणः सरूपसंयुक्तः । शेषाणां गुणकारः सैकोऽन्त्यः करणमेतत्स्यात् ।। २५९ ॥
अत्रोद्देशकः ।
वैश्यात्मजास्त्रयस्ते मार्गगता ज्येष्ठमध्यमकनिष्ठाः । स्वधने ज्येष्ठो मध्यमधनमात्रं मध्यमाय ददौ ॥ २६०३ ॥ स तु मध्यमो जघन्यजधनमात्रं यच्छति स्मास्य । समधनिकाः स्युस्तेषां हस्तगतं ब्रूहि गणक संश्चिन्त्य ॥ २६१, वैश्यात्मजाश्च पञ्च ज्येष्ठादनुजः स्वकीयधनमात्रम् । लेभे सर्वेऽप्येवं समवित्ताः किं तु हस्तगतम् ॥ २६२ ।।

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मिश्रक व्यवहारः
वणिजः पश्च स्वस्वादर्थं पूर्वस्य दत्त्वा तु ।
समवित्ताः सञ्चिन्त्य च किं तेषां ब्रूहि हस्तगतम् ।। २६३ ३ ।। वणिजष्ट्र स्वधनाद्द्द्दित्रिभागमात्रं क्रमेण तज्ज्येष्ठाः । स्वस्वानुजाय दत्त्वा समवित्ताः किं च हस्तगतम् || २६४ ॥
परस्परहस्तगतधनसङ्ख्यामात्रधनं दत्त्वा समधनानयनसूत्रम्वाञ्छाभक्तं रूपं पदयुतमादावुपर्युपर्येतत् । संस्थाप्य सैकवाञ्छागुणितं रूपोनमितरेषाम् ॥ २६५ ॥
अत्रोंदेशकः ।
वणिजस्त्रयः परस्परकरस्थधनमेकतोऽन्योन्यम् । दत्त्वा समवित्ताः स्युः किं स्याद्वस्तस्थितं द्रव्यम् ।। २६६ ।।
वणिजश्चत्वारस्तेऽप्यन्योन्यधनार्धमात्रमन्यस्मात् ।
स्वीकृत्य परस्परतः समवित्ताः स्युः कियत्करस्थवनम् || २६७
जयापजययोली मानयनसूत्रम्-
स्वस्वच्छेदांशयुती स्थाप्योध्वधिर्यतः क्रमोत्क्रमशः । अन्योन्यच्छेदांशकगुणितौ वज्रापवर्तनक्रमशः || २६८ ॥
छेदांशक्रमवत्स्थिततदन्तराभ्यां क्रमेण मम्भक्तौ ।
स्वांशहरघ्नान्यहरौ वाञ्छाघ्नौ व्यस्ततः करस्थमितिः || २६९ ॥
अत्रोद्देशकः ।
दृष्ट्वा कुकुटयुद्धं प्रत्येकं तौ च कुक्कुटको ।
उक्तौ रहस्यवाक्यैर्मन्त्रौषधशक्तिमन्महापुरुषेण ॥ २७० ॥

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गणितसारसग्रहः
जयति हि पक्षी ते मे देहि स्वर्ण ह्यविजयोऽसि दद्यां ते । तड्वियंशकमद्येत्यपरं च पुनः स संसृत्य ।। २७१३ ॥ त्रिचतुर्थ प्रतिवाञ्छत्युभयस्माद्द्वादशैव लाभः स्यात् । . तत्कुकुटिककरस्थं ब्रूहि त्वं गणकमुवतिलक !! २७२ ॥ राशिलब्धच्छेदमिश्रविभागसूत्रम्--- मिश्रादूनितसङ्ख्या छेदः सैकेन तेन शेषस्य । भागं हत्वा लब्धं लाभोनितशेष एव राशिः स्यात् ।। २७३,
अत्रोद्देशकः । केनापि किमपि भक्तं सच्छेदो राशिमिश्रितो लाभः । पश्चाशत्रिभिरधिका तच्छेदः किं भवेल्लब्धम् ॥ २७४३ ॥ इष्टसङ्ख्यायोज्यत्याज्यवर्गमूलराश्यानयनसूत्रम्--- योज्यत्याज्ययुतिः सरूपविषमाग्रनार्धिता वर्गिता व्यग्रा बन्धहता च रूपसहिता त्याज्यैक्यशेषाग्रयोः । शेषैक्यार्धयुतोनिता फलमिदं राशिर्भवेद्वाञ्छयोस्त्याज्यात्याज्यमहत्त्वयोरथ कृतेर्मूलं ददात्येव सः ।। २७५ ॥
अत्रोद्देशकः । राशिः कश्चिदशभिः संयुक्तः सप्तदशभिरपि हीनः । मूलं ददाति शुद्धं तं राशि स्यान्ममाशु वद गणक ॥ २७६ ।। राशिस्सप्तभिरूनो यः सोऽष्टादशभिरन्वितः कश्चित् । मूलं यच्छति शुद्धं विगणय्याचक्ष्व तं गणक ॥ २७७ ॥

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101
राशिदिव्यशोनस्त्रिसप्तभागान्वितस्स एव पुनः । मूलं यच्छति कोऽसौ कथय विचिन्त्याश तं गणक ॥ २७८ ॥ इष्टसङ्ख्याहीनयुक्तवर्गमूलानयनसूत्रम्उद्दिष्टो यो राशिस्त्व(लतवर्गितोऽथ रूपयुतः । यच्छति मूलं खेष्टात्संयुक्त चापनीते च ॥ २७९ ॥
अत्रोद्देशकः । दशमिस्सम्मिश्रोऽयं दशभिस्तैर्वर्जितस्तु संशुद्धम् । यच्छति मूलं गणक प्रकथय सश्चिन्त्य गशि में ।। २८० ॥ इष्टवर्गीकृतराशिद्वयादिष्टनादन्तरमूलादिष्टानयन सूत्रम्--- सैकेष्टव्येकष्टावर्षीकृत्याथ वर्गितौ राशी। एताविष्टनावथ तहिश्लेषस्य मूलमिष्टं स्यात् ।। २८१ ॥
अत्रोद्देशकः । यौकौचिद्वर्गीकृतराशी गुणिती तु सैकसप्तत्या । सहिश्लेषपदं स्यादेकोत्तरसप्ततिश्च गशी को ।। विगणय्प चित्रकुट्टिकगणितं यदि वेत्ति गणक मे बेहि ॥ २८३ ।। युतहीनप्रक्षेपकगुणकारानयनसूत्रम्-- . संवर्गितेष्टशेषं द्विष्ठं रूपेष्टयुतगुणाभ्यां नत् । विपरीताभ्यां विमजेत्प्रक्षेपौ तत्र हीनी वा ॥ २८५ ॥
अत्रोद्देशकः । त्रिक पञ्चकसंवर्गः पञ्चदशाष्टादशैव चंष्टमपि । इष्टं चतुर्दशात्र प्रक्षेपः कोऽत्र हानिर्वा ॥ २८५ ।।

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गणितसारसङ्ग्रहः
विपरीतकरणानयनसूत्रम्प्रत्युत्पन्ने भागो भागे गुणितोऽधिके पुनश्शोध्यः । वर्गे मूलं मूले वर्गों विपरीतकरणमिदम् ॥ २८६ ॥ .
अत्रोद्देशकः । सप्तहते को राशिस्त्रिगुणो वर्गीकृतः शरैर्युक्तः ।
त्रिगुणितपञ्चांशहतस्त्वर्धितमूलं च पञ्चरूपाणि ॥ २८७ ॥ साधारणशरपरिध्यानयनसूत्रम्
शरपरिधित्रिकमिलनं वर्गितमेतत्पुनस्त्रिभिस्सहितम् । द्वादशहतेऽपि लब्धं शरसङ्ख्या स्यात्कलापकाविष्टा ॥२८॥
अत्रोद्देशकः । परिधिशरा अष्टादश तूणीरस्थाः शराः के स्युः । गणितज्ञ यदि विचित्रे कुट्ठीकारे श्रमोऽस्ति ते कथय ।। २८९ ।। इति मिश्रकव्यवहारे विचित्रकुट्टीकारः समाप्तः ॥
श्रेढीबद्धसङ्कलितम् । इतःपरं मिश्रकगणिते श्रेढीबद्धसङ्कलितं व्याख्यास्यामः । हीनाधिकचयसङ्कलितधनानयनसूत्रम् ...
व्येकार्धपदोनाधिकचयघा नोनान्वितः पुनः प्रभवः । गच्छाभ्यस्तो हीनाधिकचयसमुदायसङ्कलितम् ॥ २९० ॥ ।
अत्रोद्देशकः । चतुरुत्तरदश चादिर्हानचयस्त्रीणि पञ्च गच्छः किम् । द्वावादिद्धिचयः षट् पदमष्टौ धनं भवेदत्र ॥ २९१ ॥

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मिश्रकव्यवहारः. अधिकहीनोत्तरसङ्कलितधने आद्युत्तरानयनसूत्रम् - गच्छविभक्ते गणिते रूपोनपदार्धगुणितचयहीने । आदिः पदहतवित्तं चाचूनं व्येकपददलहतः प्रचयः ॥ २९२ ।।
अत्रोद्देशकः । चत्वारिंशदाणितं गच्छः पञ्च त्रयः प्रचयः ।
न ज्ञायतेऽधुनादिः प्रभवो द्विः प्रचयमाचक्ष्व ॥ २९३ ।। श्रेढीसङ्कलितगच्छानयनसूत्रम्----
आदिविहीनो लाभः प्रचयार्धहतस्स एव रूपयतः । गच्छो लाभेन गुणो गच्छस्सङ्कलितधनं च सम्भवति ।। २९४ ।।
अत्रोद्देशकः । त्रीण्युत्तरमादि वनिताभिश्चोत्पलानि भक्तानि । एकस्या भागोऽष्टौ कति वनिताः कति च कुसुमानि ॥ २९१ ॥ वर्गसङ्कलितानयनसूत्रम्
सैकष्टकतिर्दिना सैकेष्टोनेष्टदलगुणिता । कृतिघनचितिसङ्घातस्त्रिकभक्तो वर्गसङ्कलितम् ॥ २९६ ॥
अत्रोद्देशकः । अष्टाष्टादशविंशतिषष्टयेकाशीतिषदकनीनां च । कृतिघनचितिसङ्कलितं वर्गचिति चाश मे काय ॥ २९७ ॥ इष्टाधुत्तरपदवर्गसङ्कलिनधनानयनसूत्रम् .. हिगुणैकोनपदोत्तरकृतिहतिषष्ठांशमुखचाहतयुतिः । व्येकपदांना मुरवकृतिसहिना पदताडितष्टकृतिचितिका ॥ २९८ ॥

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गणितसारसहः . पुनरपि इष्टाद्युत्तरपदवर्गसङ्कलितानयनसूत्रम्--
द्विगुणैकोनपदोत्तरकृतिहतिरेकोनपदहताङ्गहता । व्येकपदादिचयाहतिमुखकृतियुक्ता पदाहता सारम् ॥ २९९ ॥
अत्रोद्देशकः । त्रीण्यादिः पञ्च चयो गच्छ: पञ्चास्य कथय कृतिचितिकाम् । पश्चादिस्त्रीणि चयो गच्छः सप्तास्य का च कृतिचितिका ॥ ३०० ॥ घनसङ्कलितानयनसूत्रम् ---- गच्छार्धवर्गराशी रूपाधिकगच्छवर्गसङ्गणितः । घनसङ्कलितं प्रोक्तं गणितेऽस्मिन् गणिततत्वज्ञैः ॥ ३०१ ॥
अत्रोद्देशकः । षण्णामष्टानामपि सप्तानां पञ्चविंशतीनां च ।
षट्पञ्चाशन्मिश्रितशतद्दयस्यापि कथय घनपिण्डम ॥ ३०२ ।। इष्टाद्युत्तरगच्छघनसङ्कलितानयनसूत्रम ... चित्यादिहतिमरवचयशेषघ्ना प्रचयनिम्नचितिवर्गे । आदौ प्रचयादूने विद्युता युक्ताधिके तु घनचितिका ॥ ३०३ ॥
अत्रोद्देशकः । आदिस्त्रयश्चयो द्वौ गच्छ: पञ्चास्य घनचितिका ।
पश्चादिस्सप्तचयो गच्छप्पट का भवेच्च घनचितिका ॥ ३०४ ॥ सङ्कलितसङ्कलितानयनसूत्रम् - द्विगुणैकोनपदोत्तरकृतिहतिरङ्गाहता चयार्धयुता । आदिचयाहतियुक्ता व्येकपदघ्नादिगुणितेन || सैकप्रभवेन युता पददलगुणितैव चितिचितिका ।। ३०५६ ॥

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मिश्रक व्यवहारः
अत्रोद्देशकः ।
आदिषट् पथ चयः पदमप्यष्टादशाथ सन्दृष्टम् ।
एकाद्येकोत्तरचिग्सिङ्कलितं किं पदाष्टदशकस्य ॥ ३०६ ॥
चतुरसङ्कलितानयनसूत्रम् -
सैक पदार्थपदातिरश्वैर्निहता पदांनिता व्याप्ता । सैकपदन्ना चितिचितिचितिकृतिघनसंयुतिर्भवति ॥ ३०७ ॥ अत्रोद्देशकः ।
सप्ताष्टनवदशानां षोडशपश्चाशदेकषष्टीनाम् ।
ब्रूहि चतुःसङ्कलितं सूत्राणि पृथक पृथक् कृत्वा ।। १०८ ।।
सङ्घातसङ्कलितानयनसूत्रम् -
गच्छस्त्रिरूपसहितो गच्छचतुर्भागताडितस्सैकः । सपदपदकृतिविनिघ्नो भर्वात हि सङ्घातसङ्कलितम् ॥ १०९ ॥ अत्रोद्देशकः ।
सप्तकृतेः पट्षष्ट्यास्त्रयोदशानां चतुर्दशानां च । पश्चाग्रविंशतीनां किं स्यात् सङ्घातसङ्कलितम् ॥ ११० ॥
105
भिन्नगुणसङ्कलितानयनसूत्रम् ---
समदलविषमग्वरूपं गुणगुणितं वर्गताडितं द्विष्टम् ।
अंशाप्तं व्येकं फलमाद्यन्नघ्नं गुणोन रूपहृतम् ॥ ३११६ ॥ अत्रोद्देशकः ।
दीनारार्ध पश्चसु नगरेषु चयस्त्रिभागोऽभूत् । आदिस्वयंशः पादो गुणोत्तरं सप्तभिन्नगुणचितिका ।
10

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गणित सारसङ्ग्रहः
का भवति कथय शीघ्रं यदि तेऽस्ति परिश्रमो गणिते ॥ ३१३ ॥
अधिक हीनगुणसङ्कलितानयनसूत्रम्-
गुणचितिरन्यादिहता विपदाधिकही नसगुणा भक्ता ।
व्ये गुणेनान्या फलरहिता हीनेऽधिके तु फलयुक्ता ॥ ३१४ ॥ अत्रोद्देशकः ।
पथ गुणोत्तरमादि त्रीण्यधिकं पदं हि चत्वारः । अधिकगुणोत्तरचितिका कथय विचिन्त्याशु गणिततत्त्वज्ञ ॥ ३११ ॥ आदिस्त्रीणि गुणोत्तरमष्टी हीनं द्वयं च दश गच्छः । गुणोत्तरचितिका का भवति विचिन्त्य कथय गणकाशु ॥ ३११ ।। आद्युत्तरगच्छधनमिश्राद्युत्तरगच्छानयनसूत्रम्
मिश्रादुद्धृत्य पदं रूपोनेच्छाधनेन सैकेन । लब्धं प्रचयः शेषः सरूपपदभाजितः प्रभवः ॥ ३१७ ॥
अत्रोद्देशकः ।
आद्युत्तरपदमिश्रं पञ्चाशद्धनमिहैव सन्दृष्टम् । गणितज्ञाचक्ष्व त्वं प्रभवोत्तरपदधनान्याशु || ३९८ ॥
सङ्कलितगतिध्रुवगतिभ्यां समानकालानयनसूत्रम्
ध्रुवगतिरादिविहनिश्चयदलभक्तस्सरूपकः कालः । द्विगुणो मार्गस्तद्गतियोगहृतो योगकालस्स्यात् ॥ ३१९ ॥
अत्रोद्देशकः ।
कचिन्नरः प्रयाति त्रिभिरादा उत्तरैस्तथाष्टामिः । नियतगतिरेकविंशतिरनयोः कः प्राप्तकालः स्यात् ॥ ३२० ॥

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मिश्रकव्यवहारः 107
अपरा|दाहरणम्। पड़ योजनानि कश्चित्पुरुषस्त्वपरः प्रयाति च त्रीणि। . उभयोरभिमुखगत्योरप्टोत्तरशतकयोजनं गम्यम् । प्रत्येकं च तयोः स्यात्कालः किं गणक कथय मे शीघ्रम् ।। ३२१ ॥ सङ्कलितसमागमकालयोजनानयनसूत्रम्--
उभयोराद्योश्शेषश्चयशेषहतो द्विसङ्गणः सैकः । .. युगपत्प्रयाणयोस्स्यान्मार्गे तु समागमः कालः ॥ ३२२ ॥
अत्रोद्देशकः । चत्वार्याद्यष्टोत्तरमेको गच्छत्यथो द्वितीयो ना।
हो प्रचयश्च दशादिः समागम कस्तयोः कालः । ३२३ ॥ हृद्युत्तरहीनोत्तरयोस्समागमकालानयनसूत्रम्
शेषश्राद्योरुभयोश्चगयुतदलभक्तरूपयुतः । युगपत्प्रयाणकृतयोर्मार्गे संयोगकालः स्यात् ॥ ३२४ ॥
अत्रोद्देशकः । पश्चाद्यष्टोत्तरनः प्रथमो नाथ द्वितीयनरः ।
आदिः पनघ्ननव प्रचना हीना ट योगकालः कः ॥ ३२५ ॥ शघिगतिमन्दगत्योस्समागमकालानयनसुत्रम्
मन्दगतिशीघ्रगत्योरेकाशागमनमत्र गम्यं यत् । तदत्यनरभक्तं लब्धदिनैनः प्रयाति शीघ्रोल्पम् ॥ ३२६ ।।
अत्रोद्देशकः । नवयोजनानि कश्चित्प्रयाति योजनशनं गतं तेन । प्रतिदूतो वननि पुनस्त्रयोदशामांति कैदिवसः ॥ ३२७ ।। 10-A

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गणितसारसङ्ग्रहः विषमबाणैस्सूणीरबाणपरिधिकरणसूत्रम्
परिणाहस्त्रिभिरधिको दलिलो वर्गीकृतस्त्रिभिर्भक्तः । सैकश्शरास्तु परिधेरानयने तत्र विपरीतम् ॥ ३२८६ ॥
अत्रोद्देशकः । नव परिधिस्तु शराणां सङ्ख्या न ज्ञायते पुनस्तेषाम् ।
व्युत्तरदशबाणास्तत्परिणाहशरांश्च कथय मे गणक ॥ ३२९ ।। श्रेढीबद्धे इष्टकानयनसूत्रम् --- तरवर्गों रूपोनस्त्रिभिविभक्तस्तरेण सङ्गणितः । तरसङ्कलिते खेष्टप्रताडिते मिश्रतः सारम् ॥ ३३०॥
अत्रोद्देशकः । पञ्चतरैकेनाग्रं व्यवघटिता गणितविन्मिश्रे । समचतुरश्रश्रेढी कतीष्टकास्स्युर्ममाचक्ष्व ॥ ३३१६ ॥ नन्द्यावर्ताकारं चतुस्तराः षष्टिसमघटिताः ।
सर्वेष्टकाः कति स्युः श्रेढीबद्धं ममाचक्ष्व ॥ ३३२ ॥ छन्दश्शास्त्रोक्तषट्प्रत्ययानां सूत्राणि --- समदलविषमस्वरूपं द्विगुणं वर्गीकृतं च पदमङ्ख्या । सङ्ख्या विषमा सैका दलनो गुरुरेव समदलतः ॥ ३३३, ॥ स्यालघुरेवं क्रपशः प्रस्तारोऽयं विनिर्दिष्टः । नष्टाङ्का लघुरथ तत्सैकदले गुरुः पुनः पुनः स्थानम् ॥ ३३४ ॥ रूपाहिगुणोत्तरतस्तद्दिष्टे लाङ्कसंयुतिः सैका । एकाद्यकोत्तरतः पदमूर्ध्वाधर्यतः क्रमोत्क्रमशः ॥ ३३५ ॥

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मिश्रकव्यवहारः
स्थाप्य प्रतिलोमनं प्रतिलोमन्नेन भाजितं सारम् ।
स्याल्लघुगुरुक्रियेयं सङ्ख्या द्विगुणैकवर्जिता साध्वा ॥ ३३६ ॥
अत्रोद्देशकः ।
सङ्ख्यां प्रस्तारविधिं नष्टोद्दिष्टे लगक्रियाध्वानौ ।
षट्प्रत्ययांश्च शीघ्रं व्यक्षरवृत्तस्य मे कथय ॥ ३३७३ ॥
100
इति मिश्रकव्यवहारे श्रेढीबद्धसङ्गतिं समाप्तम् ॥ इति सारसङ्घहे गणितशास्त्रे महावीरायास्य को मिश्रकगणितं
नाम पश्चमव्यवहारः समाप्तः ॥

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षष्ठः
क्षेत्रगणितव्यवहारः. सिद्धेभ्यो निष्ठितार्थेभ्यो वरिष्ठेभ्यः कृतादरः ।
अभिप्रेतार्थसिद्धयर्थ नमस्कुर्वे पुनः पुनः ॥ १ ॥ इतः परं क्षेत्रगणितं नाम षष्ठगणितमुदाहरिष्यामः । तद्यथा
क्षेत्रं निनप्रणीतं फलाश्रयाद्व्यावहारिकं सूक्ष्ममिति । भेदाद् द्विधा विचिन्त्य व्यवहारं स्पष्टमेतदभिधास्ये ॥ २ ॥ त्रिभुजचतुर्भुजवत्तक्षेत्राणि स्वस्वभेदभिन्नानि । गणितार्णवपारगतैराचार्यैस्सम्यगुक्तानि ॥ ३ ॥ त्रिभुज त्रिधा विभिन्नं चतुर्भुजं पञ्चधाष्टधा वृत्तम् । अवशेषक्षेत्राणि ह्येतेषां भेदभिन्नानि ॥ ४ ॥ त्रिभुजं तु समं द्विसमं विषमं चतुरश्रमाप समं भवति । द्विद्विसमं द्विसमं स्यात्रिसमं विषमं बुधाः प्राहुः ॥ ५ ॥ समवृत्तमर्धवृत्तं चायतवृत्तं च कम्बुकावृत्तम् । निनोन्नतं च वृत्तं बहिरन्तश्चक्रवालवृत्तं च ॥ १ ॥
व्यावहारिकगणितम् । त्रिभुजचतुर्भुजक्षेत्रफलानयनसूत्रम्त्रिभुजचतुर्भुजबाहुप्रतिबाहुसमासदलहतं गणितम् । नेमे जयुत्यर्थं व्यासगुणं तत्फलार्धमिह बालेन्दोः ॥ ७ ॥
अत्रोद्देशकः । त्रिभुजक्षेत्रस्याष्टौ बाहुप्रतिबाहुभूमयो दण्डाः । तयावहारिकफलं गणयित्वाचक्ष्व मे शीघम् ॥ ८ ॥

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क्षेत्रगणितव्यवहारः.
111 द्विसमत्रिभुनक्षेत्रस्यायामः सप्तसप्ततिर्दण्डाः । विसारो द्वाविंशतिरथ हस्ताभ्यां च सम्मिश्राः ॥ ९ ॥ त्रिभुनक्षेत्रस्य भुजस्त्रयोदश प्रतिभुजस्य पश्चदश । ' भूमिश्चतुर्दशास्य हि दण्डा विषमस्य किं गणितम् ॥ १० ॥ गजदन्तक्षेत्रस्य च पृष्ठेऽष्टाशीतिरत्र सन्दृष्टाः । द्वासप्ततिरुदरे तन्मूलेऽपि त्रिंशदिह' दण्डाः ॥ ११ ॥ क्षेत्रस्य दण्डषष्टिर्बाहप्रतिबाहकस्य गणयित्वा । समचतुरश्रस्य त्वं कथय सम्वे गणितफलमाशु ॥ १२ ॥ आयतचतुर श्ररुप व्यायामः सैकषष्टिरिह दण्डाः । विस्तारो द्वात्रिंशद्यवहारं गणितमाचक्ष्व ॥ १३ ॥ दण्डास्तु सप्तषष्टिर्दिसमचतुर्बादकस्य चायामः । - व्यासश्चाष्टत्रिंशत् क्षेत्रस्यास्य त्रयस्त्रिंशत् ॥ १४ ॥ क्षेत्रस्याष्टोत्तरशतदण्डा बाहवयं मरवे चाष्टौ । हस्तैस्त्रिभिर्युताम्तत्रिसमचतुर्बाहुकस्य वद गणक |॥ १५ ॥ विषमक्षेत्रस्याप्टत्रिंशद्दण्डाः क्षितिर्मुरवे द्वात्रिंशत् । पभाशत्प्रति बाहुः पटिम्वन्यः किमस्य चतुरश्रे ॥ १६ ॥ परिघांदरस्तु दण्डास्त्रिंशन्टष्ठं शतत्रयं दृष्टम् । नवपश्चगुणां व्यासा नेमिक्षेत्रस्य किं गणितम् ॥ १७ ॥ हस्ती ही विष्कम्भः पृष्ठेऽष्टापटिरिह च सन्दृष्टाः । उदरे तु द्वात्रिंशद्वालेन्दोः किं फलं कथय ॥ १८ ॥
JThe reading in both BandMin विशातः utnetha in erronsomitin porrooted into a man to out the requirements of the metro aldo.
• Broads देक for प्रति

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उत्तक्षेत्रफलानयनसूत्रम्-- त्रिगुणीकृतविष्कम्भः परिधिासार्धवर्गराशिरयम् । त्रिगुणः फलं समेऽर्धे वृत्तेऽधैं प्राहुराचार्याः ॥ १९ ॥
अत्रोद्देशकः । व्यासोऽष्टादश उत्तस्य परिधिः कः फलं च किम् । व्यासोऽष्टादश सत्ताधे गणितं किं वदाशु मे ॥ २० ॥ आयतवृत्तक्षेत्रफलानयनसूत्रम्व्यासार्धयुतो द्विगुणित आयतत्तस्य परिधिरायामः । विष्कम्भचतुर्मागः परिवेषहतो भवेत्सारम् ॥ २१ ॥
अत्रोद्देशकः । क्षेत्रस्यायतवृत्तस्य विष्कम्भो द्वादशैव तु ।
मायामस्तत्र षट्त्रिंशत् परिधिः कः फलं च किम् ॥ २२ ॥ शलाकारवृत्तस्य फलानयनसूत्रम् ----- वदना|नो व्यासस्त्रिगुणः परिधिस्तु कम्बुकावृत्ते । वलयार्धकृतित्र्यंशो मुरवार्धवर्गत्रिपादयुतः ।। २३ ॥
अत्रोद्देशकः । व्यासोऽष्टादश हस्ता मुरवविस्तारोऽयमपि च चत्वारः ।
कः परिधिः किं गणितं कथय त्वं कम्बुकाइत्ते ॥ २४ ॥ निम्नोन्नतत्तयोः फलानयनसूत्रम्
परिधेश्च चतुर्भागो विष्कम्भगुणः स विद्धि गणितफलम् । चत्वाले कूर्मनिमे क्षेत्रे निम्नोन्नते तस्मात् ॥ २१ ॥

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अत्रोद्देशकः । चत्वालक्षेत्रस्य व्यासस्तु भसयकः परिधिः । षट्पञ्चाशदष्टं गणितं तस्यैव किं भवति ॥ २६ ॥ कूर्मनिभस्योन्नतवृत्तस्योदाहरणम्विष्कम्भः पश्चदश दृष्टः परिधिश्च पत्रिंशत् । कूर्मनिभे क्षेत्रे किं तस्मिन् व्यवहारजं गणितम् ॥ २७ ॥
अन्तश्चक्रवालवृत्तक्षेत्रस्य बहिश्चक्रवालवृत्तदोत्रस्य च व्यवहारफलानयनसूत्रम्
निर्गमसहितो व्यासस्त्रिगुणो निर्गमगुणो बहिणतम् । रहिताधिगमव्यासादभ्यन्तरचक्रवालवृत्तस्य ॥ २८ ॥
अत्रोद्देशकः । व्यासोऽष्टादश हताः पुनर्बहिनिर्गतास्त्र यस्तत्र । व्यासोऽष्टादश हम्ताश्चानः पुनरधिगतास्त्रयः किं स्यात् ।। २९ ।। समवत्तक्षेत्रस्य व्यावहारिकफलं च परिधिप्रमाणं च व्यासप्रमाणं च संपोज्य एतत्संयोगसङ्ख्यामेव स्वीकृत्य तत्संयोगप्रमाणराशेः सकाशात पथक् परिधिव्यासफलानां सङ्यानयनसूत्रम् ....
गणिते द्वादशगुणिते मिश्रप्रक्षेपकं चतुःषष्टिः । तस्य च मुलं कृत्वा परिधिः प्रक्षेपकपदोनः ॥ ३० ॥
अत्रोदेशकः । परिधिव्यासफलानो मिश्र पोडशशतं सहस्रयुतम् । कः परिधिः किं गणितं व्यासः को वा ममाचक्ष्व ॥१॥

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यवाकारमर्दलाकारपणवाकारवजाकाराणां क्षेत्राणां व्यावहारिक फलानयनसूत्रम्
यवमुरजपणवशक्रायुधसंस्थानप्रतिष्ठितानां तु । मुरवमध्यसमासार्धं त्वायामगुणं फलं भवति ॥ ३२ ॥ "
अत्रोद्देशकः । यवसंस्थानक्षेत्रस्यायामोऽशीतिरस्य विष्कम्भः । मभ्यश्चत्वारिंशत्फलं भवत्किं ममाचक्ष्व ॥ ३३ ॥ आयामोऽशीरियं दण्डा मुखमस्य विशतिमध्ये । चत्वारिंशत्क्षेत्रे मृदङ्गसंस्थानके ब्रूहि ॥ ३४ ॥ पणवाकारक्षेत्रस्यायामः सप्तसप्ततिर्दण्डाः । मुरवयोर्विस्तारोऽष्टी मध्ये दण्डास्तु चत्वारः ॥ ३५ ॥ बजाकतेस्तथास्य क्षेत्रस्य षडग्रनवतिरायामः ।
मध्ये सूचिमुरवयोस्त्रयोदश व्यंशसंयुता दण्डाः ।। ३१ ॥ उभयनिषेधादिक्षेत्रफलानयनसूत्रम्
व्यासात्खायामगुणादिकम्भार्धपदीर्घमुत्सृज्य । त्वं वद निषेधमुभयोस्तदर्धपरिहीणमेकस्य ॥ ३७ ॥
अत्रोद्देशकः । आयामः षट्त्रिंशद्विस्तारोऽष्टादशैव दण्डास्तु । उभयनिषेधे किं फलमेकनिषेधे च किं गणितम् ॥ ३८ ॥ बहुविधवजाकाराणां क्षेत्राणां व्यावहारिकफलानयनसूत्रम्-. रज्ज्वर्धकतिव्यशो बाहुविभक्तो निरेकबाहुगुणः । सर्वेषामश्रवा फलं हि बिम्बान्तरे चतुर्थाशः ॥ ३९ ।'

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क्षेत्रगणितव्यवहारः 115
अत्रोदेशकः । षड्बाहुकस्य बाहोविष्कम्भः पञ्च चान्यस्य । व्यासस्त्रयो भुनस्य त्वं षोडशबाहुकस्य वद ॥ ४० ॥ त्रिभुनक्षेत्रस्य भुजः पञ्च प्रतिबाहरपि च सप्त धरा षट् । अन्यस्य षडअस्य टेकादिषडन्तविस्तारः ॥ ४१ ॥ मण्डलचतुष्टयस्य हि नवविष्कम्भस्य मध्यफलम् ।
षट्पश्चचतुव्योसा वृत्तत्रितयस्य मध्यफलम् ॥ ४२ ॥ धनुराकारक्षेत्रस्य व्यावहारिकफलानयनसुत्रम् ..... कृत्वेषुगुणसमासं बाणार्धगुणं शरासनं गणितम् । शरवर्गात्पश्चगुणाज्यावर्गयुतात्पदं काष्ठम् ॥ ४३ ॥
__ भत्रोदेशकः । ज्या षाविंशतिरेषा त्रयोदशेषश्च कार्मुकं दृष्टम् ।
किं गणितमस्य काठं किं वाचक्ष्वाशु मे गणक || ४४ ॥ बाणगुणप्रमाणानयनसूत्रम् ---
गुणचापकृतिविशेषात् पाहतात्पदमिषुः समुदिष्टः । शरवर्गात्पश्चगुणादना धनुषः कृतिः पदं नीवा ॥ १५ ॥
अत्रोदेशकः । भस्य धनुःक्षेत्रस्य शरोऽत्र न ज्ञायते परस्यापि ।
न ज्ञायते च मौर्वी तद्यमाचक्ष्व गणितज्ञ ॥ ११ ॥ बहिरन्तश्चतुर अकसत्तस्य व्यावहारिकफलानयनसूत्रम्
बाहो वृत्तस्येदं क्षेत्रस्य फलं त्रिसंगुणं दलितम् । अभ्यन्तरे तदर्ध विपरीते तत्र चतुर श्रे ॥ १७ ॥

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अत्रोद्देशकः । पश्चदशबाहुकस्य क्षेत्रस्याभ्यन्तरं बहिर्गणितम् । चतुरश्रस्य च वृत्तव्यवहारफलं ममाचक्ष्व ॥ ४८ ॥
इति व्यावहारिकगणितं समाप्तम् ।
अथ सूक्ष्मगणितम्. इतः परं क्षेत्रगणिते सूक्ष्मगणितव्यवहारमुदाहरिष्यामः । तद्यथा' -- आबाधावलम्बकानयनसूत्रम् -----
मुजकृत्यन्तरभूतभूसङ्कमणं त्रिबाहुकाबाधे । तद्भुनवर्गान्तरपदमवलम्बकमाहुराचार्याः ॥ ४९ ॥ सूक्ष्मगणितानयनसूत्रम्भुजयुत्यर्धचतुष्काढुजहीनाद्धातितात्पदं सूक्ष्मम् । अथवा मुरवतलयुतिदलमवलम्बगुणं न विषमचतुरश्रे ॥ ५० ॥
अत्रोद्देशकः । त्रिभुजक्षेत्रस्याष्टौ दण्डा मूर्बाहको समस्य त्वम् । सूक्ष्मं वद गणितं मे गणितविदवलम्बकाबाधे ॥ ५१ ॥ द्विसमत्रिभुजक्षेत्रे त्रयोदश स्युर्भुजद्वये दण्डाः । दश भूरस्याबाचे अथावलम्ब च सूक्ष्मफलम् ॥ १२ ॥ विषमत्रिभुजस्य भुजा त्रयोदश प्रतिभुजा तु पश्चदश । भूमिश्चतुर्दशास्य हि किं गणितं चावलम्बकाबाधे ॥ १३ ॥
I after this M adds the following :-त्रिभुजक्षेत्रस्य अपस्थितभामिसंस्पृष्ठरवाया नाम अवलम्पक: स्यात् ।
भुजद्वयसंयोगस्थानमारभ्य

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क्षेत्रगणितावहारः
117 इतः परं पश्चप्रकाराणां चतुरश्रक्षेत्राणां कर्णानयनसूत्रम्--- क्षितिहतविपरीतभुजौ मुरवगुण मुजमिश्रितो गुणच्छेदो । छेदगुणौ प्रतिभुजयोः संवर्गयुनेः पदं कर्णौ ॥ १४ ॥
अत्रोद्देशकः । समचतुरश्रस्य स्वं समन्ततः पञ्चबाहुकस्याशु । कर्ण च सूक्ष्मफलमपि कथय सरवं गणिततत्त्वज्ञ ॥ १५ ॥ आयतचतुरश्रस्य द्वादश बाहश्च कोटिरपि पश्च । कर्णः कः सक्ष्मं किं गणित चाचक्ष्व में शीराम ॥ १६ ॥ द्विममचतुर अभूमिः पत्रिशद्वाहरेकषष्टिश्र। सोऽन्यश्चतर्दशास्यं कर्णः कः सूक्ष्मगणितं किम् ॥ ५५ ॥ वर्गस्त्रयोदशानां त्रिसमचतुर्बाहकं पनभूमिः। " सप्त चतुश्शतयुक्तं कर्णावाधावलम्बगणितं किम् ॥५८ ॥ विषमचतर श्रबाहू त्रयोदशाभ्यस्तपश्चदविंशतिकी।
पक्षघना वदनमधस्त्रिशतं कान्यन कर्णमुखफलानि ॥ ५९॥ इतः परं वृत्तक्षेत्राणां सुक्ष्मफलानगनमत्राणि । तत्र समवृसक्षेत्रस्य सूक्ष्मफलानयन सूत्रम--
वृत्तक्षेत्रव्यासो दशपदणता मवेत्परिक्षपः । व्यासचतुर्भागगुणः परिधिः फलमर्धमर्धे नत् ॥ ६ ॥
अत्रांद्देशकः । समवृत्तव्यासोऽष्टादश विष्कम्मश्च पष्टिग्न्यस्य ।
शांतिपरस्त क्षेत्र। हि के च परिधिफलं ॥ ६१ ॥ बादशविष्कम्भस्थ क्षेत्रमा हि चार्धवृत्तम्य । षटुशियासस्य कः परिधिः कि फलं पति ॥ ॥ १२ ॥

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गणितसारसज्जन्दः
आयतवृत्तक्षेत्रस्य सूक्ष्मफलानयनसूत्रम्व्यासकृतिष्षणिता द्विसङ्गणायामकृतियुता (पदं) परिधिः ।
व्यास चतुर्भाग गुणश्चायतवृत्तस्य सूक्ष्मफलम् ॥ १३ ॥ अत्रोद्देशकः ।
आयतवृत्तायामः षट्त्रिंशङ्खादशास्य विष्कम्भः । कः परिधिः किं गणितं सूक्ष्मं विगणय्य मे कथय || १४ |
118
शङ्खाकारक्षेत्रस्य सूक्ष्मफलानयनसूत्रम्
वदनानो व्यासो दशपदगुणितो भवेत्परिक्षेपः । मुखदलरहितव्यासार्ध वर्गमुखचरणकृतियोगः || १५ || दशपदगुणितः क्षेत्रे कम्बुनिभे सूक्ष्मफलमेतत् ॥ ६५ ॥ अत्रोद्देशकः ।
व्यासोऽष्टादश दण्डा मुखविस्तारोऽयमपि च चत्वारः । कः परिधिः किं गणितं सूक्ष्मं तत्कम्बुकावृत्ते ॥ ६६६३ ॥ बहिश्चक्रवालवृत्तक्षेत्रस्य चान्तश्चक्रवालवृत्तक्षेत्रस्य च सूक्ष्मफलानय
नसूत्रम्—
निर्गमसहितो व्यासो दशपदनिर्गमगुणो बहिर्गणितम् । रहितोऽधिगमेनासावभ्यन्तरचक्रवालवृत्तस्य ॥ ६७ ॥
अत्रोद्देशकः ।
व्यासोऽष्टादश दण्डाः पुनर्बहिर्निर्गतास्त्रयो दण्डाः । सूक्ष्मगणितं वद त्वं बहिरन्तश्चक्रवालवृत्तस्य ॥ ६८ ॥
व्यासोऽष्टादश दण्डा अन्तः पुनरधिगताश्च चत्वारः । सूक्ष्मगणितं वद त्वं चाभ्यन्तरचक्रवालवृत्तस्य ॥ ६९ ॥
"

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क्षेत्रगणितव्यवहारः
110 यवाकारक्षेत्रस्य च धनराकारक्षेत्रस्य च सूक्ष्मफलानयनसूत्रम्इषुपादगुणश्च गुणो दशपदगुणितश्च भवति गणितफलम् । यवसंस्थानक्षेत्रे धनुराकारे च विज्ञेयम् ॥ ७० ॥
अत्रोद्देशकः । द्वादशदण्डायामो मुखद्वयं सूचिरपि च विसारः । चत्वारो मध्येऽपि च यवसंस्थानस्य किं तु फलम् ॥ ७१, । धनुराकारसंस्थाने ज्या चतुर्विशतिः पुनः । चत्वारोऽस्येपुरुद्दिष्टसूक्ष्मं किं नु फलं भवेत् ॥ ७२ ॥ पनाकारक्षेत्रस्य धनःकाष्ठवाणप्रमाणानयनमूत्रम्
शरवर्गः षड्गुणितो ज्यावर्गसमन्वितस्तु यस्तम्य ।
मूलं धनुर्गुणेषुप्रसाधने तत्र विपरीतम् ॥ ७३, ।। विपरीतक्रियाया सूत्रम्-- गुणचापकृतिविशंपात्तकहनात्पदमिपुः समुद्दिष्टः । शरवर्गात् पणितादूनं' धनुषः कृतेः पदं जीवा ।। ७४ ॥
अत्रोइंशकः । धनुराकारक्षेत्रे जपा द्वादश षट् शर: काष्ठम् । न ज्ञागते सरवे त्वं का जीवा कश्शरस्तस्य ।। ७५ ॥
मृदङ्गनिभक्षेत्रस्य च पणवाकारले स्य च वत्राकारक्षेत्रस्य च सूक्ष्मफलांनयनसूत्रम् --
मुग्वगुणितायामफलं स्वधनुःफलमंयुतं मृदङ्गनिभ ।
तत्पणववजनिपयोर्धनुःफलोनं तयोरुभयोः ॥ ६ ॥ | The reading in thern IB and Mir ki vivah in ro : Int पहाणतादनाया पनुष्कृत: wfat given the required morning

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गणितसारसहः
फलान
अत्रोद्देशकः । चतुर्विशतिरायामो विस्तारोऽष्टी मुरवद्वये । क्षेत्रे मृदङ्गसंस्थाने मध्ये षोडश किं फलम् ॥ ७० ॥ चतुर्विशतिरायामस्तथाष्टौ मुरवयोईयोः । चत्वारो मध्यविष्कम्भः किं फलं पणवाकतौ ॥ ७८६ ॥ चतुर्विंशतिरायामस्तथाष्टौ मुरवयोईयोः । मध्ये सूचिस्तथाचक्ष्व वज्राकारस्य किं फलम् ॥ ७९ ॥ नेमिक्षेत्रस्य च बालेन्डाकारक्षेत्रस्य च इभदन्ताकारक्षेत्रस्य च सूक्ष्मफलानयनसूत्रम्
पृष्ठोदरसंक्षेपः षड्भक्तो व्यासरूपसङ्गणितः ।। दशमूलगुणो नेमेर्बालेन्द्रिभदन्तयोश्च तस्यार्धम् ॥ ८ ॥
अत्रोद्देशकः । पृष्ठं चतुर्दशोदरमष्टौ नेम्याकृती भूमौ । मध्ये चत्वारि च तद्वालेन्दोः किमिभदन्तस्य ॥ ८१६ ॥ चतुर्मण्डलमध्यस्थितक्षेत्रस्य सूक्ष्मफलानयन सूत्रम्-- विष्कम्भवर्गराशेर्वृत्तस्यैकस्य सूक्ष्मफलम् । त्यक्त्वा समवृत्तानामन्तरजफलं चतुर्णा स्यात् ॥ ८२ ॥
अत्रोद्देशकः । गोलकचतुष्टयस्य हि परस्परस्पर्शकस्य मध्यस्य। । सक्ष्म गणितं किं स्याञ्चतुष्कवि'कम्भयुक्तस्य ॥ ३ ॥

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क्षेत्रगणितव्यवहारः. वृत्तक्षेत्रत्रयस्थान्योऽन्यस्पर्शनाजातस्यान्तरास्थितक्षेत्रस्य सूक्ष्मफलानपनसूत्रम
विष्कम्ममानसमकत्रिभुजक्षेत्रस्य सूक्ष्मफलम् । वृतफलार्धविहीनं फलमन्नरजं त्रयाणां स्यात् ॥ ८४ ॥
भत्रोद्देशकः । विष्कम्भचतुष्काणां वृत्तक्षेत्रत्रयाणां च ।
अन्योऽन्यस्दृष्टानामन्तरनक्षेत्रसूक्ष्मगणितं किम् ॥ ८५६ ॥ परप्रक्षेत्रस्य कर्णावलम्बकसूक्ष्मफलानपनसूत्रम् --
मुजभुजकृतिकृतिवर्गा द्वित्रित्रिगुणा यथाक्रमेणैव । श्रुत्यवलम्बकतिधनकृतयश्च षडके क्षेत्रे ॥ ८ ॥
अत्रोद्देशकः। मुजषटक्षेत्रे द्वौ हौ दण्डौ प्रतिभुजं स्याताम् ।
अस्मिन् श्रुत्यवलम्बकमूक्ष्मफलानां च वर्गाः के ॥ ७॥ वर्गस्वरूपकराणिराशीनां युतितक्यानयनस्य च नेषां वर्गस्वरूप. करणिराशीनां यथाक्रमेण परम्पवियुतिनः शेषसाख्यानयनस्य । सूत्रम्
केनाप्यपवर्तितफलपदयोगवियोगकृतिहताच्छेदात् । मूलं पदयुतिवियुती राशीनां विद्धि करगिगणितमिदम् ॥ ८॥
अत्रोद्देशकः। षोडशषट्त्रिंशच्छतकरणीनां वर्गमूलपिण्डं मे । मथ चैतत्पदशेषं कथय सरवे गणिततस्यज्ञ ॥ ८९ ॥
हात सूक्ष्मगणितं समाप्तम् ॥

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गणितसारसाहा.
जन्यव्यवहारः. इतः परं क्षेत्रगणिते जन्यव्यवहारमुदाहरिप्यामः । इष्टसजाय बीजाभ्यामायतचतुरश्रक्षेत्रानयनसूत्रम्
वर्गविशेषः कोटिसंवर्गो द्विगुणितो भवेद्बाहुः । वर्गसमासः कर्णश्चायतचतुरश्रजन्यस्य ॥ ९०॥
अत्रोद्देशकः । एकहिके तु बीजे क्षेत्रे जन्ये तु संस्थाप्य । कथय विगणय्य शीघ्रं कोटिभुजाकर्णमानानि ॥ ९१ ॥ बीजे हे त्रीणि सरेव क्षेत्रे जन्ये तु संस्थाप्य । कथय विगणय्य शीघ्रं कोटिभुजाकर्णमानानि ॥ ९२ ॥ पुनरपि बीजसंज्ञाभ्यामायतचतुर श्रक्षेत्रकल्पनायाः सूत्रम्बीजयुतिवियुतिघातः कोठिस्तद्वर्गयोश्च सङ्क्रमणे । बाहुश्रुती भवेतां जन्यविधी करणमेतदपि ॥ ९३ ॥
अत्रोद्देशकः । त्रिकपञ्चकबीजाभ्यां जन्यक्षेत्रं सरवे समुत्थाप्य । कोठिभुजाश्रुतिसङ्ख्याः कथय विचिन्त्याशु गणिततत्त्वज्ञ ॥ ९४ । इष्टजन्यक्षेत्राद्वीजमंज्ञसङ्ख्ययोरानयनसूत्रम् -
कोठिच्छेदावाप्त्योरसङ्क्रमणे बाहुदलफलच्छेदौ । बीने श्रुतीष्टकृत्योर्योगवियोगार्धमूले ते ॥ ९५ ॥
अत्रोद्देशकः । कस्यापि क्षेत्रस्य च षोडश कोटिन बीजे के। त्रिंशदथवान्यबाहुबर्षीने के ते श्रुतिश्चतुस्त्रिंशत् ।। ९६ ॥

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क्षेत्रगणितव्यवहारः. कोटिसङ्ख्या ज्ञात्वा मुजाकर्णसङ्ख्यानयनस्य च भुजसङ्ख्या ज्ञात्वा कोटिकर्णसङ्ख्यानधनस्य च कर्णसङ्ख्यां ज्ञात्वा कोठिभुजासल्यानयनस्य च सूत्रम् -- , कोटिकृतेश्छेदाप्योस्सङ्कमणे श्रुतिभुजौ भुजलतेर्वा ।। अथवा श्रुतीष्टकृत्योरन्तरपदमिष्टमपि च कोटिभुजे ॥ ९७॥
भत्रोद्देशकः । कस्यापि कोटिरेकादश बाहुष्पष्टिरन्यस्य ।
श्रुतिरेकषष्टिरन्यस्यानुक्तान्यत्र मे कथय ॥ ९८॥ द्विसमचतुर क्षेत्रस्यानयनप्रकारस्य सूत्रम्
जन्यक्षेत्रभुजार्धहारफलजप्राग्जन्यकोव्योर्यतिभूरास्यं वियुनिर्भुजा श्रुतिस्थाल्पाल्पा हि कोटिर्भवेत् । आबाधा महती श्रुतिः श्रुतिरभूज्येष्ठं फलं स्या फलं बाहुस्स्यादवलम्बको द्विसमकक्षेत्र चतुर्बाहके ।। ९९ ॥
अत्रोद्देशकः । चतुरश्रक्षेत्रस्य द्विसमस्य च पर षटूबीजस्य । मुरवभूभुजावलम्बककर्णावाधाधनानि वद ॥ १० ॥ त्रिसमचतुरश्रक्षेत्रमा मुरवभूभुनावलम्बककर्णावाधाधनानगनस्
त्रम्
मुजपदहतबीजान्तरहतजन्यधनातभागहाराभ्याम् । तनकोटिभ्या च हिसम इव त्रिसमचतुर अं ॥ १०१।।
अत्रोद्देशकः । चतुर प्रक्षेत्रस्य त्रिसमस्यास्य द्विकत्रिकस्वबीजस्य । मुरव मुभुजावलम्बककर्णाबाधाधनानि वद ॥ १०२ ।
11-A

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124
गणितसारसाहा. विषमचतुर क्षेत्रस्य मुरवभूमुजावलम्बककर्णाबाधाधनानपनसू . त्रम्
ज्येष्ठाल्पान्योन्यहीनश्रुतिहतभुजकोठी मुजे भुमुरवे ते कोयोरन्योन्यदोभ्या हतयुतिरथ दोर्घातयुक्कोठिघातः । कर्णावल्पश्रुतिघ्रावनधिकभुजकोट्याहतौ लम्बको तावाबाधे कोटिदो ववनिविवरके कर्णघातार्धमर्थः ॥ १०३ ॥
अत्रोद्देशकः । एकहिकहिकत्रिकजन्ये चोत्थाप्य विषमचतुरश्रे। मुरव पुजाबलम्बककर्णाबाधाधनानि वद ॥ १०४ ॥ पुनरपि विषमचतुरश्रानयनसूत्रम्---- द्वस्व श्रुतिकृतिगुणितो ज्येष्ठभुजः कोटिरपि धरा वदनम् । कर्णाभ्यां सङ्गुणितावुभयभुजावल्पभुजकोटी ॥ १०५६ ॥ ज्येष्ठभुजकोटिवियुनिधिाल्पमुजकोटिताडिता युक्ता । हस्वभुजकोठियुतिगुणपृथुकोव्याल्पश्रुतिनको कर्णौ ॥ १० ॥ अल्पश्रुतिहतकर्णाल्पकोटिभुजसंहती पृथग्लम्बौ । तगुजयुतिवियुतिगुणात्पदमाबाधे फलं श्रुतिगुणार्धम् ॥ १०७ ।। एकस्माजन्यागतचतुरश्राहिसमात्रभुजानगनसूत्रम् -- कणे मुजद्वयं स्याद्वाहुढिगुणीकृतो भवभूमिः । कोटिरवलम्बकोऽयं द्विसमत्रिभुजे धनं गणितम् ॥ १० ॥
अत्रोद्देशकः । त्रिकपश्चकबीजोत्थद्विसमत्रिभुजस्य गणक जाहू दो। भूमिमवलम्बकं च प्रगणय्याचक्ष्व मे शीघ्रम् ।। १०९ ।।

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क्षेत्रगणितव्यवहारः.
विषमत्रिभुजक्षेत्रस्य कल्पनाप्रकारस्य सूत्रम्-जन्यभुजार्थं छित्वा केनापिच्छेदलब्धजं चाभ्याम् । कोटियुतिर्भूः कर्णौ भुजौ भुजा लम्वका विषमे ॥ ११० ॥ अत्रोद्देशकः ।
हे द्वित्रिबीजकस्य क्षेत्रभूजार्थेन चान्यमुत्थाप्य । तस्माद्विषमत्रिभुजे भुजभूम्यवलम्बकं ब्रूहि ॥ १११ ॥
इति जन्यव्यवहारः समाप्तः ॥
पैशाचिक व्यवहारः.
इतः परं पैशाचिकव्यवहारमुदाहरिष्यामः ।
समचतुरश्रक्षेत्रे वा आयतचतुरश्रक्षेत्रे वा क्षेत्रफले रज्जुसङ्ख्या समे सति, क्षेत्रफले बाहुसङ्ख्या समे सति क्षेत्रफले कर्णसङ्खचया समे सति, क्षेत्रफले रज्ज्वर्धसङ्खचया समे सति, क्षेत्रफले वाहोस्तृतीयांशसङ्ख्या समे सति, क्षेत्रफले कर्णसङ्ख्यायाश्चतुर्थांशसङ्ख्या समे सति, द्विगुणित कर्णस्य त्रिगुणितवाहोश्च चतुर्गुणितकोटेश्व रज्जो संयोगसङ्ख्यां द्विगुणीकृत्य तद्द्द्दिगुणितसङ्ख्या क्षेत्रफले समाने सति, इत्येव - मादीना क्षेत्राणां कोटिभुजा कर्णक्षेत्रफलरज्जुषु इष्टराशिद्वयसाम्यस्य चेष्टराशिद्वयस्यान्यमिष्टगुणकारगुणित फलवत् क्षेत्रस्य भुजाकोटिसह ख्यानयनस्य सूत्रम् -
स्वगुणेष्टेन विभक्तास्वेष्टानां गणक गणितगुणितेन । गुणिता मुना भुजाः स्युः समचतुरश्रादिजन्यानाम् ॥। ११२३ ॥
अत्रोद्देशकः ।
125
रज्जुर्गणितेन समा समचतुरश्रस्य का तु मुजसङ्ख्या । अपरस्य बाहुसदृशं गणितं तस्यापि मे कथय ॥ ११३ ॥

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126
गणितसारसङ्ग्रहः.
कर्णो गणितेन समः समचतुरश्रस्य को भवेद्वाहुः । रज्जुर्द्विगुणोऽन्यस्य क्षेत्रस्य धनाच्च मे कथय ॥ ११४ ॥ आयतचतुरश्रस्य क्षेत्रस्य च रज्जुतुल्यमिह गणितम् । गणितं कर्णेन समं क्षेत्रस्यान्यस्य को बाहुः || १९९३ ॥ कस्यापि क्षेत्रस्य त्रिगुणो बाहुर्धनाच्च को बाहु: । कर्णश्वतुर्गुणोऽन्यः समचतुरश्रस्य गणितफलात् ॥ ११६ ॥ आयतचतुरश्रस्य श्रवणं द्विगुणं त्रिसङ्गुणो बाहुः । कोटिश्चतुर्गुणा तै रज्जुयुतैर्द्विगुणितं गणितम् ॥ ११७ ॥ आयतचतुरश्रस्य क्षेत्रस्य च रज्जुरत्र रूपसमः । कांटिः को बाहुर्वा शीघ्रं विगणय्य मे कथय ॥ ११८ ॥ कणों हिगुणो बाहुत्रिगुण. कोटिचतुर्गणा मिश्रः । रज्वा सह तत्क्षेत्रस्यावतचतुरश्रकस्य रूपसमः || १९९३ ॥ पुनरपि जन्वायतचतुरश्रक्षेत्रस्य बीज सङ्ख्यानयने करणसूत्रम् -- कोट्यून कर्णदलतत्कर्णान्तरमुभययोश्ध पदे । आयतचतुरश्रस्य क्षेत्रस्येयं क्रिया जन्ये ॥ १२० ॥ अत्रोद्देशकः ।
आयतचतुरश्रस्य च कोटिः पञ्चाशदधिकपव भुजा । साष्टाचत्वारिंशत्रिसप्ततिः श्रुतिरथात्र के बीजे ॥ १२१ ॥ इष्टकल्पितसङ्ख्याप्रमाणवत्कर्णसहितक्षेत्रानयनसूत्रम् - पद्यत्क्षेत्रं जातं बीजैस्संस्थाप्य तस्य कर्णेन । इष्टं कर्णं विभजेल्लाभगुणाः कोठिदोः कर्णाः ॥ १२२३ ॥ अत्रोद्देशकः ।
एकद्विकाहिक त्रिकचतुष्कसप्तैकसाष्टकानां च । गणक चतुर्णां शीघ्रं बीजैरुत्थाप्य कोठिभुजाः ॥ १२३ ॥

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क्षेत्रगणितव्यवहारः.
आयतचतुरश्राणां क्षेत्राणां विषमबाहुकानां च । कर्णोऽत्र पञ्चषष्टिः क्षेत्राण्याचक्ष्व कानि स्युः ॥ १२४ ॥ इष्टजन्यायतचतुरश्रक्षेत्रस्य रज्जुसां च कर्णसङ्ख्यां च ज्ञात्वा तज्जम्यायतचतुरश्रक्षेत्रस्य भुजकोटिसङख्यानयनसूत्र
कर्णकृतौ द्विगुणायां रज्वर्धकृति विशोध्य तन्मूलम् । रज्वर्थे सङ्क्रमणीकृते भुजा कोटिरपि भवति ॥ १२१ ॥ अत्रोद्देशकः ।
परिधिः स चतुस्त्रिंशत् कर्णश्चात्र त्रयोदशो दृष्टः । जन्यक्षेत्रस्यास्य प्रगणय्याचक्ष्व कोटि भुजौ || १२६ ॥ क्षेत्रफलं कर्णसख्यां च ज्ञात्वा भुजकोटिसङख्यानयनसूत्रम्-कर्णकृतौ द्विगुणीकृतगणितं हीनाधिकं कृत्वा । मूलं कोटिभृजौ हि ज्येष्ठ इखेन मङ्क्रमणे ॥ १२७ ॥ अत्रोद्देशकः । आयतचतुरश्रस्य हि गणितं पष्टिस्त्रयोदशास्यापि ।
कर्णस्तु कोटिभुजयोः परिमाणं श्रोतुमिच्छामि ॥ १२८ ॥ क्षेत्रफलसख्यां रज्जुसख्यां च ज्ञात्वा आयतचतुरश्रस्य भुजकोटिसख्यानयनसूत्रम्-
रज्ज्वर्धवर्गराशेर्गणितं चनुराहतं विशोध्याय ।
मूलन हि रज्ज्वर्थे सङ्क्रमणे सति भुजाकोटी ।। १२९ ॥
अत्रोद्देशकः ।
सप्ततिशतं तु रज्जुः पश्वशतोत्तरसहस्त्र मिष्टधनम् । नन्यायतचतुरश्रे कोटिभुजौ में समाचक्ष्व ॥ १३० ॥
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198
गणितसारसाहः. आयतचतुरश्रक्षेत्रद्वये रज्जुसङ्ख्यायां सदृक्षायां सत्यां द्वितीयक्षेत्रफलात् प्रथमक्षेत्रफले द्विगुणिते सति, अथवा क्षेत्रद्वयेऽपि क्षेत्रफले सदशे सति प्रथमक्षेत्रस्प रज्जुसङ्ख्याया अपि द्वितीयक्षेत्ररज्जुसङ्ख्यायां द्विगुणायां सत्यम्, अथवा क्षेत्रद्वये प्रथमक्षेत्ररज्जुसङ्ख्याया अपि द्वितीयक्षेत्रस्य रज्जुसङ्ख्यायां द्विगुणायां सत्यां द्वितीयक्षेत्रफलादपि प्रयमक्षेत्रफले द्विगुणे सति, तत्तत्क्षेत्रद्वयस्थानयनसूत्रम्--
स्खाल्पहतरज्जुधनहतकृतिरिष्ट त्रैव कोटिस्स्यात् । व्येका दोस्तुल्यफलेऽन्यत्राविकगणितगुणितेष्टम् ॥ १३१ ।। व्येकं तदूनकोटिः त्रिगुणा दोः पादथान्यस्य । रज्ज्वर्धवर्गराशेरिति पूर्वोक्तेन सूत्रेण । तद्गणितर ज्जमितितः समानयेत्तगुजाकोटी ॥ १३३ ॥
___अत्रोद्देशकः । असमव्यासायामक्षेत्रे द्वे द्वावथेष्टगुणकारः । प्रथमं गणितं द्विगुणं रज्जू तुल्ये किमत्र कोटिभुजे ॥ १३४ ॥ आयतचतुर श्रे हे क्षेत्रे द्वयमेव गुणकारः । गणितं सदृशं रजढिगणा प्रथमात् द्वितीयस्य ॥ १३५ ॥ आयतचतुरश्रे द्वे क्षेत्रे प्रथमस्य धनमिह द्विगुणम् । द्विगुणा द्वितीयरज्जुस्तयोर्भुजां कोठिमपि कथय ॥ १३ ॥ द्विसमत्रिभुजक्षेत्रयोः परस्पररज्जधनसमानसङ्ख्ययोरिष्टगुणकगुणितरजुषनवतो; द्विसमत्रिभुजक्षेत्रद्वयानयनसूत्रम्
रज्जुकृतिनान्योन्यधनाल्पाप्तं षद्विघमल्पमेकोनम् । तच्छेषं द्विगुणास्पं बीजे तज्जन्ययोर्भुजादयः प्राग्वत् ॥ १३७ ॥
अत्रोद्देशकः । द्विसमत्रिभुनक्षेत्रहयं तयोः क्षेत्रयोस्समं गणितम् । रज्जू समे तपोस्स्यात् को बाहुः का भवेद्रूमिः ॥ १३ ॥

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क्षेत्रगणितव्यवहारः.
129 द्विसमत्रिभुजक्षेत्रे प्रथमस्य धनं द्विसङ्गणितम् । . रज्जुः समा द्वयोरपि को बाहुः का भवेद्भमिः ॥ १३९॥
द्विसमत्रिभुजक्षेत्रे हे रज्जदिगणिता द्वितीयस्य । • गणिते द्वयोस्समाने को बाहः का भवेद्भूमिः ।। १४० ॥ द्विसमत्रिभुजक्षेत्रे प्रथमस्य धनं हिसङ्गणितम् । द्विगुणा द्वितीयरज्जुः को बाहुः का भवेमिः ।। १४१ ।।
एकट्यादिगणनातीतसङ्ख्यासु इष्टसण्यामिष्टवस्तुनो भाग - सयां परिकल्प्य तदिष्टवस्तुभागसङ्ख्यायाः सकाशात् समचतुर श्रक्षेत्रानयनस्य च समवृत्तक्षेत्रानयनस्य च समत्रिभुजक्षेत्रानानस्य चायत. चतुर क्षेत्रानयनस्य च सूत्रम
वसमीकृतावधृतिहतवनं चतुघ्नं हि वृत्तसमचतुर श्रयासः । षड्गुणितं त्रिभुजायतचतुर श्रभुजार्धमपि कोटिः ॥ ११२ ॥
अत्रौद्देशकः । स्वान्तःपुर नरेन्द्रः प्रासादतले निजाननामध् । दिव्यं स रनकम्बलमपीपतत्तच्च समवृत्तम् ॥ १४३ ।। ताभिर्देवीभिधृतमभिर्मुजयोश्च मुष्टिभिर्लब्धम् । पञ्चदशैकस्याः स्युः कति वनिताः कोऽत्र विष्कम्भः ।। १४ ४ ॥ समचतुर श्रभुजाः के समत्रिबाही भुजाधात्र ।
आयतचतुरश्रस्य हि तत्कोटिभुजो सरख कथय ।। १४५॥
क्षेत्रफलसङ्ख्या ज्ञात्वा समचतुर क्षेत्रानयनस्य चायतचतुरश्रक्षेत्रानयनस्य च सूत्रम्
सूक्ष्मगणितस्य मूलं समचतुरश्रस्य बाहुरिष्टहतम् । धनमिष्टफले स्यातामायतचतुरश्रकोटिमुजौ ॥ १४ ॥

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गणितसारसबहः.
अत्रोद्देशकः । कस्य हि समचतुरश्रक्षेत्रस्म फलं चतुष्षष्टिः । फलमायतस्य सूक्ष्मं षष्टिः के वात्र कोटिभुजे ॥ १४७ ।।
इष्टद्विसमचतुर क्षेत्रस्य सूक्ष्मफलसङ्ख्यां ज्ञात्वा, इष्टसख्यां गुणकं परिकल्प्य, इष्टसख्याङबीजाभ्यां जन्याचतचतुर श्रक्षेत्रं परिकल्प्य, तदिष्टदिसमचतुरश्रक्षेत्रफलवादिष्टद्विसमचतुर श्रानयनसूत्रम् --
तडनगुणितष्टकतिर्जन्यधनोना भुजाहता मरवं कोटिः ।। द्विगुणा समुरवा मुदलिम्वः कर्णो भने ादिष्टहनाः ॥ १४८ ॥
अत्रोद्देशकः। सूक्ष्मधनं सप्तेष्टं त्रिकं हि बीजे द्वि के त्रिक दृष्टे । हिसमचतुरश्रबाहू मुरवभूम्यवलम्बकान् हि ।। १४९ ।। इष्टसूक्ष्मगणितफलवत्रिसमचतुर क्षत्रान नसूत्रम् ... इष्टघनभक्तधनकृतिरिष्टयुताधं भुजा द्विगुणितष्टम । विभुजं मुरवामिष्टाप्तं गणितं ह्यवलम्बकं त्रिसमजन्य ।। १५ ॥
अत्रोद्देशकः । कस्यापि क्षेत्रस्य त्रिसमचतुर्बाहुकस्य सूक्ष्मधनम् । षण्णवतिरिष्टमष्टौ भूवाहमुरवावलम्बकानि वर ॥ १५१ ॥
सूक्ष्मफलसङ्ख्यां ज्ञात्वा चतुर्भिरिष्टच्छेदैश्च विषमचतुर क्षेत्रस्य मुरवभूमुजाप्रमाणसङ्ख्यानयनसूत्रम् धनकृतिरिष्टच्छेदै श्रतुभिराप्तैव लब्धानाम् । युतिदलचतुष्टयं तैरूना विषमाख्यचतुर श्रभुजसङ्ख्या ॥ १५२ ॥
अत्रोद्देशकः । नवतिर्हि सूक्ष्मगणितं छेदः पञ्चैव नवगुणः । दशधृतिविंशतिषतिहतः क्रमाद्विषमचतुरश्रे॥ मुरवभूमिमुनासङ्ख्या विगणय्य ममाशु सङ्कथय ।। १५३॥

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क्षेत्रगणितव्यवहारः.
131 सूक्ष्मगणितफलं ज्ञात्वा तत्सूक्ष्मगणितफलवत्समत्रिबाहुक्षेत्रस्य बाहुसयानयनसूत्रम्
गणितं तु चतुर्गुणितं वर्गीकृत्वा' भजेत् त्रिभिर्लब्धम् । . त्रिभुजस्य क्षेत्रस्य च समस्य बाहाः रनवर्गः ॥ १५४ ॥
अत्रोद्देशकः । कस्यापि समध्य क्षेत्रस्य व गणितमुद्दिष्टम् ।
रूपाणि त्रीण्येव ब्रूहि प्रगणय्य मे बाहुम् ॥ १५५ ॥ सूक्ष्मगणितफलसख्यां ज्ञात्वा तत्सूक्ष्मगणितफलवाहसमत्रिबाहु. क्षेत्रस्य मुजभूम्यवलम्बकसला नयनसतम
इच्छाप्तवनेच्छाकृति(तिमूलं दोः क्षितिदिगणतंच्छा। इच्छाप्तधनं लम्बः क्षत्रे हिसमविवाह जन्ये स्यात् ॥ १५६ ॥
अत्रोद्देशकः । कस्यापि क्षेत्रस्य हिरामत्रिभुजम्य सूक्ष्मगणितमिनाः । त्रीणीच्छा कथय सरव भुजम्मवलम्बकानाशु ॥ १५७ ॥
सूक्ष्मगणितफलसळ्यां ज्ञात्वा तत्सूक्ष्मगणितफलवद्विपमत्रिभुजानयनस्य सूत्रम्
भष्टगुणितेष्टकृतियुतधनपदघनमिष्टपद हदिष्टार्थम् । भूः स्यानं द्विपदाहनेष्टवर्ग भुजे च सङ्कमणम् ।। १५८ ॥
अत्रोद्देशकः । . कस्यापि विषमवाहाम्य अक्षत्रस्य सूक्ष्मगणितमिदम् । हे रूपे निर्दिष्टे त्रीणीष्टं मिबाहवः के स्युः ॥ ११९ ॥ पुनरपि सूक्ष्मगणितफलसख्या ज्ञात्वा तत्फलवहिषमत्रिमुजानयनसूत्रम्
'वीकृत्वा ought to be वर्गीकृत्य : but this form will not suit the require. ments of the metre.

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गाणतसारसग्रहः. खाष्टहतात्सेष्टकतेः कृतिमूलं चेष्टमितरदितरहतम् । ज्येष्ठं स्वल्पा!नं स्वल्पार्ध तत्पदेन चेष्टेन ।। १६० ॥ क्रमशो हत्वा च तयोः सङ्कमणे भूभुजौ भवतः । इष्टार्थमितरदोः स्याद्विषमत्रकोणके क्षेत्रे ॥ १६१ ॥ '
__ अत्रोद्देशकः । हे रूपे सूक्ष्मफलं विषमत्रिभुजस्य रूपाणि ।
त्रीणीष्टं भूदोषो कथय सरवे गणिततत्त्वज्ञ ॥ १६२ ।। सूक्ष्मगणितफलं ज्ञात्वा तत्सूक्ष्मगणितफलवत्समवृत्तक्षेत्रानयनत्रम्
गणितं चतुरभ्यस्तं दशपदभक्तं पदे भवेद्यासः । सूक्ष्मं समवृत्तस्य क्षेत्रस्य च पूर्ववत्फलं परिधिः ॥ १६३ ॥
__ मत्रोद्देशकः । समवृत्तक्षेत्रस्य च सूक्ष्मफलं पञ्च निर्दिष्टम् । विष्कम्भः को वास्य प्रगणय्य ममाशु तं कथय ॥ १६४६ ।।
व्यावहारिकगणितफलं च सूक्ष्मफलं च ज्ञात्वा तयावहारिकफलवतत्सूक्ष्मगणितफलवहिसमचतुरश्रक्षेत्रानयनस्य त्रिसमचतुरश्रक्षेत्रानयनस्य च सूत्रम
धनवर्गान्तरपदयुतिवियुतीष्टं भमुरवे भुजे स्थूलम् । हिसमे सपदस्थूलात्पदयुतिवियुतीष्टपदहतं त्रिसमे ॥ १६५३ ॥
अत्रोद्देशकः । गणितं सूक्ष्म पञ्च त्रयोदश व्यावहारिकं गणितम् । द्विसमचतुरश्रममुरवदोषः के षोडशेच्छा च ॥ ११९॥

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क्षेत्रगणितव्यवहारः.
त्रिसमचतुर श्रस्योदाहरणम्। गणितं सूक्ष्मं पश्च त्रयोदश व्यावहारिकं गणितम् । त्रिसमचतुरश्रबाहून सश्चिन्त्य सरवे ममाचक्ष्व ।। १६७ ॥ व्यावहारिकस्थूलफलं सूक्ष्मफलं च ज्ञात्वा तद्यावहारिकस्थूलफलवत्सूक्ष्मगणितफलवत्समत्रिभुजानयनस्य च समवृत्तक्षेत्रव्यासानयनस्य च सूत्रम्
धनवर्गान्तरमूलं यत्तन्मूलाट्विसङ्गणितम् । बाहुस्त्रिसमत्रिभुने समस्य वृत्तस्य विष्कम्भः ॥ १६८॥
अत्रोद्देशकः । स्थूलं धनमष्टादश सूक्ष्मं त्रिघनो नवाहतः करणिः । . विगणय्य सम्वे कथय त्रिसमत्रिभुजप्रमाणं मे ॥ ११९ ॥ पश्चकृतेर्वों दशगणितः करणिर्भवदिदं सूक्ष्मम । स्थूलमपि पश्चसप्ततिरेतको वृत्तविकम्भः ॥ १७० ॥
व्यावहारिकस्थूलफलं च सूक्ष्मगणितफलं च ज्ञात्वा तद्यावहारिकफलवत्तत्सूदमफलवडिसमत्रिभुजक्षेत्रस्य भभजाप्रमाणसङ्ख्ययोरानयनस्य सूत्रम्
फलवर्गान्तरमूलं द्विगुणं भावहारिकं बाहः । मम्यर्धमूलभक्ते द्विसमत्रिभुजस्य करणमिदम् ।। १७१ ॥
अत्रोदेशकः । मूल्मधनं षष्टिरिह स्थूलधनं पचपष्टिरुद्दिष्टम् । गणयित्वा हि सरवे हिसमत्रिभुजस्य पुजसङ्ख्याम ।। १७२ ॥

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गणितसारसाहः. इष्टसङ्ख्यावाविसमचतुरश्रक्षेत्रं ज्ञात्वा तट्विसमचतुर क्षेत्रस्य सू. क्ष्मगणितफलसमानसूक्ष्मफलवदन्यद्विसमचतुरश्रक्षेत्रस्य भभुजमुरवसा - ख्यानयनसूत्रम्लम्बकताविष्टेनासमसङ्कमणीकृते भुजा ज्येष्ठा । हस्वयुतिवियुति मुरव भयुतिदलितं तलमुरवे द्विसमचतुरश्रे॥ १७३ ॥
भत्रोद्देशकः । भूरिन्द्रा दोर्विश्वे वक्रं गतयोऽवलम्बको रवयः । इष्टं दिक् सूक्ष्मं तत्फलवडिसमचतुरश्रमन्यत् किम् ॥ १७४ ॥
द्विसमचत्र क्षेत्रव्यावहारिकस्थूलफलसख्यां ज्ञात्वा तद्यावहारिकस्थूलफले इष्टसङ्ख्याविभागे कृते सति तद्विसमचतुर क्षेत्रमध्ये तत्तदागस्य मूमिसङ्ख्यानयनेऽपि तत्तत्स्थानावलम्बकसहख्यानयनेऽपि सू. त्रम् ---- रवण्डयुतिभक्ततलमुवि कत्यन्तरगुणितरवण्डमुरववर्गयुतम् । मूलमधस्तलमुखयुतदलहतलब्धं च लम्बकः क्रमशः ॥ १७५॥
अत्रोद्देशकः । वदनं सप्तोक्तमधः क्षितिस्त्रयोविंशतिः पुनस्त्रिंशत् । बाहू द्वाभ्यां भक्तं चैकैकं लब्धमत्र का भूमिः ॥ १७६॥ ममिर्दिषष्ठिशतमथ चाष्ठादश वरनमत्र सन्दृष्टम् । लम्बश्चतुश्शतीदं क्षेत्रं भक्तं नरेश्चतुर्भिश्च ॥ १७७ ॥ एकद्विकत्रिकचतुःरवण्डान्यकैकपुरुषलब्धानि । प्रक्षेपतया गणितं तलमप्यवलम्बकं ब्रूहि ॥ १७८॥ भूमिरशीतिर्वदनं चत्वारिंशञ्चतर्गणा षष्टिः । अवलम्बकप्रमाणं त्रीण्यष्टौ पञ्च रवण्डानि ॥ १७९ ।।

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क्षेत्रगणितव्यवहारः.
135 स्तम्भहयप्रमाणतयां ज्ञात्वा तत्स्तम्भवयाग्रे सूत्रद्वयं बद्धा तत्सू. त्रद्वयं कर्णाकारेण इतरेतरस्तम्भमूलं वा तत्स्तम्भमूलमतिक्रम्य वा संस्पृ. श्य तत्कर्णाकारसूत्रद्वयस्पर्शनस्थानादारभ्य अधःस्थितभूमिपर्यन्तं तन्मध्ये एक सूत्रं प्रसार्य तत्सूत्रप्रमाणसङ्ख्यैव अन्तरावलम्बकसंज्ञा भवति । अन्तरावलम्बकस्पर्शनस्थानादारभ्य तस्यां भूम्याम्भवपार्श्वयोः कर्णाकारसूत्रद्वयस्पर्शनपर्यन्तमावाधाप्तंज्ञा स्यात् । तदन्तररावलम्बकसङ्ख्यानय. नस्य माबाधासख्यानयनस्य च सुत्रम -----
स्तम्भौ रज्ज्वन्तर भूहती स्खयोगाहनौ च भगुणितौ । आवाधे ने वामप्रक्षेपगुणोऽन्तरवलम्बः ॥ १८० ॥
___ अत्रोद्देशकः । षोडशहनोच्छायो स्तम्भव पनिश्च पाडशोद्दिष्टौ ।
आबाधान्तरसहरू पामत्राप्यवलम्बकं वार्ड ॥ १८१ ॥ स्तम्भकस्योच्छ्रापः षट्त्रिंशदिशा मदिनीयस्य । मिदिश हस्ताः काचाधा कोऽयमवलम्ब ॥ १८२ ॥ द्वादश च पणश च सम्पान्न भूमिगपि च चत्वारः । द्वादशकस्तम्भापादजः पनि गन्यतो मला ।। १८३ ।। आक्रम्य चतुर्ह नापास्य मूलं तथैक :म्नाञ्च । पतिनाग्रात्काबाधा को मिन्नवलम्बको भवति ॥ १८४ ॥ बाहुप्रतिबाह हो त्रयोदशानियिं चतुश च । वदनेऽपि चतुर्हस्ताः कावाधा का लरावलम्बश्च ।। १८५ ॥ क्षेत्रमिदं मुरव भूम्योरेककोनं पास्पगग्राञ्च । रज्जुः पतिता मूलात्वं ब्रह्मवलम्बकाबाधं ॥ १८ ॥

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गणितसारसाहः. बाहुपयोदशैकः पञ्चदश प्रतिभुजा मुरवं सप्त । भूमिरियमेकविंशतिरस्मिन्नवलम्बकाबाधे ॥ १८॥ समचतुर प्रक्षेत्रं विंशतिहस्तायतं तस्य । कोणेभ्योऽश चतुभ्यो विनिर्गता रज्जवस्तत्र ॥ १८ ॥ मुजमध्यं द्वियुगभुजे' रज्जुः का स्यात्सुसंवीता ।
को वावलम्बकः स्यादाबाधे केऽन्तरे तस्मिन् ॥ १८९६ ॥ स्तम्भस्योन्नतप्रमाणसङ्ख्या ज्ञात्वा तस्मिन् स्तम्भे येनकेनचित्कार. णेन भग्ने पतिते सति तत्स्तम्भारमूलयोर्मध्ये स्थितौ भूसङ्ख्यां ज्ञात्वा तत्स्तम्भमूलादारभ्य स्थितपरिमाणसङ्ख्यानयनस्य सूत्रम्- .
निर्गमवर्गान्तरमितिवर्गविशेषस्य यद्भवेदर्धम् । निर्गमनेन विभक्तं तावस्थित्वाय भमः स्यात् ॥ १९०९ ॥
___अत्रोद्देशकः । स्तम्भस्य पञ्चविंशतिरुच्छ्रायः कश्रिदन्तरे भग्नः । स्तम्भारमूलमध्ये पच स गत्वा कियान् भनः ॥ १९१ ॥ वेगृच्छ्राये हस्ताः सप्तकृतिः कश्चिदन्तरे भग्नः । भूमिश्च सैकविंशतिरस्य स गत्वा कियान् मनः ॥ १९२ ॥ वृक्षोच्छ्रायो विंशतिरग्रस्थः कोऽपि तत्फलं पुरुषः । कर्णाळत्या व्यक्षिपदथ तरूमूलस्थितः पुरुषः ।। १९३ ॥ तस्य फलस्वाभिमुरवं प्रतिभुनरूपेण गत्वा च । फलमग्रहीञ्च तत्फलनरयोर्गतियोगतङ्ख्यैव ॥ १९४३॥ पञ्चाशदभूत्तत्फलगतिरूपा कर्णसङ्ख्या का । तहसमूलगतनरगतिरूपा प्रतिभुजापि कियती स्यात् ॥ १९५॥
'भुजचतुर्ष च 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.

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क्षेत्रगणितव्यवहारः. ज्येष्ठस्तम्भसङ्ख्यां च अल्पस्तम्भसङ्ख्यां च ज्ञात्वा उभयस्त. म्भान्तरभूमिसङ्ख्यां ज्ञात्वा तज्ज्येष्ठसङ्ख्ये भग्ने सति ज्येष्ठस्तम्भाग्रे अस्पस्तम्भाग्रं स्टशति सति ज्येष्ठस्तम्भस्य भनसङ्ख्यानयनस्य स्थित. शेषसङ्ख्यानयनस्य च सूत्रम्
ज्येष्ठस्तम्भस्य कृतेर्हस्वावनिवर्गयुतिमपोह्यार्धम् । स्तम्भविशेषेण हृतं लब्धं भग्नोन्नतिर्भवति ॥ १९६ ॥
अत्रोद्देशकः । स्तम्भः पञ्चोच्छ्रायः परस्त्रयोविंशतिस्तथा ज्येष्ठः । मध्यं द्वादश भग्नज्येष्ठाग्रं पतितमितराग्रे ॥ १९७ ॥
आयतचतुर श्रक्षेत्रकोटिसङ्ख्यायास्तृतीयांशद्वयं पर्वतोत्सेधं परि. कल्प्य तत्पर्वतोत्सेधसङ्ख्यायाः सकाशात् तदायतचतुरश्रक्षेत्रस्य भुज. सङ्ख्यानयनस्य कर्णसङ्ख्यानयनस्य च सूत्रम् -..
गिर्युत्संधो द्विगुणो गिरिपुरम यक्षितिगिररर्धम् । गगनं तत्रोत्पतितं गिर्यर्धव्याससंयुतिः कर्णः ॥ १९८ ॥
अत्रोद्देशकः । षड़योजनो शिरवरिणि यतीश्वरी तिष्ठतस्तत्र । एकोऽनिचर्ययागात्तत्राप्याकाशचार्यपरः ।। १९९ ॥ श्रुतिवशमुत्पत्य पुरं गिरिशिग्वरान्मूलमवरुह्यान्यः । समगतिको सातौ नगरव्यासः किमुत्पतितम् । २०० ॥
डोलाकारक्षेत्रे सम्भद्वयस्य वा गिरिद्वयस्य वा उत्सेधपरिमाण. सख्यामेव आयतचतुरश्रक्षेत्रहये भुजद्वयं परिकल्प्य तद्विग्हियान्तर. मुम्यां वा नस्तम्भद्यान्नर पूम्यां वा आबाधाहगं परिकल्प्य तदाबावा.
12

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गणितसारसग्रहः. इयं व्युत्क्रमेण निक्षिप्य तद्युत्क्रमं न्यस्ताबाधाद्वयमेव आयतचतुरश्रक्षेत्र हये कोटिद्वयं परिकल्प्य तत्कर्णद्वयस्य समानसङ्ख्यानयनसूत्रम्
डोलाकारक्षेत्रस्तम्भद्वितयोर्ध्वसङ्ख्ये वा। शिखरिद्वयोर्ध्वसङ्ख्ये परिकल्प्य भुजद्वयं त्रिकोणस्य ।। २०१३॥ तहोतियान्तरगतभूसङ्ख्यायास्तदाबाधे । आनीय प्राग्वत्ते व्युत्क्रमतः स्थाप्य ते कोटी ॥ २०२ ।। स्यातां तस्मिन्नायतचतुरश्रक्षेत्रयोश्च तद्दोभ्याम् । कोठिभ्यां कर्णौ द्वौ प्राग्वत्स्याता समानसङ्ख्यौ तौ ॥ २०३ ॥
अत्रोद्देशकः । स्तम्भस्त्रयोदशैकः पश्चदशान्यश्चतुर्दशान्तरितः । रज्जुर्बदा शिरवरे भूमीपतिता क' आबाधे ।। ते रज्जू समसये स्यातां तद्रजुमानमपि कथय ॥ २० ॥ द्वाविंशतिरुत्सेधो गिरेस्तथाष्टादशान्यशैलस्य । विंशतिरुभयोर्मध्ये तयोश्च शिरवयोस्स्थितौ साधू ॥ २० ॥ आकाशचारिणौ तौ समागतौ नगरमत्र भिक्षायै । समगतिको सञ्जातौ तत्राबाधे कियत्सङ्ख्ये ॥ समगतिसङ्ख्या कियती डोलाकारेऽत्र गणितज्ञ ॥ २०७४ ।। विंशतिरेकस्योन्नतिरद्रेश्च जिनास्तथान्यस्य । तन्मध्यं द्वाविंशतिरनयोरयोश्च शृङ्गयोः स्थित्वा ॥ २०८॥
आकाशचारिणौ द्वौ तन्मध्यपुरं समायाती। भिक्षायै समगतिको स्यातां तन्मध्यशिरवरिमध्यं किम् ॥ २०९॥
विषमत्रिकोणक्षेत्ररूपेण हीनाधिकगतिमतोर्नरयोः समागमदिन. सयानयनसूत्रम
1. 31 is grammatioally incorrect since there onn be no candhi hotwroon in the dual number and a ; vide footnoto on page 136.

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क्षत्र गाणतव्यवहारः.
दिनगतिकृतिसंयोगं दिनगतिकृत्यन्तरेण हत्वाथ । हत्वोदग्गतिदिवसैस्तल्लब्धदिने समागमः स्यान्नोः ॥ २१० ॥
•
अत्रोद्देशकः ।
योजने प्रयाति हि पूर्वगतिस्त्रीणि योजनान्यपरः ।
उत्तरतो गच्छति यो गत्वासौ तद्दिनानि पञ्चाथ ॥ २११ ॥ गच्छन् कर्णाकृत्या कतिभिर्दिवसैर्नरं समाप्नोति । उभयोर्युगपद्गमनं प्रस्थानदिनानि सदृशानि ॥ २१२ ॥ पश्चविधचतुरश्रक्षेत्राणां च त्रिविधत्रिकोणक्षेत्राणां नेत्यष्टविधबाह्य
103
वक्तव्यास सख्यानयनसूत्रम् ----
श्रुतिरवलम्बकभक्ता पार्श्वशुजन्ना चतुर्भुजे त्रिभुजे । भुजघातो लम्बहृतो भवेद्बहिर्वृत्तविष्कम्भः ॥ २१३३ ॥ अत्रोद्देशकः ।
समचतुरश्रस्य त्रिकबाहुप्रतिबाहुकस्य चान्यस्य । कोटिः पच द्वादश भुजास्य किं वा बहिर्वृत्तम् || २१४ ॥ बाहू त्रयोदश मुखं चत्वारिं धरा चतुर्दश प्रोक्ता । द्विसमचतुरश्र बाहिरविष्कम्भः को भवेद ।। २९९ ।। पचकृतिर्वदनमुनाश्रत्वारिंशम् भूमिरेकोना । त्रिसमचतुरश्रबाहिरवृत्तव्यासं ममाचक्ष्व ॥ २१६ ॥ व्येका चत्वारिंशद्वाहुः प्रतिबाहुको द्विपचाशत् । षष्टिर्भूमिर्वदनं पश्ञ्चकृतिः कोऽत्र विष्कम्भः ॥ २१७ ॥ त्रिसमस्य च षट् बाहुस्त्रयोदश द्विसमबाहुकस्यापि । भूमिर्दश विष्कम्भावनयोः कौ बाह्यत्तयोः कथय ॥ २९८ ॥
18

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गणितसारसङ्गहः. बाहु पञ्चव्युत्तरदशको भूमिश्चतुर्दशो विषमे । त्रिभुजक्षेत्रे बाहिरवृत्तव्यासं ममाचक्ष्व ॥ २१९ ॥ द्विकबाहुषडश्रस्य क्षेत्रस्य भवेद्विचिन्त्य कथय त्वम् । बाहिरविष्कम्भं मे पैशाचिकमत्र यदि वेत्सि ।। २२० ॥ इष्टसङख्याव्यासवत्समवृत्तक्षेत्रमध्ये समचतुरश्राद्यष्टक्षेत्राणां मुरव. ममुजसख्यानयनसूत्रम् --
लब्धव्यासेनेष्टव्यासो वृत्तस्य तस्य भक्तश्च । लब्धेन भुजा गुणयेद्भवेच्च जातस्य भुजसख्या ॥ २२१ ॥
अत्रोद्देशकः । वृत्तक्षेत्रव्यासस्त्रयोदशाभ्यन्तरेऽत्र सञ्चिन्त्य । समचतुर श्राद्यष्टक्षेत्राणि सरवे ममाचक्ष्व ॥ २२२ ॥
भायतचतुरश्रं विना पूर्वकल्पितचतुरश्रादिक्षेत्राणां सूक्ष्मगणितं प रज्जुसङ्ख्यां च ज्ञात्वा तत्तत्क्षेत्राभ्यन्तरावस्थितवृत्तक्षेत्रविष्कम्मानयन - सूत्रम्----
परिधेः पादेन अजेदनायतक्षेत्रसूक्ष्मगणितं तत् । क्षेत्राभ्यन्तरवृत्ते विष्कम्भोऽयं विनिर्दिष्टः ।। २२३ ।।
अत्रोद्देशकः। समचतुर श्रादीनां क्षेत्राणां पूर्वकल्पितानां च । कृत्वाभ्यन्तरवृत्तं ब्रह्मधुना गणिततत्त्वज्ञ ॥ २२४ ॥
समवृत्तव्याससङ्ख्यायामिष्ट सङ्ख्या बाणं परिकल्प्य तद्वाणपरिमाणस्य ज्यासख्यानयनसूत्रम---

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क्षेत्रगणितव्यवहारः. व्यासाधिगमोनस्स च चतुर्गुणिताधिगमेन सङ्गणितः । यत्तस्य वर्गमूलं ज्यारूपं निर्दिशेत्प्राज्ञः ॥ २२५ ॥
अत्रोद्देशकः । • व्यासो दश वृत्तस्य हाम्यां छिन्नो हि रूपाभ्याम् ।
छिन्नस्य ज्या का स्यात्प्रगणथ्याचक्ष्व तां गणक ॥ २२६॥
समवृत्तक्षेत्रव्यासस्य च मौाश्च सङ्ख्या ज्ञात्वा बाणसख्या. नयनसूत्रम्
व्यासज्यारूपकयोर्वर्गविशेषस्य भवति यन्मूलम् । तद्विष्कम्भाच्छोध्यं शेषामिषं विजानीयात् ॥ २२७ ॥
___ अत्रोद्देशकः। दश वृत्तस्य विष्कम्भः शिअिन्यभ्यन्तरे सरवे ।
उष्टाष्टो हि पुनस्तस्याः कः स्यादधिगमो वद ॥ २२८॥
ज्यासङ्ख्यां च बाणसङ्ख्यां च ज्ञात्वा समवृत्तक्षेत्रस्य मध्यव्यात. सख्यानयनसूत्रम्
भक्तश्चतुर्गुणेन च शरेण गुणवर्गराशिरिषुसहितः । समवृत्तमध्यमस्थितविष्कम्भोऽयं विनिर्दिष्टः ॥ २२९ ॥
अत्रोद्देशकः । . कस्यापि च समवृत्तक्षेत्रस्याभ्यन्तराधिगमनं हे । ज्या दृष्टाष्टो दण्डा मध्यव्यासो भवेत्कोऽत्र ॥ २३०॥
समवृत्तद्वयसंयोगे एका मत्स्याकतिर्भवति । तन्मत्स्यस्य मुरखपुच्छ. विनिर्गतरेरवा कर्तव्या । तया रेवया अन्योन्यामिमुरवधनुर्बयालतिर्म

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गणितसारसाहः. वति । तन्मुखपुच्छविनिर्गतरेरवैव तहनु यस्यापि ज्याकतिर्मवति । तद्धनुईयस्य शरद्वयमेव वृत्तपरस्परसम्पातशरौ ज्ञेयो । समवृत्तद्वयसंयोगे तयोः सम्पातशरयोरानयनस्य सूत्रम्
ग्रासोनव्यासाभ्यां ग्रासे प्रक्षेपकः प्रकर्तव्यः । वृत्ते च परस्परतः सम्पातशरौ विनिर्दिष्टी ॥ २३१॥
अत्रोद्देशकः । समवृत्तयोईयोर्हि द्वात्रिंशदशीतिहस्तविस्तृतयोः । ग्रासेऽष्टौ को बाणावन्योन्यभवौ समाचक्ष्व ॥ २३२ ॥ इति पैशाचिकव्यवहारः समाप्तः॥
इति सारसहे गणितशास्त्रे महावीराचार्यस्य कृतौ क्षेत्रगणितं नाम षष्ठव्यवहारः समाप्तः ॥

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सप्तमः
खातव्यवहारः, सर्वामरेन्द्रमकटार्चितपादपीठं सर्वज्ञमव्ययमचिन्त्यमनन्तरूपम् । भव्यप्रजासरसिजाकरबालभानु भक्त्या नमामि शिरसा जिनवर्धमानम् ॥ १ ॥ क्षेत्राणि यानि विविधानि पुगेदितानि तेषां फलानि गुणितान्यवगाहनानि (नेन) । कर्मान्तिकौण्ड्रफलसूक्ष्मविकल्पितानि वक्ष्यामि सप्तममिदं व्यवहाररवातम् ॥ २ ॥
सूक्ष्मगणितम्. अत्र परिभाषाश्लोकः--
हस्तघने पांसूनां द्वात्रिंशत्पलशतानि पूर्याणि ।
उत्कीर्यन्ते तस्मात् षट्विंशत्पलश नानीह ॥ ३ ॥ खातगणितफलानयनसूत्रम् ---
क्षेत्रफलं वेधगुणं समरवाने व्यावहारिकं गणितम् । मुरवतलयुतिदलमय सत्सङ्ख्याप्तं स्यात्समीकरणम् ॥ ४ ॥
अत्राशकः. समचतुर श्रस्याष्टी बाहुः प्रतिबाहुकश्च वेधश्च । क्षेत्रस्य वातगणितं समवाते किं भवेदत्र ॥ ५॥ त्रिभुजस्य क्षेत्रस्य द्वात्रिंशद्वाहकस्य वेधे तु । षट्त्रिंशदृष्टास्ते षडलान्यस्य किं गांणतम् ॥ ६॥ साष्टशतव्यासस्य क्षेत्रस्य हि पत्रषष्टिसहितशतम् । वेधो उत्तस्य त्वं समरवाते किं फलं कथय ॥ ७ ॥ आयतचतुरश्रस्य व्यासः पञ्चायविंशतिर्बाहुः । षष्टिवैषोऽष्टशतं कथयाशु समस्य वातस्य ॥ ८ ॥

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गणितसारसहः. ____ अस्मिन् वातगणिते कर्मान्तिकसंज्ञफलं च औण्डूसंज्ञफलं च ज्ञात्वा ताभ्यां कर्मान्तिकोण्डूसंज्ञफलाम्यां सूक्ष्मखातफलानयनसूत्रम् –
बाह्याभ्यन्तरसंस्थिततत्तत्क्षेत्रस्थबाहुकोटभुवः । खप्रतिबाहुसमेता भक्तास्तत्क्षेत्रगणनयान्योन्यम् ॥ ९॥ गुणिताश्च वेधगुणिताः कान्तिकसंज्ञगणितं स्यात् । तद्बाह्यान्तरसंस्थिततत्तत्क्षेत्रे फलं समानीय ॥ १० ॥ संयोज्य सङ्ख्ययाप्तं क्षेत्राणां वेधगणितं च । औण्ड्रफलं तत्फलयोर्विशेषकस्य त्रिभागेन ।। संयुक्तं कर्मान्तिकफलमेव हि भवति सूक्ष्मफलम् ॥ ११॥
अत्रोद्देशकः । समचतुरश्रा वापी विंशतिरुपरीह षोडशव तले। वेधो नव किं गणितं गणितविदाचक्ष्व मे शीघ्रम् ।। १२ ।। वापी समत्रिबाहुविंशतिरुपरीह षोडशैव तले । वेधो नव किं गणितं कर्मान्तिकमौण्डुमपि च सूक्ष्मफलम् ।। १३ ॥ समवृत्तासी वापी विंशतिरुपरीह षोडशैव तले । वेषो द्वादश दण्डाः किं स्यात्कर्मान्तिकौण्डूसूक्ष्मफलम् ॥ १४ ॥ आयतचतुरश्रस्य त्वायामप्पष्टिरेव विस्तारः । हादश मुरवे तलेऽधं वेधोऽष्टौ किं फलं भवति ॥ १५ ॥ नवतिरशीतिः सप्ततिरायामश्चोर्ध्वमध्यमूलेषु । विस्तारो द्वात्रिंशत् षोडश दश सप्त वेधोऽयम् ।। १५ ।। व्यासः षष्टिर्वदने मध्ये त्रिंशत्तले तु पञ्चदश । समवृत्तस्य च वेधः षोडश किं तस्य गणितफलम् ॥ १७ ॥

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खातव्यवहारः.
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त्रिभुजस्य मुरवेऽशीतिः षष्टिर्मध्ये तले च पश्चाशत् । बाहुत्रयेऽपि वेधो नव किं तस्यापि भवति गणितफलम् ॥ १८ ॥
खातिकायाः स्वातगणितफलानयनस्य च स्वातिकाया मध्ये सूची मुरवाकारवत् उत्सेधे सति खातगणितफलानयनस्य च सूत्रम्
परिरखामुरवेन सहितो विष्कम्भस्त्रिभुजवृत्तयोस्त्रिगुणात् । आयामश्चतुरश्रे चतुर्गुणो व्याससङ्गुणिनः ॥ १९ ॥ सूचीमुरववद्वेधे परिरवा मध्ये तु परिरवार्धम् । मुरवसहितमथो करणं प्राग्वत्तलसचिवेधे च ।। २० ॥
अत्रोद्देशकः. त्रिभुजचतुर्भुजवृत्तं पुरोदितं परिरवया परिक्षिप्तम् । ' दण्डाशीत्या व्यासः परिग्वाश्चतुरुर्विकास्त्रिवेधाः स्युः ॥ २१ ॥ आयतचतुरायामो विंशत्युत्तरशतं पुनासः । चत्वारिंशत् परिरवा चतुरुर्वीका त्रिवधा स्नात् ॥ २२ ॥
उत्सेधे बहुप्रकारवति सति वातफलानयनस्य च, गस्य कस्यचित् स्वातफलं ज्ञात्वा तत्त्वातफलात् अन्यक्षत्रस्य खातफलानयनस्य च सूत्रम्---
वेधयुतिः स्थानहना वेधो मुग्वफलगुणः स्व वातफलम् । त्रिचतुर्भुजवृत्तानां फलमन्यक्षेत्रफलहतं वेधः ॥ २३ ॥
अत्रोद्देशकः । समचतुर क्षेत्रे भूमिचतुर्हस्तमात्रविस्तारे । तत्रैकद्वित्रिचतुर्हस्तनिखाते कियान् हि समबंधः ॥ २४ ॥
14-A

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गणितसारसहः. समचतुरश्राष्टादशहस्तभुजा वापिका चतुर्वेधा ।
वापी तज्जलपूर्णान्या नवबाहात्र को वेधः ।। २५ ॥ यस्य कस्यचित्खातस्य उर्ध्वस्थितभुजासङ्ख्यां च अवस्स्थित . भुजासङ्ख्यां च उत्सेधप्रमाणं च ज्ञात्वा, तत्खाते इष्टोत्सेधसङ्ख्याया भुजासङ्ख्यानयनस्य, अधस्सूचिवेधस्य च सङ्ख्यानयनस्य सूत्रम्
मुखगुणवेधो मुरवतलशेषहतोऽत्रैव सूचिवेधः स्यात् । विपरीतवेधगुणमुरवतलयुत्यवलम्बहयासः ॥ २६ ॥
अत्रोद्देशकः । समचतुरश्रा वापी विंशतिरूवे चतुर्दशाधश्च । वेधो मुखे नवाधस्त्रयो भुजाः केऽत्र सूचिवेधः कः ॥ २७ ॥ गोलकाकारक्षेत्रस्य फलानयनसूत्रम् - व्यासार्धधनार्धगुणा नव गोलव्यावहारिकं गणितम् । तदशमांशं नवगुणमशेषसूक्ष्मं फलं भवति ॥ २८ ॥
भत्रोद्देशकः । षोडशविष्कम्भस्य च गोलकवृत्तस्य विगणय्य । किं व्यावहारिकफलं सूक्ष्मफलं चापि मे कथय ॥ २९॥
शृङ्गाटकक्षेत्रस्य रखातव्यावहारिकफलस्य खातसूक्ष्मफलस्य च सूत्रम् ---
भुजकतिदलघनगुणदशपदनवहद्यावहारिकं गणितम् । त्रिगुणं दशपदभक्तं शृङ्गाटकसूक्ष्मघनगणितम् ॥ ३०॥
अत्रोद्देशकः । व्यश्रस्य च शृङ्गाठकषड्बाहुघनस्य गणपित्वा । किं व्यावहारिकफलं गणितं सूक्ष्मं भवेत्कथय ॥ ३१ ॥

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खातव्यवहारः.
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वापप्रिणालिकानां विमोचने तत्तदिष्टप्रणालिकासंयोगे तज्जलेन वाप्यां पूर्णायां सत्यां तत्तत्कालानयनसूत्रम् --
वापीप्रणालिकाः स्वस्वकालभक्ताः सवर्णविच्छेदाः । तद्युतिभक्तं रूपं दिनांशकः स्यात्प्रणालिकायुत्या ॥ तदिनभागहतास्ते तज्जलगतयो भवन्ति तद्वाप्याम् ॥ ३३ ॥
अत्रोद्देशकः । चतस्त्रः प्रणालिकाः स्युस्तत्रैकैका प्रपूरयति वापीम् । द्वित्रिचतुःपञ्चांगैर्दिनस्य कतिभिर्दिनांशैस्ताः ॥ ३४ ॥
त्रैराशिकाख्यचतुर्थगणितव्यवहारे सूचनामात्रोदाहरणमेव ; अत्र सम्यग्विस्तार्य प्रवक्ष्यते -
समचतुरश्रा वापी नवहस्तघना नगस्य तले । तच्छिरवराजलधारा चतुरश्राङ्गुलममानविष्कम्मा ॥ ३५ ॥ पतिताग्रे विच्छिन्ना तया घना सान्नरालजलपर्णा । शैलोत्सेधं वाप्यां जलप्रमाणं च मे बृहि ॥ ३१ ॥ वापी समचतुरश्रा नवहस्तघना नगस्य तले ।। अलसमवृत्तघना जलधारा निपतिता च तच्छिखरात् ।। ३७ ॥ अग्रे विच्छिन्नाभत्तस्या वाप्या मुखं प्रविष्टा हि । सा पूर्णान्तरगतजलधारोत्सेधेन शैलस्य। उत्सेधं कथय सावे जलप्रमाणं च विगणय्य ॥ ३८ ॥ समचतर श्रा वापी नवहस्सघना नगम्य तले । तचिलखगजलधारा पतिताङ्ग रायनत्रिकोणा सा ॥ ३० ॥ वापीमुरवप्रविष्ठा साग्रं छिन्नान्त गलजलपुर्णा । कथय सरवे विगणय्य च गिर्युत्सेधं जलप्रमाणं च ॥ ४० ॥

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गणितसारसङ्ग्रहः. समचतुरश्रा वापा नवहस्तघना नगस्य तले । अङ्गुलविस्ताराङ्गुलरवाताङ्गुलयुगलदीर्घजलधारा ॥ ४१ ॥ पतिताने विच्छिन्ना वापीमुरवसंस्थितान्तरालजलैः । सम्पूर्णा स्याद्वापी गिर्युत्सेधो जलप्रमाणं किम् ॥ ४२ ॥ इति वातव्यवहारे सूक्ष्मगणितं सम्पूर्णम् ।
चितिगणितम्. इतः परं वातव्यवहारे चितिगणितमुदाहरिष्यामः । अत्र परि
भाषा--
हस्तो दीर्घो व्याप्तस्तदर्धमङ्गलचतुष्कमुत्सेधः ।
दृष्टस्तथेष्टकायास्ताभिः कर्माणि कार्याणि ।। ४३ ॥ इष्टक्षेत्रस्य वातफलानयने च तस्य खातफलस्य इष्टकानयने च सूत्रम्--
मुरवफलमुदयेन गुणं तदिष्टकागणितभक्तलब्धं यत् । चितिगणितं तद्विद्यात्तदेव भवतीष्टकासङ्ख्या ॥ ४४ ॥
अत्रोद्देशकः। वेदिः समचतुरश्रा साष्टभुजा हस्तनवकमुत्सेधः । घटिता तदिष्टकाभिः कतीष्टकाः कथय गणितज्ञ ॥ ४५ ॥ अष्टकरसमत्रिकोणनवहस्तोत्सेधवेदिका रचिता। पूर्वेष्टकाभिरस्यां कतीष्टकाः कथय विगणय्य ॥ ४६॥ समवृत्ताकृतिवेदिर्नवहसो; कराष्टकव्यासा। घटितेष्टकाभिरस्यां कतीष्टकाः कथय गणितज्ञ ।। ४७ ।। आयतचतुश्रस्य त्वायामः षष्टिरेव विस्तारः । पञ्चकतिः षड़ वेधस्तदिष्टकाचितिमिहाचक्ष्व ॥ ४८॥

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खातव्यवहारः.
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प्राकारस्य व्यासः सप्त चतुर्विशतिस्तदायामः । घटितेष्टकाः कति स्युश्चोच्छ्रायो विंशतिस्तस्य ॥ ४९ ॥ व्यासः प्राकारस्योर्चे पडघोऽथाष्ट तीर्थका दीर्घः । घटितेष्टकाः कति स्युनोच्छ्रायो विंशतिस्तस्य ।। ५०॥ द्वादश षोडश विंशतिरुत्संधाः सप्त षट् च पञ्चाधः । व्यासा मुरवे चतुस्त्रिहिकाश्चतुर्विंशतिर्दीर्घः ॥ ११ ॥ इष्टवेदिकायां पतितायां सत्यां स्थितस्थाने इष्टकासङ्ख्यानयनस्य च पतितस्थाने इष्टकासङ्ख्यानयनस्य च सत्रम् ----
मुरवतलशेषः पतितोत्सेधगुणः सकलवेधहत्समुरवः । मुखभूम्योभूमिमुखे पूर्वोक्तं करणमवशिष्टम् ॥ ५२ ॥
अत्रोद्देशकः। द्वादश देयं व्यासः पश्चाधश्चोर्ध्वमेकमुत्सेधः । दश तस्मिन् पञ्च करा भग्नास्तत्रेप्टकाः कति स्युस्ताः ॥ ५३॥ प्राकारे कर्णाकारेण भग्ने सति स्थितष्टकानयनस्य च पतितष्टकानयनस्य च सूत्रम्
भूमिमुरवे द्विगुणे मुग्वभूमियते मन्मभूदगयतोने । दैयोदयपष्ठांशने स्थितपतितेप्टकाः क्रमेण स्युः ॥ ५४॥
अत्रोद्देशकः । प्राकारोऽयं मूलान्मध्यावर्तन वागुना विद्वः । कर्णाकृत्या मनस्तत्स्थितपतितप्टकाः कित्यः स्युः ॥ ५५ ॥ प्राकारोऽयं मूलान्मध्यावर्तन चैकहम्तं गन्वा । कर्णाकृत्या भग्नः कतीप्टकाः स्युः स्थिताश्च पतिताः काः ॥ १६ ॥

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गणित सारसङ्ग्रहः.
प्राकारमध्यप्रदेशोत्सेधे तरवृद्धयानयनस्य प्राकारस्य उभयपार्श्वयोः
तरहानेरानयनस्य च सूत्रम् -
150
इष्टेष्टको हृतो वेधश्च तर प्रमाणमे कोनम् ।
मुखतलशेषेण हृतं फलमेव हि भवनि तरहानिः ॥ ५७ अत्रोद्देशकः ।
प्राकारस्य व्यासः सप्त तले विंशतिस्तदुत्सेधः । एकेनायें घटितस्तरबृद्धयूने करोदयेष्टकथा ॥ ९८३ ॥ समवृत्तायां वाप्यां व्वासच तुप्के ऽर्धयुक्तकर भूमिः । घटितेष्टकाभिरभितस्तस्यां वेधस्त्रयः काः स्युः । घटितेष्टकाः सरखे मे विगणय्य ब्रूहि यदि वेत्सि ॥ ६० ॥ इष्टकाघटितस्थले अधस्तलव्यासै सति ऊर्ध्वतलव्यासे सति च
गणितन्यायसूत्रम्—
द्विगुणनिवेश व्यासायामयुतो द्विगुणितस्तदायामः । आयतचतुरश्रे स्यादुत्सेधव्याससङ्गणितः ॥ ६१ ॥
11
अत्रोद्देशकः ।
विद्याधरनगरस्य व्यासोऽष्टौ द्वादशैव चायामः । पञ्च प्राकारतले मुखे तदेकं दशोत्सेधः ॥ ६२ ॥ इति स्वातव्यवहारे चितिगणितं समाप्तम् ।
*
क्रकचिकाव्यवहारः.
इतः परं क्रकचिकाव्यवहारमुदाहरिष्यामः । तत्र परिभाषा - हस्तद्वयं षडङ्गुलहीनं किष्क्वायं भवति । इष्टाद्यन्तच्छेदनसङ्ख्यैव हि मार्गसंज्ञा स्यात् ॥ ६३ ॥ अथ शाकाख्यद्यादिद्रुमसमुदायेषु वक्ष्यमाणेषु । व्यासोद्दयमार्गाणामङ्गुलसंख्या परस्परन्नाप्ता ॥ ६४ ॥

Page #189

151
रवातव्यवहारः. हस्ताङ्गुलबर्गेण क्राकचिके पटिकाप्रमाण स्यात् । शाकाहयद्रुमादिद्रुमेषु परिणाहदैर्घ्यहस्तानाम् ॥ १५ ॥ संख्या परस्परम्ना मार्गाणां संख्यया गुणिता । तत्पट्टिकासमाप्ता क्रकचकना कर्मसंख्या स्यात् ॥ ६६ ॥ शाकार्जुनाम्लवेतसतरलासितसर्जडुण्डुकाख्येषु । श्रीपर्णीउलाख्यद्रुमेश्वमीचेकमार्गस्य ।। षण्णवतिरङ्गुलानामायामः कि कुरेव विस्तारः ॥ १७॥
अत्रोद्देशकः । शाकायतरी दीर्घः षोडश हस्ताश्च विस्तारः । सार्धत्रयश्च मार्गाश्चाष्टौ कान्यत्र कर्माणि ॥ ६८ ॥ इति वानव्यवहारे क्रांचकाव्यवहारः समाप्तः ॥
इति सारसङ्गाहे गणितशास्त्रे महावीराचास्य कृती सप्तमः वातव्यवहारः समाप्तः ॥

Page #190

अष्टमः.
छायाव्यवहारः,
शान्तिर्जिनः शान्तिकरः प्रजानां जगत्प्रभुतिसमस्तभावः' । . यः प्रातिहार्याष्टविवर्धमानो नमामि तं निर्जितशत्रुसङ्गम् ॥ १ ॥ आदी प्राच्याद्यष्टदिक्साधनं प्रवक्ष्यामः
सलिलोपरितलवस्थितसमभूमितले लिरवेदत्तम् । बिम्बं स्वेच्छाशद्विगुणितपरिणाहसूत्रेण ॥ २ ॥ तवृत्तमध्यस्थतदिष्टशकोश्छाया दिनादौ च दिनान्तकाले । तहत्तरेरवां स्पृशति क्रमेण पश्चात्पुरस्ताच ककुप् प्रदिष्टा ॥ ३ ॥ तग्रियान्तर्गततन्तुना लिरवेन्मत्स्याकर्ति याम्यकुबेरदिस्थाम् । तत्कोणमध्ये विदिशः प्रसाध्याश्छायैव याम्योत्तरदिग्दिशार्धजाः ॥ ४ ॥ अजधठरविसङ्कमणधुदलजभैक्यार्धमेव विषुवदा ॥ ४ ॥ लङ्कायां यवकोव्यां सिद्धपुरीरोमकापुर्योः । विषुवगा नास्त्येव त्रिंशद्वठिकं दिनं भवेत्तस्मात् ॥ ५ ॥ देशेष्वितरेषु दिनं त्रिंशन्नाज्याधिकोनं स्यात् । मेषधटायनदिनयोस्त्रिंशदठिकं दिनं हि सर्वत्र ॥ ६॥ दिनमानं दिनदलभां ज्योतिश्शास्त्रोक्तमार्गेण । ज्ञात्वा छायागणितं विद्यादिह वक्ष्यमाणसूत्रीधैः ॥ ७॥
IM reads तत्त्व:.

Page #191

छायान्यवहारः.
153
विषुवच्छाया यत्रयत्र देशे नास्ति तत्रतत्र देशे इष्टशङ्कोरिष्टकाल. च्छायां ज्ञात्वा तत्कालानयनसूत्रम
छाया सैका द्विगुणा तया हृतं दिनमितं च पूर्वाद्धे । अपराह्ने तच्छेषं विज्ञेयं सारसङ्गाहे गणिते ॥ ८॥
अत्रोद्देशकः । पूर्वाद्धे पौरुषी छाया त्रिगुणा वद किं गतम् ।
अपरालेऽवशेषं च दिनस्यांशं वद प्रिय ॥ ९ ॥ दिनांशे जाते सति घटिकानयनसूत्रम
अंशहतं दिनमानं छेदविभक्तं दिनांशके जाते । पूर्वा गतनाज्यस्त्वपराद्धे शंपनाज्यस्तु ॥ १८ ॥
अत्रोद्देशकः । विषुवच्छायाविरहितदेशेऽष्टांशो दिनस्य गाः ।
शेषश्चाष्टांशः का घटिकाः स्युः ग्वाग्निनाड्योहः ॥ ११ ॥ मल्लयुद्धकालानयनसूत्रम् -- कालानयनाद्दिनगतशेषसमासोनितः कालः । स्तम्भच्छाया स्तम्भप्रमाणभक्तैव पीरुपी छाया ॥ १२ ॥
अत्रोद्देशकः । पूर्वाद्दे शङ्कसमच्छायायां मल्लयुद्धमारब्धम् । अपराह्ने द्विगुणायां समाप्तिगतीच्च गुढकालः कः ॥ १३ ॥
अपगर्धस्यो हाहरणम् । द्वादशहस्तस्तम्भच्छाया चतुरुत्तरेव विंशतिका । तत्काले पौरुषिकच्छाया कियती भवेद्गणक ॥ १४ ॥

Page #192

154
गणितसारसङ्ग्रहः. विषुवच्छायायुक्ते देशे इष्टच्छायां ज्ञात्वा कालानयनस्य सूत्रम्'
शक्युतेष्टच्छाया मध्यच्छायोनिता द्विगुणा । तदवाप्ता शङ्कमितिः पूर्वापरयोदिनांशः स्यात् ॥ १५ ॥
अत्रोद्देशकः । द्वादशाङ्गुलशङ्कोवुदलच्छायाङ्गुलद्वयी । इष्टच्छायाष्टाङ्गुलिका दिनांशः को गतः स्थितः । व्यंशो दिनांशो घठिकाः कास्त्रिंशन्नाडिकं दिनम् ॥ १७ ॥ इष्टनाडिकानां छायानयनसूत्रम्द्विगुणितदिनभागहृता शङ्कमितिः शङ्कमानोना । धुदलच्छायायुक्ता छाया तत्वेष्टकालिका भवति ॥ १८ ॥
अछोद्देशकः । द्वादशाङ्गुलशङ्कोर्युदलच्छायाङ्गुलद्वयी। दशानां घठिकानां मा का छिंशन्नाडिकं दिनम् ॥ १९ ॥ पादच्छायालक्षणे पुरुषस्य पादप्रमाणस्य परिभाषासूत्रम्
पुरुषोन्नतिसप्तांशस्तत्पुरुषाङ्गेस्तु दैर्ध्य स्यात् ।
यद्येवं चेत्पुरुषः स भाग्यवानाङ्गिभा स्पष्टा ॥ २० ॥ आरूढच्छायायाः सङ्ख्यानयनसूत्रम् -
नृच्छायाहतशङ्कभित्तिस्तम्भान्तरोनितो भक्तः । नृच्छाययैव लब्धं शङ्कोभित्त्याश्रितच्छाया ॥ २१ ॥
____ अछोद्देशकः । विंशतिहसः स्तम्भो भित्तिस्तम्भान्तरं करा अष्टौ । पुरुषच्छाया हिना भित्तिगता सम्भभा किं स्यात् ॥ २२ ॥
Not found in any of the M88.

Page #193

छायाव्यवहारः.
स्तम्भप्रमाणं च भिच्यारूढस्तम्भच्छायासयां च ज्ञात्वा भित्ति
स्तम्भान्तरसङ्ख्यानयनसूत्रम् —
पुरुषच्छायानिम्नं स्तम्भारूढान्तरं तयोर्मध्यम् । स्तम्भारूढान्तरहृततदन्तरं पौरुषी त्राया ॥ २३ ॥
अत्रोद्देशकः ।
विंशतिहस्तः स्तम्भ: षोडश मित्त्याश्रितच्छाया । द्विगुणा पुरुषच्छाया भित्तिस्तम्भान्तरं किं स्यात् ॥ २४ ॥
अपरार्धस्योदाहरणम् ।
विशनिहतः स्तम्भः षोडश भिच्याश्रितच्छाया । किती पुरुषच्छाया भित्तिस्तम्भान्तरं चाष्टौ ॥ २५ ॥ आरूढच्छायायाः सख्यां च भित्तिस्तम्भान्तर भूमिसख्यांच पुरुषच्छायायाः सङ्ख्यां च ज्ञात्वा स्तम्भप्रमाणसङ्गख्यानयनसूत्रम-
नृच्छायानारूढा भित्तिस्तम्भान्तरेण संयुक्ता । पौरुषभाहृतलब्धं विदुः प्रमाणं बुधाः स्तम्भं ॥ २६ ॥
अत्रोद्देशकः ।
षोडश भिच्यारूढच्छाया द्विगुणैव पौरुषी छाया । स्तम्भोत्सेधः कः स्याद्भित्तिस्तम्भान्तरं चाष्टौ || २७ ||
155
शङ्कप्रमाणशङ्कच्छायामिश्रविभक्तसूत्रम् -
शङ्कुप्रमाणशङ्कच्छायामिश्रं तु सैकपीरुण्या |
मक्तं शङ्कुमितिः स्याच्छङ्कुच्छाया तदूनमिश्रं हि ॥ २८ ॥
अत्रोद्देशकः । शङ्कुप्रमाणशङ्कुच्छायामिश्रं तु पञ्चाशत् । शङ्कत्सेधः कः स्याच्चतुर्गुणा पौरुषी छाया || २९ ॥

Page #194

156
गणितसारसग्रहः. शङ्कच्छायापुरुषच्छायामिश्रविभक्तसूत्रम्--
शङ्कनरच्छाययतिर्विभाजिता शङ्कुसैकमानेन । लब्धं पुरुषच्छाया शङ्कुच्छाया तदूनमिश्रं स्यात् ॥ ३०॥
___ अत्रोद्देशकः। शङ्कोरुत्सेधो दश नृच्छायाशङ्कभामिश्रम् ।
पश्चोत्तरपञ्चाशनृच्छाया भवति कियती च ॥ ३१ ॥ स्तम्भस्य अवनतिसङ्ख्यानयनसूत्रम्- ---
छायावर्गाच्छोध्या नरभाकतिगुणितशङ्ककृतिः । सैकनरच्छायाकृतिगुणिता छायाकृतेः शोध्या ॥ ३२ ॥ तन्मूलं छाया शोध्यं नरभानवर्गरूपेण'। भागं हृत्वा लब्धं स्तम्भस्यावनतिरेव स्यात् ॥ ३३ ॥
अत्रोद्देशकः । द्विगुणा पुरुषच्छाया व्युत्तरदशहस्तशङ्कोर्भा । एकोनत्रिंशत्सा स्तम्भावनतिश्च का तत्र ।। ३४ ॥ कश्चिद्राजकुमारः प्रासादाभ्यन्तरस्थस्सन् । पूर्वाह्ने जिज्ञासुर्दिनगतकालं नरच्छायाम् ॥ ३५ ॥ द्वात्रिंशद्धस्तो जाले प्राभित्तिमध्य आयाता । रविभा पश्चादित्ती व्यकत्रिंशत्करोर्ध्वदेशस्था ॥ ३६ ।। तद्भित्तिद्वयमध्यं चतुरुत्तराविंशतिः करास्तस्मिन् । काले दिनगनकालं नृच्छायां गणक विगणय्य । कथयच्छायागणिते यद्यस्ति परिश्रमसव चेत् ॥ ३७ ॥ समचतुर श्रायां दशहस्लघनायां नाच्छाया। पुरुषोत्सेधद्विगुणा पूर्वाद्दे प्राक्तटच्या ॥ ३ ॥
is the rending given in the 21SS, for AT HTT; but it is metrically
innorradt.

Page #195

छायाव्यवहारः.
तस्मिन् काले पश्चात्ताश्रिता का भवेद्गणक ।
आरूढच्छायाया आनयनं वेत्सि चेत्कथय ॥ ३९३ ॥ शङ्कोर्दीपच्छायानयनसूत्रम्
•
शङ्कनितदीपोन्नतिराप्ता शङ्कप्रमाणेन । तल्लब्धहृतं शङ्कोः प्रदीपशङ्कुन्तरं छाया ॥ ४० ॥ अत्रोद्देशकः.
शङ्कप्रदीपयोर्मध्ये षण्णवस्यङ्गलानि हि ।
द्वादशाङ्गलशकोस्तु दीपच्छायां बाशु मे ।
षष्टिर्दीपशिखोत्सेधो गणितार्णवपारग ।। ४२ ।।
दीपशकुन्तरानयनसूत्रम् -
शङ्कनितदीपोन्नतिराप्ता शङ्कप्रामाणेन ।
तलब्धहता शङ्कुच्छाया शङ्कुप्रदीपमध्यं स्यात् ॥ ४३ ॥ अादेशकः ।
शङ्कुच्छायाङ्गुलान्यष्टौ षष्टिर्दीपशिखोदयः । शङ्कुदीपान्तरं ब्रूहि गणितार्णवपारंग ॥ ४४ ॥
दीपोन्नतिसङ्ख्यानयनमूत्रम्
-
शङ्कुच्छायामक्तं प्रदीपशङ्कन्तरं सैकम् । शङ्कु प्रमाणगुणितं लब्धं दीपोन्नतिर्भवति ।। ४१ ।।
अत्रोद्देशकः ।
शत्रुच्छायाद्विनिभैव द्विशतं शङ्कुदीपयोः । मन्तरं ह्यङ्गुलान्यत्र का दीपस्य समुन्नतिः ॥ ४६ ॥
157

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168
गणितसारसमहः. शाप्रमाणमत्रापि हादशाङ्गुलकं गते ।
ज्ञात्वोदाहरणे सम्यग्विद्यात्सूत्रार्थपद्धतिम् ।। ४७ ॥ पुरुषस्य पादच्छायां च तत्पादप्रमाणेन वृक्षच्छायां च ज्ञात्वा वृक्षो. नतेः सङ्ख्यानयनस्य च, वृक्षोन्नतिसङ्ख्यां च पुरुषस्य पादच्छायायाः सङ्ख्या च ज्ञात्वा तत्पादप्रमाणेनैव वृक्षच्छायायाः सङ्ख्यानयनस्य च सूत्रम्स्वच्छायया भक्तनिजेष्टवृक्षच्छाया पुनस्सप्तभिराहता सा । वृक्षोन्नतिः साद्रिहता स्वपादच्छायाहता स्यामभैव ननम् ॥ ४८ ॥
अत्रोद्देशकः । आत्मच्छाया चतुःपादा वृक्षच्छाया शतं पदाम् ।
वृक्षोच्छ्रायः को भवेत्स्वपादमानेन तं वद ॥ ४९ ॥ वृक्षच्छायायाः सङ्ख्यानयनोदाहरणम् ।
आत्मच्छाया चतुःपादा पञ्चसप्ततिपिर्यंतम् । शतं वृक्षोन्नतिवृक्षच्छाया स्यात्कियती तदा ॥ ५० ॥ . पुरतो पोजनान्यष्टौ गत्वा शैलो दशोदयः । स्थितः पुरे च गत्वान्यो योजनाशीतितस्ततः ॥ ५१ ॥ तदग्रस्थाः प्रदृश्यन्ते दीपा रात्रौ पुरे स्थितैः । पुरमध्यस्थशैलस्यच्छाया पूर्वागमूलयुक् ।
अस्य शैलस्य वेधः को गणकाशु प्रकथ्यताम् ॥ ५२ ।। इति सारसङ्गहे गणितशास्त्रे महावीराचार्यस्य कृतौ छायाव्यवहारो नाम अष्टमः समाप्तः ।
समाप्तोऽयं सारसङ्गहः ॥

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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 ... Bhá gå pura ha fractions Bhd yandts fractions

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CONTENT $.
CHAPTER IV.
PAGE
70
MISCELLANEOUS PROBLEMS (on fractions) ...
Bhd ga and Sēra varieties Mala variety .. ... Sēramula variety ... Sāpamala variety involving two known quantities Ambamdla variety ... ... ... " Bhd gasa marga variety ... Avitavarga variety characterized by the subtraction or addition of
known quantities .. Malamiéra variety ... .. Bhinnadráya 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-SĀRA-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 Amõghavarna (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 Cubrikůbhanjanın by name, in in reality & cnkrikābhanjana (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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GANITABĀRA 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 syadrāda-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 triprašna + 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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GANITASĀRASANGRAHA.
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 trasarēnu which is derived therofrom, the rathurēnu, 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 (rynvaharāngula) coustitutos (another magula known as) pramāạo. 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, wyarahāra and pramāna 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 stökie ; soven stokrx make one lura, 'aud with thirty-eight and a half of this the ghuli in formed.
34. Two ghatis make oue muhürlu; 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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GANITABĀRASANGRAHA.
Terminology relating to the measurement of)
Other Metals. 12. What is known as a kala consists of four pādas; six and a quarter kalis make one yava; four yavas make one amsa; four ambas mako one bhāya.
43. Six bhägax make one drukšūna ; twice that (draksūna) is one dinära; two dināras make one satèra. 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 tulás 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 kötikā. Next I give the names of the (principal) operations (in arithmetic).
Names of the Operations in Arithmetic. 46. The first among these oporations) is gunakära (multiplication), and it is also called) pratyutpanna ; the second is what is known as bhägahāra (division); and krti (squuring) is said to be the third
47. The fourth, as a matter of courso, is varga-müla (square root), and the fifth is said to be ghana (cubing); then ghanamüla (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 ēka (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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GANITASĀRASANGRAHA.
65. The ninth is lasa-koti (ten-croro) and the tenth is kataköļi (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 mahā. Kharva. Similarly the fifteonth is padma and the sixteenth mahā. padma,
67. Agnin the seventeenth is knonā, the eighteenth maha-ksonā. The nineteenth place is kmkha anul the twentieth is maha-sankha.
68. The twenty-first place is hşityā, the twonty-second mahāksitya. Then the twenty-tbird is ksõbhu 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 Sāra. sangraha, which is a work on arithmetic by Mahā. virācārya.

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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
GANTTABĀRASANGRAHA. 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 parlmarāga 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 pusyaräga 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
GAŅITASĀBABANGBABA.
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. Dināras (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. Dināras (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. Jambû fruits amounting to) 31317 bave been divided between 39 persons. Tell me the share of each.
25. Jambù 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.

Page #215

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* + 200€ + 80€ + 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+2° +81 +4*) + 2(1 x 2+1 3+1x 4+2 3+%** +8 x 4).

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GANITASÁRASANGRAHA.
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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GANITASĀRASANGRAHA.
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.

Page #220

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

Page #221

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

Page #222

GANITASĀRASANGRAHA.
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,

Page #223

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) + (%)
(*)

Page #224

22
GANITASĀRASANGRAHA.
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 fråraka 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

Page #225

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

Page #226

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

Page #227

OHAPTER 11 - ARITHMETICAL OPERATIONS.
The common difference (in respect of another series) is 5, the number of terms is 8, and the sum is 15€. 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

Page #228

GANITASĀRASANGRAHA.
difference. (In splitting up the namber of terms from the miéradhana), 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-miúradhanu, 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,

Page #229

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 = ša, 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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GANITASĀRASANGRAHA.
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 «ertain quantity in which (product) that (quantity) occurs as often as the pumber of terms (in the series); and this (quantity) is the required) common ratio. The gunadhana, when divided by that (self-multiplied) produot of the common ratio in which (product the frequenoy of the occurrence of this common ratio) is measured by the number of terms (in the series), gives rise to the first term.
The rule for finding out in relation to a given gunadhana the number of terms in the corresponding geometrically progressive series) :
98. Divide the yunadhana (of the series) by the first term (thereof). Then divide this (quotient) by the common ratio (time after time) so that there is nothing left (to carry out such a division any further); whatever happens here) to be the number of vertical strokes, (oach representing a single such division), so much is the value of the number of terms in relation to the (given) gunadhana.
Examples in illustration thereof. 99. A certain man (in going from city to city) carned money (in a geometrically progressive series) having 5 linärus for the first term (thereof) and 2 for the common ratio. Ho (thus) entered 8 cities. How many are the dinarax (in) his (possession)?
100. What is the value of) the wealth owned by a merchant (when it is ineasured by the sun of a geometrically progressive series), the first term whereof is 7, the common ratio 3, and the number of terms (wherein) is 9: and again (when it is measured by the sum of another geometrically progressive series), the first
97 and 88. It is clear that ar, when divided by a given : And this is disia. iblo by as many times AA #, which is accordingly the measure of the number of terms in the norius. Sirailarly r *r*?...... up to times gives r*; and the gunadhana i.e., ar divided by this rgives a, which is the required first terin of the series.

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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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GANITASĀRASANGRAHA.
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 ļa 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
GANITASĀRASANGRAHA.
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
GANITASĀRASANGRAHA.
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 palı of long peppor for 1 pana. O arithmetician, mention, after multiplying, what (he gets) for в рапав.
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.

Page #241

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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GANITABĀRASANGRAHA.
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

Page #243

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.

Page #244

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

Page #245

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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GANITASĀRASANGRAHA.
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 ksēpapada 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
GANITABĀRABANGRAHA.
(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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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 ļ, 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 ļ. () 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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GANITASĀRASANG 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. Bhäga (or simple fractions), Prabhāga (or fractions of fractions), then Bhagabhága (or complex fractions), then Bhāgānubandha (or fractions in association), Bhagāpavāha (or fractions iu dissociation), together with Bhāyamātr (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

Page #253

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
GANITASĀRASANGRAHA.
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 srāvaka 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

Page #256

54
GANITABĀRASANGRAUA.
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 mitrinovávett on var? = z{ckstatoest...+ aco', +]

Page #258

50
GANITARĀRASANGRAHA.
(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

Page #259

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

Page #260

GANITASĀBASANGRAHA.
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
も
up+b ท
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
.
GAÑITABĪBASANGKAHÁ.
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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GANITASĪBASANGBAHA.
Examples in compound fractions. 100 to 102. To offer in worship at the feet of Jina, lotuses, jasamines, kėtakis and lilies were prrchased in return for the payment of f of 1, of }, of of }, } of fof, of } of ș, 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 ó 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.
Bhāgānubandha Fractions.
The rule in respect of the (simplification of) Bhagānubandha or associated fractions:
113. In the operation concerning (the simplification of) the Bhägänubandha 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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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 māsas) 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
GANITASĀRASANGRAHA.
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 Bhāgānubandha fractions), and (then) diminished by one, become the unknown (fractional quantities) in the required places of our choice.
Thus ends the Bhāgānubanıtha class (of fractions).
Bhagapavāha Fractions. Thon (comes) the rule for the (simplification of) Bhägapaviha (or the dissociated) variety (in fractions) :
124. In the operation concerning (the simplification of the Bhāgāpavāha 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 Bhigīpaväha 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 Bhagánu 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 Bhagapavāha fractions), and (then) subtracted from one (severally), become the unknown (fractional quantities) in the (required) places of our choice.
Thus ends the Bhāgāpaváha variety of fractions.
The rule for finding out the unknown fractions in all the places in relation to a Bhaganubandha or Bhāyāpaväha 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.
Bhagamät! 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, .., Bhigamalı, 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 jārasangraha which is a work on arithmetic by Mahāviräcärya.
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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GAŅITASĀRASANGRAHA.
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 syāulvvida, 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, Māli, Sesamüla, 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 Sumülle 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
Dvirayrašėramila and Arukamüla, and then Bhagabhyasa, then Ankararga, Mūlamiira and Bhinnad, sya.
Thứ rule relating to the Bhöga 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 Mülawira 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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GANITASĀRASANGRAHA.
6. Out of a collection of excellent bees, * took delight in pătali 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 srūvaka, 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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GANITASĀRASANGRAHA,
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 axöka trecs, respectively, multiplied by 6, hecame hidden in a crowd of big patali trees. The difference between those that hid themselves in the patalī trees and the axöka 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 müla 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 verscıl 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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GANITASĀRASANGRAHA..
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 Sesamīla variety of miscellaneous problems on fractions).
The rule relating to the Sesamūla 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 Sésamūla 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 saunīla 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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GANITASĀRASANGRAHA.
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 tamāla 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 Bhāyasamvarya 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 Malamišra 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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GANITASĀRASANGRAHA.
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 Sārasangraha which is a work on arithmetic by Mahāvīrācārya.
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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GANITASĀRASANGRAHA.
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 Techá and the Promāna 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 kvöxer 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 karsē 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 yöjunas 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 dinārus 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) é 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 į 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
GANITASĀRASANGRAHA.
(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 Váha + 1 Kumbha = 1+ Väha. 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 Puränas 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 ydrņas "36. A inan, by carrying 9 minix of wheat over 3 yöjunus, obtained 60 panas. How much (would lo obtain) w currying one kumbha along with one rūka (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 ở 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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GANITASĀRASANGRABA.
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 vāhas 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 Sārasangraha which is a work on arithmetic by Mahāvīrācārya.
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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GAŅITASĀRASANGRAFA.
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 Icchă, 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. Purānas, 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 kārsīponas 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 purānas 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 kårsrīpanas 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
GANITASĀRASANGRAHA.
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
GANITASĀRASANGRAHA.
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 Purānas 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
GANITASĀRASANGRAHA.
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 cæpitals, 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 ē **+ ( 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

Page #306

104
GAŅITASĀRASANGRAHA.
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
схрх I 67. Symbolically,
сх т 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

Page #308

106
GANITASĀRASANGRAHA.
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

Page #310

108.
GANITASĀRASANGRABA.
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.

Page #311

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+. ..)

Page #312

110
GANITASÁRASANGRAHA.
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 kuttiköra 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 dināras 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 xrävaka 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 Päráva, and th to Jinapati, and to sage Suvrata ; he dovotedly gave to Ariştanémi who destroyed all the eight kinds of karmas and who was beloved by the world, and I of 1 to Jinasånti: 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 praksēpaka, 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.

Page #314

112
GANITASĀRASANG 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 purānas. 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

Page #316

114
GANITASĀBASANGRAHA.
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 lināras 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.

Page #317

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 ļ, , $ (respectively), and having bought and sold (the same commodity), and with as the remnant price, three merchants became pos8olgors of equal sale-firoccods

Page #318

116
GANITASĀRASANGRAHA.
(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.

Page #319

CHAPTBR TI-MIXED PROBLEMS.
117
Prillika-kuttikära. Hereafter we shall explain the process of calculation known as Vallika-kuttikära":--
The rulo underlying the process of calculation known as Vallikä, in relation to Kuttikör (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 stanzı 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
۲۱
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
۲۰
Thus we have, 72 P1+ Pri 98 P+Pa
P1
Pa
۲۰ 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+ Pε.
"P-b ۲۱
-i
and is the

Page #321

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

Page #322

120
GANITASĀRASANGRAHA.
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, hintäla 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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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 jambū fruits. Of them, 2 (heape) were equally divided among 9

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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 sëpánba.

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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-kuttikāra).
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 sēxa 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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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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GANITASĀRASANGRAHA.
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 Bhinnakuttikära.*
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-kuttıkara. The rulo in relation to sakala-kuttikära.
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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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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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-kuṭṭikara--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-kuṭṭīkdra 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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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
.
GANITASĀRASANGRARA.
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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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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GANITASĀRASANGRAHA.
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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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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GANITASĀRASANGRAHA.
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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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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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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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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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-kuttikäru in the chapter on mixed problems.
Suvarna-kuttikära. Hereafter wg ahall explain that kuttikära 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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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 varnııs, 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 purūnas (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
GAŅITABĀRABANGRAHA. 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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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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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 andò 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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GANITABIRABANGRAHA.
(weight of all the sticks, this is to be subtracted. What remains is (the quantity of) the prapūranika (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 măsu 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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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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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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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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GANITABĀRASAKURAHA.
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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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 Sucarṇa-kuṭṭikära in the chapter on mixed problems
Vicitra-kuttikära.
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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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) flowerskétaki, 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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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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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 yöjuna, 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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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 yöjunas, and 720 dināras form their wagen. "I'wo men stop away after going over two krošus ; 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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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) (€ +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 setī is greater than the third.

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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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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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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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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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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 citrakuttikära-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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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+α=/
=1; where, a, b, c, d, fare the amounts on hand with the 5 merchants.

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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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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 vajrāpavartana 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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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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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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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-kutsikära, 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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GAŅITABĀRASANGRAHA.
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 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 vicitrakuttikära.
Thus ends vicitra-kuttikära 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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GANITASÁRASANGRAHA.
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 lābha, 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 läbha becomos the sum of the series.
An example in illustration thereof. 295. (There were a nuinber of utpalı 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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GANITASĀRASANGRAHA.
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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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

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GANITASİRASANGRAHA.
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
GANITASÃ 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 dināra and the common ratio is $. (Find out the sum of the dināras 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
GANITASĀRASANGRAHA,
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 nanlyávarta. 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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"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
GANITASĀRASANGRAHA.
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.
GANITASĀRABANGBAHA.
6. (The ourvi-linear area may be) a circle, à 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.
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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
GANITASĀRASANGRAHA.
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 pañuva, 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:
Yaväkirakaótra.
Morajäkärakpôtra.
Papavilārak 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-nisëdha 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

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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
ا.

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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.

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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
GANITASĀRASANGRAHA.
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

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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 :
लम्बयोः कर्णयोर्वेकमनिर्दिश्यापरान् कथम् । पृच्छत्यनियतत्वेऽपि नियतं चापि तत्फलम् !! स पृच्छकः पिशाचो वा वक्ता वा नितरां ततः । यो न वेति चतुर्बाहुक्षेत्रस्यानियतां स्थितिम् ।।

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GANITASĀRASANGRAHA.
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 figuré 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. Bhänkaricdrys also given to it the mme valde, and represents it in reduced tormi sa

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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

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GANITASĀRASANGRAHA,
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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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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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 Mṛdanga. 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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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

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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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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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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 vā(v +19+w25), vi{w25- (Vò-VA)}.
=V 4 (2+8+8); =v7{6 - 18 - 2). EN 4 (10),
-Vix 100; =v7x18. . = 100;

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✓
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

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GANITASĀRASANGRAHA.
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 bējas).
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 -6° 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,.

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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

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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 construêted. 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

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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 4að (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.

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•
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® -d") + babed} × (n*+b2)
Perpendiculars= {(a* — b*) × 2cd + (c2 —da) 2ab } » x 2ah; and
{(a* − b )(c" —d") + 4abcd} × (a*b*)
+ 4abcd
Segmenta
{ (a*—b®) × 2cd + (ca —d1) × 2ab | (a —b"); and {(a*—b) (c*—*)
cd}
x 2ab.
•

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GANITASĀRASANGRABA.
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:

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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 perpendiculør-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")}(20–0°)+ {2ab + (a1 +00)}(co-4"}]x (2"+W). [{2cd=(09–")} * 2ab + { 2ab +(29–60, } (c– d") ]ca+61) .
[{2cd=(co-d") }(2° -6° +) 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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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 bējas) 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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'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 lījas, 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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devilishly difficult problems. Hereafter we shall expound the subject of treatment known as Paikācika.
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.
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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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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 bējax 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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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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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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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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GANITASĀRASANGRAHA.
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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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 à + 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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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.
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(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
тх тахь =
; 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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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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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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GANITASĀRASANGRAHA.
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 bājas 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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(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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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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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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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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GANITAPÄRASANGRAHA.
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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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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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 visamasankramaņa, 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
GANITASĀRASANGBAHA.
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 antarávalambaka 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 âbâdhā, 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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GANITASĀBARANGRAHA.
(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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(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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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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(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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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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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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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 yõjanas (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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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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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 paikācika (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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.
GAŅITASABASANGRAHA.
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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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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GANITASĀRASANGRAHA.
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 praksépaka 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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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 Sārasangraha, a work on arithmetic by Mahāvirācārya.

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GANITABĀRASANGRAHA,
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 sūksmaphala (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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GANITASĀRABANGRAHA.
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 karmäntika 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 karmāntika 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
GANITASĪBASANGBAHA. three, gives rişe 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 Karmäntskaphala) 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.

Page #465

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.

Page #466

264
GAŅITASABASANGRAHA.
measure of the depth also is known, in relation to å 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-CALÇULATIONS 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 İRABANGBAHA.
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,

Page #469

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
QANITASİRASANGBAHA.
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

Page #471

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 märga (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 märgas 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) patřikās. 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 pattikäs 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
GANITASĀBASANGRANA.
67-67. In relation to (logs of wood obtained from) trees named saka, arjuna, amla-větasa, sarala, asita, sarja and dunḍuka, and also (in relation to varieties of wood) named riparni and plaksa, the märga 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 märgas (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 Mahaviracārya.

Page #477

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
GAŅITASARASANGBAKA.
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 Romakāpuri, there is no (such) eqninootial shadow at all; and, therefore, the day-time is of 30 ghatīs.
04. In other regions, the day-time happens to be longer or shorter hy 30 ghaťīs. On the days of the entrance (of the sun) into Aries and into Libra, the day-time is everywhere of 30 ghatīs (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 90°. 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 ghație (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).
•

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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

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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,

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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

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282
GAŅITASĀRASANGRAHA,
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.

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CHAPTER IX-CALOULATIONS RELATING TO SHADOWS.
283
square of the (given) shadow. This (remainder) is to be multiplied by the sum of the square of the human shadow and one. (The quantity so arrived at) is to be subtracted from the square of the (given) shadow. The square root of this resulting 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.

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284
QAŅITABĀRABANGBAKA.
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.

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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.

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286
GANITASĀRABANGRAHA.
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 Särasangraha, which is a work on arithmetic by Mahāvirācārya.
80 ENDS TAIS 8 ĀRASANGRA RA.
61-685. This example is intended to illustrato the rulo given in stanna 46
above.

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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 Porūnas, 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
अश्व
अश्विन्
आकाश
इन
इन्दु इन्द्र
इन्द्रिय
इभ
इषु
इक्षण
उदधि उपेन्द्र
ऋतु
...
...
...
...
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
...
...
...
...
कर
The hand
करणीय 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 ऐरावत, पुण्डरीक, वामन, कुमुद, अञ्जन, पुष्पदन्त, सार्वभौम, and सुप्रतीक.
The arrows of Manmatha or the Indian Cupid are declared to be five, viz. अरविन्द, अशोक, चूत, नवमल्लिका, Bud नीलोत्पल.
8
Vide अक्षि.
4 Vide अब्धि.
9
5
8
The horses of the sun's chariot are supposed to be seven.
Vide अश्व.
Vide अनन्त.
The number of suns is reckoned to be 12 corresponding to the 12 months of &
मित्र, अर्यमनू, रुद्र, year, viz., धातृ, वरुण, सूर्य, भग, विवस्वत्, पूषन्, सवितृ, त्वष्टृ and विष्णु. They are called
the twelve ādityan.
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.
&
वसन्त, प्रीष्म, वर्ष, शरद, हेमन्त, and शिशिर.
Human beings have two hands.
There are 5 vratas or austerities to be observed according to the Jaina religion, vis., अहिंसा, सूनृत अस्तेय, ब्रह्म, and अपरिग्रह.

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APPENDIX I.
289
करिन् ... An elephant ... 8 Tide इम. कर्मन् ... Aotion : the effect_8 Acoording to Jainaa there are eight
of action as its kinds of karma, vir. ज्ञानावरणीय, karma.
दर्शनावरणीय, मोहनीय, अन्तराय,
वेदनीय, नामिक, गोत्रिक and आयक. कलाधर, ... The moon ... 1 Vide इन्दु. कषाय ... Attachment to 4 According to the Jains religion there are wordly objects. four causes for such attachment, viz.,
क्रोध, मान, माया, लोभ. कुमारवदन The faces of Ru. G This War-god ia gupposed to have aix
mara or the faces. C. षण्मुख.
Hindu war-god. केशव ... A name of Vian. vide उपेन्द्र. . क्षपाकर ... The moon ... | Vide इन्दु. ख ... Bky ... ... o ride अनन्त. खर .... ....... 6 गगन ... sky ... ... 0 Vide अनन्त. गज Elephant ... 8 Vide इभ. गति ... Passago : passage 4 According to the Jaina roligion nouin may into re-birth.
have four kinds of embodimen", vis.,
___४ देव, तिर्यक, मनुष्य, नरक. गिरि ... Monntain ... 7 Vide अचल. गुण ... Quality ... 3 Primordial matter is maid to have three
__ qualition', vir., सत्व, रजस्, तमस्. ह ... A planet ... 9 Nine planets are recognised in llinda
Astronomy, viz., Mars, Moroury, Jupi. ter, Venus, Hatorn, Rahul, Kotu the
8nn and the Moon. चक्षुस ... The eye ... ! Vide अक्षि. चन ... The moon ... 1 Hide इन्दु. चन्द्रमस ... The moon ... 1 Vide इन्दु. जलपरपथ 8y ... .. o Vide अनन्त. जलाधि ... Ocean ... ... . Vide अधि . जलनिषि.. Ooosn ... ... - Vids आन्ध.

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290
GANITASĀRAMANGRALA.
जिन
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
Åar, 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
यजमान. at ... Evidence ... 6 The six kinds of evidence are 97
39417, JUHTA, 375, suffer, and
भनुपलब्धि. arealet. Viępu ... ..., 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 Durgă 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

द्विप
द्विरद
द्वीप
धातु
धृति
ཝཱ,
नग
मन्द
नभस्
नय
नयन•
नाग
faध
नेत्र पदार्थ
पचन
...
...
...
...
...
...
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
पयोधि Ocean पयोनिधि... Ocean पावक 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 इभ.
Vide इभ.
There are seven such divisions, vis. जम्बू,
प्रक्ष, शाल्मली, कुश, क्रीश्च, शाक, पौष्कर
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 अचल.
Nine Nanda kings are said to have reigned in Magadha.
Vide अनन्त.
According to Jains there are two Nayas: द्रव्यार्थिकनय and पर्यायार्थिकनय
Vide अक्षि.
Vide इभ.
Nine famous treasures are maid to belong to Kabers, the god of wealth, viz.,
पद्म, महापद्म, शङ्ख, मकर, कच्छप, मुकुन्द, कुन्द, नील, खर्व.
Vide अक्षि.
The Jainas recognize nine categories of things.
Sometimes eight and sometimes seven principal serpents are reckoned in Hindu mythology.
$ Vide आग्ध.
4 Vide अब्धि.
8 Vide अग्नि.

Page #494

292
GANITAS RASANGBABA.
go
... City
...
...
3
Three oitios representing three Asuras are said in the Parāmas 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 अचल. 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

रख
रव
रन्ध्र
रस
रुद्र
रूप
लग्घ
लब्धि
लेख्य
लोक
लोचन
वर्ण
वसु
...
·
⠀⠀
BBBB
...
...
...
वडि
वारण
वर्ष
विधु विषाध विषनिधि.. विषय
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
वियत् Bky
Ocean
Осевд
Object of sense...
APPENDIX Í.
...
...
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, सम्यग्दर्शन, सम्यग्ज्ञान, सम्यक्चारित.
Nine gems are usually recokned, viz.,
वज्र, वैदूर्य, गोमेद, पुष्पराग, पद्मराग,
93
मरतक, नोल, मुक्ता, प्रवाल.
There are nine chief openings in the human body.
The six principal tastes are मधुर, अम्ल, लवण, कटुक, तिक्त, कषाय.
Vide मृड.
Everything has its one only shape.
The nine powers to be attained are अनन्तदर्शन, अनन्तज्ञान क्षायकसम्य, क्स्त्वक्षायकचारित्र, अनन्तदान, अनन्तकाभ, अमन्तभोग, अनन्तोपभोग and अनन्तवीर्य. Vide लब्ध.
Vide भुवन.
Vido अक्षि.
Those deities are considered to be eight in
number.
Vido अग्नि.
Vide इभ.
Vide अब्धि.
Vide इन्दु.
Vide अब्धि.
Vide effort.
The objects oognisable by the five organs
of sense are five, viz. गन्ध, रस, रूप, स्पर्श, शब्द.
Vide अनन्त.

Page #496

294
GAŅITASİBASANGRAHA.
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. व्यसन 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 ... ..... सलिलाकर 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.
हिमकर

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
Aristanēmi
Arbula
Arjuna
Asita
Aička
Avali
Ayana Bija
Bhaga
...
...
Aundra-Aundraphala
Bhayabhdga Bhd gabhyd sa
Bhajya
***
Bhd gandra... Bhagamatr
BMra
Bhaganubandha Bhd gd pavá 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 Bhagipaváha 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
GANITARŠRASANGRAHA.
Bhinnakufftkdra ...
Cakrikabhañijana ...
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 ódnti ... ... 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 Kööbha .. Koori Kudaha or Kudaba
...
Kwmbha ... Kunkuma ...
Kuravaka ... Kufaja Kutfikara ... Labha Lakra Lanka
Lανα .. Madhuka ... Madhyadhana
...
Mahakharga Μαλάkoitud Mahaksőbha Mahaksoni Mahápadma Mahá 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 prógrce
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
Märga
χάρα
คน
Mifradhana Urdanga ...
Muharta Mukha Nola
Xolamitra...

Page #503

APPENDIX 11.
801
Muraja .. Nandyd varla
Narapala ... Nilótpala Niruddha ... Nipka Nyerbuda ... Pada Padma ... Padmard ga. Paisdcika ...
...
..
.
Pakya Paal
Papa
'. ,
.
Panava
...
Paramánu.. Parikarman Pariva Påfali
... 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 ... Prakpöpaka-karana Prawda ... ...
Prepargid
...
Prastha
Tabaco
Pratyuipenna
...

Page #504

Bö
.
GANITABİRASLAGBAHA.
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 90° 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 ... Särasa ... 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
Tramarānu ... 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
...
...
Ubhayenipõdha Ucohod na Utpala .. Uttaradhana
... ...
Utaramitradhana
...
...
tha Vafra
Vajrd parartona
Vakula
...
.
VIH Voluwmf filed na ...

Page #506

804
GANITASI RASANGRALA
Vardhamana
...
Vargamela Varna
...
· Vicitra-kutfikára ...
Vidyddhara-nagara
Vivamakuifikára ...
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 90° to the East of Lanka. ... Penance; practice of meditation and mental oan.
centration. A mesure of length. Vide Table 1, Appendix II.
Vitasti ... Vrracha ... ... Vyaváhard ngula ... Vyutkalita
Κανα
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 õi 18° 26° 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, Tõ' 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
GANITASIRAKANGRAÐA.
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 dröras 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. ģ; 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 drönas 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 ; 110€; 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
GANITASĪBASANGRAHA.
(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) äici 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 kráda 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+). í of a day. (141). 2. (167 to 17). Ā 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
GANITASĀBAKANGBANA.
3. MEASURES OF CAPACITY (GRAIN MEASUREMENT).
4 fodaftada 4 Xudahas 4 Prasthas 4 Adhakas 4 Dronas 4 Manis 6 KMris 4 Pravartikdo 6 Pravartikás
= 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 Satóra. = 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

Page #528


Page #529


Page #530


Page #531
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..