The Bakhshali Manuscript: An Ancient Treatise Of Indian Arithmetic

41

THE

BAKfISHALI MANUSCRIPT

Svami Satya Prakash Sarasvati

Dr. Usha Jyotishmati

THE BAKHSHALI

P 1'

AN ANCIENT TREATISE OF INDIAN Alt I'I'I I N 4F,'

EDITED BY

Svami Satya Prakash Sarasvat I and

00 Usha,jyotishmati, M. Sc., D. Phil.

Dr. Rataa Kumari Svadhyaya Sansthani ALLAHABAD.

PUBLISHE S BY

Dr. Ratn Kumari Svadhyaya Sansthana Vijnaua Parishad Buildings Maharshi Dayanand Marg Allahabad-211002 Phone : 54413

FIRST EDITION

1979

Price Rs. 50/- ( £ 3.5 or $ 7 )

Printed at :ARVIND PRINTERS 20-1), Bell Road, Allahabad, Phone Not 3711

CDr `-Patna Born 20-3- 19 12

umari Died 2-12-1964

PREFACE

Dr. Ratna Kumari, M. A., D. Phil. was deeply interested in education, higher research and scholarship, and when she died in 1964, the Director of the Research Institute of Ancient Scientific Studies, Now Delhi, graciously agreed to publish in her commemoration a Series to be known as the "Dr. Raffia Kumari PubllcaNnu Series", and tinder this arrangement, the five volumes published were: Satapatha Brahmanain. Vol. I, 11 and III (1967, 1969, 19711): ltaudhayana Sulba Sutram (1968) and the Apastanrba Sulba Sutruun (1968). It is to be regretted that in 1971, Pundit Rain Swarnp

Sharnta, the Director of the Institute died and shortly afterwards, the activities of the Institute came to a close. In 1971, from an endowment created by the relations of late Dr. Ratna Kurnari, Dr. Ratna Kumari Svadhyaya Sansthana, a research orgunicution for promotion of higher studies amongst ladies, was ostahll. shed at Allahabad, with Sri Anand Prakash, the younger son of 1)r. Ratna Kumari as the first President. Svumi Satya Prukunlt (formerly, Prof. Dr. Satya Prakash) has authorised Dr. Ratna Kumari Svadhyaya Sansthana to publish several of his works, plirti. cularly, all of them which were published by the Rcicarch Institute of Ancient Scientific Studies, West Patel Nagar, New Delhi, and has assigned the right of publication of these works to the' Sansthana.

We are pleased to offer to the public for the first time, the classified text of the Bakhshall Manuscript in Devanagari script. The

manuscript was literally dug out of a mound at Bakhshuli on the north-west frontier of India in 1881, and Dr. Hoornlc eurefully examined the manuscript and published an English translation ol'tt few of the leaves in 1888. This being the earliest manuscript on Indian Arithmetic, it aroused considerable interest in the West. '11w Archaeological Survey of India published the manuscript in Paris I, 11 and III. as edited by 0. R. Kaye. In 1902, Dr. Hoernle presented the manuscript to the Bodleian Library. 1 his distinguished work of Kaye is now not available in market. Dr. Aibhuti Bhusun

Datta of Calcutta University also made detailed study of this manuscript,

Vi

Dr. I Jsha Jyotishmati, a distinguished member of the Sansthana, took onsiderable.pains in editing the text, and we are obliged to Svami Saty,t Prakash Sarasvati for his supervision and critical introduction. We r re also obliged to Sri Jagdish 111rasad Misra,

B A., LL. B., for his dedicated services to the Sansthana, particularly in the publication programmes.

We regret to announce the premature death of our first President Sri Anand Prakash on December 13, 1976. He was a distin-

guished graduate of the University of Allahabad, and took his graduation in Mechanical Engineering from the University of Glasgow. and since then he had been working in industry. To cherish his memory, we have the privilege of dedicating this little volume.

The Sansthana Allahabad Ramanavanti April 5, 1979.

Prabha Grover

M Sc., D. Phil. Director

Anand'Prakaok Born t-4-1:)38

Died 13-12-1976

To Our President ANAND PRAKASH Who cliedat

au early age of 38.

CONTENTS

N Pltol)UCTION Chajakputra, the scribe

Ilow the text was found? The manuscript The script The language

7

Ortlor and arrangement of folios

7

Contents of the manuscript (ioncralized arithmetic and algebra

8 111

't'ext is of Indian form

Il

IN the work homogeneous ?

12

The script (details) An indigenous Hindu treatise Various opinions about its age

12 13

14

Ilakhshali manuscript older than the present manuscript

15

Ilakltshali work, a commentary

1l

Present manuscript-a copy

18

Exposition and method

20

fundamental operations

32

Square root and surds

35

Rule of three

40

Calculation of errors and process of reconciliation

41

Negative sign

45

Least common multiple

47

Arithmetical notation-word numerals

SO

Regula falsl

52

Known still earlier in India

54

The rupona method

55

ii Proofs

59

Solutions Symbols for the unknown Plan of writing equations Certain complex series Systems of linear equations Quadratic indeterminate equations Happy collaboration of the east and the west. Appendix : Weights and measures The arithmetical problems incorporated in the Bakhshali Text The Bakhshali Text (Transliterated in Devanagari

65

Script) Index

66 70 71

72

77

78 81

85

... 1-71 ...

109

INTRODUCTION Chajakputra -The Scribe :

While we pay homage to the unknown author of it text cause to be known as the Bakhshali Manuscript, we should express our gratitude to another person of unknown name who scribed tho manuscript from a text quite lost to us, and not only that, since thI't scribe was himself a mathematician of no less repute, being known as it prince amongst calculators, he added illustrations and otien added something to the i unning commentary of this Text. 1houph we no independent source to corroborate, it is definite that u

Text c.isted as early as the second or third century A. I). it' not earlier; this I ext was traditionally used by the lovers of calculations

(teachers and taughts) in some parts of the country (specially the north-western parts of the Aryavartta), and as it was handed down to us generation after generation, new worked and unworkcd problems were added to the existing material. At the end of the mown. eript (or somewhere on Folio 50, recto) we find a colophon mention-

ing that the work was scribed by a certain Brahmana, a prince of calculators the son of Chajaka (rq;). Thus we know the sunk of the father of the scribe only. I would like to call the scribe of the present manuscript himself as Chajaka-Putra (tqq; qa), the son of C'hajaka. He himself is not an author of the Text; he merely copies it out from some other text already current. Even if the date of copying this text were so late as the tenth or eleventh century and even it' some illustrations included in the manuscript were not of it very early origin, the Bakhshali Text has its great importance, this being one of the earliest Texts in history available to us on the science of calculations. Blow the Text was Found ?

In 181, a mathematical work written on birchbark was found at Bakhshali near Mardan on the north-west frontier of India. 1 his manuscript was supposed to be of great age and its discovery urouned considerable interest. Part of it was cxumincd by Dr. H ocrMe,

i

Bakhshali Manuscript

who published a short account of it in 18831, and a fuller account in 1886x, whic&i together with the translation of a few of the leaves was republished in 18883. Dr. Hoernle had intended, in due course, to publish a complete edition of the Text, but was unable to do so. The work was later on printed and published in 1927, under the title "The Bakhshali Manuscript", Part I and 11, by the Government of India with photographic fascimiles and transliteration of the Text together with a very comprehensive introduction by G. R. Kaye4. This was followed by the publication of the Bakhshali Manuscript, Part 111, in 1933 as the Text Rearranged shortly after Kaye's death. Dr. Bibhutibhushan Datta, a distinguished worker in the field of Indian mathematics, also published a critical review on the Bakhshali Manuscripts. Bakhshali (or Bakhshalai, as it was written in the official maps) is a village of the Yusufzai subdivision of the district of Peshawar of the North-Western Frontier of India ( now in Pakistan ). It is

situated on, or near, the river Mukham, which eventually joins the Kabul river near Nowshers, some twenty miles further south. Six miles W. N. W. of Bakhshali is Jamalgarhi, twelve miles to the West of Takht-i-Bhai and twenty miles W. S. W. is Charsada, famous for their Indo-Greek art treasures. Bakhshali is about 150 miles from Kabul, 160 from Srinagar, 50 from Peshawar, 350 from Balkh and 70 from Taxila. It is in the Trans-Indus country and in ancient times was within Persian boundaries -in the Arachosian satrapy of the Achaemenid kings.

It is within that part of the country to which the name Gandhara 1.

Indian Antiquary, XII (1883), pp.

89-90.

2. Verhandtungen des VII lnternationalen Orientalisten Congresses, Arische Section, pp. 127 et. seq. 3. Indian Antiquary, XVII (1888), pp. 38-48, 275-279.

4. The Bakhshali Manuscript--A Study in Medieval Mathematics, Part I and II, Calcutta, 1927. Kaye made two previous communications on the subject matter of the Bakhshali work: (i) Notes on Indian Mathematics-Arithmetical Notation (J. Asiat. Soc. Beng., 111, (1907) and (ii) The Bakhshall Manuscript, ibid.

VIII, 1912.

5. Datta : Bulletin of the Calcutta Mathematical Society, 1929, XXI, p. 1, entitled the Bakhshall Manuscript. Also rcforoncos in the History of Hindu Math,. matics, By Datta and A. N. Singh, 1935 and 1962 editions.

I low the'I'ext wit'; found

3

has been given, and was subject to Iliose wc';tern inflpenccu whit It IIIN e0 bountifully illustrated in the so-cnllcd (7andhara nil. I

The authentic record of the discuveiy of the ntnnuNcript nl'pr. gra to be contained in the following letter dated the 5th ofJuly, I1tiil I. from the Assistant Commissioner at Martian.

"In reply to your No. 1306. dated 20111 ultimo, and iln coololttree, I have the honour to inform you that the remains of Ihtl papyrus MS. referred to were brought to me by the lnspcctttr of police. Mien An-Wan-Udin. The finder, a tenant of the latter, Hnid he had round the manuscript while digging in it ruined stone enOlasure on one of the mounds near Bakhshali2. These mound,, lie on the west side of the Mardan and I3akhshtlli roads and nro evidaltly the remains of a former village. Close to the same spot the Mite round a triangular-shaped 'diva', it soap-stone i'cncil, rued it large Iota of baked clay with a perforated bottom. I had march made but nothing else was found.

it

fuilhty

"According to the finder's statement the greater part of Ihn Manuscript had been destroyed in taking it up from (lie place %%hviv III lay between stones. The remains when brought to inc weir Ill,v

dry tinder, and there may be about fifty pages left sonic of which -Would be certainly legible to any one who knew the characters, The litters on some of the pages are very clear and look like some kiln) of Prakrlla (sTTr

), but it is most difficult to separate the pages

Without injuring them. I had intended to forward the manuscript to the Lahore Museum in the hope that it might be sent on thence to

fiMe Scholar, but I was unable to have a proper tin box made 1'brr it before I left Mardan. I will see to this on my return from Inve. The papyrus will require very tender manipulation. 'I'lle fSNlt will be interesting if it enables us to judge the age.of the tutu whore the manuscript was founds." I.

Apparently iha manuscript was found in May 1881.

General ('unnigham in a private letter to hr. Hocrnic, dated Sinda,

Kilt

June, 1882, nays : "13akhrhali is 4 miles north of Shuhbaxgarhi. II Is it nunnnl with the village on the top of it The birch-hark manuscript was found la it field near a well without trace of any building near the spot, which Is autnklu

the ntonnd vlllag.......".

3. kayo hits, however, expressed doubts reltardling the uulhontitily of thin aISOUnt, for, he thinks. It was written app:ucatty from menmoty, it munlh nt $ e for the discovery of the rnnnuscript.

4

Bakhshali Manuscript

In the meantime notices of the discovery had found their way into the Indian newspapers. Professor Buhler, who had read of the discovery in the "Bombay Gazette,' communicated the announcement to Professor Weber, who brought it to the notice of the Fifth International Congress of Orientalists then assembled in Berlin. In

Professor Buhler's letter to Prof.

Weber it was stated that the

manuscript had been found "carefully enclosed in a stone chamber"

and it was thought that the newly discovered manuscript might prove to be "one of the Tripitakas which Kaniska ordered to be deposited in Stupas".

Kaye, however, says that there is nothing whatever in the record of the find to justify Buhler's sta'ement, which seems to have

originated in a rather strange interpretation of the words "while digging in a stone enclosure" that occur in the letter quoted above, and which are themselves of doubtful reliab;lity. The manuscript was subsequently sc nt to the Lieutenent-Gover-

nor of the Punjab, who, on the advice of General Cunningham, directed it to be transmitted to Dr. Hoernle, then head of the Calcutta Nladrasa, for examination and publication. In 1882, Dr. Hoernle gave a short description of the manuscript before the Asiatic Society of Bengal, and this description was published in the Indian Antiquary of 1883. At the Seventh Oriental Conference, held at Vienna in 1886, he gave a fuller account which was published in the proceedings of the Conference, and also with some additions, in the 1. This accont appears in the Bombay Gazette of Wednesday, August 13th, 1881 and is as follows : "The remains of a very ancient papyrus manuscript have been found near Bakjtshali, in the Mardan Tahsil, Peshawar District. On the west side of the

Mardan and Bakhshali road are some mounds, believed to be the remains of a former village, though nothing is known with any certainty regarding them, and it was while digging in a ruined stone enclosure on one of these mounds the discovery was made. A triangular-shaped stone 'Diwa', and a soap-stone pencil and a large lotah of baked clay, with a perforated bottom, were found at the same place. Much of the manuscript was destroyed by the ignorant finder in taking it up from the spot where it lay between the stones; and the remains are described as being like dry tinder, in sonic of the pages However, the character, which somewhat resembles 'prakrita'(A(0), i s clear. and it is hoped it may be deciphered when it reach^.s Lahore, whither we understand it is shortly to tc sonl".

Manuscript

hallrrn Anllquary of 1888. In 1902 1)r. n Library.

$

Iloernle pichcnlvd

IIn'

t

The Manuscript

The manuscript consists of some 70 loaves of birchhark, but some of these are mere scraps. The largest loaf measures about 5.75 by 3.5 inches or 14.5 by 8.9 centimetres. The loaves, which are numbered according to the Bodleian Library arrangement from I III 70, may be classified according to their size and condition as follows

In fair condition but broken at the edges-sivc, not less flame 5 by 3 inches (13 by 8 cros.)

I, 4, 5. 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 211, ?2, 23, 24, 32, 33, 34, 36, 37, 43, 44, 47, 49, 59, 60, 61, 62, 63

.

Total 35.

Rather more damaged but otherwise in fair condition not lemv than 41 by 2 inches (12 by 5 ems.) 2, 25, 26, 42. 45, 46, 48, 50, 51, 52, 55, 56, 57, 58, 64. 69

Total 16. Much damaged 21, 31, 35, 41, 5', 66, 68-Total 7.

Scraps. 27, 28,

29, 33,

35, 38, 39, 40, 54, 68, 70-Total 11.

One folio (19) is entirely blank.

Certain folios consist of two leaves stuck together, namely 7, ,11 and 65, and possibly others. It would not be difficult to separate these double leaves without damaging the manuscript.

The leaves are now mounted between sheets of mica and placed within an album. The mica sheets arc about 7.4 by 4.6 inches and are fixed together by strips of gummed paper at the edges leaving it ulcer area of 61 by 3j inches,

Bakhshali Manuscript

6

Possibly the original strip of birchbark from which the leaves of Bakhshali

manuscript were taken was roughly of the shape of the annexed diagram and was cut up into the oblongs indicated. If A, B, C, etc., represent the upper layer, and A', B', C', etc. the lower layer, then, according to the evidence of the leaves themselves, they were arranged for purposes of writing upon in the order A, A'; B, B'; C, C' ; etc., or A, A' D' ; etc.

A

B

C

f1

F

F

G

H

I

L

J

Nl

P

Q

P.

j

Regarding the format of the Bakhshali manuscript (ratio 1.7) as a criterion of age, Kaye says that he could come to no positive conclusion from the evidence before him. Dr. Hoernle, however, writes as follows :"It is noteworthy that the two oldest (Indian paper) manuscript known to us point to their having been made in imitation of such a birch bark prototype as the Bakhshali manuscript."

Kaye, however, does not accept this argument, for it would be quite as reasonable to cocclude that the Bakhshali format was determined by the paper manuscript formats, and that it is of later date than the introduction into India of paper as a writing material; and this, according to Kaye, would place the Bakhshali manuscript about the twelfth century of our era at the earliest. The Script

The Bakhshali text is writ',en in the Sarada (V1TT-6,T) script, which flourished on the north-west borders of India from about the Ninth Century until within recent times. Its distribution in space is fairly definitely limited to a comparatively small area lying between

longitudes 72 and ;8 east of Greenwich and latitudes 32 and 36 north

Dr. Vogel distinguishes between Sarada proper, of which the

latest examples are of the early Thirteenth Century, and modern

Sarada.

The writing of the Bakhshali manuscript is of the earlier period and is generally very good writing indeed, It was written by at least

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A page of the Bakhshali Manuscript ( 17 verso )

The Script

,)

two scribes. Kaye gives a detailed account of it, anti ham clnortilk

t

the writing styles also (a and R as he denotes

Style it it divided into four sub-sections which possibly belong to the woo 1, 401' four separate scribes, although it is not easy to point out any fluid: mental differences between these styles. Folio 65 possibly exhibit:. the writings of two separate scribes on the two sides, which do not belong to the same original leaf.

The style Q is distinguished by its boldness by 4 tails" or flonri.

shes (including very long virama marks), the methods of writing mediale, ai and o, thj looped six, etc , etc. A particular section, which Kaye calls as M, has no example of ai and contains practically all the examples of "o". Sections 1. (ar) and G (a3) show marked differences in the methods of writing e, while section F is, in this matter, very much like section L.

Stie.

tion C (a2) has no example of either the JIHVAMULIYA (Nt.jtqor UPADHMANIYA, (34szrt'Pr), and so on; but it must he borne in mind that these statistics are only of value in the mums. The Language

The language of the text may be described as an irregular Sanskrit. Nearly all the words used are Sanskrit, and the rules of Sanskrit grammar and prosody are followed with some laxity, The peculiarities of spelling, .sandhi, grammar, etc, that occur in the text

are exceedingly common in the inscriptions of the Eleventh and Twelfth Centuries found in the north-west of India. Dr. however, implies that the language is much older. He states that the text "is written in the socalled Gatha dialect, or in thty literary form the North-Western Prakrita which which preceded the employment, in secular composition, of the classical Sanskrit". He also

states that this dialect "appears to have been in general use, in North-Western India, for literary purposes, till about the end of the Third Century A. D". Order and Arrangement of Folios

The Bodleian Library order of the folios, which was definitely

fixed by circumstances not within the control of the edttorr has

9

Bakhshali Manuscript

necessarily bden followed by Kaye in his edition, in the arrangement of the facsimilqs and the first transliteration of the taxt (Part I and Part II). The whole material was rearranged in the posthumous publication : The Bakhshali Manuscript, Part Ill. The leaves were disarranged to some extent before they reached

Dr. Hoernle, but unfortunately he did not leave any proper record of the order in which the leaves reached him.

Dr. Hoernle attempted to arrange the leaves on the basis of the numbered sutras; but the numbered sutras were two few and too unevenly distributed to serve this purpose, and in some ways they were even misleading. The chief criterion of order is, of course, the nature of the contents of the leaves, but Kaye made an attempt to utilise all available criteria as help towards the solution of the problem of order. Had all the leaves been extant, even if fragments, the problem could have been completely solved; but some leaves are completely missing and many fragments have disappeared altogether, so that the problem could be only partially soluble Sometimes, even

the knots in the birch-bark were of assistance, and besides this natural aid there was the accidental one of the effects of the method of storage. Possibly for some hundreds of years the bundle of leaves was subject to a certain amount of pressure, and was exposed

particularly at the edges, to chemical and other disintegrating actions. Some of the leaves stuck together, the edges of all became

frayed and certain leaves became so frail as to break up into scraps when handled. On the principle that contiguous leaves would be affected approximately to the same extent, we might, if no disintegrating effects had taken place since the find, rebuild in layers the original bundle. But we know that further disintegration has taken place; however, similarity in size and shape and mechanical makings were of distinct help in rearranging the leaves. Contents of the Manuscript

The portions of the manuscript that have been preserved are wholly concerned with mathematics. Dr. Hoernle described the work in 1888 in the following words :"The beginning and end of the manuscript being lost, both the name of the work and its author are unknown. The subject of the

Contents of the Manuscript

4)

work, however, is arithmetic. It contains it great variety of lun hlems relating to daily life. The following are examples : (i) in a carriage, instead of 10 horses, their are yoked ow distance traversed by the former v as one hundred; how much will the other horses be able to accomplish" ?

(ii) The following is more complicated :- 'A certain travels 5 yojanas on the first day, and 3 more on each succerdinlt day ; another who travels 7 yojanas on each day, has it start of S days ; in what time will they meet ?'a (iii) The following is still more complicated :--'Of 3 merehnnln, the first possesses 7 horses, the second 9 ponies, the third 10 e.ich of them gives away 3 animals to be equally distributed anions themselves. The result is that the value of their respective tics become equal : how much was the value of each animal ?'s "The method prescribed in the rules for the solution of thene problems is extremely mechanical and reduces the labour of thinking to a minimum".

The following is a summary list of the contents of the work its far as its present state allows of such analysis :Section A

Problems involving systems of linear equations Indeterminate equations of the second degree Arithmetical progressions Quadratic equations Approximate evaluations of square-roots Complex series

A and K B and C O C

F

0

Problems of the type x (1-al) t.

rf rvu

r

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rri gar: a;' (VI j#Tzrm...gtkt q gxfgt pit wt nfomt rv# art srrforat ggq; gfq ......"%* a&I -Folio Wfffsa dvor. I i

T

3, verso,

2

Bakhshali Manuscript

'1o

The computation of the fineness of gold Problems oR income and expenditure, and profit and loss Miscellaneous problems Mensuration

H

L, D and E M

Such is a very rough outline of the work as it now stands. Perhaps the most interesting sections are C, A and M; and of these C is the most complete and was evidently treated as of cosiderable importance. Section A is also of special interest as it contains examples which may be described as of the epanthcm type. Section M is of interest principally on account of the methods of expressing the numerous measures involved and also because of its literary and social references. Generalized Arithmetic and Algebra

Although the work is arithmetical in form it would be misleading to describe it as a simple arithmetical textbook No algebraical symbolism is employeJ, but the solutions are often given in such a general form as to imply the complete general solution, i. e., the solutions, though arithmetical in form, are really generalized arithmetic, or algebra. First of all a particular rule is given, which is intended to apply to the particular set of examples that follows. These

rules are often expressed in language that would be impossible to interpret without the light thrown upon them by the solutions..The examples are themselves sometimes trivial, but the .solutions, often expressed, with what at first glance appears to be meticulous care, often redeem the examples from their apparent triviality. Proofs or verifications are often given with some elaboration and on occasions are multiplied.

Kaye has said that the work may be divided roughly into algebraic, arithmetical and geometrical sections, knowing of course that the boundaries of these sections are not clear, but according to him it would be more correct to classify the problems as (a) academic, (b) commercial, and (c) miscellaneous. Judging by the manuscript as it now stands, the problems involving geometric notions were comparatively very few, and we can only guess at the meanings of the remaining fragments dealing with this branch of mathematics.

Generalized Arithmetic and Algebra

One pleasing feature is the small space occupied by om,-,ier offal problems. There is only one problem on and a unobtrusive section containing problems on profit and los,,.

t

Those problems classed as academic are concerned with cular mathematical notions that in early mediaeval times hn,l it traditional value and interest, such as the ('pantheina, the rr;,;uhi virginum, certain indeterminate equations of the second degree, and certain sets of linear equations.

The miscellaneous problems include examples where the :iiel' interest is rather in the illustrative material than in the notions involved; e. g., there are problems concerned with the abduetion of Sita by Ravana, the prowess of Haihaya, the constitution of an army, the Sun's chariot, the daily journey of the planet Satin ii. gifts to Siva, etc. etc.

Such, or similar features, are, however, common to many mediaeval mathematical works. The l3akhshali manucript is, however. almost unique in at least two respects of some mathematical importance: (i) The first of these is the employment of a special sign in tin, form of a cross-exactly like our plus sign but placed after the

quantity it effects- to indicate a negative quantity. (ii) The second special characteristic consists of the set of methods for indicating the change-ratios of certain measures.

Text is of Indian form Whether of a purely Indian origin or not, Kayc says that the work is Indian in form. It is written in a sort of Sanskrit and generally conforms to the Indian text-book fashion but there are certain apparent omissions. Perhaps the most noteworthy feat ture of the classical Hindu texts is their treatment of indeterminate equa. tions of the first degree, while their greatest achievement is the full solution of the so-called Pellian equation. A great part of the texts of Brahmagupta, Mahavira and I3haskara are devoted to one or both of these topics, but there is no evidence of either in what remains of the Bakhshali text; and this apparent omission is the more noticeable, inasmuch as there is evidence of considerable skill in the treatment of systems of linear equations and certain indeterminates of the second degree. Another omission to note is of a different character altogethor. Every early Hindu work of this kind has a section rclatinlj to

Bakhsbali Manuscript

12

the "shadow of a gnomon", but in our text there is no evidence of such a section.,, We must not, however, as Kaye says, "pay too much attention to these apparent omissions. The possibility of entire sections of the manuscript being destroyed is not great, but negative evidence and a mutilated manuscript do not carry us very far.

Is the Work Homogeneous ? On the first examination of the manuscript it would be noticed that the writing was not uniform, and that, in particular, certain leaves different'ated themselves from the rest by a bolder and,on the whole, a better style of writing; and Kaye has distinguished this set of leaves as the "M" section. This early differentiation was a most useful one for it marked not only a difference in style of writing but also one of matter. Indeed this "M" section proved to have to many peculiarities that the idea that it was possibly a separate work could not be ignored. But the rest of the manuscript is by no means uniform in style of writing or anything else, and Kaye is not convinced that the "M" section is the work of a separate author, although, he rather suspects that it is. It is, however, pretty certain that it was the work

of a separate scribe; but, as there are slight indications that the other portions of the manuscript were dictated, this does not affect the question of authorship conclusively.

The peculiarities of the "M" section, although they may not prove heterogeneity of workmanship, call for some special mention and are here summarized. 1.

The Script :

(a) The writing is bolder and, on the whole, more uniform than that of the rest of manuscript.

(b) Flourishes or extensions of the bottom endstrokes are common. These flourishes occur particularly at the end of ligatures,

but also in the cases of the numeral figures "5", "7" and "9". and even in the case of the stop bars, and occasionally they even occur at the ends of the frame-works of the "cells" (e. g., see fol. 47, etc.).

(c) The numerical symbols of section "M" have been shown in a table by Kaye [not reproduced here, Table IV (7) line 1], where the looped ' 6" should be noted. This is useful Ltzt not an infallible criterion,

13

Is the Work Homogeneous ?

(d) The following table relating to the formation of modinl vowels a, i, and o taken from Part 11. Medial

'M' section Whole, manuscript

Medial of

ai

ai

ui

o'

0%

100%

32%,

32%

24%,

76%

75%,

15%

o'

n' 1n';;,

Here the indication appears to be very definite indeed, but it must be borne in mind that such criteria only apply in tho unnN'i, that the total number of ai examples is only 19, and so on. i II. It is curious that all the mythological and semi historical references (see Sections 47-48) occur in the "M" section. IIIderd this section is peculiarly Hindu in contrast with the remainder of the manuscript.

III. The mathematical contents of the "M" section may ho described as miscellaneous problems, which are generally solved by "rule of three", but a special feature is the occurrence of numerous "measures" and a special method of exhibiting their change-ratio'. But, of course, these points in no way indicate work-rather other-

wise for if section "M" were an entirely different work we aright expect some duplication, and there is none here. IV. The method of exposition is somewhat dif erent. The example is followed by a statement and the answer is then given, generally, without any detailed working, and generally there is no "proof" or verification. There are, however, exceptions. V.

There are differences in language. Certain technical terms

that are extremely common in the rest of the manuscript -do not occur' e g , pratyaya and yuta. An Indigenous Hindu Treatise

Dr. Hoernle believes that arithmetic and algebra developed in India entirely on indigenous lines. According to him "in the Bakhshali manuscript, there has been preserved to us a fragment of an early Buddhist or Jain work on arithmetic (perhaps a portion of larger work on astronomy), which may have been one of the sourccy (I) The symbols used hero are merely mnemonic,

14

Bakhshali Manuscript

from which the later Indian astronomers took their arithmetical in. formation." Again according to Hoernle, the arithmetic in India developed without any Greek influence, "Of the Jains, it is wellknown, says Hoernle, "that they possess astronomical books of a very ancient type, showing no tracees of westeren or Greek influence" (Here is a probable reference to the Sur)'aprajnapti,). In India arithmetic and algebra are usually treated as portions of works on astronomy. Tn any case, it is impossible that the Jains should not have

possessed their own treatises on arithmetic, when they possessed such on astronomy. The early Buddhists too are known to have been proficients in mathematics. The prevalence of Buddhism in North-Western India, in the early centuries of our era, is a wellknown fact." Kaye, however definitely asserts that there is not the slightest evidence in the manuscript itself of its being connected either, with the Jains or Buddhists. It is Hindu (Saivite); the author was a

Brahmana (fol. 50); to Siva, he attributes the gift of calculation to the human race (fo1.50); offerings to Siva are mentioned on more than one occasion (fols. 34. 44); references are made to certain incidents recorded and persons named in the Hindu epics (fol. 32 etc.) and there

is not a single reference that could be construed as indicating any connection with Buddhism and Jainism. Various Opinions about its Age.

The original Bakhshali work has been assigned various dates by several scholars. Hoernle says : "I am disposed to believe that the composition of the former ( the Bakhshali work) must be referred to the earliest centuries of our era, and that it may date from the third or fourth century A. D.I. " This estimation about the age of the original Bakhshali work has been accepted as fair by emirent orientalists like Buhlera and historians of mathematics like Cantores and Cajori4. Thibaut has followed Hoernle in accepting the date of the present manuscript to l e lying between 700 and 900 A. D.5

But Kaye would refer the work to a period about the

twelfth century. "The script, the language. the contents of the work", says he, "as far as they can give any chronological evidence, (1) Indian Antiquary, XV11, p. 36. (2) Indian Palaeography, p. 82. (3) M. Cantor, Gcschichle der Math, 1. P. 598. (4) F. Cajori, "history of Mathematics", 2nd ed. Boston, 1922 p. 85.

(5) G. Thibaut, Astronoiie, astrolo;ie and inathematlk, p.

71.

Various Opinions about its Ago

I'

all point to about this period, and there is no cvid nca wbtitlgltrir incompatible with it"', Bibhutibhusan Datta agrees with Hograle that the work was written towards the beginning oil the

era, and he has discussed this issue in his paper, "The fic Manuscript" published in the Bulletin of the Calcutta Malhenuili'u11 Society, 1929, XXI, p. 1. We shall present his arguments hero.

Bakhshali Mathematics Older Than the Present I unuscript Hoernle thinks that the mathematical treatise contained in Iho Bakhshali manuscript is considerably older than the present niammcript itself. "Quite distinct from the question of the ago of tin,

manuscript", says he, "is that of the work contained in it. 'Photo is every reason to believc that the Bakhshali arithmetic is of it vemy earlier date than the manuscript in which it has come down to uH,"a. This conclusion has been disputed and rejected by Kaaye who thinks it to be based on unsatisfactory grounds. Ile then adds, "ill' course it will be impossible to say dctinitely that the manuscript hi

the original and only copy of the work but we shall be able Iu show that there is no good reason for estimating the age of the work

as different from the age of the manuscript to any considerable degree..' a Kaye has adversely criticised the linguistic and palaeo-

graphic evidence of Hoernle. Datta, however, thinks that Kttyo's arguments, if proved sound and sufficient, will establish at tile 111olit that the present manuscript was written about the twelfth century, as is contended by him.`t Hoernle himself considers it to be not

much older, belonging probably to a period about the ninth century of the Christian era.5 Most of the other reasons of Kayo against Hoernlc's vicw, based on certain internal evidence, such as (i) the the general use of the decimal place value notation, (ii) the occurcnco of the approximate square-root rule and (iii) the employment of the regulafalsi, on imperfect knowledge of the scope and develppmcnt (1) Bakli.

Ms., § 13S.

(2) Indian Antiquary, XVII, p. 36 (4) Bakh. (3) Bakes. Ms. § 122 SS) Indian Antiquary, XVII, p. 36.

Ms, § 1.45

§. In support of this opinion, Kay. states : "There is evidence that Me. Is not a copy at all. It is not the work of a single scribe : there are crony references to leaves of the manuscript; there is a case of wrongly numlic. ring a Sutra and the mistake is noted in another hand-writing" (p. 74 fn ). The facts noted in the latter part of this statement cannot possibly suppom I

what Is stated in the beginning. On the contrary they strongby lend to show that the present manuscript Is a copy.

Bakhshali Manuscript

16

of Indian mathematics. There is, however, other internal evidence of unquestionable value to show that the Bakhshali mathematics cannot belong to so late a period in which Kaye would like to place it. Bakhshali Work a Commentary

There is another noteworthy fact about the work contained in the present Bakhshali From the method of its treatment, Hoernle thinks it to be a Karana work .1 In the opinion of Datta, the Bakhshali work is not a treatise on mathematics in its true sense, but a commentary' a running commentary, of course-on such an earlier work. The manner of its composition and particular'y the very elaborate, rather over- elaborated details with which the various workings of the solution are most carefully recorded, without trying to avoid even unnecessary repetiti,)ns, strongly tend to such a conclusion. Here and there are given explanatory notes of passages, literary synonyms of words and technical terms, some of which will in no way be considered difficult, or which are already well established. For instance, on folio 3 verso the word parasparakrtam ( q-Tx:rtW, ) has been explained by the word gunitam (tata parasparakrtam gunitam U1341gtf?f !I bj f ); again on a subsequent occasion, this latter term has been interpreted as equivalent to another more difficult and less

known term abhyasa (tatra guna abhyasam n jq alvgrtt; folio 27, recto ) s. On another occasion we have avrttipravrttigunanam, aTtjft-kofri (folio 12, recto)S. It is stated in several instances that (1) Indian Antiquary, XII, p. 89.

(2) c c qMzF ic' jkid WI M (folio 3, verso);

q&,- r: (folio 15, verso) ul zrw Vr I gerv vt f7t art 7qw, vt faf y...a }z 13 a m# I cq ;1 fftm (C;

I

i

Y%I araft wrd (3)

T4M,-Mq)j ...6q:

(folio 27, recto) C; 9I am. rff -

u Tff : V . Ar (folio 12, recto)

lakhshali Work it ('ontnuntury ksayampastcum,

rTdTrf;,f

Here is a very

t?

typical gas*ap 60-0

the works. 11.........lwighnamadi' adidviguna 12 i chayo,iihftnm uttaram ato uttaram i patayitva ekam bhnvati ....."

a (yt;;

cltar-i-4eri,,dt. :;l a The above steyle of composition is very commentary. And the whole work is written more or Ie4r in lho

same style.

Then are some cross-references in the Bakhshall Kinuetatipt which are of immense significance. For instance it has been ub vr'v#d about the 10th sutra (rule), which refers to a method of mul!!nlir.rrtrrttt,

that "this rule is explained on the second page"a A similar rrt rk has been made about the 14th sutra that it is "written on t14 Yeventtt page"4 The importance of these two observations in dctcrmiitt31ll

the character of the Bakhshali work cannot be overeat imated.1", It will he easily recognized that those observ.ttion cq,ttM01 in any way be due to the author of the original truati4. For evidently those sutras occur at two places in the work. No author is likely to retain consciously such recurrences in his work and pass them merely by giving a cross-reference. So the duplicationc

as also the observations, must be attributed to a second person, the commentator. And they happened in this way : The original llukltshali treatise was not a systematic work. It was an ordinary eotnpon(1)

(I)

cu

eT T trim =Tfcf

-Folio 12, recto (ii) weirr vg im4 QTf

T1r

-Folio 10, verso

0") TM rT V# qlTff -Folioj4, recto fd r I %rgfa...... i

(2) firs; rrfa i arrf 'T1&ITew

met

i %r ?t i

ark

t

-Foll6 50, vcreo

(3) t r y q$ fg*qq4 far

Tff -Folio 1, recto

(4) RqM ......g#fvr

fRfqM-..ffq (fr)'

-Folio 3, rcctor

(5) 4iT w 0w Trrcq rt 1 tEtftY3it cOsfq rf r g fa -Folio 4, verso 3

Bakhshali Manuscript

18

diurn of mtth!matical rules and examples in which the rules relating to the same topic of discussion even, were not always put together

at the same plane. We notice this irregularity of treatment to a certain extent in the portion of the treatise which has been left to us, It may be pointed out that such irregular treatment is not at all unusual in case of early works and we find another instance o! the kind in the Aryabhatiya of Aryabhata (466 A. D.). The commentator very properly attempted to improve upon the order (rather disorder) of the author, here and there, as far as posssble, without disturbing it too much, by noticing and commenting upon at the same time the sutras which are very closely connected. So that he had

sometimes to explain a sutra earlier than its turn according to the plan of the author. Sometimes a commentator is compelled to refer to a subsequent sutra before time owing to indiscretion of the author.

Therefore, when there comes the proper turn for the explanation of such a sutra, he simply passes it over, very naturally, by giving the cross-reference to previous pages. Thus there will remain very little doubt that the present l3akhshali work is a running commentary

on an earlier work. Further there are found other cross-references which very strongly suggest that the illustrative examples are also due to the original author .1

Present Manuscript-A Copy

Inspite of what is stated on the contrary by Kaye2 there are many things to make one believe that the present manuscript is not the original of the Bakhsbali work, but is a copy from another manuscript. For it exhibits writings of more than one scribe, possibly

of fives. This can be explained most satisfactorily only on the (1) For instance, the author may give an illustrative example which may involve a mathematical principle which is yet to be explained. An instance

of the kind is found in the Trisatika of Sridhara where the auteor very indiscretely gives two examples (Ex.7) in illustration of the Rule 16,which involves mathematical principle explained in the Rules 23 and 24. In this

work the commentator, who is no other than the author of the treatise himself gives the cross-references.

(2) Bakh, Ms., p. 74 footnote. "There is evidence that the Ms is not a copy at all. It is not the work of a single scribe : there are crossreferences to leaves of the Ms: there is a case of wrongly numbering Sutra and the mistake is noted in another hand-writing."

-Kayo (3) Bakh. Ms. p. 11, 97.

Present Manuscript--A Copy

assumption that Is a copy. Further on folio 4,

11 v( r c>,

is

t itvt ;:ri

observation as regards a certain sutra (rule) that ` Eh r is a 1_1 tgt:4 c; in the rule" (.sutra b/:rantineasti).* The style of wriurt ,it t,tir 011twtt!. vation is same as that of other writings on the leaf. So tiic.r(, IN

utely no doubt that all the writings on the leaf are dr:r the flair scribe. Moreover, though this observation is pin:; til )ci tevo 1V. lines of writings, it is not an ordinary case of interlining. From tbn apportionment of space in and about the remark, It is apparent tint the remark was introduced at the time of in tking the copy, but nut on any subsequent occasion. Now that observation cannot be duty t ) the author of the original treatise. For no author would p,144 over a mistake in his work with a mere observation that it is wrong So it must be from another person, possibly the scribe. There IN also another possibility, and there are reasons to believe it to ho more probable, that the scribe found it in the copy which he urged. In any case, it will follow that the present manuscript is it copy. A more conclusive proof of this is furnished by the colophon that the work is "written (likhitam) by a Brahmana mathematician, son of Chajaka, for the education of the son Vasista."1. 1-lad Chajakn been the author of the work, the more appropriate and usual word for this colophon to begin with would have been kriam or vlrae!lam ("composed ').

The scribe seems to be a careless one. For the manuscript is full of slips and mistakes. Here are a few of them :(1) On folio 4, verso, occurs the passage "sodasamasutram 179. Evidently the figure should be 16.

2. On folio 8, recto, a portion, ' uttarardhcnabhajayct" i.. c l)

...... afara: fkTq; mii gar

1'r *;T I

q

q'

' 3quhr4 w

: fwfta-59a nor ".

rarr 'rfma ;fsk faaaf 6 i

a >r i

#t* 3ar-q twink a twii* furl tr

Obi d wezv mum r,

, 04"'

(2) gfOr Ja` * %awr9wiff

"'

I

11-Folio 50, recto, 4, vcrso

tr

arrt!rrz rr t

q

Bakhshali Manuscript

20

deleted'. This was written by mistake for "uttarenabhajet" which

is the relevant portion of the sutra meant for quotation there. The deleted portion can be tracced to a preceeding sutra (folio '1, verso, where we have "uttarardhena bhajitan". 3. On folio 11, verso, 158 1

5 1

64

is twice miswritten for 158. This

latter fraction is once again

13

64

wrongly written as "158 to 1 se l". Another mistake on the leaf 64

is "93 to ...masa 9;" what is meant 931.

(4) In the Bakhshali manuscript, the end of a sutra is usually marked by a special design. Q) Owing to the carelessness of the scribe,

the sign has been put many times at an intermediate place in the sutra. These illustrations are sufficient to show the carelessness of the scribe. Exposition and Method

(i) The text consists of rules, sutras, and examples. There is no explanation whatever of the processes by which the rules were obtained, that is, there is no m.thematical theory at all. In this the work follows the usual Indian fashion, as exhibited in all early texts. But there is a good deal of mathematical theory implied and the rules and examples are often set-forth in such a way as to convey the principles followed quite clearly to the student. (1) (3
%rrW4(T), this is deleted in the MS; this was written by

mistake for

3rZrbi wF

-Folio 7, verso

(2) We have denote.' this design by

exposition and h1cthod

(ii) The rules (sturus) are written in verse, and are ROOM* f+h+. +:s tF i+ numbered; and often in the solutions of the rrrr.,rtg t io the sutras are quoted. When Dr. I-locrnlc tried leaves of the manuscript, he took the numbered miras as tltae 4A5i* too few in number to lend 1(, n Hnti'l of his order, but they scc4 These sutras do not iepresent, as might he factory result. the most valuable part of the text. They are usually of partkuilas application rather than general and are often very ohsoitr.Iy

expressed.

(iii) The examplesgiven are generally formally stated In foil, without the use of notation or abbreviation of any kind; and in n tt A cases they are stated in verse. They are introduced by the terra ufa*2 "an example", After the qucstbp an abbreviation for sometimes comes a formal statcnneat with numeric.ll symboll and abbreviations. often arranged in cells. Then comes the hole intt or working (karuna,*3 and here, sometimes, fragments of the ,u;r+S` are quoted. Finally conies demonstration-often mote of the Harem of verification than proof. Generally these demonstrations, h the

aid of the answer found to the question, rediscover one of the original elements of the problem; and sometimes several such' demonstrations are attached to an individual pioblcm, but times the variation is merely a matter of the form of statement. The full scheme of exposition is therefore

or rule.

Udaharnam*s or "example"; indicated its abbreviation uda*'t sometimes the example is called prasnal*v.

1.

Folios 46, verso, and 65 recto.

2.

Folios, 23, recto 25 verso ; 29, recto, and ,55, verso. Compare folio 35, recto (a) and verso (b).

3.

Folios 32, 36 44, verso and 46, recto.

i hcse references must him: escaped the notice of Hoernic who remarks otherwise.

*9 (W); *k (34TQvri );

*;(

* (n);

r );

(' vq)

i

22

Bakhshali Manuscript Sthapanam

or "statement". Sometimes

it

is

also

called

nyasa2*1 ol nyasa.stl,apanu3*a.

Karanam*3 or -solution".

Pratyayam*4 or .'verification".

(Some

time; a solution

is verified in more than one way.

The end of such sutra is marked after last example by the device and the number of the sutra is also given at the end. The above will explain in general the method of the exposition of the Bakhshali work. But there are also occasional deviations from it. For it is not always that a sutra is illustrated by examples and an example is followed by its solution. There are at least two sutras in the surviving portion of the Bakhshali work which have no examples attached to them. They have been passed over as having been explained or written on preceeding pages. Two examples are left without solution with similar remarks. Again solutions of examples in the beginning of the work have got no verifications, and so also a few others2 .

The above method of exposition differs considerably from what is now commonly met with in other Hindu mathematical treatises. (1) For exawple, on folio 11, verso, there is mention of verification by the fourth method (anvam caturthapratyayam krlyantc) (2) Vide folios 1-3: also folios 23-25.

(3) BrSpSi. ed. by Sudhakara Dvivedi Benaras, 1912, Solutions of some oC the examples h.sva been included as "authentic in Colebrooke'- Sid-

dhanta (H. T. Colebrooke, Algebra with Arithmetic and Mensuration from the Sanskrit of Brahmagupta and Bhaskara, London, 1817) translation of the arithmetical and algebraical portion of Brahmagupta's. But they are not found in Dvivedi's edition of the work. Dvivedl also procured for himself a transcript of the copy of the commentary by Prthudakasvami which was originally secured by Colebrook (vide Dvivedi's Preface, p. 3). In these circumstances, we accept Dvivedi's version of Brahmagupta's above mentioned work to be more authentic. So it wiil have to be said that Brahmagupta did not give any solut on of his exam. pies.

See also the Brahrnasphuta-Siddhanta, New Delhi, 1968.

* t (mmr); * i (mrrt

itmt); * i (q ur ); * it (sr

)I

xpositlon and Method

Brahmagupta (628 A.D.) gives a very few examplt t a. illustration of a limited number of his rules, but not heft

?1 IQ

sclkt1teonp,

This want has been aml.ly made up by his ;nmar,,, comtat trail Prthudakasvatni (860A.D.) who hai supplied sufficient }t farts trative examples with soluiions uuder each rule.Muhuviru(ed.M:1ll,\ 1) ) gives a copious number of examples for each rule". flu calls thetas uddesaka*. But he too does not give solution The first writer Io

give statements (nyasa) as well as answers of his examples called, udaharana is Sridliara (c. 750 A. D.)°. Then comes Bhatihanta. These writers have not recorded working of their solution. And no other Hindu writer is known to have given any verification of the solution of their examples. We do not find this even in the works o:' the latter commentators. The method of grouping sets of figures is of interest, and shows features in common with mediaeval Sanskrit mathematical manuscripts, where also it is the practice to place groups of numbers (iv)

in cells.

In the Bakhshali manuscript, however, this fashion is rather more elaborated than in any Sanskrit manuscript so for examined. T he mathematical possibilities of this scheme do not appear to have been realized and the student must always be careful to interpret any group of figures from the context, and not from any similarity with other groupings. However, there is a certain amount of consistency in the arrangements, as the examples exhibited below will show. The

real purpose of the arrangements appears to prevent confusion by demarcating the numerical figures from the text itself. The text is (1) GSS., edited-with English translation by T. Rangacharya Madras 1912.

OUR ;R"#q (2) Trisatika, edited by Sndhkara Dvivedi, Benaras, 1899.

f?FRRM (3) Lilavati edited by Sudhakara Dvivedi, Benaras, 1910, BiJaganlta, edited by Sudhakara Dvivcdi and revised by Muralidhara Jha, Bcnaras, 1927. English translation of these works are incorporated in Colebrook's Hindu Algebra,

q1' r rfbrir ffhi 7-f?

wow

Bakhshali Manuscript

24

often written almost independently of the figure groups. and a word may be arbitrarily divided by the cell araangement, which may also cut into several lines of the text not necessarily connected with it. The economic neeessity of utilizing the whole of the writing surface of the birch-bark availabe seems to have been the determining factor. The text itself should be consulted but the following examples may be helpful : (a)

Integral numbers occasionally occur without any marking off by lines or cells, but often.

(b) each integral number has a cell to itself, e. g.

I11 (c)

1421

1391

Sometimes an integer is marked off by two vertical bars

:

thus 1 14 1 and invariably a series of integers is thus de. marcated, e. g.

1 20 1 40 1 60 1 80 1

evam

2001

(d) Fractions and groups of fractions are placed in cells or

groups of cells, e. g. (i)

1 132 1 33 1 1

(ii) 16055040625 1 13227520000 I

(vi)

21

20 21

(iii)

t

1

1

1

11

1436 121 (vii)

7 12 12

Exposition and Method

2

te) Complete sets of operations are sometimes murky-,:' For example, (d) (iii) meant. ; I i) (t °Y°.J), (d) (v) 40 (1-(d) (iv) means (1 1-.) (1.1 4)

a similar manner.

tl'

( f) Series of operations may he connected together by cell arrangments; for example 2 di°

1 di°

1

1

2

2

3 di°

1 di°

1

1

2

3

4 di°

I di°

1

1

2

4

I00000 di°

phalam di° 60((X)

947

947

157500 di° 947

phalam di° 60000

216000 di°

phalam di° 60000

947

947

947

which means,

2-21

dinaras: :

1

12 days

:

:

1

1

1 3 days

3

2

:

60000 100000 dinaras : 947 947 days 157600 947

216000 1

60000 947 604147 000

'o

4

(g) The data of the problem and the solution may be indicated in one combined statement; e.g. 16

4 a° che° 2187* 1

phalam sara 2(2440(1

1

1

4 1

3

This is a statement of properties whore the second term means 4

Bakhshali Manuscript

16 0

1a (1-i-i) (1+1) and the number marked with an asterisk is a charge-ratio.

(h) 10 1

12

1

3

3 112 41

I

1

1

1

1

1

dr°

300

1

may be roughly expressed by means of x + 2x + 3 x 3x + 12 x 4x = 300.

(v) The use in our text of the sexagesimal notation in the form in which it occurs is of rather special interest, for there is, as far as we know no other example of the kind in any of the classical Sanskrit

works. The Hindus, from Aryabhata onwards, were well aware of

the advantages of the sexagesimal notation for astronomical purposes. but they never used it for an arithmetical purpose. Apparently there is only one purely arithmetical example of

the use in the text and this example occurs, in connection with a problem in arithmetical progression, on folio 6, verso, and 7. recto, where the fraction 178/29 is expressed as 6 +8' + 16" +33"+61v. This sexagesimal fraction is actually written thus6 8

60 16 cha° 60 33 1i° 60

6 vi° 60 se° 6 29

The upper three figures are missing in the manuscript but the restoration is certain. Of the abbreviation, 1i° stands for lipta, ft" (Gk.

lipte) which in Sanskrit words ordinarily means a minute of arc, or the sixtieth part of a degree; vi° stands for villpta fa' err, ordinarily a second of arc : while se° stands for sesam, or "remainder".

Exposition and Method

Here one may not agree with Kaye that the purely (and perfectly legitima'e) use of the notation points I.

e

teat

influence (Kaye thinks that aithough such a use Sanskrit works, it was extremely common in nirdiauval Munluu

is uii)Mns ;YiA it

works).

It will be noticed that the term lipta hero applies to "third parts instead of 'first parts', and vilipta to 'fourth parts'. abbreviation cha° has not yet been traced to its origin.

[ lw

(vi) The modern place-value arithmetical notation is employed is not the slightest in .lication that to the author, it was a new or strange invention.

throughout the text and Kaye says that there

(vii) As already indicated algebraic symbols are not generally employed, but a symbol for the unknown quantity is. This symbol is the arithmetical symbol for 'nought' or "zero' and on sevciul occasions it is referred to by the term sunya-sthana (> +ql'r) or 'empty place'. The symbol occurs some twenty times but only in sections It,

Co F, G, H, K. Its employment is illustrated in the following exmple ;(a)

( a°

1

I

u°

1

I

pa°

0

I

labdham

10

This is a 'statement' of an arithmetical progression where the first term is 1, the common difference is 1, the number of terms unknown, and the 'quotient' is 10 (where lOx=the sum of the series). Here the symbol simply indicates that the number of 'terms pada is unknown, i.e., that the place in the statement is empty. The symbol does not enter into any operation here or elsewhere. The giving it a denominator of unity is curious and really indicates that it is an integral number. (b)

a°

5

u°

6

1

a°

10 1

pa°

0

pa°

0

1

u°

3 1

I

1

Bakhshali Manuscript

28

Here are two equivalent arithmetical progressions in which the number of terms are equal and the sums also are the same in both cases, but both are unknown. (c)

0

1

I

I

1

1

1

1

1

1

3

3

3

bha°

se

16 1

3

which-means x=16/(1- 13) (1-,:') (t -;) (1- 1) and since this is a

deduction from a problem it is a distinct step towards a proper algebraic symbolism. (d)

0

2

3

4

1

1

1

1

drsya

200 1

which may be represented by x+2xd-3x+4x=2('0 But the method of solution is by the regulafalsi. Any number is put in the place of the 0, and the sum of the series is obta;red. Here the given sum multiplied by the assumed value and divided by the (false) sum is the correct solution... (e)

0

5

yu mu 0

1

1

1

sa I

0

7

1

1

which means x+5-s2, x-7= t. Here the symbol

mu

0 7

0

1 stands for three different unknown quantities. It simply indicates in each case an unknown number.

Negative Sign : the only distinct mathematical symbol employed is the sign for the negative quantity, which takes the form of a cross + placed after the number affected. This is peculiar and has given rise to discussions. In Sanskrit manuscript a dot before the quantity affected is the usual method of indicating a negative quantity. In the transliteration of the text the original negative sign (-i-) is, perhaps illogically preserved; but it is a rather special feature of the manuscript, easily printed, and leading to no ambi(viii)

guity.

On this negative sign Dr. Hoernle writes (1) (1) Indian 4nNyuary,

XVII (1888), p. 34,

Exposition and Method

?1)

"Here, therefore, there appears to he a mark of great antitluity As to its origin I am unable to suggest any satisfactory"cxplan,owiu

I have been informed by Dr. Thibaut of Benares that I)ioplt.ioor. in his Greek arithmetic uses the letter 1s ( short for 41cit»RR reversed (thus T) to indicate the negative quantit t. There is un,lonle tedly a slight resemblance between the two signs; but considering that the Hindus did not get their elements or the arithmetical Neiviicn from the Greeks (a), a native Indian origin of the negative sign t;er"t more probable. It is not uncommon in Indian arithmetic to indicate

a particular factum by the initial syllable of a word of that iinpoit subjoined to the terms which compose it ..............The only plausiltlo suggestion I can make is, that it is the abbreviation (ka) of the wotd kanita, diminished". (ix) Abbreviations :-Abbreviations are employed to such an extent as to become, at times embarrassing, and they embrace almost every type of term. None of these abbreviations is used in an al. get;laic sense, although, at first sight, when we find a°, u", pa" used

consistently for the elements of an arithmetical' progression, an algebraic symbolism does not seem very far off. But a'), hao.. U0, 1101 are the abbreviations of names of animals ya°, go" and sa" of pl.itttH,

etc., ect. The names of measures which are numerous, arc nearly Certain common terms of operation are '.lien abbreviated, and of these the following occur most often-

always abbreviated.

bhai for bhaga2, placed after a term to indicate that it is it divisor.

se°S for sesam4, a remainder.

mu°5 for mulamc, a root, a quantity that has a root, capital, pha°7 for phalam", an answer. etc.

etc..

Indeed the writers seem to have used abbreviations whenever there was no ambiguity incurred in their own minds. We arc no- tm fortunately placed in this matter and occasionally it has been inii on. sible to rediscover the term implied. (2) They got a good deal from them, and particularly all their later us. tronomy came from the Greeks. q

3

a1')

tis

alrtrq)

t3 (qi) Q. (¢qst)

30

Bakhshali Manuscript Illustration

Ai

Just to show how the problems me treated in the Bakhshali Manuscript, we shall reproduce one it ustration: (Folio I, verso and 11, rect -)).

to

The sutra means change b

b+a

and quotations from it

a-e given on Folios 1, verso and 2, verso.

The example solved on I

verso and 2 recto is son.ewhat as follows :

The combined capitals of five merchants less one-half of that of the first, one-third that of the second, one-fourth that of the third one-fifth that of the fourth, or one-sixth that of the fifth is equal to the cost of a jewel. Find the cost of jewel. and the capital of each merchants

(1)VM;Wa;( 10

q-lI Sri fa w

k

II

fafaf

1

T

... wO aff sTht

:mrfta er

' Wfar MAO fr

3111V91 --11 4311t afam

era ,r wfar ft 1w 8far

it fn irkq

arfdrrir4: T8t4?a9irgt ra ***6& Air: T 120 1 90 60 60 1

( 80 160

i

75

60

1

f"sFA..' mm 72 60

si

I

b' arTd 1 .0 190

" t r w& wra 427 3T O alq 377 qq =TfoT

agrh# W Lama( Ii

9

90 1 80 1 75 1 72 WTdq sra=rfq utf att srgff ti I gfai:f a 377 qq sTc r r

r aIJ

;ga

347 fi

cl

80 175

317 Ar4:

fdar 30 qq

377 Qq fhxn aq ww* i c9nt4 on next page

172

An 11llustration

.11

Verso 1 appears to give part of the solution and proofs of itir question on Folio 1, recto. Since.

< Ex-- 4x1 = Ix- gx2 = .x-1X3 = 2x- 1x4

=Ex-axs=C, We have

=x$ = 54x4 = $ xa = k>

X = 120+90+80+75+7 a 437 k

Whence

60

LLL

60

437C 43-7--'60

Sq. VM ft ft 94 SIT... tr 357

q q qTd 20 q.t

rs* 377, qg i* rc tTr waf r I SM f zr ftzr wv=r(nr 362 qeM qs

q 15 qq

Ti 377, tag ajift:i q4 %TarRr Folio 1 verso.

*zm 377

am amgf

q

rd377fgc*V't1

120

%jgFZT

30 80 75 12

erq a1

rf

120

1

t

377

tf vr'tT

90 80 75 70

WOW ftk

120

t# 377

fir t

vmRr

90 80 15 72

'i

et qr fto mqqlt

120

0ash377

90 80 75 12

(Folio 2, recto)

Bakhshali Manuscript

32

1 f k = 60, then C = 377, and xl -120, x2 = 75 and x, =72.

90y x3

80, x4

Then follows a proof which may be expressed,

90+80+75+72-317 and 31'+ts10=377; 120+80+75+72=374 and 347+90 =377: 120+90+75+72=375 and 357+0.0=377;.

120+90+80+72=362 and 362+ a -377; 120-90+80+75=365 and 365+7C2--- 377

Again on Folio 11, recto, there appears to be another verification of the example which we had on I, recto and verso.

'9°+90+80+75f7?= 377 120+ 9 'D+ 80 {-75+7 '=377

120+90+

+75+72=377

120+90+80+-,,r- +75=377 120+90+80+75+'a2 = 377

and this is the measure of the price of the jewel.

Fundamental Operations

In the classical Sanskrit works there is generally very little principles and =methods. formal information about Axioms or postulates seldom or ne er occur and strict definition is seldom attempted. Rules are of particular application rather than general - order appears to have been a matter of convenience rather than logic; and the fundamental operations receive scanty attention. (i) Examples of addition (1)

i

(a)

960 164 yutam jatam 1024

(b)

10 ( sa rupam 113 I 3 31

Fundamental Operations

12 rupa samyutam 13 I

(c)

(d)

datva jatam 889.

840 1 49 1

(e)

924 1 836 1 798 i esham yutim kriyate 2358

(f)

120 1 90 1 80 1 75 1 72 I esham yoga krite jata 437.

(g)

120 90 80 75

evam 377

12

(h)

110 1 30 1 90 1

ekatram ` 1301 I

(i)

29 48 58 I

(1)

36 48 58

I

7 anena yutam

5 she 1

(k)

16

1

10

I

1

15 16

se

I

45 sardha traya yutam 2

I

evam 16

52

2

Examples of subtraction (ii)

(ii)

(a)

15 19 1 vishesham 14

b) (12 3

2 1

vishesham I

I

1

2

(1) (a) 960 and 64 added : 1024 is produced.

(b) t$ plus unity

(c) 2 and unity added together = 3. (d) 49 having been given to 840, 889 is produced. (e) 924, 836, 798 : the sum of these may be determined: they produce 2558. (f) 120, 90, 80, 75, 72 ; the sum of these is made : they produced 437. (g) 120, 90, 80, 75, 12 : thus 377. (h) 10, 30,90 : altogether 130, (i) 291-s. by this (7) increased=36W. . (k) 5110.._ 1016 thus 16.

(1)

with three and a half added = 52/2.

2 : the difference is 1/2, (0) 5, 3 : subtracted, 2 is produced; (d) 6, 3 : the difference is 3; (0) 42 t , t i less three is 39; (f) 3, 7 : having subtracted, 4, (g) T having subtracted the difference is 91ti

(ii) (a) 5, 9 : the difference is 4. (b)

S

Bakhshali Manuscript

34

(c)

(d)

13 I rahitam jatam

15 I

6 13 1 shuddhi 13 I

(e)

1 42 1 tryunam 1 39 I

(f)

131

(g)

1 771

171 vishodhya 14 1 294 patya shesham 11

11

(ii) (a)

121

I

I 217 12

I

Examples of multiplication : 121

dvigunann 1 4 1 , i. e., 2 x 2= 4.

(b) 130 I asta gunam 1240, i.e. 30 x 8 = 240,

(c) I 2 40 I gunita jatam 16, i. e. a x'r° = 16. gunita jata 1277 I i. e. (4+})

anena gunitam jatam

i. e. 6 X (1-1)=44 (f)

2

13

3 I

4

I

4 5

gunita jatam

i.e.IX-x,=s (g)

40

phalam 16, i.e. 40(1-s) (1-.1) (1-1)=16

1+ 3 1

4+ 1

5-1-

(h) 1 3

4 abhyasam 12, i.e., 3 x 4- 12. (i) 18 I atma-gunam 64, i. c. 8 x 8 x 64.

15

Fundamental Operations

(j)

gunita jatam 880 964 84 I 168 I 880 x 964 _ 848320

I

i. e.

anena gunitam jatam

178

I

58

I

841

65593 i.e. 58 x 29= 841 pada-ghna 108625 656001 1 108625

65593

I

29

1

737

(1)

848320 14112

168= 14112

84

737

(k)

I

178

6455040625 3227520000

I

multiplied by number of terms 149200, 6455040625 3227520000

(m)

405280

samgunya jatam

44074 1

387241 77448)

44004

405280

i. e.,

179945781120 I

2999096352

179945781120_

38724 x 77448 - 2999096352

(iv) Examples of Division : Fraction quantities are expressed in the usual Indian way with the numerator written above the denominator with out any dividing line, and they are usually placed in cells. This indic ition of the operation of division often does away with the necessity for an explanatory word or phrase.

Examples of division : (a) I 168 I 168 6 4 I.e.

168

4

168

=42;

(b) 14

6

1 1

- 28;

168 7

I

7 168

35 I

gunitam

41

i.e. 4 divided = 1, which multplied into 35= I vibhaktvam (c) 10 I 3 I

i e. 10, 3 having divided 447 (d) 1447 I dalita 29

i.e.

429

132 33

(e) 1,e.

58

halved =

3

4587

vartyam jatam 1

redused gives 4,

1

28 j 24

=24

vibhaktam

I

labdham 42

I

35

4

Bakhshali Manuscript (f)

j I

798 1463

I

projjhya 798

i e. having described the denominater

798 1463

becomes 798

(g) 2558 chchheda projjhyam 1095

i.e, 2558-1463=1095, where 1463

the denominator of

is

the fraction. 160 anena drishyam1 1 300 I jets j 5 60 bhajitam i.e. by this (60) the known quantity (300) is divided and as (h)

of 300 = 5 Square-root and Surds

The method of extracting square-roots of non-square numbers exhibited in the text is of special interest. The rule is much mutilated but as it is given on three separate occasions the fragments pieced together enable us to give the complete sutra 1 "In case of a non-square (number), subtract the nearest square number; divide the remainder by twice (the root of that number). Half the square of that (that is, the fraction just obtained) is divided by the sum of the root and the fraction2 and substract; (this will be

the approximate value of the root) less the square of the last term."2 (1)

The rule is not preserved in its entirety at any one place in the mutilated manuscript of tha Bakhshali mathematics that has come down to us. But from references and quotations one can restore the rule completely : 21

20 21

NO fee Jq;TT 8'tq*qt f": 1 .a

#f$m: fir lgf $f is

mar:

Folio 56, recto

a1 f3 rc wi r 91R.vt f t

r:

Folio 57 verso. (2) English translation of this rule given by Kaye (Bakh. Ms Sec. .68) is discarded as being wrong and meaningless. He has admittedly failed to grasp the true significnce of the text. "The rule as it stands," observes Kaye,

"is cryptic and hardly translatable." He further thinks that "no rule for approximations' is preserved". It is not true, I he method of 'second (foot-note carried to next page)

Square-root and Surds

' Ti

Then follows a note to the effect that "by means of thin t oil an approximation (aT) to the proper root of a mixed quantity ju

found.....". The rule as it stands is cryptic and hardly translatable, bul fortunately there are examples given in some detail, and these trltot I that the rule was extended to "second approximations". The rule means that the first approximation to

Q = V A' +b is A+b/2A or q1 ; but

q12- Q a (b/2A)1k

= e1. No rule for second approximation' is preserved but there are several examples; and, of course, no fresh. rule is really rcquircil, for

er "'q1 - e1 12g1 = 4A2 A + 8As-{-

+4Ab A

elk =

el

z-

and the second error' may be indicated by,

b ]4/4A i 2A

2q1

b

z

+ 2A I

All the examples preserved belong to section 'C and appear to be merely subsidiary to the solution of certain quadratic equations finding the third approximation is indicated in the second line of the verso. lit. deed the only two difficult words in the sutra are samshlista (8f$ isa) and krill. kshaya (offer) We have interpreted the form as referring to "the Hum

of the root and the fraction," That that is the interpretation aimed by the author will be corroborated by what is written in the folio 56, recto; while evaluating the surd %/481. 21

20

400

21

441

9)

1

2

(faf8 s ;)

20 (wr) 21

Compare also folio, verso. Kriti-kshya (f4w) literally means " Iusa

of the squares" or less the "square". It means that the square of the upproxlmate value thus found loss the square of the last term of it will be equal to tho given quantity. The commentator has attempted an explanation of It on full, 57, recto but it is much mutilated. He says: kritiksbayam krltam osamulam (ig;(tt), Our explanation seems to be the only rational one possiblle. tt1R Wff q

38

Bakhshali Manuscript

arising out of, problems in arithmetical progressions. These problems have been fully worked out by Kaye in Sec.68 and below are given merely a summary of the square-root evaluations. There are no less than six such applications of this rule in the Bakhsbali manuscript.' (I) x/41

= 36-{-5

and b1= 6;'2 while e1= 14:51

and qs = 6; 4a

105 = -4 106-+_5 and q1=10} ; e1=1 s ;

(ii)

t q2 l04 2X10=10

81

; e2=

328

r

T8

\2 x 104

\

1

107584

(1)' Folios 57 and 64, verso; 45, recto; 56, recto and 56, verso: 45 and

or 46, recto. Note the expression mulam shlistakaranya ( "the root by the method of approximation" folio 65, verso). 64 verso, 57, verso, and 57 recto: are all (except the last line) concerned with one example, the beginning of which is lost. The example is [a = 1, d- It s = 5; therefore,

1=

41-11 The first approximation to 41 is 2

ql =

sl

,,JJ

= 77 and t1 = 24

5 6 12 65

L(24

-

1)

89. 65

1] 263

1

+

Therefore.

=

4

5785

24 =

48

.

25

1152 = 5 1152 =

65

41

1

(2'24+ 1)24 5

1

1

+8

12)

>'

The second approximation is introduced by the square-root rule (as previously on folio 55, recto) and is given by : 5 11858-25 11833 5 1 (5 2 qa = 6 82- 2 1848 = 1848 12 = 1 9985 9985 1 11833 _ 1

12) / 6

and to =

2

1848

=2'

1848

3696

and

,,this is the number of terms when the sum is five)" Part III, p. 181.'

-Kaye, Bhakh, Ms.

Square-root and Surds

l `401'

(iii) -4481= -4212+40 ; qt=2I4II ; el

8 8 42 /

_

1600

14112 ' 2 20 q2 = 2121 - (21) /2 x

9020

ea

= 219681

2120

_ 140 4 I

' 8-8

42)

40 2

( 2 x 21 22)

160,000 2,999,096,352

(iv) x/889 = /29a } 48 ; qi

r 19

29L8

24 = 24\58/

24 841

/336009 = / 5792+768 ; qi = 579

(v)

384 57

q2

579

e2_

9'

515,225,088

777,307,500

_

21,743, 271, 936

8= 7,250,483,394,675,000,000 384``2

(v')

384

339009

= 579 + 579 -

0579/ 2

579-1-

384\

There is not much doubt about the exegesis of the ru'e. It wan neither connected with continued fractions, nor with the so called Pellian equation. Brahmagupta gave the converse of the rule,

namely (A+x)2' A2+2Ax from which the square-root rule given in our text is immediately deducible. In this connection, Kaye has been of opinion that the square-root rule itself was not used by the Indians and was not even noticed by them until the sixteenth century. Indeed the Hindus had a very good practical rule of their own, which was given by Sridhara (Trishallka, 46) and Bhaskara 11

Bakhshali Manuscript

46

(Liitvati, 138),' namely-"Multiply the quantity whose square-root cannot be found by any large square number, take the square-root of the product-leaving out of account the remainder-and divide by the square-root of the multiplier". For example, 1471- x/41 x 1000000 -1-1000^-6.403+

The above approximate formula is now generally attributed to the Greek Heron (c. 200 A.D.)', and it is restated by the Arab AlHassarar (c. 1175 A.D.) and other mediaeval algebraists2. But it was known, as has been shown el3ewhere, to the second order of approximation, to the ancient Indians several centuries befores. The values of these surds are summarised below : Q

91

q2

q (approximately correct).

41

6.41667

6.40313

6.403124

481

21.9524

21.9307

21.9317122

889

29.828

-

29.8161030

33609

579.664

57966615

We have in the 'Sulba sutras (Baudhayana, c 800 B. C.) the value of V-27 and the other surds, not only obtained by geometric

devices but also by the process of continued fraction. Kaye is awefully mistaken in asserting that the "Square-root" rule was not used by the Hindus and was not even noticed by them until the sixteenth century.

Rule of Three

The term trairasika*1 "relating to three quantities" occurs in the Bakhshali Manuscript about a dozen times-always employed in the sense of arithmetical proportion. Most often it occurs in the (1)

T. Health, "History of Greek Mathematics", Vol. 11, p. 324

Heron's time is uncertain.

He may have livcd in the 3rd century. A. D.

(2) D. B. Smith, History of Mathematics" in two volumes, 1972, Vol. 11, p. 254.

(3) Bibhutibhusan Datta "Hindu Contribution to Mathematics", Bulletin of the Mathematical Association, University of Allahabad, (1927-28), 269),

Rules of Three

41

phrase pratyaya tratrasikena*a or "proof by the rule of three- null the proportion is generally set out in the following manner, Tqrr phafam

18 1

which means

:

3

1 4.:

:

4 : 18.

The term trairasika is orthodox but the term phalam*n Is used throughout our text quite appropriately as equivalent to -answer". and is not applied to the second term of proportion as in the Lilavati*4. There are very many examples of the rule of the three' used either as the direct method of solution or as a proof. There are

also some very much damaged examples of what appears to In, problems in so-called compound proportion. There is nothing of the nature of a theory of proportion discussed but several cxnmplcn of the following principle occurIf,

bi

b2=...=

bn' then

lb=B

Calculation of Errors and Process of Reconciliation

The Bakhshali mathematics exhibits an accurate method of calculating errors and an interesting process of reconciliation, the like of which are not met elsewhere. They are necessitated by the application of the foregoing approximate square-root formula. There are certain examples whose solution leads to the determination

of the number of terms of an arithmetical progression whose number of terms is unknown but first term (a), common difference

(d), and the sum (s) are known. If the number of terms be t, then according to the Bakhshali work,

so [(t_1).4 + a, 6

1,

Bakhshali Manuscript

42

whence t - - [2a-d]+ [2a- jdn+ 8ds

The negative sign of the radical has been overlroked in the Bakhshali work. Putting p - 2a- d and Q = [2a-dj'+8d, we have,

t _

-p+V Q ld --

2dt 4 P=V Q or =tad+pt also 2s

...(2) r

Oftentimes in the examples given the value V-Q does not come out

in exact terms, so that a method of approximation has to be adopted.

Let q1, g2.......... be the successive approximations to

the value of Q and let the values of t obtained from them be tl, t .....Neither of these values will evidently give the original quantity s when substituted in the equation [1] for the purpose of verification of the results obtained. Suppose the values of s corresponding to the values of t be s1, s; ...... Then,

2 sl = tl 2d + ptl 8 dsi +p2 _

or

(2tid+p)

a

8ds-j-p2 - (2td+p)2 Therefore. 8 d(sl-s) _ (2tid + p)2 -- (2 td + p)9

now,

2 td 1 + p = ql (1) The Krltt-ksaya (FftrWq), probably refers to this second error.

Calculation of Errors and Process of Reconciliation

st - s= 91-Q 8d

Hence,

a+ Za =91

Since

up to the first approximation,

( r 2a) a

we have, q1- Q =

=E1, say

Then ( , will denote the first error. Therefore,

si--s

8d

Similarly, for the second approximation, the error will be'

(rl2a)2 Ea. 2(a+ r 2a

J

CIL and ss -s=8d

We shall now refer to a specific instance, in whichl

a=1,

d=1,

s=60.

(I) Folio 56,verso and 64, recto. Portions of the detail workings are not

8tt

preserved in the existing manuscript. But they can be easily restored. Folio 56, verso and recto as preserved are,

I8II

14112

846720 34Z1`Qrc"t4 114112

1600

....

l

21

afr

60

watt

20 21

VFW firm WTff i

r *t fa'

ar'. W4 WW OR66: 'ITc Igfi' Wf r Pr: 11

fir W...

(carried over to next page)

it

Bakhshali Manuscript

44

The detailed workings given are

8ds=480, 2a-d=2.1-1=1.480+1=481 40 _ 882+40 x/481 = 2142

And

si

l 42

°

ei

42`

x922 - 1 _

Then, t1=

He ce,

= 922

42

J

t1(t1+1)

-

880 84

880

= 84

2

1 (40112

X

_964 168

848.320 = 60. 14,112

=

1600

- 842/ =14112

8d

1600 _ ei 848:20 s=s1 - 8d= 14112- 14212

846720 14112

=60

Again for the second approximation, x/481 =

21

_ 425,042-400

(20/21)2

20

l 21-

21 1

21

ir1t'grq

"

21+

441

r %T, few ain t 3'qt

irrau O

_

... 425042 I

400 itl'q 19362 , 193621

ai 42464 19362

(Folio 56, recto and verso)

Folio, 64, recto, is405280 444004 387241 387241

405280 444004 177448 IWSW 38724 .,, 1179945941120

2999096352

3;Ef aT

1-i

k;q

°i

160000-f-

--Folio 64, recto.

Calculation of Errors and Process of Reconeilintion t

Hence,

405,280

1 (424,642 s = '2 19,362

ss

t2(12+1)

=

'1"

J -"

38,724'

405,280

444,004

- 38,724 X 71,448

2

179,945,941,120 2,999,096,352

ez

gd

160,001

404

= s x 214 x (21;, )a =2,999,06,352 e2

s = s a - 8d =

179,945,941,120- 160,000 2,999,096, 352

179,945,781,120 = 60 2,999,096,352

Negative Sign

In the Bakhshali manuscript, a negative quantity is denoted by a cross (-I-) placed after the number affected. Thus, 11 7 i inea lift 11-7. This is very remarkable. For in the manuscripts of Prillutdakasvami (600 A.D.) and later Indian writers a dot is usually pineal above the quantity for the same purpose, so that according to them 11-7 is denoted by 11 7°. The origin of the use of a cross for the negative sign has been the subject of much conjecture. Thibaut Iuts

suggested its probable connexion with the Diophantine negative sign fi ( reversed 0, abbreviation for Xeo'is, meaning "wanting",' This has been accepted by Kaye2. But in the opinion of Datta such a conjecture seems to be hardly reliable. For firstly, the Greek sign or minus is not ' but an arrow-head (T) and "it is now certain", observes Heath, that the sign has nothing to do with 0." And arrow-

head and a cross are too much different to be connected together, or too distinct to be confused for each other. Secondly, the Greek symbol itself is of doubtful origins. And above all, we are not sure if it is as old as it wili have to be for being the precursor of the Bakhshali cross. For there is no manuscript of the Arithmetica of (1) Indian Antiquary, XVII, p. 34 (2) Bakh. Ms. 127, JASB, VIII (1912), p. 357. (3) Heath, Greek Mathematics, Vol. 11, p. 459.

46

Bakhshali Manuscript

Diophantus 'which is older than the Madrid copy of the thirteenth century A. D. and again in many cases in this work, the negative quantity is indicated by writing the full Greek word for "wanting" in its different case ending. So we cannot be sure if Diophantus did actually use that symbol for the minus signs. Under such circumstances it will not be proper to assume the possibility of Greek connec-

tion for the negetive symbol of the Bakhshali manuscript, Hoernle thinks it-though he is not quite confident in this respect-to be the abbreviation ka of the word kanita or nit (t or una, an) of the word nyuna, ft q), both of which means "diminished" and both of which abbreviations, in the Brahmi characters, would be denoted by a cross9. Datta thinks that this supposition has got a very notable

point in its favour. In the Bakhshali manuscript all other arithmetical operations are generally indicated by the abbreviations (initial syllables of the words of that import, though often the words are written in full and occasionally nothing is indicated at all).' So it will be very natural to search for the origin of its negative sign in that direction. In this way, Hoernle's hypothesis appears to be a very probable one. But its principal drawback is that neither the word kanita (wfr r) nor the word nyuna (t9) is found to have been used in the Bakhshali work in connection with the subtractive operation. The nearest approach to that sign is that of ksha, FR, abbreviated from kshaya (41zr, "decrease") which has been used several times, indeed more than any other word indicative of subtraction. The sign for ksha, (a), whether in the Brahmi characters or in the Bakhshali characters, differs from the simple cross (+) only in having a little flourish at the lower end of the vertical line. The flourish might have besn dropped subsequently for convenient simplification. (1)' D. E. Smith, History of Mathematics, Vol. 11, p. 396. (2) D. E. Smith, History of Mathematics, Vol. 11, p. 396. (3) Indian Antiquary, XVII, p. 34. (4) It may be noted that abbreviations of all sorts of things, mathematical

as well as non-mathematical, have been freely used in the Bakbshali work (vide 62 of Kaye)

al', a° (for asva, a, horse. q, ha° for hasti, Zrmc, elephant)

U;, u° (for ustra, 3, camel)

(carried over to next pager

Least Common Multiple

4'1

Least Common Multiple

The plan of reducing fractions to the lowest common denoniiu ator before adding or subtracting is known correctly to the author of of the Bakhshali mathematics, We have a few instances of itn ap, plication in the work. In one instance', it is required to find II

sum of the fractions.

T,1'E,13,I l16 They are first reduced to a common denominator (i

fr;nti)

so as to become, 120

60

'

90 60

80

72 60

75

60 '

b0 '

respectively. Finally, the sum is stated to be

437.

go° (for go, rft, cow) t, di (for dinar, 'Tart, a coin) fi ', li (for lipta, fcTr,, a minute of an arc) fir, vi (for vilipta, frf , a second of an arc) v(;r, or zero or empty space, a cypher q, y, pa° (for pads, w , a term) and similarly of plants. In mathematical operations, %ri (bha) for %rM, a divisor; i)r (she) for *I, a remainder, 1(mu) for IJ 4 a root, and T (pha) fore for, an answer. (1) ant fawrw grain gadwr;r R' 8r a ... a

51 w: q0 f ry

3rrlc r

120 90 80 7.i 72 qar pj@(tt 3rr' 120 60160 160 160160 I

90 1 80 1 15 , 72

qqf q)-;T fW wrcr 437 arcrr ij 377 qtr Yrfv1' " I Folio I, recto and verso,

Bakhshali Manuscript

48

In a different instance', it became necessary to add up, 2

1

*

3

It is stated that the sum, after having reduced to a common denominator (q

zrcl'), will be

03. On reducing the fractions.

T8, T,IT to a common denominator, they are stated to be respectively2, 798 1463'

836

924

1463 ' 1463 '

(i)

11

zr jar eg

I

2132143 1 4

5

cTic : ti;TT $

cq

:..

l

1J11

163 60

Folio 17, recto,

(2) 34 To a WOr N ft

fa'9R'd...... qRM

for...

94W, i

aerr

7+ 13+ 12 12

12 12

5+

6

v r fat t ffqff r i 'M ...f

ararr f

19 1 7 111 12 4 6

yftff gf -4,d

... W"M

924

4

1 121 4( 6 I tQft

19 7 8361 798

11

1463 (14631 1463 924 1 836 Aj

1095

798

jrff f

qaq

c

T11c

2558 s Folio 2, verso.

Least Common Multipie A fairly difficult case is to simplify2,

3t

+

"

1

+5

8

121j

5 133'

11

1807

The result is correctly obtained as 240

.........armor

;r

tai gI T*ara vft

kc

-4T;T aaf?48f fa4 wfor:t

rg1

Tr' T :ram r a aruort ff+4 T

q

t

WMfat

X R>

ITaersT 1E

T41

f;; qf'h

WrM T?SfkMk1Ur

2 wrr 13

3 W 13 8 V 1 3 4r

1

l irr 1

1

1

1

1

2

1

2

3

3

8

( 2

1

10

vrr

11 2

1

1

4+2 31 sir

1 l 9rr

[4 4 4

1

3

5

2

1

S 1

3

360 I M°gi

I 1

2

1

3

Translation ; --and also twelve and a half in thirty three, and one third days for the best wine for the consumption of merchants. In the treasure house was stored twelvc hundred. Say 0 Pandita, how long can this expenditure continue. The statemcnt means-

10, -

daily income =

21s 11

Daily expenditure= -

p 3T

+13; 'ff

11s

3a

f51+?5+331 g

3

1807

240 1807

9

727

,The daily loss is, therefore, 240--2-f4-0 8001

and 7270-240 x 360- 7 7

is the period)

f1

Bakhshali Manuscript

50

The fiethod of finding the least common multiple is found in the Ganitz-Sara-Samgraha of Mahavira (c. 650 A. D.1) and probably also in Prithudakasvami's commentary on the Brahma-Sphuta-Sid-

dhantas, but not in works of Aryabhata, Brahmagupta and Bhaskara. Arithmetical Notation-Word Numerals

The arithmetical notation generally empolyed throughout the Bakhshali work is the decimal place-value notation. This fact has been differently utilised by different writers. On the one hand, Hoernle3 and Buhler4, who believe in the antiquityaof the Bakhshali

mathematics, consider it as evidence of the earlier date of the discovery of that notation by the Indians. On the contrary, Kaye6 who believes in the non-Indian origin of the place-value notation and in its late introduction into India, considers its general adoption in theBakhshali work as proof against the hypothesis of the previous writers about the early date of this work. It is now definitely known that Kaye's notions about the origin of the place-value notation is wrong. It was invented in India about the beginning of the Christian era, probably a few centuries earlier. For this, one may consult the writings of Dattae. But apart from that controversy it should be noted that the nearly exclusive application of the notation in the Bakhshali work is very much noteworthy in asmuch in almost all the available Indian mathematical treatises, save and except the AryaBhatiya of Aryabhata (499 A. D.), we find copious use of the wordnumerals. There is, however, evidence to show that the author of the Bakhshali work did know the principle of the word numeral system of arithmetical notation. In it we find the use of the word (1) OSS. Ganita Sara-Samgraha,111, 56.

(2) Colebrooke, Hindu Algebra, p. 281, footnote; p. 289 fn. Also pp. 118-9. (3) Indian Antiquary, XVII, p. 38.

(4) Indian Paleography, p. 82, (5) Bakh. Ms., Sec. 131.

(6) Datta, "A Note on the Hindu-Arabic Numerals I,( Amer. Math. Month., Vol. 33, 1926, pp. 220-1), "Early literary evidence on the use of the zero in India" (ibid., pp, 449-54); :the present mode of expressing numbers" (Ind. History Quart., Vol. 3, 1927, pp. 530-40 "Al Biruni and the origin ofthe Arabic Numerals" (Proc. Benares Math. Soc., Vol. 7,1928)

Arithmetical Notation -Word Numerals

'%I

Fq,rupa(=1),-m,rasa (= 6,1 andgM,pada (_;)' with numelirnl significance. The use of the last word is as old as the Veda N, 'I he first occurs as early as in the Jyotisa Vedanga" kc. 1200 13.C.) vial second in the Chandah-suira of Pingala (before 200 B. C.)4. Again, in speaking of a very large number, 2653296226447064994 ... 83218

the Bakhshali mathematics writese :

Sadvimsasca tripancasa ekonatrimsa eva ca Dvasa

sadvimsa

(sti)

catuhcatvalimsa

saptatI msanamturant

Catuhsastina (va)...

pakam,

Trirasiti ekavimsa asta.......

Clearly the principle of the word-numeral system has been followed in this instance. The only departure from its popular features lies in (I) the use of the number names in the place of the word-names and (2) the adoption of the left-to-right system fit the arrangement of the figures. But, according to Datta°, these features,

(1) faf

f

wtWw of i : i g- va'W Z T1 Tc Sa: II

3r1

(2) ... %

'd'g}a

1

c,(arrcr $ k r

gf

-Folio 60, verso. i 135 l

4

1

41

-Folio 4, recto. (3) Yajus-Jyautisa, 23: Arsa-Jyautisa, 31.

(4) Chandah-Sutra, VI, 34: VIII, 2, 3.

(S) 3Zfo molar 1W fagsaI* tf War a i gm ...Wsfgr W a1Tfrl'8f it rid sfi

ffQ3%fT m49i

q it

(2653) 296226447064994 ... (83218)

-Folio 58, recto. (6) Sec Datta's papers in the Aanglya Saliltya Prarika, 13351 D. S. (1925), pp 8.30 on the comprehensive history of the orlein and development of the Word-numerals,

Bakhshali Manuscript

52

though not common, are not altogether foreign to the system. Once at least the author has followed the right-to-left sequence. For the compound word 'Catuh panca' (aT._gs;w) has been used to denote (Folio 27, recto). once 14 15 j and again 1, 5 14 1. Regula falsi

The rule of "false position" or "supposition" is used in two ways in our text' in solving linear equations.

(i) f (x)=p is solved by assuming a value e for x. f(e)=p' and the correct value of x is ep p'.

This gives

An example is,

The amount received by the first is not known. The second receives twice as much as the first, the third thrice as much as the first two and the fourth four times as all the others. Altogether they receive 300. How much did the first receive ?

Suppose the first receives one, then second rcceives 2, the third 9, and the fourth 48 ; or altogether they receive 60. Actually therefore, the first received 300/60=5.

(ii` If f (x)=p, reduces to bx+c- p and f (e`=p' reduces to be +c=p', then x=(p=p')/(b+e), which appears to have been consider. ed useful when it was desired to keep b and c unchanged.

For example, if x1+x,=a1, x,+x3-a; x3-+1 =as, we have 2x1+(a,-a1) =a3 ; and if in place of x we put e then 2e+(a,-at) =a'3 and the correct value of x1 is (a3-a,')/(2+e). This rule of 'false position' is interesting as being, in a way, a precursor of algebraic symbolism. It connotes the idea of an un(1) No rule is preserved in the text but Bhaskara ti gives the following (Lila. vatl, 50) :

"Any number assumed at pleasure is treated as specific in the particular question, being multiplied and divided, raised and diminished by fractions,

then the given quantity, being multipled by the assumed number and divided by the result yields the number sought.

Rcgulu falsi

I

known quantity and even of a symbol for that quantity (c. g., w+ in our texts but it does not embrace the notion of such a quantity la'ma' subject to operation or being isolated. As soon as we irnttrnlnr algebraic symbols, the rule, as it were, disappears. Algebraically. tliv

rule solves bx =p by x=ep/p'. where p'-be, which shows that the transformation from bx=p to x=p/b was not conceived. There is no algebraic symbolism in our text but it may be noted

that Bhaskara gives both the regula falsi and also an early form of algebraic symbolism. Neither Aryabbata, Brahmagupta nor Sridlttu'n gives this rule or makes use of the principle. On this, Kaye says that its occurrence in the Lilavati, therefore seems to indicate that it watt introduced into northern India after the time of Sridbara (Xltl cent.). Mahavira (IXth cent.), however, uses the method in rather a special way in connection with a geometrical problem'.

But such a surmise according to Datta is wholly baseless, Wo shall sum up here the views advanced by Datta : The rule of 'false position' has been applied in certain eases in the Ganita-Sara-Samgraha of Mahavira. For instance, for finding out unknown quantity (avyakta, arzn; ajnata area'), the sum of the various fractional parts of which is known, Mahavira says : "The given sum, when divided by whatever happens to bo alto sum arrived at in accordance with the rule (mentioned) before by

putting down one in the place of the unknown (element in

tito

compound fractions), gives rise to the (required) unknown (elements) in (the summing up of) compound fractions'".

We have a few other instances of this kind in the work". Further Mahavira has applied the method of supposition in solving certain geometric problems4. Hence Kaye is not truely accurate when (1) Ganita-Sara-Samgraha., VII. 112. [21 Ganita-Sara-Samgraha 111. 107. Compare the original expression in the work Vq-,gM;rT etc. with the passage 1grg Mi' '94 WEIT and similar oihrr passages in the Bakhshali work [folios 25, verso and 26, rccto; compare also folios 23, 23] [31 Ibid III, 122, 182, 135--7 [4J Ibid. VII, 112, 221. This should be more accurately culled the geometric prototype of the regulu finst of algebra.

54

Bakhshali Manuscript

he says : "Mahavira (9th century), however, uses the method in rather a special way in connection with a geometrical problem"'. In fact Mahavira has made more extensive use of the method in connection with certain algebraical as well as geometrical problems. Still it is quite true that he has not made as general use of the method as is found in the Bakhshali work, or even in the Lilaval,'. In any case, Kaye's hypothesis of the foreign import of the rule of false position into India after the eleventh century must be abandoned. It should be further noted that Mahavira (c. 850) is a contemporary of the earliest Arab algebraist to use that rule, namely, Al khowarrizmi (c. 825). Hence it is quite certain that the Hindus have not taken the regula jalsi from the Arab scholars, f they have done so at all from a foreign nations. Known still earlier in IndiaIt should be observed that the rule of false position was report-

ed to by the Arab and Furopean algebraists at the early stage of development of their science when there were no symbols. It almost

disappeared from amongst them, as it is bound to do so, with the introduction of a system of notations'. It will be nothing unreasonable to expect that such had been the case with that rule in India too, if it was ever followed here. Now it is a well established fact that the Hindus reached Nessel-mann's third and the last stage of development of the science of algebra long before all the other nations of the world. They invented a good system of notations (1) Bakh. Ms., Sec. 72, This statement of Kaye followed by another of the same kind, "it (the regula falsi] occurs in no Indian work until the time of Mahavir a" Sec. 134], v. ill obviously contradict his previous statement, "Its occurrence in the Lilavati, therefore, seems to indicate that it was introduced intonorthern in India after the time of Sridhara [11th century]" Sec. 721. Thus it appear that Kaye is not sure of his own grounds. (2) Certain problems in the Bakhshali work, the Lilavati [pp. 10 et. seq], the Trishatika [pp. 13 et seq] and GSS [IV, 5-32] which are of the same kind but differ only in detail, have been solved in the first two works expressly by the regulafalsi, but not so in the other two works, thought in them the unknown quantity has been tacitly assumed to be one.

(3) It should be noted in this 'connection that while there are ample proofs in

the writings of the early Arab scholars of their heavy indebtedness to Indian Mathematics it is still to be proved that the Indians took anything in return from them.

known still earlier in India

5o

by the beginning of the seventh century of the Christiatl era. It lsn' been laid down by Brahmagupta (628 A D ) that a thorough know ledge of algebraic symbols (varna) is an essential qualification for it

good algebraist9. We find mention of a symbol for tl'e unknown even in the Aryabhatiya of Aryabhata (499 A. D.). So the method of false position must have disappeared from India before that time.

Or it is at least bound to have been regulated to a very inferior position from that time. This will account for the absence of tlw method from the works of Aryabbata and Brahmagupta as well an for its limited application in the Ganita-Sara-Samgraha of Mahaviru. There is now left no direct evidence from the Hindu source to show that that method was followed in India before the fifth century A. 1). Unfortunately no Hindu treatise on arithmetic or on algebra w'he't can be definitely referred to that period has survived and come down to this day with the exception of the Bakhshali manuscript. Thorn in, however, external evidence. A mediaeval, Arabic writer of note, possibly Rabbi Ben Ezra (b. 1095) refers the origin of the rule of false position to India'. And if our hypothesis about the origin of the term yavat tavat (zrri err r) for the unknown in Hindu algoina be true, it is in all probability so, then there will remain very little to doubt that the rule was known in India much earlier. is

The rupona method

There are several references to the rupona (tatr"*) method, and the

phrases rupona karanena (Ingot

,t

i) or pratyaya rupot,u

karanena (g q V*at TTY) occur. In all the cases in arithmetical progression according to the rule, 0

s+I (t-1) 2 +a] t [I] Smith, D. E, History of Mathematics, Vol. Ii, p. 437. ]2] Brahmagupta says : By the pulverizer, cipher, negative and afrmaliva quantities, elimination of Inc middle term, colours [or symbols and well understood, a man becomes a teacher among the learned and by tile affected square' (Cole brook, Hindu Algebra, p. 325). [1] Smith, D. B., History of Mathematics, 11, p, 437, foolnot u

Bakhshali Manuscript

56

The recurrence of the phrase rupona karanena (q ur w(4 0 seems to imply that the rule in question began with the term rupona

which corresponds to the (t-1) of the formula'. The rule is not preserved in our text but we find the following in the Ganita-SaraSamgraha of Mahavira2.

"The number of terms is diminished by one, halved and multiplied by the increment. This when combined with the first term of the series and multiplied by the number of terms becomes the sum of all". The rule is exemplified in two ways in our text of which the following are particular cases : (i)

aTr

3°

1

I

q°

1

19

I

1

g wK4q qTI

(Z)

qo 1

air I

T:grq

1,

1

qO

19 *'q wzi;T

nR

1

I

19

1I

Folio 9, recto, verso. aTro 3 I ;To 4 ' qo 1I

3I

g

21

grTsT

1

Folio 7, verso.

q: ' jqr ag' fgrfq' fW sr fgfr

21

TT Vqlnlw.T T qrq7

STc=TG

1

TTfiZ'rc

I

65 I rgq-*q . fk 65

(1)

fg

217

11

Folio 4, verso

Folio, 5 recto.

Is this a faint echo of Pythogorcan style ? Anatolius writes : The Pytha goreans state that their master, in conncctiun with numbers that form a right-angled triangle, showed how such could be composed by means of unity'. i. c.,

#I (a2- l))2+a2= [I (a2+1)]2 See

Tannery

:

Scientifq,,es, 111, 14.

v4*;ft wrT3t '1 a: sT wrrfq aY f=raT: I ST'T$q qqrTi'ffffti*fgd q-gfcr ti gTN 11 GSS. 11, 63.

Rupona Method

S'l

which means : a-1, d-1, t=19; by the rupona method,

tier

answer is s=1 0.

(ii) In section 'C the application of the rupona method k always worked out step by step. For example, when a - 5, d. :1, t=178/29; then s=[(t-1)-1-5] t; and the rupona method is applied as follows' :-

(t-1)= 2149 9;

149 _ 447

3x

447

29- 29

447

737

58+5

737 58

58 '

447

29x x

58-- ;

178

65593

29

841

_ 77 836 841

The origin of that name is supposed by Hoernle' and Kaye" to be lying in the fact that "the rule in question began with the ), i.e. one less] which corresponds to tho

term rupona [ (-qzq -;-

(t-1) of the formula". The term rupona literally means, "deducting one". As the rule is not preserved in the available portion of the Bakhshali mathematics, it is not possible to verify this (1)

I

447 29

6 1

'

': Wid

dqff 447 58

a

737 58

1

q I

8

60*

r 1 ?ff Qv

178

16a 60*

ate jf46 wvt

291

33fr fall -M W f4 1

-

!

65569 I

65593 941

slit t[841

60*

6 fW

k;r 1

60* We 6 29 Otaiw 42*r

1

7 ufto 1

28

I

fff#aWq

178 q 9

... 77

29 (Folio 7, recto.) (2) Indian Antiquary, XVII, p. 47.

(3) Bakh. Ms., Sec. 73

$

Bakhshali Manuscript

S$

supposition. Kaye, however, points out that the rule has very nearly the same beginning in the Ganita-sara-samgrahal of Mahavira : Ruponena gaccho d<

It may be noted here that in the Bakhshali mathematics, the word rupa (r-q) occurs also with different significance than unity. For instance, we finds

"...... (bha) jitam

jatam 12 , labdham sarupa j esa

1

24

rupadhikam

3

!

esa kala...

a°

phalam ru°

...I1

di°

3

7

1

`

21

di°

3

u° 4

Fa°

3

1 dvitiyasya trairasikena...

pha° ru°

21"

II

I

or ' ...divided becomes 2; 'quotient plus 1 (r upa),' this increased by 1 bec mes 3; which times... by the ruponakarana, the result is ru° 21. Of the second, by the rule of three, the result is ru° 21". (1)

ii, 63.

(2) Folio 7, verso. The problem is

7t- [(t -1)- 2-+3 , t, whence t. 2-(4 3) +1 =3. By the rupona method, ,q = [(3 - l) +3] t - 21 and by the 'rule of three' 1:7::3:21.

I'1'ool's

Y1

In this passage'. the number 21 has twice been marked =ul ru , abbreviated from rupa., :Again in a sutra (folio 8, recto) iclutud too an arithmatical progression, we find the passage i rh4hom +y m' ylnirdiset, (qmq iFq faf ai ), that is, the quotient should be indi cated as rupa'. Here again the term rupa seems to have it pnivlY technical significance. There are other instance, in which ruptt Is,e'i not mean one, but is used in connexion with an integer'. Similar urn ,of the word rupa is found in later Indian mathematical treatise whrle it denotes, besides 1, an integer or the integral part of it mixed fractions. I venture to amend the word rup ma (>ar)i) to ruiuutrt (Vgor) a Then it will mean 'making rup.' which means 'known 411' absolute number,' 'known quantity as having specific form''. it, rupana-karana (iFgvf zv() will mean the method of making absolute number', that is, 'totalization' or 'summation'. This hypothesis will be strongly supported by the expression 'rupana karancna pltttla,tt

rupam lupa 21' (v-qur q;zvt;r tp Vq rq 21, or 'by the method of making rupa the result is rupa 21'). Proofs

To many solutions are attached proofs, which are generally sfcgrr-T.'proof' or 'verification'. introduced by the term Th`s is sometimes amplified into pratvava-trairasikena (>ttgrt dCIfbF (1)

Folios 21, recto; 60, recto; etc.

faW1T1"' f

aqr I falm gz:~qt ; " -Folio 21, recto.

q5I 91

rfic =r

faafzarf r

rri l 1

10

1

.

-Folio 60, recto. (2)

See Br.Sp. Si, XII, 2; Trisatika, pp. 7 et seq; Lilavatl, pp. 6.7; B(/a., pp. 2 et seq; (Colebrooke : Hindu Algebra, p. 149).

()

Rupona (Vq'Qr = qzggq) may be an archaic form of rupana (Vqur)

(4)

See Monier-Williams, Sanskrit English Dictionary, revised by Cappaller and Lcumnnn, on rupa. Compare this use of the word rupa with Its use in algebra in the sense of nhgoluto known number v a Ogvation..

;,,i

60

Bakhshali Manuscript

rd) 'proof by the rule of three', or pralyaya-rupona-karanena sT:q vc

pzor r

'proof by the rupona method'.

Sometimes the 'proof' seems to be merely a matter of rewriting a statement in another form, but generally the answer is utilized and one of the original terms of the problem is rediscovered. Occasion-

ally the 'proof' consists of a solution of the original problem in another way, e. g., by steps; but sometimes it deals with a subsidiary aspect of the original problem. In certain problems approximations are employed and then the 'proof' may be described as a process of i econciliation. These 'reconciliations' entain a comparatively high degree of mathematical skill. Occasionally several different proofs are attached to a problem. The following are specimens. (1) Problem

t-7=s t+7

Solution t = 2 x 7j(3--a) = 30.

Proof'--3:5 :: 30 : 50, 5:6 :: 30:36 and 50-7=36+7 -Folio 46I, recto. (2) Problem-A gives 21; in 11- days,

B gives 3; in I' days. C gives 41 in 1; days.

In what time they have given 500 dinaras altogether ?2

(1)"'aTf

r r

r

fq

STTT f0ZrfRRM

3 1 5 1 30 1 q o 50

--1I6I30

t$ 43 1 ..................43

43Iq

36

r 17

wrrwT T

-Folio 61, recto. (2)m +4735C0 I affird I 947

rr'ic T

Mt *l 500

(carrkd ovor to aoxt pap)

Proofs

Solution,

216,000 947

OOOdays _ 60. 947

500 33

t°2

And A gives

(,1

+z+14

100,000/947,

B gives 157,000/947, C

dinaras.

Proof. 21 :

1 }y

100,000 947

60,000 947

11

00 157,000 -

600 00 947

31

4:

1a

2166000: 60000

;

9

Folio 22, recto. (3)

Problem. One earns e and spends f daily. How 1onit

will a capital of C last ?

Solution, = t -C

(f--e)

Proof. 1 : f :: t : F (the total expenditure). I : e : : t : E (the total earnings) and F-E= C. (4) Problem, If 7 are bought for 2 and 6 sold for 3, and 11w capital is 24, what will be the profit ? Solution.

p-c'C 124(x=-1)-18

Proof. 2 : 7

84 (the number of articles)

24

and

84 : 42 ( the total proceeds), and 42-24=18.

or

...1 :c::C:n, c:l .:n:C+p

6:3

C+p--C =P,

and 2

o

1 ifto 1

2

2

3 t.^to

I Zfto

1

1

2

3

4 Rto

I

1

1

2

4

gff ifto 60000

100000 41o

947

947

1

157000 Efto 947

qMr

947

I

21600000

'o

947

ro 60000

i ;fto 600011 (

947

I

Polio 28, recto.

Bakhshali Manuscript

62

(I-;) (1--1)=x-280

Problem.

(5)

Solution.'

9=200, 400-200- 200 200

4_ = 50, 200 - 50 =150

150

= 30, 150--10 = 120

and 400 -120= 280. -Folio 52, verso. Problem.

(6)

(l +& (2+

(3+s (4+" (5A- )))))

Solution = ; Go .

Proof. (((((io-1) 2-2) 2-3)-4) '-5-1)00. Problem. Solve x+5 = s2, x-7=12

(7)

Solution. x=[ I (L+-7- 2, +7=11

Proof. 11+5=4°; 11-7=2. -Folio 59, recto. (8) Problem. F= Solution. F -

Proof. 10:

(1

'1+

---

(2) + 41 (3) + ti (4) 1+2+31-4

163--. 60

163

10

600

603::1

16633

600

gfq i1rTF1t m it it Qt .. .

wr w fa 150 3mTfcT q-- q%r T 3 o 1

1

1

2

4

5

1

1

I

.. .

I q;9 fqur 280

1

2+ 4+ 2+ 28 I qfg' 140 gwlm^t rrcT tot $cr...280 atergfa 140 28 1

9 280 fM# W14 400 qq q*

fr -Folio 52, verso.

63

Pr6dili

i

10:

1

163

60 ::3

200 163

03 6

10:

63

NO

1

10:

or, since F=

3

60

4

If x w

1

50

w : 2f x w:: w, : w,l

((t_1) Z -d-{-a

(9) Problem. Dt =- l Solution.

t

t

--?(DI a) + --

Prcof. By the rupona method, s = C(t -1) 2 +a] t and Dt = s.

(10) Problem. s =

rkt_ i)d

+a] t

Solution. 1={,1(2a_d)2.. 8s-(2a-d)}+2d -Folio 65, verso. Generally the solution is an approximation ti or t2 depending on the method of evaluation of the surd quantity; and the rupona method gives sl or s2, neither of which is the same as the original s. A process of reconciliation is therefore calved for, and this is given by

s-s' = 8d where s' is the approximation to s given by the proof and e may be termed the square-root error. Problem. a =1, d =1, s=60. First solution, t1={x/(2.1-1)2+480-1}j-2.1 40

(2142 -1)

880

-Folio 65, verso.

First proof. si = tl (tl+1)+2- 8 0 x 84

' e i = C 40 ) and 4 0

964 168

e

sa

l

=

Q

848,320 1411

848 , 320- 1600 14112

-Folio 56, verso.

Bakhshall Manuscript

64

Second Solution, ta.s (424,642 19362

Second Proof. sa =

38724

X

405,280

+2=

444,004

77448

38724

179,945,941,120

=

2,999,096,352

40 a 40 41) -t-4 (21+ -42 )'

Now e. and

405.280

1

-

_=

ea

160,000 2,999,096,352

and s = s2 -

179,945,781.120

ea

d 8 2,989.096,352

= 60

-Folio 64, recto. (11) The problems in section 'G' are characterised by numerous proofs-as many as five separate proofs being attached to one example. The following is fairly typical'

Problem. x=500(1-}) (1-;) (1-1)(1-1) Solution. x=158 's;. Proof (i). x' (1-a) (1-'s) (1-1) (1-1).1581;. whence x' = 500.

This is written horizontally. Proof (ii). X, (1-1) (1-1)

(1-3) (1-;)=1588;,

whence

x'=500. This is written vertically.

(1 W o a9 ffir 0

1

1

5* fo

4+

1

1

64

1+

4

500 11 Tft wmm Or

158

1

4+

IM Tr4 wff-4-d

500

1

1

1

1-}- i+ 1+I +4 4 4 1

1+f 4

afkk

ig158cte lWrb 1 64

-Folio 11, verso.

Solutions

Proof (iii), x'= (1-a, Proof (iv)-By steps,

65

158;, F=

500-i-4=125, 500-125=375; 375=4 = 934, 375 - 934 :1 - 2814 i 2814' =4 = 701!`9, 2814 - 701"c = 2 10,

210iao =4=5284, 210;4-5284.. 158,11-14.

-Folio 11, verso. Solutions.

The solutions are sometimes very detailed, proceeding mostcarefully step by step, so that they become expositions in general terms

of the processes involved. Also they often give actual quotations from the rules,-to such an extent sometimes that the original wording of the rule can be reconstructed. Unfortunately, however, such helpful suggestions are more often missing than not : in many eases the particular portions of the manuscripts are lacking or damaged beyond repair. In some sections, e. g. section 'M', nothing but the bare answer is generally given, and in others the outline only of the working is given. The following examples illustrate these remarks (i) The problem is DT+Dt = {(t-1)+a} t where a and d it re, respectively the first term and common difference of an arithmetical

progression, and t the number of terms, to be found. The rule

is

t={2(D-a)+d+./t2(D-a)+d)a-}

and the ptttticu lar example gives D = 5, T=6, a =3, d=4., The actual working Of the solution, translated as literally as possible, is as follows

:

'The daily rate diminished by the first term;* the daily rate is five yojanas, 5; the first term is 3; their difference is 2: this doubled is 4: *this increased by the common difference is 8; and squat cd is (4 which is known as the ksepa quantity.* *Multiplied by eight* ; the fixed term (30) multiplied by eight is 240; and multiplied by the coinmon difference-multiplied 960. *The quantity known as ksepa is added*. Now the quantity known as kshepa is 64, which added gives 1024. The root of this is 32; the quantity set aside is 8 and this added gives 40. *Divided by twice the common difference: *twice the common difference is 8-divided 5." The phrases marked off by asterisks are quotations from the sutra or rule. 9

Bakhshali Manuscript

66

The phrases placed between asterisks are quotations from the A perfectly literal rcndering would not be very intelligible to the ordinary reader and one or two gaps have been filled in; but the translation is a perfectly fair representation of the original. (ii) The problem may be represented by x(1- 4) (1-11) (1- a)= x- 24 and the solution is given as follows :*Having calculated the loss on unity* the terms become ., and these multiplied together give a. This subtracted from unity gives 6, which, inverted and multiplied' by the given amount rule.

24, is r, of 24 = 40.

(iii) Something travels 3 yavas a day. How long will it take to go 5 yojanas ? Here the solution is indicated by the following proportion only :

3 ya° : 360 years :: 5 x 4,608,000 : 21,333 years 4 months. Symbols for the Unknown

In the Bakhshali mathematics the unknown quantity is referred to by the symbol 0, which is called 'sunya,' Tq 'void' or 'empty') 2. Datta thinks that strictly speaking it is not a symbol for the 1.

Literally "divided and multiplied" into. This is generally used to

indicate division by a fraction. 2 (1) fk i FCZrj ifr t 1T"'

Ira

eff4r1 a RmW

W..J 0 1213 41 jgq 200

II

I

1

T

r

I err

I

f

a

jd pM

1

601 80 J 2CO 11 20 13 20 q 111FghlVX j;r v-j 20 1

The problem of this type :

-Folio 22, verso.

"A certain amount was given to the first, twice that to the second, thrice it to the third and four times to the fourth. State the amount given to the first and the shares of others, if the total amount given was 200." The shares are represented by 0, 1, 2, 3, 4. (Here zero stands as a symbol for the unknown). Now having added 1 to the unknown (nought), the

sum is 1+2+3+4= 10, which shows the proper share of the first is 200/10 or 20, and the series is 20+40+60+80° 200.. The proof by the rupona method gives,

(Contd. on the next page.

Symbols for the Unknown

67

unknown as has been supposed by Hoernle1 and Kayo'. For same symbol has also been used for the zero' (sunya) for the dcchmil arithmetical notation. This is, indeed, its true significance. Its iwo in connexion with an algebraic equation, in a sense other than f4)1arithmetical notation, is simply to indicate that the quantity which should be there is absent or not knowns. Hence its place in ilto equation is left vacant and this is clearly indicated by putting the sign of emptiness there. Or in short, the use of the truely arithmetical symbol for zero in an algebraic equation is a clear proof of the want of a symbol for the unknown in the Bakhshali mathematics. Correctness of this interpretation will be borne out by the facts [1] that this symbol does nowhere enter into any operation, its It ought to have done had it been truely a symbol for the unknown, and [2] that often times it is referred to as sunya, sthana or the 'empty place', proving thereby that nothing is in that place". This

[ (4-1)22 +20]4=200.

2(ii)..9 fkt-... srqfim 3 fw 2

0

I

19 i

9sur fku 12121316 ... siM !fa chi'

1111...I

...sift

33 I

f

r

1 12 16

132 I at?i 331

24 I ... 141 1

Folio 23, recto. The example may be represented by

x+2T1+3Ta+4Ts =132, whcre Ti, Ts, etc, represent the values of the first, second, etc. terms. Make x=1, then the terms are

1+2+6+24

= 33, whence x = 133 = 4 and the series becomes, 4+ 8+'4 -1.96

=132 (Kaye). (1) (2) (3)

Indian Antiquary. xii, p. 90; xvii, p 30 Journ. Asiat. Soc. Beng.. viii (1912), p. 357 ; Bakh. Ms., §9 42, 60. Compare such expression as nrularn na jnayate, itr"T'

c* (folio 13.

verso; 15 verso, prathamam na janami) 51 sli1 9 irgrfsf (24, verso), padam na jnayate tr' Wro (54, verso), etc. in each case of which the ajnata, arffTU (unknown) element has been indicated in the state(4)

ment by Tzr, Folio 25, verso ; pnd folio 26, recto,

Bakhshall Manuscript

68

hypothesis will be further supported by the fact that the similar use of the `zero' sign to denMe the unknown element in the statemen IF

(nyasa) of problems is found in the crithmelics of Sridharal and Thus we haven,

Bhaskara2.

afk 20 3 0 r(aw: 7 I Trffrc7RR 245 which is a statement of an arithmetical progression whose first term is 20, number of terms is 7, sum is 245 and whose common not known. Both these writers have well-defined notations for the unknown, and do never use the cipher in this way in their treatises on algebra. But as the use of algebraic symbols is

difference is

not permissible in arithmetic, they make use of the cipher to indicate that certain clement in a problem is waz;ting. Of course, the cipher

has wider use in the Fakhshali mathematics than in any of these works.

The lack of an efficient symbolism is bound to give rise to a certain amount of ambiguity in the representation of an algebraic equation, especially when it contains more than one unknown4. For instance, inb 5

Er

0rr 0

Zr

0

1

It 7+

0 1

Jx_ 7 _ t, different unknowns will have to be assumed at different vacant places. Again in the which

denotes

x 4 5 = s,

statements. air

5 36 q0 u0 1

1

1

1

1

1

1

1

ajr10 33 q0 q 0 refers to two arithmetical progressions whose first terms'and common differences are different but known, and whose

which

(I)

Trisatika, pp. 19 et seq.

(2)

'Lilavati', r.p. 18 scq. 7 his is not evident from Colebrooke's transla tion of the works where the cipher has been replaced by the query. Trisatika, p. 29. Nearly similar difficulty and inconvenience %ere experienced by the Greek algebraists who had only one symbol for the unknown. Folio 59, recto. }foernle and Kaye are not right in thinking that this statement represents :

(3) (4)

(5)

x-I.5=.s'= and x--7=t2. ,6)

Folio 5, recto,

Symbols for the Unknown

W4

0

sums and number of terms arc equal but unknown, stands in tlir place of two different unknowns'. To avoid such ambiguity, in ,,,ir

instance which contains as many as five unknowns, the abbreviations of ordinal numbers such as pra, A (abbreviated from pratliantu,

sr" 'first'), dvi ft (from dviliya, fgtq,'second'), ir, q (from trdya, qtq, 'third'), ca from caturth, :qer4, 'fourth') and pam, d (front pancama, gsazi, 'fifth') have been used to represent the unknowns;

e g.'. 9A

7

7

l0

I

1011 8:q

fZg

I

8a 114

I

114 9sr

Zia

450q;

(Tier)

16117,18119(201

which means,

x14-x,= 16, x,+x3= 17, xs+x4=18, X4+X8=19,

X6-+1=20. The want of a proper symbol for the unknown eventually leads

to the adoption of the method of 'false position' or 'supposition' for solution of algebraic equations. The solution generally begins with putting 'any desired quantity' (q1%v yadrecha) in the vacant places. (1)

It is not easy to say what is intended to be implied by placing the unity below the cipher. It is supposed by some to be an Indication that the unknown quantity will be an integer (Kaye, Bakh. Mi., t 60). Such a supposition is quite untrue. For in the instance cited while dhana is an integer (= 65), pada is a fraction (= 13/3). Strangely this very statement has been quoted by Kaye just after the

remark referred to. In certain instances, it is a mixed surd (vide folio 6 and 45, rectos), (2)

Folio 27, verso. In one instance in Dhaskara's Bijaganlia, Initial syllables of the names of particular things have been used as Symbols

3)

for the unknowns. (Colebrooke, Hindu Algebra, p. 195; compare also p. xi. Cf. aTa Jui pi ' : Blja., p 2, qt TW TU p Yadrecha pinyase sunye %ZT f9M1W W'W or yadrccha vinyase that is 'putting any desired quantity In the q''qmT fk,q vacant place' (Folios 22, verso, and 23, recto). On another occasion or 'the desired it is said Ka,nika,n sra,ye pinyasta,n, q,7fgq; ;Wfq quantity is placed in the vacant place (Folio 3, recto and verso). We have also such expressions as sunya-sthane rupam dattava, vjw ctitk Vq qM or 'putting one in the vacant place' (Folios 25, verso and 26, recto; compare also folio 22, vcrso).

Bakhshali Manuscript

70

Plan of Writing Equations

In the Bakhshali mathematics two sides of an equation are written down one after the other in the same line without any sign of equality being interposed.

Thus the equations,

appear asI

I0 1

s; 01m 0 1''1

7+

5 rt

1

1

r 01

The equation, x+2x+3.3x+ 12.4x= 300 is stated asp

0 2 1 3 3 12 41 dart 300 111

1I1 1I

1

1I

1

Sometimes the unknown quantity is not indicated. Thus the equation

2 3 4 is followed in the arithmetical treatise of

Sridhara and Bhaskara. According to the former' 11 it jag2 1 I

2I6 12I

means

x`( 2 + 6 +

12

2.

Bhaskara does not use the lines8. It will be noticed that in all the aforementioned works the absolute term is called drsya, (TW'.

meaning 'visible' which is sometimes abbreviated into dr (1). A distinction is sometimes made in its connotation in the different works. " The problems in connection with which the above equations

arise are of the same kind in all the works. But in the Bakhshali work, the term drsya (jar) refers to the 'gives' while in the other works it generally refers to the 'remains'e. There is, however, one Folio 59, recto. Folio 23, verso. See also folios 21, 23, 24, recto, etc. Folio 70, recto and verso (C). See also folio 69, verso.

Trisatika, pp. 13 et seq Lilavati, pp. 11 et seq.

The term drsya occurs also in the (GSS), (IV, 4) in the sense pf "remainder".

it

Plan of Writing Equations

instance in the Lilavati in which the connotation of drsya is exactly the same as in the Bakhshali work', This term is closely related to rupa (FT), meaning appearance', which is the same for the absolute term in the Indian algebra. We find thus the true significance of the

Indian name for the absolute term in an algebraic equation. It represents the visible or known portion of the equation while its remaining part is practically unknown or invisible.

The above plan of writing equations differs much from the plan found in ancient Indian algebra in which, (1) two sides tire usually written one below the other without any sign of equality and (2) the terms of similar denominations are written below one another, the terms of absent denominations from either sides being indicated by putting zero as its co-efficient (Datta).

Certain Complex Series

As already stated, the author of the Bakhshali mathematics is well acquainted with rule for the summation of series in arithmeti. cal progression. Indeed he gives considerable importance to its treatment. There are instances of the geometrical progression in the works. There are further elementary cases of a certain class of complex series. the law of formation of which is quite clear. If al, a a. . ... ... denote the succesive terms of any series, we find series of the types, (1) Lilavati, p. 11. 13 19 127 181 1 (2)

329

t

1 3r'r' 242...

y

1

1

a 1 STasT t

1

1

a) r -
iT vi

1 3 1 9 1 27 1 81 ril 1 121 1 aft;r 1 2 1 6 1 18 1 54 1 162 1 vmmft 1

=ii aTmi

M12J

2 15 1 48 147 444

1I 2

2

2

2

-Folio 51, verso,

This appears that it represents a double series : + 34'3 + 3313 tl + 3t1 + 32t1

0 + ;t3,+ 4 (11+ta) +4 (t1+ta+t3)

+;s (tl-4-trl-ta+t.)Jl -329

Set t t = 2, then the first series becomes 242 and the second 87, and the 444 147 48 15 -I- i-+ -2 = 329. combined series, 2-I- 2 1

2

(3) The series of these types occur respectively on folio 22, verso ; 23, recto ; 23, recto and verso ; 25, verso and 26, recto : 24, verso, and 25, recto ; 51, recto and verso.

71

Bakhshali Manuscript

(1) ai+2.71+3a1 + 4a1.......... +na1, l2) a1+2a1+3a2+4a3+ .....nan-1, (3) a1+2a1+3 (a1+aa)+4 (a1+as-Fa

(4) a1+(2a,±b)-f{3a1 f (b+d)}+4a,+ (b+2d)}+...... (5) a1+(2a, +b)+{3a,+(b+d)}+ {4v3+(b+2d)}+.........

(6) a1+(2a1+b)+{3 (a,+a,)f(b+d)+ {4 (a1+a2+aa)f(b+2d)+... (7) a1+(a1r+dal) +{alra+d (a,+atr)}+. {atr3+d (a1-l-air-J--air2) )-f--......

Evidently the seric; (4), (5), (6) are obtained respectively from the series (1), ('-), (3) with the help of the subsidiary series in arithmetical progression,

b+ (b+d)+(b+2d)+... Systems of Linear Equations

Examples of the following type occur :

x1+xa=a1, x+x3 =a2..........xn+xnl1=an always odd. We have,

where n is

an-'a2-a1)+......+ (an_.1-an _2)+2x1 ; and if we assume x1= p then, an'=(a2-a1)'l-'...... } (an_1an %)+2p.

Subtracting we get, x1=p+ (an-a'n)l2, which is the solution

employed in the text. The following examples occur (1) xt+xa = 13, x3+x3=14, x;,+x1=15. The value of x1 is assumed to be 5, then by 'subtraction order

x2=8, x3=6 and x3-+1=11. The correct values are therefore, x1=5+(15-11)-, 2=7, x,=6 and x3=8. (ii) x1+x5= l6, xa+x3=17, x3+x4=18, x4+xa=19, xg+x1= 20.

Here the value of x1 is assumed to be 7 and x'e+x'1=16 therefore the correct values are x1= 9, x,=7, x3 =10, x4= 8, x8=111 The following are implied, 1. Since in these two particular examples the values of at as, etc. are in arithmetical progression a, simpler solution would be x1"i where m is the mean of the series at, as etc.

7:

Systems of Linear Equations

(ill) x1+xa=9, xa+xs=5, x3+x,=8. (iv) x1+xa= 70, xa +xs = 52, xs+x 1= 66.

(v) x1+x2=1860, xa+xs=1634, xs+x1=1722. (vi)1 xa+xs+x4+x5=317 x, +xs+x4-+5 = 347 x1+xa+x4+x5 = 357 x1+x2+xs+x4= 365 xl+xa+x3+x5 = 362 and the following, which, however, is too mutilated to be sure of

(vii) x1+xa=36, xa+xs=42, x3+x4=48, x4+x5=54, xa+X1 =60.

The directions seem to indicate that we should cancel by six. We then get,

a'1=6, a9=7, a'3=8, a'4=9, a'5=10 of which the solution is x'1-4, x'2=2, etc., whence x1=24, xa=12, xs=30, etc. Further Examples

The next set of examples can be represented in the. form

;'x-xr=c-d1x1, Ex-x9=c-dsxa,...I:x-xa=c-daxa. If we set, a=1-d, these become, 1. This can be arranged in the form y1+ya= 365, ys+ys= 347, Ys+Y4 - 362. y4 +y5 = 317, y5 +y1= 357 Solving this we get x1+xa = 210,

xa+x3=170, xs+x4=155, x4+x5=147. x5+x1=192, which solved gives x 1 =120, xa =go, x3 = 80, x4 = 75, x5 = 72. (See-folio 1) Also note that most of these six examples can be expressed in the form

Y,x-x1=a1, Mx-xa=aa... Ex-xn=aa. The examples given in the text pre undoubtedly akin to the'Ephanthema', usually attributed to Thymaridas, which may be expressed by xo =

Ea-Ex

n-1

where xo+x1=a1, xo+X2=aa...xo-f-xn=an. In Aryabhata's 'Canna is a similar rule Ex= Ed

n-1

where

EX-x1=d1r

Ex-xa=da ....

Ex- xp = dn, which Cantor ( Vorlcsungen uber der Gesclilcmuc der Matlrc. milk 1. 624) considers to be a modification of the rule given by Thymaridas. See also P. Tannery Memolres Sclenifiques, Tome II, pp. 192-195. 10

Bakhshali Manuscript

'f4

x2= ....=`_x-anxn=C, and alxt.=aaxa=....anxn

=2,x-c=k. Therefore : x=

^F 1

l

l

U.

1\J k

+..

p k.

an

as

q

A solution is xi= ri ,x2= q ..xn= 1 and al

c-p=q'

an

ag

There are four examples illustrating this process which may be

tabulated thus :S.

No

da

di

I

d:,

dg

I P

ds

I

437 60

_

7 311

C

xi

I

Y,

377 111

91

X3

X4

x5

81

71

71

2118 1095 914 836 798

i2_4___6

1463

7

8

161

41

18

14

1

2

3

4

17

6

3

1

Of these only fragments remain. For example, of the first we have; (1) Compare with this the treatment of the epanthetn by Jamblichus (Heath, Greek Mathematics, 1, 93-94

(1) The nearest approach to this that I (Kaye) have came across in an Indian work is given by Mahavira in his GSS, VI. 239-240).

Five merchants saw a purse of money. They said one after another, by obtaining _, T, 1 a of the contents of the purse I shall become three times as rich as all of you. ,,Let p be the value of the purse and xis x2 etc., the original capitals:

then 3 (2x-xl)= 6 ±x1,3 11Yx1

_

Zx-

6+

6508p

r

P -i-xs,etc.whence I

8

- 9 + 10

6508p

10080 ' or

If p=110880, then x1=261.

110880' But al-Karkhi (Xlth cent.) gave (iii, 6) the following :A certain sum is divided among three people, one-half being given to the

first, one-third to the second and one-sixth to the third. But then onehalf the share of the first, one-third the share of the second and one-sixth that of the third is pooled and shared equally by the three.

(Contd. on next page)

Further Examples

2T4 1

1

1

1

120

1

-

-C

.S 7

90

75 80

75

60 + 60 + 60 + 6U

72

417

60

(dl

90+80+75+72-317, 317+ 120 = 377 120+80+75+72=347, 347+ 30 = 377

120+90+75+72-357, 357+1q = 377 120+90+80+72=362, 362-f 55 = 377 120+90+80-1-75=365, 365+ G = 377

The second of these examples may be expressed in the form

xl+xz-( +.) x3=xa+xs-(,+4) x1=xs+xl-(g+'s) x9 From this !9 1463 2558 1463 '

x,, =

7

x3 = 6 xs and . x c

12

4

6

= 19 + 7 + -11

2

The solution given

:Ex= 2558

is

and c=2558-1463

1091, whence x1= 924, x3= 836, x.,1=798.

Of the third, there is sufficient of the formal question preserved to enable it to be restored.

"One possesses seven horses, another nine mules (?), and a

third ten camels. Each gives one of his animals to each of the others and then their possessions become of equal value". In. A. D. 1225 this problem became famous as it was one of lboso propounded to Leonnrdo of Pisa and solved by him. If x1, x2, x1, be the shares and 3c be the amount pooled, then,

x1-

X -f e= -x , 2

X3 - 6 +c=

xa

X2-

x9 3

-f-c_ `x 3

and j'x-x1=2c, -2x:-2x,-3e,

6 Fx-5x3 =6c. whence x1®33,x3=13,x3 01.

-47 . Make c= 7 and

Bakhshali Manuscript

76

The fourth example of similar form'. Mutilated Examples

The following occurs on some very mutik ted scraps that have been pieced together. it is now impossible to say how it originated or what its context was.

x (x+y+z)=60 y (x+y+z)=75 z (x-}-y+z)=90

We have (x+y ; z )a=225 whence x+y+z=15, and x-4, y=5 and z=6. Solution in Positive Integers :

There is one nearly complete statement and solution of the following pair of equations :--

x+y+z = 20

3x+3y+lz=20 of which the only solution in positive integers is x= 2, y= 5, and z- 13. 4

This type of problem was a favourite in Europe and Asia in early mediaeval times. Later it was known as the 'Regula Virginum', 'Regula Potatorum', etc. It was given by Chang-ch' iu-chien (sixth century A. D.)2, by Alcuin the Englishman (eighth century) and.others. About 900 A. D. it was pretty fully treated by Abu Kamil al-Misri who gives some six problems, varying from three to five terms and attempts to find all the positive integral solutionss. (1) This set of examples is intreduccd by a rule which simply means : change the fraction A/B into B/ (B.- A). This illustrates the fact that the sutras

or rules often makes no pretencc of indicating in any way the general theory : they are merely intcnded to be helpful. (2) Yoshio Mikami. (3) H. Suter, Das Buch der Selienheil, etc. Bib. Math., (1910-11), pp. 100120. Suter's Die Mathematiker and Asironomen der Araber and Ihren Worke, p. 43.

Quadratic Indeterminate Equations

77

In the earlier Indian works problems of this type do not occur,

but exactly the same example is given by both Mahavirat and Bhaskara 112.

Quadratic Indeterminate Equations.

There are two types of quadratic indeterminates preserved. First Type : (i) x -}- a = ss, x- b = t 3.

The solution given may be represented by

f( l a+b"l3 c ),+b

x-12 which

c

makes both x+a and x-b perfect

squares.

In the

actual example preserved a= 5 and b= 7 and the solution is x= 11, which is the only possible integral (positive) solution obtainable from

the formula. No general rule is preserved but the solution itself indicates the rule. It proceeds by steps thus :

5+7-12,12-t-2-6, 6-2=4,4-+-2-2,22=4,4+7=ll. The value of this type of detailed exposition is here selfevident. It seems to have been almost necessitated by the absence of ti suitable algebraic symbolisms.

Second Type : (ii) xp-ax-kv-c=0 (1) GSS., vi, 152. (2) Vija-Gan., 158.

(3) This type of equation occurs in many medeiaeval works from the time of Diophantus onwards (c. g., see Diophantus, ii, 11ff : Brahmagupta, xvili, 84: A1-Karkhi, p. 63, etc.) and has here no very special interest beyond the indication it gives that the Bakhshali text followed the fashion. Dr. Hoernle, however, thought that it indicated a 'peculiar' connexion between the Bakhshali Ms. and Brahmagupta's work : and from this deduced that our text may have been one of the sources from whence the later astronomers took their arithmetical information.' (Indian Antiquary, xvii, 1888, p. 37).

Bakhshali Manuscript

78

The solutions which appear to be followed in the text are

x=(ab+c) -;- ,n+b, v=a+m; or y=(?b+c)--m+a, where in is any assumed number. The only example that is preserof which the solutions given are 15 and 4, ved' is and 16 and 5, i, e.,

(3.4-1) - 1+4= 15, 3 }-1 -4; (3.4 -1-1) _ 1+3=16, 4±1= 5. Happy Collaboration of the East and the West

In chemistry, astronomy, mathematics, medicine and surgery, there was always in our history a happy collaboration in the advancements, in which all great nations of the East, Middle-East and the West took part. The Bakhshali manuscript in its original form (agreeing to Dr. Hoernle's suggestion) must have had its authorship in the early centuries of our era. From generation to generation, it must have been copied out with alterations and additions. By this time the western mathematics had incorporated almost all that could

be traceable to Greek sources, and had assumed new forms and included some new notions also. On the whole the western mediaeval mathematics tended to become less rigorous and but more 1.

Unfortunately the text 'folio 27, recto' is so mutilated that the correctness of the interpretation h-re t'iren cannot b. guranteed. But the equation was well known to mediaeval mathematicians and has historical interest. Brahmagupta gave a general solution (-,viii, 61) which, however, he appears to have th,-,ught unnecessary. Bhaskara also'givec a general solution (Bija

Ganita, 212.214) together with demonstra'ion in both algebraical and geometrical forms; but he also on another occasion (ib. 208-209) gives an arbitrary solution, The general solution is also given by Al-Karkhi. Mahavira gives (vi, 284) the equation in a different form, namely

( a+

i

> (h-i- -i )=ale -}A which reduces to xr- A X-

A0 m; or y = the n x=

of which the solution is x= lA aAb a

A

r \ ° Aa

A

),=±

-}-

A

x- A -I- in. If we make m= A -{-1

A b

A

1

,

y

=

a

A

which is the solution given by Mahaviia.

His examples are : The product of 3 and 5 is 15, and the required product is 18 or 14. What arc the quantites to be added or subtracted ? i.e.

(i) (3+lJx) (5+11v)=15+3,(ii)(3-l/x,) (5-15-1. His answers are (i) 2 and 7, (ii) 6 and 17.

Happy Collaboration of the East and the West.

I`

mixed than the mathematics of the classical Greece. Practical calculation (logistic) altogether supported the earlier pure arithmetic: mensuration took the place of pure geometry, and algebra slightly developed. Such changes are, indeed, indicated in the later Alexandrian works, and what we sometimes term degeneration had already set in there. The introduction of a place-value arithmetical notation made calculation easier and more popular, and more intricate arithmetical problems than those exhibited in the Greek Antho. logy, appeared in the later mediaeval text-books. The general body of popular mathematical knowledge became more diffused. (Identical problems occur in Chinese, Indian, Arabic and European text-books of a comparatively early period). Indeed the mediaeval

mathematical works of Asia and Europe had so much in common that at first it seems almost impossible to pick out that which is definitely western or eastern in origin. In this connexion it should be remembered that the early astronomers (Aryabhata 11, and Varaha-Mihira, an astronomer of Greek or middle-east origin, settled and naturalized in India,) were among the first to collaborate with Greeks in mathematical and astronomical learning; that later the Arabs, after sampling Indian works, entered into this collabora. tion; and that it was from the Arabs that Europe received once more the learning it had previously rejected. (Kaye)

We give here a summary chronological table of the period, that may serve to recall the chief mathmatical writers and their works : Earlier than 4000 B. C. Before Christ-Rgvedic period Taittiriya Samhita 20L0 B. C. 1000 B.C. Satapatha Brahmana 500 B. C. Baudhayana Sulba Sutra Vedanga Jyautisa, Lagadha 500 B. C.

2nd to 3rd Century Bakhshali Manuscript (original) 5th Century

Hypatia Proclus Boethius Aryabhata

300 A. D. d.415 A. D. 410- 485 A. D. b.470 b.476

80

6th Century

7th Century

8th Century

9th Century

Bakhshali Manuscript Bhaskara I Eutocius Damascius Simplicius Dominus Chang Ch'-iu Chien Varahamihira Isidore of Seville

522 529

550

505-587 570-636

Brahmagupta Fall of Alexandria Papyrus of Akhmin

b.598 640/1

Bede

b.735.

Muhammad b. Musa Alcuin Sridhara

b.804

Mahavira Tabit b. Qorra

850

Aryabhat II

c. 950

Avicenna Pope Sylvester 11

d. 1003

750

835-901

10th Century

11th Century

12th Gentury

1017-1030

Albiruni visited India al-Karkhi Psellus Omar Khayyam

b. 1046

Adeland at Cordova Bhaskara II Leonardo

b. 1114 b. 1175

1120

For the contents of this chapter the author expresses his indebtedness to the critical account of the Bakhshali Mathematics given by Kaye in his publications and to the critical paper of Dr. Bibhutibhushan Datta in the Bulletin of the Calcutta Mathematical Society, 1929.

APPENDIX I WEIGHTS AND MEASURES

The measures exhibited in the manuscript are of rather special interest. As a whole they are Indian and the terminology is Sanskrit; but there are some Sanskritized western terms such as lipta, dramma dinara, satera employed. Most of the terms are well defined but the, values of some are doubtful. Money measures, however, are, as in

most early Indian works, very ill-defined and hardly show any differentiation from measures of weight.

Change ratios : The change ratios are often given with considerable care and elaboration, and are expressed in several different

The change ratio appears to be considered as a divisor for it is most frequently marked by the term chhedam which indicates ways.

the operation of division. Examples are :

chhe° 80 rakti°- su° I. e. 80 raktika =1 suvarna chhe° 24 am°-ha° I. e. 24 angula= l hasta chhe° 2 gha°-mu° i. e. 2 ghatika=l muhurta chhe° 4608000 ya°-yo° I. e. 4608000 yava- I yojana urdha chchhe° 768000 a°-yo°, i.e. 768000 angula=1 yojana, and' is

this particular example the operation of multiplication is to be performed.

urdha chchhedam 108000 viliptanam rasi, i.e. 108000 vilipta=1 rasi and multiplication is indicated. adha chchhedam 2000 pa°-bha°, i.e. 2000 pals =1 bhara and division is indicated.

Another form is illustrated in the following exampler I

tolenasti'dhane 12, i.e. I tola - 12 dhana. dhanenasti am° 4, i.e. I dhana=4 andika dinaranasti dhane 12, i.e. I dinara - 12 dhana II

13akhshali Manuscript

82

Abbreviations :

The Bakhshali abbreviation for weights

and

measures are

usually as follows :

Adh, arr= Adhaka, anW

Gav, qa= Gavyuti, gnfg

Am, aT - Amsha, aTar

Gha, q=Ghatika, gfkr

Am, at-Andika, arfsgWr

Gum, = Gunja,r

Am, aT= Angula, aTp

Ha - Hasta,

Bha, qr=Bhara, qR

Ka, q, - Kakini, WTf ?

Dha, tTr= Dhanaka, UT;1q;

Ka, ; = Kala, qMT

Dha, q _ Dhanusha, qjq

Kha, q a Khari, Arf t

Di, fk=Dins, f

Kro,

Di, ift=Dinara, wftffrZ

Ku, = Kudava, lga Li. fq= Lipta, fjr

Dram, x =Drama, tM Dra, = Drankshana, ttjur Dro

= Krosa.- %),q

Ma, g=Masa, qR{ Ma, q<= Mashaka, tTl@rq;

=Drona, f

Mu, 11= Muhurta

Sa, q-Satera, c T Si, fir= Siddhartha, fi`r

Pa, qT= Pada, qW

Su, ;g = Suvarna,

Pa, q = Pala, qff

To, cat=Tola, FiT

Mu, IX= Mudrika, lfti

Pra, sr=Prastha, M Pra, A- Prasriti, Si

Va, a= Varsha, 44

Vi, fa=Vilipta, fa1r

fic

Ra, Z- Raktika, zfrti Ra, Tr. Rasi, zTfar

Ya, q=Yava, Yo, rft=Yojana, xj

Time Measures

I Varsha=I2 Masa=360 Dina 1 Dina - 30 Muhurta = 60 Ghatika Are Measures 1 Rashi =30 Ansha= 1800 Lipta=108,000 Vilipta

I Ansha=60Lipta(7.E,rv'i) 1 Lipta - 60 Vilipta

Moncy Mcasures

HA

Money Measures

I Suvarnar=11, Dinara=22 Dramkshana=16 Dhanaka-80 Raktika I Dinara = 2 Dramkshana I Dramkshana = 6 Dhanaka

I Dhanaka

= 5 Raktika

Weight Measures

I Bhara

= 2000 Pala

I Pala

= 8 Tola

I Tola

=2 Dramkshana

1 Dramkshana=6 Dhanaka =4 Andika I Dhanaka I Andika

=I I Raktika

1 Raktika

= 3T Yava

I Yava

- 2} Siddhartha

1 Siddhartha - 21 Kala I Kala

- 4 Pada

1 Pada

- 4 Mudrika

Length Measures.

I Yojana

=2 Gavyuti

1 Gavyuti

= 8 Krosha

I Krosha

-1000 Dhanusha

1 Dhanusha =4 Hasta .24 Angula 1 Hasta 1 Angula

- 6 Yava

13akhshali Manuscript

Capacity Measures I Khari -- 16 Drona

I Drona I Adhaka

4 Adhaka = 4 Prastha

Prastha

=4 Kudava

1 Kudava

2 Prasriti

I

I

Prasriti

=2 Pala

Arithmetical Problems INCORPORATED IN

The Bakhshali Text

Hakhshali Manuscript

86

A--7 It is possible that 52 recto gives parts of the solutions cf the example on 51 recto which would make that page the reverse, (of course, doubtful).

What is left of the solution means :

X (10+ R-+ 4) -57 or

19

A proof by the rule of three

112010 12 1 :120::

15

1 :120:: 4:30 ii. The example, which is continued on 52 verso, may be (I --- ')-x-280, whence x=400. A expressed by x (1- z) (1 proof follows : 400 x n =200 and 400--200=200; 200-50=150;

150-30-120: and 400-120-280 Again,

2+ 4(1- I)+(I ..- 4)(1_2)=2+ R+

-1.

A-9 29 a, b c recto. The problem and its solution here partly preserved may be represented as x, +.v2= 13: xa+x.; -14: x,-! x, -15.

If x1=5, then x', =8, x'3=6, and hence r'.,+x, -5-1 6=11. From this, the correct value would be

x,=5+(15- 11)-2-' whence x%=6 and x3=8.

Problems

117

The phrase "shodhayet kramat" recurs in the next example and is a quotation from .a lost sutra.

29 d, b, c. verso. The example here given (continued on folio 27 verso) is formulated with exactly the same phraseology as the previous one. It may be represented by :

xi+x'= 16; x2+x3 =17; x3+x4 =18; x4+xs = 19 and xa A-xl =20. if xl=10, then xa=6; x3= 11; x4=7; xa=12 and x'5-I-x'1-22

Therefore, the correct value of x1=10- (22- 20)+2= 9, and thenx2=7,x8=10,x4=8 and x6=11. The phrases "icchu' and "shodhayet kramat" are quotations from a lost sutra.

A-10 27 verso gives the answer of the problem given on folio 29 verso, namely x1= 9, xa = 7, xs = 10, x1- 8, .rz =1 I and the sums of the pairs of 16, 17, 18, 19. 20. 27 recto. Solution of a lost problem which may have boon

xy- 3x -4y± 1=0, of which solutions are :

x_3.41 1+4=15;y=3+1=4; x=4+1=5; y =3.i+1+3-16. The quotations are from a sutra very much like the one that follows. The phrase "prathak rupam vinikshipya" 'having

added unity in each case', appears to be a quotation from a lost sutra.

A-11 The example is solved on 1 verso and 2 recto and appears to have been somewhat as follows :

The combined dapitals of five merchants less one-half of that

of'the fifth is equal to

the cost of a jewel. Find that cost of

the jewel and the capital of each merchant.

I

.

A 11. 1 verso. This appears to give part of the solution and proofs dr tihb' question on 1 verso.

88

tIakhshali Manuscript

Then follows 'proof' which may be expressed by

90+80+75+72=317 and 317+1,2,°=377 120+80+75+72=347 and 347+90-377. 120+90+75+72=357 and 357+840-377. 120+90+80+72=362 and 362+75 =377.

120{-90+801-75=365 and 365+°9 =317. Compair with sutra I I on I recto. A 12

A 12.2 recto This appears to be another 'verification' of the

example on 1 recto et verso; and means :

l3°+901 -80-7-75+72=377.

120+ '+80+75+72=377, 120+90+80+ a6 +72- 377.

120+90+80+75-} a = 377.

and this is the measure of the price of the jewel'. A 13

A 13. Example : One possesses 7 horses (a°=asva), another

9 horses (ha°=haya) and a third 10 camels (u°=ushtra). Each gives one of his animals to both the others and then their possessions are of equal value. It is required to find the capital of each merchant or the price of each animal, if thou art able, solve me this riddle.

B1

i

B 1.8 verso. i. The sutra is partially reconstructed from the quotations in the solutions below.

ii. The example is : There are ten horses of which five are yoked at a time to the chariot. How many changes should there be in a journey of one hundred yojanas and how much will each horse do ?

The solution is 110F0=10 stages and 10x5-50. Mahavira gives a similar example (vi, 158).

Problems

$9

"ravi-ratha-turagas sapta hi chatvaro'sva vahant dhuryuktnli yojana-saptati=gatyah ko vyudhah ko chaturyogah". It is well known tL at the horses of the Sun's chariot are seven. Four horses are yoked at a time. they have to perform a journey of 70 yojanas. Ilow many times are they unyoked and how many times yoked ?

B2 "For a certain feast one Brahman is invited on the first day, and on every succeeding day one more Brahman is invited. For another feast, 10 Brahmans are invited on every day. In how many days will their numbers be equal; and how many Brahmans were Invited ? The use of the term labdham is here rather curious. The phrases labdhan: dvigunitam kritva, tathadvyunam, uttarain vibhajitua! and rupadhikam are probably quotations from a sutra.

B 2. 9 verso. The example probably meant : A and B start

for a place 70 yojanas distant. A travelled at the rate of 1 yojana a day and B at the rate of 6. At what point on his return journey did B meet A ?

Since, 1

2-706-x

, where x is the distance traversed by A,

x=20 as given in the text, and since A travels we have x= 270 6+1 at the rate of one yojana a day, this is also the time. Proof by the 'rule of three'. 1 day : 6 yo° :: 20 days : 120 yon, and 70-20=50 and 70+50=120. Also 1 day : 1 yo° :: 20 days : 20 yoe.

B4 B 4. i. One goes at the rate of 5 yojanas for 7 days and then a second starts at the rate of 9 yojanas a day. When will they have traversed eqnal distances ?

The phrase gatisyaiva vishesam ca is a quotation from sutra 15 (folio 3 recto) and purva gala. is a reference to the same rule.

The solution is t=

1 :: 5 :.

55

9.5

5:

- 45 days. 'Proof by the rule of three

175 and 1 4

-4 = 35 : 9 :.354 . 315_ 4 -. 175

Bakhshali Manuscript

90

Qne travels at the rate of 18 yojanas in one day for a period of 8 days. A second goes at the rate of 25 yojanas in one day. Determine in what time......

D5 D 4 With some uncertainty : A earns 33 drammas in 2 days, B earns 2p in 3 days. A gives B7

drammas and this makes their possessions equal. How long had they been earning ? Since,

33 2

2

t-7 =

'+7. we have t =.

14 3_8

=

168 11

3

= 1511

days.

And 3 days 294 11

77

24. drammas:: iii days

:

77

140

l1 - l1 + 11 ii.

140

days

:

294

drammas and

drammas

Another example of the same kind.

D6 D 6. 67 recto. i. The example seems to relate to a game at which a certain quantity was staked and eventually all lost. The statement means, 1-4-

(2-f I (3-+(3+12 (4+ V

E1

(5+4. .7 -)))=Isi

E 1. The problem may have been something like this : The rates of purchage are one, two, three, four and six articles for one drama.

The cost of one of each would be 1+1+It+H4==p, therefore the cost of 12 of each is 27 drammas, and the numbers of articles are 12, 6, 4, 3 and 2.

Problems

91

E2 53 recto. The following conjectural restoration of the problem is offered :

One goes 11 yojana in a day and 'another 6 in 3 days. If the first had a start of 9 yojanas, when would the second overtake him ? 53 verso. The following is.merely a guess at the problem :

One goes 18 yojanas in 96 days and another 27 yojanas in 108 days. The first starts from A and the second from B and the distance Alt

is 9 yojanas. When will they meet if they go only for 1"V (or 35 ghatikas) of each day ? (60 ghatikas = 24 hours).

E3 58 verso. There is basis for the following restoration- A man earns 3 in one day, a young woman I in one day and..... i In one day. If 20 earn 20 mandas in one, let x, y, z be the numbers of each class. Then, x--y±z=20 individuals :

3x+

3y + Z = 20 mandas

of which the only solution in positive integers is that given in the text, namely x = 2, y = 5 and z = 13. This problem is known at the Hundred liens problem in China, and as the Requla Virginum, etc. in Europe.

F1 F 1.22 verso I. This appears to be the beginning of a now section. This sutra is lost.

The problem was something like this : A certain amount given to the first, twice that to the second, thrice it to the third, and four times to the fourth. State the amount given to the first and the shares of the others, if the total amount given was 200.

The shares are represented by 0, 1, 2, 3, 4 'Having added one to the nought' the sum is 1+2+3+4= 10. Then the proper share of the first is 1T °a° = 20. Having added in this value the series

becomes 20+40+60+80=200. The proof by the rupona method

gives (4 -1) '0+2014-200.

Bakhshali Manuscript

92

For the meathod of solution, the regula falsi, see Part I. Sec. 71 and 72, and for the rupona method see Sec.73. The whole section is _

dealt with in Sec. 87, and the use of the symbol for nought in Sec. (Kaye)

60.

The sutra beings. "Put what number you please in the empty place (or for the nought)." This is quoted on folio 23 recto and so is tada vargarn to karayet, etc. ii.

F2 F 22, 23 recto. The example may be represented by x+2T1

+3Ta+4T,=132, where T1, Ta, etc. represent the values of the first, second etc. terms. Make x= 1, then the terms are 1+24 6+ 244= 33 and the proper value of C is 1,31 = 4 and the series becomes

4+8+24+96 =132. All the technical terms here employed Ere of interest : ichchha an assumed number'; varga a aeries; prakshepa' something thrown in' or an interpolation'; vartya 'cancelled'; drsya the given number' etc.

ii. The term karnika (wrfm) is practically synonymous with ichchha or yadrichchha' that you please. For an assumed number, Bhaskara uses ishta much in the same way. A good deal of the sutra is quoted on folio 2, 23 verso. 23 verso. i. The example may be represented by x = 2T1+ 3 (T1+Ta) +4 (T1+Ta+T3)= 300. Put x=1, then the series becomes 1+2+9+48 = 60 and the proper value of x is v°s" = 5 and we have T1=5, Ta=10, Ts= 45, T4=240 and total T= 300. ii. The example is solved on folio 24 recto. e

F3

F 3. 24 recto. The example may be represented by x (1+14)

+ 2T1 + 21 x + 37' + 33 x +4T3+4gx=144 : Set x=1 and the series becomes :

+ 2+ 2-= 2 given sum and therefore x=1 is correct. 5

2

+

15

52

217

2

289

144

1

2

which is the same as the

t)3

Problems 24 verso. i. The example may be represented by

x (1+1?) + 2T, -14x+3 (T1+T2) + 3x+4 (Ti+Ta T,) +r-x= 222.

Set x=1 and the series becomes 2 +

15

+ 2? +

222

1'lu

same quotation shunya sthane.. rupam datva occurs on folio 25 verso, See also at the bottom of folio 26 recto. F 4.

F 4. 25 recto. The example may be represented by x (1+:I)

+2T1-cx-1-3 (Ti+Ta)-3x--4 (Tl-f Ta+T,)-- x=78. Set X-1 and the series became $ } i+ a + 43 =1-6 and x= 1. 25 verso. An example, of which only the solution remains.'

ii. The example of which the solution is given on folio 26 recto.

F 5.

F5.29 recto. i. This is the solution of the example given tit the bottom of folio 24 verso. Let x=1, then the series become"

$+$+3+ q = v and the correct value of x is

y =1,

G1.

G 1. 10 verso. i. Gives further proofs of the example on the obverse, namely

1 (1-3) (1-s) (1 - )=32, hence x1=108;

then two proportions in words and figures 287 a : 33 :: 32:108.

108 : 32 and

ii. Example-Of iron or ce refined three-tenths is lost. What is the remainder of twice seventy, tell me, 0' Pandit ? The loss on unity is T& and the remaineder is Ia. The original quantity is 140 and is of 140=98. The loss is therefore, 42 and 98+42=140. 1 :: 98 : 140 Proof. Continued on fol. ii. recto.

iii. The example may be rendered :

94

Bakhshall Manuscript

The third part of the burnt bronze in three instalments (is lost). The amount given was one-hundred and eight. State the remainder,

0' Pandit. The solution according to the rule gives 108 (1- %) (1-1) (1-Wi) = 32. -But proceeding by steps ' 0e = 36 and the remainder is 72; 11 = 24 and the remainder is 48; 48= 16 and the remainder is 32.

The proof may be represented by 32 xi _ (1-1/3) (1-1/3) (1-1/3)

G. 2

ii. Example-In purchasing one and a half palas the loss is oneState what would be the loss on eighteen. Since U1/,3= 9,, the

third.

loss on unity, the remainder is g. Now n of 18 = 14 and the loss is 4.

Proof by the rule of three

18 : 4 and 3 : 1g :: 4 : 18

Example-In refining bronze there is a loss of one-fourth. What would be the loss on 500 suvarnas four times refined? The solution is lost. It amounted to

500 (1-1) (t-1) (1-1) (1-{) = 158su° 64 - 158 su°+1 F; too (Since 5 tolas =1 Suvarna)

(11 rcrso.) This appears to have contained five proofs of the example on the obverse, for the present third proof is designated ,the fourth'. The proofs arei. Missing.

ii. x1(1-;) (1-1) (1-1)=158 su°+11-4 too, therefore, xl = 500. iii.

500(1-1) (1-4) (1-1)=x' and x=158 su°+18s too

iv.

xl - (158 su°+ 11: to°)/(1-1) (1- 3) (1-1) and x1= 500.

Problems v. The first basis is

540

Ts

su° - 125 su° and remainder

itt

375 su°. The second loss is 345 su° = 93su°+ 3 to° + 9 masha (sin u

12 mao=1 to°) and remainder is 281; su°. The third loss is 28111470.r$,; and the remainder is 21018. The fourth loss is 210,%/4, 5217 and the remainder is 15813 64 G. 3.

The example may be conjecturally restored : A traveller goes on a journey of 4 gavyutis and takes with him 4 prasthas of wine. After each gavyuti he drinks 1 prastha and then fills up his bottle with water. How much wine and how much water will there be at the end of his journey ? G. 3. (12 verso). The solution of the example on the obverse is now done by steps. The original amount of 4 prasthas is expressed in kudavas. namely 16.

Kudavas of wine drunk Originally After first gavyuti

Kudavas of wine left

0

16

4

12

Kudavas of water 0 4

After second gavyuti 1 - 3

9

7

After third gavyuti 1=24 After fourth gavyuti ; x }= 2r IV

6

9;

541

10-34

G 3. 12 verso and 13 recto. Kumkum was purchased,for 8, and at each toll office the toll'was paid one-fourth. There were four toll-offices, calculate the total toll paid, and the Kumkum left, O' Pandit.

Solution :

i. 8 x ; e 6, and 2 is paid as the first toll. ii. 6 (6 -4) a 43i and the second toll is 18. iii. 4g (1-1) = a - 38, and the third toll is 1

.

iv. 38 (1- 11-tsi-2g1, and the fourth toll is ai.

Bakhshali Manuscript

96

The total toll paid is :

-2+lW-Ig+2'-s3 (64+48±36+27)- a6 =537. 8-515 = 32 - 232'

G4. G 4. (13 recto). i. Here are four 'proofs' of the example given on folio 12 verso.

(a) 8(1_1 ) (l-h) (1

)

(c) 8(1 -) (1-;) (1-4) (1-;)=s4 (d) x1=

81 /32_

__

(1-) (1-l) (1-i) (1-a)

whence xt = 8

ii. There is a quantity of molasses weighing eigut bharakas. What will be left after giving away one-third, one-sixth and onefifth.

8(I-s) (1-4) (1.r6)= a =3v iii. Example-By a gain of five-fourth, ten dronas are obtained Let it be said. 0 best calculators, what will be the gain by three transactions.

Here the term labha seems to have meaning "capital profit", what is termed the mixed quantity' misraka on folio 62.

(b) xi (1-4) (1-;) (1-4) = 19$3 dro°-19 dro°+2a°+ 0 pra° -l- 2 ku° whence x1= 10.

The conversion table of capacity measures is given in folio 13 as follows (marked with asteriks) :

4 adhakas (a°) = 1 drona (dro°) 4 prasthas (pra°) = 1 adhaka 4 kudava (ku°) - 1 prastha.

ii. Example-The capital of a certain banker is sixty, One half of it goes in loss and then he gains by one-third; next he loses

Examples

9'/

one-fourth of it and finally gains one-fifth; so that he has two gains. What is his gain and what is his loss and what-the remain-

der and let that be stated.

Solution : 60 (1-;) (1-i) (1+s) (1-1-4)=36. Proofs : (a) x1= (1-3) (1+s)

36

(l+3)

whence x1= 60.

(b) 60(l-4.)(l+4) ,l-4.) (1+a)=36. (c)

xl (1-fi) (1+a) (l-3) (1+4.)=36. whence x1= 60.

G-5 G S. ii. Example-A known amount of molasses equal to four is increased by one-third, one-fourth, one-fifth, one-sixth and then forty is lost. No solution is preserved.

G 5. (14 verso). i. Example-An unknown quantity of lapislazuli loses one-third, one-fourth, and one-fifth, and the remainder after the three-fold operation on the original quantity is twenty seven. State what the total was, 0 wise one, and also tell me the loss.

Solution and this is the loss.

1-1-73; 27.:-.53-45 and 45-27-18

The meaning of ambha-loha, lapis-lazuli, was suggested by Dr. Hoernle.

ii. Example-Of the loss of iron the third is one-fit th of a masha. The original quantity is not known and neither is the remainder given but only the original remainder which quantity stands as twenty. Tell me what is the unknown, original quantity and what is the remainder.

The interpretation, however, is by no means certain. The solution is lost. 12

gakhshali Manuscript

G-7 ii. Example-Of a ghataka of honey two-third is given, to the second two-fifth; to the third two-sevenths; to the fourth twoninths, till only three palas are left). 0' Pandit, state how much altogether was taken away by the tax collector.

H-i H 1. ii. Rule : Having multiplied the parts of gold with the ksaya let this sum be divided by the sum of the parts of gold. The result is the average ksaya. This means :

J 3 fig1 +f282 + +fngn 8i+82+...+8n where f denotes ksaya and g gold.

iii. Example-f1=1, f= 2, f3=3, f4=4, and gi =1, ga= 2, 93 - 3, 94 =4, therefore,

fs

1.1+2.2+3.3+44 = 30_ 3

1+2+3+4

10-

H2 H2. ii. Example-Gold one, two, three, four; abandoned, the following

mashakas

one-half,

one-third,

one-fourth, and

one-fifth. I

F.

_

163

1+2+3+4

60

+10=

163

600

Proof by the rule of three g : fg :: gr : grF. H 3.

H3. ii. Example-Mashakas of one and two, gold of two and five, mashkas of three and gold unknown. All that is known is the sum of mashakas. six; and the average mashakas four. State the mashaka of the unknown gold. Statement : f, -1, fs- 2,fs a 3; gl o 2,

g5 .5, 8a=xi F= 4

Problems Solution .

99

4.6-(2.l+5.2) _ 24-2-10 24-12 3____ - 3 -- - _3-

12

3=4 U

J 1.

J 1. Example-If a man requires six drammas for his livelhood for 30 days, for how many dais will 70 men (guards of a fort) live on six drammas ? The details are, however, uncertain. K 1.

K 1. i. Example-What number with five added is a square and that same number w'th seven substracted also being a square ? What is that cumber ? is the question.

Statement : x+5=s2, x-7=t2 1

Solution : x= C 2

( (5- -i-7) 2

-2

)1a+7=11.;

by ssteps thus

Having combined the added and subtracted number 5+7=12 : that halved =6; two substracted 4; halved 2; squared 4; then that subtractive number (7) is to be added and by the addition of this 4+7=11; and this is the required quantity. Proof : 11+5=42, 11-7=22 For such problems, if x+a=sa and x-b=t2, then the solution is, l

x=[2\

ac

la

-c)] +b. L 1.

L 1. iii. Example-In two days one earns five; in three dtys he consumes nine. His store is thirty. In what time will his earnings be consumed ? Solution

:

t=

°30=60 and the amount earned in this time is S-11

2

of 60.150 dinaras. iii. Example-In three days one Pandit earns a wage of five and a second wise man earns six (rnsa) in five days. The second is given by the first seven from his store and I y this giving their possessions become equal. Let it be stttited in what time,

Bakhshali Manuscript

100

Solution : t- o"e30. 3_65

L 2.

Profit and loss.

The author gives a number of solved examples on problems concerning profit and loss. The following rules are given with illustrative examples :

[ Let C be the capital, p the profit, M= C-}- p, N- Cc, the number of articles, C the cost rate, and s the sa'e rate, C = alb, where a is the number purchased for b drammas (but money measures are not generally stated), and s=C/d, where C is sold for d money. ] p

(i) C=

(ii) p=

c-1

Ccs

-C

S

(iii) C -

M S

= Cc

p

s

;

(iv) C=

1- c s

L 2. ii. Example-Two Rajputs are the servants of a king. The wages of one are two and one-sixth a day, of the second one and one-half. The first gives to the second ten dinaras. Calculate and tell me quickly in what time there will be equality. (Indian Antiquary), 1881, p. 44).

Statement : 'as, , given 10. Solution

: The difference of the daily earnings.

ee-;j=g dinaras. Therefore, a difference of 20 dinaras would be in 30 days.

L 2. 6 verso, i. Proof of example on the obverse-

1 : a :: 30 : 65. 1 : it :: 3045 and 65-10=45+10.

ii. The rule rr.eans, C. cp

s

, where C is the capital, p the

profit, C is cost rate and s is sale rate.

Problems

101

iii. Example-One buys 7 for 2 and sells 6 for three find

I

s his profit. What was his capital 7

Solution : C= 7 186

=24. The proof

2-t- 3-I

is given on

folio

62 recto. L 3.

iii

Example-Eight articles are obtained for three and trip

are sold for four. The sum of the capital and profit is c me-hundu d and sixty. State, 0' best of calculators, what was the capital and what is the profit.

The solution is lost except for the first quotation, but part of a proof is given on folio 63, recto. The solution was,

C = 160=90. 16

Since C=

9

- "

p

--

3 4- 1 9 -1 8

/

p6

16

9

p=; C and C+p=160, and the number of articles bought was s of 90 = 240.

L 4.

iii. Example-With five four-squared are obtained by son-c man. For one-six are sold and fifty-six is the loss. Calculating purchase and sale let his capiteal be stated. 56

The solution is C= I

16

= 120 and the number of articles

30

is s of 120=384.

L 4. recto. i. Proof of example given on folio 62 verso. 8:3::240:90 and 6:4::240:160 and 160=90+70. ii. The rule means C=

c

1- S

, where 1 is the loss sustained,

i e. having investigated the selling rate multiply with the purchase

rate and having subtracted from unity divide and the capital it} Pbtaine4.

Bakhshali Manuscript

102

L 4. 63 verso. Proof of example on the obverse : s:1::384:120 then with the selling rate 6:1:384:64 and 120 - 64 = 56.

ii. Rule-That whieh is the tax on cloth is taken in cloth the tax on a piece of cloth is one-twentieth part.

On the pieces being brought to market, two pieces are taken by way of tax : ten is (?) the selling price. What is the value ? Some one sells three-hundred.

M L M I. 20 verso. i. A mere fragment : 12,000 mudrikas...

ii. Example-A snake eighteen hastas long enters its hole at the rate of one-half plus one-ninth; of that, minus one-twenty first part of an angula a day. In what time will it have completely entered its hole ? Solution : Since 24 angulas =I hasta, we have s+ V of TI- TIT: ggu :: 18 x 24 : x, whence x =' = 2 years 4 months 10II days. M 2.

M 3. 38 recto. i. Example-The earning of dinaras is difficult but consuming them is easy. One gives one-half increased by ratio on one-half (six times) for food for the poor. What is the amount consumed in 108 days ? Solution : 1 : 108:1 di°+8dba°-{-lam° i e. 1:(4)° :: 108 : I dio+ 8 dha+1 am° and 4 amsas°=1 dhanaka° and 12

dhanaka°=1 dinara°.

ii. Example-(This is not understood, but appears to refer to the number of hairs on the skin of an animal).

M3 M 3. i. A mutilated example about Ravana and Sita. When Sita had been carried up 30 yojanas into the air she dropped some-

thing to earth, which turned over 8 times in lia dhanusha How wany revolutions did it make before reaching the earth ?

(1 yojana = 8000 dhanusha )

10k

t'roblems

dha° : 8 revolutions

Solution :

30 x 8,000 dhanusha :: 210, 181, 181,", revolutions. ii. Example - A snake which is 100 yojanas, 6 krosas, 3 hast.o. aid 5 hastas and 5 angulas long sheds its skin at the-rate of 1 anl;n l

,

in 2 days. In what time will it be free. ? The solution is given (?) on the reverse. 30 verso. i.} 1 an° : 2 d° :: 100 yo°:+6kr°+3 ha°-} 511n"

429,867 years I month and 4 years, or

1

an°

: 2 :: 77, 376, 077

:

154752154 years. 360

ii. An example about some garment falling to the earth. '1 he elements are uncertain. Compare with - the problem on tlic obverse (1). M 4.

M 4. 36 recto. i. "This land measurement is completed" may refer to the fragmentary example at the bottom of folio 32 vclnlo but it is doubtful.

ii. The example appears to refer to heaps of salt. If one heap or quantity weighs 1,075 palas bow much will 56 beupn weigh ? 1

or

1075 x 56 200

__

: 1075 :: 56 : 30 bha° + 200 pn° 60200 bbara=30 bha°-{-200 pa°. 2000

iii. One-tenth of a cowry is givea in eighty eight ...of this one twenilleth and one-hundredth... 36 verso. i. The statement means 3 yo° : 1 day :: 5 ,yo° n

21, 333 years 4 months,

or 3yo° : s8a years :: 5 x 4,608,000 yo° : 21.333 years 4 months. 5 x 4, 608,000

3X360

'

3'

ii. A boat goes one-half of a third of a yojana plus one-third lcss one quarter, I of 1+,'-J, yojana in one-half of one-third of a

Bakhshali Manuscript

104

day, but then it is driven back by the wind one-half of one-fifth of a yojana, , of s yo°, in one eight of three days. In what tirre mill it travel one hundred and eight yojana. 108

I year 3 months 12; days

0

of the question and the solution are not clear. M 5.

M 5. 34 recto. i. The problem is ; Eleven birds feed on prasriti (handful) of corn; how many can feed on 8 Kharis of corn ? It ends '-Say, 0' friend, say what are the Khagas, 0' Sundari. If this be correct, the same Sundari, "beautiful one" is used in exactly the same way as Lilavati is used by Bhaskara.

The solution is 1 pra° : 11 Kha° :: 8 khai : 63,360 khagas which would make 720 prasriti I khari; but there are many elements of doubt and the application of esha hahu pramanam to this particular problem is not clear. ii. By certain persons one kala plus one pada and one yava are given in gold daily at the shrine of Shulin. What would be the

amount of the gift in five years, five months and fifteen days... I desire to know that.

Solution- I day : 1 yo° + 1 ka° + 1 pa ° :: 5 years, 5 months 15 days : x.

or 1 day ; 30 paV ;;

1

,965

day

;

1965 x 30 t o la 192 x 25

=12 tola°+3 dha°+144 am°. Sec Part 1, Sec. Ill (Kaye for measures of weight)

ii. The problem is about a diamond weighing IIT mashaka, and obtained for 7 55 satera.

The statement means 1i ku°+

ma°, 55 sa°, and indicates

that 128 ma°= I ku°= I and that 40 si°= I ma°. See part Ill (Kaye).

Problems

ION

M 6.

M 6. 37 recto. i. The question may be roughly restored : t lie Sun (Surya) traverses 500,000,000 yojanas in a day. State with

certainty the amount of the journey of the Sun (Divakara) In it ghatika.

60 ghatika = l day = 30 muhurta.

The statement means 30 mu° : 500,000,000 yo° : I gha° : 83. 333, 333$ yo° and it indicates that 2 ghatika= l muhurta (=>f , of it day). The origin of the length of the daily journey of the Sun, namely 500,000,000 yojanas, is not known.

ii. The chariot of the Sun (Bhanu) is surrounded by the groups

of Gods, great snakes, Siddhas and Vidyadharas. In a day and night, its journey is said to be half a hundred kotis. Tell me, 0 beat of calculators, how much in one muhurta ?

1 day=30 mu° 30 mu° : 500,000,000 :: 2 gha° : 16,666, 6661 yo°. 57 verso. i.

The remnant of a problem possibly related to the

daily motion of Jupiter, which according to the Surya Siddhanto amounted to very nearly 5 minutes of arc (lipta). ii. If Bhanuja (Saturn) moves through a sign in two and it half years, state, 0 knower of the truth, what will its motion in it a solar day be equal to (1 sign= 30°=108(00") ? The soluticn is 2b years : 1 sign : : d degree : x Is

x=1 sign x s8° degrees 2J years

= 30 x 60 x 60 x 2 9108.00000=120" = 2 minutes. 5

x 360

of arc (not 2 seconds as stated in the text, where vilipta appears to have been written by mistake for lipta) The terms employed are all orthodox except perhaps vasara for solar day, but its special tiso is quite intelligible. 14

Bakhshali Manuscript

106

.

M 7.

M 7. 47, recto. i. This appears to Partha, the Mahabharata hero who pierced each soldier with 16 (1-q) (1+;) arrows and slew four divisions of the army. How many arrows did he use ? I si° :: 16 (1-1s) (1+1 :: 4 :x;21,870:2,624,40)

The abbreviation si°=?; a°=anikini. See Part I. See 52 Kaye).

There is a very similar example about Partha in the Lilavati (67) which has already been quoted (Part I, 47 recto).

i. Apparently, 3 chamus= l pritana; 3 pritanas= l anikini and 10 anikinis= 1 akshauhini. The statement means that a Patti

consists of ratha+ I gaja+5 nara+3 turaga i. e., 1 elephant+5 foot soldiers- 3 horsemen) and that an akshauhini contains 3.710 of each of these, namelyx 37

chariots= 21,870 chariots (ratha).

10 x 37

elephants- 21,870 elephants (gaja).

1

5 x lOx37 footmen- 109,350 footmen (nara). 3 x 10 x 37 horsemen= 65,610 horsemen (baya).

Total 218, 700

Albiruni gives the following scheme :akshauhini has 10 anikini Each anikini 3 chamu. chamu 3 pritana. pritana 3 vahini. vahini 3 guna. guna 3 gulma. gulma 3 senamukha. senamukha 3 patti. patti

1 ratha.

Problems

1 07

M 10.

M 10. 55 verso. If tola costs thirty-five drammas (dro), whist will be the price of one and a half tolas, one and a half mashakas and one and a half andikas and one and a half yavas.

1 tola = 12 mashakas; 1 mashaha = 4 andi =16 yava.

17 to°+1"9 ma°+17 an°+1, ya°=3197 ya°. Statement - (i) 1 to° : 35 dr° :: 17 to°+17 ma°+17 an° +17 Ya° : 58192

1:35 :: 31911/192 : 58

or

761 19 2

dr°

M 11.

Example-One produces ten and a half in to and one-third days. For the sake of religion, he gives thirteen and one-third In three and one-eight days; be offers for Vasudeva one-quarter lets than thirteen in eight and a half days. Desiring reward in a future world he gives i.e. 12- to Brabmanas for food, one-third in three and

one-fifth days......... (two and a quarter in five days.......(cowd. on M 12). M 12.

M 2. 43 recto. and also twelve and a half in thirty-three and one-third days for the best wine for the consumption of merchants. In the treasure house is stored twelve hundred. Say, 0 Pandit, bow long can this expenditure continue. The statement means= Daily income=

2'1

s

=2.

Daily expenditure=13g

+ $ +3s +

2;

127

1807

+ 5 +334 = 24

!

{-

3g

Bakhshali Manuscript

108

9

727

1807 = and The daily loss is, therefore, 240 X40 - 2

727. 240

1::1200:x

where x the period. 120x=-1-270X366 1

years= 727 years.

240

M 14.

50 recto. At the top of this page is the remnant of a problem. too broken up to make out. The rest of the page is devoted to what appears to be a colophon. This is not all clear but what remains seems to state that the work was written by a certain Brahman, a prince of calculators, the son of Chhajaka. It also refers to the importance of the science of calculation which, it is said, we owe to ' hivo

The Bakhshali Text Transliterated in Devanagari Script

INDEX

The Bodleian Library Order of the Folks r = recto, v = verso

Al

A2 A3

40ar, 39ar, 38br, 40av, 39av, 38bv 39r, 4Cr, 38r, 39v, 40v, 38v. 40r, 39r, 38r, 40v, 39v, 38v.

A4 A5 A6

A7 A8 A9

Al0

All A12 A13 BI

B2 B3 B4

E4 E5

Fl

21r, 21v 22r

22v

F2

23r, 23v

F3 F4 F5

24r, 24v

54v, 54r F6 35r, 35v 51 r, 51v, 35v G1 G2 51r, 51v G3 52r, 52v G4 29r, 29v G5 27v, 29v G6 27r, 29r G7 lr, lv 2r, 2v H1 H2 3v, 3r 8r, 8v H3 J1 9r, 9v

25r, 25v 26r

26v

10r, 10v

llr, llv

12r, 12v 13r, 13v 14r, 14v 15v, 15r 16r 16v

17r, 17v 18r, 18v

30r. 30v

4r, 4v

J2 J3

K

C2

5r, 5v 6r. 6v

L1

C3

7r

C4 C5 C6 Cl C8

65v

L2 L3 L4

60r. 60v 61r, 61v 62r, 62v 63r, 63v

64r

M1

20r, 20v

57v, 57r

M2

33v, 33r

45r, 45v

32r, 32v

C4 D1

46r

M3 M4 M5 M6 M7 M8 M9

Cl

D2 D3 DD4

D5 D6 D7 El

E2 E3

7v

56v, 56r

46v

70 (a, b, c)r 70 (a,b,c,) v 69r , 69v 68r 31r

67v, 67r 28r

66r, 66v 53r, 53v 58r, 58v

M10 M11 M12 M13 M14

65r

41r, 41v 59r

36r, 36v 34r, 34v

37r 37v 47v, 47r 48r, 48v 49v, 49r

55r, 55v 44v. 44r 43r, 43v

42r, 42v 50v, 50r

(40a

recto, 39a recto, 38b recto, 39b recto 40b verso, 39a verve 38b verso)

iaagq-r...' 3 !ra-g4 4 t r urr+r-

era a. Tr...9rrT 4-r qa: ajQ-,zrrft...wr ai T fur

q}ar rt fa-ft? OgVT fiqrq fWq i

r

ar

s6

i

a;Eu

R"

ra I

FT;

writ

i

900 X

I qv

mm

...mod cTTi`

via...... oo qq qr fEr giiu

q ...

srgt

...8rk T v4 qdOgTr I

aF63... r I f f rfw =Iu t r of

ari

...ate... WM I ugqz l fm

tur a;ir afz QRri t

q$rr1ftwT'i t f4Ft i : 'cc ... c c ar°

q...

9

zxqt

o xrork arr wq, I q;...

I

A 2.

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INDEX Abbreviation 29 Addition 32, 33 Age of MS 14 Algebraic characteristics 10 Al Hassarar 40 Al Karkhi 74, 77, 80 Al Khowarizmi 54 Appendix 18 Approximation, first and second 41-45

Arc measures 26

Arithmetic, generalized 10 Aryabbata and Aryabhatiya 18 Bakbshali 1, 2 Bhaskara II of Bijaganita and unknown 23, 67, 70 Brabmagupta and Brahmasphutasiddhanta, Br. Sp. 22,

Commentary nature of ms 1( Complex series 71 Data and solution In ucllni 24, 25

Datta, B. 2, 15, 16, 40, 45, 5051, 53, 58, 66, 71, 80 Denominator, common 47, 411

Dinaras 25, 82 Diophantus 77, and negative sign 29, 46 on equation 77 Division 35

Drsya, absolute term 70 Equations, linear 72: quadratic 77 Error and reconciliation 41, 43

Example, Udaharana 21 False position. see also regula

and rupona method 55

falsi 15, 52, 53 Folios, arrangement of conte. nts of MS. 8

on equations 70, 77

Fractions in cells 24, 7

23, 80 on square roots 39

Buhler 4, 14, 50

Fundamental operations 32-35

Cajori 14

Ganita Sara

Cantor 14, 73 Capacity measures 84

Chajakaputra, the scribe 1, 19 Chronogical table 79, 80 Colebrooke 50, 69

Samgraha of Mahavira (GSS) 23, 50, 53,74,77

Heath 40, 45 Heron 40 Hindu treatise, indigenous 13

110

Historical refcrences in MS 13

Mythological references 13

History of the find of MS. 1. 4

Negative sign 23, 4 i 46 Non-square numbers 36

Hoernle 1, 4, 6, 7, 14, 15, 21, 28, 50, 78 Homogeneity of contents 12 Indeterminate quadratic equaVon 77

Notation, place value 27 Numerical symbols 12

Nyasa (statement) 22

Integral numbers 24

Operations, fundamental 32 in groups of cells 25

Karanam: solution 22

Place value of notations 15, 27

Kaye 2, 6, 7, 10, 12, 14, 15,

Positive integer, solution in 76

18, 27, 36, 38. 45. 57,

Pratvayam (verification) 22 Problems, solutions and

58, 69, 74, %9, 80

Language of MS 7

proofs 59-65

Least common multiple 47, 49

Problems and treatment 30

Length measures 83

Progression, aiithmeticai 26 Proof 32, 59-65

Lilavati 23, 40, 59, 71

Linear equations 72-75 Lipta 26, 27, 82

Mahavira (of GSS) 23, 74, 80 Manuscript, MS 2, discovery

Quadratic equations 77 Regula falsi 15, 52, 53 Regula Vorzinum 76 Rule (Sutra) 18, 21

3-4, condition 5, leavcs 5,

Rule of three 40, 41

a commentary 16, cross references in 17, Sutras

Rupona method 55

18, a copy from other

Solution, see problems and

MS 18, flourish design at

the end of a Sutra

20,

exposition and method, data, figures and solutions in cells 24. Minutes and second 26 Money measures 83 Multiplication 34 20,

Script 6, 12

proofs 59-65 Square root 36

Sridhara (of Trisatika) 23,39,59

Sthapanam (Statement) 22 Subtraction 33 Sunyam (zero) 67, 68 Surds 36-40 Sutra (rules) 18, 21

(

lit

Symbols 27, 66 (for unkuown)

tities, 27

Tannery 73

symbols for 66-69

1 hibaut 14, 29

Verification 32

Time measure 82

Vilipta 26, 27

Trairashika 41

Vowels 13

Trisatika of Si idhara) 2 39,59 Udaharana (Uda), illustration 21, 23

Weber 4

Uddeshaka, example 23

Zero 67, 68

Unknown or unknown quan-

Weight measures 83 Weights and measures 81-H4