C ONTE NT S .

— I INTROD UC TI ON — II TH E S ULVA SUTRA S The Pythagorean theor em Squares an d r ec tan gles S qu ar in g the circle — — I D . 400 600 C RC . A . III , The Pan cha Sidhan tika

Aryabhaga — D 600 —1 200 IV CIRC . A .

I n determin ate e quation s Ration al right -an gle d trian gles G eometry

V—IN DIAN MATH E MA TICAL W ORKS Topics Termin ology Illustrative examples — VI ARI TH ME TI C AL OTA TIO S E TC . N N , — VII INDIAN MA TH E MA TICI ANS — VIII F ORE IG N I N F LU E N C E i n n Poss ble co e ction with Chin ese m athematics Arabic mathematic s Greek in fl uen ce

IX—APPE N DIC E S

Extracts from texts Examples Chron ology Bibliography

X—IND EX

1 TH E who t l in iansh sor u . orientalists exploi ed d g : i t yEa d literature about a century ago w ere not always perfect in their methods of investigation and consequently promulgated w r many errors . Gradually , ho ever , sounde methods have obtained and w e are n ow able to see the facts in more

has correct perspective . In particular the early chronology been largely revised and the revision in some instances has import an t bearings on the history of mathematics and allied ' A u S z w a subjects . ccording to orthodox Hind tradition the y

S iddkdn ta w , the most important Indian astronomical ork , was tw o B w composed over million years ago ailly , to ards ' the end of the eighteenth century, considered that Indian astronomy had been founded on accurate observations made

C a . P AC thousands of years before the hristi n era LA L E , basing his arguments on figures given by BAILLY considered h m B. C t at so e years . the Indian astronomers had recorded actual observations of the planets correct to on e ’ secon d ; PLAYF AIR eloquently supported BAILLY s views Sir WILLIAM J ON E S argued that correct observations must

1 81 so have been made at least as early as 1 BC . and on w C B Y B but ith the researches of OLE ROOKE , WHITNE , WE ER,

B U vi w w u THI A T , and others more correct e s ere introd ced an d it w as proved that the records used by Bailly were qu ite modern and that the actual period of the composition of the

a S fir a i D S i dkiin ta rl ar A. 4 0 0 . origin l y d was not ea than .

n It may , indeed , be generally stated that the tende cy of the early orientalists was towards antedating and this tendency is exhibited in discussions connected with two ' ' a l w Su lvasfitms Bakhskdli i not b e orks , the and the arithmet c,

a w x the d tes of hich are not even yet definitely fi ed . 1 C INDIAN MATHEMATI S .

l 6th A . D. i a a ri 2 . In the century , , H ndu tr dition sc b the invention of the nine figures w ith the device of place ‘ to make them suffice for all numbers to the ben efic en

‘ Creator of the u n iv erse and this w as accepted as evidence of the system 2 This is a particular

was that quite general , for early ' Ihdiaii to be direc tly revealed or of divine

u origin . One conseq ence of this attitude is that we find absolutely no references to foreign o rigins or foreign in flu

* w a e nce . We have , ho ever , great deal of direct evidence that proves conclusively that foreign influence was very real — R w indeed Greek and oman coins, coins ith Greek and i G n a . Ind n inscriptions , reek tech ical terms, etc , etc . ; an d the implication of considerable foreign influence o ccurs in c ertain classes of literature and also in the archaeological

- rema in s of the north west of India . One of the few references to foreigners is given by Vahr aha Mihira who a cknowledged that the Greeks knew somethin g of astrology but although

’ ‘ he gives accounts of the Romaka an d the Pau lisa siddhiin tas he never makes any direct ackn owledgment of w estern influence .

d th v a u - It may be n oted that beyon e g e pseudo prophetic refer n th e P u r d za s n o ea Indian w rite en tion s the in v e n c es i g , rly r m as ion of a t Alexan d er the G re . II .

u 3 . For the p rpose of discussion three periods in the history of Hindu mathematics may be considered

’ S u lvasatm w h 0 (I) The period it upper limit .

A . D. 200 ; 4 — A . D . 0 6 c . 0 00 (II) The astronomical period . h t A . D (III) The Hindu mat ema ical period proper , . — 600 1 200 .

S uch a division into periods does not , of course , perfectly represent the facts , but it is a useful division and serves the

w a purposes of exposition ith sufficient ccuracy . We might

a have prefixed an e rlier , or Vedic , period but the literature

" of the Vedic age does not exhibit anything of a mathematical nature beyond a few me asures and numbers used quite in for s mally . It is a remarkable fact that the econd and third o f ou r periods have no connection w hatever w ith the first

’ a o r S u lvasfltm period . The later Indian mathem ticians ’ completely ignored the mathematical contents of the S u lva

o bu t sfitras . They not nly never refer to them do not even utilise the results given therein . We can go even further a n d state that no Indian writer earlier than the nineteenth

’ c entury is kn own to have referred to the S u lvasatm s

n i as containing anythi g of mathematical value . Th s dis c onnection will be illustrated as W e proceed and it will be

“ s een that the w orks of the third period may be considered as a direct development from those of the second .

’ — 4 TH E S V P D S u lvasfitm . UL ASUTRA ERIO The term means the r u les of the cord and is the name given to the supplements of the Kalpasatras w hich treat of the c on stru c ’ in w S u lvasfl tion of sacrificial altars . The period hich the ”as were composed has been variously fixed by various 4 INDIAN MATHEMA TIC S .

a MA X MHLLE R uthors . gave the period as lying between 0 R D 8 5 20 B . C . 00 00 C . . and UTT gave R C ; BIlH LE R. places the origin of the Apastamba school as probably somewhere w ithin the last four centuries before the Christian

Bau dha an a w M C era , and y some hat earlier A DONNELL gives 5 B A . 00 00 C . D 2 s . o the limits as and , and on . As a matter of fact the dates are n o t known and those suggested by diff erent authorities must be used w ith the greatest circumspection . It must also be borne in mind that t he

’ ’ S u lvaszitms as w contents of the , kno n to us , are taken from quite modern manuscripts ; and that in matters of

robablv detail they have p been extensively edited . The

A astamba Bau dha an a Kat a an a w editions of p , y and y y hich w ff have been used for the follo ing notes , indeed , di er from each other to a very considerable extent .

’ The S u lvasatm s are not primarily mathematical but — are ru les ancillary to religious ritual they have not a mathe

No matical but a religious aim . proofs or demonstrations ar e given an d indeed in the presentation there is nothing mathematical beyond the bare facts . Those of the rules that ‘contain mathematical notions relate to (1 ) the c on stru c

a 2 tion of squ res and rectangles , ( ) the relation of the 3 diagonal to the sides , ( ) equivalent rectangles and squares ,

(4) equivalent circles and squares .

5 w 1 2 . In connection ith ( ) and ( ) the PYTHAGOREAN theorem is stated quite generally . It is illustrated by a number of examples which may be summarised thus

MBA BAUD H APAS TA . AYAN A . 3 2 + 42 = 52 3 2 + 2 2 2 z 2 2 2 12 -l-1 6 z - 20 5 + l 2 = 1 3 152 -1 2 5 -i 7 2 -l 2 2 I5 + 3 6 ! 3Q

2 2 2 2 8 4-1 5 2 1 7 1 5 -i 2 2 2 1 2 + 35 : 37 5 INDIAN MATHEMATIC S .

K atyayan a gives no such rational examples but gives (with Apastamba an d Bau dhayan a) the hypotenuse corresponding

a i. e. to sides equal to the side and di gonal of a square , , the t a a 2 a an d riangle , 1/ , and he lone gives

’ ‘ There is no indication that the S u lvaszctra w rational examples ere obtained from any general rule . Incidentally is given an arithmetical value of the diagonal of a square which may be represented by

1 1 V36 1 L 3

a This has been much commented upon but , given scale of

U 3 4 an d 34. measures based pon the change ratios , , (and Bau dhayan a actually gives such ascale) the result is only an expression of a direct measurement and for a side of six f eet it is accurate to about éth of an inch or it is possible t hat the result was obtained by the a pproximation 1 — ’ t 7) : a by TANNERY S R process , but it is qui e m + 2 ; c ertainthat no such process w as known to the authors of the

’ S u l a fitra a v s s . The only notew orthy character of the fr c t ion is the form with its unit numerators. Neither the value itself nor this form of fraction occurs in any later w Indian ork .

There is one other point connected w ith the

via PYTHAGOREAN theorem to be noted , , the occurrence of an indication of the formation of a square by the s a uccessive addition of gnomons . The text rel ting to this is as follows

Two hundred an d tw enty-five of these bricks ' constitute the sevenfold ow n! with am tm an d ” mdesa p .

- a To these sixty four more are to be dded . With

these bricks a square is formed . The side of the

- square consists of sixteen bricks . Thirty three 6 INDIAN MATHEMATIC S .

bricks still remain an d these are pl aced on all

sides round the borders .

This subj ect is never again referred to in Indian mathe i l w mat ca orks .

The questio ns (a ) w hether the Indians of this period had completely realised the generality of the PYTHAGOREAN t an d t ha the heorem , whether hey d a sound notion of irrational have been much discussed ; bu t the ritualists ' w ho composed the S u lvasfitras w ere n ot interested in the

bev on d ow n a t a wa an d PYTHAGOREAN theorem their c u l nts , it is quite certai n that even as l ate as the 1 2th century no Indian mathematician gives evidence of a complete understan ding

at t at a . n o of the irr ion l Fur her , period did the Indians

an y t y an d a a a develop real theory of geome r , comp r tively modern Indian w ork denies the possibility of au v proof o f the PYTHAGOREAN theorem other th an experience .

The fanciful suggestion of BiiRK that possibly PYTH A GORAS obtained his geometrical knowledge from India is not

C ad ac u ain supported by an y actu al evidence . The hinese h q

a the a t a a B C an d t he t nce with theorem over hous nd ye rs ,

Egyptians as early as 2000 B C .

a a an d a 6 . Problems rel ting to equiv lent squares rect ngles ar e involved in the prescribed altar constructions and c on se ' S u lvasfitm s tr the quently the give cons uctions , by help of

PYTHAGOREAN theorem , of

( 1 ) a square equ al to the sum of two squ ares

(2) a square equ al to the difference of two squares (3 ) a rectangle equal to a given squ are ' (4) a square equal to a given rectangle

(5) the decrease of a square into a smaller square .

Again W e have to remark the significant fact that none of these geometrical constructions occur in any later Indian

two a o f work . The first are direct geometrical applic tions INDIAN MATHEMATIC S . 7

‘ ’ 2 —— ’ t i a the rule c z a l b ; the third gives in a ge o me r c l form 93 ? the sides of the rectangle as C V? an d 5 the fourth rule a —b 2 b b gives a geometrical construction for a ( T )

an d to C 5 fif corresponds EU LID , II , ; the th is not

a C 4 . perfectly cle r but evidently corresponds to EU LID , II ,

TH E C C E — A t i 7 . IR L ccording to the al ar build ng ritual

was of the period it , under certain circumstances , necessary

the an d w e to square circle , consequently have recorded

’ S u lvasfitm s t at hi in the at empts the solution of t s problem , and its connection with altar ritual reminds us of the cele u brated DELIAN problem . The sol tions offered are very

a a is a a crude lthough in one c se there pretence of ccur cy . Denoting by a the side of the square an d by d the diameter 2 of the circle whose area is supposed to be (1 the rules given may be expressed by (e d = m+axwv2 — o

w a : 1 1 1 1 _d 1 ( 7 ) a ( 3

N t two w are i ei her of the first rules , hich g ven by both Apas t a a an d Bau dha an a a a mb y , is of p rticul r value or interest . The third is given by Bau dhayan a on ly an d is evidently

obtained from ( a ) by utilisin g the value for V 2 given in 5 paragraph above . We thus have

a, 3 3 d

u s w , c t t is hich negle ing the last erm , the value given in rule (y) .

b u lva s tm s u ; for as THIB AUT says there is nothing in 8 INDIAN MATHEMATIC S . these rules w hich would justify the assumption that they were expert in long calculations and there is no indication in any other Work that the Indians Were ever acquainted with the process and in no later Works are fractions ex

in pressed this manner. It is w orthy of note that later Indian mathematicians record no attempts at the solution of the problem of squaring the circle and never refer to those recorded in the ’ m S u lvasat s . D 400 To 60 A. . 0 .

8 w . There appears to be no connecting link bet een the

’ ’ S u lvaszltm mathematics an d later Indian developments

’ f S S u lvasfitms n O the subj ect . ubsequent to the nothi g further is recorded until the introduction into India of * D . a A . western stronomical ideas . In the sixth century ' VARAH A MIH IRA wrote his Paiicha S iddhdn mkd which gives a summary account of the five most important astro

ical w S fir a S iddhdn ta n om orks then in use . Of these the y , which W as probably composed in its original form not earlier

4 0 a . A . D. 0 w than , fter ards became the standard Work

’ VARAH A MIH IRA S collection is the earliest and most authentic account w e have of what may be termed the scientific treat “ A B ment of astronomy in India . lthough , writes THI AUT , not directly stating that the Hindus learned from the

G an d reeks , he at any rate mentions certain facts points of doctrine which suggest the dependence of Indian astronomy

A w a on the science of lexandria ; and , as We kno lready

his a w s from astrologic l ritings , he freely employ terms of G ” undoubted reek origin . ' VARAB A MIH IRA writes There are t he following

’ - S a Pau lisa B Vasishtha . iddh ntas the , the omaka , the , the Saura and the Paitamaha The Siddhanta made by

Pau liéa S a a is accurate , near to it stands the iddh nt pro

B Sé vitra Su a claimed by omaka , more accurate is the ( ry ) ” The two remaining ones are far from the truth .

This has a somewhat importan t bearin g o n the d ate of the ' S u lc asfitr as. If e a e the ate t ei siti n w e e a e te , for x mpl , d of h r compo o r cc p d a s 500 R C . a eriod n ea ea s a s te an as f ar as p of rly y r , b olu ly bl k

at e ati a n ti ns are n e n ed w a e to be a n ted . m h m c l o o co c r , ould h v ccou for 1 0 M INDIAN MATHE ATI C S .

9 T e P . h a ficha S iddhdn tikd contains material of considerable mathematical interest and from the historical point of View of a v alue not surpassed by that of any later

f Indian works . The mathematical section of the Pau lisw

d n a S i dhci t is perhap s of the most interest and may be

t o ta a considered con in the essence of Indi n trigonometry . It as follows ‘

(l ) The square - root o f the tenth part o f the square

o f 3 60 the circumference , which comprises

is t n the parts , the diame er . Havi g assumed four parts of a circle the sine of the eighth

a a p rt of sign [is to be foun d] .

(2) Take the square of the radius and call it the th constant . The fourth part of it is [ e square

n b of] Aries . The consta t square is to e

a A lessened by the squ re of ries . The square

a r roots of the tw o qu ntities a e the sines . (3 ) In order to find the rest take the double of the

arc n , deduct it from the quarter , dimi ish the radius by the sine of the remainder and add to the square of half of that the square

ar The of half the sine of double the c .

- t su m the square roo of the is desired sine .

° ' [The eighth part of a sign is 3 45 an d by Aries is indicated the first sign of

The rules given may be expressed in our notation (for unit radius) as — (1 ) (2) Sin Sin i

2 — 2 ; I 90 2 Si 2 S D. 2 n 3 ( 7 ) (3) 8 111 y 2

They are followed by a table of 24 sines progressin g by in ter ° ' ’ vals of 3 45 obviously taken from PTC LE MY s table of chords .

a d n 60 as Inste d , however , of ividi g the radius into parts , 1 1 INDIAN MATHEMATIC S .

’ Pau lisa it n 1 20 a as did Ptolemy , divides i to p rts ; for _i‘_ c hord a sin v u to 2 2 this di ision of the radi s enabled him

convert the table of chords into sines without numerical

A r abh ta change . y a gives another measure for the radius which enabled the sines to be expressed in a sort o f

a circular me sure .

We thus have three distinct stages

’ a c h d a w t ( ) The chords of PTOLEMY , or , i h

Ch d a ’ 2 20 fi “ 7 1 b Pau liéa 8 111 6 0 1 s th ( ) The g z 3 h 3 L _ C (0 ) The A rvabhata i 8 111 S ne or 7 ;

’ with r = 3438 4 ) To obtain (0 ) the value of 7r actually used was i35

Thus the e arliest known record of the use of a sin e function occurs in the Indian astronomical w orks of this At w as period . one time the invention of this function Q D 877— 9 1 9 w e t n A . attributed t o el Ba ta i [ . ] and although now know this to be incorrect w e must acknowledge that the Arabs utilised the invention t o a much more scientific

end than did the Indians . — In some of the Indian works of this period an in terpo

the i lation formula for the construction of. table of s nes is

It ma e y given . y be repres nted b

i n S n . a — — : n a Sin n l A w here A Sin . ( ) n S i n a

m a bu t This is given ostensibly for the for ation of the t ble ,

a a a the t ble ctually given cannot be obt ined from the formula .

1 0 ARYA BH A — . r t AT . Tradition places A yabha a (born

D 476 at the A . . ) head of the Indian mathematicians and * was fir indeed he the st to write formally on the subject .

' ’ t Kr abhata s Gamta as fi s t is ed Ke n if Al hough y , r publ h by r ,

en e a a e te as a t en ti t e e is an e e en t d t in the . g r lly cc p d u h c , h r l m of oub atte m r . 1 2 INDIAN MATHEMATIC S .

He w as renowned as an astronomer and as such tried to int ro duce sounder views of that science but was bitterly opposed

to by the orthodox . The mathematical Work attributed him consists of thirty-thr ee couplets into which is condensed a a S a good de l of matter . tarting with the orders of numer ls

an d a a an d he proceeds to evolution and involution , re s

N - volumes . ext comes a semi astronomical section in which

i w the c irc le a set he deals ith , shadow problems , etc . then o f propositions on progressions followed by some Simple algebraic identities . The remaining rules may be termed practical applications w ith the exception of the very last which relates to indeterminate equations of the first degree .

N are either demonstrations nor examples given , the whole t ext consisting of Sixty-six lines of bare rules so condensed t As a hat it is often difficult to interpret their meaning . mathematical treatise it is of interest chiefly because it is some record of the state of knowledge at a critical period in

w as the intellectual history of the civilised orld ; because ,

w at far as We know , it is the earliest Hindu ork on pure m he maties and because it forms a sort of introduction to the school of Indian mathematicians that flourished in succeeding c enturies .

’ A ryabhata s work contains one of the earliest records

in deter known to us of an attempt at a general solution of. min ates of the first degree by the continued fraction process .

The rule , as given in the text , is hardly coherent but there “i as s no doubt as to its general aim . It may be considered forming an introduction to the somewhat marvellous develop ment of this branch of mathematics that w e find recorded A in the works of Brahmagupta and Bhaskara . nother n oteworthy rule given by Ar yabhata is the one which contains an extremely accurate value of the ratio of the circumference o f via 7: 3 it a circle to the diameter , , 1g) but is rather extraordinary that Aryabhata himself never utilised t was a m his value , that it not used by any other Indi n athe

6 — A . . 0 1 0 D 0 2 0 .

1 1 Ar abhata to . y appears to have given a definite bias

I a ndian mathematics , for following him We have series of w Works dealing ith the same topics . Of the Writers them selves w e know very little indeed beyond the mere names b u t some if not all the Works of the following authors have been preserved

a A 5 8 . D. 9 Br hmagupta born . M a i 2 9th ah v ra century . ' ara D ridh A . 9 S . 91 born . B a A D 1 1 1 4 h skara born . . .

a is c abl Bh skara the most renowned of this S hool , prob y ’ so a undeservedly , for Brahmagupt s Work is possibly sounder

mathematically and is of much more importance historically . — Generally these Writers treat of the same topics with a ’ difference— and Brahmagupta s work appears to have been

B a a used by all the others . h sk ra mentions another mathe M i a a i a . matic an , Padman bha , but omits from his list ah v r One of the chief points of difference is in the treatment w o f geometry . Brahmagupta deals fairly completely ith c yclic quadrilaterals w hile the later writers gradually drop this subject until by the time of Bhaskara it has ceased t o be understood . The most interesting characteristics of the w orks of this p eriod are the treatment of (i) indetermin ate equations ;

the rational right-angled triangle ;

iii a an d ( ) the perfunctory tre tment of pure geometry . 1 INDIAN MATHEMATIC S . 5

‘ Of these topics it w ill be noted that the second w as dealt

’ ’ with to some extent in the S u lvasattras ; bu t a close exa mination seems to Show that there is no real connection an d that the w riters of the third period w ere actually ignorant

a Bau dha an a an d A astamba o f the results chieved by y p .

1 2 . INDETERMINATE EQ UATIONS . The interesting n ames and dates connected w ith the early history of in d eter min ates in India are

Diophantus Hypatia Aryabhata

Brahmagupta Bhaskara

That w e cannot fill up the gap between Diophantus an d Anyabhata with more than the mere name of Hypatia is probably due to the fanatic ignorance and cruelty of the early Alexandrian Christians rather than the supposed

A a M a d estruction of the lex ndrian library by the uh mmadans . It Would be pleasant to conceive that in the Indian w orks

e w have some record of the advances made by Hypatia , or of the contents of the lost books of Diophantus— but We are not justified in indulging in more than the mere

is fanc y . The period one of particular interest . The D murder of Hypatia (A. . the imprisonment an d B A D execution of oethius ( . . the closing of the 53 D. 0 Athen ean schools in A . and the fall of Al exandria in 640 are events full of suggestions to the historian of

was hi mathematics . It during t s period also that

Damasciu s S , implicius (mathematicians of some repute)

an d c A at others of the s hools of thens , having heard th ' ’ PLAr o s ideal form of government was actually realised

Chosr oes A.D under I in Persia , emigrated thither (c . 1 6 C INDIAN MATHEMATI S .

They were naturally disappointed but the effect of

a their visit may h ve been far greater than historical records w Sho .

1 3 . The state of knowledge regarding indeterminate i n fi k equations the west at this period is not de nitely nown . Some of the Works of Diophantus an d all those of Hypatia are lost to us but the extant records Show that the Greeks

had explored the field of this analysis so far as to achieve rational solutions (not necessarily integral) of equations of the first and second degree and certain c ases of the third

r w deg ee . The Indian orks record distinct advances on what

is left of the Greek analysis . For example they give rational in tegral solutions of

(A) ax i by z c

2 — 2 (B) Da -l 1 = t

The solution of (A) is only roughly indicated by Arya ’ bhata but Brahmagupta s solution (for the positive Sign) is practically the same as

w : - bt = c at ; t , y ¥ p + where t is zero or any integer and p /q is the penultimate con

vergent of a/h. The Indian methods for the solution of 2 2 Du 1 t may be summarised as follows

g 2 2 2 - z = If Da H i c and Da +fi 7 then will —'' 2 ” (a) D(Ca F 7 6 0+ (0 Vi a a D)

r2- D (b D ) b where r is any suitable integer . Also 2 D + n

w here n is any assumed number . The complete integral solution is given by a combination (a) and (b) of which the former only is given by Brahma INDIAN MATHEMATIC S 1 7

a w a a a s gupt , hile both are given by Bh sk r (five centurie ‘ att a method b later) . The l er designates ( ) the y composi

ar tion an d (b) the cyclic method . These solutions e alone sufficient to give to the Indian Works an important place in the history of mathematics . Of the cyclic method

a a an d b a d the combin tion of ( ) ( ) HANKEL s ys , It is beyon all praise it is certainly the finest thing achieved in the

a a theory of numbers before L grange . He ttributed its

a i i s invention to the Indi n mathematicians , but the op n on

a A C of the best modern uthorities T NNERY , ANTOR , HEATH) ar e rather in favour of the hypothesis of ultimate G reek origin .

The following conspectus of the indeterminate problems dealt with by the Indians will give some idea of their Work in this direction ; and although few of the cases actually occur in Greek Works n ow kn own to us the conspectus sign i fi l c an t y illustrates a general similarity of treatment .

an: if: by c

(2) on by as

3 a; E a Mod 6 a Mod 6 ( ) , . 1 , . ,

(4) Ax + By + 0 xy = D (mDm+ i = n

e 2 (6) Du 1 = t

2 ? (7) Dn A: s t

i 2 (8) D n i s z t

s 2 (9) a s at

2 2 (1 0) Du i an t

2 2 (1 1 ) s Du t

2 (1 2) Du s t

2 x i b z t

Of t ese n n e s 1— 5 7 8 12 —14 e e the tw e t h o ly umb r , , , occur b for lf h n t ce ury . 1 8 INDIAN MATHEMATIC S .

2 —— 2 a 1 = s boo a z t (1 4) x + , l

2 2 — 2 2 ( 1 5) 2 (x 3 (x y ) + 3 t ' 2 2 — = 2 (1 6) c m - by 1 : t

" ar:

z — — (18) w a E fi o Mod m

Q —— (1 9 ) ax l o Mod m

x — z t x z u y , y

3 3 2 2 2 3 21 x = 3 x = t ( ) + y , + y

3 2 2 3 22 x - = s x = t ( ) y , + y

3 —— ‘z 2 x l y z t

2 —— 2 as l 1 = t (26)

3 — o b (27) x a M d. — " 4 u ,

my ? 2 ' 2 s _ ’ l 2 w (I) y l :

2 ~ - - —w — = z .s t u + v 2

3 - W a y 2 2 — - (29) g Vx + y 1/w+ y+ 2 i/x y w

9 2 g wx 1 8 = e m 1 8 = z 1 8 = + , y f , y + g ,

- — 1 4 R . . ATIONAL RIGHT ANGLED TRIANGLES The Indian mathematicians of this period seem to have been particularly attracted by the problem of the rational right-angled triangle and give a number of rules for obtaining integral solutions . The following summary of the various rules relating to this problem shows the position of the Indians fairly Well . INDIAN MATHEMA TIC S 1 9

z 1/A " -B A uthorities .

2 il m n

D a ioph ntus .

2 Brahma gupta and Maha V i ra .

t Brahmagup a , M a i ah v ra , and Bhas

kara .

Bhaskara .

g— m n l 2 mn a ( ) Bh skara . 9 z n + 1 n + 1

— 2‘ 2 2 ix 2 l mn l (m n ) l (m + n ) General for

mula .

Mahavira gives many examples in which he employs ’ whi m n formula (V) of ch he terms and the elements . From given elements he constructs triangles and fr om given fi e. . triangles he nds the elements , g , What are the ele

- 48 55 73 An w ments of the right angled triangle ( , , ) s er

3 , 20 INDIAN MATHEMATIC S .

Other problems connected with the rational right-angled triangle given by Bhaskara are of some historical interest

1 su m is 40 60 2 Th e e. . g , ( ) The of the sides and the area , ( )

u m 56 a n d 7 X 600 3 a s of the sides is their product , ( ) The are

al a 4 a s is numeric ly equ l to the hypotenuse , ( ) The are i

a a of numeric lly equ l to the product the sides .

1 5 . The geometry of this period is characterised by

1 a ( ) L ck of definitions , etc . ; (2) Angles are not dealt with at all ; (3) There is no mention of parallels and no theory of proportion ;

(4) Traditional inaccuracies ar e not uncommon ;

(5) A gradual decline in geometrical kn owledge is

a notice ble .

we On the other hand , have the following noteworthy rules relatin g to cyclic quadrilaterals

i = 8 — a 8 — ( ) Q V( ) ( ( «S d) (n)

where a: an d y are the diagonals of the cyclic quadrilateral ‘ d A B a a b o . ( , , , ) This ( ) is sometimes designated as rahm ’ ’ gupta s theorem .

6 a ff a 1 . The bsence of definitions and indi erence to logic l order sufficiently differentiate the Indian geometry from that of the early Greeks but the absence of What may be termed a theory of geometry hardly accounts for the complete

‘ a absence of any reference to parallels and angles . Where s on the one hand the Indians have been credited with the i is nvention of the sine function , on the other there no evidence to Show that they were acquainted with even th e

a most elementary theorems (as such) rel ting to angles . The presence of a number of incorrect rules side by Side w a ith correct ones is signific nt . The one relating to the area

via is of triangles and quadrilaterals , , the area equal to the

1 22 N DIAN MATHEMATIC S .

w 63 56 . a f ith diagonals and , etc He also introduces proo ’ of Ptolemy s theorem and in doing this follows Diophantus

iii 1 9 a b c a 3 ( , ) in constructing from triangles ( , , ) and ( , / , y , ) n ew a a b C an d a c 0 o a tri ngles ( y , y , y , ) ( , 8 , y , ) and uses the actu l

D 39 52 65 an d 25 examples given by iophantus , namely ( , , ) ( , 0 6 ,

1 8 An a G . ex mination of the reek mathematics of the period immediately anterior to the Indian period with w hich we ar e n ow dealing Shows that geometrical knowl edge W a s A D A . 300 a . . eomet in a state of dec y fter Pappus (c . ) no g

was e . ric al work of much valu done His successors w ere , a a a the pp rently , not interested in the gre t achievements of earlier Greeks and it is certain that they w ere often not even

a acquainted With m ny of their works . The high standard

of the earlier treatises had ceased to attract , errors crept in , the style of exposition deteriorated and practical purposes r predominated . The geometrical Work of B ahmagupta is almost What one might expect to find in the period of decay

in Alexandria . It contains one or two gems but it is not a

scientific exposition of the subj ect and the material is .

obviously taken from Western works . 9 1 . We have , in the above notes , given in outline the

historically important matters relating to Indian mathematics . For points of detail the w orks mentioned in the annexed bibliography should be consulted but W e here briefly indicate

the other contents of the Indian Works , and in the following sections W e shall refer to certain topics that have achieved

to a somewhat fictitious importance , the personalities of the Indian mathematicians and to the relations between the

C A . mathematics of the hinese , the rabs and the Indians

20 . Besides the subj ects already mentioned BRAHMA GUPTA deals very briefly with the ordinary arithmetical

- operations , square and cube root , rule of three , etc . interest ,

mixtures of metals , arithmetical progressions , sums of the squares of natural numbers geometry as already described

‘ but also including elementary notions of the circle ; w elementary mensuration of solids , shado problems , negative

iden ti and positive qualities , cipher , surds , simple algebraic ties ; indeterminate equations of the first and second

degree , which occupy the greater portion of the work , and simple equations of the first and second degrees which receive comparatively but little attention .

’ MAHAVIRA S work is fuller but more elementary on the

whole . The ordinary operations are treated with more completeness and geometrical progressions are introduced ; many problems on in determin ates are given but no mention is made of the cyclic method and it contains no formal w algebra . It is the only Indian Work that deals ith ellipses

(inaccurately) . 2 4 INDIAN MATHEMATIC S .

' ’ The only extant work by S RID H ARA is like Mahavira s but shorter but he is qu oted as having dealt with quadratic e quations , etc .

’ ' ’ Bhaskara s Lildvati is a S ridhara s an d b sed on Work ,

a a a besides the topics lre dy mentioned , deals with combin ' t w Vi a- a mta t a ions , hile his j g , being a more sys em tic expo s t a a a a a i iou of the lgebr ical topics de lt with by Br hmagupt , is a a a the most complete of the Indi n lgebr s . A D 1 1 1 a a a . 4 fter the time of Bh sk r (born A . ) no Indian m a w is k athematic l ork of historical value or interest nown . Even before his time deterioration had set in an d although “ a college W as founded to perpetuate the teaching of

a a a a a an a a . Bh sk r it , pp rently , took strological bi s

21 — . The Indian method of stating examples parti — c u larly those involving algebraic equations are of sufficient

a interest to be recorded here . The e rly Works Were rhetorical and not symbolical at all an d even in modern times the nearest approach to a symbolic algebra consists of abbrevia t a ions of special terms . The only re l symbol employed is the

whi a negative Sign of operation , ch is usually a dot pl ced above o r at a a a the Side of the quantity ff ected . In the B khsh li

Ms a as the a , cross is used in place of the dot l tter in the

S arada script is employed to indicate cipher or nought .

The fir st mention of special terms to represent unknow n

’ quantities occurs in Bhaskara s Vija-ga nvta which W as Written

a a a A S in the tw elfth century of our era . Bh skar s ys ‘ a as dvat tdvat an d a kdlaka m ny (y ) the colours bl ck ( ) , blue

n ilaka itaka an d lohvtaka an d ( ) , yellow (p ) red ( ) others besides * t hese ha ve been selected by ancient teachers for na mes of ” values of unkn own quantities .

The term ydvat tdvat is understandable an d so is the use o f colours bu t the conjunction is not easy to un derstand . The use of two such diverse types as ydvat trit an d kdlaka

* n N ot I n dia s . 25 INDIAN MATH EMATIC S .

' (generally abbreviated to yri and led) in one system suggests a the possibility of a mixed origin . It is possible th t the

’ former is connected w ith Diophantus defin ition of the

an létkos mon ddon ( Loriston i. e . unknown quantity , p , , ’ a u ndefin ed (or unlimited) number of units . To p ss from an unlimited number to as many as requires little ima

in ation D w g . iophantus had only one symbol for the unkno n an d if the u se of ydvat tdvat were of Diophantine origin the Indians w ould have had to look elsewhere for terms for the

to other unknowns . With reference the origin of the use of colours for this purpose w e may point out that the very early Chinese used calculating pieces of two colours to r epre t t sen posi ive and negative numbers .

As neither the Greeks nor the Indians used an y sign for addition they had to introduce some expression to distingu ish

a G the absolute term from the v riable terms . The reeks used ° M an abbreviation for mon ddes or units while the Indians ? M ra e . used for p , a unit

The commoner a bbreviations used by the Indians ar e a s follows

d dvat tdvat . y for y , the first unknown

kd kd‘ la a k . , the second unknown ml mi a a . p , the absolute qu ntity

va var a a a g , squ re .

ha han a g g , a cube .

ka kam y a a , surd .

It is hardly appropriate to discuss Sanskrit mathematical t erminology in detail here but it will not be out of place to

f ew mention a other terms . To denote the fourth power varga var a w g is used but it occurs only once ithin our period . ' ' In more modern times varga ghan a gkcitaj denoted the fifth

var a kan a the an o power , g g , sixth d s on .

i l dta = the produ ct. 26 INDIAN MATHEMATIC S .

' C a r dm G a e e . . z m t Gk. ia ert in reek terms used , g , j ( d met

k n on i dm n ron trik a Gk. tr C e Gk. ke t n on ) , , , li ta Gk ron ( ) ( g ) p ( , ' le té kam a k oriz Cn dramma Gk. d G . e dln m p ) , j ( ) , ( drachm ) ,

G d n ari n M k. e o . w ar e ( ) , etc any of these terms , ho ever , borrowed from Indian astrological w orks Which contain a considerable number of Greek terms such as H r vdroga (Gk ' e d o l m o ~ r os Pd a k. k v k u d oc ho rtkon G a Gk. a ( ) ( Parth nos) , p ( p lima) , etc . , etc .

' ar sva w The curious may compare p a rib , side ith

’ the Greek p leu ra kotv which primarily means a claw or horn w but is used for the perpendicular side of a triangle , ith

’ kdtketos dt a a j y which me ns legitimate , genuine , but

’ ’ is used to denote a right -angled triangle with ortkogon ea and so on .

A - e 22 . ccording to the Hindus the modern place valu

system of arithmetical notation is of divine origin . This

at a led the early orientalists to believe that , any r te the system had been in u se in India from time immemorial but an examination of the facts shows that the early notations in use were not place- value ones and that the modern place -value system was not introduced until comparatively

modern times . The early systems employed may be c on v e n ien l a t i b a i ty termed ( ) the Kharosh h , ( ) the Br hm , ’ c Ar abhata s d - l ( ) y alphabetic notation , ( ) the Word symbo notation .

(a) The Kharosklki script is written from right to left and was in use in the north-west of India an d ' Cen tral Asia

C era o is at the beginning of the hristian . The n tation shown

. was a a in the accompanying table It , pp rently , derived from the Aramaic system and has little direct connection

a with the other Indi n notations . The smaller elements ar e Written on the left .

(b) The Brdkmi notation is the most important of the old notations of India . It might appropriately be termed the Indian notation for it occurs in early inscriptions and was

in fairly common use throughout India for many centuries , and even to the present day is occasionally used . The symbols employed varied somewhat in form according to w time and place , but on the hole the consistency of form

a a to exhibited is rem rk ble . They are Written from left S right with the smaller elements on the right . everal false theories as to the origin of these symbols have been published , some of Which still continue to be recorded . The earliest

- n orientalists gave them place value , but this error soo

A - e 22 . ccording to the Hindus the modern place valu

system of arithmetical notation is of divine origin . This

led the early orientalists to believe that , at any rate . the system had been in use in India from time immemorial bu t an examination of the facts shows that the early notations in use were not place-value ones and that the modern place -value system w as not introduced until comparatively

modern times . The early systems employed may be c on v e n ien l a t i b a i ty termed ( ) the Kharosh h , ( ) the Br hm , ’ c A r abhata s a d - l ( ) y alph betic notation , ( ) the Word symbo notation .

(a) The Kharoshthi script is written from right to left and was in use in the north-west of India an d Central Asia C w at the beginning of the hristian era . The notation is sho n

was in the accompanying table . It , apparently , derived from the Aramaic system an d has little direct connection t with the other Indian notations . The smaller elemen s ar e Written on the left .

(b) The Brdkml notation is the most important of the old notations of India . It might appropriately be termed the Indian notation for it occurs in early inscriptions and w as

u se a a in fairly common throughout Indi for m ny centuries , and even to the present day is occasionally used . The symbols employed varied somewhat in form a ccording to w time and place , but on the hole the consistency of form

a a w exhibited is rem rk ble . They are ritten from left to

S a right With the smaller elements on the right . ever l false theories as to the origin of these symbols have been published , t some of which still continue to be recorded . The earlies

- n orientalists gave them place value , but this error soo 2 8 INDIAN MATHEMATIC S . d isproved itself it was then suggested that they were initial letters of numerical Words then it was propounded that the symbols were akskam s or syllables then it W as again claimed that the symbols were initial letters (this time Kharosktki) of t a he corresponding numer ls . These theories have been severally disproved .

The notation W as possibly developed on different prin c iples at diff erent times . The first three symbols are natural an d only differ from those of many other systems in consisting N of horizontal instea d of vertical strokes . o principle of “ “ ” f ormation of the symbols for four to thirty is n o w “ evident but possibly the forty W as formed from the

” ‘ thirty by the addition of a stroke and the sixty and “ ” seventy an d eighty and ninety a ppear to be

are a c onnected in this way . The hundreds (to limited

a a as B e xtent) evidently built upon such pl n , which , AYLEY

a as pointed out , is the s me that employed in the Egyptian hieratic forms but after the three hundred the Indian “ system forms the four hundred from the elements of “ ” “ ” so . n a hundred and four , and on The otation is exhibited in the table annexed .

3 0 INDIAN MATHEMATIC S .

’ (c ) Aryabhata s alphabetic notation also had no place value and diff ered from the Brahmi notation in having the

s w as w maller elements on the left . It , of course , ritten and

read from left to right . It may be exhibited thus

Letters k kk g gk a o ch j jk ii 3 4 5 6 7 8 1 Values 1 2 9 0 .

Letters t th d dk n t ih d dk n 4 1 5 1 6 1 7 1 8 1 9 20 Values 1 1 1 2 1 3 1 .

Letters k b bk m r l v s sh s k . . p p y 1 3 Values 2 22 2 24 25 30 40 50 60 70 80 90 1 00 . The vowels indicate multiplication by powers of one

h w a as undred . The first vo el may be considered equivalent t ° = l o the second vowel v 1 00 an d so on . The values of t he vowels may therefore be shown thus

Vowels a 73 u r ?) l?) e at 0 an . . Values 1 1 0 ° 1 04 1 0° 1 08 1 0 1 0 1 0 14 1 0 16

’ The following examples taken from Aryabhata s Gitvkd illustrate the application of the system

2 cayagvyinu éu lcklv= 6 +30 +3 1 0

The notation could thus be used for expressing large

n s numbers in a sort of mnemonic form . The table of si e referred to in paragraph 9 above was expressed by A rya t hi bha a in this notation w ch , by the Way , he uses only for

a n stronomical purposes . It did not come into ordi ary use

in India , but some centuries later it appears occasionally in a form modified by the place-value idea with the followmg v alues 1 2 3 4 5 6 7 8 9 0

k kk g gk n e ch j jk n t th d dk n t tk d dk n p pk b bh m y r l v é sk s k l

32 INDIAN MATHEMATIC S

w ar e w later , hile there hundreds intervening ith examples

- of the old non place value system . The references in me

diaev al Works to India do not necessarily indicate India proper but often simply refer to the East an d the u se of the term w ith regard to numbers has been further confused by the misreading by W oepc ke an d others of the Arabic term

’

dasz a . hin (geometric l , having to do with numeration , etc )

has w a . A has which nothing to do ith Indi gain , it been assumed that the u se of the abacus has been universal in

m a India from time imme ori l , but this assumption is not

a ide b ased upon fact , there being ctually no ev nce of its

a . use in Indi until quite modern times Further , there is evidence that indicates that the notation W as introduced

was a as . into Indi , it into Europe , from a right to left script

a a 7 a at a 23 . In p ragr ph bove certain attempts squ ring the circle are briefly described and it has been pointed out (in 1 0) t hat Aryabhata gives an extremely accurate value ' of W The topic is perhaps of su flic ien t interest to deserve some special ment ion . The Indian values given and used ar e not altogether consistent and the subject is wrapped in

- some mystery . Briefly put the Indians record an extremely accurate value at a very early date but seldom or never

w how actually use it . The follo ing table roughly exhibits the matter stands

a 7: Authority . V lue of

Ahmes the Egyptian Archimedes ' The n u lvasfi tras

230 Apollonius 1 20 Heron INDIAN MATHEMATIC S . 33

Authority . Value of 7 .

2 3 1 Ptolemy 41 66 .

— 3 Liu Hiu 1 4 .

Puli sa

’ Tsu Ch ung-chi

— ' A r yabhata 3 3

—

3 1 41 66 .

628 Brahmagu pta

— 3 1 4286 .

? M x/ 1 0 2 800 M. ibn usa

Mahavira 3 3 °

1 020 Sr idhara

1 1 50 Bhaskara

l l

Approximately correct value 3 4 INDIAN MATHEMATIC S .

e a a 24. The mistak s m de by the early orientalists h ve

a an d naturally misled the histori ns of mathematics , the o C W aiPC KE pinions of HASLES , , HANKEL and others founded

are n ow upon such mistakes no longer authoritative . In

w a a spite , ho ever , of the progress made in historic l rese rch there are still many errors current , of which , besides those a e w ma lready touch d upon , the follo ing y be cited as examples ( C ) The proof by casting out nines is not of Indian origin an d occurs in no Indian Work before the 1 2th century ; b ( ) The scheme of multiplication , of Which the follow ing is an a 1 6 Indi n example of the th century , w as know n much earlier to the Arabs a n d there is no evidence that it is of Indian origin ; (0 ) The Regu la du ormn f alsoru m occurs in no Indian w ork ; (d) The India ns Were not the first to give double solutions of quadratic equations Bhaskara was not the discoverer of the principle of the ff di erential calculus , etc . , etc . VII .

the 25 . Of personalities of the Indian mathematicians w e know very little indeed but Alberu n i ha s handed down ’ * Brahmagupta s opinion of Aryabhata an d -Pulisa an d his

’ o w n w a a Bh skar opinions are orth repeating . We h ve lso a a s h . T e w ta a all inscription follo ing notes con in , perh ps , that

is w orth recording .

Alberu n i writes Now it is evident that that

w a own a t t an d hich Brahm gupta relates on his u hori y , with

is bu t is which he himself agrees , entirely unfounded ; he

t e a Ar abhata w blind to his from She r h tred of y , hom he abuses

’ An t Ar t e xcessively . d in his respect yabha a an d Pulis a are

a t a w a the s me o him . I t ke for itness the pass ge of Brahma

a a Ar abhata has a e .gupt where he s ys that y subtr ct d some thing from the cycles of Cap u t Dracon is an d of the 0498 21? of the moon an d thereby confused the computation of the e a Ar abhata to a clipse . He is rude enough to comp re y worm w a a hich , e ting the wood , by ch nce describes certain charac

w aw . ters in it , ithout intending to dr them He , however , who knows these things thoroughly stands opposite to

Ar abhata Srishen a Vishn u c han dra y , and like the lion against

a gazelles . They are not c pable of letting him see their ” a a Ar abha faces . In such offensive terms he tt cks y ta and

a a m ltre ts him .

Again Aryabhata differs from the doctrine of

’ the S mritv an d who book , just mentioned , he differs from us is ” an a a a a opponent . On the other h nd , Br hmagupt pr ises

’ s w a ff Puli a for h t he does , since he does not di er from the

A a Varahamihira S rishena book g in , speaking of , ,

’ ’ in t Alb r n i i Accord g o e u Pul s a w as an I n d ian an d Pau lis a a G reek . 36 INDIAN MATHEMATIC S .

Ar abhata an d ishn u c han dra a a y V , Br hm gupta says : If a man declares these things illusory he stands outside the

’ a a w a an d w gener lly ckno ledged dogm , that is not allo ed .

arahamihira Alber u n i t : Of V , Wri es In former times the Hindus used to acknowledge that the progress of scienc e due to the Greeks is much more important than that w hich

a Vara a i a is due to t hemselves . But from this p ssage of h m hir alone (see paragraph 2 above) you see what a self-lauding man he is whilst he gives himself airs as doing justice t o

a 1 1 0 Alberu n i a : others; but in nother place (ii , ) s ys On the whole his foot stands firmly on the basis of truth an d he t o clearly speaks out the tru h . W uld to God all distin

’ gu ished men followed his example .

a a Alberu n i w 1 1 0 Of Brahm gupt , rites (ii , ) But look ,

n at B a u a wh o a for i stance , r hmag pt , is cert inly the most

a distinguished of their stronomers . he shirks the tr u th an d lends his support to imposture under the compulsion of

w a is a some mental derangement , like a man hom de th bout

a a . on to rob of his consciousness If Brahm gupt . is e

a of those of whom God s ys , They have denied our Signs ,

s although their hearts knew them clearly , from wickednes ” a w e a him nl and h ughtiness , Sh ll not argue with , but o y — “ whisper into his ear If people must under circumstances give up opposin g the religious codes (as seems to be your

W h case) , y then do you order people to be pious if you ” so am n n forget to be yourself I , for my part , i cli ed to the belief that that w hich made Brahm agupta Speak the above mentioned w ords (which involve a sin again st con

W as of a a science) something a calamitous f te , like th t of

S had a him w ocrates , which bef llen , not ithstanding the abundance of his knowledge an d the sharpness of his

t an d . intellec , notwithstanding his extreme youth at the time

’ For he w rot e the Brahmaszddhcin ta when he was only thirty

his W e a years of age . If this indeed is excuse ccept it and

’ drop the matter . INDIAN MATHEMATIC S . 37

An t a atn a a inscrip ion found in ruined temple at P , d a eserted village of Khandesh in the Bomb y Presidency , refers t o Bhaskara in the follow ing terms : Triumphant is the illustrious Bhaskarac harya whose feet ar e revered by the w n a who a w law ise , emi ently le rned . l id do n the in metrics , was Vaiseshika W as deeply versed in the system , in poetics a t - a poe , like unto the three eyed in the three br nches , the

t a a t a a a the m the . . mul if rious ri h etic and rest Bh sk_r , learned , e w w t a an d the t the ndo ed i h good f me religious merit , roo of — t w the a an t creeper rue kno ledge of Ved , omniscien seat o f a w t w t le rning hose fee ere revered by crowds of poe s , etc .

’ The inscription goes on t o tell us of Bhaskara s gr andson

Chan adev a a t n Sini han a W ho g , chief s rologer of Ki g g , , to spread the doctrines promulgated by the illustrious Bhas ' karac har a a t the S iddhcin ta y , founds college , hat in his college

’ ’ ’ s zroma u z w a a w and other orks composed by Bh skar , as ell as t a a a o her Works by members of his f mily , sh ll be necess rily

’ e xpounded .

’ Bhaskara s most popular w ork is entitled the Lild vati

’ w a a i a D a hich me ns ch rm ng . He uses the phr se e r ” t t Lildvati t t a a a n e c . an d in elligen , , hus h ve risen cert i legends a s t o a a daughter he is supposed to be ddressing . The

a t legends h ve no his orical basis .

Bhaskara at the end of his Vija ganita refers to the trea ' t a a a u ta S ridhara an d a a a a ises on lgebra of Br hm g p , P dm n bh as t oo diffusive an d states that he has compressed the

a w a t he substance of them in ell re soned compendium , for ” at at a gr ific ion of le rners . — 26 . C M C a a to a HINESE ATHEMATI S . There ppe rs be bun dant evidence of an intimate connection between Indian an d

C A a a t o hinese mathematics . number of Indi n emb ssies China and Chinese visits t o India ar e recorded in the fourth an d ar e succeeding centuries . The records of these visits not generall y found in Indian w orks an d our know ledge of

C a an d t them in most cases comes from hinese uthorities , here is no record in Indian w orks that would lead us to suppose that the Hindus were in an y way indebted to China for mathe i t mat c al . knowledge But , as poin ed out before , this silence

t the a a t t an d n o on the par of Hindus is ch r c eris ic , must on a t ta a t ccoun be ken as n indica ion of lack of influence . We have now before us a fairly complete account of Chinese * mathematics which appears t o prove a very close connection

o t betw een the tw countries . This connec ion is briefly t illustrated in he following notes .

The earliest Chinese work that deals w ith mathematical

an d s questions is said to be of the 1 2th century B C . it record

a a ta w a an cqu in nce ith the Pythagorean theorem . Perh ps the most celebrated Chinese mathematical w ork is the Chin Chang S atin -shit or Arithmetic in Nine Sections which

was at a t as a a t composed le s e rly s the second cen ury B C .

’ ’ while Chang T san g s commentary on it is know n to have

w far A . D . 263 N S been ritten in . The ine ections is more complete than any Indian Work prior t o Brahmagupta w A D 628 an d In t in a a t a . ( . . ) some respec s is dv nce of h t riter

a a a It treats of fr ctions , percent ge , p rtnership , extraction of - a an d square and cube roots , mensuration of pl ne figures

a the an d solids , problems involving equ tions of first second

By Yoshio Mikami.

40 INDIAN MATHEMATIC S .

the a an a an d of dist nce of isl nd from the shore , the solu

’ tion given occurs in Aryabhata s Gan ita some two centu

’ W a -t sao w 6th ries later . The ritten before the century a ppears t o indicate some deterioration . It contains the i e rroneous rule for areas given by Brahmagupta and Mahav ra .

’ The arithmetic of Chang-Ch iu -chien written in the 6th century contains a great deal of matter that may have been

a the basis of the later Indian Works . Indeed the l ter Indian

’ w orks seem to bear a much closer resemblance to Chang arithmetic than they do to an y earlier Indian Work .

The problem of the hundred hens is of considerable

C a the interest . h ng gives following example A cock

o 5 i 3 an d 3 1 ec es . c sts jp of money , a hen pieces chickens piece If then We buy with 1 00 pieces 1 00 of them what will be their ” respective numbers 7

No l a a a a mention of this prob em is m de by Br hm gupt , but it occurs in Mahavira an d Bhaskara in the following

I ar e had 3 drammas 7 a form Five doves to be for , cr nes

5 9 7 an 3 a 1 0 d 9 . 0 of for , geese for pe cocks for Bring these

’ ” s 1 00 drammas bird for for the prince s gratification . It is notew orthy that this problem W as also very fully treated by ' Ab ii am S a 9th an d in K il ( hog ) in the century , in Europe the

‘ a es it a a middle g cquired consider ble celebrity .

Enoug h has been said to Show that there existed a very considerable intimacy between the mathematics of the Indians an d Chinese ; an d a ssuming that the chronology i s

1 t the d t C e a oughly correc , is inct priority of the hin se m the

a a the a h a m ties is est blished . On other hand Br m gupta gives more advanced developments of indeterminate equa

a C e an d tions th n occurs in the hin se Works of his period , it

’ is not until after Bhaskara that Ch in Chu -sheo recorded (in

’ ’ A D 1 247 a t ai- en ch in - i-shu . . ) the celebr ted y y or process of

a a w w a indeterminate n lysis , hich is , ho ever , ttributed to

’ - C I hsing nearly six centuries earlier . The hinese had maintained intellectual intercourse with India s ince the 4 1 i N DIAN MATHEMATIC S .

first century and had translated many Indian (Buddhistic)

w . orks . They (unlike their Indian friends) generally give the source of their information and acknowledge their indebt e d 7th n ess with becoming courtesy . From the century Indian scholars w ere occasionally employed on the Chinese

A t a a Mr M a s ronomic l Bo rd . . Yoshio ik mi states that there

a i . is . of a C a no evi Indi n A influence on hinese m them t cs h ‘ ot a a in ;China On the her h nd he says , f the discoveries m de ma a of a h ve touched the eyes _ Hindoo schol rs . _ 0 -0 ‘ f

— A B C M C S It has a , 27 . RA I ATHEMATI often been ssumed w t l tt t at t the A a w i i h very i le jus ific ion , hat r bs o ed the r knowledge of mathematics to the Hindus .

- 82 D . 7 M a a Mfi sa cl Chowar ez mi A . uh mm d b . ( ) is the earliest Arabic writer on mathematics of note an d his best

n w Al ebra a ta a a k own ork is the g . The e rly orien lists ppe r t o have been somew hat prejudiced against Arabic scholarship

a a w a i at a a for , pp rently ithout ex m n ion , they scribed an Indi n

’ M u a was as o a . rigin to b . M s s Work The rgument used ” w COS S ALI follows There is nothing in history , rote ,

C B a M a and OLE ROOKE repe ted it , respecting uhamm d ben

Mfisa a y the i i n t at individu ll , which favours op n o h he took

the G the a w a t the Mu ham from reeks , algebr hich he t ugh to

in adan s t . History presents him in no o her light than a mathematician of a country most distant from Greece an d

t N t a a o a . o a ta contiguous Indi . . h ving ken lgebr from the

G t t a t it t a reeks , he mus ei her h ve inven ed himself or ken it

’ the a AS a tt a t a a from Indi ns . ma er of f c his lgebr shows , as t R t a an d poin ed out by ode , no Sign of Indi n influence is practically wholly based upon Greek knowledge ; an d it is now well known that the development of mathematics among t he A a was l n ot t r bs argely , if wholly , independen of Indian

an d a other ‘ han d a w influence th t , on the , Indi n riters on mathematics later than Brahmagupta w ere po ssibly in flu en c ed a t A a Alberu n i a the consider bly by he r bs . e rly in 1 1 th century w rote You mostly find that even the 42 INDIAN MATHEMATIC S . so -called scientific theorems of the Hindus are in a state of

an a . . . utter confusion , devoid of y logic l order since they cannot raise themselves to the methods of strictly scientific

. I t . t deduction . began o Show them the elemen s on which

t t o t ou t to t a this science res s , poin hem some rules of logic l ’

an d all a at . deduction the scientific method of m them ics , etc

t - 72 a t A D . 7 The fact is hat in the ime of el Mamu n ( . ) certain Indian astronomical w ork (or certain works) w as

a t A a a was assu med at transl ted in o r bic . On this b sis it th the Arabic astronomy an d mathematics was Wholly of Indian .

w the a t a a w W tra a origin , hile f c th t Indi n orks ere nsl ted is

a the a t t ai re lly only evidence of intellectu l spiri hen prev ling .

a N in B ghdad . 0 one c an deny that Aryabhata an d Brahma

t a * gup preceded M. b . Mfi sa bu t the fact remains that there is n ot the slightest resemblance between the previous Indian w t w t M. a . t as orks and hose of b Mfisa. The poin w some h obscured by the publication in Europe of an arithmetical t a M M u mero ' re tise by . b . fisa under the title Algoritmi de N

I n doru A m. S is Well know n the term India did not in mediaeval times necessarily denote the India of to -day and despite the title there is nothing really Indian in t he Work . Indeed its contents prove conclusively that it is not of

a s a a Indi n origin . The same remark pply to sever l other

a w medi eval orks .

8 th Mu ham r 2 . w e t M. Mfisa a From the ime of b . on rds

madan mathematician s made remarkable progr ess . To illustrate this fact we need only men tio n a few of their dis a t tin gu ished w riters an d their w orks on mathematics . T bi

- A a . orr M rw an 1 a . e 826 90 b Q b ( ) wrote on Euclid , the lm gest ,

the a Nic omac hu s t -a a the rithmetic of , the righ ngle tri ngle

ta . a a a a a etc . os p rabol , m gic squ res , micable numbers , Q b

’ u l- B al 9 1 2 t a at D a A . D . L ka e a beki (died 0 . ) r nsl ed ioph ntus

It s n o t be tt en w e e t at Nic o ma c h u s A . D . 100 ) hould forg o , ho v r , h ( w as a n A a ian w i e J amblic h u s Dain asc iu s an d E u t o c iu s w e e r b , h l , , r n at es ia iv of S yr . 43 INDIAN MATHEMATIC S .

an d w 0 11 an d two , rote the sphere cylinder , the rule of errors

' - ‘ D 877-9 19 w t E l zi A . . . a i a . S a M . G etc B tt n ( . b bir b in n , ) ro e a commentary on Ptolemy an d made notable advances

' 850 -930 A a . Abii a S a . in trigonometry . K mil hog b sl m (c )

w t a a an d the ta o an d a ro e on lgebr geometry , pen g n dec gon ,

' ’ Ab ii l- efa el- Bfi z an i the rule of two errors , etc . W g , born D A . D 40 . 9 a in , wrote comment ries on Euclid , iophantus ,

a an d M Mfisa w a t Hipp rchus , . b . , orks on ri hmetic , on the

' ’ l-si z i Ah an d t . a i e e c . Abii S d circle sphere , , etc , é ( med

- t . A l l the M. . 1 t bde ali A D . 951 0 4 . 2 b b g , , ) Wro e on trisec ion

an a the t c the a a an d of ngle , the sphere , in erse tion of par bol

a ata A t the hyperbol , the Lemm of rchimedes , conic sec ions ,

a a d its a t bs el- a i n . A hyperbol symp otes , etc . , etc Bekr , K rch

M el- a 1 0 t t an d in . . a 1 6 A . D ( b H s n , . ) wrote on ari hme ic i t a a a D Alberu n M. . de ermin te equ tions fter iophantus . ( b

’ ’ A - - D 7 3 an d A . 9 Abfi l R a el Bir imi was in . hmed , ih n ) born

a an d a besides Works on history , geogr phy , chronology stro n om t 0 11 at at a a a y Wro e m hem ics gener lly , and in p rticul r on

a t t t a a i c l he e c . . t ngen s , the chords of circle , Om r b Ibr h m Chai ami 1 046 D . the t W as a A . j , celebrated poe , born bout an d A D . 1 1 23 a a a a was died in . few years after Bh sk r born t He Wro e an algebra in which he deals with cubic equations , a commentary on thedifficulties in the postulates of Euclid on mixtures of metals and on arithmetical difficulties .

This very brief an d incomplete resumé of Arabic mathe matical Works Written during the period intervening between the time of Brahmagupta an d Bhaskara indicates at least considerable intellectual activity an d a great advance on the Indian Works of the period in all branches of mathematics

a a . except , perhaps , indetermin te equ tions at a r the 29 . Th the most import nt pa ts of w orks of the Indian mathematicians from Aryabhata to Bhaskara ar e e ssentially based upon w estern knowledge is 11 0 W established . A somewhat intimate connection between early Chinese and — Indian mathematics is also establishedJ but the connection in this direction is not very intimate with respect to those

t in d eter a ma G e. . a a Sections th t y be ermed reek , g , qu dr tic

i a a A a m n ates . , cyclic qu dril terals , etc That the r bic develop ment of mathematics w as pra ctically independent of Indian

i . nfluence , is also proved

The Arab mathematicians based their Work almost wholly upon Greek knowledge but the earliest of them known t M sa er a a o M. . fi a us , b , flourished aft Brahm gupt so th t the Arabs could not have been the - Intermediaries between the

G an a ed t a t reeks d Indi ns . Inde heir chronologic l posi ion has misled certain w riters to the erroneous conclusion that t hey obtained their elements of mathematics from the

Indians .

Other possible paths of communication between the Indians and Greeks are by way of China an d by way of

so as at . Persia . The former is not improbable it first seems Further information about the early silk trade with China

w a might possibly thro light on the subj ect . The intellectu l c ommunication between India an d China at the critical period is w ell know n— there being numerous references to

a such communication in Chinese literature . If sound tr ns lation s of the early Chinese mathematical w orks Were a a w e a to a v ilable might be ble dr w more definite conclusions , bu t as the evidence n ow stands there is nothing that would w a a a the a a C rr nt more th n b re suggestion of hinese source . INDIAN MATHEMATI C S . 45

W e have already mentioned the visit of certain Greek

a C Ch osroes an d are mathematici ns to the ourt of I, there certain other facts which at least justify the consideration

Sa a - A . . D 229 2 . 65 of the Persian route The ss nid period , , show s a somewhat remarkable parallelism with the age of enlightenment in India that roughly corresponds w ith the

a Gupta period . The re l missionaries of culture in the

a i w S a who Persi n empire at th s time ere the yri ns , were con i n ec ted W est an d w ho with the by their religion , in their ff G t translations, di used reek literature hroughout the ” Mr S orient . . Vincent mith discusses the probability of Sassé n ian influence on India but states that there is no direct evidence .

Al though it may be possible to offer only conjectures as to the actual route by which any particular class of Greek k a a a the nowledge reached Indi , the f ct rem ins that during period under consideration the intellectual influence of

as a is Greek on India w consider ble . It evident not only in the mathematical work of the Indians but also in

85 0 . sculpture , architecture , coinage , astronomy , astrology ,

r S M . Vincent mith refers to the cumulative proof that the remarkable intellectual and artistic output of the Gupta period w as produce d in large measure by reason of the C ontact ' between the civilization of India and that of the Roman

Empire an d research is almost daily adding to such proof .

f G r The flourishing state o the upta empire , the g eatest

A a an d w in India Since the days of sok , the ise influence of its principal rulers gave a great impetus to scholarship of

a an d all kinds . The numerous emb ssies to from foreign countries— which were means of intellectual as Well as political — communication no doubt contributed to the same end ; w Ar abhata an d the know ledge of Greek orks displayed by y ,

Mihira an d a a was l Var aha , Br hm gupta one of the natura results of this renaissance of learning . A ND X PPE I I .

X C F X E TRA TS ROM TE TS .

The Sa l iitra vas s .

* 1 the w l . In follo ing we sha l treat of the different

a a n 2 ' i. e m nners of building the g . W Shall explain h ow to

a measure out the circuit of the rea required for them . a: a:

45 C a . The ord stretched cross a square produces an

a a re of twice the size .

46 . a h . t e a T ke the measure for the breadth , diagon l of its squ are for the len gth the diagonal of that oblong is the side of a square the area of which is three times the area of t he square . as: a: a: s

48 a an t . The diagon l of oblong produces by i self both t he areas w hich the tw o sides of the oblong produce sepa r l ate y .

49 . This is seen in those oblongs whose sides are three an d an d fiv e an d an d four , twelve , fifteen eight , seven

w - w - - ix an d fiv e S . t enty four , t elve thirty , fifteen and thirty s 5 1 . If you Wish to deduct one square from another square cut off a piece from the larger square by making a mark on the ground with the Side of the smaller square w w vDraw hich you ish to deduct . one of the sides across the

so h oblong t at it touches the other Side . Where it touches

ff B -w a there cut o . y this line hich has been cut off the sm ll

is square deducted from the large one .

a: s: a: a:

’ These n umbers refer to Bau dhfiyan a s edition as t ran slated by Dr i au t Th b .

48 INDIAN MATHEMATIC S .

1 t 7 . The number of erms less one multiplied by the common difference and added to the first term is the amount

an of the last . Half the sum of the last d first terms is the

a a an d me n mount , this multiplied by the number of terms

is the sum of the Whole . s a:

an d 21 . The product of half the sides opposite Sides is

a a a the rough area of tri ngle or quadrilateral . H lf the sum

set w an d the of the sides do n four times ~ each lessened by — sides bein g multiplied together the square -root of the

a a product is the exact re . a: an: s a: >k

a an d a 40 . The di meter the square of the r dius r espec tiv ely multiplied by three are the practical circumference

a - a and area . The squ re roots extr cted from ten times the

a square of the s me are the exact values . an: s: a

n a 62 . The i teger multiplied by the sexagesim l parts of its fraction an d divided by thirty is the square of the minutes

a a and is to be dded to the squ re of the whole degrees . a: an: an:

are a ratifi a 1 01 . These questions st ted merely for g c

ma a t hou san d tion . The proficient y devise others or may

a solve by the rules t ught problems set by others .

As su n a so t 1 02 . the obscures the st rs does the exper eclipse the glory of other astronomers in an assembly of

y a a an d r people b reciting lgebr ic problems , still more by thei solution .

’ n ite -S ara-S an raha— C s a a . D Mahavira G g ( irc A . .

— 4 dia a d 1 3 1 . n i . The number , the meter the circum

a an d a ference of islands , oce ns mount ins the extensive dimensions of the row s of habitations an d halls belonging to

a a t w t a w the inh bit n s of the orld , of the in ersp ce , of the orld

an d of light , of the World of the gods to the dwellers in hell , — an d miscellaneous measurements of all sorts all these are made out by means of computation . INDIAN MATHEMATICS . 49

1 47 . D . D vi . ivide by their rate prices iminish by the least among them and then multiply by the least the mixed price of all the things and subtract from the given number

Now as of things . split up (this) into many (as there are left) an d then divide . These separated from the total price give

a the price of the de rest article of purchase . [This is a 6 solution of example 3 below . ] a:

ha n 1 9 . s vi . 6 It to be k own that the products of gold as multiplied by their colours when divided by the mixed

var n a S e gold gives rise to the resulting colour ( ) . [ e example s

24 an d 25 below . ] a: a: a:

A a has w 2 . t o vii . re been taken to be of kinds by Jina — w r in accordance ith the result namely , that which is f o

an d a m practical purposes th t which is inutely accurate . as as =l< 3 n ifli 23 . d cu lt vii . Thus e ds the section of devilishly problems .

’ ’ Trié atika— C A D d ara s . S r i h ( irca .

1 a i n . Of series of numbers beg nni g with unity and

su m increasing by one , the is equal to half the product of the

n number of terms and the number of terms together with u ity . an: di 32 . In exchange of commo ties the prices being trans posed apply the previous rule (of three) . With reference to the sale of living beings the price is inversely proportional

a e t o their g . at

5 di t 6 . If the gnomon be vided by twice the sum of he gnomon and the shadow the fraction of the day elapsed or which remains will be obtained .

' — D Bhaskara (Born A . . hi 1 . n L . I propou d t s easy process of calculation ,

w s delightful by its elegance , perspicuous ith concise term , soft and correct and pleasing to the learned . a s an: s 50 INDIAN MATHEMATIC S

-LI 3 A isf p t . mt 1 9 . L . p Fro hat multiplied by twice ' " sbme assu med n u inbei an d divided i by one less than the

um is a square of the ass ed number a . perpendicular obt ined .

’ ' This being set aside is mu ltiplied b y the arbitrary number — . t e w an d the side as put . is subtracted h remainder ill be the

S a tr an le ! . . i t r e hypotenuse . uch q g is e med genuin >l< >l< at: a;

8 w the a 1 9 . L . Thus , ith same sides , may be m ny

in u aldiilateral . Yet t i , d agonals the q , hough indeterminate , '

‘ ‘ Q ' a determin at e diagonalS H av e b eefi sou ght s by Brahma ' " a n eth gupta. er s

21 3 . are L . The circumference less the being multi

’ ’ fir t plied by the arc the product is termed s . From the

' quarter of the square of the circumference mu ltiplied by five ' the r emain der subtract that first product . By divide the

first product multiplied by four times the diameter . The

quotient will be the chord . as a: s 1 0 w V 7 . . In the like suppositions , hen the operation ,

. w t owing to restriction , disappoints the ans er mus by the intelligent be disco ver ed by the

’ ‘ A b rdiii l it iS’Shrd t —a r c p gy clea intellect ,

a Su m tion w u anIities a an d S p of unkno n q , equ tion , the rule — of three are means of op erat on iii an afyS IS

ale ik

' 4 l t r th s 22 . V. The ru e of hree terms is a i metic potless own understanding is algebra . What is there unkn to the t , se . intellig ent Therefore for the dull alone it , is forth

w e n V 225 . n . To augment isdom and str ngthe co fidence ,

d e a read , read , mathematician , this abri gement leg nt in style , h W easily understood by yout , comprising the hole essence of calculation and containing the demonstratio n of its principles —full of excellence and free A ND PPE IX II .

X P E AM LES .

1 - - - . w One half , one sixth , and one t elfth parts of a pole

a . are immersed respectively under water , cl y , and sand

Tw o hastas are visible . Find the height of the pole — S 2 3 . An swer 8 hastas . .

2 a a - . The qu rter of sixteenth of the fifth of three quar t ers of two -thirds of half a dramma was given to a b eggar by

a el how a C w a person from whom he sked alms . T l m ny o ries t he miser gave if thou be conversant in arithmetic w ith the r eduction termed sub -division of fractions 2

An swer— l 2 . . 3 . , cowrie L

ri = 1 c ow eS dramma) .

3 - f . Out of a swarm of bees one fi th settled on a blossom o f - silin dhr i t kadamba, one third on a flower of , three imes the

ff ku ta a di erence of those numbers flew to a bloom of j . One

w an d air bee , hich remained, hovered flew about in the , a llured at the same moment by the pleasing fragr ance of a

an d an dan u sfi i w a j asmine p Tell me , charm ng om n , the number of bees ?

A n swer- 1 L 5 . 5 4 . , V. 1 08 .

4 ‘ . The third part of a necklace of pearls broken in an amorous struggle fell on the Its fifth part was seen

was resting on the couch , the sixth part saved by the lady nt Six a and the te h part was taken up by her lover . pe rls

= ‘ L the Li avati V= Vi a Gan ita t BhfiS km a 31 : Ma a i a l , j , bo h by , h v r . S = S ridha = ra, C Chat u r ved a. 52 INDIAN MATHEMATIC S .

0 11 Sa remained the string . y , of how many pearls the neck lace was composed ?

A n swer— 30 . S 26 . .

5 A w . po erful , unvanquished , excellent black snake which is 32 hastas in length enters into a hole 7 angu las

a a a in of a day , and in the course of a qu rter of d y 2 an u las a . a its tail grows by 3; g O orn ment of rithmeticians , tell me by what time this same enters fully into the hole 9

— 4 A n swer 7 6 . M v 31 days . , . = (24 afigu las 1 hasta . )

A at 9 o an as 6 . certain person travels the rate of y j a day

o an as a N a and 1 00 y j have already been tr versed . ow m es

a 1 3 o an as da senger sent fter goes at the rate of y j a y . In how many days will he overtake the first person — 25 Answer . M vi 327 . , .

A w hite ~ an t 8 avas -fifth 7 . advances y less one in a day an d afi a la 3 returns the twentieth part of an y in days . In

a w w what sp ce of time ill one , hose progress is governed by these rates of advancing and receding proceed 1 00 yojan as — 9 8042553 A n swer . 283 days 0 . .

= = 8 a a 1 n 7 68000 afi u las 1 a . ( y v s a gula , g yojan )

w tw o o an as 8 . T enty men have to carry a palanquin y j an d 7 20 din aras for their wages . Two men stop after ‘ ’ o krosas two krosas going tw , after more three others give up ,

a and after going half the remaining distance five men le ve . What w ages do they earn

— 1 8 5 M vi 231 rs 7 55 4 0 . . . An swe , , 1 , 9 ,

’ sas= 1 a (4 kro yojan ) .

n w 9 . It is well k o n that the horses belonging to the

’ a sun s ch riot are seven . Four horses drag it along being

I = t o Lilavati V= Vi a Ga ita t B aska a III : Ma a i a h , j n , bo h by h r , h v r ,

a = S S ridh ar a, C Chat u r v ed a 53 INDIAN MATHEMATIC S .

70 harnessed to the yoke . They have to do a journey of

o an a a y j s . How m ny times are they unyoked and how many t imes yoked in four 2 — A n swer 1 0 a an d a t a Every yojan s , e ch horse r vels 40 a . 8 M VI 1 5 . yoj nas . ,

1 a a 0 . If fem le Slave Sixteen years of age brings thirty w w w two , hat ill one t enty cost

Answer— 25 3 6 L . 7 .

1 1 . Three hundred gold coins form the price of 9 damsels o f 1 0 a a 2 ye rs . Wh t is the price of 36 damsels of 1 6 years

A n — 5 swer 7 0 . 40 114 . . V,

1 2 . w The price of a hundred bricks , of hich the length , t k a 1 6 8 an d 1 0 hic ness and bre dth respectively are , , is settled at six din dras w e , have received of other bricks a

a s Sa w a qu rter les in every dimension . y, h t we ought to pay 2

An swer— 2531 2 5 8 . 4. C .

1 3 . l Two e ephants , which are ten in length , nine in

-Six breadth , thirty in girth and seven in height , consume one dron a a w of gr in . How much ill be the rations of ten other e lephants which are a quarter more in height and other dimensions

A n — 285 v C . . swer 1 2 3 rasthas I ku a as . dronas , p , S d

6 = = n ( 4 ku dav as 1 6 prasthas 1 dro a) .

a 1 4 . One bestows alms on holy men in the third p rt of a a a a a da a t day , nother gives the s me in h lf y and hird dis

t o t tributes three in five days . In what time , keeping hese r w 2 ates , ill they have given a hundred — An w 1 . 282 s er 7 C .

L= bbe Lilav ati V= Vi a G an ita t : B aska a ilI = Mahavira , j , bo h by h r ,

S = r d = h v S i hara, 0 C at u r ed a. 54 INDIAN MATHEMATIC S .

1 5 . Sa are y , mathematician , what the apportioned Shares of three traders whose original capitals were resp ec 5 1 68 85 tively , and , which have been raised by commerc e conducted by them in j oint stock to the aggregate amount of 3 00 ?

An swer— 75 1 00 1 25 . 93 . , , L .

1 6 . an d Six One purchases seven for two sells for three .

Eighteen is the profit . What is the capital — A n swer 1 8 4 Bakh hd 5 s li Ms . 4 457 7 .

1 7 ala If a p of best camphor may be had for two n ishkas an d ala , a p of sandal wood for the eighth part of a dramma and half a p ala of aloe wood also for the eighth of a dramma a a n ishka , good merch nt , give me the v lue of one in

1 1 6 8 w a the proportions of , and ; for I ish to prepare perfume

o An swer— — Drammas 1 4h i 8 9 . Prices w s+ a s L . = (1 6 drammas 1 nishka) .

1 8 . If three and a half man as of rice may be had for

dramma a e one and eight of beans for the same price , t ke thes

kdkin is a s thirteen , merch nt , and give me quickly two part of rice and one of beans for we must make a hasty meal

a and dep rt , Since my companion will proceed onwards 7 n 1 1 5 . A swe . 97 . r an d . T a, 5 2 L , V 64 kaki i = ( n s 1 dramma) .

1 6 rammas 9 . 200 d If the interest on for a month be , in

' what time will the same su m lent be tripled

A n swer 0 . 28 . months . 7

su m 20 . If the principal with interest at the rate of five on the hundred by the month amount in a year to one thousand , tell the principal and interest respectively — 8 9 . An swer 625 3 5 . . , 7 L

L = the Lilavati V= Vi a G a ita b t B aska a M= Ma a i a , j n , o h by h r , h v r , = = S r dh r v . S i a a, C Chat u r ed a

5 6 INDIAN MATHEMATIC S .

26 w . Of two arithmetical progressions ith equal sums an d ar e 2 3 the same number of terms the first terms and ,

3 an d 2 1 the increments respectively and the su m 5 . Find the number of terms — 3 A n wer . B dl s akhsh i Ms 1 8 . .

27 A merchant pays octroi on certain goods at three

At r d f . fi i ferent places the st he gives of the goods , at the A 24 s . . econd 4, and at the third ! The total duty paid is What was the original amount of the goods — a 5 . Bakhsh li Ms . 2 An swer 40 .

8 an d w 2 . One says Give me a hundred I Shall be t ice as rich as you , friend The other replies If you deliver w ten to me I Shall be Six times as rich as you . Tell me hat w as the amount of their respective capitals ? — 1 06 1 56 . A n w . s er 40 1 70 . and V ,

9 a B w 2 . A as gives certain amount , gives t ice much a s A 0 3 sas as B D 4 im as , gives time much , gives t es much a s C an d 1 3 the total is 2 . — An swer A 4 Bakhshali Ms 54 gives , etc . . .

30 s 8 . Four j ewellers posses ing respectively rubies ,

1 0 r 1 00 5 a e a sapphi es , pearls and di monds , pres nted e ch from his Own stock one apiece to the rest in token of regard and gratificati on at meeting ; and thus they became owners

w a of stock of precisely the same value . Tell me , friend , h t were the prices of their gems respectively — An swer 24 1 6 1 96 . , , , [These are relative values only]

1 00 . L .

3 1 The and . quantity of rubies without flaw , sapphires ,

n an d pearls belongi g to one person , is five , eight seven

L = th e Lilavati V= Vi a Ga ita t B aska a 1Y= Mahav ira , j n , bo h by h r , ,

= dh r a = hat u r veda S S ri a , C C . 57 INDIAN MATHEMATIC S .

a respectively . The number of like gems belonging to nother

’ i ix -two sev en n n e s . has is , and One ninety , the other sixty rupees . They are equally rich . Tell me quickly , then ,

a a intelligent friend , who art conversant with lgebr , the prices of each sort of gem — 1 05 1 56 . A n sw 1 4 1 . er 1 . , , , etc V

[Bhaskara assumes relative values ]

r es ec 32 . The horses belonging to these four persons p tiv el six y are five , three , and eight ; the camels belonging

two are to them are , seven , four and one their mules eight , tw o an d are , one and three the oxen C wned by them seven ,

two All . a . one , and one are equ lly rich Tell me severally , friend , the rates of the prices of horses and the rest — 1 5 8 . 7 An swer 5 7 6 31 4 . , , , , etc V

da w 33 . Sa w _ y quickly , friend , in hat portion of a y ill n i four fountai s, be ng let loose together fill a cistern, which , w if opened one by one , ould fill it in one day , half a day, the third and the sixth parts respectively — 95 . An swer T3 . L .

34 son Pritha a . The of , angered in combat , shot quiver

a n o ith w of arrows to slay K r a . W half his arro s he parried those of his antagonist ; w ith four times the square -root of the quiverful he killed his horse ; with Six arrow s he slew

S a a a an d alya with three he demolished the umbrell , st nd rd bow and with one he cut off the head of the foe . How many were the arrows w hich Arjuna let fly

w — An s r 3 . e 1 00 . 6 1 3 . 7 V. L ,

35 3 p . a ers 5 alas t For p p of ginger are ob ained , for 4 p a rlas 1 1 p alas of long pepper an d for 8 p an as 1 p a la of

L = th e Lilav ati V= Vi a G a ita t B as a a JI = Ma a i a , j n , bo h by h k r , h v r , ’ S = r d S i hara , C Ch atu r v ed a . 58 INDIAN MATHEMATIC S .

pepper . By means of the purchase money of 60 panas

a 68 la ? 5 a s M vi 1 0 . quickly obt in p . ,

An swer— G 20 44 4 . inger , long pepper , pepper

3 6 . Five doves are to be had for three drammas seven cranes for five nine geese for seven an d three peacocks for nine . Bring a hundred of these birds for a hundred drammas for the prince ’ s gratification — i 1 52 . 1 58 9 M . v V . ; , A n swer— 3 40 21 3 6 . Prices , , , 5 56 27 1 2 . Birds , , ,

’ (This class of problem was treated fully by AbiI Kamil el-M e 900 . S S isri ( ee H . uter Das Bach der S elten

h i - e t etc . M 1 1 1 1 1 9 0 . 1 , Bibliotheca athematica ( pp 00 20 .

>l< =l<

3 7 . In a certain lake swarming with red geese the tip of a bud of a lotus was seen half a hasta above the surface of

the w ater . Forced by the wind it gradually advanced and

t a t h a wo astas . C was submerged a dist nce of alculate quickly ,

0 a a w mathem tici n , the depth of the ater

1 5 A n swer— 1 53 1 25 1 . L . ; V. .

a - hastas 38 . If bamboo measurin g thirty two and stand ing upon level ground be broken in one place by the force of the wind and the tip of it meet the ground at sixteen

hastas sa a hastes the , y , m thematician , at how many from root it is broken

An sw — 1 er 2 .

L = th e Lilavati Vi a G a ita t B aska a M = Mahavira , j n , bo h by h r , , ’ S S r idhar a, C Ch at u r v ed a . 59 INDIAN MATHEMATIC S .

’ 3 A l 9 hastas 9 . snake s hole is at the foot of a pi lar S high and a peacock is perched on the summit . eeing a

a i sn ke , at a distance of thrice the pillar , glid ng towards his

Sa hole , he pounces obliquely on him . y quickly at how

’ a m ny hastas from the snake s hole do they meet , both pro c eedin g an equal distance ? — An 1 . 1 50 . swer 2 . L

40 hastas a . From a tree a hundred high , monkey

two hastas descended and went to a pond hundred distant ,

off while another monkey , jumping a certain height the tree ,

a proceeded quickly diagonally to the same spot . If the sp ce

a man tr velled by them be equal , tell me quickly , learned , the the height of leap , if thou hast diligently studied calcula tion

An — 1 26 . sw r 1 55 . e 50 . L . V

41 . The man who travels to the east moves at the rate

2 o an as an d who of y j , the other man travels northward moves at o a the rate of 3 y jan as . The l tter having journeyed for

5 a how d ys turns to move along the hypotenuse . In many days will he meet the other man ? — An sw 1 3 ii 2 1 1 . er . M v . ,

42 . The Shadow of a gnomon 1 2 angu las high is in one

1 5 afi a las an u las place g . The gnomon being moved 22 g

a w 1 further its Sh do is 8 . The difference between the tips of the Shadow s is 25 and the difference between the lengths 3 of the Shadows is . Find the height of the light — 5 An 4 . w . 2 s 1 00 . 3 8 r 6 er 1 A . 1 C . ; ; L

4 3 . The shadow of a gnomon 1 2 afigu las high being fi l lessened by a third part of the hypotenuse became 1 4 a gu as .

a Tell , quickly, m thematician , that shadow — A n w 1 41 . s er . 224. V

L= th e Lilavati V= V1 a G an ita t B aska a M= Mahav ir a . j , bo h by h r , ,

S = r = S idh ara, 0 Chat u rv ed a . 60 C INDIAN MATHEMATI S .

44 w . Tell the perpendicular dra n from the intersection of strings mutually stretched from the roots to summits of two bambo os fifteen and ten bastas high standing upon ground of unknown extent

An w r— 6 1 6 . s e . L . 0

45 . Of a quadrilateral figure whose base is the square

hastas an d w the a of four and the face two altitude t elve , fl nks

an d w a a thirteen fifteen , h t is the are — A n sw r 8 S . 77 . e 1 0 .

46 . In the figure of the form of a young moon the middle length is Sixteen and the middle breadth is three hastas . By

tw o a its Splitting it up into tri ngles tell me , quickly, area — 83 A n . swer 24 S . .

47 . The Sides of a quadrilateral with unequal Sides 1 3 1 5 1 3 20 5 are x , x and the top side is the cube of and the bottom side is 300 . What are all the values here beginning with that of the diagonals

An wer — 5 8 4 s s 31 280 4 252 1 32 1 68 224 1 89 41 00 . , , , , , , , ,

ii 59 . M . v ,

If A 2 B 2 0 2 a n d a 2 b2 0 2 t en th e a i atera A c B c , h qu dr l l , ,

a0 60 is i an d th e dia n a s ar e A b C B an d A d E h , cycl c go l , 2 ’i = th e a ea is AB C o bC & c . In th e esen t ase A I5 r 4 ( ) , pr c ,

B 20 0 25 a 5 b 12 c 1 3 . Th e ia n a s ar e , , , d go l

3 15 2 80 th e a ea 44 100 . F or etai s see th e .Lilava ti , r full d l ,

1 93 .

48 . 0 w a friend , who kno est the secret of c lculation , construct a derived figure ~ with the aid of 3 and 5 as ele

an d ments , and then think out mention quickly the numbers

a measuring the perpendicul r side , the other Side and the hypotenuse ? — A n wer vii 94. s 1 6 3 34 . . M , 0 , , 7 — 9 at is n st t a t ian e th e 2 mn 7112 —712 ”1 11 : Th co ruc r gl of form , , 4

w e e m 5 n 3 . h r ,

' L = th e Lilav ati V= Vi a Ga ita t B as a a M Ma a i a , j n , bo h by h k r , h v r , = = S S ridh ara , 0 Chat u r ved a . INDIAN MATHEMATIC S . 61

49 . In the case of a longish quadrilateral figure the 55 48 perpendicular side is , the base is and then the diagonal

3 a ar e is 7 . Wh t the elements here — 1 1 M . v i 2 8 . i . An swer 3 , ,

5 w w 0 . Intelligent friend , if thou kno est ell the spotless

Lilav ati sa a a a , y wh t is the rea of a circle the di meter of which

a a e is measured by seven , and the surf ce of a globe or are lik

a a a e a net upon b ll , the di met r being seven , and the solid content w ithin the same Sphere ?

n er— A 38 n- a 1 53 A sw rea gag33 ; surf ce volume 1 79 {436

4. L . 20

51 . a a In circle whose di meter is ten , what is the cir

c u mf er en c e ? a If thou knowest , calculate , and tell me lso the area

An swer— G2 5O S 85 J M . . .

a Ra 52 a 25 52 . The me sure of hu is , th t of the moon ,

the part devoured 7 .

An swer— Ra 2 5 The arrow of hu is , that of the moon . C’ 31 1 . .

This is an eclipse problem a n d m ean s that circles of diameters 52 an d 2 5 in t ersect so t hat the portion of the lin e join in g th e tw o e n t es n t o th e tw o i e i c r commo c rcl s s 7 . Th e common

ts t is in t se en ts 5 an d 2 chord cu h o g m of .

53 . The combined su m of the measure of the circumfer

the an d a a 1 1 1 6 ence , diameter the re is . Tell me what the

w a a circumference is , hat the calcul ted rea , and what the diameter

An sw — 2 36 er 1 08 97 . M , , . vii, 32 .

6 4 64 hi = 3 ) 1/ w ch assumes that rr .

L= th e Lilavati V= Vi a C a ita t B aska a M= Mahav ira , j n , bo h by h r , , S = = S ridhara, 0 Chatu r ved a . 62 INDIAN MATHEMATIC S

54. w 1 8 The circumferential arro s are in number . How many are the arrows in the quiver ? — An swer 37 . M 28 . 9 .

3+ + 3 i IS n w The rule g ven 1 2, here c IS the number a of arrows in the outside l yer . w 55 . Tell me , if thou kno est , the content of a spherical piece of stone whose diameter is a hasta and a half — An swer 1 . 3g S . 9 3 .

3 l v= d — The rule given is (2+ 7! iS ) ’

A a a n 56 . 6 a u las s crifici l altar is built of bricks g high ,

hasta is hasta . 6 hastas 3 half a broad and one long It long,

hasta hi hastas . w broad and half a gh Tell me rightly , ise

w its is an d how a man , hat volume many bricks it cont ins . — Answer 9 72 . , S . 96 .

= a 24 afi gu las 1 hast .

5 k w a 7 . If thou no est , tell me quickly the measure of mound of grain w hose circumference is 36 and height 4 hastas — 4 S 1 2 An swer 1 4 . . 0 .

-2 Th e rule u sed ass umes th at 71 3 .

58 . a bow In the case of a figure having the outline of , is 1 2 w the string measure , and the arro measure is 6 . The

0 . measure of the bow is not known . Find it , friend

An swer . M . v M3GO ii 75 . ,

a n 59 . In the case of figure havi g the outline of a bow

24 its the string is in measure , and arrow is taken to be 4 in measure . What is the minutely accurate value of the area

— M vii 2 An swer . 7 V5760 . , .

The rule used is S

>xc

at

L = th e Lilavati V= Vi a G an ita t B aska a M= Mahavira , j , bo h by h r , , = S S ridh ara, C Ch atu r ved a.

64 INDIAN MATHEMATIC S .

66 . What number is that which multiplied by nought an d added to half itself an d multiplied by three and divided by nought amounts to the given number Sixty-three — A n swer 1 4 . a a 4 This ssumes th t L . 6 .

67 . What four numbers are such that their product is

w su m sa a equal to t enty times their , y, learned m thematician

w ho art conversant w ith the topic of the product of unkn own quantities — 1 1 s er 5 4 2 . An w . , , , V, 21 0 i Bhaskara put s arb trary values for three of th e qu an tities an d g ets

1 1 for th e fourth .

a 68 . If you are convers nt with operations of algebra tell the number of w hich the fourth power less double the sum of the square and of tw o hundred times the simple number is ten thousand less one ? — n swer 1 1 . 1 A V. 38 .

~ — a be e ess e : l 4 2 962 - 200 = i This m y xpr d by ( 4 x ) 9999 . It s th e on ly

in w i th e u t w case h ch fo r h po er occurs .

su m tw o 69 . The square of the of numbers added to the cube of their su m is equal to twice the su m of their cubes — A n swer 1 20 5 76 . 1 8 , , , etc V. 7 .

k w 0 . a 7 Tell me , if you no , two numbers such th t the su m of them multiplied respectively by four and thr ee may when added to two be equal to the product of the same numbers

n swe — 5 1 1 1 6 2 A r 0 . V. 09 2 12 , and , , .

w 7 1 . Sa y quickly, mathematician , hat is the multiplier by w hich tw o hundred and twenty -one being multiplied and

i - v e su m s xty fi added to the product , the divided by a hundred ? an d ninety-fiv e becomes cleared — 3 5 20 5 6 . n r 253 . 5 A swe & . V , , c . L ;

= L az th e Lilav ati = Vi a G a ita t B as ka a M Ma a i a, , V j n , bo h by h r , h v r = = S S ridhara , 0 Chatu r v ed a. INDIAN MA THEMATIC S . 65

di six has 72 . What number vided by a remainder of

five , divided by five has a remainder of four , by four a remain der of three and by three one of two — 5 9 . Br An swer . wviii , 7 V. 1 60 .

73 . What square multiplied by eight and having one added to the product will be a square

V 82 . . 2 — = 2 = = 8u 1 t u 6 35 . t 1 7 99 Here and , , etc , , etc .

4 Ma n 7 . ki g the square of the residue of signs and minutes on Wednesday multiplied by nin ety-two and eighty three respectively w ith one added to the product an exact

who is a square does this in a year a mathem tician . B r . xviii 6 , 7 .

? 2 2 2 1 92 u —1 = t 2 83 u 1 = t ( ) ( ) + .

= = = = An swe u 1 20 t 1 1 51 . 2 u 9 t 8 r , ( ) , 2 .

75 . What is the square which multiplied by sixty seven and one bein g added to the product will yield a square root and what is that which mu ltiplied by “sixty-one with

so wi ? D one added to the product will do like se eclare it ,

d a friend , if the metho of the rule of the squ re be thoroughly i ? spread , like a creeper, over thy m nd V 8 . 7 .

2 2 2 = t 2 61 u 1 = t (1 ) 67 u + 1 . ( ) + .

= t= 48842 . 2 Answers u 5967 , ( ) = t 1 , 7

6 h m , 7 . Tell me quickly , mat e atician two numbers such that the cube -root of half the su m of their product and the - su m smaller number, and the square root of the of their

B = = t V= VI a Ge ita t aska a , M Mah§vira L bbe Lilava i, J n , bo h by h r ,

= a. S ridhara , C Chatu rved 6 6 INDIAN MATHEMATIC S

x su m ff squares , and those e tracted from the and di erence

x f increased by two , and that e tracted from the di ference of their squares added to eight , being all five added together

- — x w Six ? may yield a square root e cepting , ho ever , and eight 1 90 . V.

— m— 2 -z y+ 2 + V y + s= z

w s— xz 8 1 677 4 1 5128 Ans er . 4 246 1 . ; / ; , etc ; , etc

L = th e Lilav ati = V1 a Ga ita t B ask a a 111 : Ma a i a , V J n , bo h by h r , h v r ,

= = at v a . S S ridhara , 0 Ch u r ed Y CHRONOLOG . C r i o. B 530 Pythagoras C . Euclid 290 Archimedes 250

’ TH E S ULVAS UTRAS The Nine Sections 1 50 Hipparchus 1 80

Nic omac hu s A . D. 1 00 Ptolemy 1 50 S assan ian p eriod begin s 229 The Sea Island Ar ithmetic 250 Gu p ta p eriod begin s 320 Diophantus 3 60 Hypatia 41 0

Boethius 470 b .

A H T 4 6 RYAB A A 7 b .

Damasciu s 480 b . Athenian schools closed 530

’ Chang ch iu -chien 550

H A 587 . VARA A MIH IR . d

P 598 BRAHMAGU TA b . Fall of Alexandria 640 Gu p ta p eriod en ds 650 S assan ian p eriod ends 652

M Mu a 8 uhammad b . s 20 d .

MAH AviE A 850 7

’ orra 826 2 Tabit b . Q

el- a i 8 Batt n 77 b .

el-Bir u n i 9 3 7 b . Ibn Sina 980 b ' S RIDH ARA 991 b . el-Karchi 1 01 6 ‘ - . a i el Chai ami 1 046 Omar b Ibr h m j b . BH ASKH ARA 1 1 4 1 b . A BIBLIOGR PHY.

(For a more complete bibliography see that given in the

J ou rn al o the Asiatic S ociet o Ben al 1 0 f y f g , VII , ,

FIRST PERIOD . ' — n B G . O S u lvasu tras X V THI AUT , the , LI , — ’ 1 875 . Bau dha an a S u The Pan dit The y ulvas tra , 8 - — ’ ( 1 75 6 . Kat a an a S u lvasfitra Benares) The y y , I 1 h. 88 , 2 . ' — - - Bq A . Das A astamba S Sfitra 55 , p ulba , , 1 9 1 0 ; 56, 1 902 .

S C E OND PERIOD .

— Su S a G . BURGESS , E . AND WHITNEY, The rya iddh nta , 5 o r O S oc VI 1 85 . J a . Am. r . , ,

- Th fi P L. e S r a BA U DE VA SASTRI AND WILKINSON, y

‘ ’ S iddhan ta an d the S iddhan ta S iroman i C , alcutta , 1 8 1 6 .

B —The Pan ho S D H AR DVIVE DI . c . U ARK THI AUT , G AND

siddhdn tikd o Varaha Mihira 1 889 . f , Benares ,

’ R L — l a hata 1 8 9 . e ons de Calca d Ar b 7 . ODET, L § y , Paris , — R Ar abhata 1 7 1 908 . KAYE , G . . y , IV , ,

THIRD PERIOD .

C B T — A l e ra with Arithmetic an d Men . b OLE ROOKE , H . g ‘ saration f rom the S an scrit of Brahmagu p ta an d

Bhascara 1 81 7 . , London ,

R N GACARYA M— Th avita-Sdra-S anraha o Mahd A , . e G g f

viracdr a M dr s 1 908 . y , a a , INDIAN MATHEMATIC S

— Trisatika R. RAMAN UJAC H ARIA N . . , AND KAYE , G The

’ Ma h X 3 1 91 3 . S ridhar ac har a Bib . t . of y , , III , ,

N OTATIONS .

— G In dische Palaeo ra hie S 1 896 . BUHLER, . g p , trassburg ,

-0 rn Nu merals C . he od . n t Gen ealo o M e BAYLEY, E gy f , 1 882 London , .

— Mé su r a C i E PC KE F . W O , moire la propag tion des h ffres

i i u 86 i di J ou r . As at e 1 3 . n ens , q , — A . J . R. a Ar N KAYE , G . Indi n ithmetical otations , A 1 . B . 7 9 Use S . 07 , III , , The of the bacus in

32 1 908 . A a A . S . R J . ncient Indi , , IV , , Old

N S In dian An ti u ar Indian umerical ystems , q y,

1 9 1 1 .

’ F A hata s J . r ab FLEET, . y system o f expressing

1 91 1 . Use S . A . N R. J . umbers , , The of the

- 1 1 91 . A . S . R. A a in a J . b cus Indi , ,

- The H in du Ara ic C . b S D . P . MITH , E . AND KAR INSKI , L

N u merals 1 91 1 . , Boston ,

OTHER WORKS .

— ’ S C O Alberu n i s I n dia 1 91 0 . A HAU , E . . , London ,

B — A G . A M THI AUT , stronomie , strologie und athematik ,

Gru n driss der I n do-Arischen Philolo ie g , III , 9 , 1 899 .

H OE RN LE R - a i M I n An ti . ndia , The Bakhsh l anuscript ,

u ar X 1 888 g y, VIII , . R —N i M G . . M KAYE , otes on H ndu athematical ethods , I 1 1 — Bl Ma h . 4 1 9 X . b. t M , , , Hindu athematical

M In dian E du ca io — - t n 1 9 10 1 9 1 3 . ethods , The

S i Ma A . S . 1 91 0 . J . R. ource of H ndu thematics , , a i M S B II J A . V 9 The Bakhsh l anuscript , . . . , , , 1 91 2 . 70 INDIAN MATHEMATIC S .

L —Dio han tu s o Alex . n dr a . a i C HEATH , T p f , ambridge ,

1 910 .

F — The l M h R . A ebra o o ammed ben Mu sa OSEN, g f , 1 831 London , .

- Di M hema i S H . e at t ker a n d n UTER , Astro omen der

Araber u n d Ihre W rke e 1 900 . , Leipzig ,

— The develo men o c YOSHIO MIKAMI . p t f Mathemati s in

h n a an d a an 1 91 2 C i J . p , Leipzig , The general works on the history of mathematics by

C ! RAUN MUH L . B ANTOR , GUNTHER , EUTHEN , TANNERY and v

s W OE PCKE R S and the article by , ODET , VOGT , UTER and

WIEDEMANN Should also be consulted .

XI —82

72 INDE x .

‘ F au c ha S iddh n tik th 9 -1 6 d ct e 1 8 . G ee te s 9 26 . , , r k rm , , ,

G n t e S a s 22 . u . 70 , h r , , . P ppu a a e s 20 P r ll l , . P au lisa S i dh nta the 2 1 0 an k l H 4 d a , , , . H c , 1 7 , 3 .

e ian e ati n 1 6 1 7 65 . H eat S ir . L. 1 7 70 . , , , h , T , , P ll qu o - - l a e a e n tati n s 2 29 3 1 32 . rn e R 69 . H o e , . Pl c v lu o o , , , , at 1 H atia 1 5 16 . 5 1 9 . yp , , Pl o , , e s Probl m ,

In a a ies 40 . essi n s 23 48 49 5 6 . ccur c , Progr o , , , ,

- - n e te in ate e ati n s 12 1 5 1 8 t e 10 1 1 33 . I d rm qu o , , , P ol my , , , i 40 65 6 6 . sa 3 5 . , Pul , a i 2 1 l In s i ti n s 3 1 3 7 . , e . cr p o , , Pyr m d volum of , - t a ean t e e 4 6 38 . In te ati n u a 1 1 . y rpol o form l , P h gor h or m , , In te t 4 6 e s 7 5 . r , ,

- osta b. L a 42 43 . Q uq , Q a r ati e ati n s 1 6 -1 8 2 4 Jamblieh u s 42 . u d c qu o , , , , - 63 66 . J n es Sir W . . o , , l i - - a r ate a s 20 22 60 6 1 . Q u d l r l , ,

Ka a S u t as 3 . lp r , Raman i - u ac har a M. 6 9 . K a i 43 . g , , cl rch , Ra i i n ac haria M. 6 8 . K a n s L. 6 , C . 9 . g , rp k , , - ati n a t ian es 4 1 8 1 9 5 0 60 . Kat a an a 4 5 6 8 . R o l r gl , , , , y y , , ,

Re u la du oru m a lsoru rn 3 4 . Ke n 1 1 . g , r , f i t -an e t ian es 4 -5 1 8 -2 K a s t i n e a s 2 7 2 9 . 0 , h ro h h um r l , , R gh gl d r gl , ,

60 . La a e 1 . pl c ,

- et L 41 68 70 . Lette n e a s 30 3 1 . , , r um r l , Rod , ,

- - B aka Si an ta 2 9 . Lilavati, the 24 37 49 50 5 1 64 . om ddh , , , , , ,

sen 7 0 . Ma n n e 4 . cdo ll , Ro , 2 a u di e t r ee 3 2 6 a 5 0 5 3 . M co , 3 1 . Rul of h , , ( ), ,

‘ e 34 . Ma a i a 1 4 1 9 2 1 23 3 9 40 tw o e s, h v r , , . , , ; , Rul of rror

- - 48 49 52 63 6 8 . , , ‘ ’ - S ea I slan d A r ithmetic the 39 40 . Mi a i Y s i 3 8 4 1 7 0 . , k m , o h o , , , , n M a a b M s 4 -42 S iddhan ta S iroma i, the 3 7 6 8. . u a 1 70 . , , uh mm d , , ’ M a a b Si i i s 1 5 . . A e Abu l , uh mm d hm d , mpl c u ' - Sin e n ti n 9 . Rihau el Birlmi 3 5 -36 4 1 -42 fu c o , , , ,

in 1 1 . S es , ta e oi 10 43 6 9 . bl , ,

S it . 69 . M e Max 4 . ull r , , m h, D

S it Vin en t 45 . M. b M s M sa see . ai u ( a ) , m h , c , S e e e Oi 1 3 39 6 1 6 2 ph r , volum , , , .

- S a e t 5 63 . au F 3 1 . N , qu r roo , ,

a hu s 42 S a es n st ti n oi 7 46 . Nic o m c , . qu r , co ruc o , ‘ ’ n c on s A r ithmetic the S a in the i e 7 8 47 . N i e S e ti , , qu r g c rcl , , - S r idhar a 1 4 2 1 24 3 7 49 5 1 5 2 3 8 39 . , , , , , , , ,

- - N tati n s 2 2 7 32 69 . 60 62 69 . o o , , , ,

N e i a w s 3 1 . S rish en a 3 5 . um r c l ord , , S u lv as tras the 1 3 -8 46 -47 68 fi , , , , , .

- - - O a b. a i e l Chai ami 43 . S u n Tsu S u an chin the 3 9 . m r Ibr h m j , g, , '

S u r a S iddhcmta the 9 68 . y , , l , ,

ad an a a 1 4 3 7 . S s 24. P m bh , , ymbol , 7 3 INDEX .

62 V es 1 3 2 1 6 1 , . T a it b. orra 42 . olum , , , b Q ,

a e sin es 10 . T bl of ,

W itn e 68 . - 40 h y, l , Tai en ess . y proc , ebe W , 1 . an n e T 5 1 7 70 . r T ry, . , , , iedde an E 0 - W m , 7 . e in 24 26 . T rm ology ,

k 4 69 70 . W oe c e 32 3 , i a t G . l 7 9 68 69 . p , , , Th b u , , , , , ,

W n e a s 3 1 . ian e a ea of 20 2 1 47 . m , Tr gl , r , , ord u r l ’ - - - W u t sao the 40 . ian es i t an ed 4 1 8 1 9 , , Tr gl , r gh gl , , ,

50 , 60 .

' ’ - 24 2 5 26 . - Ycwat tcwat, , (a) i n et 9 10 . Tr go om ry, ' Y shi Mi a i 3 8 4 1 70 . o o k m , , , a 24 2 7 69 . Tris tika, , ,

! e 64. ar ha Mihira 2 9 3 5 3 6 68 . ro , V a , , , , ,

’ - ! e t en H 70 . i a Gamta the 24 3 7 50 5 1 66 . u h , Vj , , , , ,

an a 35 36 . Vishn u c h dr , ,