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 .