ET LES DÉBUTS
DE LA SCIENCE MODERNE
S. N. Sen (India)
INDIAN ELEMENTS IN EUROPEAN RENAISSANCE
Did “algorism ”, 1 the m edieval nam e for arithm etic based on decimal place-value H indu-A rabic num erals, play any effective role in th e E u ro pean Renaissance of the fifteenth and the sixteenth centuries? If the answ er is “yes”, as m any believe it is, the question of relationship of Indian m athem atics, arithm etic in particular, w ith European Renaissance assumes im m ediate importance.
As to algorism as one of the active prom oters of the Renaissance, one has only to look a t the sudden appearance in q u an tity of p rin ted arithm etical works, a good num ber of them in vernaculars, during the 16th century and the quick m ultiplication of th eir editions or reprints. Among th e works in circulation in Italy, m ention m ay be m ade of C ardano’s Practica arithm etice et m ensurandi singular is (Pavia 1501; Rome 1576) and T artaglia’s volum inous La Prima Parte del general
trattato di num eri e m isure (Venice 1556). In England, R obert R ecorde’s The grounde of artes, teachyng the w orke and practise of arithm etike
(London 1542) w as rep rin ted 17 times before 1601 and Digg’s A rith m eti-
call m ilitare treatise called Stratiotios (London 1579) enjoyed consider
able popularity. In G erm any, the arithm etical m ovem ent was led by Jacob Köbel, au th o r of Rechenbiechlin (Augsburg 1514), Stifel whose scholarly work Arithm etica integra (N ürnberg 1514) was rep rin ted several times, and by C hristopher Clavius whose Epitom e arithm eticae practice (Rome 1583) and its Italian translation (1586) w ere in extensive use
1 The words “algorism ” and “augurim ” (used by Chaucer) w ere originally derived from th e nam e of al-K hw arizm i (9th century). W itness th e use of th e w ords algoritm i, algorism i, algorism o in th e tw elfth century Latin translations of al K hw arizm i’s arithm etic or of other sim ilar Arabic w ork. H enri de M onde- ville, in his C yru rgia, explained “algorism ” as m eaning arithm etic. W hereas com putus dealing w ith Church calendar used Roman num erals, every algorism us used decim al p lace-valu e Hindu numerals (G. Sarton, The A p p recia tio n of A n
cien t and M edieval Science during the R enaissance (1450— 1600), P hildephia 1955,
5 6 S. N. Sen
among the Jesuit missionaries. In France w here arithm etic attained great popularity in the 16th century, th e leading w orks w ere Boissière’s
L ’art d’arythm étique contenant toute dimention... tant pour l’art m ili taire que pour autres calculations (Paris 1554) and P ierre Forcadel’s L ’A rithm étique (Paris 1556— 7); A rithm étiq ue entière et abrégée,
P aris 1565) and A rith m étiqu e dém ontrée (Paris 1570). In th e N eth er lands, Gemma Frisius’ Arithm eticae practicae m ethodus facilis (Antwerp 1540) scored a record by reappearing in p rin t 60 tim es during the Renaissance. Simon S tevin’s L ’arithm étic et la pratique d’arithm étique (Leiden 1585) and its subsequent p a rt appearing in the sam e year under a separate title La disme have been characterized as “the greatest arithm etical m onum ent of Renaissance.” 2 The convenient num erical symbolism of which arithm etic is a logical consequence proved to be a pow erful tool for the developm ent in due course of o th er branches of m athem atics.
The genesis of this new m ovem ent has been traced to the activities, among others, of A delard of Bath, John of Seville, R obert of Chester, Villedieu, Sacrobosco and Leonardo Pisano, all of whom w ere engaged —the first th ree in the tw elfth and the last three in th e th irteenth centuries, in transm itting Arabic m athem atical knowledge in Latin translations to C hristian Europe. A t this tim e and up to a much la ter date, the abacus of G erbert w as in general use for all calculations and counting purposes. A delard of B ath (c. 1142) w ho w as an abacist to s ta rt w ith and became an algorist la ter was probably the earliest L atin exponent of H indu arithm etic, trigonom etry and astronom y through his translations of al-K hw àrizm ï’s m athem atical and astro nomical works. In 1126, he tran slated from Arabic into Latin the astronom ical tables of al-K hw àrizm ï in the version of the Spanish astronom er Maslama al-M ajritï. He is also believed to have translated
L iber ysagogarum Alchorism i in artem astronomicam magistro A. com- positus, an arithm etical work attrib u ted to al-K h w àrizm ï.3 B ut the
earliest L atin version of al-K hw ârizm ï’s arithm etic A lgoritm i de
num éro Indorum , of which th e Arabic original is lost, is due to an
unknow n tra n s la to r.4 This w ork containing about 5000 words, treats a t length of num eration, using in most cases Roman num erals, and also
2 G. Sarton, op. cit., p. 157. La dism e is of special im portance as in it is discussed S tevin ’s discovery of decim al fractions.
3 G. Sarton, In trodu ction to th e H isto ry of Science, II, p. 167— 68. Also see C. H. Haskins in The English H istorical R e view , X X VI, p. 494.
4 The MS w as discovered in th e Cambridge U niversity Library by Prince B aldassare Boncam agni and transcribed and printed by him under th e title
T ra tta ti d’a ritm etica , Rome 1857, w hich also included John o f S ev ille’s L iber algorism i de pratica arism etrice. S ee also Suzan Rose Benedict, A C om parative stu d y of the ea rly tre a tise s in trodu cin g in to Europe th e H indu a rt of reckoning,
discusses fundam ental operations w ith integers and fractions. The use of Roman num erals and the c u rre n t abacus term s, e.g., caracter, erigere,
levare, indicate the tra n sla to r’s u nfam iliarity w ith the new arithm etic
based on decimal place-value num eration. Jo h n of Seville’s (c. 1135)
Liber algorismi, although based on al-K hw arizm i’s arithm etic, w as not
a verbatim translation, b u t utilized o ther Arabic sources. I t contains discussion of the extraction of roots and sum m ation of series, absent from th e Algoritm i which it strikingly resembles in the general mode of treatm ent.
R obert of C hester (c. 1141) introduced th e study of algebra in Latin Europe by his translation of al-K hw arizm i’s Hisab al-jabr w al-m uqa-
bala, w hich bears th e im press of H indu algebraic th o u g h t.5 He also
revised al-K hw arizm i’s astronom ical tables in A delard’s version of the m eridian of London and was possibly the first to tran slate the Arabic
jaib, derived from the S anskrit jiva, into sinus.
V illedieu’s (Alexandre de Villedieu, d. 1240) Carmen de algorismo, composed in hexam eter, closely followed Jo h n ’s Liber algorismi. In this work zero signifying nothing w as treated as one of th e H indu num erals, whose num ber was stated to be “twice five.” T ranslations of Carmen into English, French and Icelandic, several com m entaries produced on it in m any languages and large num ber of MS copies found in the libraries of Europe b ear testim ony to the influence it exerted and the wide circulation it enjoyed in the th irteen th century. The w ork possibly played an im portant role in the diffusion of H indu num erals in Latin Europe. 6
V illedieu’s contem porary John Sacrobosco, au th or among others, of the arithm etical treatise Algorism us vulgaris “contributed pow erfully to the diffusion of the H indu nu m erals” in L atin Europe. 7 The work, designed to be a practical exposition of the decim al place-value num e ration for use in the universities, dealt w ith, besides the four fu nda m ental operations, th e methods of extraction of square and cubic roots and arithm etical series. Its great popularity for about th ree hundred years is evident from the large n u m ber of MS copies available in the libraries of Europe, and from several editions afte r it was first com m itted to printing in Strassburg in 1488. Despite his great service in popularizing the H indu a rt of reckoning he was largely responsible
5 L. C. Karpinsky, R obert of C h ester’s L atin tran slation of th e A lgebra of
a l-K h o w a rizm i—w ith an introduction, critical notes and an English version,
M acm illan, 1915. Solomon Ganz, upon w hom Aldo M ieli largely depended for his comm ents on Arabic algebra (La, S cience A rabe, Leiden 1939) is of the opinion that al-K hw arizm l’s algebra w as derived from some ancient Babylonian or Iranian source (“The Origin and developm ent of th e quadratic equations in Babylonian, Greek and early Arabic algebra,” O risis, III, pp. 405— 557).
6 G. Sarton, Introduction..., II, p. 617.
5 8 S. N. Sen
for the la ter m istaken medieval view th a t th e A rabs w ere the in ventors of the science of calculation.
A nother date often taken to be the starting point of European m athem atical renaissance is 1202 in which year appeared Leonardo Pisano’s arithm etical classic Liber abaci, containing probably th e first com pjete exposition in L atin of H indu and M uslim arithm etic and of decimal place-value num eration. Through another w ork Flos Leonardo also initiated, in anticipation of Bachet de Meziriac (1581— 1638), the study of interg ral solutions of indeterm inate equations of the first and the second degree, in which the H indus had excelled from the time of A ryabhata (c. A.D. 499) and B rahm agupta (c. 628) and of w hich the origins have been traced to the Sulba-siitras. 8
The process of transm ission of H indu arithm etic to Latin Europe through Arabic translations and sum m aries is closely associated w ith the transm ission of H indu astronom y, of which m athem atics formed an integral part. Y et the im portance of the la tte r was relegated to the background due largely to this system being superseded by th e more sophisticated methods of Ptolemy. N evertheless the initial and early source of inspiration of Arab astronom ical renaissance never died out com pletely and received a new lease of life in Spain at a time w hen th e E astern A rabs w ere revising th e ir astronom ical tables afte r the
Almagest. Kennedy, in his survey of Islamic astronom ical tables, 9 has
listed 21 Z ijes (including 5 in the supplem entary list), whose com puta tions were either based on or influenced more or less by the Indian
Siddhantas.
The starting point was the p reparation of the Arabic version of
A z-Z ij as-Sindhind (c. 770) of one of the Indian Siddhantas, possibly
the Brahm asphuta Siddhanta of B rahm agupta. This work, through revisions and refinem ents by subsequent authors such as al-Fazari, Y a-’qub ibn Tariq, al-AdamT, al-Quasim, al-K hw arim zI, al-H asan ibn Misbah, an-N airizi, al-M ajritl, as-S affar and others exerted a profound influence, first among the Eastern A rabs and subsequently among the W estern Arabs of Spain. Maslama al-M ajritl (fl. 1000) of Cordova edited al-K hw arizm l’s tables which, as already mentioned, have su r vived in the Latin translation of A delard of Bath. 10 The Zij compiled by Abu al-Qasim Asbagh... ibn as-Sam h of G ranada (c. 1010) and the collection of tables M ukhtasar a z-Z ij by Ahmad bin ’Abdallah... ibn
8 S. N. Sen, “Study of Indeterm inate A nalysis in A ncient India,” Proceedings
of the Ten th International Congress of H isto ry of Science, Herm an 1964, pp. 493—
497.
9 E. S. Kennedy, “A Survey of Islam ic Astronom ical Tables,” T ransactions of
the A m erican Philosophical S ociety, 46, 1956, pp. 123—177.
10 O. Neugebaur, “The Astronomical Tables of Al-K hw arizm I—T ranslation w ith com m entaries of the Latin version edited by H. Suter supplem ented by Corpus Christi College MS. 283,” H isto risk -filo so fisk e sk rifte r u d g ivet af Det
as-S affar of Cordova (c. 1100) were based on the Sindhind. Some of the A rab astronom ers like ibn Yunis and al-B attani who chose to compile astronom ical Zijes afte r Ptolem y had also access to Sindhind Z ijes and occasionally used H indu param eters.
Some of the special features of H indu astronom y w hich w ere in this w ay incorporated in Islamic Zijes are th e zero m eridian of Uj-
ja yim w hich assumed the nam e A rin in Arabic, the era of K aliyuga
(February 17, 3102 B.C.) which became the “era of Flood” the H indu planetary theory, the tables of sines (R = 150) and the tables of solar declinations (E = 240°), and the m ethods of spherical trigonom etry. The sine and the solar declination tables given by al-Z arkall (c. 1050) in the Toledan tables, for example, in w hich the norm R = 150 is used, correspond to those m et w ith in B rahm agupta’s Khandakhadyaka. 11 In 1951, Lynn Thorndike published and translated an anonym ous fifteenth cen tu ry Latin MS Ashmole 191 II, in w hich com putations w ere m ade for th e geographical latitu de of N ew m inster, England for the year 1428. The study of the astronom ical param eters and tables given in the unsuspecting MS has revealed th e characteristic featu res of H indu astronom y and another interesting instance of transm ission as late as th e fifteenth century possibly through Arabic interm ediaries n . The N ew m inster MS begins the era of Flood from th e year 3102 B.C. in keeping w ith the beginning of the K aliyuga. To find the geographical latitudes, the typical H indu m ethod of determ ining the length of the equinoctial noon-shadow has been given. The sine functions have been used instead of the G reek chord, and the values have Taeen tabulated for R = 150, a norm used in the K handakhadyaka and in the Toledan
tables.
The m eeting of the two kinds of trigonom etrical functions, the Hindu sine of the Suryasiddhanta and the Khandakhadyaka and the chord of the Alm agest, in the astronom ical Zijes no doubt stim ulated in terest in this new m athem atical technique which w as fully exploited by the astronom ers and translators engaged in compiling the A lfonsine
tables. Continuance of this interest by Levi ben Gerson and Richard
W allingford in the 14th cen tu ry paved the w ay for the great trigono m etrical work of Regiomontanus, De triangulis omnimodis (1476).
In such circuitous and unexpected w ays elem ents of Indian a rith metic, algebra, trigonom etry and methods of astronom ical com putations did find a place in th e great revival of learning in L atin Europe from the th irteen th to th e fifteenth century, upon w hich the Renaissance w as u ltim ately based.
11 O. Neugebaur, “The Transmission of P lanetary Theories in A ncient and M edieval A stronom y,” S crip ta M athem atica, 22, 1956, pp. 165— 192.
,J O. Neugebaur and O laf Schmidt, “H indu Astronom y at N ew m inster in 1428,” Annals of Science, 8, 1952, pp. 221—228.
