The Status of Mathematics in India and Arabia During the "Dark Ages" of Europe Author(S): F

The Status of Mathematics in India and Arabia during the "Dark Ages" of Europe Author(s): F. W. Kokomoor Source: The Mathematics Teacher, Vol. 29, No. 5 (May 1936), pp. 224-231 Published by: National Council of Teachers of Mathematics Stable URL: http://www.jstor.org/stable/27951935 . Accessed: 21/10/2014 14:00 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to The Mathematics Teacher. http://www.jstor.org This content downloaded from 146.186.124.59 on Tue, 21 Oct 2014 14:00:31 PM All use subject to JSTOR Terms and Conditions The Status o?Mathematics in India andArabia duringthe "DarkAges o? Europe By F. W. Kokomoor University of Florida Political and economic world histori terial left us and so much of it is worthless, ans have found it convenient to divide the due largely to the overabundance of study time from the fall of Rome to the discov devoted to Scholasticism, that the search ery of America into two periods, and to for gems among the rubbish has seemed designate the first of these by the term hardly worth the effort. (3) Then, too, the "Dark Ages." One work accounts for this difficulty of accessibility involved is enor name by "the inrush into Europe of the mous. Before many Greek and Hindu barbarians and the almost total eclipse of works could be assimilated in the main the light of classical culture." The period scientific current in theWest, they had to covers, roughly, the time from 500 to be translated first into Syriac, then Arabic, 1000 A.D. Part of these "barbarians" then Latin and finally into our own lan came down from the north and the rest guage. Thus the completeness and the ac attacked from the south, the latter bound curacy of these transmissions can only be together politically and religiously by the determined by painstaking investigations great, although probably totally illiterate, of historians of science. (4) Furthermore, leader Mohammed into a vast dominion but few persons have ever been qualified that at one time or another covered all of to advance our knowledge of Muslim eastern Asia, northern Africa, Spain, mathematics, due to the rare qualifica France in part, and the European islands tions required. Not only does one need to of the Mediterranean Sea. It was during know well mathematics and astronomy, this period that Europe was dark, learning but also Sanskrit, Syriac, Arabic and at low ebb, and the development ofmathe Persian, and, in addition, one needs to matics almost negligible. The world as a have a thorough training in paleography whole was not dark, and as applied to gen and a keen historical sense. This rare com eral history the expression "Dark Ages" bination, together with a lifetime of cease is a gross misnomer. Throughout the en less work, is the price that must be paid to tire period there was considerable intellec increase our understanding of oriental tual (including mathematical) activity mathematics. among the Hindus and, beginning about Modern mathematics is easily trans 750, there developed many centers of mitted; the process is simple. Articles ap Muslim civilization which rose to the very pearing in any scientific journal are an-% peak of mathematical productivity. nounced in other principal ones, and hence A number of facts combine to account it is easily possible for a worker in any for our heretofore slight emphasis upon field to be fully informed on what is being oriental mathematical science. (1) We done throughout the world regardless of have been so enamored by the story of the the language used in the original publica Golden Age of Greece and the Modern tion. So simple is it that the modern sci Period that the Orient failed until recently entist who lacks historical training can to divert our attention. (2) There is such hardly comprehend the difficulties in an overwhelming mass of mediaeval ma volved in the handing down of knowl * In the preparation of this article the author has drawn heavily from Dr. George Sarton's Introduction to the History of Science, volume I, a work of great value to the student of the history of mathematics. 224 This content downloaded from 146.186.124.59 on Tue, 21 Oct 2014 14:00:31 PM All use subject to JSTOR Terms and Conditions INDIA AND ARABIA DURING THE "DARK AGES 225 edge from the early ages to the present. prised of the Greek and Hindu sources of For our knowledge of Greek mathemat mathematical knowledge they were fired ics we owe an unpayably large debt to two with a contagious and effective enthusi men who devoted years to the production asm that led to numerous remarkable in of numerous accurate translations: J. L. vestigations in mathematics prosecuted Heiberg of Copenhagen, and T. L. Heath, from a number of cultural centers through a great English scholar in both mathemat out the Muslim world, and that did not ics and Greek. Unfortunately we have abate until the close of the 12th century neither a Heiberg nor a Heath to enlighten when they had made a permanent im us inHindu and Muslim mathematics. In pression on mathematics as a whole. our knowledge of the former we are per What I wish to do is to give as compre haps least fortunate of all. Cantor's three hensive a survey of Hindu and Muslim chapters are quite satisfying on the works mathematics as space permits, pointing of Brahmagupta and Bh?skara, being out principal achievements of leaders in based upon Colebrooke's Algebra with the field, and indicating work still to be Arithmetic and Mensuration, from theSan done to make the history complete. scrit of Brahmegupta and Bhascara, but Our attention is first called to Hindu later historians place much dependence mathematics in one of the five Hindu upon the interpretations of G. R. Kay e scientific works on astronomy called who was formany years a resident of India Siddh?ntas, which were theoretical as as a high government official, and whose opposed to karanas which were practical. work on Indian mathematics (1915) is Its date is very uncertain, but is placed in shown to be erroneous inmany respects by the first half of the fifth century. The competent scholars of India today, such as S?rya-Siddh?nta, the only one we have in Saradakanta Ganguli. One of the ablest full, is composed of fourteen chapters of scholars in the field ofMuslim mathemat epic stanzas (slokas) which show decided ics was Carl Schoy (1877-1925), who, be knowledge of Greek astronomy but also tween the years 1911 and 1925, contrib much Hindu originality, especially the uted many valuable papers and books con consistent use throughout of sines (jy?) taining critical translations of Muslim instead of chords, and the firstmention of mathematics and astronomy. versed sines (utramadjy?). Recent scholars such as Schoy have Even more important is the Pauli?a shown us that, just as the greatest achieve Siddh?nta which we have only indirectly ments of antiquity were due to Greek through the commentator Var?hamihira genius, so the greatest achievements of the (c. 505). It contains the foundation of Middle Ages were due to Hindu and trigonometry and a table of sines and Muslim genius. Furthermore, just as, for versed sines of angles between 0? and 90? many centuries of antiquity, Greek was by intervals of 225' (kramajy?). The sine the dominant progressive language of the and the arc of 225' were taken to be equal, learned, so Arabic was the progressive and sines of multiples of 225' were ob scientific language of mankind during the tained by a rule equivalent in our sym period of the Middle Ages. We have bolism to that the fall of ancient learned further = ? sin (n+1) x 2 sin nx sin (n? l)x science and the dampening of the scien tific spirit inEurope was far less due to the Olii A/- JU- MMV . of southern the , overrunning Europe by sin barbarians than it was to the passive in difference of the Romans themselves, and The next important Hindu advance is to the theological domination of a little due to ?ryabhaja (The Elder) who wrote later time. As soon as the Arabs were ap in 499 a work, ?ryabha?yam, of four parts, This content downloaded from 146.186.124.59 on Tue, 21 Oct 2014 14:00:31 PM All use subject to JSTOR Terms and Conditions 226 THE MATHEMATICS TEACHER the second of which?the Ganitap?da? the sides and diagonals of a cyclic quadri was a mathematical treatise of 32 stanzas lateral, s the half-perimeter, and the in verse, containing essentially the con area, his results can be expressed by the tinued fraction process of solution of equations : indeterminate equations of the first de an accurate value of = gree; amazingly 7r, K V(s-a) (s-b) (s-c) (s-d); namely 3 177/1250; the solution of the in the (ab+cd) (ac+bd) quadratic equation implied prob x2 =-; lem of finding of an arithmetic series (ad+bc) when a, d, and s are known; and the sum (ad+bc) (ac+bd) =-.

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