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

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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 in part, and the European islands tions required. Not only does one need to of the . It was during know well mathematics and astronomy, this period that Europe was dark, learning but also , 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 , volume I, a work of great value to the student of the . 224

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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 , 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 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 (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 (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 and a table of sines and Muslim genius. Furthermore, just as, for versed sines of between 0? and 90? many centuries of antiquity, Greek was by intervals of 225' (kramajy?). The 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,

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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 (ad+bc) when a, d, and s are known; and the sum (ad+bc) (ac+bd) =-. ming of an arithmetic progression after y2 (ab+cd) the pth term. These are expressed now by means of the formulas: Another proposition, "Brahmagupta's = trapezium," states that if a2+b2 c2 and d-2a?V(2a-d)8ds = x2+y2 z2, then az, cy, bzy cx form a cyclic whose diagonals are at right angles. His work in combinations and (2) permutations ismuch like that now offered s=n^a+^^+p^dJ. in a first course, but is not quite complete. To him also were due other startling truths Brahmagupta used three values of : for less mathematical in character, among rough work, 3; for "neat" work, VT?] and for which was the theory that the apparent close accuracy, the finer value rotation of the heavens is due to the rota given by ?rvabhaja. tion of the earth about its axis. By the close of the period of Brahmagup ta Hindu Var?hamihira, astronomer-poet and mathematics had developed to such an extent that contemporary of ?ryabhat;a, contributed its influence reached out far toward the equivalent of trigonometric facts and both the east and the west. formulas as follows : Two Chinese authors are of special value as witnesses of the influence of Hindu sin 30? = cos 60? = 1/2; /1-1/4; mathematics in . (1) The first is Ch'?-t'an a as ? Hsi-ta, Hindu-Chinese /l cos ?= trologer of the first half of the eighth cen sin A/ 2 V 2 tury, whose work gives a detailed account of a number of ancient systems of chro = sin2 z+versin2 4 sin2?; nology, the most important of which, from 2 our point of view, is the Hindu system, sin2 2x [1-sin (90?-2x)]2 and in the explanation of which is implied sin2 x =-1-. 4 4 the Hindu decimal notation and rules. Thus we think this must have been the Then we have a leap of about a century very latest date of the introduction of the when Brahmagupta (c. 628), one of India's Hindu numerals and hence other Hindu greatest scientists, and the leading scien mathematics into China. Much more tist of his time of all races, made his study probably they were introduced earlier, of and indeterminate determinate equa about the second half of the sixth century, tions of first and second for , cyclic the catalog of the Sui dynasty (589 and combinatorial analysis. 618) lists many books devoted to Hindu solved the for roots He quadratic positive "mathematics and astronomy. Then, too, completely, and the Pellian equation it was about this time that Buddhism en in was nx2+l=y2 part (it finished by tered China from India. (2) The second If and are Bh?skara, 1150). a, b, c, d, x, y author, I-hsing (683-727), or Chang Sui

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(the first being his religious name) was Now let us set our clock back at the an astronomer of note who undertook, by second half of the 8th century. Here we order of the emperor, an investigation of find that practically all the work done in chronological and arithmetical systems of mathematics was done by Arabs. This is India, but he failed to finish the work on the beginning of a long period extending a account of premature death. to the close of the 11th century, during The westward reach of Hindu mathe the whole of which there was an over matics is equally certain. The Syrian whelming superiority of Muslim culture. philosopher and scientist, Severus Seb?kht Stimulated from the east by the Hindus (fi. 660), who studied also Greek philos and from the west by the eastward trans ophy and astronomy, was the first to men mission of , the Arabs tion the Hindu numerals outside of India, began a remarkable and altogether too and expressed his full appreciation of little emphasized flourish of activity. At Hindu learning in these words: "I will first there were mainly students and trans omit all discussion of the science of the lators of the five Hindu siddh?ntas and same as Hindus, a people not the the ohter works. Chief of these were al-Faz?r?, Syrians; their subtle discoveries in this the elder, and his son. Then with the 9th science of astronomy, discoveries that century began a series of very important are more ingenious than those of the steps forward, especially in the field of Greeks and the Babylonians; their valu trigonometry and the construction of able methods of calculation; and their astronomical tables; but there was also an computing that surpasses description. I imposing group of geometers, arithme wish only to say that this computation is ticians, algebraists and translators of done by means of nine signs. If those who Greek works. believe, because they speak Greek, that The cause of science was greatly en they have reached the limits of science hanced by the caliph al-Ma'm?n (813 should know these things they would be 833) who, although religiously exceedingly convinced that there are also others who intolerant, was one of the world's greatest know something." (Quoted from Smith: patrons of science. He collected all the History ofMathematics, v. I, p. 167.) Greek manuscripts he could, even sending But few Hindu mathematicians after a special mission into Armenia for that the day of Brahmagupta stand out prom purpose, then ordered the translation of inently. One of uncertain date but prob these into Arabic. He built two observa ably of the 9th century is Mah?v?ra, tories, had made (probably by al-Khw?riz author of the Gattita-S?ra-Sangraha, m?) a large map of the world which was which deals with arithmetic, including a much improved revision of 's, geometric progressions, the relation be organized at Bagdad a scientific academy, tween the sides of a rational sided right and stocked a library which was the finest ? (2ran, m2+n2, ra2 2), and the since the Alexandrian (3d century B.C.). solution of several types of equations in He then invited many of the world's volving the unknown and its square root. greatest scientists to his court. Among Two hundred years later, about 1030, them was al-Khw?rizm? (d.c. 850) who another Ganita-S?ra (compendium of wrote very important works on arithmetic calculation) was produced by Sr?dhara, and algebra and widely used astronomic but was quite elementary. However, he and of sines and tan wrote a work (now lost) on quadratic gents. He revised Ptolemy's geography, equations in which, according to the syncretized Hindu and Greek knowledge eminent Bh?skara of the 12th century, and is recognized by authorities as having was found our present formula for the influenced mathematical thought more quadratic solution. than any other mediaeval writer.

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A very notable astronomer of that were among the earliest to use the gar period was al-Fargh?n? (fi. 860) who dener's method of the construction of the wrote the first comprehensive treatise in ellipse (use of a string and two pins) and Arabic on astronomy, which was in wide discovered a trammel based upon the use until the close of the 15th century, conchoid for trisecting angles. Various and was translated into Latin and Hebrew problems of mechanics and geometry thus influencing European astronomy interested them most. greatly before . Perhaps chief among the many trans Among the notes and extracts left us lators is to be mentioned al-Hajj?j (c. from IJabash al-H?sib (c. 770-c. 870) we 825) who first (under H?r?n al-Rash?d) find record of the determination of time by translated 's Elements into Arabic the altitude of the and complete trigo and improved it later (under al-Ma'm?n) nometric tables of tangents, cotangents in a second translation. He also was an and cosecants which are at present pre early translator of Ptolemy's Syntaxis, served in the Staatsbibliothek in Berlin. which he called aUmdjisti (the greatest) In connection with the problem of finding from which term the name was the sun's altitude, the cotangent function derived. He was preceded, however, in = = arose. If x the sun's altitude, h the this work by the Jewish Arab al-Tabar? = height of a stick, and l the length of its of the same period. By the middle of the shadow, 9th century, then, these men and others (cos - x\ of their day had made accessible to the ?I sin xl Arabs the most important works of the Greeks and the earlier Hindus, had ex and al-H?sib constructed a table of values tended the sum total of astronomy and = of h for x l, 2, 3, . . . degrees, from trigonometry, and had given tremendous which either or h could be read if the impetus to independent investigation, other were known. According to Schoy, which bore its fruit in the century to fol the Berlin MS. also contains the equiva low. lent of There was, however, much translating tan cos y still to be done. Al-M?h?n? (fl. 860) sin x =-, sin wrote commentaries on at least the first, fifth and tenth books of the Elements, and where y is the declination and the obliq on ' Sphere and Cylinder, and uity of the ecliptic. studied also considerably the Spherics of One of the world's best general scien Menelaos which led him to an equation of = tists and the greatest philosopher of the the form xz+a2b ax2, with which he Arabs was al-Kind? (fl. 813-842), prolific wrestled long enough to cause his suc author of between 250 and 275 works on cessors in the field to refer to it as "Al astronomy, geography, mathematics and M?h?n?'s equation." He never solved it. physics. He understood thoroughly the Al-Him?? (d.c. 883) translated the first Greek mathematical works and influenced four books of the Conies of Apollonius. widely the early European scientists Al-Nair?z? (d.c. 922) wrote (both com among whom were Girolamo Cardano, of mentaries lost) most authoritatively on fame, and Roger Bacon. the Quadripartitum and Almagest of The Ban? M?s? (Sons of Moses, also Ptolemy as well as on the Elements. Th? known as the Three Brothers) were bit ibn Qurra (c. 826-901) founded within wealthy scientists and patrons of science his own family a school of translators and of this period. Through their efforts enlisted outside scholars to aid him in many Greek MSS. were collected, studied, producing translations of nearly all the translated and thus preserved for us. They Greek mathematical classics even includ

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ing the commentary of Eutocius. Without translated into Latin by Plato of Tivoli in naming others, suffice it to say that by the 12th century and into Spanish the the beginning of the 10th century there 13th, and exerted a tremendous influence was probably not a single important work for 500 years. of the Golden Age of Greece that had not Considering now the first half of the been translated and mastered by the 10th century, we note again that almost Arabs. all the original work was done by Arabs As to the results of independent inves in Arabic, but with the marked difference tigators of this period, the following must that there is a decided decline in activity. be mentioned. The best and most complete The development of mathematics may be study among the Arabs on the spherical compared with rainfall, which, after un astrolabe was produced by Al-Nair?z?. certain periods of drouth or average in The school of Th?bit ibn Qurra wrote tensity, may burst forth in torrents. Some about 50 works of independent research, sort of unknown law of rhythm seems to and 150 (about) translations. Many of hold rather than a law of uniform advance. these works are still extant. The most The outstanding men of the period are valuable ones are those on theory of num Muslim. Ibrahim ibn Sin?n (d. 946) wrote bers and the study of parabolas and parab numerous commentaries on astronomy oloids. A sample of the former on amica and geometry, but has received high ble numbers shows the fine reasoning recognition only since 1918 when Schoy's employed: 2npq and 2nr are amicable translation of his Quadrature of theParab numbers if p, q, and r are prime to each ola revealed the fact that his method was = = other and if p (3)(2*)-l, g (3)(2?-1) superior to and simpler than that of = -1, and r (9X22*-1) -1. The latter con Archimedes. Ab? K?mil (c. 925) was an tains ingenious developments of Archi able algebraist, improved essentially the medes' Method. algebra of al-Khw?rizm? by some gen In many respects the outstanding schol eralizations, development of algebraic ar of the century was al-Batt?n? (d. 929), multiplication and division, and opera Muslim's greatest astronomer, in whose tions with radicals. He also gave an alge principal work on astronomy are found braic treatment of regular inscribed poly numerous important facts. He gave the gons. inclination of the ecliptic correct to 6"; Toward the close of the century a de calculated the precession at 54.5'' a year; cided renewal of creative work is to be did not believe in the trepidation of the observed. Far more names of famous equinoxes, which Copernicus still believed mathematicians could be mentioned than many years later. In trigonometry, to space permits. But Ab?-1-Fath (fl. 982), which his fifth chapter is devoted, he gave al-Kh?zin (fl. 950), al-K?h? (fl. 988), al the equivalent of our formulas: Sijzi (fl. 1000), al-S?f? (fl. 975), al-Khu 1 1 jand? (fl. 990) and'Ab?-1-Waf?' (fl. 990) - - sin x = cos = cannot be omitted. esc sec In Florence there is an untranslated sin cos commentary on the first five books of x= X tan cot C who cosx sin Apollonius' orties, by Ab?-1-Fath, also wrote a translation of the first seven = = esc x Vl+cot2 x; see? \/l+tan2 x; books of the Conies. His work on books V-VII is considered be tan highly important sin x= cause the original Greek is no longer ex sec tant. Al-Kh?zin solved the cubic equation = cos a cos b cos c+sin b sin c cos A; and of al-M?h?n? (mentioned before in this pa the sine law (doubtful). This work was per). Al-K?h? is chiefly known for his

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 230 THE MATHEMATICS TEACHER work on the solution of higher degree of considerably more importance in the equations by means of the intersections of history of trigonometry than we have two conies. His problem on trisecting an suspected. I quote here (from French) from Noteworthy among the contributions Woepcke's UAlgebre d'Omar Alkhayyami of Ibn Y?nus is the introduction of the (1851), page 118: prosthaphaeretical (sum and difference) Let the given angle be CBE. Take on formulas of trigonometry which were so E produced points A and D, and on the useful before the time of logarithms. Al = other side a point C so that (1) AD DC, Karkh?, whose work, al-Fakhrl, Woepcke (2) AB:BC::BC:BD. Draw BP parallel has given us in French, was an algebraist = to DC. Then the angle CBP 1/3 angle of the first rank. He gave a splendid treat CBE. ment of the solution of Diophantine equa The geometry of the figure shows tions, equations of quadratic form, opera clearly that the angle is trisected, but the tions with radicals, and summation of construction involves the solution of a integro-geometric series. In connection cubic equation. with series he gives such results (not sym bolically, of course) as

/ \ 2 and ,)? Of the works of al-Sijz?, fourteen are -( Ibn S?n? was a now preserved in Cairo, Leyden, Paris, principally philosopher and the British Museum. All deal with and hence emphasized that phase of al-Husain wrote one of the conic sections, trisection problems and mathematics; few Arabic treatises on the construction the resulting cubic equations. These works of with rational and rank him as an outstanding Arabic geom right sides; eter. al-Nasaw?, an able arithmetician, ex extraction of and cube The greatest mathematician of the plained square roots a method similar to our century, however, was Ab?-l-Waf?,, who by very and furthermore decimal wrote 15 or 20 works (mostly extant) on own, anticipated fractions in the manner indicated in the geometry, geometrical solutions of special fourth degree equations, and trigonome equations try, to which he added a number of for ^_ 1 ,_ 412 of functions. VT7 =-V170000 =-, mulas and the line values the 100 100 The 11th century continues in the first half to show remarkable activity, with an but he changed them to sexagesimals for imposing array of first order mathemati the final form of his answer. cians, the principal ones being Al-B?r?n? There is a distinct decline toward the (973-1048), Ibn Y?nus (d. 1009), al close of the 11th century with a notable Karkh? (d. 1025), Ibn S?n? (980-1037), decrease in the number ofmathematicians al-Husain (?) and al-Nasaw? (c. 1025). Of of the first rank. Of these I mention but paramount importance is the work of Al one, Omar Khayyan, who, although living B?r?n?. Two of his writings are of great in a period of decline, was one of the great mathematical significance. We have est of the Middle Ages. His chief distinc known the first,A Summary ofMathemat tion results from his admirable work on ics, for some time, but the other, Al Algebra, in which he classified equations Q?n?n al-Mas?d?, has recently been given by the number and degree of terms, us in German, and proves its author to be treated 13 types of cubic equations, and

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referred to the general expansion of the on the quadrivium (arithmetic, music, ge binomial with positive integral coefficients ometry and astronomy), which, though which he treated in another work now comparatively poor, were so widely used unknown. His other mathematical writ in schools that they had tremendous in ings dealt with the assumptions of Euclid fluence. Anthemios (d. 534) is interesting and a very accurate reform of the calendar. for his history of conic sections and his use Thus closes the marvelous periods of of the focus and directrix in the construc scientific activity of the Hindus and Mus tion of the parabola. Eutocius (b. 480?) is lims in the field of mathematics. The important for his commentaries on the Hindu period began shortly after the time works of Archimedes and Apollonius. Bede of Proclus (d. 485), last straggler of the (673-735) deserves mention because our great Greek period. After the Hindu peak information on finger reckoning is almost comes the tremendous Muslim flourish. entirely dependent on his work. Alcuin or Suter in his work, Die Mathematiker und Albinus (735-804) wrote a work on puzzle Astronomen der Araber und ihre Werke, problems which furnished material for lists the works of 528 Arabic scholars who textbook writers for ten centuries. Gerbert were active in mathematics from 750 to (Pope Sylvester II) (999) was great and in 1600. Certainly 400 of these came within fluential as compared with other Euro the period of the "Dark Ages." We have pean writers of his day and doubtless did learned much about them, but there is much to popularize the Hindu-Arabic nu still much to be done. Tropfke tells us merals. But these men were all small as that "ueberreiche Schaetze" still lie un compared with the Muslim giants. translated in the large libraries of Europe. Let me repeat what I said at the begin During this same period in Europe, but ning: Europe was dark, but India and the few names deserve even feeble mention as Muslim world were not. To quote Dr. writers on mathematics. There was no George Sarton, "the 'Dark Ages' were creative work. Boetius (d. 524) wrote texts never so dark as our ignorance of them."

Does the harmony which human intelligence thinks it discovers in Nature exist apart from such intelligence? Assuredly no. A reality completely independent of the spirit that conceives it, sees it, or feels it, is an impossibility. A world so external as that, even if it existed, would be forever inaccessible to us. What we call objective reality is, strictly speaking, that which is common to several thinking beings and might be com mon to all; this common part, we shall see, can only be the harmony expressed by mathematical laws.?Poincar?. The Value of Science.

There is no subject in the entire high school curriculum which by its very nature lends itselfmore admirably to a realization of some educational objectives as does the teach ing of mathematics. I know of no other subject, save possibly foreign language, where a close day by day application of the student is so absolutely essential to success. When can the boy or girl be more impressed with the importance of doing his job day to own success a by day, and meeting his obligations that lead his than in class of mathematics? He is constantly confronted with the necessity of being alert, critical and observant. He learns to develop habits of inspection and inquiry concerning printed and spoken statements, habits worth cultivating by everyone today. He learns to take nothing for granted except certain hypotheses about which there can be no argument. He waits for all the evidence to come into the picture before drawing conclusions. He or is not finally accepts no opinion, theory notion that backed by facts. Empirical rea as soning is soon detected and labeled such. Geometry is a powerful training in logic. When Lincoln had a difficult case to try in court he resorted to Euclid as the most helpful aid to jurisprudence. The practical value of geometry was much larger to Lincoln than itsmere application to the arts and industries. From an article by S. W. Lavengood, Principal, Pershing School, Tulsa, Oklahoma in the Georgia Educational Journal, Feb. 1936, p. 27 on "Contributions of the Teaching of Mathematics."

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