Chinese and Egyptians, they did not systematically develop a subject to any considerable extent. Their schools may be

taken to have lasted in all for about 650 years, and if the work produced be compared with that of Greek or modern European writers it is, as a whole, second-rate both in quantity and quality.

164

CHAPTER X.

THE INTRODUCTION OF ARAB WORKS INTO EUROPE.

CIRC. 1150-1450.

IN the last chapter but one I discussed the development of European mathematics to a date which corresponds roughly with the end of the "dark ages"; and in the last chapter I traced the history of the mathematics of the Indians and Arabs to the same date. The mathematics of the two or three centuries that follow and are treated in this chapter are characterised by the introduction of the Arab mathematical text-books and of Greek books derived from Arab sources, and the assimilation of the new ideas thus presented.

It was, however, from Spain, and not from Arabia, that a knowledge of eastern mathematics first came into western Europe. The Moors had established their rule in Spain in 747, and by the tenth or eleventh century had attained a high degree of civilisation. Though their political relations with the caliphs at Bagdad were somewhat unfriendly, they gave a ready welcome to the works of the great Arab mathematicians. In this way the Arab translations of the writings of Euclid, Archimedes, Apollonius, Ptolemy, and perhaps of other Greek authors, together with the works of the Arabian algebraists, were read and commented on at the three great Moorish schools of Granada, Cordova, and Seville. It seems probable that these works indicate the full extent of Moorish learning, but, as

all knowledge was jealously guarded from Christians, it is impossible to speak with certainty either on this point or on that of the time when the Arab books were first introduced into Spain.

The eleventh century. The earliest Moorish writer of distinction of whom I find mention is Geber ibn Aphla, who was born at Seville and died towards the latter part of the eleventh century at Cordova. He wrote on astronomy and trigonometry, and was acquainted with the theorem that the sines of the angles of a spherical triangle are proportional to the sines of the opposite sides.1

Arzachel.2 Another Arab of about the same date was Arzachel, who was living at Toledo in 1080. He suggested that the planets moved in ellipses, but his contemporaries with scientific intolerance declined to argue about a statement which was contrary to Ptolemy's conclusions in the Almagest.

3

The twelfth century. During the course of the twelfth century copies of the books used in Spain were obtained in western Christendom. The first step towards procuring a knowledge of Arab and Moorish science was taken by an English monk, Adelhard of Bath, who, under the disguise of a Mohammedan student, attended some lectures at Cordova about 1120 and obtained a copy of Euclid's Elements. This copy, translated into Latin, was the foundation of all the editions known in Europe till 1533, when the Greek text was recovered. How rapidly a knowledge of the work spread we may judge when we recollect that before the end of the thirteenth century Roger Bacon was familiar with it, while before the close of the fourteenth century the first five books formed part of the regular curriculum at many universities. The enunciations of Euclid seem to have been known before 1 Geber's works were translated into Latin by Gerard, and published at Nuremberg in 1533.

2 See a memoir by M. Steinschneider in Boncompagni's Bulletino di Bibliografia, 1887, vol xx.

On the influence of Adelhard and Ben Ezra, see the "Abhandlungen zur Geschichte der Mathematik" in the Zeitschrift für Mathematik, vol. xxv,

+

Adelhard's time, and possibly as early as the year 1000, though copies were rare. Adelhard also issued a text-book on the use

of the abacus.

Ben Ezra.1 During the same century other translations of the Arab text-books or commentaries on them were obtained. Amongst those who were most influential in introducing Moorish learning into Europe I may mention Abraham Ben Ezra. Ben Ezra was born at Toledo in 1097, and died at Rome in 1167. He was one of the most distinguished Jewish rabbis who had settled in Spain, where it must be recollected that they were tolerated and even protected by the Moors on account of their medical skill. Besides some astronomical tables and an astrology, Ben Ezra wrote an arithmetic; 2 in this he explains the Arab system of numeration with nine symbols and a zero, gives the fundamental processes of arithmetic, and explains the rule of three.

Gerard.3 Another European who was induced by the reputation of the Arab schools to go to Toledo was Gerard, who was born at Cremona in 1114 and died in 1187. He translated the Arab edition of the Almagest, the works of Alhazen, and the works of Alfarabius, whose name is otherwise unknown to us: it is believed that the Arabic numerals were used in this translation, made in 1136, of Ptolemy's work. Gerard also wrote a short treatise on algorism which exists in manuscript in the Bodleian Library at Oxford. He was acquainted with one of the Arab editions of Euclid's Elements, which he translated into Latin.

John Hispalensis. Among the contemporaries of Gerard was John Hispalensis of Seville, originally a rabbi, but converted to Christianity and baptized under the name given above. He made translations of several Arab and Moorish works, and also wrote an algorism which contains the earliest examples of the

1 See footnote 3 on p. 165.

2 An analysis of it was published by O. Terquem in Liouville's Journal for 1841.

3 See Boncompagni's Della vita e delle opere di Gherardo Cremonese, Rome, 1851.

extraction of the square roots of numbers by the aid of the decimal notation.

The thirteenth century. During the thirteenth century there was a revival of learning throughout Europe, but the new learning was, I believe, confined to a very limited class. The early years of this century are memorable for the development of several universities, and for the appearance of three remarkable mathematicians-Leonardo of Pisa, Jordanus, and Roger Bacon, the Franciscan monk of Oxford. Henceforward it is to Europeans that we have to look for the development of mathematics, but until the invention of printing the knowledge was confined to a very limited class.

Leonardo.1 Leonardo Fibonacci (i.e. filius Bonaccii) generally known as Leonardo of Pisa, was born at Pisa about 1175. His father Bonacci was a merchant, and was sent by his fellowtownsmen to control the custom-house at Bugia in Barbary; there Leonardo was educated, and he thus became acquainted with the Arabic or decimal system of numeration, as also with Alkarismi's work on Algebra, which was described in the last chapter. It would seem that Leonardo was entrusted with some duties, in connection with the custom-house, which required him to travel. He returned to Italy about 1200, and in 1202 published a work called Algebra et almuchabala (the title being taken from Alkarismi's work), but generally known as the Liber Abaci. He there explains the Arabic system of numeration, and remarks on its great advantages over the Roman system. then gives an account of algebra, and points out the convenience of using geometry to get rigid demonstrations of algebraical formulae. He shews how to solve simple equations, solves a few quadratic equations, and states some methods for the solution of indeterminate equations; these rules are illustrated by problems on numbers. The algebra is rhetorical, but in one case letters

He

1 See the Leben und Schriften Leonardos da Pisa, by J. Giesing, Döbeln, 1886; Cantor, chaps. xli, xlii; and an article by V. Lazzarini in the Bollettino di Bibliografia e Storia, Rome, 1904, vol. vii. Most of Leonardo's writings were edited and published by B. Boncompagni, Rome, vol. i, 1857, and vol. ii, 1862.