lectually qualified. This power was assumed on the precedent of a case which arose in Paris in 1426, when the university declined to confer a degree on a student-a Slavonian, one Paul Nicholas-who had performed the necessary exercises in a very indifferent manner: he took legal proceedings to compel the university to grant the degree, but their right to withhold it was established. Nicholas accordingly has the distinction of being the first student who under modern conditions was "plucked."

Athough science and mathematics were recognised as the standard subjects of study for a bachelor, it is probable that, until the renaissance, the majority of the students devoted most of their time to logic, philosophy, and theology. The subtleties of scholastic philosophy were dreary and barren, but it is only just to say that they provided a severe intellectual training.

We have now arrived at a time when the results of Arab and Greek science became known in Europe. The history of Greek mathematics has been already discussed; I must now temporarily leave the subject of medieval mathematics, and trace the development of the Arabian schools to the same date; and I must then explain how the schoolmen became acquainted with the Arab and Greek text-books, and how their introduction affected the progress of European mathematics.

THE story of Arab mathematics is known to us in its general outlines, but we are as yet unable to speak with certainty on many of its details. It is, however, quite clear that while part of the early knowledge of the Arabs was derived from Greek sources, part was obtained from Hindoo works; and that it was on those foundations that Arab science was built. I will begin by considering in turn the extent of mathematical knowledge derived from these sources.

Extent of Mathematics obtained from Greek Sources.

According to their traditions, in themselves very probable, the scientific knowledge of the Arabs was at first derived from the Greek doctors who attended the caliphs at Bagdad. It is

1 The subject is discussed at length by Cantor, chaps. xxxii-xxxv; by Hankel, pp. 172-293; by A. von Kremer in Kulturgeschichte des Orientes unter den Chalifen, Vienna, 1877; and by H. Suter in his "Die Mathematiker und Astronomen der Araber und ihre Werke," Zeitschrift für Mathematik und Physik, Abhandlungen zur Geschichte der Mathematik, Leipzig, vol. xlv, 1900. See also Matériaux pour servir à l'histoire comparée des sciences mathématiques chez les Grecs et les Orientaux, by L. A. Sédillot, Paris, 1845-9; and the following articles by Fr. Woepcke, Sur l'introduction de l'arithmétique Indienne en Occident, Rome, 1859; Sur l'histoire des sciences mathématiques chez les Orientaux, Paris, 1860; and Mémoire sur la propagation des chiffres Indiens, Paris, 1863.

said that when the Arab conquerors settled in towns they became subject to diseases which had been unknown to them in their life in the desert. The study of medicine was then confined mainly to Greeks and Jews, and many of these, encouraged by the caliphs, settled at Bagdad, Damascus, and other cities; their knowledge of all branches of learning was far more extensive and accurate than that of the Arabs, and the teaching of the young, as has often happened in similar cases, fell into their hands. The introduction of European science was rendered the more easy as various small Greek schools existed in the countries subject to the Arabs: there had for many years been one at Edessa among the Nestorian Christians, and there were others at Antioch, Emesa, and even at Damascus, which had preserved the traditions and some of the results of Greek learning.

The Arabs soon remarked that the Greeks rested their medical science on the works of Hippocrates, Aristotle, and Galen; and these books were translated into Arabic by order of the caliph Haroun Al Raschid about the year 800. The translation excited so much interest that his successor Al Mamun (813-833) sent a commission to Constantinople to obtain copies of as many scientific works as was possible, while an embassy for a similar purpose was also sent to India. At the same time a large staff of Syrian clerks was engaged, whose duty it was to translate the works so obtained into Arabic and Syriac. To disarm fanaticism these clerks were at first termed the caliph's doctors, but in 1851 they were formed into a college, and their most celebrated member, Honein ibn Ishak, was made its first president by the caliph Mutawakkil (847-861). Honein and his son Ishak ibn Honein revised the translations before they were finally issued. Neither of them knew much mathematics, and several blunders were made in the works issued on that subject, but another member of the college, Tabit ibn Korra, shortly published fresh editions which thereafter became the standard texts.

In this way before the end of the ninth century the Arabs

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obtained translations of the works of Euclid, Archimedes, Apollonius, Ptolemy, and others; and in some cases these editions are the only copies of the books now extant. It is curious, as indicating how completely Diophantus had dropped out of notice, that as far as we know the Arabs got no manuscript of his great work till 150 years later, by which time they were already acquainted with the idea of algebraic notation and processes.

Extent of Mathematics obtained from Hindoo Sources.

The Arabs had considerable commerce with India, and a knowledge of one or both of the two great original Hindoo works on algebra had been thus obtained in the caliphate of Al Mansur (754-775), though it was not until fifty or sixty years later that they attracted much attention. The algebra and arithmetic of the Arabs were largely founded on these treatises, and I therefore devote this section to the consideration of Hindoo mathematics.

The Hindoos, like the Chinese, have pretended that they are the most ancient people on the face of the earth, and that to them all sciences owe their creation. But it would appear from all recent investigations that these pretensions have no foundation; and in fact no science or useful art (except a rather fantastic architecture and sculpture) can be traced back to the inhabitants of the Indian peninsula prior to the Aryan invasion. This invasion seems to have taken place at some time in the latter half of the fifth century or in the sixth century, when a tribe of the Aryans entered India by the north-west frontier, and established themselves as rulers over a large part of the country. Their descendants, wherever they have kept their blood pure, may still be recognised by their superiority over the races they originally conquered; but as is the case with the modern Europeans, they found the climate trying and gradually degenerated. For the first two or three centuries they, however, retained their

intellectual vigour, and produced one or two writers of great ability.

Arya-Bhata. The first of these is Arya-Bhata, who was born at Patna in the year 476. He is frequently quoted by Brahmagupta, and in the opinion of many commentators he created algebraic analysis, though it has been suggested that he may have seen Diophantus's Arithmetic. The chief work of Arya-Bhata with which we are acquainted is his Aryabhathiya, which consists of mnemonic verses embodying the enunciations of various rules and propositions. There are no proofs, and the language is so obscure and concise that it long defied all efforts to translate it.1

The book is divided into four parts of these three are devoted to astronomy and the elements of spherical trigonometry; the remaining part contains the enunciations of thirtythree rules in arithmetic, algebra, and plane trigonometry. It is probable that Arya-Bhata, like Brahmagupta and Bhaskara, who are mentioned next, regarded himself as an astronomer, and studied mathematics only so far as it was useful to him in his astronomy.

In algebra Arya-Bhata gives the sum of the first, second, and third powers of the first n natural numbers; the general solution of a quadratic equation; and the solution in integers of certain indeterminate equations of the first degree. His solutions of numerical equations have been supposed to imply that he was acquainted with the decimal system of enumeration.

In trigonometry he gives a table of natural sines of the angles in the first quadrant, proceeding by multiples of 32°, defining a sine as the semichord of double the angle. Assuming that for the angle 33° the sine is equal to the circular measure, he takes for its value 225, i.e. the number of minutes in the

1 A Sanskrit text of the Aryabhathiya, edited by H. Kern, was published at Leyden in 1874; there is also an article on it by the same editor in the Journal of the Asiatic Society, London, 1863, vol. xx, pp. 371-387; a French translation by L. Rodet of that part which deals with algebra and trigonometry is given in the Journal Asiatique, 1879, Paris, series 7, vol. xiii, pp. 393