bridge; unfortunately no text-books' are mentioned. We can however form a reasonable estimate of the range of mathematical reading required, by looking at the statutes of the universities of Prague founded in 1350, of Vienna founded in 1364, and of Leipzig founded in 13893.

By the statutes of Prague3, dated 1384, candidates for the bachelor's degree were required to have read Holywood's treatise on the sphere, and candidates for the master's degree to be acquainted with the first six books of Euclid, optics, hydrostatics, the theory of the lever, and astronomy. Lectures were actually delivered on arithmetic, the art of reckoning with the fingers, and the algorism of integers; on almanacks, which probably meant elementary astrology; and on the Almagest, that is on Ptolemaic astronomy. There is however some reason for thinking that mathematics received there far more attention than was then usual at other universities.

At Vienna in 1389 the candidate for a master's degree was required* to have read five books of Euclid, common perspective, proportional parts, the measurement of superficies, and the Theory of the planets. The book last named is the treatise by Campanus, which was founded on that by Ptolemy. This was a fairly respectable mathematical standard, but I would remind the reader that there was no such thing as "plucking" in a mediæval university. The student had to keep an act or give a lecture on certain subjects, but whether he did it well or badly he got his degree, and it is probable that it was only the few students whose interests were mathematical who really mastered the subjects mentioned above.

1 See p. 81 of De l'organisation de l'enseignement...au moyen âge by C. Thurot, Paris, 1850.

2 The following account is taken from Die Geschichte der Mathematik, by H. Hankel, Leipzig, 1874.

3 See vol. I. pp. 49, 56, 77, 83, 92, 108, 126, of the Historia universitatis Pragensis, Prag, 1830.

4 See vol. I. p. 237 of the Statuta universitatis Wiennensis by V. Kollar, Vienna, 1839: quoted in vol. I. pp. 283 and 351 of the University of Cambridge, by J. Bass Mullinger, Cambridge, 1873.

At any rate no test of proficiency was imposed; and a few facts gleaned from the history of the next century tend to shew that the regulations about the study of the quadrivium were not seriously enforced. The lecture lists for the years 1437 and 1438 of the university of Leipzig (the statutes of which are almost identical with those of Prague as quoted above) are extant, and shew that the only lectures given there on mathematics in those years were confined to astrology. The records' of Bologna, Padua, and Pisa seem to imply that there also astrology was the only scientific subject taught in the fifteenth century, and even as late as 1598 the professor of mathematics at Pisa was required to lecture on the Quadripartitum, a spurious astrological work attributed to Ptolemy. According to the registers of the university of Oxford the only mathematical subjects read there between the years 1449 and 1463 were Ptolemy's astronomy (or some commentary on it) and the first two books of Euclid. Whether most students got as far as this is doubtful. It would seem, from an edition of Euclid published at Paris in 1536, that after 1452 candidates for the master's degree at that university had to take an oath that they had attended lectures on the first six books of Euclid.

The only Cambridge mathematicians of the fifteenth century of whom I can find any mention were Holbroke, Marshall, and Hodgkins. No details of their lives and works are known. John Holbroke, master of Peterhouse and chancellor of the university for the years 1428 and 1429, who died in 1437, is reputed to have been a distinguished astronomer and astrologer. Roger Marshall, who was a fellow of Pembroke, taught mathematics and medicine; he subsequently moved to London and became physician to Edward IV. John Hodgkins, a fellow of King's, who died in 1485 is mentioned as a celebrated mathematician.

1 See pp. 15, 20 of Die Geschichte der mathematischen Facultät in Bologna by S. Gherardi, edited by M. Kurtze, Berlin, 1871.

2 Quoted in the Life of bishop Smyth (the founder of Brazenose College) by Ralph Churton, Oxford, 1800.

At the beginning of the sixteenth century the names of Master, Paynell, and Tonstall occur. Of these Richard Master, a fellow of King's, is said to have been famous for his knowledge of natural philosophy. He entered at King's in 1502, and was proctor in 1511. He took up the cause of the holy maid of Kent and was executed for treason in April, 1534. Nicholas Paynell, a fellow of Pembroke Hall, graduated B.A. in 1515. In 1530 he was appointed mathematical lecturer to the university. The date of his death is unknown.

Cuthbert Tonstall' was born at Hackforth, Yorkshire, in 1474 and died in 1559. He had entered at Balliol College, Oxford, but finding the philosophers dominant in the university (see p. 243), he migrated to King's Hall, Cambridge. We must not attach too much importance to this step for such migrations were then very common, and his action only meant that he could continue his studies better at Cambridge than at Oxford. He subsequently went to Padua, where he studied the writings of Regiomontanus and Pacioli. His arithmetic termed De arte supputandi was published in 1522 as a "farewell to the sciences" on his appointment to the bishopric of London. A presentation copy on vellum with the author's autograph is in the university library at Cambridge. The work is described by De Morgan in his Arithmetical Books as one of the best which has been written both in matter, style, and for the historical knowledge displayed. Few critics would agree with this estimate, but it was undoubtedly the best arithmetic then issued, and forms a not unworthy conclusion to the medieval history of Cambridge. It is particularly valuable as containing illustrations of the medieval processes of computation. Several extracts from it are given in the Philosophy of arithmetic by J. Leslie, second edition, Edinburgh, 1820.

That brings me to the close of the middle ages, and the above account-meagre though it is contains all that I have

1 See vol. 1. p. 198 of the Athenae Cantabrigienses by C. H. and T. Cooper, Cambridge, 1858-61.

been able to learn about the extent of mathematics then taught at an English university. About Cambridge in particular I can give no details. The fact however that Tonstall and Recorde, the only two English mathematicians of any note of the first half of the sixteenth century, both migrated from Oxford to Cambridge in order to study science makes it probable that it was becoming an important school of mathematics.

CHAPTER II.

THE MATHEMATICS OF THE RENAISSANCE.

CIRC. 1535-1630.

THE close of the medieval period is contemporaneous with the beginning of the modern world. The reformation and the revival of the study of literature flooded Europe with new ideas, and to these causes we must in mathematics add the fact that the crowds of Greek refugees who escaped to Italy after the fall of Constantinople brought with them the original works and the traditions of Greek science. At the same time the invention of printing (in the fifteenth century) gave facilities for disseminating knowledge which made these causes incomparably more potent than they would have been a few centuries earlier.

It was some years before the English universities felt the full force of the new movement, but in 1535 the reign of the schoolmen at Cambridge was brought to an abrupt end by "the royal injunctions" of that year (see p. 244). Those injunctions were followed by the suppression of the monasteries and the schools thereto attached, and thus the whole system of medieval education was destroyed. Then ensued a time of great confusion. The number of students fell, so that the entries for the decade ending 1547 are probably the lowest in the whole seven centuries of the history of the university.

The writings of Tonstall and Recorde, and the fact that most of the English mathematicians of the time came from Cambridge seem to shew that mathematics was then regularly taught, and of course according to the statutes it still con