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stituted the course for the M.A. degree. But it is also clear that it was only beginning to grow into an important study, and was not usually read except by bachelors, and probably by only a few of them. The chief English mathematician of this time was Recorde whose works are described immediately hereafter; but John Dee, Thomas Digges, Thomas Blundeville, and William Buckley were not undistinguished.

The period of confusion in the studies of the university caused by the break-up of the medieval system of education was brought to an end by the Edwardian statutes of 1549 (see p. 153). These statutes represented the views of a large number of residents, and it is noticeable that they enjoined the study of mathematics as the foundation of a liberal education. Certain text-books were recommended, and we thus learn that arithmetic was usually taught from Tonstall and Cardan, geometry from Euclid, and astronomy from Ptolemy. Cosmography was still included in the quadrivium, and the works of Mela, Strabo, and Pliny are referred to as authorities on it.

The Edwardian code was only in force for about twenty years. Fresh statutes were given by Elizabeth in 1570, and except for a few minor alterations these remained in force till 1858. The commissioners who framed them excluded mathematics from the course for undergraduates-apparently because they thought that its study appertained to practical life, and had its place in a course of technical education rather than in the curriculum of a university. These opinions were generally held at that time' and it will be found that most of the English books on the subject issued for the following sixty or seventy years the period comprised in this chapter-were chiefly devoted to practical applications, such as surveying, navigation, and astrology. Accordingly we find that for the next half century mathematics was more studied in London than at the universities, and it was not until it became a

1 See for example vol. 1. pp. 382-91 of the Orationes of Melanchthon, and the autobiography of Lord Herbert of Cherbury (born in 1581 and died in 1648) which was published in London in 1792.

science (under the influence of Wallis, Barrow, and Newton) that much attention was paid to it at Cambridge.

It must however be remembered that though under Elizabethan statutes mathematics was practically relegated to a secondary position in the university curriculum, yet it remained the statutable subject to be read for the M.A. degree. That was in accordance with the views propounded by Ramus1 who considered that a liberal education should comprise the exoteric subjects of grammar, rhetoric, and dialectics; and the esoteric subjects of mathematics, physics, and metaphysics for the more advanced students. The exercises for the degree of master were however constantly neglected, and after 1608 when residence was declared to be unnecessary (see p. 157) they were reduced to a mere form.

I think it will be found that in spite of this official discouragement the majority of the English mathematicians of the early half of the seventeenth century were educated at Cambridge, even though they subsequently published their works and taught elsewhere.

Among the more eminent Cambridge mathematicians of the

1 See p. 346 of Ramus; sa vie, ses écrits, et ses opinions by Ch. Waddington, Paris, 1855. Another sketch of his opinions is given in a monograph of which he is the subject by C. Desmaze, Paris, 1864. Peter Ramus was born at Cuth in Picardy in 1515, and was killed at Paris at the massacre of St Bartholomew on Aug. 24, 1572. He was educated at the university of Paris, and on taking his degree he astonished and charmed the university with the brilliant declamation he delivered on the thesis that everything Aristotle had taught was false. He lectured first at le Mans, and afterwards at Paris; at the latter he founded the first chair of mathematics. Besides some works on philosophy he wrote treatises on arithmetic, algebra, geometry (founded on Euclid), astronomy (founded on the works of Copernicus), and physics which were long regarded on the continent as the standard text-books on these subjects. They are collected in an edition of his works published at Bâle in 1569. Cambridge became the chief centre for the Ramistic doctrines, and was apparently frequented by foreign students who desired to learn his logic and system of philosophy: see vol. 1. pp. 411-12 of the University of Cambridge, by J. Bass Mullinger, Cambridge, 1884.

latter half of the sixteenth century I should include Sir Henry Billingsley, Thomas Hill, Thomas Bedwell, Thomas Hood, Richard Harvey, John Harvey, and Simon Forman. These were only second-rate mathematicians. They were followed by Edward Wright, Henry Briggs, and William Oughtred, all of whom were mathematicians of mark: most of the works of the three last named were published in the seventeenth century.

After this brief outline of my arrangement of the chapter I return to the Cambridge mathematicians of the first half of the sixteenth century.

The earliest of these-if we except Tonstall—and the first English writer on pure mathematics of any eminence was Recorde. Robert Recorde' was born at Tenby about 1510. He was educated at Oxford, and in 1531 obtained a fellowship at All Souls' College; but like Tonstall he found that there was then no room at that university for those who wished to study science beyond the traditional and narrow limits of the quadrivium. He accordingly migrated to Cambridge, where he read mathematics and medicine. He then returned to Oxford, but his reception was so unsatisfactory that he moved to London, where he became physician to Edward VI. and afterwards to Queen Mary. His prosperity however must have been shortlived, for at the time of his death in 1558 he was confined in the King's Bench prison for debt.

His earliest work was an arithmetic published in 1540 under the title the Grounde of artes. This is the earliest English scientific work of any value. It is also the first English book which contains the current symbols for addition,

1 See the Athenae Cantabrigienses by C. H. and T. Cooper, two vols. Cambridge, 1858 and 1863. To save repetition I may say here, once for all, that the accounts of the lives and writings of such of the mathematicians as are mentioned in the earlier part of this chapter and who died before 1609 are founded on the biographies contained in the Athenae Cantabrigienses.

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subtraction, and equality. There are faint traces of his having used the two former as symbols of operation and not as mere abbreviations. The sign for equality was his invention. He says he selected that particular symbol because than two parallel straight lines no two things can be more equal, but M. Charles Henry has pointed out in the Revue archéologique for 1879 that it is a not uncommon abbreviation for the word est in mediæval manuscripts, and this would seem to point to a more probable origin. Be this as it may, the work is the best treatise on arithmetic produced in that century.

Most of the problems in arithmetic are solved by the rule of false assumption. This consists in assuming any number for the unknown quantity, and if on trial it does not satisfy the given conditions, correcting it by simple proportion as in rule of three. It is only applicable to a very limited class of problems. As an illustration of its use I may take the following question. A man lived a fourth of his life as a boy; a fifth as a youth; a third as a man; and spent thirteen years in his dotage: how old was he? Suppose we assume his age to have been 40. Then, by the given conditions, he would have spent 83 (and not 13) years in his dotage, and therefore

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hence his actual age was 60. Recorde adds that he preferred to solve problems by this method since when a difficult question was proposed he could obtain the true result by taking the chance answers of "such children or idiots as happened to be in the place."

Like all his works the Grounde of artes is written in the form of a dialogue between master and scholar. As an illustration of the style I quote from it the introductory conversation on the advantages of the power of counting "the only thing that separateth man from beasts."

Master. If Number were so vile a thing as you did esteem it, then need it not to be used so much in mens communication. Exclude Number and answer me to this question. How many years old are you?

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Master. How many days in a week? How many weeks in a year? What lands hath your father? How many men doth he keep? How long is it sythe you came from him to me?

Scholar. Mum.

Master. So that if Number want, you answer all by Mummes. How many miles to London?... Why, thus you may see, what rule Number beareth and that if Number be lacking, it maketh men dumb, so that to most questions, they must answer Mum.

Recorde also published in 1556 an algebra called the Whetstone of witte. The title, as is well known, is a play on the old name of algebra as the cossic art: the term being derived from cosa, a thing, which was used to denote the unknown quantity in an equation. Hence the title cos ingenii, the whetstone of wit. The algebra is syncopated, that is, it is written at length according to the usual rules of grammar, but symbols or contractions are used for the quantities and operations which occur most frequently. In this work Recorde shewed how the square root of an algebraical expression could be extracted—a rule which was here published for the first time.

Both these treatises were frequently republished and had a wide circulation. The latter in particular was well known, as is shewn by the allusion to it (as being studied by the usurer) in Sir Walter Scott's Fortunes of Nigel. To the belated traveller who wanted some literature wherewith to pass the time, the maid, says he, "returned for answer that she knew of no other books in the house than her young mistress's bible, which the owner would not lend; and her master's Whetstone of Witte by Robert Recorde.” So too William Cuningham' in his Cosmographicall glasse, published in 1559, alludes to

1 William Cuningham (sometimes written Keningham) was born in 1531 and entered at Corpus College, Cambridge, in 1548. The Cosmographicall glasse, is the earliest English treatise on cosmography. Cuningham also published some almanacks, but his works have no intrinsic value in the history of the mathematics. He practised as a physician in London, under the license conferred by his Cambridge degree.

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