The only one of those immediately above mentioned who came from Cambridge was Brook Taylor', who was born at Edmonton on Aug. 18, 1685, and died in London on Dec. 29, 1731. He entered at St John's College in 1705, and graduated as LL.B. in 1709. After taking his degree he went to live in London, and from the year 1708 onwards he wrote numerous papers in the Philosophical transactions, in which among other things he discussed the motion of projectiles, the centre of oscillation, and the forms of liquids raised by capillarity. He wrote on linear perspective, two volumes, 1715 and 1719. But the work by which he is generally known is his Methodus incrementorum directa et inversa published in 1715. This contained the enunciation and a proof of the well-known theorem

by which any function of a single variable can be expanded. He did not consider the convergency of the series, and the proof, which contains numerous assumptions, is not worth reproducing. In this treatise he also applied the calculus to various physical problems, and in particular to the theory of the transverse vibrations of strings.

Regarded as mathematicians, Whiston, Laughton, and Saunderson barely escape mediocrity, but their contemporary Cotes, of whom I have next to speak, was a mathematician of exceptional power, and his early death was a serious blow to the Cambridge school. The remark of Newton that if only Cotes had lived "we should have learnt something” indicates the opinion of his abilities generally held by his contemporaries.

2

Roger Cotes was born near Leicester on July 10, 1682. He entered at Trinity in 1699, took his B.A. in 1703, and in

1 An account of his life by Sir William Young is prefixed to the Contemplatio philosophica, London, 1793.

2 See the Biographia Britannica, second edition, London, 1778-93, and also the Dictionary of national biography.

1705 was elected to a fellowship. In 1704 Dr Plume, the archdeacon of Rochester and formerly of Christ's College (bachelor of theology, 1661), founded a chair of astronomy and experimental philosophy. The first appointment was made in 1707, and Cotes was elected'. Whiston was one of the electors, and he writes, "I was the only professor of mathematics directly concerned in the choice, so my determination naturally had its weight among the rest of the electors. I said that I pretended myself to be not much inferior in mathematics to the other candidate's master, Dr Harris, but confessed that I was but a child to Mr Cotes: so the votes were unanimous for him3." Newton, to whom Bentley had introduced Cotes, also wrote a very strong testimonial in his favour.

Bentley at once urged the new professor to establish an astronomical observatory in the university. The university gave no assistance, but Trinity College consented to have one erected on the top of the Great Gate, and to allow the Plumian professor to occupy the rooms in connection with it; considerable subscriptions were also raised in the college to provide apparatus. The observatory was pulled down in 1797.

In 1709 Newton was persuaded to allow Cotes to prepare the long-talked-of second edition of the Principia. The first edition had been out of print by 1690; but though Newton had collected some materials for a second and enlarged edition, he could not at first obtain the requisite data from Flamsteed, the astronomer-royal, and subsequently he was unable or unwilling to find the time for the necessary revision. The second edition was issued in March 1713, but a considerable part of the

1 The successive professors were as follows. From 1707 to 1716, Roger Cotes of Trinity; from 1716 to 1760, Robert Smith of Trinity (see p. 91); from 1760 to 1796, Anthony Shepherd of Christ's (see p. 103); from 1796 to 1822, Samuel Vince of Caius (see p. 103); from 1822 to 1828, Robert Woodhouse of Caius (see p. 118); from 1828 to 1836, Sir George B. Airy of Trinity (see p. 132); from 1836 to 1883, James Challis of Trinity (see p. 132); who in 1883 was succeeded by G. H. Darwin of Trinity, the present professor.

2 See p. 133 of Whiston's Memoirs.

new work contained in it was due to Cotes and not to Newton. The whole correspondence between Newton and Cotes on the various alterations made in this edition is preserved in the library of Trinity College, Cambridge: it was edited by Edleston for the college in 1850. This edition was sold out within a few months, but a reproduction published at Amsterdam supplied the demand. Cotes himself died on June 5, 1716, shortly after the completion of this work.

He is described as possessing an amiable disposition, an imperturbable temper, and a striking presence; and he was certainly loved and regretted by all who knew him.

His writings were collected and published in 1722 under the titles Harmonia mensurarum and Opera miscellanea. His professorial lectures on hydrostatics were published in 1738. A large part of the Harmonia mensurarum is given up to the decomposition and integration of rational algebraical expressions; that part which deals with the theory of partial fractions was left unfinished, but was completed by de Moivre. Cotes's theorem in trigonometry which depends on forming the quadratic factors of "-1 is well known. The proposition that "if from a fixed point O a line be drawn cutting a curve in Q1, Q2.. Qn, and a point P be taken on it so that the reciprocal of OP is the arithmetic mean of the reciprocals of OQ, OQ...OQn, then the locus of P will be a straight line" is also due to Cotes. The title of the book was derived from the latter theorem. The Opera miscellanea contains a paper on the method for determining the most probable result from a number of observations: this was the earliest attempt to frame a theory of errors. It also contains essays on Newton's Methodus differentialis, on the construction of tables by the method of differences, on the descent of a body under gravity, on the cycloidal pendulum, and on projectiles.

It was unfortunate for Cotes's reputation that his friend Brook Taylor stated the property of the circle which Cotes had discovered as a challenge to foreign mathematicians in a manner which was somewhat offensive. John Bernoulli solved

the question proposed in 1719, and his friends seized on his triumph as a convenient opportunity for shewing their dislike of Newton by depreciating Cotes.

The study of mathematics in the different colleges received at this time a considerable stimulus by the establishment in 1710 of certain lectureships by Lady Sadler. On the advice of William Croone (born about 1629 and died in 1684), a fellow of Emmanuel and professor of rhetoric at Gresham College, she gave to the university an estate of which the income was to be divided amongst the lecturers on algebra at certain colleges. This no doubt helped to promote the interest in that subject during the seventeenth century. With the advance in the standard of education it ceased to be productive of much benefit, and in 1860 it was changed into a professorship of pure mathematics; in 1863 Arthur Cayley of Trinity was appointed professor.

Cotes was succeeded as Plumian professor by his cousin Robert Smith. Robert Smith was born in 1689, entered at Trinity in 1707, took his B.A. in 1711, and was elected to a fellowship in the following year. He held the office of master of mechanics to the king. As Plumian professor he lectured on optics and hydrostatics, and subsequently he wrote textbooks on both those subjects. His Opticks published in 1728 is one of the best text-books on the subject that has yet appeared, and with a few additions might be usefully reprinted now. He also published in 1744 a work on sound, entitled Harmonics, which contains the substance of lectures he had for many years been giving. He edited Cotes's works. He was made master of Trinity in 1742, and died at Cambridge on Feb. 2, 1768. He founded by his will two annual prizes for proficiency in mathematics and natural philosophy, to be held by commencing bachelors and known by his name. They proved productive of the best results, and at a later time they enabled the university to encourage some of the higher branches of mathematics which did not directly come into the university examinations for degrees.

The labours of Laughton, Bentley, Whiston, Saunderson, Cotes, and Smith were rewarded by the definite establishment about the year 1730 of the Newtonian philosophy in the schools of the university. The earliest appearance of that philosophy in the scholastic exercises is the act kept by Samuel Clarke in 1694 and above alluded to. Ten years later it was not unusual to keep one act from Newton's writings; but from 1730 onwards it was customary to require at least one disputation to be on a mathematical subject—usually on Newton— and in general to expect one to be on a philosophical thesis, although after 1750 it was possible to propose mathematical questions only. The decade from 1725 to 1735 is an important one in a history of mathematics at Cambridge, not only for the reasons given above, but because the mathematical tripos, which profoundly affected the subsequent development of mathematics in the university, originated then. The history of the origin and growth of that examination may be left for the present. The death of Newton and the retirement or death of nearly all those who had been brought under his direct influence also fall within this decade, and it thus naturally marks the conclusion of this chapter.

The effect of the teaching of the above-mentioned mathematicians in extending the range of reading is shewn by the following list of mathematical text-books which were in common use by the year 1730. The dates given are those of the first editions, but in most cases later editions had been issued incorporating the discoveries of subsequent writers.

First, for the subjects of pure mathematics. The usual text-books on pure geometry were the Elements of Euclid (editions of Barrow, Gregory, or Whiston), the Conics of Apollonius (Halley's edition, 1710), or of de Lahire (1685), to which we may perhaps add the fourth and fifth sections of the first book of the Principia. [Simson's Conics was published in 1735, and became the recognized text-book for that subject for the