601 



WALLIS, JOHN. 



WALMESLEY, CHARLES. 



02 



ing liberal sentiment : " The seuior proctor, according to Lis usual 

 perfidy (which he frequently used in his office, for he was lorn and bred 

 a Presbyterian), did pronounce," &c. &c. (' Ath. Oxon.,' ii. 1045.) 



AVullis, in his literary character, is to be considered as a theologian, 

 a scholar, and a mathematician. As a divine, he would probably not 

 have been remembered, but for his eminence in the other characters. 

 Ilia discourses on the Trinity are still quoted in the histories of 

 opinions ou that subject. At the time of South and Sherlock, much 

 was written on the Athanasian Creed which was meant to be of an 

 explanatory character : those who read South and Sherlock on the 

 Trinity, may also read Wallis, who will be found inferior to neither; 

 but many have considered him scarcely orthodox. If the character of 

 "\Vallis has been elevated as a divine by his celebrity as a philosopher, 

 his services as a scholar have for the same reason been, if not under- 

 rated, at least thrown into shade. He was the first editor of Ptolemy's 

 Harmonics, of the commentary on it by Porphyrius, and of the later 

 work of Brienuius ; as also of Aristarchus of Samos. His editions contain 

 collateral information of the most valuable character, tending to throw 

 light upon his author, and exhibit an immense quantity of labour. 



As a mathematician Wallis is the most immediate predecessor of 

 Newton, both in the time at which he lived and the subjects on which 

 he worked. Those who incline to the opinion that scientific dis- 

 coveries are not the work of the man, but of the man and the hour, 

 that is, who regard each particular conquest as the necessary conse- 

 quence of the actual state of things, and as certain to come from one 

 quarter or another when the time arrives, will probably say that if 

 Wallis had not lived, Newton would but have filled his place, as far 

 as the pure mathematics are concerned. By far the most important 

 of hia writings is the ' Arithmetica Infinitorum,' a slight account of 

 which we shall preface by some mention of the others. The ' Mathesis 

 Universalis ' was intended for the beginner, and contains copious dis- 

 cussions on fundamental points of algebra, arithmetic, and geometry, 

 mixed with critical dissertations. The tract against Meibomiub's 

 dialogue on the fifth book of Euclid is wholly controversial. The 

 treatise on the cycloid is that which was sent in answer to Pascal's 

 prize questions, revised. The work on mechanics is the largest and 

 most elaborate which had then appeared, though now principally 

 remarkable from the use of the principle of virtual velocities. The 

 voluminous treatise which it contains on the centre of gravity, though 

 showing in every page how near Wallis approached to the Differential 

 Calculus, is not so interesting, even in that particular, as the ' Arith- 

 metica Infiuitorum.' The treatise on algebra, which first appeared in 

 English in 1685, was reprinted in Latin (in the collected edition) in 

 1693, with additions. It is the first work in which a copious history 

 of the subject was mixed with its theory. The defect of this history 

 has been adverted to in Vieta, col. 370; but when this is passed over, 

 it may safely be said that the algebra of Wallis is full of interest, even 

 at the present time, not only as an historical work, but as one of inven- 

 tion and originality. The tracts on the angle of contact, on the tides, 

 on gravitation, &c., are now completely gone by, and are only useful 

 as showing the state of various points of mathematics and physics 



The ' Arithmetica Infinitorum ' is preceded by a treatise on Conic 

 Sections, in which the geometrical and algebraical methods are both 

 exemplified. At the commencement, though it is not immediately 

 connected with any application to these curves, he opens with a 

 declaration of his adherence to the method of CAVALIERI, that of 

 indivisibles, but preferring the juster notion of compounding an area 

 out of an infinite number of infinitely small parallelograms. At the 

 beginning of the work Wallis arrives by this method at the areas of 

 various simple curves and spirals. Those who understand how either 

 the method of Cavalieri is employed, or that of differentials, without 

 the use of the organised methods, will easily see how close an approach 

 ia made to the integral calculus, from one instance : In the latter 



science Ix-dx, beginning at x = 0, is ^a? : the corresponding theorem 



of Wallis is that the limit of I 2 + 2 2 + . . . . + n z divided by n 3 is the 

 fraction . He then proceeds step by step until he is able to repre- 

 sent the whole or part of the area of any curve whose equation is 

 y = (a 2 i a; 2 )"," n being integer : having previously found the arcp ol 

 any curve contained under y = ax n , n being positive or negative, whole 

 or fractional. And it is here to be remarked that, though he does not 



absolutely exhibit such symbols as x~~ 2 , x*, he makes use of fractional 

 and negative indices, applying the fractions and negative quantities, 

 though not explicitly writing them in the modern manner. This 

 step was a most important one, as it put under his control, in effect, 

 all that the integral calculus can do in the case of monomial terms 

 and their combinations. Wallis was eminently distinguished by this 

 power of comparison and generalisation, and he had a large portion of 

 the faith in the results of algebra which has led to its complete 

 modern establishment, in which hardly any of that sort of faith is 

 wanted. And those who would smile at his idea of negative quantities 

 which are greater than infinity, should remember what results patience 

 and inquiry have produced out of the equally absurd notion of 

 those same quantities being less than nothing. It is not quite certain 

 that the former phraseology will not yet take its place, under defini- 

 tions, by the side of the latter. 

 This talent of generalisation, in which Wallis was superior to any pre- 



ceding mathematician, enabled him to avail himself of ideas which the 

 ordinary processes of arithmetic and algebra had offered for centuries 

 without results. Having, by his use of fractional indices, been able 



to supply every case of / x m dx, or an equivalent result, it struck him 



y / 



(a 2 x-) " dx, still using modern symbols, must bo capable of 



a similar interpolation. The case of n = \ obviously gives the circle, 

 and after making various attempts, he was enabled to present the 

 well-known result, which is still remembered as a result; but the 

 method which produced it is, though anything but forgotten, not 

 always duly remembered as belonging to Wallis. This result is as 

 follows, in modern terms : tr being the ratio of the circumference to 

 the diameter, \ir lies between 



and 



1 2 .3 2 .5 2 

 2 2 .4 2 .6 2 



2 I) 2 



(2M,) 2 



1 2 .3 2 .5 2 .... (2n !) 

 whatever integer n may be. It is frequently expressed thus : 



IT 2 4 4 6 6 8 



7=5 x-x-x-x-x- x .,..ad mfinitum. 



4 o o 5 7 7 



The works of Wallis contain many other results which must be 

 considered as advanced specimens of the integral calculus in every- 

 thing but form ; such as the rectification of the parabola, whiuh he 

 showed to depend upon the quadrature of the hyperbola. The 

 Binomial Theorem was a corollary of the results of Wallis on the 

 quadrature of curves, the sagacity of Newton supplying that general 

 mode of expression which it is extraordinary that Wallis should have 

 missed. 



We have not spoken of the work on logic, which is not only of the 

 highest excellence, but ia perhaps, owing to the change of notation 

 and methods in mathematics, the only work of Wallis on the elements 

 of a subject which we could now recommend a student to read. In 

 conclusion, we may say of the subject of this article, that it rarely 

 happens that there is so singular a union of originality and labour. 



WALLIS, SAMUEL, the first navigator after Quiros (assuming that 

 Quiros's Sagittaria is Tahiti) who discovered the island of Tahiti. 

 The date of Wallis's birth and his parentage are unknown. In 1755 he 

 was lieutenant of the Gibraltar, a twenty-gun ship, from which he was 

 promoted to be lieutenant of the Torbay seventy-four, Vice- Admiral 

 Boscawen's flag-ship. On the 8th of , April 1757, he received his com- 

 mission as captain of the Port Mahon, of twenty guns, and was sent 

 to North America with Holburne, who commanded the expedition 

 against Louisburg. In 1760 he was sent to Canada in command of 

 the Prince of Orange, a reduced third-rate ; and on his return was 

 employed on the home station. There is no account of him from 

 this time till his being appointed to the Dolphin in August 1766. He 

 was sent with the Dolphin (24 guns) and the Swallow (14 guns, 

 Captain Carteret) to continue and extend tho discoveries of Com- 

 modore Byron in the Pacific. They sailed on the 22nd of August 

 1766, from Plymouth. The Dolphin and Swallow parted company 

 on the llth of April 1767, as they were clearing the western end of 

 the Straits of Magalhaens ; the Dolphin returned to the Downs on the 

 19th of May 1768; the Swallow did not arrive at Spithead till the 

 20th of March 1679. After parting company with his consort, Wallis 

 discovered Easter Island on the 3rd of June 1767 ; and on the 19th 

 of June, Tahiti, which he called King George's Island, and Cook 

 called Otaheite. He left the island on the 27th of July, reached 

 Tinian on the 17th of September, Batavia on the 30th of November, 

 the Cape of Good Hope on the 4th of February 1768, and the Downs, 

 as mentioned above, on the 19th of May. The only record preserved 

 of Wallis's circumnavigation of the globe is that printed in Hawkes- 

 worth's 'Voyages to the Pacific.' It appears to be a literal tran- 

 script of the navigator's diary. It indicates a painstaking, sensible, 

 and veracious man. He was the first to bring down the fabulous 

 stature of the Patagonians to its real altitude. It was Wallis who 

 recommended Tahiti as the station for observing the transit of Venus 

 over the sun's disc in 1769. 



After his arrival in England, Wallis remained without employment 

 till 1771, when, on the equipping of a naval force in consequence of 

 the rupture with Spain about the Falkland Islands, he was appointed 

 to the Torbay seventy-four. He retired from active service in the 

 following year, and never again commanded a ship, except for a short 

 time in 1780. In that year he was appointed extra-commissioner of 

 the navy, an office which he held till the peace, when it was for a 

 time discontinued. It was revived in 1787, and Wallis was again 

 nominated to fill it, which he did till his death, in 1795. 



WALMESLEY, CHARLES, an English mathematician and astro- 

 nomer, was born in 1721 : being a member of the Roman Catholic 

 church, he became a monk of the Benedictine order in this country, 

 and he took the degree of doctor in theology in the Sorbonne. In 

 1750 he was elected a Fellow of the Royal Society of London, and six 

 years afterwards he was made a bishop, and apostolical vicar of t 

 western district of England. 

 His principal work, which is an extension of the ' Harmoma ALen- 



