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Rh vacated his fellowship; but the death of his mother had left him in possession of a handsome fortune. In 1645 he attended those scientific meetings which led to the establishment of the Royal Society. When the Independents obtained the superiority Wallis adhered to the Solemn League and Covenant. The living of St Gabriel he exchanged for that of St Martin, Ironmonger Lane; and, as rector of that parish, he in 1648 subscribed the Remonstrance against putting Charles I. to death. Notwithstanding this act of opposition, he was in June 1649 appointed Savilian professor of geometry at Oxford. In 1654 he there took the degree of D.D., and four years later succeeded Gerard Langbaine (1609–1658) as keeper of the archives. After the restoration he was named one of the king's chaplains in ordinary. While complying with the terms of the Act of Uniformity, Wallis seems always to have retained moderate and rational notions of ecclesiastical polity. He died at Oxford on the 28th of October 1703.

The works of Wallis are numerous, and relate to a multiplicity of subjects. His Institutia logicae, published in 1687, was very popular, and in his Grammatica linguae Anglicanae we find indications of an acute and philosophic intellect. The mathematical works are published, some of them in a small 4to volume (Oxford, 1657) and a complete collection in three thick folio volumes (Oxford, 1693–1699). The third volume includes, however, some theological treatises, and the first part of it is occupied with editions of treatises on harmonics and other works of Greek geometers, some of them first editions from the MSS., and in general with Latin versions and notes (Ptolemy, Porphyrius, Briennius, Archimedes, Eutocius, Aristarchus and Pappus). The second and third volumes include also his correspondence with his contemporaries; and there is a tract on trigonometry by Caswell. Excluding all these, the mathematical works contained in the first and second volumes occupy about 1800 pages. The titles in the order adopted, but with date of publication, are as follows: “Oratio inauguralis,” on his appointment (1649) as Savilian professor (1657); “Mathesis universalis, seu opus arithmeticum philologice et mathematice traditum, arithmeticam numerosam et speciosam aliaque continens” (1657); “Adversus Meibomium, de proportionibus dialogus” (1657); “De sectionibus conicis nova methodo expositis” (1655); “Arithmetica infinitorum, sive nova methodus inquirendi in curvilineorum quadraturam aliaque difficiliora matheseos problemata” (1655); “Eclipsis solaris observatio Oxonii habita 2° Aug. 1654” (1655); “Tractatus duo, prior de cycloide, posterior de cissoide et de curvarum tum linearum  tum superficierum  (1659); “Mechanica, sive de motu tractatus geometricus” (three parts, 1669–1670–1671); “De algebra tractatus historicus et practicus, ejusdem originem et progressus varios ostendens” (English, 1685); “De combinationibus alternationibus et partibus aliquotis tractatus” (English, 1685) “De sectionibus angularibus tractatus” (English, 1685); “De angulo contactus et semicirculi tractatus” (1656); “Ejusdem tractatus defensio” (1685); “De postulate quinto, et quinta definitione, lib. VI. Euclidis, disceptatio geometrica” (? 1663); “cunocuneus, seu corpus partim conum partim cuneum representans geometrice consideratum” (English, 1685); “De gravitate et gravitatione disquisition geometrica” (1662; English, 1674); “De aestu maris hypothesis nova” (1666–1669).

The Arithmetica infinitorum relates chiefly to the quadrature of curves by the so-called method of indivisibles established by Bonaventura Cavalieri in 1629 (see ). He extended the “law of continuity” as stated by Johannes Kepler; regarded the denominators of fractions as powers with negative exponents; and deduced from the quadrature of the parabola y = x'm, where m is a positive integer, the area of the curves when m is negative or fractional. He attempted the quadrature of the circle by interpolation, and arrived at the remarkable expression known as Wallis’s Theorem (see ). In the same work Wallis obtained an expression for the length of the element of a curve, which reduced the problem of rectification to that of quadrature.

The Mathesis universalis, a more elementary work, contains copious dissertations on fundamental points of algebra, arithmetic and geometry, and critical remarks.

The De algebra tractatus contains (chapters lxvi.-lxix.) the idea of the interpretation of imaginary quantities in geometry. This is given somewhat as follows: the distance represented by the square root of a negative quantity cannot be measured in the line backwards or forwards, but can be measured in the same plane above the line, or (as appears elsewhere) at right angles to the line either in the plane, or in the plane at right angles thereto. Considered as a history of algebra, this work is strongly objected to by Jean Etienne Montucla on the ground of its unfairness as against the early Italian algebraists and also Franciscus Vieta and René Descartes and in favour of Harriot; but Augustus De Morgan, while admitting this, attributes to it considerable merit. The symbol for infinity, ∞, was invented by him.

The two treatises on the cycloid and on the cissoid, &c., and the Mechanica contain many results which were then new and valuable. The latter work contains elaborate investigations in regard to the centre of gravity, and it is remarkable also for the employment of the principle of virtual velocities.

Among the letters in volume iii., we have one to the editor of the Acta Leipsica, giving the decipherment of two letters in secret characters. The ciphers are different, but on the same principle: the characters in each are either single digits or combinations of two or three digits, standing some of them for letters, others for syllables or words,—the number of distinct characters which had to be deciphered being thus very considerable.

For the prolonged conflict between Hobbes and Wallis, see.

 WALLIS ARCHIPELAGO,, or , a group of islands in the Pacific Ocean, N.E. of Fiji, about 13° S., 176° W., with a land area of 40 sq. m., belonging to France. It was placed under the French protectorate on the 5th of April 1887, and connected for administrative purposes with New Caledonia by decree of the 27th of November 1888. There is a French Resident in the islands, which are connected by a regular service with Nouméa, New Caledonia. The principal islands are Uvea, of volcanic formation and surrounded with coral, and Nukuatea. The islands were discovered by Samuel Wallis in 1767, and it was a missionary, Father Bataillon, who in 1837 first brought the influence of France to bear on the natives. These, about 4500 in number, are of Polynesian race, gentle and industrious. The trade of the islands is mainly with Samoa, whence cottons and iron goods are imported, and to which copra and roots are exported. The Horne Islands (Fotuna and Alofa), S.W. of the Wallis Islands, were discovered by Jacob Lemaire and Willem Cornelis Schouten in 1616, and placed under the French protectorate by decree of the 16th of February 1888. They have 1500 inhabitants.  WALLON, HENRI ALEXANDRE (1812–1904), French historian and statesman, was born at Valenciennes on the 23rd of December 1812. Devoting himself to a literary career, he became in 1840 professor at the École Normale Supérieure under the patronage of Guizot, whom he succeeded as professor at the Faculté des Lettres in 1846. His works on slavery in the French colonies (1847) and on slavery in antiquity (1848; new edition in 3 vols., 1879) led to his being placed, after the Revolution of 1848, on a commission for the regulation of labour in the French colonial possessions, and in November 1849 he was elected to the Legislative Assembly by the department of the Nord. He resigned in 1850, disapproving of the measure for the restriction of the suffrage adopted by the majority. In the same year he was elected a member of the Académie des Inscriptions, of which he became perpetual secretary in 1873. Under the empire he withdrew altogether from political life, and occupied himself entirely with his duties as a professor of history and with historical writings, the most original of which is a biography, Richard II, épisode de la rivalité de la France et de l’Angleterre (2 vols., 1864). Although remaining a republican, he exhibited decided clerical leanings in his Jeanne d’Arc (2 vols., 1860; 2nd ed., 1875); La Vie de Noire Seigneur Jesus (1865)—a reply to the Vie de Jésus of E. Renan; and Saint Louis et son temps (1871; 4th ed., 1892), which still ranks among hagiographical works. Returning to politics after the Franco-German War, Wallon was re-elected by the department of the Nord in 1871, took an active part in the proceedings of the Assembly, and finally immortalized himself by carrying his proposition for the establishment of the Republic with a president elected for seven years, and then eligible for re-election, which, after violent debates, was adopted by the Assembly on the 30th of January 1875. “Ma proposition,” he declared, “ne proclame pas la République, elle la fait.” Upon the definitive establishment of the Republic, Wallon became Minister of Public Instruction, and effected many useful reforms, but his views were too conservative for the majority of the Assembly, and he retired in May 1876. He had been chosen a life senator in December 1875. Returning to his historical studies, Wallon produced four works of great importance, though less from his part in them as author than from the documents which accompanied them: La Terreur (1873); Histoire du tribunal