WALLIS, JOHN (1616-1703), an eminent English mathematician, logician, and grammarian, was born on the 23d November 1616 at Ashford, in Kent, of which parish his father was then incumbent. Having been previously instructed in Latin, Greek, and Hebrew, he was in 1632 sent to Emmanuel College, Cambridge, and afterwards was chosen fellow of Queen's College. Having been admitted to holy orders, he left the university in 1641 to act as chaplain to Sir William Darley, and in the following year accepted a similar appointment from the widow of Sir Horatio Vera. It was about this period that he displayed surprising talents in deciphering the intercepted letters and papers of the Royalists. His adherence to the Parliamentary party was in 1643 rewarded by the living of St Gabriel, Fenchurch Street, London. In 1644 he was appointed one of the scribes or secretaries of the Assembly of Divines at Westminster. During the same year he married Susanna Glyde, and thus 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 Dr Langbaine 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, in the eighty-seventh year of his age.

The works of Wallis are numerous, and relate to a multiplicity of subjects. His Institutio Loglex, published in 1687, was very popular, and in his Grammatica Lingua: AnglicanEe we find indications of an acute and philosophic intellect. The mathematical works arc published some of them in a small 4to volme, Oxford, 1657, and a complete collection in three thick folio volumes, Oxford, 1695-93-99. 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, Porphyries, Briennius, Archimedes, Eutocius, Aristarchus, and Pappus). The second and third volumes include also two collections of letters to and from Brouucker, Freniele, Leibnitz, Newton, Oldenburg, Schooten, and others; and there is a tract on trigonometry by Caswell. Excluding all these, the mathematical works contained ha 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, scu Opus Arithmeticum Philologice et Mathematice Traditurn, Arithineticam Nume•osam et Speciosarn Aliaque Coutinens," 1657 ; " Adversus .Meibomium, de Proportionibus Dialogus," 1657 ; "De Sectionibus Conicis Nova Method° Expositis," 1655 ; "Arithmetica Infinitorum, sive Nova Methodus Inquirendi in Curvilincorum Quadraturain Alian1ue Diflieiliora l∎lathescos Problemata," 1655 ; " Eclipsis Solaris Observatio Oxonii Ilabita 2(1 Aug. 1651," 1655 ; "Tractatus Duo, prior de Cycloide, posterior de Cissoidc et de Curvarum ten Linear= Eign:vo-Ei turn Superficiernin fixaTucri.46)," 1659 ; " Meelianica, sive de Motu Tractatus Geometricus," three parts, 1669-70-71 ; "De Algebra Tractatus Ilistoricus et Praeticus, ejusdem originem et progressus varios ostendens," English, 1685 ; "De Cmnbinationibus Alternationibus et Partibus Aliquotis Tractatus," English, 1685 ; " De Sectionibus Angularibus Tractatus," English, 1685; "De Angulo Contactus et Semicirculi Tractatus," 1656; " Ejnsdem Tractatus Defensio," 1685 ; "De Postulate Quinto, et Quinta Definitione, Lib. VI. Enclidis, Disceptatio Geometrica," ? 1663 ; "Cuno-Cunens, seu Corpus partim Couum partim Cancun' Bepresentans Geometrice Consideratum," English, 1685 ; "De Gravitate et Gravitatione Disqnisitio Geometrica," 1662 (English, 10i4); "De iEstu Maris Hypothesis Nova," 1666-69.

The ilrithmetica Infinitoram relates chiefly to the quadrature of curves by the so-called method of indivisibles established by Cavalleri, 1629. and cultivated in the interval by. him, Fermat, Descartes, and Roberval. The method is substantially that of the integral calculus ; thus, e.g., for the curve y-sx2 to find the area from x-0 to x=l, the base is divided into a equal parts, and the area is obtained as=-13(12+22 . . . +a2), =6i3 so(n +1)(2n+ 1), , load been previously solved. Wallis made the important 110+1 enabled to extend the theorem to certain hyperbolic curves, but the case in a negative value larger than 1 presented a difficulty which he did not succeed in overcoming. It should be noticed that Wallis, although not using the notation x'" in the case of a positive or negative fractional value, nor indeed in the case of a negative integer value of in, deals continually with such powers, and speaks of the positive or negative integer or fractional value of no as the index of the power. The area of a curve, p-stim of a finite number of terms Ax"3, was at once obtained irom that for the case of a single term ; and Wallis, after thus establishing the several of ❑ (=-4 ), the term halfway between 1 and 2, in the series of terms 1, 2, 6, 20, 70, . ; and he thus obtained the remarkable expression 7r -3 9.5 . 5.7 .7 .9 , together with a succession of superior and inferior limits for the number a.

In the same work Wallis obtained the expression which would now be written cis-dx \/1 + CIP(Txy3 for the length of the element of a curve, thus reducing the problem of rectification to that of quadrature. An application of this formula to an algebraical curve was first made a few years later by W. Neil ; the investigation is reproduced in the " Tractatus de Cissoide, &c." (1659, as above), and Wallis adds the remark that the curve thus rectified is in fact the semicubieal parabola.

The Matkesis Universalis is a more elementary work intended for learners. It contains copious dissertations on fundamental points of algebra, arithmetic, and geometry, and critical remarks.

The De Algebra Traetatas contains (chapters 66-69) 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 Montucla on the ground of its unfairness as against the early Italian algebraists and also Pieta and Descartes, and in favour of Harriet ; but De Morgan, while admitting this, attributes to it considerable merit.

The two treatises on the cycloidl and on the eissoid, &c., and the Alteehaniea 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. The euno-cunens is a highly interesting surface ; it is a ruled quartic surface, the equation of which may be written e2y2-(e- z)2(a2 - af-).

Among the letters in volume iii., we have one to the editor of the Leipsic Acts, giving the decipherment of two letters in secret characters. The ciphers are different, but on the canoe 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 Hernias, vol. xii. pp. 36-38. (A. CA.)