JOHN WALLIS (16161703), English mathematician, logician and grammarian, was born on the 23rd of November 1616 at Ashford, in Kent, of which parish his father, Rev. John Wallis (15671622), was incumbent. After being at school at Ashford, Tenterden and Felsted, and being instructed in Latin, Greek and Hebrew, he was in 1632 sent to Emmanuel College, Cambridge, and afterwards was chosen fellow of Queens' 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 Vere. 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 Gerard Langbaine (16091658) 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 Institutio 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, 16931699). 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 Pappas). 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 traditurn, arithmeticum 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 curvilineoruin quadraturam aliaque difficiliora matheseos problemata " (1655); " Eclipsis solaris observatio Oxonii habita 2° Aug. 16544 ' (1655); " Tractatus duo, prior de cycloide, posterior de cissoide et de curvarum turn linearum € lmost turn superficierum nXaruaµca" (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 postulato quinto. et quinta definitione, lib. VI. Euclidis, disceptatio geometra " (? 1663); "cunocuneus, seu corpus partim conum partim cuneum repres('ntans geometrice consideratum " (English, 1685) ; " De gravitate et gravitatione disquisitio geometrica " (1662; English, 1674); " De aestu marls hypothesis nova " (1666–1669).
The Arithmetica infinitorum relates chiefly to the quadrature of curves by the socalled method of indivisibles established by Bonaventura Cavalieri in 1629 (see INFINITESIMAL CALCULUS). 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
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 CIRCLE, SQUARING OF). 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 Ixvi.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 l'vlontucla on the ground of its unfairness as against the early Italian algebraists and also Franciscus Vieta and Rene Descartes and in favour of Harriot; but Augustus De Morgan, while admitting this, attributes to it considerable merit. The symbol for infinity, co, 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 HOBBES, THOMAS.
End of Article: JOHN WALLIS (16161703) 

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