WALLIS, JOHN (1616-1703), English mathematician, logician and grammarian, was born on Nov. 23, 1616, at Ashford, Kent, where his father was rector. He went up to Emanuel col lege, Cambridge, in 1632, became a fellow of Queen's, and took holy orders. He gained much credit with the parliamentarians by his talent in deciphering intercepted Royalist documents, and was presented in 1643 to the living of St. Gabriel, Fenchurch street, London, exchanged later for that of St. Martin, Iron monger lane. Although he signed the Remonstrance against the execution of Charles I. he was appointed Savilian professor of geometry at Oxford in 1649, a chair which he held for over 5o years, until his death at Oxford on Oct. 28, 1703.
Works.—The works of Wallis relate to a multiplicity of sub jects. His Institutio logicae, published in 1687, was very popular, and his Grammatica linguae Anglicanae indicates 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, 1693-99). The third volume includes, however, some theological and other ma terial. The mathematical works contained in the first and second volumes occupy about 1,80o pages.
The Arithmetica infinitoruni (1655) is the most important of his works. It relates chiefly to the quadrature of curves by the so-called method of indivisibles established by Bonaventura Cava lieri 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 = xm, where in is a positive integer, the area of the curves when in is negative or fractional. As he was unacquainted with the binomial theorem,
he attempted the quadrature of the circle by interpolation, and arrived at the remarkable expression known as Wallis's Theorem. (See article on CIRCLE.) 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 vniversalis (1658) a more elementary work, con tains dissertations on algebra, arithmetic and geometry.
The De algebra tractatus (Eng. 1685) contains (chapters lxvi.– lxix.) the idea of the interpretation of imaginary quantities in geometry. This is given somewhat as follows : the distance repre sented by the square root of a negative quantity cannot be meas ured 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 scrupu lously fair to his predecessors in all cases where he was able to trace original discoveries.
The two treatises on the cycloid and on the cissoid, etc., and the Mechanica (three parts 1669-71) contain many results which were then new and valuable. The latter work contains elaborate investigations in regard to the centre of gravity, and in it Wallis employs the principle of virtual velocities.
For the prolonged conflict between Hobbes and Wallis, see HOBBES, THOMAS.