**WALLIS, JOHN** (1616–1703), 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 (1567–1622), 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 (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 εὐθύνσει tum superficierum
πλατυσμῶ (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 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* = *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 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 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 Hobbes, Thomas.