‘for the honour done me in placing so noble a picture of me in so eminent a place’ (ib. pp. 401, 411). Kneller also drew a half-length of his venerable sitter, whom he represented holding a letter in his hand, with the adjuncts of a gold chain and medal given to him by the king of Prussia for deciphering it. Both pictures were engraved by Faber, the former by David Loggan [q. v.] and William Faithorne, junior [q. v.], as well. His portrait, by Zoest, belongs to the Royal Society. Portraits of him by Loggan (1678) and by Sonmans (1698) were engraved by Michael Burghers [q. v.] to form the frontispieces of the first and third volumes of his ‘Opera Mathematica.’ A portrait after Kneller is in the National Portrait Gallery, London, and a sixth portrait is in the Uffizi Gallery, Florence.
Wallis lost his wife on 17 March 1687. His only son, John Wallis, born on 26 Dec. 1650, graduated B.A. from Trinity College, Oxford, on 9 Nov. 1669, was called to the bar in 1676, and married, on 1 Feb. 1682, Elizabeth, daughter of John Harris of Soundess House, Oxfordshire. By the death of her brother, Taverner Harris, she inherited a fine estate, and she died in 1693, leaving three children. Wallis had two daughters, ‘handsome young gentlewomen,’ according to John Aubrey (Lives of Eminent Men, p. 568), of whom the younger married William Benson of Towcester, and died childless in 1700; the elder, born in 1656, married in 1675 Sir John Blencowe [q. v.]
Wallis was endowed with ‘a hale and vigorous constitution of body, and a mind that was strong, serene, calm, and not soon ruffled and discomposed’ (Life of Wallis, by John Lewis, Add. MS. 32601). ‘It hath been my lot,’ he wrote in 1697, ‘to live in a time wherein have been many and great changes and alterations. It hath been my endeavour all along to act by moderate principles, between the extremities on either hand, in a moderate compliance with the powers in being.’ ‘Hereby,’ he added, ‘I have been able to live easy and useful, though not great.’ He was indeed thoroughly acceptable to neither royalists nor republicans, but compelled respect by his mastery of a dangerous art. He steadily refused Leibnitz's requests for information as to his mode of deciphering. In mathematical history Wallis ranks as the greatest of Newton's English precursors. He was as laborious as he was original; and, by the judicious use of his powers of generalisation, he prepared all the subsequent discoveries of that age. The principles of analogy and continuity were introduced by him into mathematical science. His interpretation of negative exponents and unrestricted employment of fractional exponents greatly widened the range of the higher algebra. Finally, he invented the symbol for infinity, ∞. His memory for figures was prodigious. He often whiled away sleepless nights with exercises in mental arithmetic. On one occasion he extracted the square root of a number expressed by fifty-three figures, and dictated the result to twenty-seven places next morning to a stranger. It proved exact. He made use of no special technique in performing such feats, working merely by common rules on the blackboard of his own tenacious mind (Phil. Trans. xv. 1269). ‘Dr. Wallis,’ Hearne wrote (Collections, ed. Doble, 1885, i. 46), ‘was a man of most admirable fine parts, and great industry, whereby in some years he became so noted for his profound skill in mathematics that he was deservedly accounted the greatest person in that profession of any in his time. He was withal a good divine, and no mean critic in the Greek and Latin tongues.’ ‘An extraordinary knack of sophistical evasion’ was unjustly attributed to him by those to whom his trimming politics were obnoxious.
Wallis's collected mathematical works were published, with a dedication to William III, in three folio volumes at the Sheldonian Theatre, Oxford, in 1693–9. The second (1696) contained Sir Isaac Newton's first published account of his invention of the fluxional calculus. In the third was inserted a statement by John Flamsteed [q. v.] regarding an ostensible parallax for the pole-star—‘a noble observation if you make it out,’ Wallis wrote to him on 9 May 1695. He fully believed that the astronomer royal had ‘made it out,’ thereby showing complete ignorance of technical astronomy. His learned and laborious editions of ancient authors were reprinted in the same volume. He began with Archimedes, whose ‘Arenarius’ and ‘Dimensio Circuli’ he corrected from manuscript copies, and published in 1676. Ptolemy's ‘Harmonicon,’ until then inedited, followed in 1680. In 1688 he unearthed and sent to the press a fragment of Pappus's second book, together with Aristarchus's ‘De Magnitudinibus et Distantiis Solis et Lunæ.’
Wallis edited in 1673 the posthumous works of Jeremiah Horrocks [q. v.] In 1687 he published his celebrated ‘Institutio Logicæ,’ reprinted for the fifth time in 1729. His various theological writings were gathered into a single volume in 1691, and Charles Edward de Coetlogon [q. v.] pub-
