John Wallis (1616–1703)

Secretary for the Westminster Assembly and a brilliant Mathematician.

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His Works:

  1. A brief and easie explanation of the shorter catechism (1652) by John Wallis
  2. A brief letter from a young Oxonian to one of his late fellow-pupils upon the subject of magnetism (1697) by John Wallis
  3. A defence of the Royal Society, and the Philosophical transactions, particularly those of July, 1670. (1678) by John Wallis
  4. A defense of infant-baptism (1697) by John Wallis
  5. A defense of the Christian Sabbath (1692) by John Wallis
  6. A defense of the Christian Sabbath. Part the first. (1693) by John Wallis
  7. A defense of the Christian Sabbath. Part the second. (1694) by John Wallis
  8. A discourse of gravity and gravitation, grounded on experimental observations (1675) by John Wallis
  9. A fifth letter, concerning the sacred Trinity (1691) by John Wallis
  10. A fourth letter, concerning the sacred Trinity (1691) by John Wallis
  11. A most useful collection of the principal sermons of Christ, with his prophets & apostles, which is the greatest light in the world (1675) by John Wallis
  12. A proposal about printing A treatise of algebra, historical and practical written by the Reverend and learned Dr. John Wallis (1683) by John Wallis
  13. A second letter concerning the Holy Trinity (1691) by John Wallis
  14. A seventh letter, concerning the sacred Trinity (1691) by John Wallis
  15. A sixth letter, concerning the sacred Trinity (1691) by John Wallis

16. A treatise of algebra, both historical and practical. (1685) by John Wallis
17. A treatise of angular sections (1684) by John Wallis
18. An answer to Dr. Sherlock’s examination of the Oxford decree (1696) by John Wallis
19. An answer to three papers of Mr. Hobs (1671) by John Wallis
20. An eighth letter concerning the sacred Trinity (1692) by John Wallis
21. An explication and vindication of the Athanasian Creed (1691) by John Wallis
22. Cono-cuneus: or, The shipwright’s circular vvedge (1684) by John Wallis
23. Cui præfigitur, de loquela sive sonorum formatione, tractatus grammatico-physicus. (1652) by John Wallis
24. De algebra tractatus; historicus & practicus. Anno 1685 Anglice editus; nunc auctus Latine (1693) by John Wallis
25. Due correction for Mr Hobbes (1656) by John Wallis
26. Elenchus geometriæ Hobbianæ (1655) by John Wallis
27. Grammatica linguae anglicanae cui praefigitur de loquela sive sonorum formatione tractatus grammatico-physicus (1674) by John Wallis
28. Hobbiani puncti dispvnctio (1657) by John Wallis
29. Hobbius heauton-timorumenos (1662) by John Wallis
30. Institutio logicæ (1687) by John Wallis
31. Item, tractatus grammatico-physicus de loquela sive sonorum formatione (1688) by John Wallis
32. Johannis Wallis, S.T.D. Geometriæ Professoris Saviliani, in celeberrima academia Oxoniensi; atque Regalis Societatis Londini sodalis; Grammatica linguæ Anglicanæ (1674) by John Wallis
33. Mechanica: sive, De motu, tractatus geometricus (1670) by John Wallis
34. Mens sobria seriò commendata (1657) by John Wallis
35. On the sad losse of the truly honourable Robert Lord Brook an elegie, to his vertuous and noble lady (1643) by John Wallis
36. Opera quædam miscellanea: quæ versa pagina indicabit. (1699) by John Wallis
37. Operum mathematicorum pars altera: qua continentur de angulo contactus & semicirculi, disquisitio geometrica (1656) by John Wallis
38. Operum mathematicorum pars prima (1657) by John Wallis
39. Operum mathematicorvm Johannes Wallisi. (1656) by John Wallis
40. Reasons shewing the consistency of the place of Custos Archivorum with that of a Savilian Professor. (1658) by John Wallis
41. Saviliani Exercitationes tres (1678) by John Wallis
42. Serenissimo Regi Carolo, regni anno decimo quarto, cum celsissima principe Katharina, nuptias consummanti (1662) by John Wallis
43. The doctrine of the Blessed Trinity briefly explained (1690) by John Wallis
44. The greatest light in the world, far exceeding the light of the Quakers (1674) by John Wallis
45. The life of faith (1684) by John Wallis
46. The necessity of regeneration: in two sermons at the University of Oxford (1682) by John Wallis
47. The resurrection asserted: in a sermon preached to the University of Oxford, on Easter-day, 1679 (1679) by John Wallis
48. Theological discourses and sermons on several occasions (1692) by John Wallis
49. Theological discourses containing VIII letters and III sermons (1692) by John Wallis
50. Thomæ Hobbes quadratura circuli, cubatio sphæræ, duplicatio cubi; confutata (1669) by John Wallis
51. Three sermons concerning the sacred Trinity (1691) by John Wallis
52. Tractatus duo prior, de cycloide et corporibus inde gentis (1659) by John Wallis
53. Tractatus duo. Prior, de cycloide et corporibus inde gentis. (1659) by John Wallis
54. Truth tried: or, Animadversions on a treatise published by the Right Honorable Robert Lord Brook (1643) by John Wallis


Biography of John Wallis:

John Wallis (1616–1703), mathematician, was born at Ashford in Kent on 23 Nov. 1616. His father, the Rev. John Wallis (1567–1622), son of Robert Wallis of Finedon, Northamptonshire, graduated B.A. and M.A. from Trinity College, Cambridge, and was minister at Ashford from 1602 until his death on 30 Nov. 1622. He married in 1612, as his second wife, Joanna, daughter of Henry and Mary Chapman of Godmersham, Kent, and had by her three daughters and two sons, John and Henry.

Wallis’s education was begun at Ashford; but, on an outbreak there of the plague, he was removed in 1625 to a private school at Ley Green, near Tenterden, kept by James Mouat, a Scot. When it broke up in 1630 Wallis ‘was as ripe for the university,’ by his own account, ‘as some that have been sent thither.’ ‘It was always my affectation even from a child,’ he wrote, ‘not only to learn by rote, but to know the grounds or reasons of what I learn; to inform my judgment as well as furnish my memory.’ When placed in 1630 at Felsted school, Essex, he wrote and spoke Latin with facility, knew Greek, Hebrew, French, logic, and music. During the Christmas vacation of 1631 his brother taught him the rules of arithmetic, and the study ‘suited my humour so well that I did thenceforth prosecute it, not as a formal study, but as a pleasing diversion at spare hours,’ when works on the subject ‘fell occasionally in my way. For I had none to direct me what books to read, or what to seek, or in what method to proceed. For mathematics, at that time with us, were scarce looked on as academical studies, but rather mechanical—as the business of traders, merchants, seamen, carpenters, surveyors of lands, and the like.’ He was admitted to Emmanuel College, Cambridge, at Christmas 1632, gained a scholarship on the foundation, and became noted as a dialectician. His course of study embraced ethics, physics, and metaphysics, besides medicine and anatomy; he being the first pupil of Francis Glisson to maintain publicly the circulation of the blood. He graduated B.A. and M.A. in 1637 and 1640 respectively, was ordained in the latter year, and became chaplain, first to Sir Richard Darley at Buttercrambe, Yorkshire, then (1642–4) to the widow of Horatio, lord Vere, alternately at Castle Hedingham, Essex, and in London. Here, one evening at supper, a letter in cipher was brought in, relating to the capture of Chichester on 27 Dec. 1642, which Wallis within two hours succeeded in deciphering. The feat made his fortune. He became an adept in the cryptologic art, until then almost unknown, and exercised it on behalf of the parliamentary party. He was rewarded in 1643 with the sequestrated living of St. Gabriel, Fenchurch Street, which he exchanged in 1647 for that of St. Martin in Ironmonger Lane. In 1644 he acted as secretary to the assembly of divines at Westminster, and obtained by parliamentary decree a fellowship in Queens’ College, Cambridge. This, however, he speedily vacated by his marriage, on 14 March 1645, with Susanna, daughter of John and Rachel Glyde of Northiam, Sussex. He now came to live in London. Already zealous for the ‘new’ or experimental philosophy, he associated there with Robert Boyle and other reformers of scientific method, whose weekly meetings, divided after 1649 between Oxford and London, led to the incorporation, in 1663, of the Royal Society (for Wallis’s account of its origin, see Weld’s History of the Royal Society, i. 30, 36). Having contributed effectively to found it, he long helped to sustain its reputation by imparting his own inventions and expounding those of others.

He was well off, his mother at her death in 1643 having left him a substantial estate in Kent, and the course pursued by him in politics, although devious, does not appear to have been dishonest. He gave evidence against Archbishop Laud in 1644 (Prynne, Canterburies Doome, 1646, p. 73), but in 1648 signed the remonstrance against the king’s execution, and in 1649 the ‘Serious and Faithful Representation.’ ‘Oliver had a great respect for him,’ according to Anthony Wood, and he showed it by appointing him in 1649 Savilian professor of geometry in the university of Oxford, of which he was incorporated M.A. from Exeter College in the same year. He further took a degree of D.D. on 31 May 1654, confirmed by diploma on 25 June 1662. His succession in 1658 to Gerard Langbaine the elder as keeper of the university archives, elicited Henry Stubbe’s hostile protest, ‘The Savilian Professor’s Case stated’ [see Stubbs or Stubbes, Henry, (1632–1676)]. In 1653 Wallis deposited in the Bodleian Library a partial collection of the letters deciphered by him, with an historical preface, published by John Davys in 1737 in his ‘Essay on the Art of Decyphering.’ Wallis was afterwards accused by Prynne and Wood of having interpreted the correspondence of Charles I captured at Naseby; but ‘he had this in him of a good subject, that at this time, in 1645, he discovered nothing to the rebels which much concerned the public safety, though he satisfied some of the king’s friends that he could have discovered a great deal’ (Life of Dr. John Barwick, p. 251). That this was his plan of action he himself expressly states in a letter to Dr. John Fell, dated 8 April 1685; and the details of the services rendered by him in this line to the royal cause during some years before the Restoration were doubtless authentically known to Charles II. He was accordingly confirmed in his posts in 1660, was nominated a royal chaplain, and obtained an appointment among the divines commissioned in 1661 to revise the prayer-book.

Wallis published, in 1643, ‘Truth Tried; or Animadversions on the Lord Brooke’s Treatise on the Nature of Truth.’ The perusal in 1647 of Oughtred’s ‘Clavis Mathematicæ’ may be said to have started his mathematical career, and his genius took its special bent from Torricelli’s writings on the method of indivisibles. Applying to it the Cartesian analysis, Wallis arrived at the new and suggestive results embodied in his ‘Arithmetica Infinitorum’ (Oxford, 1655), the most stimulating mathematical work so far published in England. Newton read it with delight when an undergraduate, and derived immediately from it his binomial theorem. It contained the germs of the differential calculus, and gave, ‘in everything but form, advanced specimens of the integral calculus’ (De Morgan, in the Penny Cyclopædia). The famous value for π, here made known, was arrived at by the interpolation (the word was of his invention) of terms in infinite series. In the matter of quadratures, first by him investigated analytically, Wallis generalised with consummate skill what Descartes and Cavalieri had already done. The book promptly became famous, and raised its author to a leading position in the scientific world.

He prefixed to the ‘Arithmetica Infinitorum’ a treatise in which analysis was first applied to conic sections as curves of the second degree. In a long-drawn controversy, begun in 1655, he exposed the geometrical imbecility of Thomas Hobbes It excited much public interest; but after the death of his adversary, Wallis declined to reprint the scathing pamphlets he had directed against him while alive (cf. Hobbes’s Works, ed. Molesworth, 1839–45, passim). A numerical problem sent to him by the French mathematician Fermat led to a correspondence, in which Lord Brouncker, Sir Kenelm Digby, Frénicle, and Schooten took part, published under the title ‘Commercium Epistolicum’ (Oxford, 1658). In a tract, ‘De Cycloide,’ issued in 1659, Wallis gave correct answers to two questions proposed by Pascal, and treated incidentally of the rectification of curves. His ‘Mathesis Universalis’ (Oxford, 1657) embodied the substance of his professorial lectures.

In 1655 Christian Huygens sent to the Royal Society a cryptographic announcement of his discovery of Titan. Wallis retorted with an ingenious pseudo-anagram, capable of interpretation in many senses, which eventually enabled him to claim for Sir Paul Neile and Sir Christopher Wren anticipatory observations of the new Saturnian satellite. Huygens surrendered his priority in all good faith, but was irritated to find that he had been taken in by a practical joke. ‘Decepisse me puto si potuisset,’ was his private note on Wallis’s letter to him of 17 April 1656. One dated 1 Jan. 1659 gave at last the requisite explanation (Œuvres Complètes de Christiaan Huygens, i. 335, 396, 401, ii. 306). Wallis was partial to his countrymen. In his ‘History of Algebra’ he attributed to Thomas Harriot much that belonged to Vieta. This narration, the first of its kind, made part of his ‘Treatise on Algebra’ (London, 1685). Roger Cotes said of the volume: ‘In my mind there are many pretty things in that book worth looking into’ (Correspondence of Newton and Cotes, ed. Edleston, p. 191).

Wallis’s ‘Grammatica Linguæ Anglicanæ’ (Oxford, November 1652) has been tacitly commended by many imitators, and often reprinted. To it was appended a remarkable tract, ‘De Loquela,’ describing in detail the various modes of production of articulate sounds. The study led him to the invention of a method for imparting to deaf-mutes the art of speech. ‘I am now upon another work,’ he wrote to Robert Boyle on 30 Dec. 1661, ‘as hard almost as to make Mr. Hobbes understand a demonstration. It is to teach a person deaf and dumb to speak’ (Boyle, Works, vi. 453). His patient was a youth named Daniel Whalley, exhibited in 1663 as a triumph of the novel curative process before Charles II, Prince Rupert, and the Royal Society. His next success was with Alexander, son of Admiral Edward Popham, previously experimented upon by Dr. William Holder Their respective shares in his instruction occasioned some dispute.

On 26 Nov. 1668 Wallis laid before the Royal Society a correct theory of the impacts of inelastic bodies, based upon the principle of the conservation of momentum (Phil. Trans. iii. 864). It was more fully expounded in his ‘Mechanica,’ issued in three parts, 1669–71, the most comprehensive work on the subject then existing. Wallis’s ‘De Æstu Maris Hypothesis Nova,’ appeared in 1668. The essential part of the tract had been communicated to the Royal Society on 6 Aug. 1666 (ib. ii. 263, see also iii. 652, v. 2061, 2068). It is worth remembering chiefly for the sagacious assumption made in it that the earth and moon may, for purposes of calculation, be regarded as a single body concentrated at their common centre of gravity.

After the Revolution, Wallis was employed as decipherer, on behalf of William III, by Daniel Finch, second earl of Nottingham. Some of the correspondence submitted to him related to the alleged supposititious birth of the Prince of Wales (James III). On one of these letters he toiled for three months, on another for ten weeks; and he wrote piteously to Nottingham asking for ‘some better recompense than a few good words; for really, my lord, it is a hard service, requiring much labour as well as skill’ (Monthly Magazine, 1802, vols. xiii. xiv.). Consulted in 1692 about the adoption of the Gregorian calendar, he strongly discountenanced the step, mainly on the ground that it would imply subserviency to Rome; and his authority prevailed.

At Sir Paul Neile’s on 16 Dec. 1666, Samuel Pepys met ‘Dr. Wallis, the famous scholar and mathematician; but he promises little.’ The acquaintance, however, continued, and Wallis wrote to Pepys, after the lapse of thirty-five years: ‘Till I was past fourscore years of age, I could pretty well bear up under the weight of those years; but since that time, it hath been too late to dissemble my being an old man. My sight, my hearing, my strength, are not as they were wont to be’ (Pepys, Diary, ed. Braybrooke, v. 399). He died at Oxford on 28 Oct. 1703, aged 86, and was buried in St. Mary’s Church, where his son placed a mural monument in his honour.

A full-length portrait of him in his robes was painted in 1701 by Kneller, who was sent to Oxford by Pepys for the purpose. Designed as a gift to the university, it was hung in the gallery of the schools, where it remains. Kneller declared to Pepys: ‘I never did a better picture, nor so good an one in my life, which is the opinion of all as has seen it.’ Wallis expressed his gratitude ‘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 and William Faithorne, junior, 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 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.

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 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 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 published his ‘Sermons’ from the original manuscripts in 1791.

[Wallis’s Account of some Passages in his own Life, in a letter to Dr. Thomas Smith, appended to Hearne’s preface to Peter Langtoft’s Chronicle; Hearne’s Works, vol. iii. p. cxl; Biogr. Brit.; Wood’s Fasti Oxon. (Bliss), ii. 124, 184, 264; Wood’s Hist. of the University of Oxford (Gutch), ii. 866, 962; General Dict.; Thomson’s Hist. of the Roy. Society, p. 271; Rigaud’s Correspondence of Scientific Men, passim; Mayor in Notes and Queries, 2nd ser. ix. 95; Sargeaunt’s Hist. of Felsted School, pp. 37–40; Foster’s Alumni; Granger’s Biogr. Hist. of England, iii. 285; Brewster’s Life of Newton, ii. 202; Europ. Mag. xxxiv. 308, xxxvi. 91, xlix. 345, 427, 429; Œuvres de C. Huygens, passim; Edleston’s Corr. of Newton and Cotes, p. 300; Calamy’s Own Times, i. 272; Neal’s Puritans (Toulmin), iv. 389; Life of Dr. J. Barwick, pp. 61, 251; Cajori’s Hist. of Mathematics, p. 192; Rouse Ball’s Hist. of Mathematics, p. 256; Montucla’s Hist. des Mathématiques, ii. 68, 348, iii. 301; Gerhardt’s Geschichte der höheren Analyse, pp. 34, 76; Marie’s Hist. des Sciences, iv. 149; Evelyn’s Diary (Bray), i. 352, 461; Allibone’s Dict. of Engl. Literature; Watt’s Bibl. Brit.; Morel’s De J. Wallisii Grammatica Linguæ Anglicanæ, Paris, 1895; Bromley’s Cat. of Engraved Portraits, p. 228; Evans’s Portraits, i. 364; Le Neve’s Monumenta Anglicana, iv. 58; Lansdowne MSS. 987 ff. 91, 251, 258, 1181 contains an analysis of Wallis’s writings, 763, f. 124, a letter by him on ancient music; Addit. MS. 32449 includes his correspondence with Nottingham, 1691–2. In Dunton’s Life and Errors (Nichols), ii. 658, is a copy of verses on Wallis’s funeral, beginning:

‘I’ll have the solemn pomp and stately show
In geometrical progression go.’]


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