Wallis, John

(1616-1703), mathematician and cryptographer

by Domenico Bertoloni Meli

© Oxford University Press 2004 All rights reserved

Wallis, John (1616-1703), mathematician and cryptographer, was born on 3 December 1616 at Ashford in Kent. He was the third of five children of the Revd John Wallis (1567-1622), son of Robert Wallis of Finedon, Northamptonshire, Cambridge BA and MA (Trinity College), and minister at Ashford from 1602 until his death on 30 November 1622, and his second wife, Joanna (d. 1643), daughter of Henry and Mary Chapman of Godmersham, Kent. She raised Wallis and his two brothers and two sisters after their father's death.

Education and early life
Wallis's education was begun at Ashford. Following an outbreak of the plague in 1625 he studied Latin grammar at a private school in Tenterden in Kent. The school had been started by a Mr Finch who had enrolled a schoolmaster to instruct his children, and it closed in 1630 when the eldest child was sent to university and the other children were sent elsewhere. Although Wallis claimed to be already fit for university, in 1631 he was sent to another school, run by Martin Holbeach at Felsted in Essex, where he continued his studies of Latin and learned Greek and Hebrew, as well as some logic. Mathematics was not taught, but Wallis gained some knowledge in that field at home during the Christmas vacation of 1631, when his brother passed on to him the knowledge he had acquired 'in Order to a Trade' (Scriba, 'Autobiography of John Wallis', 26).

At Christmas 1632 Wallis was admitted as a pensioner to Emmanuel College, Cambridge, where he became noted as a dialectician. He studied natural philosophy, ethics, metaphysics, theology, anatomy, and medicine, and graduated BA in 1637 and MA in 1640. His tutors were Anthony Burgess, divine and later member of the Westminster assembly, Thomas Horton, later president of Queens' College and professor of divinity at Gresham College, and Benjamin Whichcote, later provost of King's College and vice-chancellor of the university, and to these three men Wallis addressed the dedicatory letter of his De angulo contactus et semicirculo disquisitio geometrica (1656). Wallis claimed that, as a student of the anatomist and physician Francis Glisson, he was the first to defend publicly at Cambridge Harvey's doctrine of the circulation of the blood. At the end of his life he maintained to have been self-taught, after the informal instruction by his brother, with regard to the study of mathematics:

I did thenceforth prosecute it, (at school and in the University) not as a formal Study, but as a pleasing Diversion, at spare hours; as books of Arithmetick, or other Mathematical 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 Mathematicks, (at that time, with us) were scarce looked upon as Accademical studies, but rather Mechanical; as the business of Traders, Merchants, Seamen, Carpenters, Surveyors of Lands, or the like; and perhaps some Almanak-makers in London. And amongst more than Two hundred students (at that time) in our College, I do not know of any Two (perhaps not any) who had more of Mathematicks than I, (if so much) which was then but little; And but very few, in that whole University. For the Study of Mathematicks was at that time more cultivated in London than in the Universities. (Scriba, 'Autobiography of John Wallis', 27)
In 1640 Wallis was ordained by the bishop of Winchester and became chaplain first to Sir Richard Darley at Buttercrambe, Yorkshire, then to the widow of Horatio, Lord Vere, alternately at Castle Hedingham in Essex and in London. One evening in late 1642, sitting at supper at Lady Vere's in London, he was able to decipher in about two hours a letter in code that was shown to him. Cryptography remained one of Wallis's activities during the civil war, when he exercised it on behalf of the parliamentary party, and later in his life, as he wrote in his autobiography:
being encouraged by this success, beyond expectation; I afterwards ventured on many others (some of more, some of less difficulty) and scarce missed of any, that I undertook, for many years, during our civil Wars, and afterwards. But of late years, the French Methods of Cipher are grown so intricate beyond what it was wont to be, that I failed of many; tho' I have master'd divers of them. (Scriba, 'Autobiography of John Wallis', 38)
In 1643, following the death of his mother, Wallis was left a substantial estate in Kent. In the same year he published Truth Tried, or, Animadversions on the Lord Brooke's Treatise on the Nature of Truth. From 1644 he acted as secretary to the assembly of divines at Westminster, London, which was charged with proposing a new form of church government. In his autobiography he provided an interesting retrospective account of the activities and debates of the Westminster assembly. Also in 1644 he obtained a fellowship at Queens' College, Cambridge, but he resigned after about a year when, on 24 March 1645, he married Susanna (1622-1687), daughter of John and Rachel Glyde of Northiam, Sussex. In 1644 he gave evidence against Archbishop Laud, but in 1648 he signed the remonstrance against the execution of Charles I and in 1649 the 'Serious and Faithful Representation'.

Anticipating the Royal Society
During this period in London Wallis became engaged in experimental philosophy, and his circle was one of those whence the Royal Society emerged after the Restoration. His account of those early events is of considerable interest:

About the year 1645, while I lived in London (at a time, when, by our Civil Wars, Academical Studies were much interrupted in both our Universities:) beside the Conversation of divers eminent Divines, as to matters Theological; I had the opportunity of being acquainted with divers worthy Persons, inquisitive into Natural Philosophy, and other parts of Humane Learning; And particularly of what hath been called the New Philosophy or Experimental Philosophy. We did by agreement, divers of us, meet weekly in London on a certain day, to treat and discours of such affairs. Of which number were Dr. John Wilkins (afterward Bishop of Chester), Dr. Jonathan Goddard, Dr. George Ent, Dr. Glisson, Dr. Merret (Drs. in Physick,) Mr. Samuel Foster then Professor of Astronomy at Gresham College, Mr Theodore Haak (a German of the Palatinate and then Resident in London, who, I think, gave the first occasion, and first suggested those meetings) and many others. These meetings we held sometimes at Dr. Goddards lodgings in Woodstreet (or some convenient place near) on occasion of his keeping an Operator in his house, for grinding Glasses for Telescopes and Microscopes; and sometime at a convenient place in Cheap-side; sometime at Gresham College or some place near adjoyning. Our business was (precluding matters of Theology and State Affairs) to discours and consider of Philosophical Enquiries, and such as related thereunto; as Physick, Anatomy, Geometry, Astronomy, Navigation, Staticks, Magneticks, Chymicks, Mechanicks, and Natural Experiments; with the State of these Studies, as then cultivated, at home and abroad. We there discoursed of the Circulation of the Blood, the Valves in the Veins, the Venae Lacteae, the Lymphatick vessels, the Copernican Hypothesis, the Nature of Comets, and New Stars, the Satellites of Jupiter, the Oval Shape (as it then appeared) of Saturn, the spots in the Sun, and its Turning on its own Axis, the Inequalities and Selenography of the Moon, the several Phases of Venus and Mercury, the Improvement of Telescopes, and grinding of Glasses for that purpose, the Weight of Air, the Possibility or Impossibility of Vacuities, and Natures Abhorrence thereof; the Torricellian Experiment in Quicksilver, the Descent of heavy Bodies, and the degrees of Acceleration therein; and divers other things of like nature. Some of which were then but New Discoveries, and others not so generally known and imbraced, as now they are; With other things appertaining to what hath been called The New Philosophy; which, from the times of Galileo at Florence, and Sir Francis Bacon (Lord Verulam) in England, hath been much cultivated in Italy, France, Germany, and other Parts abroad, as well as with us in England. About the year 1648, 1649, some of our company being removed to Oxford (first Dr. Wilkins, then I, and soon after Dr. Goddard) our company divided. Those in London continued to meet there as before (and we with them, when we had occasion to be there;) and those of us at Oxford; with Dr. Ward (since Bishop of Salisbury), Dr. Ralph Bathurst (now President of Trinity College in Oxford) Dr. Petty (since Sir William Petty) Dr. Willis (then an eminent Physician in Oxford) and divers others, continued such meetings in Oxford; and brought those Studies into fashion there; meeting first at Dr. Petties Lodgings, (in an Apothecaries house) because of the convenience of inspecting Drugs, and the like, as there was occasion; and after his remove to Ireland (tho' not so constantly) at the Lodgings of Dr. Wilkins, then Warden of Wadham Coll. And after his removal to Trinity College in Cambridge, at the Lodgings of the Honorable Mr. Robert Boyle, then resident for divers years in Oxford. Those meetings in London continued, and (after the Kings return in 1660) were increased with the accession of divers worthy and Honorable Persons; and were afterwards incorporated by the name of the Royal Society, etc. and so continue to this day. (Scriba, 'Autobiography of John Wallis', 39-40)
Wallis was a founding member of the Royal Society and one of its most active fellows. He contributed more than sixty papers on several subjects and reviews of mathematical books to the society's Philosophical Transactions.

Savilian professorship
In June 1649 Wallis was appointed Savilian professor of geometry at Oxford, and was incorporated MA from Exeter College in the same year. He held his professorship for over half a century. His predecessor, Peter Turner, was a royalist and had been removed by parliament. Wallis's mathematical accomplishments were hitherto very limited. In 1648 he had composed a Treatise of Angular Sections (which remained unpublished until 1685), in which he developed results from William Oughtred's Clavis mathematicae (1631), a book he read about 1647 or 1648. In 1648 he provided an explanation of Descartes's treatment of fourth-degree equations to the Cambridge Platonist John Smith (1618-1652), lecturer of mathematics and fellow of Emmanuel College and later of Queens'. Within a few years Wallis was to become one of the leading mathematicians of his time.

In 1653 Wallis published Grammatica linguae Anglicanae, with a treatise De loquela and a Praxis grammatica. This work was often reprinted and has been praised by linguists for its deep insights and careful attention to sounds. De loquela served as a theoretical basis for Wallis's attempts to teach the profoundly deaf how to speak. He reported on his attempts to teach Alexander Popham in the Philosophical Transactions, though failure to mention Popham's previous instructor, William Holder, led to a bitter confrontation.

Wallis was admitted doctor of divinity in 1654. In 1658 he succeeded, by a somewhat doubtful procedure, Gerard Langbaine the elder as keeper of the university archives. His election elicited the protest of Henry Stubbe in The Savilian Professor's Case Stated. Despite such inauspicious beginnings, Wallis rendered valuable services to the university archives by putting the records and other papers under his care into such exact order, and managing its lawsuits with such success, that he apparently convinced even those opposed to his election that he had been an excellent choice for that post. His repertory of the entire collection was not replaced until the twentieth century.

Wallis was confirmed in his posts by Charles II in 1660, was made a royal chaplain, and in 1661 was appointed among the divines commissioned to revise the prayer book. He remained a loyal member of the official church throughout his life. Royal favour, together with his strenuous denials, are sufficient grounds for doubting the accusations that during the civil war he deciphered important letters relating to the royal family and public safety, making them available to the parliamentarians. In 1653 Wallis had deposited in the Bodleian Library a partial collection of the letters he had deciphered; these were later published by John Davys in his Essay on the Art of Decyphering (1737). Wallis continued to decipher intercepted letters on behalf of the government for many years. In old age he passed the art to his grandson William Blencowe, but refused to impart it to Leibniz, who had requested information on it on behalf of his rulers.

As Savilian professor, Wallis had to lecture on the Elements by Euclid, Conics by Apollonius, the works by Archimedes, and teach introductory courses in arithmetic. The statutes suggested also lectures on subjects such as cosmography, trigonometry, applied geometry, mechanics, and the theory of music. In the mid-1650s Wallis published several mathematical treatises within a couple of years, under the headings Operum mathematicorum pars prima (1657) and Operum mathematicorum pars altera (1656). They include, in the first volume, his Oratio inauguralis: mathesis universalis, seu opus arithmeticum, and other minor works. The second volume was an outcome of his university lectures, and in it Wallis stressed the importance of a unified notation and was influenced by William Oughtred. It also contained De angulo contactus et semicirculo disquisitio geometrica, already mentioned, and De sectionibus conicis, which dealt with a classical subject in a new fashion, namely, following the treatment introduced by Descartes. In this work Wallis introduced the sign for infinity, ∞, and used 1/∞ to represent an infinitesimal height. Moreover, Wallis employed the method of indivisibles in the version developed by Evangelista Torricelli, the Italian mathematician and pupil of Galileo. The method was and still is attributed to another pupil of Galileo, Bonaventura Cavalieri, whose books Wallis repeatedly sought in vain in bookshops.

Wallis's main work, Arithmetica infinitorum, was also included in the second volume of Opera mathematicorum, though it had already been printed separately and distributed in 1655 with a dedication to William Oughtred. Newton was greatly influenced by this work when he studied it in the winter of 1664-5. Wallis was able to find an infinite series expressing the value of 4/π by an ingenious series of interpolations. Indeed, the very term for the method was given by Wallis, who relied in this work on techniques that appear closely related to those he had devised for deciphering coded letters. As a result of his position and publications, Wallis was engaged in many mathematical challenges and disputes, notably with Pierre de Fermat, Blaise Pascal, and especially Thomas Hobbes, who in 1655 had claimed to possess an absolute quadrature of the circle. The virulent controversy, in which Hobbes also claimed erroneously to have duplicated the cube, thus solving another of the great mathematical problems, lasted for about a quarter-century, ending with Hobbes's death in 1679.

Following the discussion on the laws of impact at the Royal Society in 1668, Wallis, Christopher Wren, and Christiaan Huygens submitted papers. Wallis's was published in the Philosophical Transactions for that year and was followed by Mechanica, sive, De motu tractatus geometricus (1670-71), the major and most comprehensive work on a range of mechanical problems--such as statics, impact laws, and centres of gravity--published in England to that point. His last great mathematical work, A Treatise of Algebra, both Historical and Practical (1685) in 100 chapters, combined technical and historical exposition. An expanded Latin edition was included in the second volume of Wallis's Opera mathematica of 1693. The historical account was heavily biased towards English achievements and went as far as to claim that Descartes had gained knowledge in algebra from Thomas Harriot. The Algebra also included a discussion of the methods of exhaustion, of indivisibles, and of infinite series. Moreover, the work included the first printed account, expanded in the later edition, of some of Newton's achievements. Wallis's determination to defend English priority made him press his case with Newton, lest the glory of his inventions went to a foreigner. The third and last volume of Wallis's Opera mathematica (1693-9) contained a collection of letters on the priority dispute between Newton and Leibniz, the main items being Newton's so-called epistola prior and epistola posterior of 1676. In Institutio logicae (1687) Wallis put forward influential views concerning ways of treating singular terms as if they were general and conditional statements.

Wallis was also an active editor of works in the mathematical disciplines, including Archimedes' Arenarius et dimensio circuli (1676), Jeremiah Horrocks's Opera posthuma (1672-3 and 1678), Ptolemy's Harmonicorum libri tres (1682), Aristarchus's De magnitudinibus et distantiis solis et lunae (1688), and a fragment from Pappus (1688). His editorial work stimulated discussions at the Royal Society on musical theory, astronomy, and the theory of tides, and publications in the Philosophical Transactions.

Wallis's wife died on 27 March 1687, leaving him with one son and two daughters: John, Anne, and Elizabeth. John married Elizabeth Harris and had three children, Anne married Sir John Blencowe and had one child, while Elizabeth married William Benson and had no children.

Between 1690 and 1692 Wallis published eight letters and three sermons in defence of the Trinity against unitarian doctrines. He employed a mathematical analogy, arguing that the mystery of the Trinity could be grasped with the help of a cubical body with three dimensions: 'This longum, latum, profundum, (Long, Broad, and Tall), is but One Cube; of Three Dimensions, and yet but One Body. And this Father, Son, and Holy-Ghost: three persons, and yet but One God' (Archibald, 36). Wallis's theological works were often praised for their clarity and straightforward language. In 1692 he successfully opposed the introduction of the Gregorian calendar in Britain.

Throughout his life Wallis enjoyed excellent health and remarkable intellectual powers, with a prodigious memory for figures. Only in his eighties did he complain that his sight, hearing, and strength were not as they used to be. He died at Oxford on 8 November 1703, aged eighty-six, and was buried in St Mary's Church, Oxford, where his son placed a mural monument in his honour.


C. J. Scriba, 'The autobiography of John Wallis', Notes and Records of the Royal Society, 25 (1970), 17-46
C. J. Scriba, Studien zur Mathematik des John Wallis (1616-1703) (1966)
J. F. Scott, The mathematical work of John Wallis, D.D., F.R.S. (1616-1703) (1938)
C. J. Scriba, 'A tentative index of the correspondence of John Wallis', Notes and Records of the Royal Society, 22 (1967), 58-93
C. J. Scriba, 'Wallis, John', DSB
R. C. Archibald, 'Wallis on the Trinity', American Mathematical Monthly, 43 (1936), 35-7
D. E. Smith, 'John Wallis as a cryptographer', Bulletin of the American Mathematical Society, 24 (1917), 82-96
R. S. Westfall, Force in Newton's physics (1979), 231-44
J. L. Subbiondo, 'John Wallis' Grammatica linguae Anglicanae', Diversions of Galway, ed. A. Ahlqvist and others (1992)
L. Maierù, Fra Descartes e Newton: Isaac Barrow e John Wallis (1994)
K. Hill, 'Neither ancient nor modern: Wallis and Barrow on the composition of continua', Notes and Records of the Royal Society, 50 (1996), 165-78
A. R. Hall, Philosophers at war: the quarrel between Newton and Leibniz (1980)

Archbishop Marsh's Library, Dublin, paper on theory of music
BL, corresp. and papers, Sloane MSS 2284, 4025; Add. MS 32499
Bodl. Oxf., corresp. and papers
Bodl. Oxf., political corresp. in cipher [copies]
CKS, autobiography
RS, treatise on logic |  Bibliotheek der Rijksuniversiteit Leiden, corresp. with Christiaan Huygens
BL, letters to Sir Hans Sloane, Sloane MS 4025
Bodl. Oxf., corresp. with Sir Thomas Smith
Christ Church Oxf., MSS relating to the Paschal and Calendar
Leics. RO, corresp. with Lord Nottingham
NRA, priv. coll., letters to John Collins
RS, letters to Henry Oldenburg
RS, letters to Sir Hans Sloane

D. Loggan, line engraving, 1678, BM, NPG
G. Soest, oils, before 1681, RS
W. Sonmans, portrait, 1698
M. Burghers, line engraving, 1699 (after W. Sonmans), BM, NPG, V&A; repro. in M. Burghers, Opera mathematica (1699)
G. Kneller, oils, 1701, Bodl. Oxf. [see illus.]
W. Townesend, bust, 1703, St Mary's Church, Oxford
W. Faithorne, line engraving, BM, NPG; repro. in J. Wallis, Mechanica (1670)
G. Kneller, portrait
oils (after G. Kneller, 1701), NPG

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