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Home , Johann Heinrich Lambert, John Wallis, Quadrature (mathematics), Treatise

(1616–1703), ’s Savilian Professor of from 1649 to 1703, was the most influential English before the rise of Isaac . His most important works were his Arithmetic of and his treatise on Conic Sections, both published in the 1650s. It was through studying the former that Newton came to discover his version of the . Wallis’s last great mathematical work A Treatise of Algebra was published in his seventieth year. John Wallis Wallis’ time-line

“In the year 1649 I removed to 1616 Born in Ashford, Kent Oxford, being then Publick 1632–40 Studied at Emmanuel College, Professor of Geometry, of the Cambridge Foundation of Sr. Savile. 1640 Ordained a priest in the And Mathematicks which Church of England had before been a pleasing 1642 Started deciphering secret codes diversion, was now to be for Oliver Cromwell’s intelligence my serious Study.” service during the English Civil Wars John Wallis 1647 Inspired by ’s Clavis Mathematicae (Key to ) which considerably extended his mathematical knowledge 1648 Initiated mathematical correspondence with continental scholars (Hevelius, van Schooten, etc.) 1649 Appointed Savilian Professor of Geometry at Oxford 31 October: Inaugural lecture 1655–56 Arithmetica Infinitorum (The Arithmetic of Infinitesimals) and De Sectionibus Conicis (On Conic Sections) 1658 Elected Oxford University Archivist 1663 11 July: Lecture on ’s 1655–79 Disputes with 1685 Treatise of Algebra 1693–99 Opera Mathematica 1701 Appointed England’s first official decipherer (alongside his grandson William Blencowe) 1703 Died in Oxford John Wallis (1616–1703) Savilian Professor

During the English Civil Wars John Wallis was appointed Savilian Professor of Geometry in 1649.

Wallis is appointed Oxford in the 1650s The introduction of John Wallis Prior to taking up the Savilian The Savilian statutes obliged The pedagogical concerns of the into the Chair John Wallis had little John Wallis to give introductory Savilian professors, Wallis and was caused by politics. During mathematical experience and courses in practical and theoretical Ward, were exemplified in their the early part of the Civil Wars enjoyed no public reputation arithmetic. In his courses Wallis enthusiastic promotion of Oughtred’s the University had been the as a mathematician. However, also lectured on Euclid’s Elements, Clavis from which both had learned Royalist headquarters, and in the a more far-sighted mathematical the Conics of Apollonius, and their algebra. With their support subsequent reckoning most college appointment on flimsier evidence the works of . a new Latin edition was published heads and fellows were deposed. is difficult to imagine. Wallis’s in Oxford in 1652. It contained as appointment to the Savilian Several Oxford an appendix a work of Oughtred’s The Savilian professors were Chair, which he held until his of the time learned their trade on sundials, translated by the expelled in 1648 for Royalist death fifty-four years later, marked from the writings of William 20-year-old , sympathies, and the Parliamentary the beginning of an intense period Oughtred, whose influential a student at Wadham College Commissioners replaced them of activity which established algebra text Clavis Mathematicae ‘from whom we may expect great by John Wallis as Professor of Oxford as the mathematical (Key to Mathematics) had been things’, in the prescient words of Geometry and Seth Ward as powerhouse of the nation, for a published in London in 1631. Oughtred’s preface. Professor of . Wallis time at least, and promoted John had been a moderate supporter Wallis as the most influential of of the revolutionary cause during English mathematicians before the Civil Wars. .

Wallis’s inaugural lecture as Savilian Professor of Geometry was A solar eclipse was observed at Oxford on 2 August 1654 by John Wallis, given on 31 October 1649 in the Geometry Lecture Room at the east Christopher Wren and Richard Rawlinson. end of the Schools quadrangle, now part of the Bodleian Library. John Wallis (1616–1703) Intellectual and scientific life

Throughout his career Wallis was closely involved with England’s intellectual and scientific activities.

Oxford contemporaries The Royal Society “Oxford’s achievements in the 17th Robert Hooke (1635–1703) was John Wallis was an active member Wallis was an early and active century were founded upon a rich interested in the mathematical of the Oxford experimental Fellow of the Royal Society. Along culture of experimental , principles underlying many of group which met with many of his contemporaries and between 1648 and 1660 the city his experiments. In this extract frequently at Wadham College in the Royal Society, he had housed probably the most dynamic from his diary for 21August and Oxford coffee houses and led remarkably broad interests. scientific community in Europe.” 1678 he records a visit to a coffee to the formation of the Royal Allan Chapman house with Christopher Wren, Society. The Royal Society was The pages of the Society’s where they exchanged information founded on 28 November 1660 Philosophical Transactions reflect At the centre of this community on their recent inventions, at Gresham College in London the range of things he had views on was the Warden of Wadham including Hooke’s ‘philosophicall following a lecture given by and wanted to communicate, as well College, , another spring scales’. Christopher Wren, Gresham as topics reflecting his interest in the Civil War appointment. Wilkins Professor of Astronomy. Wren . attracted Robert Boyle to Oxford, was later appointed Savilian had Christopher Wren as a Professor of Astronomy in Oxford. In 1685 Wallis, in the Philosophical student, and Robert Hooke as Transactions, supported his an illustrious protégé. William Brouncker, the first argument that one’s memory is President of the Royal Society, is the better at night by reporting that he left-hand figure in the frontispiece calculated the square root of 3 in of Thomas Sprat’s 1667 history his head to twenty decimal places, Although Sir Christopher Wren of the Society. Wallis encouraged arriving at the correct answer (1632–1723) is mainly remembered Brouncker’s mathematical 1.73205 08075 68877 29353, and as an architect, his early career was researches and published some retained it in his mind before as an , and he was one of Brouncker’s results in his books. writing it down the next day. of the outstanding geometers of the age.

Wren’s blend of mathematical and practical insights is seen in his A plaque commemorates the design of an engine for grinding High Street site of Boyle and hyperbolic lenses, shown to the Hooke’s laboratory. Royal Society in July 1669. John Wallis (1616–1703) Publications

Wallis’s main impact was through his publications. Two of his most important ones are featured here.

Arithmetica Infinitorum De Sectionibus Conicis John Wallis’s Arithmetica Infinitorum “… Wallis drew on ideas originally Another important contribution Even where useful notations were (Arithmetic of Infinitesimals, or developed in , Italy and the that Wallis made in the 1650s was yet to be introduced, such as those a New Method of Inquiring into Netherlands: his investigation of conic sections. for fractional and negative indices, the of Curves, and and the method of indivisibles. The conics and their properties had Wallis went far towards laying Other More Difficult Mathematical He handled them in his own been known from antiquity, but the the groundwork, writing in his Problems) appeared at a critical time way, and the resulting method of curves had been viewed as sections Arithmetica Infinitorum: for the development of mathematics. quadrature, based on the summation of a cone, arising from three- “1/x whose index is –1” and of indivisible or dimensional geometry. “√x whose index is ½”. Approaches using geometrical , was a crucial step towards indivisibles had previously been the development of a fully fledged In 1656 Wallis published his Wallis’s concern for mathematical used to find under curves, and some ten years later.” investigations of conic sections, symbolism is but one facet of a had been successful for any curve Jacqueline Stedall called De Sectionibus Conicis (On lifelong exploration of issues of of the form y = x n, where n is a Introduction to her translation Conic Sections). He regarded them language and communication. positive whole , but Wallis of Arithmetica Infinitorum as plane curves, with no reference to One of his first and most successful associated numerical values to the the cone after the initial derivation, books was not a mathematical indivisibles, allowing him to extend Among the mathematicians and obtained their properties treatise but an ‘English grammar’. the result to the case when n is influenced byArithmetica through the use of the techniques fractional. The word ‘interpolation’ Infinitorum was Isaac Newton: of algebraic analysis introduced by (in its mathematical sense) was it was through his study of this René Descartes. introduced by Wallis in this work. work in the mid-1660s that he came to discover the general binomial Although Wallis was often It was here that Wallis produced theorem. Newton was attracted conservative in his use of his celebrated exact formula for to Wallis’s fundamental engine he did 4/π – namely, of discovery, the exploration and introduce two new symbols that recognition of pattern. Wallis’s are still in current use: the ‘’ career had been set in motion by his sign and the symbol for ‘greater cryptological skills, and they seem to than or equal to’. have characterised his mathematical style as well. Wallis deciphered throughout his professional career, right up to the final days before his death; he was justly regarded as Europe’s greatest code-breaker, working mainly on complex French numerical substitution ciphers.

Wallis introduced the ‘infinity’ sign and the symbol for ‘greater than or equal to’ in his 1656 book on conic sections. “At the beginning of my mathematical studies, when I had met with the works of our celebrated Wallis, on considering the by the intercalation of which he himself exhibits the of the and the . . .” Isaac Newton on how he came to discover the general binomial theorem

Wallis’s Arithmetica Infinitorum Jacqueline Stedall’s translation was published in 1656. and commentary. John Wallis (1616–1703) Controversy and collaboration

John Wallis was involved with many fierce disputes, as well as productive collaborations.

Disputes with Thomas Hobbes The Sheldonian Theatre Oxford’s Sheldonian Theatre, An alternative approach to supporting designed on an ancient Roman a ceiling by beams that were much model by the Savilian Professor shorter than the length or width of the of Astronomy, Christopher Wren, ceiling had been worked out earlier by exemplifies the creative tension John Wallis in the 1650s. between antiquity and innovation that characterised the Wallis era. His interlocking beam structure In particular, the expanse of the needed support only where its Sheldonian’s flat ceiling, supported edges rested on the walls. Wallis by trusses, caused a sensation. had worked out the mathematics of these interlocking beams in an John Wallis Thomas Hobbes innovative calculation involving his solution of no fewer than twenty- Wallis was blessed with a formidable Hobbes’s new-found enthusiasm for five simultaneous equations. intellect, a prodigious memory, mathematics subsequently pervaded and a robust constitution. A man of his philosophical approach and style short temper and robust dialogue he of writing. His materialist and anti- also possessed a highly contentious clerical Leviathan (1651) created nature and created many enemies. widespread controversy, even before he aroused the ire of both Ward Wallis quarrelled with the French and Wallis through his attacks on mathematicians the post-revolutionary state and and Blaise , but his most performance of the universities, virulent dispute, lasting nearly a which he saw as riddled with quarter of a century from the mid- priestcraft and outmoded learning. 1650s, was with the philosopher, Thomas Hobbes (1588–1679). Thomas Hobbes’s claim in 1655 Oxford’s Sheldonian A diagram of Wallis’s interlocking A friend of , Galileo, that he had solved the ancient Theatre ceiling flat ceiling structure. Descartes, and Mersenne, Hobbes Greek problem of ‘squaring the was one of the outstanding circle’ (constructing a square intellectual figures of the age. equal in area to a given circle) drew a fierce reaction and rebuttal The following account from from John Wallis. Aubrey’s Brief Lives tells how The Parallel Postulate Hobbes became interested in The Parallel Postulate states that given any straight line and a point not mathematics. on it, there exists one and only one straight line which passes through that point and never intersects the first line, no matter how far they are “He was 40 years old before he looked extended. It is equivalent to Euclid’s Fifth Postulate. on Geometry; which happened accidentally. Being in a Gentleman’s As Wallis observed, his argument Library, Euclid’s Elements lay assumes that similar figures can take open, and ’twas the 47 El. libri I [the different sizes. Wallis found this in Book I]. assumption very plausible, and if it He read the Proposition. By G—, Euclid’s Fifth Postulate states that were true then the Parallel Postulate sayd he (he would now and then if the sum of the interior angles α would be a consequence of the other sweare an emphaticall Oath by way and β is less than 180°, then the two axioms of Euclid. of emphasis) this is impossible! So straight lines, produced indefinitely, he reads the Demonstration of it, meet on that side. It does, however, imply a remarkable which referred him back to such result: in any geometry in which a Proposition; which proposition On the evening of 11 July 1663 the parallel postulate does not hold, he read. That referred him back to Wallis lectured in Oxford on the similar figures must be identical in another, which he also read. Et sic parallel postulate, and presented a size as well as in shape, and so scale deinceps [and thus, in succession] seductive argument purporting to copies can never be made. that at last he was demonstratively derive it from Euclid’s other axioms. It would also have the consequence, convinced of that trueth. This made The title page from one of Wallis’s as Johann Heinrich Lambert him in love with Geometry.” tracts against Hobbes This lecture by Wallis was the first observed a century later, that mature Western attempt to derive there would have to be an absolute the parallel postulate as a theorem. measure of length. John Wallis (1616–1703) Final years

In his final years John Wallis published his influentialA Treatise of Algebra. In 1969 Oxford University established the Wallis Chair of Mathematics in his memory.

A Treatise of Algebra John Wallis’s portrait The idea of presenting modern Wallis retained his vigour to the mathematics together with an end. In 1699 an Oxford colleague account of its development was wrote to Samuel Pepys, a friend of itself a novel one – this was the first Wallis for over thirty years: substantial history of mathematics in the English language. “He says 83 is an incurable distemper. I believe Death will TheTreatise provides a full no more surprise him than a and judicious survey of the Proposition in Mathematicks.” achievements of past centuries. But when he came to his own Some measure of the impression century his past difficulties with that Wallis left on contemporaries French mathematicians influenced may be seen in the remarkable full- his judgement in rather a startling length portrait of him at the age of way as he left the reader in no doubt 86, by the court painter Sir Godfrey that recent French achievements had Kneller. This painting – in which a solidly English basis. the aged Wallis, swathed in scarlet like some Renaissance prince- Not only had Thomas Harriot prelate, stares out at the viewer in already made Descartes’s cold disdain – was commissioned John Wallis, by Sir Godfrey Kneller discoveries, Wallis asserted, but by Pepys for presentation to the (1702). The portrait hangs in the Descartes had actually plagiarized University of Oxford. Examination Schools in Oxford. Harriot’s results. Wallis’s last great mathematical work, A Treatise of Algebra, Both Historical and Practical, Wallis Professors of Mathematics was published in 1685, in his seventieth year. Of all his vast The Wallis Professorship of Mathematics was established in 1969 output, it was this work that was in honour of John Wallis. John Kingman held the professorship most widely read over the next from 1969 to 1985, followed by Simon Donaldson. Terry hundred years, and it remains Lyons succeeded Simon Donaldson in 1999 and has held the an extraordinary work. professorship since then.

John Kingman Simon Donaldson

John Wallis’s Collected Works were published in the 1690s. The third volume contained Wallis’s editions of works by , Archimedes, Aristarchus, Pappus and others. The first volume of an eight-volume collection of the correspondence of John Wallis. Terry Lyons

These posters were conceived by Raymond Flood and Robin Wilson with help from Philip Beeley, Adrian Rice and Dyrol Lumbard