John Wallis (1616–1703), Oxford’s Savilian Professor of Geometry from 1649 to 1703, was the most influential English mathematician before the rise of Isaac Newton. His most important works were his Arithmetic of Infinitesimals 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 binomial theorem. 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. Henry 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 William Oughtred’s Clavis Mathematicae (Key to Mathematics) 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 Euclid’s parallel postulate 1655–79 Disputes with Thomas Hobbes 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 University of Oxford 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 Archimedes. 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 mathematicians 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 Christopher Wren, 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 Astronomy. 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. Isaac Newton. 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 science, principles underlying many of philosophy 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, John Wilkins, another spring scales’. Christopher Wren, Gresham as topics reflecting his interest in the Civil War appointment. Wilkins Professor of Astronomy. Wren history of mathematics. 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 astronomer, 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 France, Italy and the that Wallis made in the 1650s was yet to be introduced, such as those a New Method of Inquiring into Netherlands: analytic geometry his investigation of conic sections. for fractional and negative indices, the Quadrature 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 infinitesimal dimensional geometry. “√x whose index is ½”. Approaches using geometrical quantities, 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 areas under curves, and integral calculus 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 number, 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 mathematical notation 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 ‘infinity’ 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.