John Wallis

John Wallis was an English clergyman and mathematician, who is given partial credit for the development of infinitesimal calculus.
Between 1643 and 1689 Wallis served as chief cryptographer for Parliament and, later, the royal court. He is credited with introducing the symbol ∞ to represent the concept of infinity. He similarly used 1/∞ for an infinitesimal. He was a contemporary of Isaac Newton and one of the greatest intellectuals of the early modern mathematics.

Biography

Educational background

Cambridge, M.A., Oxford, D.D.
Grammar School at Tenterden, Kent, 1625–31.
School of Martin Holbeach at Felsted, Essex, 1631–2.
Cambridge University, Emmanuel College, 1632–40; B.A., 1637; M.A., 1640.
D.D. at Oxford in 1654.
Family

On 14 March 1645, he married Susanna Glynde. They had three children:

Anne, Lady Blencowe, married Sir John Blencowe in 1675, with issue
, MP for Wallingford 1690–1695, married Elizabeth Harris on 1 February 1682, with issue: one son and two daughters
Elizabeth Wallis, married William Benson of Towcester, died with no issue
Life

John Wallis was born in Ashford, Kent. He was the third of five children of Revd. John Wallis and Joanna Chapman. He was initially educated at a school in Ashford but moved to James Movat's school in Tenterden in 1625 following an outbreak of plague. Wallis was first exposed to mathematics in 1631, at Felsted School ; he enjoyed maths, but his study was erratic, since "mathematics, at that time with us, were scarce looked on as academical studies, but rather mechanical". At the school in Felsted, Wallis learned how to speak and write Latin. By this time, he also was proficient in French, Greek, and Hebrew. As it was intended he should be a doctor, he was sent in 1632 to Emmanuel College, Cambridge. While there, he kept an act on the doctrine of the circulation of the blood; that was said to have been the first occasion in Europe on which this theory was publicly maintained in a disputation. His interests, however, centred on mathematics. He received his Bachelor of Arts degree in 1637 and a Master's in 1640, afterwards entering the priesthood. From 1643 to 1649, he served as a nonvoting scribe at the Westminster Assembly. He was elected to a fellowship at Queens' College, Cambridge in 1644, from which he had to resign following his marriage.
Throughout this time, Wallis had been close to the Parliamentarian party, perhaps as a result of his exposure to Holbeach at Felsted School. He rendered them great practical assistance in deciphering Royalist dispatches. The quality of cryptography at that time was mixed; despite the individual successes of mathematicians such as François Viète, the principles underlying cipher design and analysis were very poorly understood. Most ciphers were ad hoc methods relying on a secret algorithm, as opposed to systems based on a variable key. Wallis realised that the latter were far more secure – even describing them as "unbreakable", though he was not confident enough in this assertion to encourage revealing cryptographic algorithms. He was also concerned about the use of ciphers by foreign powers, refusing, for example, Gottfried Leibniz's request of 1697 to teach Hanoverian students about cryptography.
Returning to London – he had been made chaplain at St Gabriel Fenchurch in 1643 – Wallis joined the group of scientists that was later to evolve into the Royal Society. He was finally able to indulge his mathematical interests, mastering William Oughtred's Clavis Mathematicae in a few weeks in 1647. He soon began to write his own treatises, dealing with a wide range of topics, which he continued for the rest of his life. Wallis wrote the first survey about mathematical concepts in England where he discussed the Hindu-Arabic system.
Wallis joined the moderate Presbyterians in signing the remonstrance against the execution of Charles I, by which he incurred the lasting hostility of the Independents. In spite of their opposition he was appointed in 1649 to the Savilian Chair of Geometry at Oxford University, where he lived until his death on. In 1650, Wallis was ordained as a minister. After, he spent two years with Sir Richard Darley and Lady Vere as a private chaplain. In 1661, he was one of twelve Presbyterian representatives at the Savoy Conference.
Besides his mathematical works he wrote on theology, logic, English grammar and philosophy, and he was involved in devising a system for teaching a deaf boy to speak at Littlecote House. William Holder had earlier taught a deaf man, Alexander Popham, to speak "plainly and distinctly, and with a good and graceful tone". Wallis later claimed credit for this, leading Holder to accuse Wallis of "rifling his Neighbours, and adorning himself with their spoyls".

Wallis' appointment as Savilian Professor of Geometry at the Oxford University

The Parliamentary visitation of Oxford, that began in 1647, removed many senior academics from their positions, including in November 1648, the Savilian Professors of Geometry and Astronomy. In 1649 Wallis was appointed as Savilian Professor of Geometry. Wallis seems to have been chosen largely on political grounds ; while Wallis was perhaps the nation's leading cryptographer and was part of an informal group of scientists that would later become the Royal Society, he had no particular reputation as a mathematician. Nonetheless, Wallis' appointment proved richly justified by his subsequent work during the 54 years he served as Savilian Professor.

Contributions to mathematics

Wallis made significant contributions to trigonometry, calculus, geometry, and the analysis of infinite series. In his Opera Mathematica I he introduced the term "continued fraction".

Analytic geometry

In 1655, Wallis published a treatise on conic sections in which they were defined analytically. This was the earliest book in which these curves are considered and defined as curves of the second degree. It helped to remove some of the perceived difficulty and obscurity of René Descartes' work on analytic geometry.
In Treatise on the Conic Sections, Wallis popularised the symbol ∞ for infinity. He wrote, "I suppose any plane to be made up of an infinite number of parallel lines, or as I would prefer, of an infinite number of parallelograms of the same altitude; and the altitude of all to make up the altitude of the figure."

Integral calculus

Arithmetica Infinitorum, the most important of Wallis's works, was published in 1656. In this treatise the methods of analysis of Descartes and Cavalieri were systematised and extended, but some ideas were open to criticism. He began, after a short tract on conic sections, by developing the standard notation for powers, extending them from positive integers to rational numbers:
Leaving the numerous algebraic applications of this discovery, he next proceeded to find, by integration, the area enclosed between the curve y = x^m, x-axis, and any ordinate x = h, and he proved that the ratio of this area to that of the parallelogram on the same base and of the same height is 1/, extending Cavalieri's quadrature formula. He apparently assumed that the same result would be true also for the curve y = ax^m, where a is any constant, and m any number positive or negative, but he discussed only the case of the parabola in which m = 2 and the hyperbola in which m = −1. In the latter case, his interpretation of the result is incorrect. He then showed that similar results may be written down for any curve of the form
and hence that, if the ordinate y of a curve can be expanded in powers of x, its area can be determined: thus he says that if the equation of the curve is y = x⁰ + x¹ + x² +..., its area would be x + x²/2 + x³/3 +... . He then applied this to the quadrature of the curves,,, etc., taken between the limits x = 0 and x = 1. He shows that the areas are, respectively, 1, 1/6, 1/30, 1/140, etc. He next considered curves of the form and established the theorem that the area bounded by this curve and the lines x = 0 and x = 1 is equal to the area of the rectangle on the same base and of the same altitude as m : m + 1. This is equivalent to computing
He illustrated this by the parabola, in which case m = 2. He stated, but did not prove, the corresponding result for a curve of the form y = x^p/''q.
Wallis showed considerable ingenuity in reducing the equations of curves to the forms given above, but, as he was unacquainted with the binomial theorem, he could not effect the quadrature of the circle, whose equation is, since he was unable to expand this in powers of x''. He laid down, however, the principle of interpolation. Thus, as the ordinate of the circle is the geometrical mean of the ordinates of the curves and, it might be supposed that, as an approximation, the area of the semicircle which is might be taken as the geometrical mean of the values of
that is, and ; this is equivalent to taking or 3.26... as the value of π. But, Wallis argued, we have in fact a series... and therefore the term interpolated between and ought to be chosen so as to obey the law of this series. This, by an elaborate method that is not described here in detail, leads to a value for the interpolated term which is equivalent to taking
.
In this work the formation and properties of continued fractions are also discussed, the subject having been brought into prominence by Brouncker's use of these fractions.
A few years later, in 1659, Wallis published a tract containing the solution of the problems on the cycloid which had been proposed by Blaise Pascal. In this he incidentally explained how the principles laid down in his Arithmetica Infinitorum could be used for the rectification of algebraic curves and gave a solution of the problem to rectify the semicubical parabola x³ = ay², which had been discovered in 1657 by his pupil William Neile. Since all attempts to rectify the ellipse and hyperbola had been ineffectual, it had been supposed that no curves could be rectified, as indeed Descartes had definitely asserted to be the case. The logarithmic spiral had been rectified by Evangelista Torricelli and was the first curved line whose length was determined, but the extension by Neile and Wallis to an algebraic curve was novel. The cycloid was the next curve rectified; this was done by Christopher Wren in 1658.
Early in 1658 a similar discovery, independent of that of Neile, was made by van Heuraët, and this was published by van Schooten in his edition of Descartes's Geometria in 1659. Van Heuraët's method is as follows. He supposes the curve to be referred to rectangular axes; if this is so, and if are the coordinates of any point on it, and n is the length of the normal, and if another point whose coordinates are is taken such that η : h = n : y, where h is a constant; then, if ds is the element of the length of the required curve, we have by similar triangles ds : dx = n : y. Therefore, h ds = η ''dx. Hence, if the area of the locus of the point can be found, the first curve can be rectified. In this way van Heuraët effected the rectification of the curve y''³ = ax² but added that the rectification of the parabola y² = ax is impossible since it requires the quadrature of the hyperbola. The solutions given by Neile and Wallis are somewhat similar to that given by van Heuraët, though no general rule is enunciated, and the analysis is clumsy. A third method was suggested by Fermat in 1660, but it is inelegant and laborious.