John Wallis

John Wallis (3 December 1616 – 8 November 1703) was an English mathematician who is given partial credit for the development of infinitesimal calculus. Between 1643 and 1689 he served as chief cryptographer for Parliament and, later, the royal court. He is credited with introducing the symbol ∞ for infinity. He similarly used 1/∞ for an infinitesimal.

Wallis was born in Ashford, Kent, the third of five children of Reverend 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 Martin Holbeach's school in Felsted; 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" (Scriba 1970).

As it was intended that 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 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 28 October 1703 (O.S.). 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 deaf mutes. 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".

Opera mathematica, 1699

Wallis made significant contributions to trigonometry, calculus, geometry, and the analysis of infinite series. In his Opera Mathematica I (1695) he introduced the term "continued fraction".

Wallis rejected as absurd the now usual idea of a negative number as being less than nothing, but accepted the view that it is something greater than infinity. (The argument that negative numbers are greater than infinity involves the quotient ${\displaystyle {\frac {1}{x}}}$ and considering what happens as x approaches and then crosses the point x = 0 from the positive side.) Despite this he is generally credited as the originator of the idea of the number line, in which numbers are represented geometrically in a line with the negative numbers represented by lengths opposite in direction to lengths of positive numbers.

Analytical 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 the Treatise on the Conic Sections Wallis popularised the symbol ∞ for infinity. He wrote, “I suppose any plane (following the Geometry of Indivisibles of Cavalieri) 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; (let the altitude of each one of these be an infinitely small part 1/∞ of the whole altitude, and let the symbol ∞ denote Infinity) 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:

${\displaystyle x^{0}=1}$

${\displaystyle x^{-1}={\frac {1}{x}}}$

${\displaystyle x^{-n}={\frac {1}{x^{n}}}{\text{ etc.}}}$

${\displaystyle x^{1/2}={\sqrt {x}}}$

${\displaystyle x^{2/3}={\sqrt[{3}]{x^{2}}}{\text{ etc.}}}$

${\displaystyle x^{1/n}={\sqrt[{n}]{x}}}$

${\displaystyle x^{p/q}={\sqrt[{q}]{x^{p}}}}$

Leaving the numerous algebraic applications of this discovery, he next proceeded to find, by integration, the area enclosed between the curve y = xm, the axis of x, 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/(m + 1), extending Cavalieri's quadrature formula. He apparently assumed that the same result would be true also for the curve y = axm, 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

${\displaystyle y=\sum _{m}^{}ax^{m}}$

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 y = (x − x2)0, y = (x − x2)1, y = (x − x2)2, 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 y = x1/m 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

${\displaystyle \int _{0}^{1}x^{1/m}\,dx.}$

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 = xp/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 ${\displaystyle y={\sqrt {1-x^{2}}}}$, 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 ${\displaystyle y={\sqrt {1-x^{2}}}}$ is the geometrical mean of the ordinates of the curves ${\displaystyle y=(1-x^{2})^{0}}$ and ${\displaystyle y=(1-x^{2})^{1}}$, it might be supposed that, as an approximation, the area of the semicircle ${\displaystyle \int _{0}^{1}{\sqrt {1-x^{2}}}\,dx}$ which is ${\displaystyle {\begin{matrix}{\frac {1}{4}}\end{matrix}}\pi }$ might be taken as the geometrical mean of the values of

${\displaystyle \int _{0}^{1}(1-x^{2})^{0}\,dx{\text{ and }}\int _{0}^{1}(1-x^{2})^{1}\,dx}$

that is, 1 and ${\displaystyle {\begin{matrix}{\frac {2}{3}}\end{matrix}}}$; this is equivalent to taking ${\displaystyle 4{\sqrt {\begin{matrix}{\frac {2}{3}}\end{matrix}}}}$ or 3.26... as the value of π. But, Wallis argued, we have in fact a series ${\displaystyle 1,{\begin{matrix}{\frac {1}{6}}\end{matrix}},{\begin{matrix}{\frac {1}{30}}\end{matrix}},{\begin{matrix}{\frac {1}{140}}\end{matrix}},}$... and therefore the term interpolated between 1 and ${\displaystyle {\begin{matrix}{\frac {1}{6}}\end{matrix}}}$ 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

${\displaystyle {\frac {\pi }{2}}={\frac {2}{1}}\cdot {\frac {2}{3}}\cdot {\frac {4}{3}}\cdot {\frac {4}{5}}\cdot {\frac {6}{5}}\cdot {\frac {6}{7}}\cdots }$

(which is now known as the Wallis product).

In this work also the formation and properties of continued fractions are 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 (i.e., find the length of) 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 (necessarily) 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 (other than the circle) 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 be so, and if (x, y) be the coordinates of any point on it, and n be the length of the normal, and if another point whose coordinates are (x, η) be taken such that η : h = n : y, where h is a constant; then, if ds be 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 (x, η) can be found, the first curve can be rectified. In this way van Heuraët effected the rectification of the curve y3 = 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.

Collision of bodies

The theory of the collision of bodies was propounded by the Royal Society in 1668 for the consideration of mathematicians. Wallis, Christopher Wren, and Christian Huygens sent correct and similar solutions, all depending on what is now called the conservation of momentum; but, while Wren and Huygens confined their theory to perfectly elastic bodies (elastic collision), Wallis considered also imperfectly elastic bodies (inelastic collision). This was followed in 1669 by a work on statics (centres of gravity), and in 1670 by one on dynamics: these provide a convenient synopsis of what was then known on the subject.

Algebra

In 1685 Wallis published Algebra, preceded by a historical account of the development of the subject, which contains a great deal of valuable information. The second edition, issued in 1693 and forming the second volume of his Opera, was considerably enlarged. This algebra is noteworthy as containing the first systematic use of formulae. A given magnitude is here represented by the numerical ratio which it bears to the unit of the same kind of magnitude: thus, when Wallis wants to compare two lengths he regards each as containing so many units of length. This perhaps will be made clearer by noting that the relation between the space described in any time by a particle moving with a uniform velocity is denoted by Wallis by the formula

s = vt,

where s is the number representing the ratio of the space described to the unit of length; while the previous writers would have denoted the same relation by stating what is equivalent to the proposition

s₁ : s₂ = v₁t₁ : v₂t₂.

Geometry

He is usually credited with the proof of the Pythagorean theorem using similar triangles. However, Thabit Ibn Qurra (AD 901), an Arab mathematician, had produced a generalisation of the Pythagorean theorem applicable to all triangles six centuries earlier. It is a reasonable conjecture that Wallis was aware of Thabit's work.

Wallis was also inspired by the works of Islamic mathematician Sadr al-Tusi, the son of Nasir al-Din al-Tusi, particularly by al-Tusi's book written in AD 1298 on the parallel postulate. The book was based on his father's thoughts which presented one of the earliest arguments for a non-Euclidean hypothesis equivalent to the parallel postulate. After reading this, Wallis then wrote about his ideas as he developed his own thoughts about the postulate, trying to prove it also with similar triangles.

He found that Euclid's fifth postulate is equivalent to the one currently named "Wallis postulate" after him. This postulate states that "On a given finite straight line it is always possible to construct a triangle similar to a given triangle". This result was encompassed in a trend trying to deduce Euclid's fifth from the other four postulates which today is known to be impossible. Unlike other authors, he realised that the unbounded growth of a triangle was not guaranteed by the four first postulates.

Calculator

Another aspect of Wallis's mathematical skills was his ability to do mental calculations. He slept badly and often did mental calculations as he lay awake in his bed. One night he calculated in his head the square root of a number with 53 digits. In the morning he dictated the 27-digit square root of the number, still entirely from memory. It was a feat that was considered remarkable, and Henry Oldenburg, the Secretary of the Royal Society, sent a colleague to investigate how Wallis did it. It was considered important enough to merit discussion in the Philosophical Transactions of the Royal Society of 1685.

Controversy with Hobbes

A long-running debate between Wallis and Thomas Hobbes arose in the mid-1650s, when mathematicians criticised errors in the work De corpore by Hobbes. It continued into the 1670s, having gathered in the later claims of Hobbes on squaring the circle, and the wider beliefs on both sides.

Wallis translated into Latin works of Ptolemy, Bryennius, and Porphyrius's commentary on Ptolemy. He also published three letters to Henry Oldenburg concerning tuning. He approved of equal temperament that was being used in England's organs.

His Institutio logicae, published in 1687, was very popular. The Grammatica linguae Anglicanae was a work on English grammar, that remained in print well into the eighteenth century. He also published on theology.

On 14 March 1645 he married Susanna Glynde (16?? – 16 March 1687), They had three children:

1. Anne Blencoe (4 June 1656 – 5 April 1718), married Sir John Blencowe (30 November 1642 – 6 May 1726) in 1675, with issue
2. John Wallis (26 December 1650 – 14 March 1717),) MP for Wallingford 1690–1695, married Elizabeth Harris (d. 1693) on 1 February 1682, with issue: one son and two daughters
3. Elizabeth Wallis (1658–1703), married William Benson (1649–1691) of Towcester, died with no issue

• Joan Thirsk, ‘Blencowe , Anne, Lady Blencowe (1656–1718)’, Oxford Dictionary of National Biography, Oxford University Press, Oct 2005; online edn, Jan 2007 accessed 16 Nov 2016
• WALLIS, John (1650-1717), of Soundness, Nettlebed, Oxon. | History of Parliament Online
• Early Modern Letters Online : Person