John Wallis (23 November 1616 – 28 October 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 also credited with introducing the symbol {infty} for infinity. He similarly used 1/∞ for an infinitesimal. Asteroid 31982 Johnwallis was named after him.

John Wallis was born in Ashford, Kent, the third of five children of Reverend John Wallis and Joanna Chapman. He was initially educated at a local Ashford school, 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 1643–49, he served as a non-voting scribe at the Westminster Assembly. Wallis was elected to a fellowship at Queens’ College, Cambridge in 1644, which he however 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, continuing throughout 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 be the Savilian Chair of Geometry at Oxford University, where he lived until his death on 28 October 1703.

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.

Although 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 made significant contributions to trigonometry, calculus, geometry, and the analysis of infinite series. In his Opera Mathematica I (1695) Wallis 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 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 where numbers are represented geometrically in a line with the negative numbers represented by lengths opposite in direction to lengths of positive numbers.

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. It was in the Treatise on the Conic Sections that John 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.”

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.

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.

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.

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