John Wallis (; Latin: Wallisius; 3 December [O.S. 23 November] 1616 – 8 November [O.S. 28 October] 1703) 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 extended Cavalieri's method of indivisibles through the use of interpolation. Applying Kepler's principle of continuity, he developed techniques for the evaluation of integrals. 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.
Contents
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 (c. 1600 – 16 March 1687). They had three children:
Anne, Lady Blencowe (4 June 1656 – 5 April 1718), married Sir John Blencowe (30 November 1642 – 6 May 1726) in 1675, with issue
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
Elizabeth Wallis (1658–1703), married William Benson (1649–1691) 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 (then known as 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). 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' 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 (as perhaps had been his Royalist predecessor Peter Turner, who despite his appointment to two professorships never published any mathematical works); 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 (1695) 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 (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:
x
0
=
1
{\displaystyle x^{0}=1}
x
−
1
=
1
x
{\displaystyle x^{-1}={\frac {1}{x}}}
x
−
n
=
1
x
n
etc.
{\displaystyle x^{-n}={\frac {1}{x^{n}}}{\text{ etc.}}}
x
1
/
2
=
x
{\displaystyle x^{1/2}={\sqrt {x}}}
x
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 Christiaan 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.
Number line
Wallis has been credited as the originator of the number line "for negative quantities" and "for operational purposes." This is based on a passage in his 1685 treatise on algebra in which he introduced a number line to illustrate the legitimacy of negative quantities:
Yet is not that Supposition (of Negative Quantities) either Unuseful or Absurd; when rightly understood. And though, as to the bare Algebraick Notation, it import a Quantity less than nothing: Yet, when it comes to a Physical Application, it denotes as Real a Quantity as if the Sign were
+
{\displaystyle +}
; but to be interpreted in a contrary sense...
+
3
{\displaystyle +3}
, signifies
3
{\displaystyle 3}
Yards Forward; and
−
3
{\displaystyle -3}
, signifies
3
{\displaystyle 3}
Yards Backward.
It has been noted that, in an earlier work, Wallis came to the conclusion that the ratio of a positive number to a negative one is greater than infinity. The argument involves the quotient
1
x
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 1298 on the parallel postulate. The book was based on his father's thoughts and 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.
Prince Rupert's Cube
Wallis demonstrated a solution to the Prince Rupert's cube, which involves finding the largest cube that can pass through a hole cut through a unit cube without splitting it into separate pieces.
Music theory
Wallis translated into Latin works of Ptolemy and Bryennius, and Porphyrius's commentary on Ptolemy. He also published three letters to Henry Oldenburg concerning tuning. He approved of equal temperament, which was being used in England's organs.
Other works
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.
Wallis as a cryptographer
While employed as Lady Vere's chaplain in 1642, Wallis was given an enciphered letter about the fall of Chichester which he managed to decipher within two hours. This started his career as a cryptographer. He was a moderate supporter of the Parliamentarian side in the First English Civil War and therefore worked as a decipherer of intercepted correspondence for the Parliamentarian leaders. For his services, he was rewarded with the Livings of St. Gabriel and St. Martin's in London.
Because of his Parliamentarian sympathies, Wallis was not employed as a cryptographer after the Stuart Restoration; after the Glorious Revolution, he was sought out by Lord Nottingham and frequently employed to decipher encrypted intercepted correspondence, though he thought that he was not always adequately rewarded for his work. King William III from 1689 also employed Wallis as a cryptographer, sometimes almost on a daily basis. Couriers would bring him letters to be decrypted and waited in front of his study for the product. The king took a personal interest in Wallis' work and well-being as witnessed by a letter he sent to Dutch Grand pensionary Anthonie Heinsius in 1689.
In these early days of the Williamite reign, directly obtaining foreign intercepted letters was a problem for the English, as they did not have the resources of foreign Black Chambers as yet, but allies like the Elector of Brandenburg without their own Black Chambers occasionally made gifts of such intercepted correspondence, like the letter of king Louis XIV of France to king John III Sobieski of Poland that king William in 1689 used to cause a crisis in French-Polish diplomatic relations. He was quite open about it and Wallis was rewarded for his role. But Wallis became nervous that the French might take action against him.
Wallis' relationship with the German mathematician Gottfried Wilhelm Leibniz was cordial. But Leibniz also had cryptographic interests and tried to get Wallis to divulge some of his trade secrets, which Wallis declined to do as a matter of patriotic principle.
Smith gives an example of the painstaking work that Wallis performed, as described by himself in an August 1689 letter to Richard Hampden. In it he gives a detailed account of his work on a particular letter and the parts he found difficult.