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John Wallis (; ; ) was an English clergyman and
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
, who is given partial credit for the development of infinitesimal calculus. Between 1643 and 1689 Wallis served as chief cryptographer for
Parliament In modern politics and history, a parliament is a legislative body of government. Generally, a modern parliament has three functions: Representation (politics), representing the Election#Suffrage, electorate, making laws, and overseeing ...
and, later, the royal court. He is credited with introducing the
symbol A symbol is a mark, Sign (semiotics), sign, or word that indicates, signifies, or is understood as representing an idea, physical object, object, or wikt:relationship, relationship. Symbols allow people to go beyond what is known or seen by cr ...
∞ to represent the concept of
infinity Infinity is something which is boundless, endless, or larger than any natural number. It is denoted by \infty, called the infinity symbol. From the time of the Ancient Greek mathematics, ancient Greeks, the Infinity (philosophy), philosophic ...
. He similarly used 1/∞ for an infinitesimal. He was a contemporary of Newton and one of the greatest intellectuals of the early renaissance of
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
.


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 ( – 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 Latin ( or ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken by the Latins (Italic tribe), Latins in Latium (now known as Lazio), the lower Tiber area aroun ...
. By this time, he also was proficient in French, Greek, and
Hebrew Hebrew (; ''ʿÎbrit'') is a Northwest Semitic languages, Northwest Semitic language within the Afroasiatic languages, Afroasiatic language family. A regional dialect of the Canaanite languages, it was natively spoken by the Israelites and ...
. 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 In mathematics and computer science, an algorithm () is a finite sequence of Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algo ...
, 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 Gottfried Wilhelm Leibniz (or Leibnitz; – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat who is credited, alongside Isaac Newton, Sir Isaac Newton, with the creation of calculus in ad ...
'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 The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, re ...
. 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.4 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 Presbyterianism is a historically Reformed Protestant tradition named for its form of church government by representative assemblies of elders, known as "presbyters". Though other Reformed churches are structurally similar, the word ''Pr ...
representatives at the Savoy Conference. Besides his mathematical works he wrote on
theology Theology is the study of religious belief from a Religion, religious perspective, with a focus on the nature of divinity. It is taught as an Discipline (academia), academic discipline, typically in universities and seminaries. It occupies itse ...
,
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...
,
English grammar English grammar is the set of structural rules of the English language. This includes the structure of words, phrases, clauses, Sentence (linguistics), sentences, and whole texts. Overview This article describes a generalized, present-day Standar ...
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 (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 The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, re ...
, 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 Trigonometry () is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. The fiel ...
,
calculus Calculus is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the ...
,
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, 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 René Descartes ( , ; ; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and Modern science, science. Mathematics was paramou ...
' 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 number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set of all ...
s: :x^0 = 1 :x^ = \frac :x^ = \frac \text :x^ = \sqrt :x^ = \sqrt \text :x^ = \sqrt /math> :x^ = \sqrt /math> Leaving the numerous algebraic applications of this discovery, he next proceeded to find, by integration, the
area Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...
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/(''m'' + 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 mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exactl ...
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 :y = \sum_^ ax^ 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'' = ''x''0 + ''x''1 + ''x''2 + ..., its area would be ''x'' + x2/2 + ''x''3/3 + ... . 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 :\int_0^1 x^\,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'' = ''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 y = \sqrt, since he was unable to expand this in powers of ''x''. He laid down, however, the principle of
interpolation In the mathematics, mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points. In engineering and science, one ...
. Thus, as the ordinate of the
circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
y = \sqrt is the geometrical mean of the ordinates of the curves y = (1 - x^2)^0 and y = (1 - x^2)^1, it might be supposed that, as an approximation, the area of the semicircle \int_0^1 \!\sqrt\, dx which is \tfrac\pi might be taken as the geometrical mean of the values of :\int_0^1 (1 - x^2)^0 \, dx \ \text \int_0^1 (1 - x^2)^1 \, dx, that is, 1 and \tfrac; this is equivalent to taking 4 \sqrt or 3.26... as the value of π. But, Wallis argued, we have in fact a series 1, \tfrac, \tfrac, \tfrac,... and therefore the term interpolated between 1 and \tfrac 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 :\frac = \frac\cdot\frac\cdot\frac\cdot\frac\cdot\frac\cdot\frac\cdots (which is now known as the Wallis product). 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 (i.e., find the length of) the semicubical parabola ''x''3 = ''ay''2, which had been discovered in 1657 by his pupil William Neile. Since all attempts to rectify the
ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
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 A logarithmic spiral, equiangular spiral, or growth spiral is a self-similarity, self-similar spiral curve that often appears in nature. The first to describe a logarithmic spiral was Albrecht Dürer (1525) who called it an "eternal line" ("ewi ...
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 Sir Christopher Wren FRS (; – ) was an English architect, astronomer, mathematician and physicist who was one of the most highly acclaimed architects in the history of England. Known for his work in the English Baroque style, he was ac ...
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 (''x'', ''y'') are the coordinates of any point on it, and ''n'' is the length of the normal, and if another point whose coordinates are (''x'', ''η'') 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 (''x'', ''η'') can be found, the first curve can be rectified. In this way van Heuraët effected the rectification of the curve ''y''3 = ''ax''2 but added that the rectification of the parabola ''y''2 = ''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 Pierre de Fermat (; ; 17 August 1601 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is recognized for his d ...
in 1660, but it is inelegant and laborious.


Collision of bodies

The theory of the collision of bodies was propounded by the
Royal Society The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, re ...
in 1668 for the consideration of mathematicians. Wallis,
Christopher Wren Sir Christopher Wren FRS (; – ) was an English architect, astronomer, mathematician and physicist who was one of the most highly acclaimed architects in the history of England. Known for his work in the English Baroque style, he was ac ...
, and
Christiaan Huygens Christiaan Huygens, Halen, Lord of Zeelhem, ( , ; ; also spelled Huyghens; ; 14 April 1629 – 8 July 1695) was a Dutch mathematician, physicist, engineer, astronomer, and inventor who is regarded as a key figure in the Scientific Revolution ...
sent correct and similar solutions, all depending on what is now called the
conservation of momentum In Newtonian mechanics, momentum (: momenta or momentums; more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. ...
; 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 Statics is the branch of classical mechanics that is concerned with the analysis of force and torque acting on a physical system that does not experience an acceleration, but rather is in mechanical equilibrium, equilibrium with its environment ...
(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 :''s''1 : ''s''2 = ''v''1''t''1 : ''v''2''t''2.


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 +; but to be interpreted in a contrary sense... +3, signifies 3 Yards Forward; and -3, signifies 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 \tfrac and considering what happens as x approaches and then crosses the point x = 0 from the positive side. Wallis was not alone in this thinking:
Leonhard Euler Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...
came to the same conclusion by considering the geometric series \tfrac = 1 + x + x^2 + \cdots, evaluated at x=2, followed by reasoning similar to Wallis's (he resolved the paradox by distinguishing different kinds of negative numbers).


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 Muḥammad ibn Muḥammad ibn al-Ḥasan al-Ṭūsī (1201 – 1274), also known as Naṣīr al-Dīn al-Ṭūsī (; ) or simply as (al-)Tusi, was a Persians, Persian polymath, architect, Early Islamic philosophy, philosopher, Islamic medicine, phy ...
, 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.


Musical theory

Wallis translated into Latin works of
Ptolemy Claudius Ptolemy (; , ; ; – 160s/170s AD) was a Greco-Roman mathematician, astronomer, astrologer, geographer, and music theorist who wrote about a dozen scientific treatises, three of which were important to later Byzantine science, Byzant ...
and Bryennius, and Porphyrius's commentary on Ptolemy. He also published three letters to Henry Oldenburg concerning tuning. He approved of
equal temperament An equal temperament is a musical temperament or Musical tuning#Tuning systems, tuning system that approximates Just intonation, just intervals by dividing an octave (or other interval) into steps such that the ratio of the frequency, frequencie ...
, 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 English grammar is the set of structural rules of the English language. This includes the structure of words, phrases, clauses, Sentence (linguistics), sentences, and whole texts. Overview This article describes a generalized, present-day Standar ...
, that remained in print well into the eighteenth century. He also published on theology.


Wallis as cryptographer

While employed as lady Vere's chaplain in 1642 Wallis was given an enciphered letter about the fall of Chicester 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 London is the Capital city, capital and List of urban areas in the United Kingdom, largest city of both England and the United Kingdom, with a population of in . London metropolitan area, Its wider metropolitan area is the largest in Wester ...
.Smith, p. 83 Because of his Parliamentarian sympathies Wallis was not employed as a cryptographer after the Stuart Restoration,De Leeuw (1999), p. 138 but after the
Glorious Revolution The Glorious Revolution, also known as the Revolution of 1688, was the deposition of James II and VII, James II and VII in November 1688. He was replaced by his daughter Mary II, Mary II and her Dutch husband, William III of Orange ...
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 Poland, officially the Republic of Poland, is a country in Central Europe. It extends from the Baltic Sea in the north to the Sudetes and Carpathian Mountains in the south, bordered by Lithuania and Russia to the northeast, Belarus and Ukrai ...
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.Smith, p. 87 But Wallis became nervous that the French might take action against him.De Leeuw (1999), p. 139 Wallis relationship with the German mathematician
Gottfried Wilhelm Leibniz Gottfried Wilhelm Leibniz (or Leibnitz; – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat who is credited, alongside Sir Isaac Newton, with the creation of calculus in addition to ...
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, pp. 83-84 Smith gives an example of the painstaking work that Wallis performed, as described by himself in a letter to Richard Hampden of 3 August 1689. In it he gives a detailed account of his work on a particular letter and the parts he had encountered difficulties with.Smith, pp. 85-87 Wallis' correspondence also shows details of the way he stood up for himself, when he thought he was under-appreciated, financially or otherwise. He lobbied enthusiastically, both on his own behalf, and that of his relatives, as witnessed by letters to Lord Nottingham, Richard Hampden and the MP Harbord Harbord that Smith quotes.Smith, pp. 89-93 In a letter to the English envoy to Prussia, James Johnston Wallis bitterly complains that a courtier of the Prussian Elector, by the name of Smetteau, had done him wrong in the matter of just compensation for services rendered to the Elector. In the letter he gives details of what he had done and gives advice on a simple substitution cipher for the use of Johnston himself.Smith, pp. 94-96 Wallis' contributions to the art of cryptography were not only of a "technological" character. De Leeuw points out that even the "purely scientific" contributions of Wallis to the science of
linguistics Linguistics is the scientific study of language. The areas of linguistic analysis are syntax (rules governing the structure of sentences), semantics (meaning), Morphology (linguistics), morphology (structure of words), phonetics (speech sounds ...
in the field of the "rationality" of Natural language as it developed over time, played a role in the development of cryptology as a science. Wallis' development of a model of English grammar, independent of earlier models based on Latin grammar, is a case in point of the way other sciences helped develop cryptology in his view.De Leeuw (2000), p. 9 Wallis tried to teach his own son John, and his grandson by his daughter Anne, William Blencowe the tricks of the trade. With William he was so successful that he could persuade the government to allow the grandson to get the survivorship of the annual pension of £100 Wallis had received in compensation for his cryptographic work. William Blencowe eventually succeeded Wallis as official Cryptographer to Queen Anne after Wallis' death in 1703.De Leeuw (1999), p.143


See also

* 31982 Johnwallis, an asteroid that was named after him * Invisible College * John Wallis Academy – former ChristChurch school in Ashford renamed in 2010 * Wallis's conical edge * Wallis' integrals


Notes


References


Sources

* The initial text of this article was taken from the
public domain The public domain (PD) consists of all the creative work to which no Exclusive exclusive intellectual property rights apply. Those rights may have expired, been forfeited, expressly Waiver, waived, or may be inapplicable. Because no one holds ...
resource: * * * * * * Stedall, Jacqueline, 2005, "Arithmetica Infinitorum" in Ivor Grattan-Guinness, ed., ''Landmark Writings in Western Mathematics''. Elsevier: 23–32. * Guicciardini, Niccolò (2012) "John Wallis as editor of Newton's Mathematical Work", ''Notes and Records of the Royal Society of London'' 66(1): 3–17
Jstor link
* Stedall, Jacqueline A. (2001) "Of Our Own Nation: John Wallis's Account of Mathematical Learning in Medieval England", Historia Mathematica 28: 73. * Wallis, J. (1691). A seventh letter, concerning the sacred Trinity occasioned by a second letter from W.J. / by John Wallis ... (Early English books online). London: Printed for Tho. Parkhurst ...


External links


The Correspondence
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John Wallis (1685) ''A treatise of algebra''
- digital facsimile, Linda Hall Library * * {{DEFAULTSORT:Wallis, John 1616 births 1703 deaths 17th-century English mathematicians Alumni of Emmanuel College, Cambridge British cryptographers British historians of mathematics Calculus Deaf education English logicians English male non-fiction writers English mathematicians English music theorists English Presbyterian ministers of the Interregnum (England) English Protestants Fellows of Queens' College, Cambridge Keepers of the Archives of the University of Oxford History of calculus Infinity Linguists of English Mathematics of infinitesimals Original fellows of the Royal Society Participants in the Savoy Conference People educated at Felsted School People from Ashford, Kent Savilian Professors of Geometry Westminster Divines