John Wallis (; la, Wallisius; ) was an English clergyman and

^{''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 ^{0} + ''x''^{1} + ''x''^{2} + ..., its area would be ''x'' + x^{2}/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
:$\backslash int\_0^1\; x^\backslash ,dx.$
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 ^{3} = ''ay''^{2}, which had been discovered in 1657 by his pupil ^{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

_{1} : ''s''_{2} = ''v''_{1}''t''_{1} : ''v''_{2}''t''_{2}.

A Short Account of the History of Mathematics

'' 4th ed. * * 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 ...

The Correspondence

o

John Wallis

i

EMLO

* *

* * *

John Wallis (1685) ''A treatise of algebra''

- digital facsimile,

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, structure, space, models, and change.
History
On ...

who is given partial credit for the development of infinitesimal calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...

. Between 1643 and 1689 he served as chief cryptographer
Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of adver ...

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, or word that indicates, signifies, or is understood as representing an idea, object, or relationship. Symbols allow people to go beyond what is known or seen by creating linkages between otherwise very different conc ...

∞ to represent the concept of infinity
Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol .
Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions amo ...

. He similarly used 1/∞ for an infinitesimal
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...

. John Wallis was a contemporary of Newton and one of the greatest intellectuals of the early renaissance of mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in 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 1654Family

On 14 March 1645 he married Susanna Glynde ( – 16 March 1687). They had three children: #Anne Blencoe
Anne Blencowe or Anne, Lady Blencowe, née Anne Wallis (4 June 1656 – 6 April 1718) was a British compiler of recipes. Her book was first published more than 200 years after her death.
Life
Anne Wallis was born to Susanna Glyde and her husband P ...

(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 inAshford, Kent
Ashford is a town in the county of Kent, England. It lies on the River Stour, Kent, River Great Stour at the southern or Escarpment, scarp edge of the North Downs, about southeast of central London and northwest of Folkestone by road. In the ...

. He was 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
Tenterden is a town in the borough of Ashford in Kent, England. It stands on the edge of the remnant forest the Weald, overlooking the valley of the River Rother. It was a member of the Cinque Ports Confederation. Its riverside today is not ...

in 1625 following an outbreak of plague
Plague or The Plague may refer to:
Agriculture, fauna, and medicine
*Plague (disease), a disease caused by ''Yersinia pestis''
* An epidemic of infectious disease (medical or agricultural)
* A pandemic caused by such a disease
* A swarm of pe ...

. 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 branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...

. By this time, he also was proficient in French, Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece, a country in Southern Europe:
*Greeks, an ethnic group.
*Greek language, a branch of the Indo-European language family.
**Proto-Greek language, the assumed last common ancestor ...

, and Hebrew
Hebrew (; ; ) is a Northwest Semitic language of the Afroasiatic language family. Historically, it is one of the spoken languages of the Israelites and their longest-surviving descendants, the Jews and Samaritans. It was largely preserved ...

. As it was intended he should be a doctor, he was sent in 1632 to Emmanuel College, Cambridge
Emmanuel College is a constituent college of the University of Cambridge. The college was founded in 1584 by Sir Walter Mildmay, Chancellor of the Exchequer to Elizabeth I. The site on which the college sits was once a priory for Dominican mon ...

. While there, he kept an ''act'' on the doctrine of the circulation of the blood
The blood circulatory system is a system of organs that includes the heart, blood vessels, and blood which is circulated throughout the entire body of a human or other vertebrate. It includes the cardiovascular system, or vascular system, tha ...

; 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
The Westminster Assembly of Divines was a council of Divinity (academic discipline), divines (theologians) and members of the English Parliament appointed from 1643 to 1653 to restructure the Church of England. Several Scots also attended, and ...

. He was elected to a fellowship at Queens' College, Cambridge
Queens' College is a constituent college of the University of Cambridge. Queens' is one of the oldest colleges of the university, founded in 1448 by Margaret of Anjou. The college spans the River Cam, colloquially referred to as the "light s ...

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
François Viète, Seigneur de la Bigotière ( la, Franciscus Vieta; 1540 – 23 February 1603), commonly know by his mononym, Vieta, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to i ...

, 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 rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specificat ...

, as opposed to systems based on a variable key
Key or The Key may refer to:
Common meanings
* Key (cryptography), a piece of information that controls the operation of a cryptography algorithm
* Key (lock), device used to control access to places or facilities restricted by a lock
* Key (map ...

. 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 (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathem ...

'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
William Oughtred ( ; 5 March 1574 – 30 June 1660), also Owtred, Uhtred, etc., was an English mathematician and Anglican clergyman.'Oughtred (William)', in P. Bayle, translated and revised by J.P. Bernard, T. Birch and J. Lockman, ''A General ...

'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 Charles I may refer to:
Kings and emperors
* Charlemagne (742–814), numbered Charles I in the lists of Holy Roman Emperors and French kings
* Charles I of Anjou (1226–1285), also king of Albania, Jerusalem, Naples and Sicily
* Charles I of ...

, 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
The position of Savilian Professor of Geometry was established at the University of Oxford in 1619. It was founded (at the same time as the Savilian Professorship of Astronomy) by Sir Henry Savile, a mathematician and classical scholar who was ...

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
A chaplain is, traditionally, a cleric (such as a Minister (Christianity), minister, priest, pastor, rabbi, purohit, or imam), or a laity, lay representative of a religious tradition, attached to a secularity, secular institution (such as a hosp ...

. In 1661, he was one of twelve Presbyterian
Presbyterianism is a part of the Reformed tradition within Protestantism that broke from the Roman Catholic Church in Scotland by John Knox, who was a priest at St. Giles Cathedral (Church of Scotland). Presbyterian churches derive their nam ...

representatives at the Savoy Conference
The Savoy Conference of 1661 was a significant liturgical discussion that took place, after the Restoration of Charles II, in an attempt to effect a reconciliation within the Church of England.
Proceedings
It was convened by Gilbert Sheldo ...

.
Besides his mathematical works he wrote on theology
Theology is the systematic study of the nature of the divine and, more broadly, of religious belief. It is taught as an academic discipline, typically in universities and seminaries. It occupies itself with the unique content of analyzing the ...

, logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...

, 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.
This article describes a generalized, present-day Standard English ...

and philosophy, and he was involved in devising a system for teaching a deaf boy to speak at Littlecote House
Littlecote House is a large Elizabethan country house and estate in the civil parishes of Ramsbury and Chilton Foliat, in the English county of Wiltshire, about northeast of the Berkshire town of Hungerford. The estate includes 34 hectares of hi ...

. William Holder
William Holder FRS (1616 – 24 January 1698) was an English clergyman and music theorist of the 17th century. His most notable work was his widely known 1694 publication ''A Treatise on the Natural Grounds and Principles of Harmony''.
Life
He ...

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

TheParliamentary visitation of Oxford
The parliamentary visitation of the University of Oxford was a political and religious purge taking place from 1647, for a number of years. Many Masters and Fellows of Colleges lost their positions.
Background
A comparable but less prominent parli ...

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 totrigonometry
Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. T ...

, calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...

, geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, and the analysis of infinite series
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...

. In his ''Opera Mathematica'' I (1695) he introduced the term "continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer ...

".
Analytic geometry

In 1655, Wallis published a treatise onconic section
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a specia ...

s 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
Degree may refer to:
As a unit of measurement
* Degree (angle), a unit of angle measurement
** Degree of geographical latitude
** Degree of geographical longitude
* Degree symbol (°), a notation used in science, engineering, and mathematics
...

. It helped to remove some of the perceived difficulty and obscurity of René Descartes
René Descartes ( or ; ; Latinized: Renatus Cartesius; 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 science. Mathem ...

' work on analytic geometry
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.
Analytic geometry is used in physics and engineerin ...

.
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 torational 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 (e.g. ). The set of all ration ...

s:
:$x^0\; =\; 1$
:$x^\; =\; \backslash frac$
:$x^\; =\; \backslash frac\; \backslash text$
:$x^\; =\; \backslash sqrt$
:$x^\; =\; \backslash sqrt;\; href="/html/ALL/l/.html"\; ;"title="">$
:$x^\; =\; \backslash sqrt;\; href="/html/ALL/l/.html"\; ;"title="">$integration
Integration may refer to:
Biology
*Multisensory integration
*Path integration
* Pre-integration complex, viral genetic material used to insert a viral genome into a host genome
*DNA integration, by means of site-specific recombinase technology, ...

, the area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape
A shape or figure is a graphics, graphical representation of an obje ...

enclosed between the curve ''y'' = ''x''parabola
In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.
One descript ...

in which ''m'' = 2 and the hyperbola
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, cal ...

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
:$y\; =\; \backslash sum\_^\; ax^$
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''binomial theorem
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial into a sum involving terms of the form , where the ...

, he could not effect the quadrature of the circle
Squaring the circle is a problem in geometry first proposed in Greek mathematics. It is the challenge of constructing a square with the area of a circle by using only a finite number of steps with a compass and straightedge. The difficulty ...

, whose equation is $y\; =\; \backslash sqrt$, 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
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...

$y\; =\; \backslash sqrt$ 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 $\backslash int\_0^1\; \backslash !\backslash sqrt\backslash ,\; dx$ which is $\backslash tfrac\backslash pi$ might be taken as the geometrical mean of the values of
:$\backslash int\_0^1\; (1\; -\; x^2)^0\; \backslash ,\; dx\; \backslash \; \backslash text\; \backslash int\_0^1\; (1\; -\; x^2)^1\; \backslash ,\; dx,$
that is, $1$ and $\backslash tfrac$; this is equivalent to taking $4\; \backslash sqrt$ or 3.26... as the value of π. But, Wallis argued, we have in fact a series $1,\; \backslash tfrac,\; \backslash tfrac,\; \backslash tfrac,$... and therefore the term interpolated between $1$ and $\backslash tfrac$ 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
:$\backslash frac\; =\; \backslash frac\backslash cdot\backslash frac\backslash cdot\backslash frac\backslash cdot\backslash frac\backslash cdot\backslash frac\backslash cdot\backslash frac\backslash cdots$
(which is now known as the Wallis product
In mathematics, the Wallis product for , published in 1656 by John Wallis, states that
:\begin
\frac & = \prod_^ \frac = \prod_^ \left(\frac \cdot \frac\right) \\ pt& = \Big(\frac \cdot \frac\Big) \cdot \Big(\frac \cdot \frac\Big) \cdot \Big(\fr ...

).
In this work also the formation and properties of continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer ...

s 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
In geometry, a cycloid is the curve traced by a point on a circle as it rolls along a straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette, a curve generated by a curve rolling on another cu ...

which had been proposed by Blaise Pascal
Blaise Pascal ( , , ; ; 19 June 1623 – 19 August 1662) was a French mathematician, physicist, inventor, philosopher, and Catholic Church, Catholic writer.
He was a child prodigy who was educated by his father, a tax collector in Rouen. Pa ...

. 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''William Neile
William Neile (7 December 1637 – 24 August 1670) was an English mathematician and founder member of the Royal Society. His major mathematical work, the rectification of the semicubical parabola, was carried out when he was aged nineteen, and w ...

. 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-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" ("ewige Linie"). Mor ...

had been rectified by Evangelista Torricelli
Evangelista Torricelli ( , also , ; 15 October 160825 October 1647) was an Italian physicist and mathematician, and a student of Galileo. He is best known for his invention of the barometer, but is also known for his advances in optics and work o ...

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 PRS FRS (; – ) was one of the most highly acclaimed English architects in history, as well as an anatomist, astronomer, geometer, and mathematician-physicist. He was accorded responsibility for rebuilding 52 churches ...

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
In Euclidean geometry, two objects are similar if they have the same shape, or one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling (enlarging or reducing), possibly wit ...

''ds'' : ''dx'' = ''n'' : ''y''. Therefore, ''h ds'' = ''η'' ''dx''. Hence, if the area of the locus
Locus (plural loci) is Latin for "place". It may refer to:
Entertainment
* Locus (comics), a Marvel Comics mutant villainess, a member of the Mutant Liberation Front
* ''Locus'' (magazine), science fiction and fantasy magazine
** ''Locus Award' ...

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''Fermat
Pierre de Fermat (; between 31 October and 6 December 1607 – 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 ...

in 1660, but it is inelegant and laborious.
Collision of bodies

The theory of the collision of bodies was propounded by theRoyal 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 PRS FRS (; – ) was one of the most highly acclaimed English architects in history, as well as an anatomist, astronomer, geometer, and mathematician-physicist. He was accorded responsibility for rebuilding 52 churches ...

, and Christiaan Huygens
Christiaan Huygens, Lord of Zeelhem, ( , , ; also spelled Huyghens; la, Hugenius; 14 April 1629 – 8 July 1695) was a Dutch mathematician, physicist, engineer, astronomer, and inventor, who is regarded as one of the greatest scientists of ...

sent correct and similar solutions, all depending on what is now called the conservation of momentum
In Newtonian mechanics, momentum (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. If is an object's mass an ...

; but, while Wren and Huygens confined their theory to perfectly elastic bodies (elastic collision
In physics, an elastic collision is an encounter ( collision) between two bodies in which the total kinetic energy of the two bodies remains the same. In an ideal, perfectly elastic collision, there is no net conversion of kinetic energy into ...

), 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 (also called moment) acting on physical systems that do not experience an acceleration (''a''=0), but rather, are in static equilibrium with ...

(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''Number line

Wallis has been credited as the originator of thenumber line
In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a po ...

"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 also 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 $\backslash tfrac$ 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 mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...

came to the same conclusion by considering the geometric series $\backslash tfrac\; =\; 1\; +\; x\; +\; x^2\; +\; \backslash cdots$, 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 thePythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...

using similar triangles
In Euclidean geometry, two objects are similar if they have the same shape, or one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling (enlarging or reducing), possibly wit ...

. However, Thabit Ibn Qurra Thabit ( ar, ) is an Arabic name for males that means "the imperturbable one". It is sometimes spelled Thabet.
People with the patronymic
* Ibn Thabit, Libyan hip-hop musician
* Asim ibn Thabit, companion of Muhammad
* Hassan ibn Sabit (died 674 ...

(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
In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry:
''If a line segment ...

. 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
In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry:
''If a line segme ...

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, andHenry Oldenburg
Henry Oldenburg (also Henry Oldenbourg) FRS (c. 1618 as Heinrich Oldenburg – 5 September 1677), was a German theologian, diplomat, and natural philosopher, known as one of the creators of modern scientific peer review. He was one of the fo ...

, 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 ofPtolemy
Claudius Ptolemy (; grc-gre, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific treatises, three of which were of importanc ...

and Bryennius, and Porphyrius's commentary on Ptolemy. He also published three letters to Henry Oldenburg
Henry Oldenburg (also Henry Oldenbourg) FRS (c. 1618 as Heinrich Oldenburg – 5 September 1677), was a German theologian, diplomat, and natural philosopher, known as one of the creators of modern scientific peer review. He was one of the fo ...

concerning tuning. He approved of equal temperament
An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave (or other interval) into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, wh ...

, which was being used in England's organs.David Damschoder and David Russell Williams, ''Music Theory from Zarlino to Schenker: A Bibliography and Guide'' (Stytvesant, NY: Pendragon Press, 1990), p. 374.
Other works

His ''Institutio logicae'', published in 1687, was very popular. The ''Grammatica linguae Anglicanae'' was a work onEnglish 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.
This article describes a generalized, present-day Standard English ...

, that remained in print well into the eighteenth century. He also published on theology.
See also

* 31982 Johnwallis, an asteroid that was named after him *Invisible College
Invisible College is the term used for a small community of interacting scholars who often met face-to-face, exchanged ideas and encouraged each other. One group that has been described as a precursor group to the Royal Society of London consis ...

* John Wallis Academy
The John Wallis Church of England Academy is a Mixed-sex education, mixed all-through school with Academy (English school), academy status in Ashford, Kent. It was known as Christ Church Church of England High School. On 1 September 2010 it bec ...

– former ChristChurch school in Ashford renamed in 2010
* Wallis's conical edge
In geometry, Wallis's conical edge is a ruled surface given by the parametric equations
: x=v\cos u,\quad y=v\sin u,\quad z=c\sqrt
where , and are constants.
Wallis's conical edge is also a kind of right conoid. It is named after the English ...

* Wallis' integrals
In mathematics, and more precisely in analysis, the Wallis integrals constitute a family of integrals introduced by John Wallis.
Definition, basic properties
The ''Wallis integrals'' are the terms of the sequence (W_n)_ defined by
: W_n = \int_ ...

Footnotes

References

* The initial text of this article was taken from thepublic domain
The public domain (PD) consists of all the creative work
A creative work is a manifestation of creative effort including fine artwork (sculpture, paintings, drawing, sketching, performance art), dance, writing (literature), filmmaking, ...

resource:
* W. W. Rouse Ball
Walter William Rouse Ball (14 August 1850 – 4 April 1925), known as W. W. Rouse Ball, was a British mathematician, lawyer, and fellow at Trinity College, Cambridge, from 1878 to 1905. He was also a keen amateur magician, and the founding ...

(1908) A Short Account of the History of Mathematics

'' 4th ed. * * 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

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EMLO

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John Wallis (1685) ''A treatise of algebra''

- digital facsimile,

Linda Hall Library
The Linda Hall Library is a privately endowed American library of science, engineering and technology located in Kansas City, Missouri, sitting "majestically on a urban arboretum." It is the "largest independently funded public library of scien ...

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