John Wallis (; la, Wallisius; ) 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, 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 1654
Family
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 in Ashford, 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
The Parliamentary 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 to trigonometry
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 on conic 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 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 (e.g. ). The set of all ration ...
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