William Allen Whitworth (1 February 1840 – 12 March 1905) was an English mathematician and a priest in the

Church of England The Church of England (C of E) is the established Christian church in England and the mother church of the international Anglican Communion. It traces its history to the Christian church recorded as existing in the Roman province of Britain ...

Education and mathematical career

Whitworth was born in

Runcorn Runcorn is an industrial town and cargo port in the Borough of Halton in Cheshire, England. Its population in 2011 was 61,789. The town is in the southeast of the Liverpool City Region, with Liverpool to the northwest across the River Mersey. ...

; his father, William Whitworth, was a school headmaster, and he was the oldest of six siblings. He was schooled at the Sandicroft School in Northwich and then at

St John's College, Cambridge St John's College is a Colleges of the University of Cambridge, constituent college of the University of Cambridge founded by the House of Tudor, Tudor matriarch Lady Margaret Beaufort. In constitutional terms, the college is a charitable corpo ...

, earning a B.A. in 1862 as 16th Wrangler. He taught mathematics at the Portarlington School and the Rossall School, and was a professor of mathematics at Queen's College in

Liverpool Liverpool is a city and metropolitan borough in Merseyside, England. With a population of in 2019, it is the 10th largest English district by population and its metropolitan area is the fifth largest in the United Kingdom, with a popul ...

from 1862 to 1864. He returned to Cambridge to earn a master's degree in 1865, and was a fellow there from 1867 to 1882.

Mathematical contributions

As an undergraduate, Whitworth became the founding editor in chief of the ''

Messenger of Mathematics The ''Messenger of Mathematics'' is a defunct British mathematics journal. The founding editor-in-chief was William Allen Whitworth with Charles Taylor and volumes 1–58 were published between 1872 and 1929. James Whitbread Lee Glaisher was the ...

'', and he continued as its editor until 1880. He published works about 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 ...

and about

trilinear coordinates In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is t ...

, but his most famous mathematical publication is the book ''Choice and Chance: An Elementary Treatise on Permutations, Combinations, and Probability'' (first published in 1867 and extended over several later editions). The first edition of the book treated the subject primarily from the point of view of arithmetic calculations, but had an appendix on

algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...

, and was based on lectures he had given at Queen's College. Later editions added material on

enumerative combinatorics Enumerative combinatorics is an area of combinatorics that deals with the number of ways that certain patterns can be formed. Two examples of this type of problem are counting combinations and counting permutations. More generally, given an infin ...

(the numbers of ways of arranging items into groups with various constraints),

derangement In combinatorial mathematics, a derangement is a permutation of the elements of a set, such that no element appears in its original position. In other words, a derangement is a permutation that has no fixed points. The number of derangements of ...

frequentist probability Frequentist probability or frequentism is an interpretation of probability; it defines an event's probability as the limit of its relative frequency in many trials (the long-run probability). Probabilities can be found (in principle) by a repea ...

life expectancy Life expectancy is a statistical measure of the average time an organism is expected to live, based on the year of its birth, current age, and other demographic factors like sex. The most commonly used measure is life expectancy at birth ...

, and the fairness of bets, among other topics. Among the other contributions in this book, Whitworth was the first to use

ordered Bell number In number theory and enumerative combinatorics, the ordered Bell numbers or Fubini numbers count the number of weak orderings on a set of ''n'' elements (orderings of the elements into a sequence allowing ties, such as might arise as the outcome o ...

s to count the number of

weak ordering In mathematics, especially order theory, a weak ordering is a mathematical formalization of the intuitive notion of a ranking of a set, some of whose members may be tied with each other. Weak orders are a generalization of totally ordered ...

s of a set, in the 1886 edition. These numbers had been studied earlier by

Arthur Cayley Arthur Cayley (; 16 August 1821 – 26 January 1895) was a prolific United Kingdom of Great Britain and Ireland, British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics. As a child, C ...

, but for a different problem. He was the first to publish

Bertrand's ballot theorem In combinatorics, Bertrand's ballot problem is the question: "In an election where candidate A receives ''p'' votes and candidate B receives ''q'' votes with ''p'' > ''q'', what is the probability that A will be strictly ahead of B throu ...

, in 1878; the theorem is misnamed after

Joseph Louis François Bertrand Joseph Louis François Bertrand (; 11 March 1822 – 5 April 1900) was a French mathematician who worked in the fields of number theory, differential geometry, probability theory, economics and thermodynamics. Biography Joseph Bertrand was the ...

, who rediscovered the same result in 1887. He is the inventor of the E 'X''notation for the

expected value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...

of a random variable ''X'', still commonly in use, and he coined the name "subfactorial" for the number of

s of ''n'' items. Another of Whitworth's contributions, in

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 ...

, concerns

equable shape A two-dimensional equable shape (or perfect shape) is one whose area is numerically equal to its perimeter. For example, a right angled triangle with sides 5, 12 and 13 has area and perimeter both have a unitless numerical value of 30. Scaling and ...

s, shapes whose area has the same numerical value (with a different set of units) as their perimeter. As Whitworth showed with D. Biddle in 1904, there are exactly five equable triangles with integer sides: the two

right triangle A right triangle (American English) or right-angled triangle (British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right an ...

s with side lengths (5,12,13) and (6,8,10), and the three triangles with side lengths (6,25,29), (7,15,20), and (9,10,17)..

Religious career

Whitworth was ordained as a

deacon A deacon is a member of the diaconate, an office in Christian churches that is generally associated with service of some kind, but which varies among theological and denominational traditions. Major Christian churches, such as the Catholic Churc ...

in 1865, and became a priest in 1866. He served as the

curate A curate () is a person who is invested with the ''care'' or ''cure'' (''cura'') ''of souls'' of a parish. In this sense, "curate" means a parish priest; but in English-speaking countries the term ''curate'' is commonly used to describe clergy w ...

of St Anne's Church in

Birkenhead Birkenhead (; cy, Penbedw) is a town in the Metropolitan Borough of Wirral, Merseyside, England; historically, it was part of Cheshire until 1974. The town is on the Wirral Peninsula, along the south bank of the River Mersey, opposite Liver ...

in 1865, of the

Church of St Luke, Liverpool St Luke's Church, more commonly known by locals as the bombed-out church, is a former Anglican parish church in Liverpool, England. It stands on the corner of Berry Street and Leece Street, at the top of Bold Street. The church was built betw ...

from 1866 to 1870 and of Christ Church in Liverpool from 1870 to 1875. He was then a

vicar A vicar (; Latin: ''vicarius'') is a representative, deputy or substitute; anyone acting "in the person of" or agent for a superior (compare "vicarious" in the sense of "at second hand"). Linguistically, ''vicar'' is cognate with the English pref ...

in London at St John the Evangelist's in

Hammersmith Hammersmith is a district of West London, England, southwest of Charing Cross. It is the administrative centre of the London Borough of Hammersmith and Fulham, and identified in the London Plan as one of 35 major centres in Greater London. ...

. From 1886 to 1905 he was vicar of

All Saints, Margaret Street All Saints, Margaret Street, is a Grade I listed Anglo-Catholic church in London. The church was designed by the architect William Butterfield and built between 1850 and 1859. It has been hailed as Butterfield's masterpiece and a pioneering buil ...

. He was the

Hulsean Lecturer The Hulsean Lectures were established from an endowment made by John Hulse to the University of Cambridge in 1790. At present, they consist of a series of four to eight lectures given by a university graduate on some branch of Christian theology. ...

in 1903.

References

External links

* {{DEFAULTSORT:Whitworth, William Allen 1840 births 1905 deaths English mathematicians 20th-century English Anglican priests 19th-century English Anglican priests Alumni of St John's College, Cambridge People from Northwich People from Runcorn