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Percy Alexander MacMahon (26 September 1854 – 25 December 1929) was an English
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 ...
, especially noted in connection with the partitions of numbers and
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 inf ...
.


Early life

Percy MacMahon was born in Malta to a British military family. His father was a colonel at the time, retired in the rank of the
brigadier Brigadier ( ) is a military rank, the seniority of which depends on the country. In some countries, it is a senior rank above colonel, equivalent to a brigadier general or commodore (rank), commodore, typically commanding a brigade of several t ...
. MacMahon attended the Proprietary School in
Cheltenham Cheltenham () is a historic spa town and borough adjacent to the Cotswolds in Gloucestershire, England. Cheltenham became known as a health and holiday spa town resort following the discovery of mineral springs in 1716, and claims to be the mo ...
. At the age of 14 he won a Junior Scholarship to
Cheltenham College Cheltenham College is a public school ( fee-charging boarding and day school for pupils aged 13–18) in Cheltenham, Gloucestershire, England. The school opened in 1841 as a Church of England foundation and is known for its outstanding linguis ...
, which he attended as a day boy from 10 February 1868 until December 1870. At the age of 16 MacMahon was admitted to the
Royal Military Academy, Woolwich The Royal Military Academy (RMA) at Woolwich, in south-east London, was a British Army military academy for the training of Officer (armed forces), commissioned officers of the Royal Artillery and Royal Engineers. It later also trained officers o ...
and passed out after two years.


Military career

On 12 March 1873, MacMahon was posted to
Madras Chennai, also known as Madras ( its official name until 1996), is the capital and largest city of Tamil Nadu, the southernmost state of India. It is located on the Coromandel Coast of the Bay of Bengal. According to the 2011 Indian ce ...
, India, with the 1st Battery 5th Brigade, with the temporary rank of lieutenant. The Army List showed that in October 1873 he was posted to the 8th Brigade in
Lucknow Lucknow () is the List of state and union territory capitals in India, capital and the largest city of the List of state and union territory capitals in India, Indian state of Uttar Pradesh and it is the administrative headquarters of the epon ...
. MacMahon's final posting was to the No. 1 Mountain Battery with the Punjab Frontier Force at Kohat on the North West Frontier. He was appointed Second Subaltern on 26 January and joined the Battery on 25 February 1877. In the ''Historical Record of the No. 1 (Kohat) Mountain Battery, Punjab Frontier Force'' it is recorded that he was sent on sick leave to Muree (or Maree), a town north of Kohat on the banks of the Indus river, on 9 August 1877. On 22 December 1877 he started 18 months leave on a medical certificate granted under GGO number 1144. The nature of his illness is unknown. This period of sick leave was one of the most significant occurrences in MacMahon's life. Had he remained in India he would undoubtedly have been caught up in Roberts's War against the Afghans. In early 1878 MacMahon returned to England and the sequence of events began which led to him becoming a mathematician rather than a soldier. The Army List records a transfer to the 3rd Brigade in Newbridge at the beginning of 1878, and then shows MacMahon as 'supernumerary' from May 1878 until March 1879. In January 1879 MacMahon was posted to the 9th Brigade in
Dover Dover ( ) is a town and major ferry port in Kent, southeast England. It faces France across the Strait of Dover, the narrowest part of the English Channel at from Cap Gris Nez in France. It lies southeast of Canterbury and east of Maidstone. ...
, moving to
Sheerness Sheerness () is a port town and civil parish beside the mouth of the River Medway on the north-west corner of the Isle of Sheppey in north Kent, England. With a population of 13,249, it is the second largest town on the island after the nearby ...
in 1880. In the same year he enrolled in the Advanced Class for Artillery Officers at Woolwich. This was a two-year course covering technical subjects and a foreign language. Successful completion of the course resulted in the award of the letters "p.a.c" (passed advanced class) after MacMahon's name in the Army List. After he passed the Advanced Course and had been promoted to the rank of captain on 29 October 1881, MacMahon took up a post as instructor at the Royal Military Academy on 23 March 1882. Here he met
Alfred George Greenhill Sir Alfred George Greenhill (29 November 1847 in London – 10 February 1927 in London), was a British mathematician. George Greenhill was educated at Christ's Hospital School and from there he went to St John's College, Cambridge in 1866. In ...
, professor of mathematics at the Royal Artillery College.
Joseph Larmor Sir Joseph Larmor (; 11 July 1857 – 19 May 1942) was an Irish mathematician and physicist who made breakthroughs in the understanding of electricity, dynamics, thermodynamics, and the electron theory of matter. His most influential work was ...
, in a letter to ''The Times'' published after MacMahon's death, wrote, 'The young Captain threw himself with indomitable zeal and insight into the great problems of the rising edifice of algebraic forms, as was being developed by Cayley,
Sylvester Sylvester or Silvester is a name derived from the Latin adjective ''silvestris'' meaning "wooded" or "wild", which derives from the noun ''silva'' meaning "woodland". Classical Latin spells this with ''i''. In Classical Latin, ''y'' represented a ...
and
Salmon Salmon (; : salmon) are any of several list of commercially important fish species, commercially important species of euryhaline ray-finned fish from the genera ''Salmo'' and ''Oncorhynchus'' of the family (biology), family Salmonidae, native ...
.’ In 1891 MacMahon took up a new post as military instructor in electricity at the Royal Artillery College, Woolwich. Some sources (e.g. his three obituarists) have said that this post was 'professor of physics', but this is not correct, as Greenhill held that post until his own retirement. MacMahon retired from the military in 1898.


Mathematical career

MacMahon was elected a fellow of 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 1890. He received the Royal Society Royal Medal in 1900, the
Sylvester Medal The Sylvester Medal is a bronze medal awarded by the Royal Society for the encouragement of mathematical research, and accompanied by a £1,000 prize. It was named in honour of James Joseph Sylvester, the Savilian chair of geometry, Savilian Prof ...
in 1919, and the Morgan Medal by the
London Mathematical Society The London Mathematical Society (LMS) is one of the United Kingdom's Learned society, learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh ...
in 1923. MacMahon was the President of the London Mathematical Society from 1894 to 1896. MacMahon is best known for his study of
symmetric function In mathematics, a function of n variables is symmetric if its value is the same no matter the order of its arguments. For example, a function f\left(x_1,x_2\right) of two arguments is a symmetric function if and only if f\left(x_1,x_2\right) = f\ ...
s and enumeration of
plane partition In mathematics and especially in combinatorics, a plane partition is a two-dimensional array of nonnegative integers \pi_ (with positive number, positive integer indices ''i'' and ''j'') that is nonincreasing in both indices. This means that : \pi ...
s; see MacMahon Master theorem. His two volume ''Combinatory analysis'', published in 1915/16, is the first major book in
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 inf ...
. MacMahon also did pioneering work in recreational mathematics and developed several successful puzzle games. His 1921 treatise ''New Mathematical Pastimes'' extended the linear edge-matching puzzle game of
dominoes Dominoes is a family of tile-based games played with gaming pieces. Each domino is a rectangular tile, usually with a line dividing its face into two square ''ends''. Each end is marked with a number of spots (also called ''Pip (counting), pips ...
to two- and three-dimensional shapes including
equilateral triangle An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...
s (a set of 24, with each edge coloured one of four possible colours was patented by MacMahon in 1892; similar triangle-tile domino games have since been published commercially, including '' Contack'' 939 '' Triominoes'' 965 and '' Trioker'' 970, squares ( MacMahon Squares; a set of 24 unique patterns results from colouring each of the four edges one of three possible colours), and cubes (a set of 30 is made by assigning each face one of six possible colours without repeating a colour).


Tribute

A reviewer in "Science Progress in the Twentieth Century", writes: : ''It is, I believe, a loss to England and to mathematics that Major MacMahon has not directed a great school of research; the gain to the youthful mathematicians of such a leader is obvious; they would have received an impetus which the printed page will only give to a few. Is it not possible also that the quality of work done in such circumstances may not, like mercy, be doubly blest? .it is impossible to resist the feeling that there are countries in which mathematical teaching is better organised than it is in England.'' Richard P. Stanley considers MacMahon as the most influential mathematician in enumerative combinatorics pre-1960.Enumerative and Algebraic Combinatorics in the 1960s and 1970s
(17 June 2021)


Portrayal in film

In the movie '' The Man Who Knew Infinity'' Kevin McNally plays as MacMahon. The film accurately depicts the first meeting of MacMahon and
Srinivasa Ramanujan Srinivasa Ramanujan Aiyangar (22 December 188726 April 1920) was an Indian mathematician. Often regarded as one of the greatest mathematicians of all time, though he had almost no formal training in pure mathematics, he made substantial con ...
, where Ramanujan successfully completes some mathematical calculations.
Gian-Carlo Rota Gian-Carlo Rota (April 27, 1932 – April 18, 1999) was an Italian-American mathematician and philosopher. He spent most of his career at the Massachusetts Institute of Technology, where he worked in combinatorics, functional analysis, proba ...
notes in his introduction to Volume I of MacMahon's Collected Papers:


See also

*
Cairo pentagonal tiling In geometry, a Cairo pentagonal tiling is a tessellation of the Euclidean plane by congruent convex pentagons, formed by overlaying two tessellations of the plane by hexagons and named for its use as a paving design in Cairo. It is also called Ma ...
, a tiling of the plane by pentagons also called "MacMahon's net"


Notes


References

*


External links

*PhD thesis by Dr Paul Garcia

* * P.A. MacMahon,
Combinatory analysis
', 2 vols, Cambridge University Press, 1915–16.
Obituary Notices
– ''
Monthly Notices of the Royal Astronomical Society ''Monthly Notices of the Royal Astronomical Society'' (MNRAS) is a peer-reviewed scientific journal in astronomy, astrophysics and related fields. It publishes original research in two formats: papers (of any length) and letters (limited to ...
'' 90, 373–378.
MacMahon's Coloured Cubes
– a puzzle with coloured cubes {{DEFAULTSORT:Macmahon, Percy Alexander 19th-century English mathematicians 20th-century English mathematicians British Indian Army officers 1854 births 1929 deaths Combinatorialists People educated at Cheltenham College Graduates of the Royal Military Academy, Woolwich Fellows of the Royal Society Royal Medal winners De Morgan Medallists Presidents of the Royal Astronomical Society Maltese military personnel Academics of the Royal Military Academy, Woolwich Crown Colony of Malta people