Major Percy Alexander MacMahon

by Dr Paul Garcia

Major MacMahon was the subject of my PhD thesis. I have put this version of my thesis on the web in case anyone wants all the detail. The main chapters and appendices are all in PDF format. The text below is chapter 1, and sets out the content of the other chapters.

Link to stuff about MacMahon not in the thesis.

Link to more stuff about MacMahon not in the thesis.

Link to the bibliography.


The conventional view of Victorian combinatorics in Britain is that it was the province of amateurs - barristers, clergymen, military men - and it might be argued that MacMahon was the among the most important of these. However, this thesis will argue that MacMahon was indeed a most important Victorian combinatorialist but was by no means an amateur.

Percy Alexander MacMahon was an unusual mathematician. He did not come from a family of mathematicians - his father and brothers were soldiers - and he did not have the university education usually associated with a prominent mathematician. Further, he entered the world of mathematics at the relatively late age of 26, and did not become well known until he was 30. Before he joined the advanced class in mathematics at the Royal Military Academy in 1880 he had shown no hint of any particular interest in, or talent for, doing mathematics.

The speed with which MacMahon rose to prominence in the mathematical community of the late nineteenth century is remarkable. From the start of his career he was held in high regard by several famous and well-established mathematicians; James Joseph Sylvester, for example, specifically mentioned MacMahon three times in his inaugural speech as Savilian Professor of Geometry at Oxford. A review of MacMahon's two-volume work, Combinatory Analysis, compared him with such luminaries as Fermat, Pascal, Euler, Lucas and Sylvester, at least in the field of combinatory analysis, and described his mathematical style as 'impeccable'.

MacMahon did pioneering work in invariant theory, symmetric function theory, and partition theory. He brought all these strands together to bring coherence to the discipline we now call combinatorial analysis. According to Peter J. Cameron, combinatorial analysis was not well thought of in the first half of the twentieth century. MacMahon was ahead of his time, since in the second half of the twentieth century, attitudes to combinatorial analysis changed dramatically, as illustrated by this quotation from Professor Steven Pinker, referred to by Cameron:

It may not be a coincidence that the two systems in the universe that most impress us with their open ended complex design - life and mind - are based on discrete combinatorial systems.

MacMahon also carried out research in recreational mathematics, creating visual puzzles, but again in a pioneering and non-traditional way. This thesis will describe only as much of the mathematics as is necessary to make the context and extent of MacMahon's work clear; it is not intended that this will be a detailed technical exposition of the topics he worked on.

How did MacMahon come from a non-mathematical background to a position of such prominence so quickly ? What was special about his work ? Why did he fade from prominence ? What was his legacy to mathematics ? To answer these questions, his personal life, which was also unconventional, and his working life will be described. MacMahon worked as a soldier, a teacher and a government official, whilst simultaneously pursuing his mathematical researches.

During his career he published over 125 papers and four books. Combinatory Analysis is still in print and New Mathematical Pastimes was reprinted in 2004 as part of the work for this thesis.

Chapter 2 outlines the type of mathematical education that MacMahon received as a Gentleman Cadet at the Royal Military Academy in Woolwich, the training institution for the Royal Artillery and Royal Engineers, and his subsequent military service in India. MacMahon's education was not particularly mathematical, and there is no evidence that his gifts were recognised by his teachers.

The beginnings of MacMahon's mathematical career are covered in Chapter 3. Whilst working as an instructor at his alma mater, the Royal Military Academy at Woolwich, MacMahon made a significant discovery in the algebra of forms (also known as invariant theory), which brought him to the attention of the wider mathematical community in London. He also began to study partition theory, in which field he was to become the most important researcher of the late nineteenth and early twentieth centuries.

During the 1890s, MacMahon's reputation as a mathematician grew. Chapter 4 chronicles his career in the Royal Artillery, his continuing work on partitions, and his election to the Royal Society and to the Presidency of the London Mathematical Society. He also became involved with the Royal Astronomical Society, received his first honorary degree and experienced a disappointment at the hands of the electors of the Savilian Professor of Geometry in Oxford.

The material in Chapter 5 covers the period between MacMahon's retirement from the Royal Artillery and his appointment as Deputy Warden of the Standards at the Board of Trade. It shows the breadth of MacMahon's involvement with the mathematical and scientific community, and the respect he commanded from the distinguished organisations he had joined whilst teaching at Woolwich.

Chapter 6 describes one of the most eventful periods of MacMahon's life. He was by now an established figure enjoying a distinguished reputation, with the Presidency of the London Mathematical Society and two honorary degrees already to his credit. A further two honorary degrees were awarded in 1911. He also wrote four books in this period, consolidating his work in partitions and symmetric functions into Combinatory Analysis, published in two volumes in 1915 and 1916 , and very favourably reviewed. An introductory volume to Combinatory Analysis followed in 1920 , and in 1921 MacMahon wrote New Mathematical Pastimes, developing the puzzle work he had carried out in the early 1890s. MacMahon also wrote more than 30 papers between 1906 and 1922, in which he developed the ideas he had expounded earlier in symmetric function theory and partition theory. In 1906 he took up the post of Deputy Warden of the Standards with the Board of Trade.

Finally, Chapter 7 records MacMahon's work at St John's College, Cambridge, his influence on some other mathematicians, and his continuing mathematical work. Ill-health forced him to move to the south coast in 1928, where he died in 1929.

There are nine appendices:

Appendix 1 Sample timetable from RMA

Appendix 2, Map of military service in India

Appendix 3 MacMahon family tree

Appendix 4 Ballistics to invariant theory

Appendix 5 Partitions

Appendix 6 Correspondence with D'Arcy Thompson

Appendix 7 Correspondence with Parker Smith

Appendix 8 Royal Society - letters and applications

Appendix 9 Correspondence with Sir Ronald Ross

If you have any questions, please e-mail me

page updated 13 December 2007