MacTutor
In MAJOR P. A. MACMAHON the Society loses one of that series of distinguished pure mathematicians who, without being themselves astronomers, have shown devoted attachment to the Society and have served long on its Council and been chosen as Presidents. Arthur Cayley was President 1872-74 and was editor of the Society's publications 1860-82, except during the two years of his Presidency. He was on the Council 1858-93 and a very regular attendant. J. W. L. Glaisher was President twice, 1886-88 and 1901-3, and served on the Council continuously 1874-1928. The older Fellows will remember also Sir James Cockle, who was a Fellow from 1854-94 and served on the Council 1888-92. Major MacMahon became a Fellow on 1897 January 8, and was elected on Council almost at once [on 1898 February 11. Of the twenty-one forming that Council only four now survive]. He remained on Council, serving several terms as Vice-President, until 1911, when his name disappeared from the list for a few years, to return as President in 1917-18. After three more years' service he retired, partly for reasons of failing health. But this brief statement of facts is sufficient to show his attachment to the Society and the esteem in which it held him.
Percy Alexander MacMahon was born at Malta, 1854 September 26, the second son of Brigadier-General P. W. MacMahon. He was sent to Cheltenham College, joined the Royal Artillery from the R.M.A. Woolwich in 1872, and saw some service in India. Ten years later he returned to the Royal Military Academy as Instructor in Mathematics, and in 1890 he was appointed Professor of Physics at the Ordnance College, holding that post until 1897. He retired from the Army in 1898. From 1906 to 1920 he was Deputy Warden of the Standards, under the Board of Trade.
The details of his military service have been kindly supplied from official quarters as follows :-
Gazetted Lieutenant, Royal Artillery, 12th September 1872, and was posted to the 5th Brigade at St. Thomas's Mount, Madras. Was trans-ferred the same year to the 8th Brigade at Lucknow, which moved that year to Meerut, Punjab, where he remained for three years, until he was posted on 26th January 1877, to No. 1 Mountain Battery, Punjab Frontier Force, at Kohat, North-West Frontier. Took part with this Battery in a punitive expedition against the Jawaki Afridis, penetrating into their territory, and capturing several villages. He had leave on medical certificate for 18 months from 21st December 1877, after which he was posted to the 9th Brigade at Dover.
"At that time the Theory of Algebraic Forms was in the full flight of development by the activities of Cayley and Sylvester and Salmon, this being the one great domain in the vast range of modern abstract mathematics whose creation may be claimed to be predominantly British. The young captain threw himself with indomitable zeal and insight into the great problems of this rising edifice of science; and in a very short time he was to be counted as conspicuous among the leaders, largely by invention of new methods of approach. So complete was his scientific absorption then, and during successive tenures as Instructor and Professor at Woolwich, that one was accustomed to hear his military friends refer in chaff to him as a good soldier spoiled.' Anyhow they were proud of him; and equally proud was the scientific world into which he had so thoroughly forced an entrance. His most remarkable gift was great native insight into the theory of permutations, a sort of glorified chess practice, and the harvest that it could yield in other domains, at first sight unlikely. This mass of work was recognised by the award of one of the Royal Medals by the Royal Society in 1900."
One phrase in this sympathetic appreciation may perhaps be emphasised, viz. that referring to "the harvest that [the theory of permutations] could yield in other domains, at first sight unlikely." MacMahon would have specially welcomed this view of his work: for the ordinary view of pure mathematics as an isolated subject was thoroughly distasteful to him. He had an earnest faith that its relationships to other domains, though they might be obscured for the moment, would ultimately emerge into the clear light of day. He gave eloquent expression to this view in his Presidential Address to Section A at the Glasgow meeting of the British Association in 1901.
"It will be gathered from remarks made above," he said, "that I regard any department of scientific work, which seems to be narrow or isolated, as a proper subject for research. I do not believe in any branch of science, or subject of scientific work, being destitute of con-nection with other branches. If it appears to be so, it is especially marked out for investigation by the very unity of science. There is no necessarily pathless desert separating different regions. Now a department of pure mathematics which appeared to be somewhat in this forlorn condition a few years ago, was that which included problems of the nature of the magic square of the ancients."
And he proceeds to give a masterly summary of the rescue of this forlorn department. Earlier in the address he had led up to his avowal of faith by giving illustrations from the theory of invariants and the combinatorial analysis "with which I have a more special acquaintance"; but this is scarcely the place in which to follow him further.
A still earlier part of the same address, however, is worth recalling because of its relationship to our own Society. After some introductory remarks on the history of mathematics in England, he paused a moment to "rescue from an oblivion which seems to threaten it" the history of the Mathematical Society of Spitalfields, founded in 1717 by Joseph Middleton, a writer of mathematical text-books, and absorbed into the Royal Astronomical Society, with its valuable library, in 1845. The number of members was originally limited to " the square of seven," but later it was successively increased to "the square of eight" and "the square of nine." The subscription was sixpence a week, and entrance was gained by the production of a metal ticket which had the proposition of Pythagoras on one side and a sighted quadrant with level on the other. The members dined together twice annually, viz. on the second Friday in January in London to commemorate the birth of Sir Isaac Newton, and on the second Friday in July, somewhere in the country, in commemoration of the birth of the Founder. It was found necessary to introduce a rule fining members sixpence for letting off fireworks in the place of meeting. Every member was entitled to a pint of beer at the common expense, and further, every five members were entitled to call for a quart for consumption at the meeting. All these details and much more, MacMahon rescued from the Minute Books of the Society, still in our possession, and recounted for the delight of his audience at Glasgow.
Soon after his Presidency of Section A, MacMahon was elected one of the General Secretaries of the British Association (1903-13), remaining in office until just before the Australian visit. On his retirement he was elected one of the three Trustees of the Association, his colleagues being Lord Rayleigh and Sir Arthur Rücker.
Though the amount of MacMahon's published work is considerable, there is but one paper to his name in our Monthly Notices (69, 126). It is interesting, however, to note that it concerns a possible "determination of the apparent diameter of a fixed star," and attracted comments from Professor Eddington. But it reads now like ancient history!
The seriousness and importance of his researches in pure mathematics are so thoroughly well established that it can cause no misapprehension to remark that a part of them grew out of a pastime, and another part of them suggested reduction to a pastime.
We may first quote, in illustration of the aforesaid importance, the following opening sentence from his paper on "Combinatorial Analysis. The Foundations of a New Theory," in Phil. Trans. R.S., A, 194 (1900):
"In the Trans. Camb. Phil. Soc. (vol. xvi., pt. iv., p. 262) I brought forward a new instrument of research in Combinatorial Analysis and applied it to the complete solution of the great problem of the Latin Square, which had proved a stumbling-block to mathematicians since the time of Euler. The method was equally successful in dealing with the general problem of which the Latin Square was but a particular case, and also with many other questions of a similar character."
Perhaps we may quote also the mere title of one of his "Memoirs on the Theory of the Partitions of Numbers" (Phil. Trans., vol. ccix. A, 1909) which runs: "On the Probability that the Successful Candidate at an Election by Ballot may never at any time have Fewer Votes than the One who is Unsuccessful: on a Generalisation of this Question, and on its Connexion with other Questions of Partition, Permutation, and Combination."
It will be seen that these erudite researches had distinct points of contact with practical life, and we may now recall, as stated above, their connection with pastimes. Simon Newcomb was very fond of the card games which go by the name of "Patience," and in playing one of them a question suggested itself which he referred to MacMahon, who had already dealt with a somewhat similar problem in the Phil. Trans, for 1893. The answer appears in the Phil. Trans. for 1908 (vol. ccvii.) as a learned memoir of seventy pages!
The converse transition, from philosophical thought to pastime, is represented by MacMahon's New Mathematical Pastimes, published by the Cambridge Press in 1921. It opens with the remark of Edwards to Dr. Johnson, "You are a philosopher, Dr. Johnson. I have tried too in my time to be a philosopher, but I don't know how, cheerfulness was always breaking in." And other literary quotations are scattered through the book in the happy manner to which Eddington had introduced us the year before. The argument starts with the game of dominoes, and it is at once suggested that there should be on each domino not two numbers but three and later, four. The various possible arrangements, with due attention to boundary conditions, are studied, needless to say with skill and beauty. One of the quotations is from Pope's Essay on Man:-
He received many honours. He was elected F.R.S. in 1890. In 1894 he was President of the London Mathematical Society which in 1923 conferred upon him the De Morgan Medal. From the Royal Society he received a Royal Medal in 1900, and the Sylvester Medal in 1919; and he was elected Vice-President in 1917. He was an Honorary Member of the Royal Irish Academy and of the Cambridge Philosophical Society. Honorary Degrees were conferred on him by Cambridge, Dublin, Aberdeen, and St. Andrews Universities, and he represented the Royal Society as a Fellow of Winchester College.
H. H. Τ.
Percy Alexander MacMahon was born at Malta, 1854 September 26, the second son of Brigadier-General P. W. MacMahon. He was sent to Cheltenham College, joined the Royal Artillery from the R.M.A. Woolwich in 1872, and saw some service in India. Ten years later he returned to the Royal Military Academy as Instructor in Mathematics, and in 1890 he was appointed Professor of Physics at the Ordnance College, holding that post until 1897. He retired from the Army in 1898. From 1906 to 1920 he was Deputy Warden of the Standards, under the Board of Trade.
The details of his military service have been kindly supplied from official quarters as follows :-
Gazetted Lieutenant, Royal Artillery, 12th September 1872, and was posted to the 5th Brigade at St. Thomas's Mount, Madras. Was trans-ferred the same year to the 8th Brigade at Lucknow, which moved that year to Meerut, Punjab, where he remained for three years, until he was posted on 26th January 1877, to No. 1 Mountain Battery, Punjab Frontier Force, at Kohat, North-West Frontier. Took part with this Battery in a punitive expedition against the Jawaki Afridis, penetrating into their territory, and capturing several villages. He had leave on medical certificate for 18 months from 21st December 1877, after which he was posted to the 9th Brigade at Dover.
Student on Advanced Class at Artillery College, 1880-1882.
Instructor in Mathematics, R.M.A., 1882-1888.
Inspector W. Stores, 1888-1891.
Instructor and Professor, Artillery College, 1891-1897, in the subjects of Electricity and Optics.
It was his return to Woolwich in 1882, and his friendship with Sir George Greenhill (then a Professor at the Artillery College, Woolwich), that changed the current of his life. To quote the words of Sir Joseph Larmor in a letter to the Times of 1929 December 31:-
Instructor in Mathematics, R.M.A., 1882-1888.
Inspector W. Stores, 1888-1891.
Instructor and Professor, Artillery College, 1891-1897, in the subjects of Electricity and Optics.
"At that time the Theory of Algebraic Forms was in the full flight of development by the activities of Cayley and Sylvester and Salmon, this being the one great domain in the vast range of modern abstract mathematics whose creation may be claimed to be predominantly British. The young captain threw himself with indomitable zeal and insight into the great problems of this rising edifice of science; and in a very short time he was to be counted as conspicuous among the leaders, largely by invention of new methods of approach. So complete was his scientific absorption then, and during successive tenures as Instructor and Professor at Woolwich, that one was accustomed to hear his military friends refer in chaff to him as a good soldier spoiled.' Anyhow they were proud of him; and equally proud was the scientific world into which he had so thoroughly forced an entrance. His most remarkable gift was great native insight into the theory of permutations, a sort of glorified chess practice, and the harvest that it could yield in other domains, at first sight unlikely. This mass of work was recognised by the award of one of the Royal Medals by the Royal Society in 1900."
One phrase in this sympathetic appreciation may perhaps be emphasised, viz. that referring to "the harvest that [the theory of permutations] could yield in other domains, at first sight unlikely." MacMahon would have specially welcomed this view of his work: for the ordinary view of pure mathematics as an isolated subject was thoroughly distasteful to him. He had an earnest faith that its relationships to other domains, though they might be obscured for the moment, would ultimately emerge into the clear light of day. He gave eloquent expression to this view in his Presidential Address to Section A at the Glasgow meeting of the British Association in 1901.
"It will be gathered from remarks made above," he said, "that I regard any department of scientific work, which seems to be narrow or isolated, as a proper subject for research. I do not believe in any branch of science, or subject of scientific work, being destitute of con-nection with other branches. If it appears to be so, it is especially marked out for investigation by the very unity of science. There is no necessarily pathless desert separating different regions. Now a department of pure mathematics which appeared to be somewhat in this forlorn condition a few years ago, was that which included problems of the nature of the magic square of the ancients."
And he proceeds to give a masterly summary of the rescue of this forlorn department. Earlier in the address he had led up to his avowal of faith by giving illustrations from the theory of invariants and the combinatorial analysis "with which I have a more special acquaintance"; but this is scarcely the place in which to follow him further.
A still earlier part of the same address, however, is worth recalling because of its relationship to our own Society. After some introductory remarks on the history of mathematics in England, he paused a moment to "rescue from an oblivion which seems to threaten it" the history of the Mathematical Society of Spitalfields, founded in 1717 by Joseph Middleton, a writer of mathematical text-books, and absorbed into the Royal Astronomical Society, with its valuable library, in 1845. The number of members was originally limited to " the square of seven," but later it was successively increased to "the square of eight" and "the square of nine." The subscription was sixpence a week, and entrance was gained by the production of a metal ticket which had the proposition of Pythagoras on one side and a sighted quadrant with level on the other. The members dined together twice annually, viz. on the second Friday in January in London to commemorate the birth of Sir Isaac Newton, and on the second Friday in July, somewhere in the country, in commemoration of the birth of the Founder. It was found necessary to introduce a rule fining members sixpence for letting off fireworks in the place of meeting. Every member was entitled to a pint of beer at the common expense, and further, every five members were entitled to call for a quart for consumption at the meeting. All these details and much more, MacMahon rescued from the Minute Books of the Society, still in our possession, and recounted for the delight of his audience at Glasgow.
Soon after his Presidency of Section A, MacMahon was elected one of the General Secretaries of the British Association (1903-13), remaining in office until just before the Australian visit. On his retirement he was elected one of the three Trustees of the Association, his colleagues being Lord Rayleigh and Sir Arthur Rücker.
Though the amount of MacMahon's published work is considerable, there is but one paper to his name in our Monthly Notices (69, 126). It is interesting, however, to note that it concerns a possible "determination of the apparent diameter of a fixed star," and attracted comments from Professor Eddington. But it reads now like ancient history!
The seriousness and importance of his researches in pure mathematics are so thoroughly well established that it can cause no misapprehension to remark that a part of them grew out of a pastime, and another part of them suggested reduction to a pastime.
We may first quote, in illustration of the aforesaid importance, the following opening sentence from his paper on "Combinatorial Analysis. The Foundations of a New Theory," in Phil. Trans. R.S., A, 194 (1900):
"In the Trans. Camb. Phil. Soc. (vol. xvi., pt. iv., p. 262) I brought forward a new instrument of research in Combinatorial Analysis and applied it to the complete solution of the great problem of the Latin Square, which had proved a stumbling-block to mathematicians since the time of Euler. The method was equally successful in dealing with the general problem of which the Latin Square was but a particular case, and also with many other questions of a similar character."
Perhaps we may quote also the mere title of one of his "Memoirs on the Theory of the Partitions of Numbers" (Phil. Trans., vol. ccix. A, 1909) which runs: "On the Probability that the Successful Candidate at an Election by Ballot may never at any time have Fewer Votes than the One who is Unsuccessful: on a Generalisation of this Question, and on its Connexion with other Questions of Partition, Permutation, and Combination."
It will be seen that these erudite researches had distinct points of contact with practical life, and we may now recall, as stated above, their connection with pastimes. Simon Newcomb was very fond of the card games which go by the name of "Patience," and in playing one of them a question suggested itself which he referred to MacMahon, who had already dealt with a somewhat similar problem in the Phil. Trans, for 1893. The answer appears in the Phil. Trans. for 1908 (vol. ccvii.) as a learned memoir of seventy pages!
The converse transition, from philosophical thought to pastime, is represented by MacMahon's New Mathematical Pastimes, published by the Cambridge Press in 1921. It opens with the remark of Edwards to Dr. Johnson, "You are a philosopher, Dr. Johnson. I have tried too in my time to be a philosopher, but I don't know how, cheerfulness was always breaking in." And other literary quotations are scattered through the book in the happy manner to which Eddington had introduced us the year before. The argument starts with the game of dominoes, and it is at once suggested that there should be on each domino not two numbers but three and later, four. The various possible arrangements, with due attention to boundary conditions, are studied, needless to say with skill and beauty. One of the quotations is from Pope's Essay on Man:-
So from the first eternal order ran,
And creature link'd to creature, man to man
...
The link dissolves, each seeks a fresh embrace,
Another love succeeds, another race.
MacMahon was very fond of billiards and was a good player, often to be seen playing after lunch at the Athenæum Club, to which he was elected (under Rule II.) in 1903. The present writer once received a surprising compliment, worth recording here because of its sequel. The editor of a journal asked him to review a book on billiards, and it was natural to refer the editor to MacMahon as the right person for the undertaking. But it was bewildering to learn, in response, that the editor's application had been suggested by MacMahon himself; and the next step was clearly to go and seek an explanation. It appeared that, perhaps from diffidence or amiable laziness, he had really made this suggestion of putting the review into much less worthy hands: though he had the friendliness to help with a few important suggestions: and it is one of these which seems worth noting here. The importance of the "half-ball" stroke in making a losing hazard is well known. MacMahon remarked that it also had a distinct value for the winning hazard, though this had not received proper attention. If the eye were trained to recognise the angle at which a half-ball stroke would send the object ball into a pocket, the winning hazard could be made oftener and more easily. Certainly in MacMahon's play this facility could be observed, though whether his principle is accepted widely seems doubtful. Is it permissible to recall another experience of his? He at one time took lessons in billiards from a well-known expert who after a time professed to have conveyed all the regular information available or nearly all. "There is one more piece of advice I could give you, which would improve your play considerably," he said, "if you care to pay another guinea; for it lies outside the regular lessons." The guinea was paid cheerfully. "Always chalk your cue before you play, and not after." On being co-opted by the University of Cambridge to the honorary degree of Doctor of Science in 1904" (to quote again Sir Joseph Larmor), "MacMahon attached himself by invitation to St. John's College, where he had acquired many friends. Some years ago he and his wife withdrew from London to Cambridge" (he had married, in 1907, Grace Elizabeth, daughter of the late C. R. Howard of 32 Gloucester Place, W.), " and as a member of the Society of St. John's College he became a most welcome and, as we thought, a happy figure in its life, armed with the knowledge that he brought of the external world. He continued the pursuit of mathematics, by public lectures as well as investigation. His absence was keenly felt when about a year ago a breakdown in health compelled him to retire to Bognor; his memory will be cherished as one of the most gracious accessions of recent times." After a brief illness he died somewhat suddenly at Bognor on Christmas Day last.
And creature link'd to creature, man to man
...
The link dissolves, each seeks a fresh embrace,
Another love succeeds, another race.
He received many honours. He was elected F.R.S. in 1890. In 1894 he was President of the London Mathematical Society which in 1923 conferred upon him the De Morgan Medal. From the Royal Society he received a Royal Medal in 1900, and the Sylvester Medal in 1919; and he was elected Vice-President in 1917. He was an Honorary Member of the Royal Irish Academy and of the Cambridge Philosophical Society. Honorary Degrees were conferred on him by Cambridge, Dublin, Aberdeen, and St. Andrews Universities, and he represented the Royal Society as a Fellow of Winchester College.
H. H. Τ.
Percy Alexander MacMahon's obituary appeared in Journal of the Royal Astronomical Society 90:4 (1930), 373-378.