Hidden away in the village of Croft (near Warrington) where he was rector, Thomas P. Kirkman (1806-1895) wrote a remarkable mathematical paper on Combinatorics which has since become in a landmark in the subject. Today this kind of mathematics has applications in coding theory and the science of ‘arrangements’ ¾ or how symbols combine and how their schemes can be designed. This was his first substantial academic paper and was published in his fortieth year, quite unusual in mathematics, which is often regarded as a young man’s game. Full recognition of Kirkman’s all round mathematical accomplishments came much later. It was only in the 1960s and the advent of the Computer Revolution that he became widely known as a formidable mathematician, and no less than the father of ‘mathematical design’.
A member of the Manchester Literary and Philosophical Society in the 1840s Kirkman was elected an Honorary member in 1852 and became its standard bearer for mathematics. In these early days he also gave lectures to the Society on his idiosyncratic system of Mnemonics for learning mathematics, and later, with his broad interests gave talks on philosophy. Kirkman was highly active in the intellectual scene of the industrialising city of Manchester for much of his life.
Kirkman was born in Bolton in 1806 the eldest child of an entrepreneur in the Lancashire Cotton Industry. Thomas proved himself an outstanding pupil at Bolton Grammar School, and the headmaster recommended university thinking an academic career would be suitable. His father though had little understanding of Academia, and Thomas left school at the age of fourteen in 1821 to work in the family cotton business. The boy was a natural auto-didact and pushed ahead with his studies so time in the back office was not wasted. He taught himself French and German, read philosophy, and identified his religious position. In 1829, he was admitted to Trinity College Dublin, a popular academic destination for students from the northern cities of England. It offered an alternative destination to Oxford and Cambridge in the south, and unlike those universities did not require the expense of residence. In 1832 he was awarded a top honours degree, a senior moderator, and along the way he gained a handful of prestigious academic prizes.
After graduation Kirkman entered the Anglican Church and served as curate in Bury and Lymm before a permanent appointment to the country parish of All Saints in Southworth-with-Croft in 1839. In 1845 he became its rector and there he was to remain for his entire career. Little is known of his mathematical education. At school the boys were taught the bare rudiments of the subject and at Trinity mathematics was taught as a part of a broad course which included logic, science, and the arts. It was very different from the Cambridge undergraduate experience where the honours degree course, the famed Mathematical Tripos, was one hundred per cent mathematics.
Settled in Croft, Kirkman and his wife raised seven children. As a diversion he contributed to the Ladies and Gentleman’s Diary. This annual almanac could enlighten its gentlefolk audience on such things as the times of sunrise and sunset, details of eclipses in the coming year and the best times to see the planets. It also supplied a compendium of challenging verbal and mathematical puzzles. In addition, the editor was keen to support articles which would educate its readership in different branches of mathematical activity.
People in the early 1840s would not have heard of Kirkman the mathematician but in the edition of the Diary of 1845, he sent in solutions to twelve problems (out of eighteen proposed) which had been set out to challenge the readership. As far as we know, this burst of activity signalled his entrance to the world of mathematics – and he would not repeat such a flurry again. One of the problems led to a rich vein which will forever be associated with his name, the tantalising ‘Schoolgirls problem’: Fifteen schoolgirls walk in threes on each of the seven days of the week. How should the arrangements be made so that each pair of girls walk together exactly once in the week? To address this issue, readers should develop a daily walking schedule for each day of the week. This challenge is far from trivial.
While problems of this type lead to interesting developments, Kirkman went on to a more substantial problem in the field of geometry, and this he announced at a meeting of the Manchester Society in 1853. The problem was to count the possible number of distinct geometrical solids (polyhedra) which have a stipulated number of faces. For example, the ordinary cube in three dimensions is one solid with six faces but there are six other solids with this number of faces. On this and cognate problems he published four papers in the Transactions of the Royal Society and, based on his increasing mathematical activity, he was elected FRS in 1857. From this platform he sought even wider recognition.
In 1858 the Académie Française des Sciences in Paris offered two subjects for prize competition, and these attracted Kirkman’s attention. They were given with closing dates of 1860 and 1861 so there was no time to lose. The first prize was on the French subject of group theory in algebra and the second on polyhedra in geometry. In his submission for the first prize, the jury noted Kirkman’s ‘ingenuity’ but they did not award him the prize (or indeed to any competitor). Kirkman took the rejection personally and despite a newspaper campaign in the Manchester Courier written in terms of a ‘national rebuff’ the jury did not change their decision.
For the polyhedra prize of 1861 Kirkman drafted a voluminous paper of twenty-three sections amounting to his magnum opus. In the end he decided not to enter the Académie contest, perhaps not chancing a second rejection. Instead, he sent his paper to the Royal Society, but alas this did not find favour and only two sections were published in their Transactions. The whole saga of the prizes caused Kirkman consternation and resulted in antagonism towards the French Academy while blaming the Royal Society referee for his paper’s rejection in England. This treatment left him embittered from which he never really recovered though it did not cause him to abandon mathematics.
Kirkman revelled in tackling the very difficult mathematical problems. His isolation by being based in Croft was both a strength and weakness. He developed ideas himself, but they were often couched in his idiosyncratic notation which posed a difficulty for other researchers. As in Philosophy in which he published his only book, Philosophy without Assumptions (1876), he prided himself in thinking from first principles in a highly individual way. His singular mode of expression was undoubtedly the reason for the judgement of the French Academy jury and the referee of the Royal Society paper.
Following his death in 1895 his parishioners provided a memorial window in remembrance of his work in the parish of Croft for over fifty years. As a mathematician he went into the literature as the father of the Fifteen Schoolgirls problem though he had done so much more. One co-worker acknowledged him as one of the most penetrating mathematicians of the nineteenth century, while a historian described him as one of the more intriguing figures in the history of mathematics.
Tony Crilly, member of Manchester Lit & Phil