Thomas Penyngton Kirkman

Excerpt from The Fifteen Schoolgirls by Dick Tahta


Kirkman was born in 1806 in Bolton, Lancashire. His father was a cotton merchant who wanted his only son to join the family business. Thomas went to Bolton Grammar School, but left unwillingly at the age of fourteen, to work with his father for the next nine years. This must have been a difficult time for him, but he continued to study on his own and he finally rebelled and enrolled at the age of 23 as a student at Trinity College, Dublin. He paid for himself by taking on some tutoring and after he graduated in 1833 he stayed in Ireland for another year as a private tutor.
On his return to England in 1834, he joined the church and obtained a
curate’s post at Bury, though he was not in fact ordained until two years
later, when he moved to be a curate in Lymm, Cheshire. Then in 1839, at the age of 33, he moved to Croft, a village near Warrington. This turning point in his life and what he made of it was later described by his eldest son, William, in an obituary:
He was enticed by fair words, by the then rector of Winwick, to bury himself for life as the rector of the newly-formed Parish of Southworth with Croft, where he remained for 52 years. Here, by perseverance and his gift of teaching, he formed, out of the roughest material, a parish choir of boys and girls who could sing at sight any four-part song set before them. Here also, with an expenditure of mental labour that only the finest of physical constitutions could have sustained, he devoted, practically, the whole of his time (for the parochial work was small) to the study of pure mathematics, the higher criticism of the Old Testament, and questions of first principles.
Kirkman married Eliza Wright in 1841 and they were to have seven children (five sons and two daughters) over the next thirteen years. In the circumstances it may seem surprising that in his fortieth year, 1846, he presented a mathematical paper to a well-known Literary and Philosophical Society in Manchester. This was the first indication that he had any specific mathematical interests, but it was to be the first of a series of articles he published for the rest of his life. His paper involved a combinatorial problem which arose from a puzzle that had been circulated in an annual publication some years before. He submitted his own puzzle about  the fifteen schoolgirls, which was a particular case of the problem he had been working on, a few years later in 1850.
He published a number of further mathematical papers on combinatorial
problems over the next few years in specialist mathematical journals,
and corresponded with many of the leading mathematicians of the time. He also worked at some related geometrical issues, including a study of the arrangements of Pascal lines associated with a hexagon inscribed in a conic, and some lengthy papers on the classification of certain polyhedra. The latter topic seems to have been stimulated by a Prize Question proposed by the French Académie des Sciences in 1855. There had been no response to this challenge and the prize had been held over for a few years. Kirkman wrote a lengthy memoir which he submitted to the Royal Society which published part of it; he had been appointed a Fellow in 1857.
He had not responded directly to the Prize Question on polyhedra because his attention had been caught by another question proposed for the prize which was to be awarded in 1860. This was on an algebraic problem on what would now be called permutation groups. Kirkman prepared and submitted a lengthy memoir on the subject, written in French over a period of two years in which he was also writing his even more lengthy paper on polyhedra. But in the event the prize was not awarded. Kirkman was disappointed and became quite obsessional about the failings of the Académie. This paralleled his feelings about what he felt were the failings of the Royal Society in not publishing all of his work on polyhedra. He continued to work on various topics, but began to feel somewhat isolated from other mathematicians. He had fallen out with Cayley whom he blamed for the neglect of his work on polyhedra. He also fell out with Sylvester over an argument about priority in work associated with the puzzle about the fifteen schoolgirls. At one stage he thought of trying to get some appointment at Cambridge and enlisted the help of de Morgan who wrote on his behalf to Trinity College. Part of this letter is quoted in an account of Kirkman’s life and work by Norman Biggs: He is buried at Croft and very much desires a better field of action in which he may be able to see a little more of intellectual life, over and above his clerical doings. He has barely £180 a year, and is an active man, moderate and of orthodox repute .. . He has worked for many years at subjects which will not bring him before the general eye, and is a staunch enthusiast and con amore mathematician. Nothing came of this. Kirkman’s mathematical work remained relatively unknown. According to Biggs, “It must have been convenient for the mathematical establishment to look upon him as a crazy cleric who had  invented an amusing puzzle about schoolgirls. And it must be said that he himself did little to dispel this illusion.” Apart from his mathematical work, Kirkman wrote on other issues that he felt passionately about. One early book that was published in 1852 was called First mnemonic lessons in geometry, algebra and trigonometry. This arose from his private tutoring of young schoolchildren in Croft and offered some curious condensed phrases that were supposed to make it easier to recall certain mathematical rules. Thus Pythagoras’ theorem was to be recalled as ‘ qua poth is both qua sides’ where qua(drate) = square of, and poth =(hy)poth(eneuse). The cosine formula was coded as ‘sq b is DUQ ac le CoBang two ac’ where sq b = square of b, le = ‘less’, DUQ = duo quadrata = sum of squares of, and CoBang = cosine of B angle; this mnemonic was meant to be read as ‘squib is duck ac le cobang two ac’.
There were lots of these mnemonics introduced in the course of discussionsbetween ‘Uncle Penyington’ and two fictional children, Richard and Jane. (These have always been favourite names for educational writers – some readers may recall their re-incarnation in Ladybird reading schemes.) One aside to the girl suggests that Kirkman did not think mathematics was a suitable subject for women: “Take some pains to master them, my dear Jane, and you will save your time, of which you have very little to spare for mathematics, a study in which I should be sorry to see you deeply engaged”. On the other hand, he gives her some interesting insights; for example, her comment on imaginaries: “So whatever mystery or appearance of contradiction there may be here, it springs not from the answer or the oracle, but from the ignorance of the interrogator. His duty is not to cavil at the response, but to go away ashamed of himself and wondering”. Moreover, Jane is shown as understanding the lesson when the teacher suggests that Richard has been finding it rather difficult. Richard: Indeed I do, and have left to Jane the honour of following you all through this lesson. How do you feel, dear sister of mine, after this feast of trigonometry? Jane: A little fatigued; but more by the variety than the difficulty of the subject. More interesting perhaps was his introductory explanation of the proposed method of making mathematics more accessible: If the experience of others in tuition agrees with my own, I may perhaps look to reap a little praise – not mathematical, on ground like this, but simply didactical – the praise of teaching well, of which I confess myself covetous. It appears to me that distaste for mathematical study often springs, not so much from any abstruseness in the subject at any point, to the student who has mastered the approaches, as from the difficulty generally felt in retaining the previous results and reasoning. This difficulty is closely connected with the unpronounceableness of formulae: the memory of the tongue and of the ear are not easily turned to account; nearly everything depends on the thinking faculty, or on the practice of the eye alone. . . My object is to enable the learner to talk to himself in rapid, rigorous and suggestive syllables, about the matters which he must digest and remember. There cannot have been many sales of this extraordinary book. The author himself doubted “that any suggestion of his, for the improvement of established methods of tuition, will be favourably received, or confessed to have any value”.
Kirkman also wrote on other than mathematical issues. He was critical of the orthodox church opinion that had been shocked by the views of John Colenso, Bishop of Natal, who had suggested, in 1862, that the Pentateuch, the first five books of the Old Testament, were not literally true. He gave two public lectures in Croft in defense of the Bishop and these were published as pamphlets in 1865. In the first, Truth against tradition, he wrote: “ I believe that it is a dangerous error to say that every word, every syllable, every letter of the Bible is just what it would be if God had spoken it from heaven to us without human intervention”. Later he was to feel that his stance had affected his chances of promotion in the Church.
He was a fanatical opponent of what he called materialist philosophers who were, he felt, peddling dangerous evolutionary ideas. He wrote more than a dozen pamphlets over two decades. He was particularly critical of Herbert Spencer whom he accused of talking nonsense by dealing with “ abstracts in the clouds, instead of building on the witness of his own selfconsciousness”. He made fun of Spencer’s definition of evolution (“a change from an indefinite incoherent homogeneity to a definite heterogeneity .. . ”) by writing that, “Evolution is a change from a nohowish untalkaboutable alllikeness, to a somehowish and in-general-talkaboutable not-all-likeness, by continuous somethingelsifications and sticktogetherness”. He published more than a dozen short pamphlets in his sixties, a decade when he does not seem to have been so involved with mathematics as he had been. These pamphlets were on various issues, with titles such as ‘The ordeal of jealousy’, ‘Philosophy without assumptions’, ‘Clerical dishonesty’ and ‘An address at the twenty-first soiree of the Brighouse Mechanics’ Institute’.
Though he was never to be again so prolific as in his first two decades at Croft, he continued to work and publish papers on the various mathematical topics he had been interested in, as well as embarking on new ones, such as the theory of knots in which he published papers in his eighties. He had been introduced to this topic by the mathematician, Peter Tait, who shared his views of Spencer. He also continued to submit numerous problems and solutions to a half-yearly collection of mathematical puzzles after his retiremen in 1892 when he moved to Bowdon. These included two problems in the theory of equations published in the year of his death in 1895. His wife Eliza died twelve days later.
A final remark is taken from one of Kirkman’s letters to a correspondent: “ What I have done in helping busy Tait in knots is, like the much more difficult and extensive things I have done in polyhedra or groups, not at all likely to be talked about intelligently by people as long as I live. But it is a faint pleasure to think it will one day win a little praise.”

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