by Robin J. Wilson

© Oxford University Press 2004 All rights reserved

**Kirkman, Thomas Penyngton** (1806-1895), mathematician and philosopher, was born on 31 March 1806 in Bolton, Lancashire, the only son of John Kirkman *(d.* 1839), cotton dealer, and his wife, Elizabeth. Baptized Pennington, he adopted the spelling Penyngton in later life. He was educated at Bolton grammar school, where he was the best scholar in the school, but in spite of protestations by the headmaster was removed at the age of fourteen to work in his father's business. During his years in the cotton trade he studied the classics and taught himself French and German. At the age of twenty-three he broke away and enrolled as a student at Trinity College, Dublin, supporting himself financially by private tutoring. He studied the required subjects of philosophy, classics, mathematics, and science, and graduated BA in 1833.

After a year as private tutor to an Irish baronet Kirkman returned to England and took holy orders in 1835. Following curacies in Bury, Lancashire, and Lymm, Cheshire, he moved in 1839 to a curacy in Croft, near Warrington, and in 1845 became rector of the newly created parish of Croft-with-Southworth, where he remained for over fifty years. At the Croft rectory:

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. (W. W. Kirkman, 238)

On 8 June 1841 he married Eliza Ann Wright of Runcorn, Cheshire, with whom he had seven children. To support his family he supplemented his parish income by private tutoring, later augmented by the income from some property inherited by his wife.
It is not known when Kirkman's interests in mathematics developed, but in 1844 he was inspired by a combinatorial prize question in the *Lady's and Gentleman's Diary,* set by the editor, the Revd William Woolhouse. This led to a pioneering paper in 1847 on the arrangements of objects into groups of three with each pair of objects appearing just once; such arrangements are now called 'Steiner triple systems', after the Swiss mathematician Jakob Steiner, even though Kirkman had priority and contributed much more than Steiner to the subject. In 1850 Kirkman presented the problem for which he is best remembered, the 'fifteen schoolgirls problem', of arranging fifteen young ladies in groups of three on each of seven days so that each young lady walks with every other one just once. He continued to write in this area for a further twelve years, and is now recognized as the founding father of design theory.

Kirkman's next area of mathematical interest was hypercomplex numbers, or 'pluquaternions', as he named them, following Sir William Rowan Hamilton's 1840s work on quaternions; Kirkman's election as a fellow of the Royal Society on 11 June 1857 arose partly from his studies in this area. About 1850 he also wrote on geometry--in particular, the sixty points of concurrence that arise from Pascal's six-points-on-a-hexagon theorem when the six points are permuted in all possible ways; this work did much to establish his reputation in British mathematical circles.

From 1853 Kirkman made pioneering contributions to the classification and enumeration of polyhedra (or 'polyedra', as he called them), and to the emerging theory of groups. Unfortunately his writings were couched in such obscure terminology that a substantial memoir on polyhedra was turned down by the Royal Society as being unreadable. Later, when the Paris Académie des Sciences proposed a *grand prix de mathématiques* for advances in these areas, Kirkman's entry for the 1860 groups prize was unsuccessful; in consequence he chose not to submit for the 1861 polyhedron prize. The resentments he felt against the Royal Society and the Paris Académie made him increasingly embittered, and his disillusionment spilled over into his relationships with leading mathematicians of the day. In spite of this he collaborated with Tait on the classification of knots with up to eleven crossings, and regularly contributed difficult mathematical questions to the *Educational Times* until he was well into his eighties.

Kirkman's contributions in other areas have barely survived. He wrote a book, *Philosophy without Assumptions* (1876), and became embroiled in theological controversies, such as his support for the rebel Bishop Colenso, who asserted that the early books of the Bible need not be taken literally. Kirkman also wrote pamphlets supporting free enquiry and free expression, but criticized the materialistic and evolutional philosophy espoused by such as Tyndall, Huxley, and Spencer. In his *Philosophy* he memorably paraphrased Spencer's description of evolution as 'a change from an indefinite, incoherent, homogeneity to a definite, coherent, heterogeneity; through continuous differentiations and integrations' (H. Spencer, *First Principles,* 1863, 216), as 'a change from a nohowish untalkaboutable all-likeness, to a somehowish and in-general-talkaboutable not-all-likeness, by continuous somethingelse-ifications and sticktogetherations' (T. P. Kirkman, 292).

Thomas Kirkman died at his home, Fernroyd, St Margaret's Road, Bowdon, near Altrincham, on 3 February 1895. In a letter he summarized his career: 'What I have done [is] not likely to be talked about intelligently by people so long as I live. But it is a faint pleasure to think it will one day win a little praise' (Macfarlane). His wife survived him by less than a fortnight.

ROBIN J. WILSON

**Sources **

N. L. Biggs, 'T. P. Kirkman, mathematician', *Bulletin of the London Mathematical Society,* 13 (1981), 97-120

S. Mills, 'Thomas Kirkman--the mathematical cleric of Croft', *Memoirs of the Literary and Philosophical Society of Manchester,* 120 (1977-80), 100-09

A. Macfarlane, *Lectures on ten British mathematicians of the nineteenth century* (1916), 122-33

H. Perfect, 'The Revd Thomas Penyngton Kirkman FRS, 1806-1895: schoolgirl parades--but much more!', *Mathematical Spectrum,* 28/1 (1955-6), 1-6

*DSB*

W. W. Kirkman, 'Thomas Penyngton Kirkman', *Memoirs of the Literary and Philosophical Society of Manchester,* 4th ser., 9 (1895), 238-43

T. P. Kirkman, *Philosophy without assumptions* (1876)

m. cert.

*CGPLA Eng. & Wales* (1895)

**Archives **

CUL, letters to Sir George Stokes

**Likenesses **

photograph, repro. in Macfarlane, *Lectures*

**Wealth at death **

£10,983 *9s. 1d.:* probate, 20 April 1895, *CGPLA Eng. & Wales*

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