Thomas attended the grammar school in Bolton where he was taught Greek and Latin but no mathematics. He did well at school but although his schoolmaster and the vicar saw that he was a potential Cambridge fellow, Thomas's father could not be persuaded and Thomas was forced to leave school at the age of 14. He worked in his father's office, continuing his study of Greek and Latin in his own time and extending his knowledge of languages by also learning French and German.
After 9 years working in the office, Thomas went against his father's wishes and he entered Trinity College Dublin to study mathematics, philosophy, classics and science for his B.A. On returning to England in 1835 he entered the Church of England. He spent five years as a curate, first in Bury, then in Lymm. By 1839 he became vicar in the Parish of Southworth in Lancashire, a position he held for 52 years.
He married Eliza Anne Young. They had one son, William (born 1843) and two daughters, Mary Eliza (born 1847) and Katherine (born 1855).
As a graduate of Dublin University, Kirkman was naturally interested when Hamilton published his work on quaternions. Kirkman's interest in mathematics was rapidly increasing and his first paper was presented in 1846 when he was 40 years old. It answered a problem which appeared in the Lady's and Gentleman's Diary of 1845 and shows the existence of 'Steiner systems' seven years before Steiner's article which asked whether such systems existed. This work of Kirkman appeared in the Cambridge and Dublin Mathematical Journal. After Steiner asked his question, a solution was given by M Reiss in 1859. Kirkman sarcastically wrote
..... how did the Cambridge and Dublin Mathematical Journal Vol II p. 191, contrive to steal so much from a later paper in Crelle's Journal Vol LVI p. 326 on exactly the same problem in combinatorics?Despite Kirkman's clear priority, we call such systems today 'Steiner systems' and not 'Kirkman systems'.
In 1848 Kirkman published a work, described by De Morgan as
the most curious crochet I ever sawin which Kirkman attempted to make mathematical formulae more memorable by asking the student
...to teach them to your ear and to your tongue, each of which has a memory of its own, by saying them again and again with a sing-song repetition...The book was not popular but it is fair to say that school teaching of mathematics today sometimes resorts to similar memory aids.
Kirkman then investigated generalisations of the quaternions. For example the Cayley numbers and generalisations are discussed. He also at this time examined certain questions in geometry. He examined points of congruence of Pascal lines and his work on this area came to be part of standard texts such as Salmon's Conics.
Kirkman is best known for the Fifteen Schoolgirls Problem. He published this in the Lady's and Gentleman's Diary of 1850.
Fifteen young ladies of a school walk out three abreast for seven days in succession: it is required to arrange them daily so that no two shall walk abreast more than once.The solution to the Fifteen Schoolgirls Problem is not particularly hard. Cayley published a solution first, then Kirkman published his own solution, which of course he knew before asking the question. Sylvester also studied aspects of this problem and later disputed with Kirkman on who had thought of it first.
There is a more general problem of when n schoolgirls can be arranged into n/3 triples on each of (n - 1)/2 days so that no two are in the same triple more than once. Clearly n must be congruent to 3 modulo 6 if such a set with n elements exists, but it was not until a paper in 1971 that it was proved that such an arrangement is possible for every such n.
As Biggs comments in  regarding the Fifteen Schoolgirls Problem,
It is unfortunate that such a trifle should overshadow the many more significant contributions which its author was to make to mathematics. Nevertheless it is his most lasting memorial.From 1853 Kirkman began a large piece of work on the enumeration of polyhedra, publishing many major papers in the Royal Society. Kirkman became a Fellow of the Royal Society in 1857, mainly for this work on polyhedra which had been communicated to the Royal Society by Cayley.
On seeing that the Académie des Sciences of Paris were awarding a prize for the study of 'group theory' in 1860, Kirkman decided to enter. This meant that he had just two years to become an expert in group theory. Indeed he achieved this and submitted a memoir of high quality. Three memoirs were submitted, the other two by Émile Mathieu and Jordan. The three submissions were praised but no prize awarded.
Kirkman continued to work on group theory, his last paper on the subject being The complete theory of groups (1863). The paper, which is an abstract of his Grand Prize Memoir, gives a recursive method for compiling lists of transitive groups and a complete list of transitive groups of degree ≤ 10 is given.
Kirkman also planned to enter for the Grand Prix of the Académie des Sciences of 1861 on the topic of polyhedra. However although much of this work had been completed, he changed his mind after his disappointment in the 1860 competition. He submitted a long work of 21 sections on polyhedra to the Royal Society in 1862. They decided to publish only the first 2 sections which themselves take up over 40 pages of the Proceedings. Again disappointed, Kirkman blamed Cayley and wrote to John Herschel suggesting Cayley wanted to prevent publication because he had a paper of his own on polyhedra.
Kirkman continued to work on combinatorial questions. Then in 1884, at the age of 78, he published his first paper on knots. This was followed by a series of papers. In joint work with Tait they produced tables of knots with 8, 9 and 10 crossings.
Kirkman continued to study mathematics until his 89th year sending questions and solutions to the Educational Times up to a few months before his death.
Article by: J J O'Connor and E F Robertson