Clifford, William Kingdon

(1845-1879), mathematician and philosopher of science

by Albert C. Lewis

© Oxford University Press 2004 All rights reserved

Clifford, William Kingdon (1845-1879), mathematician and philosopher of science, was born on 4 May 1845 at Exeter, the eldest son of William Clifford, a justice of the peace who died in 1878. His mother, whose maiden name was Kingdon, died not long after he was born. Clifford's studies at Mr Templeton's school in Exeter prepared him for entrance in 1860 to King's College, London, where he pursued a wide range of classical, literary, and scientific subjects. He developed an interest in mathematics and obtained a scholarship at Trinity College, Cambridge, in 1863. He won the college declamation prize in 1866 for an oration on Sir Walter Raleigh and in November of that year he was elected to a university debating club, the Cambridge Conversazione Society, also known as the Apostles. A common memory of his friends from student days was of Clifford's gymnastic feats. Though slight, he demonstrated considerable dexterity and strength; he could, for example, do pull-ups with only one hand on the horizontal bar. In the Apostles, Clifford met a fellow third-year student, Frederick Pollock, who was studying the classical curriculum and who became one of Clifford's closest friends. Pollock's introduction to Clifford's Lectures and Essays (1879) is a principal source of biographical information about him and, in particular, tries to document what little is known about Clifford's seemingly gradual movement away from his high-church upbringing. A major influence in this direction came from reading the authors that Clifford and Pollock most often discussed: Charles Darwin, Herbert Spencer, Benedict de Spinoza, and Giuseppe Mazzini.

In 1867 Clifford was ranked second wrangler in the university mathematical tripos and was second Smith's prizeman. His friends were surprised at his doing so well in the face of all his non-mathematical activities and the small amount of time he appeared to put into studying for the examination. The following year he was elected a fellow of Trinity College and gave his first public discourse to a large audience, 'On some of the conditions of mental development', at the Royal Institution in London. This talk, which applied Herbert Spencer's ideas on biological evolution to the development of the mind, established him as an effective and popular speaker on scientific topics. Until 1875, when failing health curtailed his public speaking, Clifford addressed general audiences on topics such as the history of the sun, the evidence for an ether, the relation between science and poetry, and Charles Babbage's calculating machines.

In 1870 Clifford received his MA and went on the English expedition to observe the solar eclipse of December that year in the Mediterranean. In spite of the wreck of his ship near Catania, Italy, this trip gave Clifford a taste for travel and a love of the Mediterranean region. He moved from Cambridge in 1871 to take up the professorship of applied mathematics at University College, London. In 1874 he was elected fellow of the Royal Society for, among other things, his distinction in 'the metaphysics of geometrical and physical science' (RS archives, election certificate, 1874). On 7 April 1875 he married Lucy [see Clifford, (Sophia) Lucy Jane (1846-1929)], daughter of John Lane. They had two daughters. Clifford's friends described him as having an amicable attitude towards everyone but enjoying himself most when planning children's parties and joining in the entertainment. Although he practised his athletic abilities beyond his student days, he was not particularly careful about his health and was known to stay up all night on occasion in order to complete a mathematical project. In 1876 he had symptoms of a pulmonary disease and reluctantly took a six-month leave of absence. His friend Leslie Stephen, later to become editor of the Dictionary of National Biography, began a subscription to pay for a trip to the warmer climate of Spain and Algiers. This mitigated Clifford's continuing weakness but in early 1878 his condition worsened to such an extent that, following medical advice, he and his wife travelled to the Mediterranean. Again he seemed improved on his return but the next year a relapse caused them to journey to Madeira where he remained for the last few months of his life.

Clifford's years of illness were at the same time among the most mathematically productive of his short life, and he was encouraged to continue by some of the leading mathematicians of the day. Olaus Henrici reported on 26 June 1878 (Royal Society Library, RR.8.84) on a paper submitted by Clifford to the Royal Society, entitled 'Classification of locii', that the work was a 'very important one and the simplicity of the methods used ... fully worthy of the genius of Clifford'. Although, 'owing to the bad state of health of its author' the work was still 'fragmentary and under ordinary circumstances' not publishable, he nevertheless recommended that it 'be printed as it stands, with the addition perhaps of one or two explanatory notes' since Clifford was in no condition to correct the manuscript. Such notes supplied by Henrici and others to this and other works in the Mathematical Papers (1882) are a useful, indeed essential, adjunct to Clifford's original presentations.

On his death, Clifford's wife gave his unfinished manuscripts to those of his colleagues who were willing to complete them and see to their publication. One such manuscript, for a book concerned with the bases of pure and applied mathematics, was completed by Karl Pearson and published in 1885 as The Common Sense of the Exact Sciences. Though Clifford intended the work to be readable by the layperson, it was more than a popularization of mathematics since it contained many original ideas. Bertrand Russell, in a preface for the 1946 edition, described his enthusiasm in reading the book at the age of fifteen in 1887. Russell had been puzzling over Euclid's geometry and was delighted to find that there was a rationale for geometry, and mathematics generally, that he found lacking in Euclid's presentation.

Most of Clifford's mathematical work centred on geometry which, considered as the study of space, he regarded as an empirical science. He believed that Immanuel Kant's argument for an a priori, universal component in mathematics was justified by discoveries about human development, perception, and language made since Kant's time by Darwin, Spencer, and Max Müller. Clifford readily adapted to his own use the work on invariant theory by the leading British mathematicians of the time, Arthur Cayley and James Joseph Sylvester. His discovery of a generalization of quaternions, which he termed 'biquaternions', and of the algebra associated with them (and later named after him), was built upon the work of the Irishman William Rowan Hamilton, who discovered quaternions, and of the German Hermann G. Grassmann, whose ideas Clifford helped to propagate in the English-speaking world. Clifford also used Hamilton's quaternions in his Elements of Dynamic: an Introduction to the Study of Motion and Rest in Solid and Fluid Bodies (1878). Though quaternions themselves did not continue to play such a direct role in physics, the notions Clifford derived from them, such as the divergence of a vector, and the vector and scalar products, continued to be useful in twentieth-century physics. Clifford's biquaternions, on the other hand, found applications (unanticipated by Clifford) in the representation of the wave function in quantum physics.

Invariant theory and biquaternions, though algebraic topics, have strong connections in Clifford's work to geometry and in particular to the non-Euclidean geometries that were discovered earlier in the nineteenth century but became incorporated into the main body of mathematics only during his lifetime. Intrigued by the possibility that one of these new geometries could describe the physical world better than the traditional Euclidean geometry, he made new discoveries concerning all of them but singled out as his favourite candidate the elliptic geometry. He discovered a surface (since named the 'Clifford surface') in elliptic space with the unexpected property of supporting parallel lines ('Clifford parallels'). In an abstract for proposed work in 1870 he posited that motion of matter could be explained by the variation of the curvature of space. This anticipation of one of the features of Albert Einstein's general theory of relativity forty years later came from his reading of the German mathematician Georg Friedrich Bernhard Riemann, whose key work in this area Clifford translated into English ('On the hypotheses which lie at the bases of geometry').

Many of Clifford's popular lectures and writings strongly upheld scientific reasoning over religious dogma. His lively, aphoristic style and authoritative and wide-ranging knowledge made him an influential spokesman for a certain view of the scientific spirit of the late nineteenth century. His creed expressed in 'The ethics of belief' (1876), that 'it is wrong always, everywhere, and for anyone, to believe anything upon insufficient evidence' (Clifford, Lectures, 1.186), was approvingly cited by others such as the psychologist William James in his Will to Believe (1897). Scientific progress for Clifford was made by having doubts, not faith. His anti-religious stance was so rigorous that, as James Sully expressed it in a review of Lectures and Essays, Clifford had a 'religious abhorrence of religion' and thus could be said to have adopted a dogmatism comparable to that of his high-church upbringing (The Academy, 17, 1880, 133-4). Clifford was the basis for William Hurrell Mallock's young materialist, Mr Saunders, in The New Republic (1877), an unflattering caricature that Mallock moderated somewhat in the version that appeared shortly after Clifford's death. One of the tenets of Clifford's alternative to the usual religious views was his version of a psychological atomism that asserted that 'Reason, intelligence, and volition are properties of a complex which is made up of elements, themselves not rational, not intelligent, not conscious' (Clifford, Lectures, 2.87). Twentieth-century theories of self-organization may have made Clifford's notion scientifically more acceptable, but his views undoubtedly appeared rather extreme in his time.

Clifford sought to formulate an ethic consistent with his philosophical and scientific positions and, like Spinoza, one that had geometrical certainty. He felt that the theory of evolution, when applied to societies, provided the best available scientific foundation for ethical beliefs and that the theory implied that the survival of the community depends on how well it treats its members. Ethics is thus a matter for the community, not for 'self-regarding' individuals, and happiness is desirable only if it makes for a more efficient citizen. Society must support creative activity and the search for truth by an individual, wherever that may lead, since a knowledge of the truth has great social importance. The formality of his ethics, however, contrasted with his more flexible and open personal manner: 'All this, by the way', he wrote to Frederick Pollock's wife, 'is only theory; my practice is just like other people's' (Clifford, Lectures, 1.49).

Clifford died at Madeira on 3 March 1879. He was buried in Highgate cemetery, Middlesex. His friends began a subscription for a pension to benefit his family which otherwise would have been without income. His wife, Lucy, maintained their social circle and became a confidante of literary figures such as George Eliot, James Russell Lowell, and Henry James. She wrote a number of novels, her first and best known being Mrs Keith's Crime (1885). She died on 21 April 1929 and was also buried in Highgate cemetery.


F. Pollock, 'Introduction', in W. K. Clifford, Lectures and essays, ed. L. Stephen and F. Pollock (1879)
W. K. Clifford, Mathematical papers, ed. R. Tucker (1882), xiii-xxix, xxxi-lxx
E. A. Power, 'Exeter's mathematician. W. K. Clifford, F.R.S., 1845-79', Advancement of Science, 26 (1970), 318-28
N. G. Annan, Leslie Stephen: the godless Victorian, rev. edn (1984)
CGPLA Eng. & Wales (1879)
The Times (17 March 1879), 6f
M. Chisholm, Such silver currents: the story of William and Lucy Clifford, 1845-1929 (2002)

UCL, letters

J. Collier, oils, 1878, RS [see illus.]
J. Collier, oils, 1899 (after his oil painting, 1878), NPG
C. H. Jeens, line engraving (after photograph by Barraud and Jerrard), repro. in Clifford, Lectures and essays, 1, frontispiece
photograph, repro. in Clifford, Lectures and essays, 2, pasted-in frontispiece

Wealth at death  
under £450: administration, 18 June 1879, CGPLA Eng. & Wales

Oxford University Press 2004 All rights reserved


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