... was distinguished for his docility and reasonableness, and regulated even his boyish amusements with a degree of gravity only expected in the more comprehensive, but not more interesting transactions of manhood.

When he had just turned his sixth year, the English teacher informed his mother that he should be sent to the grammar school. He was accordingly placed there, and, in the intervals of the classes, received lessons in writing and arithmetic at another school. His progress in Latin was not remarkable; and, incredible as it may now appear, he certainly did not evince any uncommon capacity for arithmetical calculations: but still, such was the originality of his character, that he was undoubtedly considered by his companions as "no vulgar boy."

... possessed singular attainments in the languages, and a most profound knowledge of mathematics, accompanied with the most amiable dispositions and a delightful simplicity of character. He was, indeed, in many respects, an extraordinary man. The hours of business were faithfully devoted to the counting house, and in the evenings, to a late hour, spent in literary and scientific studies.

... some half dozen or fewer of my companions formed a monthly society, at the instigation of William Spence. We read all sorts of essays about every sort of subject, from the "cedar tree that is on Lebanon to the hyssop that springeth out of the wall", and afterwards discussed philosophical topics, and then had a supper. But we were not so wise when we broke up, which was after midnight, as when the sederunt commenced at seven o'clock. I was the youngest member, and certainly not the best writer; both Spence and Park were superior; but it is curious to observe how early innate character begins to manifest itself. The essays of William Spence were very astronomical, we thought them profound; they were all about planets and comets, the cosmogony of the earth, the infinite divisibility of matter, and the boundless of nature, premundane space; any thing of this world was too gross to enter into his speculative theories.

The integral

$L_{2}(x) = -\int _{0}^{x} \log_{e}\Large\frac{1-t}{t}\normalsize dt$

has an extensive history that long predates it being named and being referred to as the dilogarithm. ... the first comprehensive detailed study of the function is the essay of Spence (1809), which was generally not referenced on the continent for several decades. (This work was, along with other manuscripts left by Spence, re-edited by John Herschel

On the appearance of the first of these Essays on the Theory of the various Orders of Logarithmic Transcendents in 1809, its merits were universally acknowledged to entitle the author to hold a high rank among mathematicians. It is to be regretted that a premature death carried him off at a time when he was making such a rapid progress in the cultivation of the sciences. Of these tracts, the first only was intended by the author to meet the public eye in its present shape, though a few copies of another of them, demominated 'Outlines of a Theory of Algebraical Equations', had been printed and distributed among the author's friends. The others were selected from Mr Spence's papers by the editor, J F W Herschel, Esq., who was pre-eminently qualified to judge the respective merits of the several pieces. These papers clearly prove, that Mr Spence possessed both genius and industry in a high degree, and a force of intellect of a very superior kind. Notwithstanding the disadvantage attending some of these essays, in common with other posthumous works, we warmly recommend them to the attention of the British student, as models of analytic papers of uncommon excellence. They will inspire him with a taste for pure analysis, which, when properly directed, in the present state of the sciences, is the most certain method of arriving at distinction.

In music he was not only a scientific harmonist and a pleasing composer, but he performed on the flute with admirable sweetness and skill. He wrote respectable verses, and had a fine tenor voice, which enabled him to sing pathetic airs, especially of his own composition, with much taste and effect.

Spence, besides being a most delicious performer was a considerable composer, and made beautiful sonatas, which had as much character as the compositions of Frederick the Great, King of Prussia. I too was a composer, but never got in practice beyond playing duets with Spence.

In general knowledge, I do not scruple to say, that in the circle of an acquaintance that embraces many of the most accomplished characters of the age, I have met with no one who, with so much information, united the same colloquial powers of unfolding what he knew. ... There was no subject of which he did not possess a liberal portion of information; in several he was profoundly versed; and with an aptitude that often surprised by its alacrity, he could show himself acquainted with the outlines of all.