[3] At school he formed a life-long friendship with John Galt, who documented much of his life and his works posthumously.

[1] Despite having received a formal education until he was a teenager, Spence never attended university, instead he moved to Glasgow where he lodged with a friend of his fathers, learning the skills of a manufacturer.

[3] Spence published An Essay on the Theory of the Various Orders of Logarithmic Transcendents: With an Inquiry Into Their Applications to the Integral Calculus and the Summation of Series in 1809.

[1][6] In his preface he derived the binomial theorem and mainly focused on the properties and analytic applications of the series:[1][5][6] which he denoted with

[3][8] Spence's work was noted to be remarkable at the time, with John Herschel, his acquaintance and one of Britain's leading mathematicians at the time, had referenced it in one of his later publications Consideration of various points of analysis, which prompted Herschel to edit Spence's manuscripts.

[1][9] Spence was held in such high regard by Galt, and later Herschel that they published a collection of his individual essays in 1819.

[1][10] Posthumously, his work was met with appreciation from his contemporaries, with a review in the ninety-fourth number of the Quarterly Review (reproduced in Galt's The Literary and Miscellanies of John Galt, Volume 1) that described his first work in 1809 as: " [The] first formal essay in our language on any distinct and considerable branch of the integral calculus, which has appeared since… Hellinsʼs papers on the ‘Rectification of the Conic Sections".