William Spence (mathematician)

Content sourced from Wikipedia, licensed under CC BY-SA 3.0.

William Spence (31 July 1777 – 20 May 1815) was a Scottish mathematician who studied logarithmic functions, algebraic equations, and how these relate to calculus.

He was born in Greenock, Scotland, the son of Ninian Spence, a copper smith, and Sarah Townsend. The Spence family were well known in Greenock. From a young age, Spence was calm and thoughtful, and he formed a lifelong friendship with John Galt, who would later write about his life and work.

Spence did not attend university. After a basic education, he moved to Glasgow to learn manufacturing with a family friend. After his father’s death in 1795, he returned to Greenock in 1797. With support from Galt and others, he started a small monthly literary society where essays on various topics were read aloud. This group met regularly until 1804, after which Spence traveled and spent several months in London.

In 1809 he published his first major work, and in 1814 his second. That same year he married and planned to live in London, but he fell ill while traveling back toward Glasgow and died there in 1815. Spence also enjoyed music and played the flute.

His first book, 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, explored special logarithmic functions and their use in calculus. He introduced ideas related to the binomial theorem and the series he denoted as Ln(1 ± x), and he worked out nine general properties of these functions. He also computed many values of the dilogarithm (now known as Spence’s function) to nine decimal places for 1 + x from 1 to 100, and created a table for arctan x.

His second work, Outlines of a Theory of Algebraical Equations, published in 1814, presented a systematic approach to solving equations up to the fourth degree using symmetric functions of the roots and connected these ideas to calculus.

After his death, John Herschel edited Mathematical Essays by the late William Spence (1819), with a short biography by John Galt. His work was praised by contemporaries, and his contributions were recognized as an early formal exploration of important parts of calculus.