His father was descended from an ancient Scottish family, represented Colchester in parliament for some years, and in 1831 became junior partner in the firm of Eyre & Spottiswoode, printers.

[4] His talent for science showed itself while he was still a schoolboy, and indeed his removal from Eton to Harrow is said to have been occasioned by an accidental explosion which occurred whilst he was performing an experiment for his own amusement.

In 1847 Spottiswoode issued five pamphlets entitled Meditationes Analyticae, that explored complex mathematical ideas through analytic methods, particularly focusing on functions and calculus.

[9] As a mathematician, he occupied himself with many branches of his favorite science, more especially with higher algebra, including the theory of determinants, with the general calculus of symbols, and with the application of analysis to geometry and mechanics.

34): "The interesting series of communications on the contact of curves and surfaces which are contained in the Philosophical Transactions of 1862 and subsequent years would alone account for the high rank he obtained as a mathematician".