Analysis and demonstration: Wallis and Newton on mathematical presentation

Emulating the Greek geometers, Newton used synthetic demonstration to present the ground-breaking arguments of the Principia. This paper argues that we can better understand Newton's reasons for using geometry by considering John Wallis's interpretation of synthetic demonstration. Wallis condemned demonstration for failing to explain the mathematical truths it presented. He opposed to it a presentation that combined symbolic analysis with a documented account of discovery. In preferring symbols, Wallis was motivated both by the nascent tradition of symbolic analysis and by contemporary interest in artificial languages. Newton maintained Wallis's characterization of Greek demonstration as adapted to common understanding rather than as strictly elucidating, but he inverted the values Wallis associated with synthesis and analysis. In Newton's new account, synthetic demonstration was preferable precisely because it could address inexpert readers without exposing them to the complications of symbols-based analysis. Newton advanced his arguments on behalf of geometry through portraits of ancient mathematicians: Archimedes and Pythagoras.