A Combination of Nonstandard Analysis and Geometry Theorem Proving, with Application to Newton's Principia

The theorem prover Isabelle is used to formalise and reproduce some of the styles of reasoning used by Newton in his Principia. The Principia’s reasoning is resolutely geometric in nature but contains “infinitesimal” elements and the presence of motion that take it beyond the traditional boundaries of Euclidean Geometry. These present difficulties that prevent Newton’s proofs from being mechanised using only the existing geometry theorem proving (GTP) techniques. Using concepts from Robinson’s Nonstandard Analysis (NSA) and a powerful geometric theory, we introduce the concept of an infinitesimal geometry in which quantities can be infinitely small or infinitesimal. We reveal and prove new properties of this geometry that only hold because infinitesimal elements are allowed and use them to prove lemmas and theorems from the Principia.