By Jacques Fleuriot PhD, MEng (auth.)
Sir Isaac Newton's philosophi Naturalis Principia Mathematica'(the Principia) includes a prose-style mix of geometric and restrict reasoning that has frequently been seen as logically vague.
In A blend of Geometry Theorem Proving and NonstandardAnalysis, Jacques Fleuriot offers a formalization of Lemmas and Propositions from the Principia utilizing a mix of tools from geometry and nonstandard research. The mechanization of the methods, which respects a lot of Newton's unique reasoning, is built in the theorem prover Isabelle. the applying of this framework to the mechanization of easy actual research utilizing nonstandard concepts is usually discussed.
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Extra info for A Combination of Geometry Theorem Proving and Nonstandard Analysis with Application to Newton’s Principia
4), we can reduce the sums of signed triangular areas in the goal to one involving the signed areas of quadrilaterals. 3 The Full-Angle Method The concept of the angle and its associated properties provide powerful tools that have been used traditionally in geometry theorem proving. In their work on 20 2. Geometry Theorem Proving producing automated readable proofs, Chou et al.  also propose a method based on the concept of full-angles that can be used to deal with classes of theorems that pose problems to the area method.
The work has involved adding concepts such as similar and congruent triangles since they are needed for formalizing Newton's proofs. Such notions have traditionally been used in geometry, though Chou et al. note that they have limitations when dealing with automated GTP . However, our proofs are not affected since we are not concerned with completely automatic proofs. To deal with some of the main types of motion analysed by Newton, definitions of ellipses, circles, tangents, and arcs amongst others have also been added to the theory.
Llen(f1 -- p)l + Ilen(f~ -- p)l = r} The ellipse is especially important since one of the major tasks of the Principia lies in providing the mathematical analysis that explains and confirms Kepler's guess that planets travelled in ellipses round the sun . llen (x -- p)l = r} The Circular Arc. The arc is an important tool in Newton's reasoning procedures. When analysing motion at a particular point on an ellipse or circle, it is the (infinitesimal) arc at that point that is usually considered.
A Combination of Geometry Theorem Proving and Nonstandard Analysis with Application to Newton’s Principia by Jacques Fleuriot PhD, MEng (auth.)