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By Jacques Fleuriot PhD, MEng (auth.)
Sir Isaac Newton's philosophi Naturalis Principia Mathematica'(the Principia) incorporates 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 mixture of tools from geometry and nonstandard research. The mechanization of the methods, which respects a lot of Newton's unique reasoning, is constructed in the theorem prover Isabelle. the applying of this framework to the mechanization of easy genuine research utilizing nonstandard thoughts 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
Example text
The problems, such as 7r = 0, that would arise if the ordinary equality for angles was used are avoided. The relation =a is an equivalence relation that is also used to express the properties that we might want. 1. c -- d = - (a -- b, c -- d) --a -2 7r The two other main properties of full-angles deal with their sign and how they can be split or joined. 2 Formulating Degenerate Conditions When dealing with geometry proofs, we often take for granted conditions that need to be stated explicitly for machine proofs: for example, two points making up a line should not coincide.
5. Geometric constructions for various circle theorems More Properties of the Ellipse. We define a few other notions associated with the ellipse that are required in the development of the geometry theory. These are all geometric notions that were well known by Newton and on whose properties he sometimes relied implicitly. The centre of the ellipse is defined as the point collinear with the foci and halfway between them: is_centre_ellipse c h 12 E == (is_ellipse h 12 E 1\ coll h 12 c 1\ lIen (h -- c)1 = lIen (c -- 12)1) If a chord p -- 9 goes through the centre of the ellipse, then it is called a diameter of the ellipse.
This relationship appears (in slightly different wording) as Lemma 12 of the Principia where it is employed in the solution of the famous Propositio Kepleriana or Kepler Problem. Newton refers us to the "writers on the conics sections" for a proof of the lemma. This is demonstrated in Book 7, Proposition 31 in the Conics of Apollonius of Perga [3]. Unlike Newton, we have to prove this result explicitly in Isabelle to make it available to other proofs. The formalization is rather involved and proceeds by a series of construction to show that the area of parallelograms cgov is equal to that of parallelogram catd and hence that areas of circumscribing parallelograms rxfYIj and ltzk are equal.