Abstract
The approach previously used to mechanise lemmas and Kepler’s Law of Equal Areas from Newton’s Principia [13] is here used to mechanically reproduce the famous Propositio Kepleriana or Kepler Problem. This is one of the key results of the Principia in which Newton demonstrates that the centripetal force acting on a body moving in an ellipse obeys an inverse square law. As with the previous work, the mechanisation is carried out through a combination of techniques from geometry theorem proving (GTP) and Nonstandard Analysis (NSA) using the theorem prover Isabelle. This work demonstrates the challenge of reproducing mechanically Newton’s reasoning and how the combination of methods works together to reveal what we believe to be flaw in Newton’s reasoning.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Apollonius. Conics. Translation by R. Catesby Taliaferro. In Great Books of the Western World, Vol. 11, Chicago, Encyclopedia Britannica (1939) 51, 56, 59
J. B. Brackenridge. Newton’s mature dynamics and the Principia: A simplified solution to the Kepler Problem. Historia Mathematica 16(1989), 36–45. 55
J. B. Brackenridge. The Key to Newton’s Dynamics: The Kepler Problem and Newton’s Principia. University of California Press, 1995. 55
H.-B. Li and M.-T. Cheng. Clifford algebraic reduction method for mechanical theorem proving in differential geometry. J. Automated Reasoning 21(1998), 1–21. 64
S. C. Chou, X. S. Gao, and J. Z. Zhang. Automated generation of readable proofs with geometric invariants, I. Multiple and shortest proof generation. J. Automated Reasoning 17 (1996), 325–347. 48
. S. C. Chou, X. S. Gao, and J. Z. Zhang. Automated generation of readable proofs with geometric invariants, II. Theorem proving with full-angles. J. Automated Reasoning 17 (1996), 349–370. 48
S. C. Chou, X. S. Gao. Automated reasoning in differential geometry and mechanics: II. Mechanical theorem proving. J. Automated Reasoning 10 (1993), 173–189. 64
S. C. Chou, X. S. Gao. Automated reasoning in differential geometry and mechanics: III. Mechanical formula derivation. IFIP Transaction on Automated Reasoning 1–12, North-Holland, 1993. 64
D. Densmore. Newton’s Principia: The Central Argument. Green Lion Press, Santa Fe, New Mexico, 1996. 48
Euclid. Elements. Translation by Thomas L. Heath. Dover Publications Inc., 1956. 49, 58
D. Wang. Geometry Machines: From AI to SMC. Proceedings of the 3rd International Conference on Artificial Intelligence and Symbolic Mathematical Computation (Stey, Austria, September 1996), LNCS 1138, 213–239. 64
S. Fevre and D. Wang. Proving geometric theorems using Clifford algebra and rewrite rules. Proceedings of the 15th International Conference on Automated Deduction (CADE-15) (Lindau, Germany, July 5–10, 1998), LNAI 1421, 17–32. 64
J. D. Fleuriot and L. C. Paulson. A combination of nonstandard analysis and geometry theorem proving, with application to Newton’s Principia. Proceedings of the 15th International Conference on Automated Deduction (CADE-15) (Lindau, Germany, July 5–10, 1998), LNAI 1421, 3–16. 47, 48, 49, 51, 52, 54, 55, 61, 64
I. Newton. The Mathematical Principles of Natural Philosophy. Third edition, 1726. Translation by A. Motte (1729). Revised by F. Cajory (1934). University of California Press. 47
D. T. Whiteside. The mathematical principles underlying Newton’s Principia Mathematica. Glasgow University Publication 138, 1970. 52
D. Wang. Clifford algebraic calculus for geometric reasoning with application to computer vision. Automated Deduction in Geometry (D. Wang et al., Eds.), LNAI 1360, Springer-Verlag, Berlin Heidelberg, 1997, 115–140. 64
W.-t. Wu. Mechanical theorem proving of differential geometries and some of its applications in mechanics. J. Automated Reasoning 7 (1991), 171–191. 64
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Fleuriot, J.D., Paulson, L.C. (1999). Proving Newton’s Propositio Kepleriana Using Geometry and Nonstandard Analysis in Isabelle. In: Automated Deduction in Geometry. ADG 1998. Lecture Notes in Computer Science(), vol 1669. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-47997-X_4
Download citation
DOI: https://doi.org/10.1007/3-540-47997-X_4
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-66672-1
Online ISBN: 978-3-540-47997-0
eBook Packages: Springer Book Archive