## Newton's Gravitational TheoremMarch 5, 2012 This article isn't a review of Isaac Newton's law of universal gravitation, although that would be a worthy topic. It's the exposition of an idea that Newton first believed to be just an approximation for calculation of the gravitational attraction between bodies that he subsequently proved as a theorem in his Principia. Newton's theorem (Principia, Proposition LXXI, Theorem XXXI) is that a uniform spherical body, as far as gravity is concerned, can be replaced by a point mass at its center. The concept is so ingrained into the physics psyche that we never give it much thought. In Newton's own words,[1]"The same things supposed as above, I say, that a corpuscle placed without the sphaerical superficies is attracted towards the centre of the sphere with a force reciprocally proportional to the square of its distance from that centre."
resolved force at the center of the sphere, so we need to multiply our values by the cosine of the angle between our random vectors and the vector pointing to the center of the sphere. This is easy to do by using the dot product.
## References:- Isaac Newton, " Principia : The Mathematical Principles Of Natural Philosophy," English translation, via Archive.org.
- Christoph Schmid, "Newton's superb theorem: An elementary geometric proof," arXiv Preprint Server, January 31, 2012.
Linked Keywords: Isaac Newton; law of universal gravitation; gravitational attraction; theorem; Principia; spherical; gravity; mass; physics; psyche; Proposition LXXI; calculus; mathematics; Gottfried Leibniz; integration; differentiation; Infinitesimal; geometrical; Christoph Schmid; Institute for Theoretical Physics; Eidgenoessische Technische Hochschule (ETH-Zurich, Switzerland); arXiv preprint server; inverse-square law; computer simulation; Samuel Johnson; Patriotism is the last refuge of a scoundrel; last refuge of the physicist; Newton_gravitation.c; random; cosine; vector; dot product; Inkscape; computer; ppm; standard deviation; sigma; six sigma; Wikimedia Commons. |
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