  ### Newton's Gravitational Theorem

March 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,
"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." Figure from Proposition LXXI of Newton's Principia. (Click to view entire proposition.)[1)]

This is easily proved using calculus, a mathematics that was developed independently by Newton and Gottfried Leibniz, Leibniz developing calculus through integration, and Newton through differentiation. Whereas Leibniz published his calculus, Newton kept his secret, using it to advantage in solving problems that none others could. Infinitesimals, the stuff of calculus, were not considered to be a genuine mathematical construction in Newton's time. He would present geometrical arguments, instead, such as for the proof of this theorem.

Since Newton's published proof is archaic, and it doesn't use calculus, it's generally unknown in the present day. Christoph Schmid of the Institute for Theoretical Physics of the Eidgenoessische Technische Hochschule (ETH-Zurich, Switzerland) has published a paper on the arXiv preprint server that reviews this theorem, which was essential to proving the inverse-square law.

Since Newton's geometrical proof is not that accessible, Schmid presents an elementary geometric proof that's easier than the use of calculus. I enjoyed my calculus courses, but I don't use calculus that often. However, one thing that I do often is write computer simulations, which are the last refuge of the physicist who forgot most of his mathematics.

I decided to tackle Newton's problem with a simulation program. The source code can be found, here. The calculation, as illustrated in the figure, is quite simple. We average many values of the inverse square of the distance between a measurement point and many random points in the interior of a sphere.

The only trick here is that we're measuring the 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. The computer simulation averages the resolved inverse-square force between a measurement point and many random points inside a sphere.

Drawing by the author, rendered with Inkscape)

The nice thing about fast computers is that the ten million random measurements done by my program are done very quickly. The force calculation using the sphere center gives 1.20846, in arbitrary units, for my particular sphere and measurement point locations.

The computer simulation yields a force of 1.20841, or a difference of just 41 ppm (0.004%). I'm quite convinced. This is about a five and a half sigma agreement. Perhaps a hundred million iterations would bring us to the magical six sigma realm. Developers of calculus, Gottfried Leibniz and Isaac Newton. Source images, left, Gottfried Wilhelm Leibniz, portrait by Christoph Bernhard Francke, c. 1700, and right, Isaac Newton, portrait by Sir Godfrey Kneller, 1689)

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