### 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,[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."

*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.[2] 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)*

### 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.

*Permanent Link to this article*

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.

### Google Search

Latest Books by Dev Gualtieri

Thanks to Cory Doctorow of BoingBoing for his favorable review of Secret Codes!

Other Books

- Tardigrades - August 14, 2017

- Roman Concrete - August 7, 2017

- Solar Spicules - July 31, 2017

- Schroeder Diffuser - July 24, 2017

- Rough Microparticles - July 17, 2017

- Robot Musicians - July 10, 2017

- Walter Noll (1925-2017) - July 6, 2017

- cosmogony - July 3, 2017

- Crystal Prototypes - June 29, 2017

- Voice Synthesis - June 26, 2017

- Refining Germanium - June 22, 2017

- Granular Capillarity - June 19, 2017

- Kirchhoff–Plateau Problem - June 15, 2017

- Self-Assembly - June 12, 2017

- Physics, Math, and Sociology - June 8, 2017

- Graphene from Ethylene - June 5, 2017

- Crystal Alignment Forces - June 1, 2017

- Martian Brickwork - May 29, 2017

- Carbon Nanotube Textile - May 25, 2017

- The Scent of Books - May 22, 2017

- Patterns from Randomness - May 18, 2017

- Terpene - May 15, 2017

- The Physics of Inequality - May 11, 2017

- Asteroid 2015 BZ509 - May 8, 2017

- Fuzzy Fibers - May 4, 2017

- The Sofa Problem - May 1, 2017

- The Wisdom of Composite Crowds - April 27, 2017

- J. Robert Oppenheimer and Black Holes - April 24, 2017

- Modeling Leaf Mass - April 20, 2017

- Easter, Chicks and Eggs - April 13, 2017

- You, Robot - April 10, 2017

- Collisions - April 6, 2017

- Eugene Garfield (1925-2017) - April 3, 2017

- Old Fossils - March 30, 2017

- Levitation - March 27, 2017

- Soybean Graphene - March 23, 2017

- Income Inequality and Geometrical Frustration - March 20, 2017

- Wireless Power - March 16, 2017

- Trilobite Sex - March 13, 2017

- Freezing, Outside-In - March 9, 2017

- Ammonia Synthesis - March 6, 2017

- High Altitude Radiation - March 2, 2017

- C.N. Yang - February 27, 2017

- VOC Detection with Nanocrystals - February 23, 2017

- Molecular Fountains - February 20, 2017

- Jet Lag - February 16, 2017

- Highly Flexible Conductors - February 13, 2017

- Graphene Friction - February 9, 2017

- Dynamic Range - February 6, 2017

- Robert Boyle's To-Do List for Science - February 2, 2017

- Nanowire Ink - January 30, 2017

- Random Triangles - January 26, 2017

- Torricelli's law - January 23, 2017

- Magnetic Memory - January 19, 2017

- Graphene Putty - January 16, 2017

- Seahorse Genome - January 12, 2017

- Infinite c - January 9, 2017

- 150 Years of Transatlantic Telegraphy - January 5, 2017

- Cold Work on the Nanoscale - January 2, 2017

### Deep Archive

Deep Archive 2006-2008

**Blog Article Directory on a Single Page**