Abstract
By relating the fall of the Moon in space to the descent of objects at the surface of the Earth, Newton showed the inverse square law applies out so far as the Moon. But could he prove that the inverse square law applied everywhere else in the solar system, as appeared to be the case with Kepler’s laws? Because the Earth does not have two moons, Newton could not directly determine whether the Keplerian proportions held true for the Moon, and thereby empirically link an inverse square law with that law. But he could inquire mathematically whether an inverse square law could account for Keplerian motion. Certainly, a theoretical connection between the two laws would more convincingly establish the universality of gravitation. This he did in the remarkably brief and intriguing corollaries to Principia’s Proposition IV of his first book.
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Notes
- 1.
Florian Cajori [1]. (Short title) Principia.
- 2.
Principia, Book I, Proposition XV, Theorem VII, 62.
- 3.
Ibid., Corollary.
- 4.
Principia, Book III, 398.
- 5.
Ibid.
- 6.
There is a 1728 translation of The System of the World (or Treatise on the System of the World). It is believed to be an earlier and more accessible draft of what eventually became Book III of the Principia. The translator of this is unknown, but may have been Andrew Motte, who also translated the Principia. An online version of the 1728 System of the World is at http://books.google.com/books?id=rEYUAAAAQAAJ&pg=PR1#v=onepage&q&f=false.
Newton had famously contentious disputes with Flamsteed. Interestingly, in this earlier account, Newton gives abundant credit to Flamsteed for his Jupiter observations and his insightful application of Kepler’s Third Law to them. But in Book III of the Principia, there is no mention whatsoever of Flamsteed’s contribution on this matter. Compare page 401 (Phenomenon I) of Book III of the Principia to pages 555–556 (section [6.]) of that publication. Here is what Newton said in the earlier version:
This proportion [Kepler’s law] has long ago been observed in those satellites [of Jupiter]; and Mr. Flamsteed, who had often measured their distances from Jupiter by the micrometer, and by the eclipses of the satellites, wrote to me, that it holds to all the accuracy that possibly can be discerned by our senses. And he sent me the dimensions of their orbits taken by the micrometer, and deduced the mean distance of Jupiter from the earth, or from the sun, together with the times of their revolutions…. Ibid., 555.
- 7.
Principia, Book III, Phenomenon I, 401.
- 8.
Principia, Book III, Phenomenon II, 402.
- 9.
Principia, Book III, Phenomenon I, 401.
- 10.
The names of the satellites in order, from Jupiter outward are: Io, Europa, Ganymede, and Callisto. Having been discovered by Galileo, they are referred to as the “Galilean” moons of Jupiter. Jupiter has over 60 moons; but these are the brightest and the most beautiful to see, being visible even in binoculars. The text here is replicated from the Principia, using Newton’s notational form. The reader should be cautioned about the notation: a number such as 82/3 does not mean 8 raised to the 2/3 power, but is 8 and the fraction 2/3.
- 11.
Principia, Book III, Proposition I, Theorem I, 406.
- 12.
Principia, Book I, Proposition II, 42. The actual text of Book I, Proposition II, Theorem II, reads: “Every body that moves in any curved line described in a plane, and by a radius drawn to a point either immovable, or moving forwards with an uniform rectilinear motion, describes about that point areas proportional to the times, is urged by a centripetal force directed to that point.” Principia, at 42.
- 13.
Principia, Book III, Proposition I, Theorem I, 406.
- 14.
See, for example, The System of the World, id., 556.
- 15.
- 16.
Simon Newcomb [2].
- 17.
Agnes M. Clerke [3].
- 18.
From the NASA/JPL link, http://ssd.jpl.nasa.gov/?sat_elem .
References
Cajori F (1949) Principia in modern English: Sir Isaac Newton’s mathematical principles of natural philosophy, Book I, Proposition IV, Theorem IV, Scholium, 46 (trans: Motte’s Revised). University of California Press, Berkeley
Newcomb S (1882) Popular astronomy. Harper & Brothers, Washington, pp 329–330
Clerke AM (2003) A popular history of astronomy during the nineteenth century. Sattre Press, Decorah, p 283; first ed., 1902
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MacDougal, D.W. (2012). Newton Demonstrates How an Inverse Square Law Could Explain Planetary Motions. In: Newton's Gravity. Undergraduate Lecture Notes in Physics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-5444-1_7
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