Is God a Mathematician_ - Mario Livio [52]
Halley indeed came to visit Newton again in November 1684. Between the two visits Newton worked frantically. De Moivre gives us a brief description:
Figure 29
Sr Isaac in order to make good his promise fell to work again but he could not come to that conclusion wch he thought he had before examined with care, however he attempted a new way which thou longer than the first, brought him again to his former conclusion, then he examined carefully what might be the reason why the calculation he had undertaken before did not prove right, &…he made both his calculations agree together.
This dry summary does not even begin to tell us what Newton had actually accomplished in the few months between Halley’s two visits. He wrote an entire treatise, De Motu Corporum in Gyrum (The Motion of Revolving Bodies), in which he proved most aspects of bodies moving in circular or elliptical orbits, proved all of Kepler’s laws, and even solved for the motion of a particle moving in a resisting medium (such as air). Halley was overwhelmed. To his satisfaction, he at least managed to persuade Newton to publish all of these staggering discoveries—Principia was finally about to happen.
At first, Newton had thought of the book as being nothing but a somewhat expanded and more detailed version of his treatise De Motu. As he started working, however, he realized that some topics required further thought. Two points in particular continued to disturb Newton. One was the following: Newton originally formulated his law of gravitational attraction as if the Sun, Earth, and planets were mathematical point masses, without any dimensions. He of course knew this not to be true, and therefore he regarded his results as only approximate when applied to the solar system. Some even speculate that he abandoned again his pursuit of the topic of gravity after 1679 because of his dissatisfaction with this state of affairs. The situation was even worse with respect to the force on the apple. There, clearly the parts of the Earth that are directly underneath the apple are at a much shorter distance to it than the parts that are on the other side of the Earth. How was one to calculate the net attraction? The astronomer Herbert Hall Turner (1861–1930) described Newton’s mental struggle in an article that appeared in the London Times on March 19, 1927:
At that time the general idea of an attraction varying as the inverse square of the distance occurred to him, but he saw grave difficulties in its complete application of which lesser minds were unconscious. The most important of these he did not overcome until 1685…It was that of linking up the attraction of the earth on a body so far away as the moon with its attraction on the apple close to its surface. In the former case the various particles composing the earth (to which individually Newton hoped to extend his law, thus making it universal) are at distances from the moon not greatly different either in magnitude or direction; but their distances from the apple differ conspicuously in both size and direction. How are the separate attractions in the latter case to be added together or combined into a single resultant? And in what “centre of gravity,” if any, may they be concentrated?
The breakthrough finally came in the spring of 1685. Newton managed to prove an essential theorem: For two spherical bodies, “the whole force with which one of these spheres attracts the other will be inversely proportional to the square of the distance of the centres.” That is, spherical bodies gravitationally act as if they were point masses concentrated at their centers. The importance of this beautiful proof was emphasized by the mathematician James Whitbread Lee Glaisher (1848–1928). In his address at the bicentenary celebration (in 1887) of Newton’s Principia, Glaisher said:
No sooner had Newton proved this superb theorem—and we know from his own words that he had no expectation of so beautiful a result