Is God a Mathematician_ - Mario Livio [51]
And the same year [1666] I began to think of gravity extending to the orb of the Moon, and having found out how to estimate the force with which [a] globe revolving within a sphere presses the surface of the sphere, from Kepler’s Rule of the periodical times of the Planets being in a sesquialternate proportion of their distances from the centres of their Orbs I deduced that the forces which keep the Planets in their Orbs must [be] reciprocally as the squares of their distances from the centres about which they revolve: and thereby compared the force requisite to keep the Moon in her Orb with the force of gravity at the surface of the earth, and found them answer pretty nearly. All this was in the two plague years of 1665 and 1666, for in those years I was in the prime of my age for invention, and minded Mathematicks and Philosophy more than at any time since.
Newton refers here to his important deduction (from Kepler’s laws of planetary motion) that the gravitational attraction of two spherical bodies varies inversely as the square of the distance between them. In other words, if the distance between the Earth and the Moon were to be tripled, the gravitational force that the Moon would experience would be nine times (three squared) smaller.
For reasons that are not entirely clear, Newton essentially abandoned any serious research on the topics of gravitation and planetary motion until 1679. Then two letters from his archrival Robert Hooke renewed his interest in dynamics in general and in planetary motion in particular. The results of this revived curiosity were quite dramatic—using his previously formulated laws of mechanics, Newton proved Kepler’s second law of planetary motion. Specifically, he showed that as the planet moves in its elliptical orbit about the Sun, the line joining the planet to the Sun sweeps equal areas in equal time intervals (figure 28). He also proved that for “a body revolving in an ellipse…the law of attraction directed to a focus of the ellipse…is inversely as the square of the distance.” These were important milestones on the road to Principia.
Figure 28
Principia
Halley came to visit Newton in Cambridge in the spring or summer of 1684. For some time Halley had been discussing Kepler’s laws of planetary motion with Hooke and with the renowned architect Christopher Wren (1632–1723). At these coffeehouse conversations, both Hooke and Wren claimed to have deduced the inverse-square law of gravity some years earlier, but both were also unable to construct a complete mathematical theory out of this deduction. Halley decided to ask Newton the crucial question: Did he know what would be the shape of the orbit of a planet acted upon by an attractive force varying as an inverse-square law? To his astonishment, Newton answered that he had proved some years earlier that the orbit would be an ellipse. The mathematician Abraham de Moivre (1667–1754) tells the story in a memorandum (from which a page is shown in figure 29):
In 1684 Dr Halley came to visit him [Newton] at Cambridge, after they had been some time together, the Dr asked him what he thought the curve would be that would be described by the planets supposing the force of attraction towards the sun to be reciprocal to the square of their distance from it. Sr Isaac replied immediately that it would be an Ellipsis [ellipse], the Doctor struck with joy and amazement asked him how he knew it, why saith he [Newton] I have calculated it, whereupon Dr Halley asked him for his calculation without