Free To Explain the World: The Discovery of Modern Science by Steven Weinberg Page B

of the distance between them.
    There were still four large holes in Newton’s arguments:
    1. In comparingthe centripetal acceleration of the Moon with the acceleration of falling bodies on the surface of the Earth, Newton had assumed that the force producing these accelerations decreases with the inverse square of the distance, but the distance from what? This makes little difference for the motion of the Moon, which is so far from the Earth that the Earth can be taken as almost a point particle as far as the Moon’s motion is concerned. But for an apple falling to the ground in Lincolnshire, the Earth extends from the bottom of the tree, a few feet away, to a point at the antipodes, 8,000 miles away. Newton had assumed that the distance relevant to the fall of any object near the Earth’s surface is its distance to the center of the Earth, but this was not obvious.
    2. Newton’s explanation of Kepler’s third law ignored the obvious differences between the planets. Somehow it does not matter that Jupiter is much bigger than Mercury; the difference in their centripetal accelerations is just a matter of their distances from the Sun. Even more dramatically, Newton’s comparison of the centripetal acceleration of the Moon and the acceleration of falling bodies on the surface of the Earth ignored the conspicuous difference between the Moon and a falling body like an apple. Why do these differences not matter?
    3. In the work he dated to 1665–1666, Newton interpreted Kepler’s third law as the statement that the products of the centripetal accelerations of the various planets with the squares of their distances from the Sun are the same for all planets. But the common value of this product is not at all equal to the product of the centripetal acceleration of the Moon with the square of its distance from the Earth; it is much greater. What accounts for this difference?
    4. Finally, in this work Newton had taken the orbits of the planets around the Sun and of the Moon around the Earth to be circular at constant speed, even though as Kepler had shown they are not precisely circular but instead elliptical, the Sun and Earth are not at the centers of the ellipses, and the Moon’s and planets’ speeds are only approximately constant.
    Newton struggled with these problems in the years following 1666. Meanwhile, others were coming to the same conclusions that Newton had already reached. In 1679 Newton’s old adversary Hooke published his Cutlerian lectures, which contained some suggestive though nonmathematical ideas about motion and gravitation:
First, that all Coelestial Bodies whatsoever, have an attraction or gravitating power towards their own Centers, whereby they attract not only their own parts, and keep them from flying from them, as we may observe the Earth to do, but that they do also attract all the other Coelestial Bodies that are within the sphere of their activity—The second supposition is this, That all bodies whatsoever that are put into a direct and simple motion, will so continue to move forward in a straight line,till they are by some other effectual powers deflected and bent into a Motion, describing a Circle, Ellipsis, or some other more compounded Curve Line. The third supposition is, That these attractive powers are so much the more powerful in operating, by how much the nearer the body wrought upon is to their own Centers. 6
    Hooke wrote to Newton about his speculations, including the inverse square law. Newton brushed him off, replying that he had not heard of Hooke’s work, and that the “method of indivisibles” 7 (that is, calculus) was needed to understand planetary motions.
    Then in August 1684 Newton received a fateful visit in Cambridge from the astronomer Edmund Halley. Like Newton and Hooke and also Wren, Halley had seen the connection between the inverse square law of gravitation and Kepler’s third law for circular orbits. Halley asked Newton what would be the actual shape of the orbit