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

the work is correct and ready to be used by other scientists. For this reason, the credit for a scientific discovery today usually goes to the first to publish. But though Leibniz was the first to publish on calculus, as we shall see it was Newton rather than Leibniz who applied calculus to problems in science. Though, like Descartes, Leibniz was a great mathematician whose philosophical work is much admired, he made no important contributions to natural science.
    It was Newton’s theories of motion and gravitation that had the greatest historical impact. It was an old idea that the force of gravity that causes objects to fall to the Earth decreases with distance from the Earth’s surface. This much was suggested in the ninth century by a well-traveled Irish monk, Duns Scotus (Johannes Scotus Erigena, or John the Scot), but with no suggestion of any connection of this force with the motion of the planets. The suggestion that the force that holds the planets in theirorbits decreases with the inverse square of the distance from the Sun may have been first made in 1645 by a French priest, Ismael Bullialdus, who was later quoted by Newton and elected to the Royal Society. But it was Newton who made this convincing, and related the force to gravity.
    Writing about 50 years later, Newton described how he began to study gravitation. Even though his statement needs a good deal of explanation, I feel I have to quote it here, because it describes in Newton’s own words what seems to have been a turning point in the history of civilization. According to Newton, it was in 1666 that:
I began to think of gravity extending to the orb of the Moon & (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 sesquialterate proportion of their distances from the center 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 centers about which they revolve & thereby compared the Moon in her Orb with the force of gravity at the surface of the Earth & found them answer pretty nearly. All this [including his work on infinite series and calculus] was in the two plague years of 1665–1666. For in those days I was in the prime of my age for invention and minded Mathematicks and Philosophy more than at any time since. 5
    As I said, this takes some explaining.
    First, Newton’s parenthesis “having found out how to estimate the force with which [a] globe revolving within a sphere presses the surface of the sphere” refers to the calculation of centrifugal force, a calculation that had already been done (probably unknown to Newton) around 1659 by Huygens. For Huygens and Newton (as for us), acceleration had a broader definition than just a number giving the change of speed per time elapsed; it is a directed quantity, giving the change per time elapsed inthe direction as well as in the magnitude of the velocity. There is an acceleration in circular motion even at constant speed—it is the “centripetal acceleration,” consisting of a continual turning toward the center of the circle. Huygens and Newton concluded that a body moving at a constant speed v around a circle of radius r is accelerating toward the center of the circle, with acceleration v 2 / r , so the force needed to keep it moving on the circle rather than flying off in a straight line into space is proportional to v 2 / r. (See Technical Note 32 .) The resistance to this centripetal acceleration is experienced as what Huygens called centrifugal force, as when a weight at the end of a cord is swung around in a circle. For the weight, the centrifugal force is resisted by tension in the cord. But planets are not attached by cords to the Sun. What is it that resists the centrifugal force produced by a planet’s nearly circular motion around the Sun? As we will see,