In 1665, in order to escape the plague, Isaac Newton evacuated Cambridge University and returned to his country home in Woolsthorpe, there to ponder the force which earth seemed to exert on the Moon. He wondered if the force might be the same one which made projectiles, such as falling apples and cannonballs, earth-bound . He combined basic knowledge of rates and the geometry of similar triangles to hypothesize that the forces might diminish as the squares of the distances. He tested his theory, and his rough estimates of the distance to the moon agreed with those of other scientists.

There the story might have ended. If Newton had been struck by one of those cannonballs at any time in the next twenty years, the Universal Law of Gravitation might have been delayed another hundred. In 1684, however, the astronomer Edmund Halley, paying Newton a call and requesting that Sir Isaac work on the problem of the Moon's orbit, was astounded when Newton gave a ready answer and showed him the propositions which clinched the matter. Newton, a bit of a perfectionist, took two years more to assemble the <Principia Mathematica>, and as a result produced a comprehensive study of the geometry and physics of bodies in mutual orbit [Chandrasekhar, pp. 1-6].

The specific question which Halley asked Newton was, as related by De Moivre, "[what] the curve would be that would be described by the planets supposing the force of attraction towards the sun to be the reciprocal to the square of their distance from it." Newton answered that the orbit would be elliptical, confirming one of Kepler's Laws. Using the calculus of vectors, Newton also established the second and third laws of Kepler [ibid, pp. 7-14]. He made his famous statement "If I have seen further [than others], it is by standing upon the shoulders of Giants." [Maor, p. 40] Newton mentioned Kepler specifically, and he credited Galileo with the inverse square insight, but assuredly his geometric facility and analytical insights evoked the spirit of Archimedes. Whereas Archimedes "fenced in" his circles with polygons, Newton had "the world on a string."

Kepler's second law made important use of areas. He claimed that the vector from the sun (one focus of the ellipse) to the orbiting body would sweep equal areas of the ellipse, in equal times. If true, one implication was that the body would have a higher tangential velocity when closer to the sun. Newton's demonstration of this law was at once simple and profound. In the figure, we have points A, B, and C on an orbit, and they are separated by equal elapsed times. Newton claimed that, at point B for example, there are only inertia and the attractive force from S, acting on the body. Inertia alone would keep the body going along the path Bc, and the centripetal force would be acting in the direction BS. Newton reasoned that the resolution of these two vectors, Bc and BF, would cause the body to "fall" toward C. The segment Bc would be equal to AB because of equal times and a constant velocity, making triangles ABS and SBc equal in area. Because we use a parallelogram to resolve the vectors, triangles SBc and SBC have the same altitudes on base SB, and therefore they have the equal areas. We have simply demonstrated the equality of areas ABS and SBC in equal time increments. What is profound is that the entire proof doesn't work without reducing the triangles to infinitesimals and having the time increments approach zero. Why? Because the force vector changes direction as the body moves and, obviously, the vector at C is not parallel to BF, unless we consider points A, B, and C to be extremely close together. [Chandrasekhar, pp. 67-69]

Uncle Bob's Puzzle Corner | Uncle Bob and Aunt Claire's Place | Math Menu | Math History Menu