B Fig. 8.2. Newton's derivation of the form of the centrifugal force.

attributed the physical cause of the irregularities in the motion of the Moon around the Earth to the pressure of the vortex of the Sun on that of the Earth.

The traditional story that Newton first thought of universal gravitation in 1666 after observing an apple falling from a tree while at home in Lincolnshire to escape the plague that had descended on Cambridge is, to say the least, misleading. There is no evidence that Newton considered the mutual attraction of heavenly bodies before 1679, or that the actual law of universal gravitation occurred to him before 1684. The theory of gravity that Newton published in 1687 was not simply a flash of inspiration - it was the result of a great deal of intellectual effort spent grappling with the fundamental principles of dynamics. He did, however, consider the possibility that the gravity of the Earth might extend as far as the Moon, and this may well have been inspired by the fall of an apple, but the terrestrial gravity he was reflecting upon was not gravity as eventually he understood it.

The key ingredient in the development of Newton's ideas that allowed him to consider the effect of gravity on the Moon came from his study of uniform circular motion. Following Descartes, he assumed that any body moving in such a manner had a centrifugal tendency away from the centre of motion that caused the body continually to try to move off on a tangent and, like Huygens, he looked for a quantitative measure of this 'force'. To get a handle on things, Newton considered a body that was moving around a circular path of radius r on the inside of a spherical surface. Then he approximated the situation by assuming that the body moved around the inscribed square ABCD shown in Figure 8.2 at a constant speed v, bouncing off the sphere at each of the vertices. At each impact, the component of the motion perpendicular to the tangent is

28 For an account of the origin of the anecdote concerning the apple, see Westfall (1980), p. 154. A detailed analysis of the claim that Newton did not believe in universal gravitation prior to 1684 is given in Wilson (1970).

reversed, which implies a change in the quantity of motion of 2m v cos n/4. The total change in the quantity of motion over the four impacts, divided by the quantity of motion the body possesses (i.e. m v), is thus 8 cos n/4.29 The length of the perimeter of the square ABCD is clearly 8r cos n/4, and so we have the result that total change in quantity of motion perimeter of square quantity of motion of body radius of circle

Newton realized that this result would hold for any inscribed regular polygon, and we can demonstrate this as follows. Half the interior angle of a polygon with n sides is n/2 - n/n (this plays the role of the marked angle in the diagram) and so at each impact the change in the component of motion perpendicular to the tangent is 2m v cos(n/2 - n/n), which must be multiplied by n to get the total change as the body moves around its polygonal path. The perimeter of the polygon is 2nr cos(n/2 - n/n) and so the result stated at the end of the previous paragraph extends to this new situation. If it is true for any regular polygon, Newton could proceed to the limit of a polygon with an infinite number of sides, i.e. the circle itself. In this way, he established that the total change in the quantity of motion of the body over the course of its circular obit is 2nm v. Then he argued that the magnitude of the instantaneous force acting on the body, f, must be this total change divided by the period, and in this way arrived at the correct form for the centrifugal force,

2nm v mv2

2nr/v r

The first use to which he put his new formula was the accurate determination of the acceleration due to gravity using a conical pendulum. His result was very accurate and roughly twice as large as the value determined by Galileo and given in the Dialogue.

Newton had read Borelli's work in which it was suggested that the centrifugal force of a planet was balanced by an attraction toward the planet and, in 1666, Newton made the crucial realization that Kepler's third law was consistent with an inverse square law for the centrifugal tendencies. Assuming circular orbits, Kepler's third law relates the periods T of the planets to their orbital radii r through the equation T2 a r3. Since T a r/v, where v is the speed of the planet, we can conclude that v2 a 1/r. The centrifugal force of the planet is

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