## Universal gravitation

A number of phenomena had caused Newton to consider that gravity was a property of all celestial bodies. In order to make the leap to universal gravitation, an understanding was required of how a large body like the Earth would attract an external object if its pull was the result of attractions from all its constituent parts. Newton answered these questions by proving some results concerning the attraction of thin spherical shells. His approach was hard-going; the description below is a modern adaptation.

The particle is at P in Figure 8.6, and simple symmetry arguments show that the force on P due to the shell of radius a centred at C must be along the line PC. In order to work out the magnitude of the force, we draw a line from P that intersects the sphere at Q and R. Each of these points can be thought of as being part of a ring of points on the sphere, the centre of which lies on the axis PC and it is not difficult to show that the attractive effect of this circle of points is given by

where d = | PC | is the distance between the particle and the centre of the shell, p = d sin0 is the perpendicular distance from C to the line PQR, and p is the mass per unit area of the shell. It follows that the ring containing Q has the same attractive effect as that containing R and, in consequence, that a circle through AB (PA and PB being tangents to the shell) with its centre on the

62 Newton Principia, Book I, Propositions 70 and 71. Here we follow Todhunter (1962), p. 2; an alternative approach is described in Littlewood (1948). Littlewood remarks that Newton's geometric proof 'must have left its readers in helpless wonder'. The geometric proof is also examined in detail in Weinstock (1984).

axis PC, divides the spherical shell into two parts, which attract the particle P equally, the total effect being

Thus, a spherical shell attracts an external particle as if all its mass were concentrated at its centre. In fact, Newton did not evaluate any equivalent of this integral, but demonstrated merely that the gravitational field exterior to a shell is inversely proportional to the square of the distance from the centre of the shell; the constant of proportionality was not determined. As no one had any idea about the densities of heavenly bodies, this is unimportant.

Newton then could conclude that the same resultwouldbe true forany sphere, the density of which was a function of radial distance alone. This implied that if the force that kept the Moon in its orbit and the force that made an apple fall to the Earth were caused solely by the gravitational pull of all the particles that make up the Earth, the calculation he had performed in 1666 should have given better agreement. Newton did the calculation again, this time using the more accurate data on the size of the Earth due to Jean Picard, and found excellent agreement.63 This demonstration was crucial to Newton's confidence in his idea of universal gravitation.

Newton realized that the combined effect of the gravitational attraction of the particles that make up the Earth and the rotation of the Earth would lead to a flattening of the Earth at its poles.64 He assumed that the Earth originally had been made up of fluid matter of constant density, and that this implied the actual shape was an oblate spheroid. In other words, any cross-section through the Earth which includes the polar axis is an ellipse, which we can suppose has a ratio of major to minor axis of 1 + e, e being the ellipticity or oblateness. Newton showed that, within the body of a homogeneous Earth, the force of gravity would be directly proportional to the distance from the centre (unlike

63 This calculation is outlined in Proposition 4 of Book III, and a more elaborate version, in which the Earth and Moon are treated as a two-body system, is given in Proposition 37. This latter calculation almost certainly was fiddled so as to give the impression of great precision, though, since Newton could not compute the centre of gravity of the Earth-Moon system accurately (see Westfall (1973), Kollerstrom (1991)).

Newton discussed the oblateness of the Earth in Proposition 19 of Book III. His method is described in modern form in Todhunter (1962).

65 An oblate spheroid is the shape formed by rotating an ellipse about its minor axis, and in the case of the Earth this axis corresponds to the polar diameter. Newton chose this shape primarily because he knew how to work with it.

The term 'ellipticity' was introduced in the eighteenth century and is sometimes defined so that the major and minor axes have lengths in the ratio 1 + e : 1 as here, but one also finds this ratio as 1 : 1 — e. Of course, to first order in e this makes no difference, since

Huygens, who thought that gravity was constant in the interior of the Earth), and if we assume that e is small, the attraction at the pole due to the combined effect of all the particles that make up the Earth divided by the attraction at a point on the equator can be shown to be 1 + 5 e. Newton considered two columns of fluid extending from the centre of the Earth, one to the pole and one to the equator (the lengths of which are thus in the ratio 1:1 + e), and showed that the force of attraction on the polar column would be greater than that on the equatorial column by the factor (1 + e)/(1 + 1 e) — 1 + 5e. But, Newton argued, the two columns must be in equilibrium, so the different forces must be balanced by the effect of rotation which, as we have seen from the work of Huygens, was to reduce the effect of gravity at the equator by a factor of 1 /289. Putting these results together, Newton concluded that

5 288

and thus that e is about 1 /23o. Hence, the ratio of the polar to equatorial axes is 229 : 23o, in contrast to Huygens' value of 577 : 578. Actually, the oblateness is closer to 1/3oo, the discrepancy being due to the fact that the Earth is not homogeneous as Newton supposed.

Newton then applied his idea of balanced fluid columns to investigate the variation in the weight of a body as a function of latitude, and showed that, since the oblateness of the Earth was small, the change in weight varied with the square of the sine of the latitude. Using this result, he calculated that a seconds pendulum would have to be shortened by about 23 mm when taken from Paris to the equator. A comparison of his result with the data he had available led Newton to believe that the oblateness was, in fact, slightly greater than predicted with his theory, and he suggested that this was due to the Earth being denser toward its centre than near its surface (this would have the opposite effect to the one he was trying to create).

Looking back, we can see in Newton's work on the shape of the Earth the beginnings of a satisfactory theory. However, his explanations were difficult to follow, and relied on numerous assumptions that were not stated clearly and not at all obvious. To most readers of the Principia, Newton's theory of the shape of the Earth was pretty incomprehensible.

There was one final element needed to characterize the nature of gravitation. Newton's third law implied that whatever force one body exerted on another, an equal and opposite force was exerted on the former by the latter. The same had to be true for gravitation. This was explained by Cotes in the preface to the second edition of the Principia, as follows:

67 Newton calculated the same value by a different method (Greenberg (1995), p. 2).

Furthermore, just as all bodies universally gravitate toward the earth, so the earth in turn gravitates equally toward the bodies; for the action of gravity is mutual and is equal in both directions. This is shown as follows. Let the whole body of the earth be divided into any two parts, whether equal or in any way unequal; now, if the weights of the parts toward each other were not equal, the lesser weight would yield to the greater, and the parts, joined together, would proceed to move straight on without limit in the direction toward which the greater weight tends, entirely contrary to experience.

The observational data showed Newton that the planets, the satellites of Jupiter and Saturn, and the Moon in its orbit around the Earth, all obeyed (as far as could be ascertained) Kepler's laws, and thus their motion was consistent with an inverse square law for gravitation. Moreover, spherical objects (like planets) would attract (as they appeared to do) as if their mass were all concentrated at their centres, if their gravitational pull was due to the pull from all the individual particles of which they were constituted. This knowledge led Newton to his law of universal gravitation:

Gravity exists in all bodies universally and is proportional to the quantity of

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