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In fact, the problem eventually was solved by the development of an accurate chronometer by John Harrison, whose story is well documented in Sobel (1996). Chronometers were, however, prohibitively expensive and the use of lunar tables was the most common method for determining longitude until the nineteenth century. The first tables that were sufficiently accurate to be of practical use were those of Tobias Mayer (1753) (see p. 304). Horrocks' mechanism is described on p. 234. Flamsteed had used |EC | cos S for the eccentricity (see Figure 7.12, p. 235), but Newton, at Halley's suggestion, used |EC |. This leads to a significant improvement. Details of Newton's early struggles with a lunar theory can be found in Whiteside (1976).

Newton Theory of the Moon's Motion. Quoted from Cohen (1975), p. 113. Details of the implementation and accuracy of Newton's procedure can be found in Kollerstrom (2000).

I wished to show by these computations of the lunar motions that the lunar motions can be computed from their causes by the theory of gravity.

In fact, Newton was not able to justify all the steps in his calculation procedure and had to resort to a certain amount of sleight of hand so as to get the numbers to agree, but the method he introduced for treating the motion of the Moon by considering the gravity of the Sun as a perturbing influence on the orbit of the Moon round the Earth was admired widely and paved the way for future developments.

Newton's attempt to deduce the perturbative effect of the Sun on the Earth-Moon system from universal gravitation was based on a diagram similar to that shown in Figure 8.8, in which E, S, and M represent the Earth, Sun, and Moon, respectively, and the Moon is assumed to move in a circular orbit around the Earth. Newton understood that the accelerative effect of the Sun on the Earth is very nearly the same as that on the Moon, since the two bodies are almost equidistant from the Sun. However, it is the small difference between these effects that leads to the irregularities in the orbit of the Moon. The Moon is attracted to the Earth by an inverse square force but is then subject to an extra force from the Sun, which can be thought of as having two components: one along the radius EM and one perpendicular to this, as indicated by the arrows in Figure 8.8. The idea is to try to relate the magnitude of these perturbing forces to the otherwise undisturbed Earth-Moon attraction. 7

The magnitude of the acceleration of the Moon due to the attraction of the Sun can be written ¡xS/\MS|2, where /¿S is the product of G and the Sun's mass, and that due to the Earth as nE/\ES\2, where ¡xE is the product of G and the mass of the Earth. Newton began by assuming that the vector ES represents a measure of the acceleration of the Earth due to the Sun's attraction (and therefore also the average acceleration of the Moon over the period of one

76 Newton Principia, Book III, Scholium to Proposition 35.

The derivation below is a modified version of that given in Wilson (1989a).

orbit) and, hence, that | ES | = /S/ | ES | 2. The magnitude of the acceleration of the Moon due to the Sun is then /s/|MS| 2 = |ES| 3/ |MS| 2. In the diagram, | MS| is less than | ES|, and so this acceleration clearly is greater than that of the Earth; we can represent it by the vector AS = AB + BS, where AB is parallel to the Earth-Moon line. Using | AS| = |ES|3/|MS|2 and the similarity of the triangles ABS and MES, we can derive the relations

and hence (neglecting quantities that are second-order in the small quantity

The net perturbing effect of the Sun in the direction ES is BS - ES, and so the outward radial component of the perturbation due to the Sun is

3|ME| cos2 a - |ME| = 11ME|(1 + 3 cos2a), the average value of which, over one orbit of the Moon, is |ME|/2. Thus, we have shown that the ratio of the mean radial perturbative acceleration to the acceleration of the Earth due to the Sun is |ME|/2|ES|. Finally, we use Kepler's third law (in the form of Eqn (8.7)), which implies that acceleration of Earth due to Sun /S/| ES|2 TM | ES |

acceleration of Moon due to Earth /E/1ME |2 T| |ME |'

where TE and TM are the periods of the Earth and Moon, respectively, and combining this with the previous result gives radial perturbative acceleration TM2 1

acceleration of Moon due to Earth 2T| 357.45'

the final figure being the one that Newton quotes in Book I, Proposition 45, Corollary 2. The effect of the perturbation is to decrease the pull on the Moon toward the Earth, and it is on the basis of this result that Newton showed that the mean effect of the Sun would be to produce an advance in the line of apsides of the orbit of the Moon and a retrogression of the nodes. His calculations predicted

78 Using the cosine rule on triangle MES, we can derive

a rate of 19° 18' 1" per sidereal year for the retrogression of the nodes, which is only about 3' too small, but his value of 1° 31' 28 ' for the monthly advance of the apsidal line was only about half the measured value or, as Newton put it,

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