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in trigonometric series and had to make numerous other approximations, but finally he obtained the equivalent of the expression k

- = 1 — e cos q9 + A cos 2n9 + B cos (2n — q) 9 + C cos (2n + q) 9, r

where n = (nM — nS)/nM, nM being the mean motion ofthe Moon and nS that of the Sun. In Eqn (9.9), the constants k, e, q, A, B, and C are all determined, with A, B, and C small, as required.24 The value predicted for q was 0.995 803 6, which implies a motion for the lunar apogee of (1 — q) x 360° « 1° 30' 39" per revolution - very close to Newton's value of 1° 31' 28'' and about half that observed.

Clairaut realized he would get a more accurate value for q if, instead of using the approximate form k/r = 1 — e cos q9, he used an expression of the form of Eqn (9.9), in which k, e, q, A, B, and C are all undetermined, but he did not anticipate that this would lead to a significant change. The calculations required for this more refined approach were extremely laborious and many more terms had to be kept in the equations. Nevertheless, Clairaut persevered and found that the extra terms in Eqn (9.9) did have an important effect on the calculated value of q and the implied motion of the lunar apse was nearly doubled, exactly what was required! Clairaut announced his sensational result to the scientific

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