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world in 1749. One of the implications of Clairaut's analysis - that had not been suspected before - was that it was not simply the radial component of the perturbing force that affected the apsidal motion, but that the tangential component also had a significant effect.

23 Detailed in Wilson (1980). 24 In fact, max(|A|, |B|, |C|) < 0.01.

It eventually was published in his La theorie de la lune (1752).

The procedure that Clairaut had used was by no means above criticism, and d'Alembert was quick to point out the flaws. A major fault with Clairaut's approach was that he had assumed the form of the solutionbased on observations rather than deducing it from the differential equation. D'Alembert showed how this couldbe avoidedby developing an iterative solution procedure forEqn(9.7). He also carried the algebraic development of the theory much further than Clairaut (who tended to substitute in numerical values appropriate to the lunar problem at every stage) so that the relationships between the derived quantities and the underlying parameters were identified more easily.

Euler and d'Alembert both subsequently managed to improve their own methods for treating the lunar problem, and found that the value for the motion of the lunar apogee predicted by theory was in accord with observations. If we denote the sidereal, anomalistic, and draconitic months by Ts, Ta, and Td, respectively, then d'Alembert showed that the effect of the Sun on the orbit of the Moon implied that

s where m « 0.0748 = nS/nM (or equivalently the length of the sidereal month divided by the sidereal year). Newton's approach, and Clairaut's first approximation, are equivalent to neglecting the cubic terms in the above expressions. This leads to a fairly accurate answer for Td and, hence, for the retrogression of the nodes, but because of the large coefficient of the m3 term in the expansion for Ta, the first term provides a very poor approximation. Inclusion of the higher-order term results in a prediction for the advance of the apsidal line which is in accord with observation. Thus, something that had appeared to threaten the validity of Newtonian gravitation eventually became an extremely powerful argument in its favour. Euler was quick to recognize the implications:

For it is very certain that it is only since this discovery that one can regard the law of attraction reciprocally proportional to the squares of the distances as solidly

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