continually varying Keplerian ellipse, and this was precisely what he did. To Horrocks' mechanism, Newton added a number of small sinusoidal oscillations so as better to fit the observations. He ended up with seven 'equations' of lunar motion; one of these accounted for the equation of centre and the evection known since Ptolemy, and the variation and the annual equation, which had been discovered by Tycho Brahe, corresponded to two of the others. The remaining four steps in Newton's algorithm were entirely new. In order to use the procedure, the position of the Moon had to be adjusted ('equated') seven times, and then, as Newton stated proudly, 'this is her place in her proper orbit'.

The accuracy of Newton's seven-step procedure was the subject of debate then as now. Flamsteed claimed that it was no better than his own calculations based on Horrocks' theory, whereas Halley made great claims for the accuracy of Newton's approach. The truth would appear to lie somewhere in-between.

In the second edition of the Principia, Newton added a completely new section, containing a revised version of the Theory of the Moon's Motion, and claiming a justification for it based on universal gravitation:

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