F 0o1 e cos 0o

This process can then be repeated until the desired accuracy is reached.6°

For the case of planetary motion, where e is very small, Newton suggests a more straightforward alternative - one clearly having its origins in Boulliau's empty focus equant hypothesis. From Kepler's first and second laws, Newton deduced that the angle subtended by the planet at the empty focus, u, is related to the mean anomaly t via u = t + 4e2 sin2t + f e3 sin31 + O(e4), and that, even for Mars, 'the error will hardly exceed one second'. Once u has been found, 0 follows from elementary geometry.

One of Newton's early breakthroughs had been the discovery that Kepler's third law was consistent with an inverse square law for gravitation, at least as

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