As yet Newton had dealt with the problem as if the sun and the planets might be regarded as heavy masses concentrated at their centres. Clearly at the best this was only an approximation, though considering the enormous distances involved it was not unreasonable. In January or February, 1685, he considered the question of the attraction of bodies of finite size, and found, to his surprise and gratification, that a sphere or spherical shell attracts an external particle as if condensed into a heavy mass at its centre. Hence the results he had already proved for the relative motion of particles were true for the solar system, save for small errors due partly to the fact that the bodies were not perfectly spherical and partly to disturbances caused by the planets attracting one another. It was no longer a question of rough approximation: the problem was reducible to mathematical analysis, subject to the introduction of minute corrections, which, given the necessary observations, could be calculated very closely. This was a new discovery of first-rate importance, and initiated the modern theory of attractions. The first book of the _Principia_ was finished before the summer of 1685. It deals with the motion of particles or bodies in free space either in known orbits or under the action of known forces. In it the law of attraction is generalized into the statement that every particle of matter attracts every other particle with a force which varies directly as the product of their masses and inversely as the square of the distance between them. Thus gravitation was brought into the domain of Science. The second book was completed by the summer of 1686. It treats of motion in a resisting medium and of various problems connected with waves. At the end of it, it is shown that the Cartesian theory of vortices is inconsistent with the laws of motion, and necessarily leads to incorrect results. This book opened another world to the application of mathematics and, in effect, created the science of hydrodynamics. The third book was finished in March 1687. In this, the theorems previously established are applied to the chief phenomena of the universe, and briefly we may say that all the facts then known about the solar system and, in particular, the motion of the moon with its various inequalities, the figure of the earth, and the phenomena of the tides, were shown to be in accord with the theory. Much of the material for these calculations was collected by Flamsteed and Halley.