2. _Formal Astronomy._--I shall not attempt here further to develop the form which such a science must assume. But I may notice one very large province which belongs to it. When men had ascertained the apparent motions of the sun, moon, and stars, to a moderate degree of regularity and accuracy, they tried to conceive in their minds some mechanism by which these motions might be produced; and thus they in fact proposed to themselves a very extensive problem in _Kinematics_. This, indeed, was the view originally entertained of the nature of the science of astronomy. Thus Plato in the seventh Book of his _Republic_[16\2], speaks of astronomy as the doctrine of the motion of solids, meaning thereby, spheres. And the same was a proper description of the science till the time of Kepler, and even later: for Kepler endeavoured in vain to conjoin with the knowledge of the motions of the heavenly bodies, those true mechanical conceptions which converted formal into physical astronomy[17\2]. [Note 16\2: P. 528.] [Note 17\2: _Hist. Induc. Sc._ ii. 130.] The astronomy of the ancients admitted none but uniform circular motions, and could therefore be completely cultivated by the aid of their elementary geometry. But the pure science of motion might be extended to all motions, however varied as to the speed or the path of the moving body. In this form it must depend upon the doctrine of limits; and the fundamental principle of its reasonings would be this: That velocity is measured by the Limit of the _space_ described, considered with reference to the _time_ in which it is described. I shall not further pursue this subject; and in order to complete what I have to say respecting the Pure Sciences, I have only a few words to add respecting their bearing on Inductive Science in general. {{159}} CHAPTER XIV. OF THE APPLICATION OF MATHEMATICS TO THE INDUCTIVE SCIENCES.