The modern mathematician will readily sympathise with Plato's delight in the properties of pure mathematics. He will not be disinclined to say with him:--Let alone the heavens, and study the beauties of number and figure in themselves. He too will be apt to depreciate their application to the arts. He will observe that Plato has a conception of geometry, in which figures are to be dispensed with; thus in a distant and shadowy way seeming to anticipate the possibility of working geometrical problems by a more general mode of analysis. He will remark with interest on the backward state of solid geometry, which, alas! was not encouraged by the aid of the State in the age of Plato; and he will recognize the grasp of Plato's mind in his ability to conceive of one science of solids in motion including the earth as well as the heavens,--not forgetting to notice the intimation to which allusion has been already made, that besides astronomy and harmonics the science of solids in motion may have other applications. Still more will he be struck with the comprehensiveness of view which led Plato, at a time when these sciences hardly existed, to say that they must be studied in relation to one another, and to the idea of good, or common principle of truth and being. But he will also see (and perhaps without surprise) that in that stage of physical and mathematical knowledge, Plato has fallen into the error of supposing that he can construct the heavens _a priori_ by mathematical problems, and determine the principles of harmony irrespective of the adaptation of sounds to the human ear. The illusion was a natural one in that age and country. The simplicity and certainty of astronomy and harmonics seemed to contrast with the variation and complexity of the world of sense; hence the circumstance that there was some elementary basis of fact, some measurement of distance or time or vibrations on which they must ultimately rest, was overlooked by him. The modern predecessors of Newton fell into errors equally great; and Plato can hardly be said to have been very far wrong, or may even claim a sort of prophetic insight into the subject, when we consider that the greater part of astronomy at the present day consists of abstract dynamics, {cxi} by the help of which most astronomical discoveries have been made.