Plato seems also to hint at the possibility of other applications of arithmetical or mathematical proportions, such as we employ in chemistry and natural philosophy, such as the Pythagoreans and even Aristotle make use of in Ethics and Politics, e.g.his distinction between arithmetical and geometrical proportion in the Ethics (Book V), or between numerical and proportional equality in the Politics.
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.