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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!