The admission that no relations of quantity are data of sense removes difficulty as to general propositions about them. 276. Here, as elsewhere, what we have to lament is not that Hume ‘pushed his doctrine too far,’ so far as to exclude ideas of those exact proportions in space with which geometry purports to deal, but that he did not carry it far enough to see that it excluded all ideas of quantitative relations whatever. He thus pays the penalty for his equivocation between a feeling of colour and a disposition of coloured points. Even alongside of his admission that ‘relations of space and time’ are independent of the nature of the ideas so related, which amounts to the admission that of space and time there are no ideas at all in his sense of the word, he allows himself to treat ‘proportions between spaces’ as depending entirely on our ideas of the spaces--depending on ideas which in the context he by implication admits that we have not. [1] If, instead of thus equivocating, he had asked himself how sensations of colour and touch could be added or divided, how one could serve as a measure of the size of another, he might have seen that only in virtue of that in the ‘general appearance’ of objects which, in his own language, is ‘independent of the nature of the ideas themselves’--_i.e._ which does not belong to them as feelings, but is added by the comparing and combining thought--are the proportions of greater, less, and equal predicable of them at all; that what thought has thus added, viz. limitation by mutual externality, it can abstract; and that by such abstraction of the limit it obtains those several terminations, as Hume well calls them--the surface terminating bodies, the line terminating surfaces, the point terminating lines--from which it constructs the world of pure space: that thus the same action of thought in sense, which alone renders appearances measurable, gives an object matter which, because the pure construction of thought, we can measure exactly and with the certainty that the judgment based on a comparison of magnitudes in a single case is true of all possible cases, because in none of these can any other conditions be present than those which we have consciously put there. [1] Part III. § 1, sub init. Hume does virtually admit this in regard to numbers.