277. To have arrived at this conclusion Hume had only to extend to proportions in space the principle upon which the impossibility of sensualizing arithmetic compels him to deal with proportions in number. ‘We are possessed,’ he says, ‘of a precise standard by which we can judge of the equality and proportion of numbers; and according as they correspond or not to that standard we determine their relations without any possibility of error. When two numbers are so combined, as that the one has always an unite answering to every unite of the other, we pronounce them equal’. [1] Now what are the unites here spoken of? If they were those single impressions which he elsewhere [2] seems to regard as alone properly unites, the point of the passage would be gone, for combinations of such unites could at any rate only yield those ‘general appearances’ of whose proportions we have been previously told there can be no precise standard. They can be no other than those unites which, not being impressions, he has to call ‘fictitious denominations’--unites which are nothing except in relation to each other and of which each, being in turn divisible, is itself a true number. We can easily retort upon Hume, then, when he argues that the supposition of infinite divisibility is incompatible with any comparison of quantities because with any unite of measurement, that, according to his own virtual admission, in the only case where such comparison is exact the ultimate unite of measurement is still itself divisible; which, indeed, is no more than saying that whatever measures quantity must itself be a quantity, and that therefore quantity is infinitely divisible. If Hume, instead of slurring over this characteristic of the science of number, had set himself to explain it, he would have found that the only possible explanation of it was one equally applicable to the science of space--that what is true of the unite, as the abstraction of distinctness, is true also of the abstraction of externality. As the unite, because constituted by relation to other unites, so soon as considered breaks into multiplicity, and only for that reason is a quantity by which other quantities can be measured; so is it also with the limit in whatever form abstracted, whether as point, line, or surface. If the fact that number can have no least part since each part is itself a number or nothing, so far from being incompatible with the finiteness of number, is the consequence of that finiteness, neither can the like attribute in spaces be incompatible with their being definite magnitudes, that can be compared with and measured by each other. The real difference, which is also the rationale of Hume’s different procedure in the two cases, is that the conception of space is more easily confused than that of number with the feelings to which it is applied, and which through such application become sensible spaces. Hence the liability to the supposition,