to the nature of the instrument by which we measure the bodies, and the care which we employ in the comparison.’ [1] [1] Pp. 351-53. [Book I, part II., sec. IV.] The universal propositions of geometry either untrue or unmeaning. 274. Such indefinite approach to exactness is all that Hume can allow to the mathematician. But it is undoubtedly another and an absolute sort of exactness that the mathematician himself supposes when he pronounces all right angles equal. Such perfect equality ‘beyond what we have instruments and art’ to ascertain, Hume boldly calls a ‘mere fiction of the mind, useless as well as incomprehensible’. [1] Thus when the mathematician talks of certain angles as always equal, of certain lines as never meeting, he is either making statements that are untrue or speaking of nonentities. If his ‘lines’ and ‘angles’ mean ideas that we can possibly have, his universal propositions are untrue; if they do not, according to Hume they can mean nothing. He says, for instance, that ‘two right lines cannot have a common segment;’ but of such ideas of right lines as we can possibly have this is only true ‘where the right lines incline upon each other with a sensible angle.’ [2] It is not true when they ‘approach at the rate of an inch in 20 leagues.’ According to the ‘original standard of a right line,’ which is ‘nothing but a certain general appearance, ’tis evident right lines may be made to concur with each other’. [3] Any other standard is a ‘useless and incomprehensible fiction.’ Strictly speaking, according to Hume, we have it not, but only a tendency to suppose that we have it arising from the progressive correction of our actual measurements. [4] [1] P. 353. [Book I, part II., sec. IV.] [2] Cf. Aristotle, Metaph. 998a, on a corresponding view ascribed to Protagoras. [3] P. 356. [Book I, part II., sec. IV.] [4] P. 354. [Book I, part II., sec. IV.] Distinction between Hume’s doctrine and that of the hypothetical nature of mathematics.