them as the parts of which extended matter consists. Thus considered as parts of a quantum and therefore themselves quanta, the infinite divisibility which belongs to all quantity belongs also to them. ‘If the point were divisible, it would be no termination of a line.’ Answer to this. 272. In this lies the answer to the most really cogent argument which Hume offers against infinite divisibility ‘A surface terminates a solid; a line terminates a surface; a point terminates a line: but I assert that if the _ideas_ of a point, line, or surface were not indivisible, ’tis impossible we should ever conceive these terminations. For let these ideas be supposed infinitely divisible, and then let the fancy endeavour to fix itself on the idea of the last surface, line, or point, it immediately finds this idea to break into parts; and upon its seizing the last of these parts it loses its hold by a new division, and so on _ad infinitum_, without any possibility of its arriving at a concluding idea’. [1] If ‘point,’ ‘line,’ or ‘surface’ were really names for ‘ideas’ either in Hume’s sense, as feelings grown fainter, or in Locke’s, as definite imprints made by outward things, this passage would be perplexing. In truth they represent objects determined by certain conceived relations, and the relation under which the object is considered may vary without a corresponding variation in the name. When a ‘point’ is considered simply as the ‘termination of a line,’ it is not considered as a quantum. It represents the abstraction of the relation of externality, as existing between _two lines_. It is these lines, not the point, that in this case are the constituents of the relation, and thus it is they alone that are for the time considered as extended, therefore as quanta, therefore as divisible. So when the line in turn is considered as the ‘termination of a surface.’ It then represents the relation of externality _as between surfaces_, and for the time it is the surfaces, not the line, that are considered to have extension and its consequences. The same applies to the view of a surface as the termination of a solid. Just as the line, though not a quantum when considered simply as a relation between surfaces, becomes so when considered in relation to another line, so the point, though it ‘has no magnitude’ when considered as the termination of a line, yet acquires parts, or becomes divisible, so soon as it is considered in relation to other points as a constituent of extended matter; and it is thus that Hume considers it, ἑκὼν ἢ ἄκων [2], when he talks of extension as ‘made up of coloured points.’ [1] P. 345. [Book I, part II., sec. IV.] [2] [Greek ἑκὼν ἢ ἄκων (hekon e akon) = like it or not. Tr.] What becomes of the exactness of mathematics according to Hume?