70. It seems that M. Descartes confesses also, in a passage of his _Principles_, that it is impossible to find an answer to the difficulties on the division of matter to infinity, which he nevertheless recognizes as actual. Arriaga and other Schoolmen make well-nigh the same confession: but if they took the trouble to give to the objections the form these ought to have, they would see that there are faults in the reasoning, and sometimes false assumptions which cause confusion. Here is an example. A man of parts one day brought up to me an objection in the following form: Let the straight line BA be cut in two equal parts at the point C, and the part CA at the point D, and the part DA at the point E, and so on to infinity; all the halves, BC, CD, DE, etc., together make the whole BA; therefore there must be a last half, since the straight line BA finishes at A. But this last half is absurd: for since it is a line, it will be possible again to cut it in two. Therefore division to infinity cannot be admitted. But I pointed out to him that one is not justified in the inference that there must be a last half, although there be a last point A, for this last point belongs to all the halves of its side. And my friend acknowledged it [113] himself when he endeavoured to prove this deduction by a formal argument; on the contrary, just because the division goes on to infinity, there is no last half. And although the straight line AB be finite, it does not follow that the process of dividing it has any final end. The same confusion arises with the series of numbers going on to infinity. One imagines a final end, a number that is infinite, or infinitely small; but that is all simple fiction. Every number is finite and specific; every line is so likewise, and the infinite or infinitely small signify only magnitudes that one may take as great or as small as one wishes, to show that an error is smaller than that which has been specified, that is to say, that there is no error; or else by the infinitely small is meant the state of a magnitude at its vanishing point or its beginning, conceived after the pattern of magnitudes already actualized.