In the form of a dialogue M. Milhaud meets the objections made to the notion of limit in Mathematics. The question whether to have a limit, for anything variable, is not synonymous with attaining a limit, is considered in connection with Zeno’s problem of Achilles and the tortoise, the strict solution of which is, not that Achilles will never overtake the tortoise, but that he will not overtake it on this side of a spot situated at a distance of 10/9 of a metre from the starting-point, within a period equal to 10/9 of a second commencing at the instant of starting. To the objection that by its very nature the limit cannot be attained, as where the limit and the variable element which indefinitely approaches it are essentially different, it is replied that when a variable element has a limit, this element is a _quantity_ and the limit is a quantity of the same kind, quality being neglected. In the proposition: the length of the circumference is the limit of the perimeters of the inscribed polygons, the limit is a quantity of the same kind, that of length. It is not necessary to know whether the definition accords with reality. M. Milhaud then shows by reference to the properties of an unlimited series of inscribed polygons and the corresponding circumscribed polygons, that two such series of geometrical lengths satisfying the required conditions can always be considered as defining a new length, superior to all the first and inferior to all the others. As to its existence, it can be said only that a length exists only as determined, as limited; and a state of length, or a particular length has a right to exist, provided that the properties of quantity which condition it are not contradictory. The essence of mathematical space, breadth, length is only the content of their definitions. Mathematics owes its existence to the condition of creating for itself a world of fictions. There is a divergence of opinion as to whether incommensurables should be represented by lengths or by numerical symbols, but the divergence is a last echo of the endless discussions which the notions of infinity and continuity have raised among mathematicians.