In the case considered, this stimulus was the notion of the physical continuum, drawn from the rough data of the senses. But this notion leads to a series of contradictions from which it is necessary successively to free ourselves. So we are forced to imagine a more and more complicated system of symbols. That at which we stop is not only exempt from internal contradiction (it was so already at all the stages we have traversed), but neither is it in contradiction with various propositions called intuitive, which are derived from empirical notions more or less elaborated. MEASURABLE MAGNITUDE.--The magnitudes we have studied hitherto are not _measurable_; we can indeed say whether a given one of these magnitudes is greater than another, but not whether it is twice or thrice as great. So far, I have only considered the order in which our terms are ranged. But for most applications that does not suffice. We must learn to compare the interval which separates any two terms. Only on this condition does the continuum become a measurable magnitude and the operations of arithmetic applicable. This can only be done by the aid of a new and special _convention_. We will _agree_ that in such and such a case the interval comprised between the terms _A_ and _B_ is equal to the interval which separates _C_ and _D_. For example, at the beginning of our work we have set out from the scale of the whole numbers and we have supposed intercalated between two consecutive steps _n_ intermediary steps; well, these new steps will be by convention regarded as equidistant. This is a way of defining the addition of two magnitudes, because if the interval _AB_ is by definition equal to the interval _CD_, the interval _AD_ will be by definition the sum of the intervals _AB_ and _AC_. This definition is arbitrary in a very large measure. It is not completely so, however. It is subjected to certain conditions and, for example, to the rules of commutativity and associativity of addition. But provided the definition chosen satisfies these rules, the choice is indifferent, and it is useless to particularize it. VARIOUS REMARKS.--We can now discuss several important questions: 1º Is the creative power of the mind exhausted by the creation of the mathematical continuum? No: the works of Du Bois-Reymond demonstrate it in a striking way. We know that mathematicians distinguish between infinitesimals of different orders and that those of the second order are infinitesimal, not only in an absolute way, but also in relation to those of the first order. It is not difficult to imagine infinitesimals of fractional or even of irrational order, and thus we find again that scale of the mathematical continuum which has been dealt with in the preceding pages.