What Kant adds concerning the necessity of recurring, in this case, to an intuition, with respect to one of the numbers, adding five to seven on the fingers, is exceedingly futile. First, in whatever way he adds the five, there will never be anything but the five that is added, and it will neither give more nor less than 7 + 5. Secondly, the successive addition on the _fingers_ is equivalent to saying 1 + 1 + 1 + 1 + 1 = 5. This transforms the expression, 7 + 5 = 12, into this other, 7 + 1 + 1 + 1 + 1 + 1 = 12; but the conception, 1 + 1 + 1 + 1 + 1, has the same relation to 5, as 7 + 5 to 12; therefore, if 7 + 5 are not contained in 12, neither are 7 + 1 + 1 + 1 + 1 + 1 contained in it. It may be replied that Kant does not speak of identity, but of intuitions. This intuition, however, is not the sensation, but the idea; and if the idea, it is only the conception explained. Thirdly, we know this method of intuition not to be even necessary for children. Fourthly, this method is impossible in the case of large numbers. 281. Kant adds that this proposition, "a right line is the shortest distance between two points," is not purely analytic, because the idea of _shortest distance_ is not contained in the idea of _right line_. Waiving the demonstrations which some authors give, or pretend to give, of this proposition, we shall confine ourselves to Kant's reasons. He forgets that here the right line is not taken _alone_, but _compared_ with other lines. The idea of right line alone neither does nor can contain the ideas of _more_ or _less_; for these ideas suppose a comparison. But from the moment the right line and the curve are compared, with respect to _length_, the relation of superiority of the curve over the right line is seen. The proposition is then the result of the comparison of two purely analytic conceptions with a third, which is _length_. 282. If Kant's reasoning were good, even this judgment, "the whole is greater than its part," would not be analytic; for the idea of _greater_ enters not into the conception of the _whole_ until the _whole_ is compared with its _part_. Thus, the judgment, four is greater than three, would not be analytic, because the idea of four until compared with three does not include the conception of greater. The axiom: "things which are equal to the same thing are equal to each other," would not be analytic, because the conception, _equal to each other_, does not enter into the conception of _things which are equal to the same thing_, until we reflect that the equality of the middle term implies the equality of the extremes. The x, of which Kant speaks, would be found in almost all judgments, if we could not form total conceptions involving comparison of partial conceptions: in this case we should have no analytic judgments except such as are wholly identical, or directly contained in this formula, A is A.