286. We will complete this explanation with an example from elementary geometry. "The surface of a rhomboid is equal to the surface of a rectangle having the same base and altitude." First: in the idea of the rhomboid, we do not see the idea of its equality with the rectangle; and this we cannot see, because the relation does not exist when there is no other term to which it may relate. The idea of the parallelogram does not contain that of the rectangle, and consequently not that of equality. Second: the relation results from the comparison of the rhomboid with the rectangle; and, consequently, it must be found in a total conception containing them both. It cannot, therefore, be said that we add any thing to the conception of the parallelogram which does not belong to it. On the contrary, we see this equality flow from the conception of the rhomboid and that of the rectangle, as partial conceptions of the total conception, formed by the combination of them both. The analysis of this total conception opens to us the relation we are now in quest of; for it must be observed that when the simple union of the conceptions compared does not suffice, we make use of another including them, and also something more; and from the new conception, duly analyzed, we deduce the relation of the parts compared. 287. In the geometrical construction, that serves for the demonstration of the above theorem, which we have used as an example, may be seen what we have just explained with regard to total conceptions containing other conceptions besides those compared. If we place the rectangle and the rhomboid upon the same base, we at once see that there is something common to both, namely, the triangle formed by the base, a part of one side of the rhomboid, and a part of one side of the rectangle. Neither synthesis nor analysis is here required, because there is perfect coincidence, and this in geometry is equivalent to perfect equality. The difficulty is in the two remaining parts, that is, in the trapezoids to which the parallelograms are reduced by the subtraction of the common triangle. The mere sight of the figures teaches nothing concerning the equivalence of the two surfaces; we see only that the two sides of the rhomboidal surface go on extending, but including a less distance in proportion as the angle becomes more oblique, under these two conditions: length of sides, and diminution of distances between two limits, of which one is infinity, and the other the rectangle. The relation of the equivalence of the surfaces may be demonstrated by prolonging the parallel opposite the base, and thus forming a quadrilateral of which the trapezoids are parts; to discover the equality of these trapezoids, it is only necessary to decompose the quadrilateral, attending to the equality of two triangles, each respectively formed by one of the trapezoids and a common triangle. Is any thing here added to the conception of each trapezoid? No. We only compare them.