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xdy + ydx = 2zdz

we know that the left-hand side is the differential of xy, and therefore that by integrating it we shall get xy; while the right side is the differential of z² which we shall get by integrating it. The relation expressed therefore is that xy = z², or, in other words, that a rectangle whose sides are x and y exactly equals a square whose side is z.

Fig. 1. Fig. 2. Fig. 3.

The use of this device in assisting calculation will be apparent if we take the case of an area bounded by a curved line. We cannot directly calculate this area, but we can easily tell that of a rectangle. Now it is evident that if we inscribe rectangles in this area ABC, the more rectangles we inscribe the less will be the error in taking their sum as equal to the curved area. This is apparent if we compare fig. 2 with fig. 3. Suppose we take a point P on the curve, call BN = x and PN = y, and suppose Nn to be dx, the differentially small increment of x, and pq = dy the corresponding small increment of y. The area of the rectangle PqnN = PN × Nn = ydx, and differs from the true curvilinear area PpnN by less than the little rectangle of Pq × pq or of dx.dy. But, as we have seen, if we push our division to the first infinitesimal order, or make Nn and pq differentials of x and y, dx.dy may be neglected—i.e. multiply the number of rectangles indefinitely, and the sum of their areas will differ from the true area inclosed by the curve by an error which is evanescent.