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Calculus Made Easy

depend one on the other. An alteration in one will bring about an alteration in the other, because of this dependence. Let us call one of the variables , and the other that depends on it .

Suppose we make to vary, that is to say, we either alter it or imagine it to be altered, by adding to it a bit which we call . We are thus causing to become . Then, because has been altered, will have altered also, and will have become . Here the bit may be in some cases positive, in others negative; and it won’t (except by a miracle) be the same size as .

Take two examples.

(1) Let and be respectively the base and the height of a right-angled triangle (Fig. 4), of which

Fig. 4.

the slope of the other side is fixed at . If we suppose this triangle to expand and yet keep its angles the same as at first, then, when the base grows so as to become , the height becomes . Here, increasing results in an increase of . The little triangle, the height of which is , and the base