Page:Calculus Made Easy.pdf/27

 way. The enlarged square is made up of the original square $$x^2$$, the two rectangles at the top and on the right, each of which is of area $$x\cdot dx$$ (or together $$2x\cdot dx$$), and the little square at the top right-hand corner which is $$(dx)^2$$. In Fig. 2 we have taken $$dx$$ as

quite a big fraction of $$x$$—about $$\tfrac{1}{5}$$. But suppose we had taken it only $$\tfrac{1}{100}$$—about the thickness of an inked line drawn with a fine pen. Then the little corner square will have an area of only $$\tfrac{1}{10,000}$$ of $$x^2$$, and be practically invisible. Clearly $$(dx)^2$$ is negligible if only we consider the increment $$dx$$ to be itself small enough.

Let us consider a simile.

Suppose a millionaire were to say to his secretary: next week I will give you a small fraction of any money that comes in to me. Suppose that the secretary were to say to his boy: I will give you a small fraction of what I get. Suppose the fraction in each case to be $$\tfrac{1}{100}$$ part. Now if Mr. Millionaire received during the next week £$$1000$$, the secretary