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

Fig. 2.	Fig. 3.

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