Page:Calculus Made Easy.pdf/26

that such quantities as $x\cdot dx$, or $x^{2}dx$, or $a^{x}dx$ are negligible. But $dx\times dx$ would be negligible, being a small quantity of the second order.

A very simple example will serve as illustration.

Let us think of $x$ as a quantity that can grow by a small amount so as to become $x+dx$, where $dx$ is the small increment added by growth. The square of this is $x^{2}+2x\cdot dx+(dx)^{2}$. The second term is not negligible because it is a first-order quantity; while the third term is of the second order of smallness, being a bit of, a bit of $x^{2}$. Thus if we took $dx$ to mean numerically, say, ${\tfrac {1}{60}}$ of $x$, then the second term would be ${\tfrac {2}{60}}$ of $x^{2}$, whereas the third term would be ${\tfrac {1}{3600}}$ of $x^{2}$. This last term is clearly less important than the second. But if we go further and take $dx$ to mean only ${\tfrac {1}{1000}}$ of $x$, then the second term will be ${\tfrac {2}{1000}}$ of $x^{2}$, while the third term will be only ${\tfrac {1}{1,000,000}}$ of $x^{2}$.

Fig. 1.

Geometrically this may be depicted as follows: Draw a square (Fig. 1) the side of which we will take to represent $x$. Now suppose the square to grow by having a bit $dx$ added to its size each