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

A very simple example will serve as illustration.

Let us think of ${\displaystyle x}$ as a quantity that can grow by a small amount so as to become ${\displaystyle x+dx}$, where ${\displaystyle dx}$ is the small increment added by growth. The square of this is ${\displaystyle x^{2}+2x\cdot dx+(dx)^{2}}$. 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 ${\displaystyle x^{2}}$. Thus if we took ${\displaystyle dx}$ to mean numerically, say, ${\displaystyle {\tfrac {1}{60}}}$ of ${\displaystyle x}$, then the second term would be ${\displaystyle {\tfrac {2}{60}}}$ of ${\displaystyle x^{2}}$, whereas the third term would be ${\displaystyle {\tfrac {1}{3600}}}$ of ${\displaystyle x^{2}}$. This last term is clearly less important than the second. But if we go further and take ${\displaystyle dx}$ to mean only ${\displaystyle {\tfrac {1}{1000}}}$ of ${\displaystyle x}$, then the second term will be ${\displaystyle {\tfrac {2}{1000}}}$ of ${\displaystyle x^{2}}$, while the third term will be only ${\displaystyle {\tfrac {1}{1,000,000}}}$ of ${\displaystyle x^{2}}$.

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