On Different Degrees of Smallness

Excerpts

• On Different Degrees of Smallness

  Obviously $1$ minute is a very small quantity of time compared with a whole week.

• On Different Degrees of Smallness

  The mathematicians talk about the second order of “magnitude” (*i.e.* greatness) when they really mean second order of *smallness*. This is very confusing to beginners.

• On Different Degrees of Smallness

  But, it must be remembered, that small quantities if they occur in our expressions as factors multiplied by some other factor, may become important if the other factor is itself large. Even a farthing becomes important if only it is multiplied by a few hundred.

• On Different Degrees of Smallness

  Then we see that the smaller a small quantity itself is, the more negligible does the corresponding small quantity of the second order become.

• On Different Degrees of Smallness

  Nowadays we call these small quantities of the second order of smallness “seconds.” But few people know *why* they are so called.

• On Different Degrees of Smallness

  Now in the calculus we write $dx$ for a little bit of $x$. These things such as $dx$, and $du$, and $dy$, are called “differentials,” the differential of $x$, or of $u$, or of $y$, as the case may be.

• On Different Degrees of Smallness

  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 $x^2 + 2x · dx + (dx)^2$.

• On Different Degrees of Smallness

  Clearly $(dx)^2$ is negligible if only we consider the increment $dx$ to be itself small enough.

• On Different Degrees of Smallness

  An ox might worry about a flea of ordinary size---a small creature of the first order of smallness. But he would probably not trouble himself about a flea’s flea; being of the second order of smallness, it would be negligible.

Equations

• On Different Degrees of Smallness

  x^2 + 2x · dx + (dx)^2

  The square of x grown by a small increment dx expands to x squared, plus twice x times dx, plus dx squared, where the last term is of the second order of smallness.

Problems

No exercises in this chapter.