# Calculus Made Easy

negligible quantity in the wealth of a millionaire.

Now if we fix upon any numerical fraction as constituting the pro-

portion which for any purpose we call relatively small, we can easily

state other fractions of a higher degree of smallness. Thus if, for the

purpose of time, 1 be called a small fraction, then 1 of 1 (being a 60 60 60

small fraction of a small fraction) may be regarded as a small quantity of the

second order of smallness.∗

Or, if for any purpose we were to take 1 per cent. (i.e. 1 ) as a 100

small fraction, then 1 per cent. of 1 per cent. (i.e. 1 ) would be a 10,000

small fraction of the second order of smallness; and 1 1,000,000

would be a small fraction of the third order of smallness, being 1 per cent. of

1 per cent. of 1 per cent. Lastly, suppose that for some very precise purpose we

should regard

1 as “small.” Thus, if a first-rate chronometer is not to lose 1,000,000

or gain more than half a minute in a year, it must keep time with an accuracy of 1

part in 1,051,200. Now if, for such a purpose, we

∗The mathematicians talk about the second order of “magnitude” (i.e. great-

ness) when they really mean second order of smallness. This is very confusing to

beginners.

DIFFERENT DEGREES OF SMALLNESS

regard 1 (or one millionth) as a small quantity, then 1,000,000

5 1 of

1,000,000 1 , that is 1 (or one billionth) will be a small quantity

1,000,000 1,000,000,000,000 of the second order of smallness, and may be

utterly disregarded, by

comparison. Then we see that the smaller a small quantity itself is, the more

negligible does the corresponding small quantity of the second order become.

Hence we know that in all cases we are justified in neglecting the small quantities

of the second—or third (or higher)—orders, if only we take the small quantity of

the first order small enough in itself.

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.

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. [You read them as dee-eks, or dee-you, or dee-wy.] If dx be a

small bit of x, and relatively small of itself, it does not follow that such quantities

as x · dx, or x2 dx, or ax dx are negligible. But dx × 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 x2

+ 2x · 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 x2. Thus if we