On Different Degrees of Smallness

We shall find that in our processes of calculation we have to deal with small quantities of various degrees of smallness.

We shall have also to learn under what circumstances we may consider small quantities to be so minute that we may omit them from consideration. Everything depends upon relative minuteness.

Before we fix any rules let us think of some familiar cases. There are  minutes in the hour,  hours in the day,  days in the week. There are therefore  minutes in the day and  minutes in the week.

Obviously  minute is a very small quantity of time compared with a whole week. Indeed, our forefathers considered it small as compared with an hour, and called it “one minute,” meaning a minute fraction—namely one sixtieth—of an hour. When they came to require still smaller subdivisions of time, they divided each minute into still smaller parts, which, in the 16th century, they called “second minutes” (i.e. small quantities of the second order of minuteness). Nowadays we call these small quantities of the second order of smallness “seconds.” But few people know why they are so called.

Now if one minute is so small as compared with a whole day, how much smaller by comparison is one second!

Again, think of a penny as compared with a ten dollar bill: it is only worth part. A penny more or less is of precious little importance compared with a ten dollar bill: it may certainly be regarded as a small quantity. But compare a penny with : relatively to this greater sum, the penny is of no more importance than of a penny would be to a ten dollar bill. Even , the secretary would receive ; but 10 cents is a small small quantity indeed, of a very secondary order. But what would be the disproportion if the fraction, instead of being , had been settled at part? Then, while Mr. Millionaire got his , and the boy 0.1 cents!

The witty Dean Swift² once wrote:

So, Nat’ralists observe, a Flea
Hath smaller Fleas that on him prey.
And these have smaller Fleas to bite ’em,
And so proceed ad infinitum

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. Even a gross of fleas’ fleas would not be of much account to the ox.