Page:Calculus Made Easy.pdf/24

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 farthing as compared with a sovereign: it is barely worth more than ${\tfrac {1}{1000}}$ part. A farthing more or less is of precious little importance compared with a sovereign: it may certainly be regarded as a small quantity. But compare a farthing with £$1000$: relatively to this greater sum, the farthing is of no more importance than ${\tfrac {1}{1000}}$ of a farthing would be to a sovereign. Even a golden sovereign is relatively a negligible quantity in the wealth of a millionaire.

Now if we fix upon any numerical fraction as constituting the proportion 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, ${\tfrac {1}{60}}$ be called a small fraction, then ${\tfrac {1}{60}}$ of ${\tfrac {1}{60}}$ (being a small fraction of a small fraction) may be regarded as a small quantity of the second order of smallness.^[1]

Or, if for any purpose we were to take $1$ per cent. (i.e. ${\tfrac {1}{100}}$) as a small fraction, then $1$ per cent. of $1$ per cent. (i.e. ${\tfrac {1}{10,000}}$) would be a small fraction of the second order of smallness; and ${\tfrac {1}{1,000,000}}$ would

1. 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.