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 ${\displaystyle {\tfrac {1}{1000}}}$ 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 £${\displaystyle 1000}$: relatively to this greater sum, the farthing is of no more importance than ${\displaystyle {\tfrac {1}{1000}}}$ 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, ${\displaystyle {\tfrac {1}{60}}}$ be called a small fraction, then ${\displaystyle {\tfrac {1}{60}}}$ of ${\displaystyle {\tfrac {1}{60}}}$ (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 ${\displaystyle 1}$ per cent. (i.e. ${\displaystyle {\tfrac {1}{100}}}$) as a small fraction, then ${\displaystyle 1}$ per cent. of ${\displaystyle 1}$ per cent. (i.e. ${\displaystyle {\tfrac {1}{10,000}}}$) would be a small fraction of the second order of smallness; and ${\displaystyle {\tfrac {1}{1,000,000}}}$ would