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.

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