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 60 minutes in the hour, 24 hours in the day, 7 days in the week. There are therefore 1,440 minutes in the day and 10,080 minutes in the week.
Obviously 1 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 60 still smaller parts, which in Queen Elizabeth's days, 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 farthing as compared with a sovereign; it is worth only a little more than 1/1,000 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 £1,000: relatively to this greater sum, the farthing is of no more importance than 1/1,000 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, x/6o be called a small fraction, then 1/60 of 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. 1/100) as a small fraction, then 1 per cent of 1 per cent (i.e. 1/10,000) would be a 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 1/1,000,000 as 'small'. Thus, if a first-rate chronometer is not to lose 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 regard 1/1,000,000 (or one millionth) as a small quantity, then 1/1,000,000 of 1/1,000,000, that is 1/1,000,000,000,000 (or one billionth) will be a small quantity 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, it does not follow that such quantities as x.dx or a2.dx or ax.dx are negligible. But dx times 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 x. Thus if we took dx to mean numerically, say 1/60 of x, then the second term would be 2/60 of x2, whereas the third term would be 1/3,600 of x2. This last term is clearly less important than the second. But if we go further and take dx to mean only 1/1000 of x, then the second term will be 2/1,000 of x2, while the third term will be only 1/1,000,000 Of x2.
Geometrically this may be depicted as follows: draw a square the side of which we will take to represent x. Now suppose the square to grow by having a bit dx added to its size each way. The enlarged square is made up of the original square x2, the two rectangles at the top and on the right, each of which is of area x. dx (or together 2x.dx), and the little square at the top right-hand corner which is (dx)2. We have taken dx as quite a big fraction of x - about 1/5. But suppose we had taken it as only 1/1000 - about the thickness of an inked line drawn with a fine pen. Then the little corner square will have an area of only 1/10,000 of x2, and be practically invisible. Clearly (dx)2 is negligible if only we consider the increment dx to be itself small enough.
(From Calculus Made Easy by Silvanus P. Thompson.)