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 con- sider 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 1440 minutes in the day and 10080 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 com- pared with an hour, and called it “one minu`te,” meaning a minute fraction—namely one sixtieth—of an hour. When they came to re- quire still smaller subdivisions of time, they divided each minute into 60 still smaller parts, which, in Queen Elizabeth’s days, they called “second minu`tes” (i.e. small quantities of the second order of minute- ness). 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 barely
1000
worth more than 1 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
1000
this greater sum, the farthing is of no more importance than 1 of a
farthing would be to a sovereign. Even a golden sovereign is relatively a negligible quantity in the wealth of a millionaire.
60
of
60
Now if we fix upon any numerical fraction as constituting the pro- portion which for any purpose we call relatively small, we can easily state other fractions of a higher degree of smallness. Thus if, for the
60
purpose of time, 1
be called a small fraction, then 1
1 (being a
small fraction of a small fraction) may be regarded as a small quantity of the second order of smallness.*
100
Or, if for any purpose we were to take 1 per cent. (i.e. 1 ) as a
10,000
small fraction, then 1 per cent. of 1 per cent. (i.e. 1 ) would be a
1,000,000
small fraction of the second order of smallness; and 1 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
* The mathematicians talk about the second order of “magnitude” (i.e. great- ness) when they really mean second order of smallness. This is very confusing to beginners.
regard
1 1,000,000
(or one millionth) as a small quantity, then 1 of
1,000,000
1 , that is 1 (or one billionth) will be a small quantity
1,000,000 1,000,000,000,000
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 of itself, it does not follow that such quantities as x · dx, or x2 dx, or ax dx are negligible. But dx × 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 x2. Thus if we
60
took dx to mean numerically, say, 1 of x, then the second term would be 2 of x2, whereas the third term would be 1 of x2. This last term
60 3600
is clearly less important than the second. But if we go further and take
dx to mean only 1 of x, then the second term will be 2 of x2, while
1000
1,000,000
the third term will be only 1
of x2.
1000
x
x
Fig. 1.
Geometrically this may be depicted as follows: Draw a square (Fig. 1) 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. In Fig. 2 we have taken dx as quite a big fraction of x—about 1 . But suppose we had taken it only 1 —about the
5 100
10,000
thickness of an inked line drawn with a fine pen. Then the little corner square will have an area of only 1 of x2, and be practically invisible. Clearly (dx)2 is negligible if only we consider the increment dx to be itself small enough.
Let us consider a simile.
x dx
dx
dx
(dx)2
x
x dx
x
x · dx
x · dx
x2
Fig. 2. Fig. 3.
Suppose a millionaire were to say to his secretary: next week I will give you a small fraction of any money that comes in to me. Suppose that the secretary were to say to his boy: I will give you a small fraction
100
of what I get. Suppose the fraction in each case to be 1 part. Now
100
if Mr. Millionaire received during the next week 1000, the secretary would receive 10 and the boy 2 shillings. Ten pounds would be a small quantity compared with 1000; but two shillings is a small small quantity indeed, of a very secondary order. But what would be the disproportion if the fraction, instead of being 1 , had been settled at
1 1000
part? Then, while Mr. Millionaire got his 1000, Mr. Secretary
would get only 1, and the boy less than one farthing!
The witty Dean Swift* once wrote:
* On Poetry: a Rhapsody (p. 20), printed 1733—usually misquoted.
“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.
