# Calculus Made Easy

instead of “a little bit of.” Just as you please. But you will find that

these little bits (or elements) may be considered to be indefinitely small.

(2) which is merely a long S, and may be called (if you like) “the

sum of.” Thus dx means the sum of all the little bits of x; or dt means

the sum of all the little bits of t. Ordinary mathematicians call this symbol “the

integral of.” Now any fool can see that if x is considered as made up of a lot of

little bits, each of which is called dx, if you add them all up together you get the

sum of all the dx’s, (which is the

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same thing as the whole of x). The word “integral” simply means “the whole.” If

you think of the duration of time for one hour, you may (if you like) think of it as

cut up into 3600 little bits called seconds. The whole of the 3600 little bits added

up together make one hour.

When you see an expression that begins with this terrifying sym- bol, you will

henceforth know that it is put there merely to give you instructions that you are

now to perform the operation (if you can) of totalling up all the little bits that are

indicated by the symbols that follow.

That’s all.

CHAPTER II.

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

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much smaller by comparison is one second! Again, think of a farthing as

compared with a sovereign: it is barely

worth more than 1 part. A farthing more or less is of precious little 1000

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 1 of a 1000

farthing would be to a sovereign. Even a golden sovereign is relatively a