Calculus Made Easy HTML version

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
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.
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
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