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CHAPTER I.
TO DELIVER YOU FROM THE PRELIMINARY TERRORS.

The preliminary terror, which chokes off most fifth-form boys from even attempting to learn how to calculate, can be abolished once for all by simply stating what is the meaning—in common-sense terms—of the two principal symbols that are used in calculating.

(1) ${\displaystyle d}$ which merely means “a little bit of.”
Thus ${\displaystyle dx}$ means a little bit of ${\displaystyle x}$; or ${\displaystyle du}$ means a little bit of ${\displaystyle u}$. Ordinary mathematicians think it more polite to say “an element of,” 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) ${\displaystyle \int }$ which is merely a long ${\displaystyle S}$, and may be called (if you like) “the sum of.”
Thus ${\displaystyle \int dx}$ means the sum of all the little bits of ${\displaystyle x}$; or ${\displaystyle \int dt}$ means the sum of all the little bits of ${\displaystyle t}$. Ordinary mathematicians call this symbol “the integral of.” Now any fool can see that if ${\displaystyle x}$ is considered as made up of a lot of little bits, each of which is called ${\displaystyle dx}$, if you add them all up together you get the sum of all the ${\displaystyle dx}$’s, (which is the same thing as the whole of ${\displaystyle x}$). 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 ${\displaystyle 3600}$ little bits called seconds. The whole of the ${\displaystyle 3600}$ little bits added up together make one hour.