"Now then what is this Integral Calculus of yours ?" "It is a branch of Mathematics that has for its object the summation of a certain infinite series of indefinitely small terms: but for the solution of which, we must generally know the function of which a given function is the differential coefficient.

In other words," continued Barbican, "in it we return from the differential coefficient, to the function from which it was deduced." "Clear as mud!" cried Ardan, with a hearty laugh.
"Now then, let me have a bit of paper and a pencil," added Barbican, "and in half an hour you shall have your formula; meantime you can easily find something interesting to do." In a few seconds Barbican was profoundly absorbed in his problem, while M'Nicholl was watching out of the window, and Ardan was busily employed in preparing breakfast.
The morning meal was not quite ready, when Barbican, raising his head, showed Ardan a page covered with algebraic signs at the end of which stood the following formula:-- 1 2 2 r m' r r -- - (v' - v ) = gr {--- - 1 + -- - (-- --- - -- ---) } 2 x m d - x d - r "Which means ?" asked Ardan.
"It means," said the Captain, now taking part in the discussion, "that the half of _v_ prime squared minus _v_ squared equals _gr_ multiplied by _r_ over _x_ minus one plus _m_ prime over _m_ multiplied by _r_ over _d_ minus _x_ minus _r_ over _d_ minus _r_ ...

that is--" "That is," interrupted Ardan, in a roar of laughter, "_x_ stradlegs on _y_, making for _z_ and jumping over _p_! Do _you_ mean to say you understand the terrible jargon, Captain ?" "Nothing is clearer, Ardan." "You too, Captain! Then of course I must give in gracefully, and declare that the sun at noon-day is not more palpably evident than the sense of Barbican's formula." "You asked for Algebra, you know," observed Barbican.
"Rock crystal is nothing to it!" "The fact is, Barbican," said the Captain, who had been looking over the paper, "you have worked the thing out very well.

You have the integral equation of the living forces, and I have no doubt it will give us the result sought for." "Yes, but I should like to understand it, you know," cried Ardan: "I would give ten years of the Captain's life to understand it!" "Listen then," said Barbican.

"Half of _v_ prime squared less _v_ squared, is the formula giving us the half variation of the living force." "Mac pretends he understands all that!" "You need not be a _Solomon_ to do it," said the Captain.