"Well, Barbican, dear boy," observed Ardan, "all I've got to say is, you might chop the head off my body, beginning with my feet, before you could make me go through such a calculation."
"Simply because you don't understand Algebra," replied Barbican, quietly.
"Oh! that's all very well!" cried Ardan, with an ironical smile. "You great x+y men think you settle everything by uttering the word Algebra!"
"Ardan," asked Barbican, "do you think people could beat iron without a hammer, or turn up furrows without a plough?"
"Hardly."
"Well, Algebra is an instrument or utensil just as much as a hammer or a plough, and a very good instrument too if you know how to make use of it."
"You're in earnest?"
"Quite so."
"And you can handle the instrument right before my eyes?"
"Certainly, if it interests you so much."
"You can show me how they got at the initial velocity of our Projectile?"
"With the greatest pleasure. By taking into proper consideration all the elements of the problem, viz.: (1) the distance between the centres of the Earth and the Moon, (2) the Earth's radius, (3) its volume, and (4) the Moon's volume, I can easily calculate what must be the initial velocity, and that too by a very simple formula."
"Let us have the formula."
"In one moment; only I can't give you the curve really described by the Projectile as it moves between the Earth and the Moon; this is to be obtained by allowing for their combined movement around the Sun. I will consider the Earth and the Sun to be motionless, that being sufficient for our present purpose."
"Why so?"
"Because to give you that exact curve would be to solve a point in the 'Problem of the Three Bodies,' which Integral Calculus has not yet reached."
"What!" cried Ardan, in a mocking tone, "is there really anything that Mathematics can't do?"
"Yes," said Barbican, "there is still a great deal that Mathematics can't even attempt."
"So far, so good;" resumed Ardan. "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:—
"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. "All these signs that you appear to consider so cabalistic form a language the clearest, the shortest, and the most logical, for all those who can read it."
"You pretend, Captain, that, by means of these hieroglyphics, far more incomprehensible than the sacred Ibis of the Egyptians, you can discover the velocity at which the Projectile should start?"
"Most undoubtedly," replied the Captain, "and, by the same formula I can even tell you the rate of our velocity at any particular point of our journey."
"You can?"
"I can."
"Then you're just as deep a one as our President."
"No, Ardan; not at all. The really difficult part of the question Barbican has done. That is, to make out such an equation as takes into account all the conditions of the problem. After that, it's a simple affair of Arithmetic, requiring only a knowledge of the four rules to work it out."
"Very simple," observed Ardan, who always got muddled at any kind of a difficult sum in addition.
"Captain," said Barbican, "you could have found the formulas too, if you tried."
"I don't know about that," was the Captain's reply, "but I do know that this formula is wonderfully come at."
"Now, Ardan, listen a moment," said Barbican, "and you will see what sense there is in all these letters."
"I listen," sighed Ardan with the resignation of a martyr.
"d is the distance from the centre of the Earth to the centre of the Moon, for it is from the centres that we must calculate the attractions."
"That I comprehend."
"r is the radius of the Earth."
"That I comprehend."
"m is the mass or volume of the Earth; m prime that of the Moon. We must take the mass of the two attracting bodies into consideration, since attraction is in direct proportion to their masses."
"That I comprehend."
"g is the gravity or the velocity acquired at the end of a second by a body falling towards the centre of the Earth. Clear?"
"That I comprehend."
"Now I represent by x the varying distance that separates the Projectile from the centre of the Earth, and by v prime its velocity at that distance."
"That I comprehend."
"Finally, v is its velocity when quitting our atmosphere."
"Yes," chimed in the Captain, "it is for this point, you see, that the velocity had to be calculated, because we know already that the initial velocity is exactly the three halves of the velocity when the Projectile