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 p