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### Chapter IV. A Little Algebra

The night passed without incident. The word “night,”
however, is scarcely applicable.

The position of the projectile with regard to the sun did not
change. Astronomically, it was daylight on the lower part, and
night on the upper; so when during this narrative these words are
used, they represent the lapse of time between rising and setting
of the sun upon the earth.

The travelers’ sleep was rendered more peaceful by the
projectile’s excessive speed, for it seemed absolutely
motionless. Not a motion betrayed its onward course through space.
The rate of progress, however rapid it might be, cannot produce any
sensible effect on the human frame when it takes place in a vacuum,
or when the mass of air circulates with the body which is carried
with it. What inhabitant of the earth perceives its speed, which,
however, is at the rate of 68,000 miles per hour? Motion under such
conditions is “felt” no more than repose; and when a
body is in repose it will remain so as long as no strange force
displaces it; if moving, it will not stop unless an obstacle comes
in its way. This indifference to motion or repose is called
inertia.

Barbicane and his companions might have believed themselves
perfectly stationary, being shut up in the projectile; indeed, the
effect would have been the same if they had been on the outside of
it. Had it not been for the moon, which was increasing above them,
they might have sworn that they were floating in complete
stagnation.

That morning, the 3rd of December, the travelers were awakened
by a joyous but unexpected noise; it was the crowing of a cock
which sounded through the car. Michel Ardan, who was the first on
his feet, climbed to the top of the projectile, and shutting a box,
the lid of which was partly open, said in a low voice, “Will
you hold your tongue? That creature will spoil my
design!”

But Nicholl and Barbicane were awake.

“A cock!” said Nicholl.

“Why no, my friends,” Michel answered quickly;
“it was I who wished to awake you by this rural sound.”
So saying, he gave vent to a splendid cock-a-doodledoo, which would
have done honor to the proudest of poultry-yards.

The two Americans could not help laughing.

“Fine talent that,” said Nicholl, looking
suspiciously at his companion.

“Yes,” said Michel; “a joke in my country. It
is very Gallic; they play the cock so in the best
society.”

Then turning the conversation:

“Barbicane, do you know what I have been thinking of all
night?”

“No,” answered the president.

“Of our Cambridge friends. You have already remarked that
I am an ignoramus in mathematical subjects; and it is impossible
for me to find out how the savants of the observatory were able to
calculate what initiatory speed the projectile ought to have on
leaving the Columbiad in order to attain the moon.”

“You mean to say,” replied Barbicane, “to
attain that neutral point where the terrestrial and lunar
attractions are equal; for, starting from that point, situated
about nine-tenths of the distance traveled over, the projectile
would simply fall upon the moon, on account of its
weight.”

“So be it,” said Michel; “but, once more; how
could they calculate the initiatory speed?”

“Nothing can be easier,” replied Barbicane.

“And you knew how to make that calculation?” asked
Michel Ardan.

“Perfectly. Nicholl and I would have made it, if the
observatory had not saved us the trouble.”

“Very well, old Barbicane,” replied Michel;
“they might have cut off my head, beginning at my feet,
before they could have made me solve that problem.”

“Because you do not know algebra,” answered
Barbicane quietly.

“Ah, there you are, you eaters of x^1; you think you have
said all when you have said ‘Algebra.’”

“Michel,” said Barbicane, “can you use a forge
without a hammer, or a plow without a plowshare?”

“Hardly.”

“Well, algebra is a tool, like the plow or the hammer, and
a good tool to those who know how to use it.”

“Seriously?”

“Quite seriously.”

“And can you use that tool in my presence?”

“If it will interest you.”

“And show me how they calculated the initiatory speed of
our car?”

“Yes, my worthy friend; taking into consideration all the
elements of the problem, the distance from the center of the earth
to the center of the moon, of the radius of the earth, of its bulk,
and of the bulk of the moon, I can tell exactly what ought to be
the initiatory speed of the projectile, and that by a simple
formula.”

“Let us see.”

“You shall see it; only I shall not give you the real
course drawn by the projectile between the moon and the earth in
considering their motion round the sun. No, I shall consider these
two orbs as perfectly motionless, which will answer all our
purpose.”

“And why?”

“Because it will be trying to solve the problem called
‘the problem of the three bodies,’ for which the
integral calculus is not yet far enough advanced.”

“Then,” said Michel Ardan, in his sly tone,
“mathematics have not said their last word?”

“Certainly not,” replied Barbicane.

“Well, perhaps the Selenites have carried the integral
calculus farther than you have; and, by the bye, what is this
‘integral calculus?’”

“It is a calculation the converse of the
differential,” replied Barbicane seriously.

“Much obliged; it is all very clear, no doubt.”

“And now,” continued Barbicane, “a slip of
paper and a bit of pencil, and before a half-hour is over I will
have found the required formula.”

Half an hour had not elapsed before Barbicane, raising his head,
showed Michel Ardan a page covered with algebraical signs, in which
the general formula for the solution was contained.

“Well, and does Nicholl understand what that
means?”

“Of course, Michel,” replied the captain. “All
these signs, which seem cabalistic to you, form the plainest, the
clearest, and the most logical language to those who know how to
read it.”

“And you pretend, Nicholl,” asked Michel,
“that by means of these hieroglyphics, more incomprehensible
than the Egyptian Ibis, you can find what initiatory speed it was
necessary to give the projectile?”

“Incontestably,” replied Nicholl; “and even by
this same formula I can always tell you its speed at any point of
its transit.”

“On your word?”

“On my word.”

“Then you are as cunning as our president.”

“No, Michel; the difficult part is what Barbicane has
done; that is, to get an equation which shall satisfy all the
conditions of the problem. The remainder is only a question of
arithmetic, requiring merely the knowledge of the four
rules.”

“That is something!” replied Michel Ardan, who for
his life could not do addition right, and who defined the rule as a
Chinese puzzle, which allowed one to obtain all sorts of
totals.

“The expression v zero, which you see in that equation, is
the speed which the projectile will have on leaving the
atmosphere.”

“Just so,” said Nicholl; “it is from that
point that we must calculate the velocity, since we know already
that the velocity at departure was exactly one and a half times
more than on leaving the atmosphere.”

“I understand no more,” said Michel.

“It is a very simple calculation,” said
Barbicane.

“Not as simple as I am,” retorted Michel.

“That means, that when our projectile reached the limits
of the terrestrial atmosphere it had already lost one-third of its
initiatory speed.”

“As much as that?”

“Yes, my friend; merely by friction against the
atmospheric strata. You understand that the faster it goes the more
resistance it meets with from the air.”

“That I admit,” answered Michel; “and I
understand it, although your x’s and zero’s, and
algebraic formula, are rattling in my head like nails in a
bag.”

“First effects of algebra,” replied Barbicane;
“and now, to finish, we are going to prove the given number
of these different expressions, that is, work out their
value.”

“Finish me!” replied Michel.

Barbicane took the paper, and began to make his calculations
with great rapidity. Nicholl looked over and greedily read the work
as it proceeded.

“That’s it! that’s it!” at last he
cried.

“Is it clear?” asked Barbicane.

“It is written in letters of fire,” said
Nicholl.

“Wonderful fellows!” muttered Ardan.

“Do you understand it at last?” asked Barbicane.

“Do I understand it?” cried Ardan; “my head is
splitting with it.”

“And now,” said Nicholl, “to find out the
speed of the projectile when it leaves the atmosphere, we have only
to calculate that.”

The captain, as a practical man equal to all difficulties, began
to write with frightful rapidity. Divisions and multiplications
grew under his fingers; the figures were like hail on the white
page. Barbicane watched him, while Michel Ardan nursed a growing
headache with both hands.

“Very well?” asked Barbicane, after some
minutes’ silence.

“Well!” replied Nicholl; every calculation made, v
zero, that is to say, the speed necessary for the projectile on
leaving the atmosphere, to enable it to reach the equal point of
attraction, ought to be——”

“Yes?” said Barbicane.

“Twelve thousand yards.”

“What!” exclaimed Barbicane, starting; “you
say——”

“Twelve thousand yards.”

“The devil!” cried the president, making a gesture
of despair.

“What is the matter?” asked Michel Ardan, much
surprised.

“What is the matter! why, if at this moment our speed had
already diminished one-third by friction, the initiatory speed
ought to have been——”

“Seventeen thousand yards.”

“And the Cambridge Observatory declared that twelve
thousand yards was enough at starting; and our projectile, which
only started with that speed——”

“Well?” asked Nicholl.

“Well, it will not be enough.”

“Good.”

“We shall not be able to reach the neutral
point.”

“The deuce!”

“We shall not even get halfway.”

“In the name of the projectile!” exclaimed Michel
Ardan, jumping as if it was already on the point of striking the
terrestrial globe.

“And we shall fall back upon the earth!”

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