A Little Algebra
: From The Earth To The MoonThe 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 trave
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!"
More
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