<h2 id="sigil_toc_id_67">CHAPTER IV.</h2>
<h3 id="sigil_toc_id_68">A LITTLE ALGEBRA.</h3>
<p>The night passed without incident. The word "night," however, is
scarcely applicable.</p>
<p>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 the rising and setting of
the sun upon the earth.</p>
<p>The travellers' 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.</p>
<p>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.</p>
<p>That morning, the 3rd of December, the travellers 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!"</p>
<p>But Nicholl and Barbicane were awake.</p>
<p>"A cock!" said Nicholl.</p>
<p>"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 honour to the
proudest of poultry-yards.</p>
<p>The two Americans could not help laughing.</p>
<p>"Fine talent that," said Nicholl, looking suspiciously at his
companion.</p>
<p>"Yes," said Michel; "a joke in my country. It is very Gallic; they
play the cock so in the best society."</p>
<p>Then turning the conversation,—</p>
<p>"Barbicane, do you know what I have been thinking of all
night?"</p>
<p>"No," answered the president.</p>
<p>"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."</p>
<p>"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
travelled over, the projectile would simply fall upon the moon, on
account of its weight."</p>
<p>"So be it," said Michel; "but, once more; how could they calculate
the initiatory speed?"</p>
<p>"Nothing can be easier," replied Barbicane.</p>
<p>"And you knew how to make that calculation?" asked Michel
Ardan.</p>
<p>"Perfectly. Nicholl and I would have made it, if the Observatory
had not saved us the trouble."</p>
<p>"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."</p>
<p>"Because you do not know algebra," answered Barbicane quietly.</p>
<p>"Ah, there you are, you eaters of <i>x</i>¹; you think you have
said all when you have said 'Algebra.'"</p>
<p>"Michel," said Barbicane, "can you use a forge without a hammer,
or plough without a ploughshare?"</p>
<p>"Hardly."</p>
<p>"Well, algebra is a tool, like the plough or the hammer, and a
good tool to those who know how to use it."</p>
<p>"Seriously?"</p>
<p>"Quite seriously."</p>
<p>"And can you use that tool in my presence?"</p>
<p>"If it will interest you."</p>
<p>"And show me how they calculated the initiatory speed of our
car?"</p>
<p>"Yes, my worthy friend; taking into consideration all the elements
of the problem, the distance from the centre of the earth to the
centre 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."</p>
<p>"Let us see."</p>
<p>"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."</p>
<p>"And why?"</p>
<p>"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."</p>
<p>"Then," said Michel Ardan, in his sly tone, "mathematics have not
said their last word?"</p>
<p>"Certainly not," replied Barbicane.</p>
<p>"Well, perhaps the Selenites have carried the integral calculus
farther than you have; and, by the bye, what is 'integral
calculus?'"</p>
<p>"It is a calculation the converse of the differential," replied
Barbicane seriously.</p>
<p>"Much obliged; it is all very clear, no doubt."</p>
<p>"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."</p>
<p>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.</p>
<p>"Well, and does Nicholl understand what that means?"</p>
<p>"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."</p>
<p>"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 to the
projectile?"</p>
<p>"Incontestably," replied Nicholl; "and even by this same formula I
can always tell you its speed at any point of its transit."</p>
<p>"On your word?"</p>
<p>"On my word."</p>
<p>"Then you are as cunning as our president."</p>
<p>"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."</p>
<p>"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.</p>
<p>"The expression <i>v</i> zero, which you see in that equation, is
the speed which the projectile will have on leaving the
atmosphere."</p>
<p>"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."</p>
<p>"I understand no more," said Michel.</p>
<p>"It is a very simple calculation," said Barbicane.</p>
<p>"Not as simple as I am," retorted Michel.</p>
<p>"That means, that when our projectile reached the limits of the
terrestrial atmosphere it had already lost one-third of its
initiatory speed."</p>
<p>"As much as that?"</p>
<p>"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."</p>
<p>"That I admit," answered Michel; "and I understand it, although
your x's and zero's, and algebraic formulæ, are rattling in my head
like nails in a bag."</p>
<p>"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."</p>
<p>"Finish me!" replied Michel.</p>
<p>Barbicane took the paper, and began again to make his calculations
with great rapidity. Nicholl looked over and greedily read the work
as it proceeded.</p>
<p>"That's it! that's it!" at last he cried.</p>
<p>"Is it clear?" asked Barbicane.</p>
<div class="illus"><ANTIMG alt="Illustration: DO I UNDERSTAND IT? CRIED ARDAN; MY HEAD IS SPLITTING WITH IT." id="splitting" src="images/splitting.jpg" /></div>
<div class="caption">"DO I UNDERSTAND IT?" CRIED ARDAN; "MY HEAD IS
SPLITTING WITH IT."</div>
<p>"It is written in letters of fire," said Nicholl.</p>
<p>"Wonderful fellows!" muttered Ardan.</p>
<p>"Do you understand it at last?" asked Barbicane.</p>
<p>"Do I understand it?" cried Ardan; "my head is splitting with
it."</p>
<p>"And now," said Nicholl, "to find out the speed of the projectile
when it left the atmosphere, we have only to calculate that."</p>
<p>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, whilst Michel Ardan nursed a growing headache
with both hands.</p>
<p>"Very well?" asked Barbicane, after some minutes' silence.</p>
<p>"Well!" replied Nicholl; "every calculation made, <i>v</i> 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—"</p>
<p>"Yes?" said Barbicane.</p>
<p>"Twelve thousand yards."</p>
<p>"What!" exclaimed Barbicane, starting; "you say—"</p>
<p>"Twelve thousand yards."</p>
<p>"The devil!" cried the president, making a gesture of despair.</p>
<p>"What is the matter?" asked Michel Ardan, much surprised.</p>
<p>"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—"</p>
<p>"Seventeen thousand yards."</p>
<p>"And the Cambridge Observatory declared that 12,000 yards was
enough at starting; and our projectile, which only started with that
speed—"</p>
<p>"Well?" asked Nicholl.</p>
<p>"Well, it will not be enough."</p>
<p>"Good."</p>
<p>"We shall not be able to reach the neutral point."</p>
<p>"The deuce!"</p>
<p>"We shall not even get half way."</p>
<p>"In the name of the projectile!" exclaimed Michel Ardan, jumping
as if it was already on the point of striking the terrestrial
globe.</p>
<p>"And we shall fall back upon the earth!"</p>
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