A Little Algebra

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

The night passed without anything happening. But the word “night” hardly applies.

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 projectile’s position relative to the sun did not change. Astronomically speaking, it was daylight on the lower part and night on the upper part. So when this narrative uses the words “day” and “night,” they refer to the passage of time between sunrise and sunset on 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.

The travelers slept more peacefully because the projectile was moving so fast that it seemed completely still. Nothing revealed its movement through space. However fast a body moves, its speed has no noticeable effect on the human body in a vacuum. The same is true when the air moves along with the body carrying the people. Who on Earth feels its speed, even though it travels at 68,000 miles per hour? Under these conditions, motion is no more “felt” than rest. A body at rest will remain at rest as long as no outside force moves it. If a body is moving, it will not stop unless an obstacle comes in its way. This indifference to motion or rest 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.

Inside the projectile, Barbicane and his companions might have thought they were completely still. They would have felt the same way outside it. The moon above them was growing larger. If not for that, they might have sworn they were floating without moving at all.

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!”

On the morning of December 3rd, a cheerful but unexpected sound woke the travelers. A rooster was crowing inside the car. Michel Ardan was the first to get up. He climbed to the top of the projectile and closed a box whose lid was partly open. Then he said quietly, “Will you be quiet? That creature will ruin my plan!”

But Nicholl and Barbicane were awake.

But Nicholl and Barbicane were awake.

“A cock!” said Nicholl.

“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.

“Why, no, my friends,” Michel answered quickly. “I wanted to wake you with this country sound.” With that, he gave a splendid cock-a-doodle-doo, worthy of the proudest barnyard.

The two Americans could not help laughing.

The two Americans couldn't help laughing.

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

“That’s a fine talent,” 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.”

“Yes,” said Michel, “it’s a joke in my country. It’s very French. They imitate a rooster like that even in the best society.”

Then turning the conversation:

Then, changing the subject:

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

“Barbicane, do you know what I’ve been thinking about all night?”

“No,” answered the president.

“No,” the president replied.

“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.”

“Of our Cambridge friends. You have already noticed that I know nothing about mathematics. And I cannot figure out how the observatory scientists calculated the speed the projectile needed when leaving the Columbiad to reach 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.”

“You mean reaching the neutral point where Earth’s and the moon’s gravitational pulls are equal,” replied Barbicane. “That point is about nine-tenths of the way there. From that point, the projectile would simply fall onto the moon because of its weight.”

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

“All right,” said Michel. “But again, how could they calculate the initial speed?”

“Nothing can be easier,” replied Barbicane.

“Nothing could be easier,” said Barbicane.

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

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

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

“Exactly. Nicholl and I would have done it if the observatory hadn't 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.”

“All right, old Barbicane,” replied Michel. “They could have cut off my head, starting at my feet, before they could have made me solve that problem.”

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

“Because you don’t know algebra,” Barbicane answered quietly.

“Ah, there you are, you eaters of _x_1; you think you have said all when you have said ‘Algebra.’”

“Ah, there you are, you eaters of _x_1. You think saying ‘Algebra’ says it all.”

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

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

“Hardly.”

“Hardly.”

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

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

“Seriously?”

“Seriously?”

“Quite seriously.”

“Completely seriously.”

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

“And can you use that tool while I’m here?”

“If it will interest you.”

“If you're interested.”

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

“And show me how they calculated our car’s initial speed?”

“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.”

“Yes, my good friend. I can account for all the factors in the problem. These include the distance between the centers of the earth and the moon, the earth’s radius, and the size of both bodies. With a simple formula, I can tell exactly what the projectile’s initial speed should be.”

“Let us see.”

“Let's 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.”

“You will see it. But I will not show you the projectile’s actual path between the moon and the earth, taking into account their motion around the sun. Instead, I will treat both bodies as perfectly still. That will serve all our purposes.”

“And why?”

“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.”

“Because it will be trying to solve what is called ‘the problem of the three bodies.’ Integral calculus is not yet advanced enough to solve it.”

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

“Then,” said Michel Ardan slyly, “mathematics hasn't said its last word?”

“Certainly not,” replied Barbicane.

“Of course not,” Barbicane replied.

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

“Well, perhaps the Selenites have taken integral calculus further than you have. By the way, what is ‘integral calculus’?”

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

“It is a calculation that reverses differentiation,” Barbicane replied seriously.

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

“Thanks. It’s 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.”

“And now,” Barbicane continued, “give me a piece of paper and a pencil, and I’ll find the formula in less than half an hour.”

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.

Less than half an hour later, Barbicane looked up and showed Michel Ardan a page covered in algebraic symbols. It contained the general formula for the solution.

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

“Well, 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.”

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

“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?”

“And you claim, Nicholl,” Michel asked, “that you can use these hieroglyphics to find the initial speed the projectile needed? They’re even more incomprehensible than the Egyptian Ibis.”

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

“Absolutely,” replied Nicholl. “And I can use this same formula to tell you its speed at any point in its journey.”

“On your word?”

“Do you promise?”

“On my word.”

“I promise.”

“Then you are as cunning as our president.”

“Then you are as clever 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.”

“No, Michel, the difficult part is what Barbicane has done. He has found an equation that satisfies all the conditions of the problem. The rest is just arithmetic, requiring only addition, subtraction, multiplication, and division.”

“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.

“That is something!” replied Michel Ardan. He could never get addition right and called it a Chinese puzzle that could produce 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.”

“The expression v zero in that equation is the speed the projectile will have when it leaves 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.”

“Exactly,” said Nicholl. “We must calculate the speed from that point. We already know that the speed at departure was exactly one and a half times greater than when leaving the atmosphere.”

“I understand no more,” said Michel.

“I don’t understand anymore,” said Michel.

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

“It’s a very simple calculation,” said Barbicane.

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

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

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

“That means our projectile had already lost one-third of its initial speed when it reached the edge of Earth's atmosphere.”

“As much as that?”

“That much?”

“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.”

“Yes, my friend, simply from friction against the layers of air. You understand that the faster it goes, the more resistance it meets 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.”

“I admit that,” Michel replied. “And I understand it, although your x’s, zeros, and algebraic formulas are rattling around 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.”

“Those are the first results of algebra,” replied Barbicane. “Now, to finish, we will verify the numbers given by these different expressions. In other words, we will work out their values.”

“Finish me!” replied Michel.

“Finish me!” Michel replied.

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

Barbicane took the paper and began calculating quickly. Nicholl looked over his shoulder and eagerly read the work as it progressed.

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

“That’s it! That’s it!” he finally cried.

“Is it clear?” asked Barbicane.

“Is it clear?” asked Barbicane.

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

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

“Wonderful fellows!” muttered Ardan.

“Wonderful people!” muttered Ardan.

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

“Do you understand it now?” asked Barbicane.

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

“Do I understand it?” cried Ardan. “My head is splitting from it.”

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

“And now,” said Nicholl, “we only need to calculate the projectile’s speed when it leaves the atmosphere.”

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.

The captain was a practical man who could handle any difficulty. He began writing at a frightening speed. Divisions and multiplications poured from his fingers. Numbers fell like hail across the white page. Barbicane watched him, while Michel Ardan held his aching head in both hands as the pain grew worse.

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

“Well?” asked Barbicane after a few minutes of 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—”

“Well!” replied Nicholl. “After all the calculations, v zero should be this: the speed the projectile needs as it leaves the atmosphere to reach the point of equal attraction.”

“Yes?” said Barbicane.

“Yes?” said Barbicane.

“Twelve thousand yards.”

“Twelve thousand yards.”

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

“What!” exclaimed Barbicane, startled. “You say...”

“Twelve thousand yards.”

“Twelve thousand yards.”

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

“The devil!” cried the president, gesturing in despair.

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

“What’s wrong?” asked Michel Ardan, very 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—”

“What is the matter? Why, if friction had already reduced our speed by one-third, our initial speed must have been...”

“Seventeen thousand yards.”

“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—”

“And the Cambridge Observatory said that twelve thousand yards was enough at the start. And our projectile only started at that speed...”

“Well?” asked Nicholl.

“Well?” asked Nicholl.

“Well, it will not be enough.”

“Well, it won't be enough.”

“Good.”

“Good.”

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

“We won’t be able to reach the neutral point.”

“The deuce!”

“Damn!”

“We shall not even get halfway.”

“We won’t 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.

“In the name of the projectile!” Michel Ardan exclaimed, jumping as if the projectile were already about to strike Earth.

“And we shall fall back upon the earth!”

“And we will fall back to Earth!”