From the Earth to the Moon; and, Round the Moon

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!”