CHAPTER IV.
A CHAPTER FOR THE CORNELL GIRLS.
No incident worth recording occurred during the night, if night indeedit could be called. In reality there was now no night or even day in theProjectile, or rather, strictly speaking, it was always _night_ on theupper end of the bullet, and always _day_ on the lower. Whenever,therefore, the words _night_ and _day_ occur in our story, the readerwill readily understand them as referring to those spaces of time thatare so called in our Earthly almanacs, and were so measured by thetravellers' chronometers.
The repose of our friends must indeed have been undisturbed, if absolutefreedom from sound or jar of any kind could secure tranquillity. Inspite of its immense velocity, the Projectile still seemed to beperfectly motionless. Not the slightest sign of movement could bedetected. Change of locality, though ever so rapid, can never revealitself to our senses when it takes place in a vacuum, or when theenveloping atmosphere travels at the same rate as the moving body.Though we are incessantly whirled around the Sun at the rate of aboutseventy thousand miles an hour, which of us is conscious of theslightest motion? In such a case, as far as sensation is concerned,motion and repose are absolutely identical. Neither has any effect oneway or another on a material body. Is such a body in motion? It remainsin motion until some obstacle stops it. Is it at rest? It remains atrest until some superior force compels it to change its position. Thisindifference of bodies to motion or rest is what physicists call_inertia_.
Barbican and his companions, therefore, shut up in the Projectile, couldreadily imagine themselves to be completely motionless. Had they beenoutside, the effect would have been precisely the same. No rush of air,no jarring sensation would betray the slightest movement. But for thesight of the Moon gradually growing larger above them, and of the Earthgradually growing smaller beneath them, they could safely swear thatthey were fast anchored in an ocean of deathlike immobility.
Towards the morning of next day (December 3), they were awakened by ajoyful, but quite unexpected sound.
"Cock-a-doodle! doo!" accompanied by a decided flapping of wings.
The Frenchman, on his feet in one instant and on the top of the ladderin another, attempted to shut the lid of a half open box, speaking in anangry but suppressed voice:
"Stop this hullabaloo, won't you? Do you want me to fail in my greatcombination!"
"Hello?" cried Barbican and M'Nicholl, starting up and rubbing theireyes.
"What noise was that?" asked Barbican.
"Seems to me I heard the crowing of a cock," observed the Captain.
"I never thought your ears could be so easily deceived, Captain," criedArdan, quickly, "Let us try it again," and, flapping his ribs with hisarms, he gave vent to a crow so loud and natural that the lustiestchanticleer that ever saluted the orb of day might be proud of it.
The Captain roared right out, and even Barbican snickered, but as theysaw that their companion evidently wanted to conceal something, theyimmediately assumed straight faces and pretended to think no more aboutthe matter.
"Barbican," said Ardan, coming down the ladder and evidently anxious tochange the conversation, "have you any idea of what I was thinking aboutall night?"
"Not the slightest."
"I was thinking of the promptness of the reply you received last yearfrom the authorities of Cambridge University, when you asked them aboutthe feasibility of sending a bullet to the Moon. You know very well bythis time what a perfect ignoramus I am in Mathematics. I own I havebeen often puzzled when thinking on what grounds they could form such apositive opinion, in a case where I am certain that the calculation mustbe an exceedingly delicate matter."
"The feasibility, you mean to say," replied Barbican, "not exactly ofsending a bullet to the Moon, but of sending it to the neutral pointbetween the Earth and the Moon, which lies at about nine-tenths of thejourney, where the two attractions counteract each other. Because thatpoint once passed, the Projectile would reach the Moon's surface byvirtue of its own weight."
"Well, reaching that neutral point be it;" replied Ardan, "but, oncemore, I should like to know how they have been able to come at thenecessary initial velocity of 12,000 yards a second?"
"Nothing simpler," answered Barbican.
"Could you have done it yourself?" asked the Frenchman.
"Without the slightest difficulty. The Captain and myself could havereadily solved the problem, only the reply from the University saved usthe trouble."
"Well, Barbican, dear boy," observed Ardan, "all I've got to say is, youmight chop the head off my body, beginning with my feet, before youcould make me go through such a calculation."
"Simply because you don't understand Algebra," replied Barbican,quietly.
"Oh! that's all very well!" cried Ardan, with an ironical smile. "Yougreat _x+y_ men think you settle everything by uttering the word_Algebra_!"
"Ardan," asked Barbican, "do you think people could beat iron without ahammer, or turn up furrows without a plough?"
"Hardly."
"Well, Algebra is an instrument or utensil just as much as a hammer or aplough, and a very good instrument too if you know how to make use ofit."
"You're in earnest?"
"Quite so."
"And you can handle the instrument right before my eyes?"
"Certainly, if it interests you so much."
"You can show me how they got at the initial velocity of ourProjectile?"
"With the greatest pleasure. By taking into proper consideration all theelements of the problem, viz.: (1) the distance between the centres ofthe Earth and the Moon, (2) the Earth's radius, (3) its volume, and (4)the Moon's volume, I can easily calculate what must be the initialvelocity, and that too by a very simple formula."
"Let us have the formula."
"In one moment; only I can't give you the curve really described by theProjectile as it moves between the Earth and the Moon; this is to beobtained by allowing for their combined movement around the Sun. I willconsider the Earth and the Sun to be motionless, that being sufficientfor our present purpose."
"Why so?"
"Because to give you that exact curve would be to solve a point in the'Problem of the Three Bodies,' which Integral Calculus has not yetreached."
"What!" cried Ardan, in a mocking tone, "is there really anything thatMathematics can't do?"
"Yes," said Barbican, "there is still a great deal that Mathematicscan't even attempt."
"So far, so good;" resumed Ardan. "Now then what is this IntegralCalculus of yours?"
"It is a branch of Mathematics that has for its object the summation ofa certain infinite series of indefinitely small terms: but for thesolution of which, we must generally know the function of which a givenfunction is the differential coefficient. In other words," continuedBarbican, "in it we return from the differential coefficient, to thefunction from which it was deduced."
"Clear as mud!" cried Ardan, with a hearty laugh.
"Now then, let me have a bit of paper and a pencil," added Barbican,"and in half an hour you shall have your formula; meantime you caneasily find something interesting to do."
In a few seconds Barbican was profoundly absorbed in his problem, whileM'Nicholl was watching out of the window, and Ardan was busily employedin preparing breakfast.
The morning meal was not quite ready, when Barbican, raising his head,showed Ardan a page covered with algebraic signs at the end of whichstood the following formula:--
1 2 2 r m' r r--- (v' - v ) = gr {--- - 1 + --- (----- - -----) } 2 x m d - x d - r
"Which means?" asked Ardan.
"It means," said the Captain, now taking part in the discussion, "thatthe half of _v_ prime squared minus _v_ squared equals _gr_ multipliedby _r_ over _x_ minus one plus _m_ prime over _m_ multiplied by _r_ over_d_ minus _x_ minus _r_ over _d_ minus _r_ ... that is--"
"That is," interrupted Ardan, in a roar of laughter, "_x_ stradlegs on_y_, making for _z_ and jumping over
_p_! Do _you_ mean to say youunderstand the terrible jargon, Captain?"
"Nothing is clearer, Ardan."
"You too, Captain! Then of course I must give in gracefully, and declarethat the sun at noon-day is not more palpably evident than the sense ofBarbican's formula."
"You asked for Algebra, you know," observed Barbican.
"Rock crystal is nothing to it!"
"The fact is, Barbican," said the Captain, who had been looking over thepaper, "you have worked the thing out very well. You have the integralequation of the living forces, and I have no doubt it will give us theresult sought for."
"Yes, but I should like to understand it, you know," cried Ardan: "Iwould give ten years of the Captain's life to understand it!"
"Listen then," said Barbican. "Half of _v_ prime squared less _v_squared, is the formula giving us the half variation of the livingforce."
"Mac pretends he understands all that!"
"You need not be a _Solomon_ to do it," said the Captain. "All thesesigns that you appear to consider so cabalistic form a language theclearest, the shortest, and the most logical, for all those who can readit."
"You pretend, Captain, that, by means of these hieroglyphics, far moreincomprehensible than the sacred Ibis of the Egyptians, you candiscover the velocity at which the Projectile should start?"
"Most undoubtedly," replied the Captain, "and, by the same formula I caneven tell you the rate of our velocity at any particular point of ourjourney."
"You can?"
"I can."
"Then you're just as deep a one as our President."
"No, Ardan; not at all. The really difficult part of the questionBarbican has done. That is, to make out such an equation as takes intoaccount all the conditions of the problem. After that, it's a simpleaffair of Arithmetic, requiring only a knowledge of the four rules towork it out."
"Very simple," observed Ardan, who always got muddled at any kind of adifficult sum in addition.
"Captain," said Barbican, "_you_ could have found the formulas too, ifyou tried."
"I don't know about that," was the Captain's reply, "but I do know thatthis formula is wonderfully come at."
"Now, Ardan, listen a moment," said Barbican, "and you will see whatsense there is in all these letters."
"I listen," sighed Ardan with the resignation of a martyr.
"_d_ is the distance from the centre of the Earth to the centre of theMoon, for it is from the centres that we must calculate theattractions."
"That I comprehend."
"_r_ is the radius of the Earth."
"That I comprehend."
"_m_ is the mass or volume of the Earth; _m_ prime that of the Moon. Wemust take the mass of the two attracting bodies into consideration,since attraction is in direct proportion to their masses."
"That I comprehend."
"_g_ is the gravity or the velocity acquired at the end of a second by abody falling towards the centre of the Earth. Clear?"
"That I comprehend."
"Now I represent by _x_ the varying distance that separates theProjectile from the centre of the Earth, and by _v_ prime its velocityat that distance."
"That I comprehend."
"Finally, _v_ is its velocity when quitting our atmosphere."
"Yes," chimed in the Captain, "it is for this point, you see, that thevelocity had to be calculated, because we know already that the initialvelocity is exactly the three halves of the velocity when the Projectilequits the atmosphere."
"That I don't comprehend," cried the Frenchman, energetically.
"It's simple enough, however," said Barbican.
"Not so simple as a simpleton," replied the Frenchman.
"The Captain merely means," said Barbican, "that at the instant theProjectile quitted the terrestrial atmosphere it had already lost athird of its initial velocity."
"So much as a third?"
"Yes, by friction against the atmospheric layers: the quicker itsmotion, the greater resistance it encountered."
"That of course I admit, but your _v_ squared and your _v_ prime squaredrattle in my head like nails in a box!"
"The usual effect of Algebra on one who is a stranger to it; to finishyou, our next step is to express numerically the value of these severalsymbols. Now some of them are already known, and some are to becalculated."
"Hand the latter over to me," said the Captain.
"First," continued Barbican: "_r_, the Earth's radius is, in thelatitude of Florida, about 3,921 miles. _d_, the distance from thecentre of the Earth to the centre of the Moon is 56 terrestrial radii,which the Captain calculates to be...?"
"To be," cried M'Nicholl working rapidly with his pencil, "219,572miles, the moment the Moon is in her _perigee_, or nearest point to theEarth."
"Very well," continued Barbican. "Now _m_ prime over _m_, that is theratio of the Moon's mass to that of the Earth is about the 1/81. _g_gravity being at Florida about 32-1/4 feet, of course _g_ x _r_ mustbe--how much, Captain?"
"38,465 miles," replied M'Nicholl.
"Now then?" asked Ardan.
MY HEAD IS SPLITTING WITH IT.]
"Now then," replied Barbican, "the expression having numerical values, Iam trying to find _v_, that is to say, the initial velocity which theProjectile must possess in order to reach the point where the twoattractions neutralize each other. Here the velocity being null, _v_prime becomes zero, and _x_ the required distance of this neutral pointmust be represented by the nine-tenths of _d_, the distance between thetwo centres."
"I have a vague kind of idea that it must be so," said Ardan.
"I shall, therefore, have the following result;" continued Barbican,figuring up; "_x_ being nine-tenths of _d_, and _v_ prime being zero, myformula becomes:--
2 10 r 1 10 r rv = gr {1 - ----- - ---- (----- - -----) } d 81 d d - r "
The Captain read it off rapidly.
"Right! that's correct!" he cried.
"You think so?" asked Barbican.
"As true as Euclid!" exclaimed M'Nicholl.
"Wonderful fellows," murmured the Frenchman, smiling with admiration.
"You understand now, Ardan, don't you?" asked Barbican.
"Don't I though?" exclaimed Ardan, "why my head is splitting with it!"
"Therefore," continued Barbican,
" 2 10 r 1 10 r r2v = 2gr {1 - ----- - ---- (----- - -----) } d 81 d d - r "
"And now," exclaimed M'Nicholl, sharpening his pencil; "in order toobtain the velocity of the Projectile when leaving the atmosphere, wehave only to make a slight calculation."
The Captain, who before clerking on a Mississippi steamboat had beenprofessor of Mathematics in an Indiana university, felt quite at home atthe work. He rained figures from his pencil with a velocity that wouldhave made Marston stare. Page after page was filled with hismultiplications and divisions, while Barbican looked quietly on, andArdan impatiently stroked his head and ears to keep down a risinghead-ache.
"Well?" at last asked Barbican, seeing the Captain stop and throw asomewhat hasty glance over his work.
"Well," answered M'Nicholl slowly but confidently, "the calculation ismade, I think correctly; and _v_, that is, the velocity of theProjectile when quitting the atmosphere, sufficient to carry it to theneutral point, should be at least ..."
"How much?" asked Barbican, eagerly.
"Should be at least 11,972 yards the first second."
"What!" cried Barbican, jumping off his seat. "How much did you say?"
"11,972 yards the first second it quits the atmosphere."
"Oh, malediction!" cried Barbican, with a gesture of terrible despair.
"What's the matter?" asked Ardan, very much surprised.
"Enough is the matter!" answered Barbican excitedly. "This velocityhaving been diminished by a third, our initial velocity should have beenat least ..."
"17,958 yards the first second!" cried M'Nicholl, rapidly
flourishinghis pencil.
"But the Cambridge Observatory having declared that 12,000 yards thefirst second were sufficient, our Projectile started with no greatervelocity!"
"Well?" asked M'Nicholl.
"Well, such a velocity will never do!"
"How??" }"How!!" } cried the Captain and Ardan in one voice.
"We can never reach the neutral point!"
"Thunder and lightning"
"Fire and Fury!"
"We can't get even halfway!"
"Heaven and Earth!"
"_Mille noms d'un boulet!_" cried Ardan, wildly gesticulating.
"And we shall fall back to the Earth!"
"Oh!"
"Ah!"
They could say no more. This fearful revelation took them like a strokeof apoplexy.