Complete Works of Lewis Carroll (93 page)

KNOT IX.

A SERPENT WITH CORNERS.

"Water, water, every where, Nor any drop to drink."

"It'll just take one more pebble."

"What ever
are
you doing with those buckets?"

The speakers were Hugh and Lambert.
Place, the beach of Little Mendip.
Time, 1.30, P.M.
Hugh was floating a bucket in another a size larger, and trying how many pebbles it would carry without sinking.
Lambert was lying on his back, doing nothing.

For the next minute or two Hugh was silent, evidently deep in thought.
Suddenly he started.
"I say, look here, Lambert!"
he cried.

"If it's alive, and slimy, and with legs, I don't care to," said Lambert.

"Didn't Balbus say this morning that, if a body is immersed in liquid, it displaces as much liquid as is equal to its own bulk?"
said Hugh.

"He said things of that sort," Lambert vaguely replied.

"Well, just look here a minute.
Here's the little bucket almost quite immersed: so the water displaced ought to be just about the same bulk.
And now just look at it!"
He took out the little bucket as he spoke, and handed the big one to Lambert.
"Why, there's hardly a teacupful!
Do you mean to say
that
water is the same bulk as the little bucket?"

"Course it is," said Lambert.

"Well, look here again!"
cried Hugh, triumphantly, as he poured the water from the big bucket into the little one.
"Why, it doesn't half fill it!"

"That's
its
business," said Lambert.
"If Balbus says it's the same bulk, why, it
is
the same bulk, you know."

"Well, I don't believe it," said Hugh.

"You needn't," said Lambert.
"Besides, it's dinner-time.
Come along."

They found Balbus waiting dinner for them, and to him Hugh at once propounded his difficulty.

"Let's get you helped first," said Balbus, briskly cutting away at the joint.
"You know the old proverb 'Mutton first, mechanics afterwards'?"

The boys did
not
know the proverb, but they accepted it in perfect good faith, as they did every piece of information, however startling, that came from so infallible an authority as their tutor.
They ate on steadily in silence, and, when dinner was over, Hugh set out the usual array of pens, ink, and paper, while Balbus repeated to them the problem he had prepared for their afternoon's task.

"A friend of mine has a flower-garden—a very pretty one, though no great size—"

"How big is it?"
said Hugh.

"That's what
you
have to find out!"
Balbus gaily replied.
"All
I
tell you is that it is oblong in shape—just half a yard longer than its width—and that a gravel-walk, one yard wide, begins at one corner and runs all round it."

"Joining into itself?"
said Hugh.

"
Not
joining into itself, young man.
Just before doing
that
, it turns a corner, and runs round the garden again, alongside of the first portion, and then inside that again, winding in and in, and each lap touching the last one, till it has used up the whole of the area."

"Like a serpent with corners?"
said Lambert.

"Exactly so.
And if you walk the whole length of it, to the last inch, keeping in the centre of the path, it's exactly two miles and half a furlong.
Now, while you find out the length and breadth of the garden, I'll see if I can think out that sea-water puzzle."

"You said it was a flower-garden?"
Hugh inquired, as Balbus was leaving the room.

"I did," said Balbus.

"Where do the flowers grow?"
said Hugh.
But Balbus thought it best not to hear the question.
He left the boys to their problem, and, in the silence of his own room, set himself to unravel Hugh's mechanical paradox.

"To fix our thoughts," he murmured to himself, as, with hands deep-buried in his pockets, he paced up and down the room, "we will take a cylindrical glass jar, with a scale of inches marked up the side, and fill it with water up to the 10-inch mark: and we will assume that every inch depth of jar contains a pint of water.
We will now take a solid cylinder, such that every inch of it is equal in bulk to
half
a pint of water, and plunge 4 inches of it into the water, so that the end of the cylinder comes down to the 6-inch mark.
Well, that displaces 2 pints of water.
What becomes of them?
Why, if there were no more cylinder, they would lie comfortably on the top, and fill the jar up to the 12-inch mark.
But unfortunately there
is
more cylinder, occupying half the space between the 10-inch and the 12-inch marks, so that only
one
pint of water can be accommodated there.
What becomes of the other pint?
Why, if there were no more cylinder, it would lie on the top, and fill the jar up to the 13-inch mark.
But unfortunately——Shade of Newton!"
he exclaimed, in sudden accents of terror.
"When
does
the water stop rising?"

A bright idea struck him.
"I'll write a little essay on it," he said.

Balbus's Essay.

"When a solid is immersed in a liquid, it is well known that it displaces a portion of the liquid equal to itself in bulk, and that the level of the liquid rises just so much as it would rise if a quantity of liquid had been added to it, equal in bulk to the solid.
Lardner says, precisely the same process occurs when a solid is
partially
immersed: the quantity of liquid displaced, in this case, equalling the portion of the solid which is immersed, and the rise of the level being in proportion.

"Suppose a solid held above the surface of a liquid and partially immersed: a portion of the liquid is displaced, and the level of the liquid rises.
But, by this rise of level, a little bit more of the solid is of course immersed, and so there is a new displacement of a second portion of the liquid, and a consequent rise of level.
Again, this second rise of level causes a yet further immersion, and by consequence another displacement of liquid and another rise.
It is self-evident that this process must continue till the entire solid is immersed, and that the liquid will then begin to immerse whatever holds the solid, which, being connected with it, must for the time be considered a part of it.
If you hold a stick, six feet long, with its end in a tumbler of water, and wait long enough, you must eventually be immersed.
The question as to the source from which the water is supplied—which belongs to a high branch of mathematics, and is therefore beyond our present scope—does not apply to the sea.
Let us therefore take the familiar instance of a man standing at the edge of the sea, at ebb-tide, with a solid in his hand, which he partially immerses: he remains steadfast and unmoved, and we all know that he must be drowned.
The multitudes who daily perish in this manner to attest a philosophical truth, and whose bodies the unreasoning wave casts sullenly upon our thankless shores, have a truer claim to be called the martyrs of science than a Galileo or a Kepler.
To use Kossuth's eloquent phrase, they are the unnamed demigods of the nineteenth century."[B]

"There's a fallacy
somewhere
," he murmured drowsily, as he stretched his long legs upon the sofa.
"I must think it over again."
He closed his eyes, in order to concentrate his attention more perfectly, and for the next hour or so his slow and regular breathing bore witness to the careful deliberation with which he was investigating this new and perplexing view of the subject.

"HE REMAINS STEADFAST AND UNMOVED."

FOOTNOTE:

[B]
Note by the writer.
—For the above Essay I am indebted to a dear friend, now deceased.

KNOT X.

CHELSEA BUNS.

"Yea, buns, and buns, and buns!"

Old Song.

"How very, very sad!"
exclaimed Clara; and the eyes of the gentle girl filled with tears as she spoke.

"Sad—but very curious when you come to look at it arithmetically," was her aunt's less romantic reply.
"Some of them have lost an arm in their country's service, some a leg, some an ear, some an eye——"

"And some, perhaps,
all
!"
Clara murmured dreamily, as they passed the long rows of weather-beaten heroes basking in the sun.
"Did you notice that very old one, with a red face, who was drawing a map in the dust with his wooden leg, and all the others watching?
I
think
it was a plan of a battle——"

"The battle of Trafalgar, no doubt," her aunt interrupted, briskly.

"Hardly that, I think," Clara ventured to say.
"You see, in that case, he couldn't well be alive——"

"Couldn't well be alive!"
the old lady contemptuously repeated.
"He's as lively as you and me put together!
Why, if drawing a map in the dust—with one's wooden leg—doesn't prove one to be alive, perhaps you'll kindly mention what
does
prove it!"

Clara did not see her way out of it.
Logic had never been her
forte
.

"To return to the arithmetic," Mad Mathesis resumed—the eccentric old lady never let slip an opportunity of driving her niece into a calculation—"what percentage do you suppose must have lost all four—a leg, an arm, an eye, and an ear?"

"How
can
I tell?"
gasped the terrified girl.
She knew well what was coming.

"You can't, of course, without
data
," her aunt replied: "but I'm just going to give you——"

"Give her a Chelsea bun, Miss!
That's what most young ladies likes best!"
The voice was rich and musical, and the speaker dexterously whipped back the snowy cloth that covered his basket, and disclosed a tempting array of the familiar square buns, joined together in rows, richly egged and browned, and glistening in the sun.

"No, sir!
I shall give her nothing so indigestible!
Be off!"
The old lady waved her parasol threateningly: but nothing seemed to disturb the good-humour of the jolly old man, who marched on, chanting his melodious refrain:—

Listen

"Far too indigestible, my love!"
said the old lady.
"Percentages will agree with you ever so much better!"

Clara sighed, and there was a hungry look in her eyes as she watched the basket lessening in the distance: but she meekly listened to the relentless old lady, who at once proceeded to count off the
data
on her fingers.

"Say that 70 per cent.
have lost an eye—75 per cent.
an ear—80 per cent.
an arm—85 per cent.
a leg—that'll do it beautifully.
Now, my dear, what percentage,
at least
, must have lost all four?"

No more conversation occurred—unless a smothered exclamation of "Piping hot!"
which escaped from Clara's lips as the basket vanished round a corner could be counted as such—until they reached the old Chelsea mansion, where Clara's father was then staying, with his three sons and their old tutor.

Balbus, Lambert, and Hugh had entered the house only a few minutes before them.
They had been out walking, and Hugh had been propounding a difficulty which had reduced Lambert to the depths of gloom, and had even puzzled Balbus.

"It changes from Wednesday to Thursday at midnight, doesn't it?"
Hugh had begun.

"Sometimes," said Balbus, cautiously.

"Always," said Lambert, decisively.

"
Sometimes
," Balbus gently insisted.
"Six midnights out of seven, it changes to some other name."

"I meant, of course," Hugh corrected himself, "when it
does
change from Wednesday to Thursday, it does it at midnight—and
only
at midnight."

"Surely," said Balbus.
Lambert was silent.

"Well, now, suppose it's midnight here in Chelsea.
Then it's Wednesday
west
of Chelsea (say in Ireland or America) where midnight hasn't arrived yet: and it's Thursday
east
of Chelsea (say in Germany or Russia) where midnight has just passed by?"

"Surely," Balbus said again.
Even Lambert nodded this time.

"But it isn't midnight, anywhere else; so it can't be changing from one day to another anywhere else.
And yet, if Ireland and America and so on call it Wednesday, and Germany and Russia and so on call it Thursday, there
must
be some place—not Chelsea—that has different days on the two sides of it.
And the worst of it is, the people
there
get their days in the wrong order: they've got Wednesday
east
of them, and Thursday
west
—just as if their day had changed from Thursday to Wednesday!"

"I've heard that puzzle before!"
cried Lambert.
"And I'll tell you the explanation.
When a ship goes round the world from east to west, we know that it loses a day in its reckoning: so that when it gets home, and calls its day Wednesday, it finds people here calling it Thursday, because we've had one more midnight than the ship has had.
And when you go the other way round you gain a day."

"I know all that," said Hugh, in reply to this not very lucid explanation: "but it doesn't help me, because the ship hasn't proper days.
One way round, you get more than twenty-four hours to the day, and the other way you get less: so of course the names get wrong: but people that live on in one place always get twenty-four hours to the day."

"I suppose there
is
such a place," Balbus said, meditatively, "though I never heard of it.
And the people must find it very queer, as Hugh says, to have the old day
east
of them, and the new one
west
: because, when midnight comes round to them, with the new day in front of it and the old one behind it, one doesn't see exactly what happens.
I must think it over."

So they had entered the house in the state I have described—Balbus puzzled, and Lambert buried in gloomy thought.

"Yes, m'm, Master
is
at home, m'm," said the stately old butler.
(N.B.—It is only a butler of experience who can manage a series of three M's together, without any interjacent vowels.) "And the
ole
party is a-waiting for you in the libery."

"I don't like his calling your father an
old
party," Mad Mathesis whispered to her niece, as they crossed the hall.
And Clara had only just time to whisper in reply "he meant the
whole
party," before they were ushered into the library, and the sight of the five solemn faces there assembled chilled her into silence.

Her father sat at the head of the table, and mutely signed to the ladies to take the two vacant chairs, one on each side of him.
His three sons and Balbus completed the party.
Writing materials had been arranged round the table, after the fashion of a ghostly banquet: the butler had evidently bestowed much thought on the grim device.
Sheets of quarto paper, each flanked by a pen on one side and a pencil on the other, represented the plates—penwipers did duty for rolls of bread—while ink-bottles stood in the places usually occupied by wine-glasses.
The
pièce de resistance
was a large green baize bag, which gave forth, as the old man restlessly lifted it from side to side, a charming jingle, as of innumerable golden guineas.

"Sister, daughter, sons—and Balbus—," the old man began, so nervously, that Balbus put in a gentle "Hear, hear!"
while Hugh drummed on the table with his fists.
This disconcerted the unpractised orator.
"Sister—" he began again, then paused a moment, moved the bag to the other side, and went on with a rush, "I mean—this being—a critical occasion—more or less—being the year when one of my sons comes of age—" he paused again in some confusion, having evidently got into the middle of his speech sooner than he intended: but it was too late to go back.
"Hear, hear!"
cried Balbus.
"Quite so," said the old gentleman, recovering his self-possession a little: "when first I began this annual custom—my friend Balbus will correct me if I am wrong—" (Hugh whispered "with a strap!"
but nobody heard him except Lambert, who only frowned and shook his head at him) "—this annual custom of giving each of my sons as many guineas as would represent his age—it was a critical time—so Balbus informed me—as the ages of two of you were together equal to that of the third—so on that occasion I made a speech——" He paused so long that Balbus thought it well to come to the rescue with the words "It was a most——" but the old man checked him with a warning look: "yes, made a speech," he repeated.
"A few years after that, Balbus pointed out—I say pointed out—" ("Hear, hear"!
cried Balbus.
"Quite so," said the grateful old man.) "—that it was
another
critical occasion.
The ages of two of you were together
double
that of the third.
So I made another speech—another speech.
And now again it's a critical occasion—so Balbus says—and I am making——" (Here Mad Mathesis pointedly referred to her watch) "all the haste I can!"
the old man cried, with wonderful presence of mind.
"Indeed, sister, I'm coming to the point now!
The number of years that have passed since that first occasion is just two-thirds of the number of guineas I then gave you.
Now, my boys, calculate your ages from the
data
, and you shall have the money!"

"But we
know
our ages!"
cried Hugh.

"Silence, sir!"
thundered the old man, rising to his full height (he was exactly five-foot five) in his indignation.
"I say you must use the
data
only!
You mustn't even assume
which
it is that comes of age!"
He clutched the bag as he spoke, and with tottering steps (it was about as much as he could do to carry it) he left the room.

"And
you
shall have a similar
cadeau
," the old lady whispered to her niece, "when you've calculated that percentage!"
And she followed her brother.

Nothing could exceed the solemnity with which the old couple had risen from the table, and yet was it—was it a
grin
with which the father turned away from his unhappy sons?
Could it be—could it be a
wink
with which the aunt abandoned her despairing niece?
And were those—were those sounds of suppressed
chuckling
which floated into the room, just before Balbus (who had followed them out) closed the door?
Surely not: and yet the butler told the cook—but no, that was merely idle gossip, and I will not repeat it.

The shades of evening granted their unuttered petition, and "closed not o'er" them (for the butler brought in the lamp): the same obliging shades left them a "lonely bark" (the wail of a dog, in the back-yard, baying the moon) for "awhile": but neither "morn, alas," (nor any other epoch) seemed likely to "restore" them—to that peace of mind which had once been theirs ere ever these problems had swooped upon them, and crushed them with a load of unfathomable mystery!