marble, and tastefully decorated with a pigsty in each corner. Soldiers,
carrying pigs, were marching in all directions: and in the middle stood
a gigantic officer giving orders in a voice of thunder, which made
itself heard above all the uproar of the pigs.
"It is the Commander-in-Chief!" the Governor hurriedly whispered to his
companions, who at once followed his example in prostrating themselves
before the great man. The Commander gravely bowed in return. He was
covered with gold lace from head to foot: his face wore an expression of
deep misery: and he had a little black pig under each arm. Still the
gallant fellow did his best, in the midst of the orders he was every
moment issuing to his men, to bid a courteous farewell to the departing
guests.
"Farewell, oh old one--carry these three to the South corner--and
farewell to thee, thou young one--put this fat one on the top of the
others in the Western sty--may your shadows never be less--woe is me, it
is wrongly done! Empty out all the sties, and begin again!" And the
soldier leant upon his sword, and wiped away a tear.
"He is in distress," the Governor explained as they left the court. "Her
Radiancy has commanded him to place twenty-four pigs in those four
sties, so that, as she goes round the court, she may always find the
number in each sty nearer to ten than the number in the last."
"Does she call ten nearer to ten than nine is?" said Norman.
"Surely," said the Governor. "Her Radiancy would admit that ten is
nearer to ten than nine is--and also nearer than eleven is."
"Then I think it can be done," said Norman.
The Governor shook his head. "The Commander has been transferring them
in vain for four months," he said. "What hope remains? And Her Radiancy
has ordered up ten thousand additional----"
"The pigs don't seem to enjoy being transferred," the old man hastily
interrupted. He did not like the subject of bamboos.
"They are only _provisionally_ transferred, you know," said the
Governor. "In most cases they are immediately carried back again: so
they need not mind it. And all is done with the greatest care, under the
personal superintendence of the Commander-in-Chief."
"Of course she would only go _once_ round?" said Norman.
"Alas, no!" sighed their conductor. "Round and round. Round and round.
These are Her Radiancy's own words. But oh, agony! Here is the outer
gate, and we must part!" He sobbed as he shook hands with them, and the
next moment was briskly walking away.
"He _might_ have waited to see us off!" said the old man, piteously.
"And he needn't have begun whistling the very _moment_ he left us!" said
the young one, severely. "But look sharp--here are two what's-his-names
in the act of starting!"
Unluckily, the sea-bound omnibus was full. "Never mind!" said Norman,
cheerily. "We'll walk on till the next one overtakes us."
They trudged on in silence, both thinking over the military problem,
till they met an omnibus coming from the sea. The elder traveller took
out his watch. "Just twelve minutes and a half since we started," he
remarked in an absent manner. Suddenly the vacant face brightened; the
old man had an idea. "My boy!" he shouted, bringing his hand down upon
Norman's shoulder so suddenly as for a moment to transfer his centre of
gravity beyond the base of support.
Thus taken off his guard, the young man wildly staggered forwards, and
seemed about to plunge into space: but in another moment he had
gracefully recovered himself. "Problem in Precession and Nutation," he
remarked--in tones where filial respect only just managed to conceal a
shade of annoyance. "What is it?" he hastily added, fearing his father
might have been taken ill. "Will you have some brandy?"
"When will the next omnibus overtake us? When? When?" the old man cried,
growing more excited every moment.
Norman looked gloomy. "Give me time," he said. "I must think it over."
And once more the travellers passed on in silence--a silence only broken
by the distant squeals of the unfortunate little pigs, who were still
being provisionally transferred from sty to sty, under the personal
superintendence of the Commander-in-Chief.
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.
[Illustration: "HE REMAINS STEADFAST AND UNMOVED."]
FOOTNOTE:
[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:--
[Music: Chel-sea buns! Chel-sea buns hot! Chel-sea buns!
Pi-ping hot! Chel-sea buns hot! Chel-sea buns!]
"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
_piece 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!
"It's hardly fair," muttered Hugh, "to give us such a jumble as this to
work out!"
"Fair?" Clara echoed, bitterly. "Well!"
And to all my readers I can but repeat the last words of gentle Clara--
FARE-WELL!
APPENDIX.
"A knot!" said Alice. "Oh, do let me help to undo it!"
ANSWERS TO KNOT I.
_Problem._--"Two travellers spend from 3 o'clock till 9 in walking along
a level road, up a hill, and home again: their pace on the level being 4
miles an hour, up hill 3, and down hill 6. Find distance walked: also
(within half an hour) time of reaching top of hill."
_Answer._--"24 miles: half-past 6."
* * * * *
_Solution._--A level mile takes 1/4 of an hour, up hill 1/3, down hill
1/6. Hence to go and return over the same mile, whether on the level or
on the hill-side, takes 1/2 an hour. Hence in 6 hours they went 12 miles
out and 12 back. If the 12 miles out had been nearly all level, they
would have taken a little over 3 hours; if nearly all up hill, a little
under 4. Hence 3-1/2 hours must be within 1/2 an hour of the time taken
in reaching the peak; thus, as they started at 3, they got there within
1/2 an hour of 1/2 past 6.
* * * * *
Twenty-seven answers have come in. Of these, 9 are right, 16 partially
right, and 2 wrong. The 16 give the _distance_ correctly, but they have
failed to grasp the fact that the top of the hill might have been
reached at _any_ moment between 6 o'clock and 7.