"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 gla.s.s jar, with a scale of inches marked up the side, and fill it with water up to the 10-inch mark: and we will a.s.sume 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 s.p.a.ce 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 quant.i.ty 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 quant.i.ty 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 mult.i.tudes who daily perish in this manner to attest a philosophical truth, and whose bodies the unreasoning wave casts sullenly upon our thankless sh.o.r.es, 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 demiG.o.ds 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.

[Ill.u.s.tration: "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 pa.s.sed 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 pa.s.sed by?"

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