FOOTNOTE:

"One piecee thing that my have got,Maskee[A]that thing my no can do.You talkee you no sabey what?Bamboo."

They landed, and were at once conducted to the Palace. About half way they were met by the Governor, who welcomed them in English—a great relief to our travellers, whose guide could speak nothing but Kgovjnian.

"I don't half like the way they grin at us as we go by!" the old man whispered to his son. "And why do they say 'Bamboo!' so often?"

"It alludes to a local custom," replied the Governor, who had overheard the question. "Such persons as happen in any way to displease Her Radiancy are usually beaten with rods."

"WHY DO THEY SAY 'BAMBOO!' SO OFTEN?""WHY DO THEY SAY 'BAMBOO!' SO OFTEN?"

The old man shuddered. "A most objectional local custom!" he remarked with strong emphasis. "I wish we had never landed! Did you notice that black fellow, Norman, opening his great mouth at us? I verily believe he would like to eat us!"

Norman appealed to the Governor, who was walking at his other side. "Do they often eat distinguished strangers here?" he said, in as indifferent a tone as he could assume.

"Not often—not ever!" was the welcome reply. "They are not good for it. Pigs we eat, for they are fat. This old man is thin."

"And thankful to be so!" muttered the elder traveller. "Beaten we shall be without a doubt. It's a comfort to know it won't be Beaten without the B! My dear boy, just look at the peacocks!"

They were now walking between two unbroken lines of those gorgeous birds, each held in check, by means of a golden collar and chain, by a black slave, who stood well behind, so as not to interrupt the view of the glittering tail, with its network of rustling feathers and its hundred eyes.

The Governor smiled proudly. "In your honour," he said, "Her Radiancy has ordered up ten thousand additional peacocks. She will, no doubt,decorate you, before you go, with the usual Star and Feathers."

"It'll be Star without the S!" faltered one of his hearers.

"Come, come! Don't lose heart!" said the other. "All this is full of charm for me."

"You are young, Norman," sighed his father; "young and light-hearted. For me, it is Charm without the C."

"The old one is sad," the Governor remarked with some anxiety. "He has, without doubt, effected some fearful crime?"

"But I haven't!" the poor old gentleman hastily exclaimed. "Tell him I haven't, Norman!"

"He has not, as yet," Norman gently explained. And the Governor repeated, in a satisfied tone, "Not as yet."

"Yours is a wondrous country!" the Governor resumed, after a pause. "Now here is a letter from a friend of mine, a merchant, in London. He and his brother went there a year ago, with a thousand pounds apiece; and on New-Year's-day they had sixty thousand pounds between them!"

"How did they do it?" Norman eagerly exclaimed. Even the elder traveller looked excited.

The Governor handed him the open letter. "Anybody can do it, when once they know how," so ran this oracular document. "We borrowed nought: we stole nought. We began the year with only a thousand pounds apiece: and last New-Year's-day we had sixty thousand pounds between us—sixty thousand golden sovereigns!"

Norman looked grave and thoughtful as he handed back the letter. His father hazarded one guess. "Was it by gambling?"

"A Kgovjnian never gambles," said the Governor gravely, as he ushered them through the palace gates. They followed him in silence down a long passage, and soon found themselves in a lofty hall, lined entirely with peacocks' feathers. In the centre was a pile of crimson cushions, which almost concealed the figure of Her Radiancy—a plump little damsel, in a robe of green satin dotted with silver stars, whose pale round face lit up for a moment with a half-smile as the travellers bowed before her, and then relapsed into the exact expression of a wax doll, while she languidly murmured a word or two in the Kgovjnian dialect.

The Governor interpreted. "Her Radiancy welcomes you. She notes the Impenetrable Placidityof the old one, and the Imperceptible Acuteness of the youth."

Here the little potentate clapped her hands, and a troop of slaves instantly appeared, carrying trays of coffee and sweetmeats, which they offered to the guests, who had, at a signal from the Governor, seated themselves on the carpet.

"Sugar-plums!" muttered the old man. "One might as well be at a confectioner's! Ask for a penny bun, Norman!"

"Not so loud!" his son whispered. "Say something complimentary!" For the Governor was evidently expecting a speech.

"We thank Her Exalted Potency," the old man timidly began. "We bask in the light of her smile, which——"

"The words of old men are weak!" the Governor interrupted angrily. "Let the youth speak!"

"Tell her," cried Norman, in a wild burst of eloquence, "that, like two grasshoppers in a volcano, we are shrivelled up in the presence of Her Spangled Vehemence!"

"It is well," said the Governor, and translated this into Kgovjnian. "I am now to tell you," he proceeded, "what Her Radiancy requires of youbefore you go. The yearly competition for the post of Imperial Scarf-maker is just ended; you are the judges. You will take account of the rate of work, the lightness of the scarves, and their warmth. Usually the competitors differ in one point only. Thus, last year, Fifi and Gogo made the same number of scarves in the trial-week, and they were equally light; but Fifi's were twice as warm as Gogo's and she was pronounced twice as good. But this year, woe is me, who can judge it? Three competitors are here, and they differ in all points! While you settle their claims, you shall be lodged, Her Radiancy bids me say, free of expense—in the best dungeon, and abundantly fed on the best bread and water."

The old man groaned. "All is lost!" he wildly exclaimed. But Norman heeded him not: he had taken out his note-book, and was calmly jotting down the particulars.

"Three they be," the Governor proceeded, "Lolo, Mimi, and Zuzu. Lolo makes 5 scarves while Mimi makes 2; but Zuzu makes 4 while Lolo makes 3! Again, so fairylike is Zuzu's handiwork, 5 of her scarves weigh no more than one of Lolo's; yet Mimi's is lighter still—5 of hers will but balance3 of Zuzu's! And for warmth one of Mimi's is equal to 4 of Zuzu's; yet one of Lolo's is as warm as 3 of Mimi's!"

Here the little lady once more clapped her hands.

"It is our signal of dismissal!" the Governor hastily said. "Pay Her Radiancy your farewell compliments—and walk out backwards."

The walking part was all the elder tourist could manage. Norman simply said "Tell Her Radiancy we are transfixed by the spectacle of Her Serene Brilliance, and bid an agonized farewell to her Condensed Milkiness!"

"Her Radiancy is pleased," the Governor reported, after duly translating this. "She casts on you a glance from Her Imperial Eyes, and is confident that you will catch it!"

"That I warrant we shall!" the elder traveller moaned to himself distractedly.

Once more they bowed low, and then followed the Governor down a winding staircase to the Imperial Dungeon, which they found to be lined with coloured marble, lighted from the roof, and splendidly though not luxuriously furnished with a bench of polished malachite. "I trust you willnot delay the calculation," the Governor said, ushering them in with much ceremony. "I have known great inconvenience—great and serious inconvenience—result to those unhappy ones who have delayed to execute the commands of Her Radiancy! And on this occasion she is resolute: she says the thing must and shall be done: and she has ordered up ten thousand additional bamboos!" With these words he left them, and they heard him lock and bar the door on the outside.

"I told you how it would end!" moaned the elder traveller, wringing his hands, and quite forgetting in his anguish that he had himself proposed the expedition, and had never predicted anything of the sort. "Oh that we were well out of this miserable business!"

"Courage!" cried the younger cheerily. "Hæc olim meminisse juvabit!The end of all this will be glory!"

"Glory without the L!" was all the poor old man could say, as he rocked himself to and fro on the malachite bench. "Glory without the L!"

FOOTNOTE:[A]"Maskee," in Pigeon-English, means "without."

[A]"Maskee," in Pigeon-English, means "without."

"Base is the slave that pays."

"Aunt Mattie!"

"My child?"

"Wouldyou mind writing it down at once? I shall be quitecertainto forget it if you don't!"

"My dear, we really must wait till the cab stops. How can I possibly write anything in the midst of all this jolting?"

"ButreallyI shall be forgetting it!"

Clara's voice took the plaintive tone that her aunt never knew how to resist, and with a sigh the old lady drew forth her ivory tablets and prepared to record the amount that Clara had just spent at the confectioner's shop. Her expenditure was always made out of her aunt's purse, but the poor girl knew, by bitter experience, that sooner or later"Mad Mathesis" would expect an exact account of every penny that had gone, and she waited, with ill-concealed impatience, while the old lady turned the tablets over and over, till she had found the one headed "PETTY CASH."

"Here's the place," she said at last, "and here we have yesterday's luncheon duly entered.One glass lemonade(Why can't you drink water, like me?)three sandwiches(They never put in half mustard enough. I told the young woman so, to her face; and she tossed her head—like her impudence!)and seven biscuits.Total one-and-two-pence.Well, now for to-day's?"

"One glass of lemonade——" Clara was beginning to say, when suddenly the cab drew up, and a courteous railway-porter was handing out the bewildered girl before she had had time to finish her sentence.

Her aunt pocketed the tablets instantly. "Business first," she said: "petty cash—which is a form of pleasure, whateveryoumay think—afterwards." And she proceeded to pay the driver, and to give voluminous orders about the luggage, quite deaf to the entreaties of her unhappy niece that she would enter the rest of the luncheon account."My dear, you really must cultivate a more capacious mind!" was all the consolation she vouchsafed to the poor girl. "Are not the tablets of your memory wide enough to contain the record of one single luncheon?"

"Not wide enough! Not half wide enough!" was the passionate reply.

The words came in aptly enough, but the voice was not that of Clara, and both ladies turned in some surprise to see who it was that had so suddenly struck into their conversation. A fat little old lady was standing at the door of a cab, helping the driver to extricate what seemed an exact duplicate of herself: it would have been no easy task to decide which was the fatter, or which looked the more good-humoured of the two sisters.

"I tell you the cab-door isn't half wide enough!" she repeated, as her sister finally emerged, somewhat after the fashion of a pellet from a pop-gun, and she turned to appeal to Clara. "Is it, dear?" she said, trying hard to bring a frown into a face that dimpled all over with smiles.

"Some folks is too wide for 'em," growled the cab-driver.

"I TELL YOU THE CAB-DOOR ISN'T HALF WIDE ENOUGH!""I TELL YOU THE CAB-DOOR ISN'T HALF WIDE ENOUGH!"

"Don't provoke me, man!" cried the little oldlady, in what she meant for a tempest of fury. "Say another word and I'll put you into the County Court, and sue you for aHabeas Corpus!" The cabman touched his hat, and marched off, grinning.

"Nothing like a little Law to cow the ruffians, my dear!" she remarked confidentially to Clara. "You saw how he quailed when I mentioned theHabeas Corpus? Not that I've any idea what it means, but it sounds very grand, doesn't it?"

"It's very provoking," Clara replied, a little vaguely.

"Very!" the little old lady eagerly repeated. "And we're very much provoked indeed. Aren't we, sister?"

"I never was so provoked in all my life!" the fatter sister assented, radiantly.

By this time Clara had recognised her picture-gallery acquaintances, and, drawing her aunt aside, she hastily whispered her reminiscences. "I met them first in the Royal Academy—and they were very kind to me—and they were lunching at the next table to us, just now, you know—and they tried to help me to find the picture I wanted—and I'm sure they're dear old things!"

"Friends of yours, are they?" said Mad Mathesis. "Well, I like their looks. You can be civil to them, while I get the tickets. But do try and arrange your ideas a little more chronologically!"

And so it came to pass that the four ladies found themselves seated side by side on the same bench waiting for the train, and chatting as if they had known one another for years.

"Now this I call quite a remarkable coincidence!" exclaimed the smaller and more talkative of the two sisters—the one whose legal knowledge had annihilated the cab-driver. "Not only that we should be waiting for the same train, and at the same station—thatwould be curious enough—but actually on the same day, and the same hour of the day! That's what strikesmeso forcibly!" She glanced at the fatter and more silent sister, whose chief function in life seemed to be to support the family opinion, and who meekly responded—

"And me too, sister!"

"Those are notindependentcoincidences——" Mad Mathesis was just beginning, when Clara ventured to interpose.

"There's no jolting here," she pleaded meekly. "Wouldyou mind writing it down now?"

Out came the ivory tablets once more. "What was it, then?" said her aunt.

"One glass of lemonade, one sandwich, one biscuit—Oh dear me!" cried poor Clara, the historical tone suddenly changing to a wail of agony.

"Toothache?" said her aunt calmly, as she wrote down the items. The two sisters instantly opened their reticules and produced two different remedies for neuralgia, each marked "unequalled."

"It isn't that!" said poor Clara. "Thank you very much. It's only that Ican'tremember how much I paid!"

"Well, try and make it out, then," said her aunt. "You've got yesterday's luncheon to help you, you know. And here's the luncheon we had the day before—the first day we went to that shop—one glass lemonade,four sandwiches,ten biscuits.Total, one-and-fivepence." She handed the tablets to Clara, who gazed at them with eyes so dim with tears that she did not at first notice that she was holding them upside down.

The two sisters had been listening to all thiswith the deepest interest, and at this juncture the smaller one softly laid her hand on Clara's arm.

"Do you know, my dear," she said coaxingly, "my sister and I are in the very same predicament! Quite identically the very same predicament! Aren't we, sister?"

"Quite identically and absolutely the very——" began the fatter sister, but she was constructing her sentence on too large a scale, and the little one would not wait for her to finish it.

"Yes, my dear," she resumed; "we were lunching at the very same shop as you were—and we had two glasses of lemonade and three sandwiches and five biscuits—and neither of us has the least idea what we paid. Have we, sister?"

"Quite identically and absolutely——" murmured the other, who evidently considered that she was now a whole sentence in arrears, and that she ought to discharge one obligation before contracting any fresh liabilities; but the little lady broke in again, and she retired from the conversation a bankrupt.

"Wouldyou make it out for us, my dear?" pleaded the little old lady.

"You can do Arithmetic, I trust?" her aunt said, a little anxiously, as Clara turned from one tablet to another, vainly trying to collect her thoughts. Her mind was a blank, and all human expression was rapidly fading out of her face.

A gloomy silence ensued.

"This little pig went to market:This little pig staid at home."

"By Her Radiancy's express command," said the Governor, as he conducted the travellers, for the last time, from the Imperial presence, "I shall now have the ecstasy of escorting you as far as the outer gate of the Military Quarter, where the agony of parting—if indeed Nature can survive the shock—must be endured! From that gate grurmstipths start every quarter of an hour, both ways——"

"Would you mind repeating that word?" said Norman. "Grurm——?"

"Grurmstipths," the Governor repeated. "You call them omnibuses in England. They run both ways, and you can travel by one of them all the way down to the harbour."

The old man breathed a sigh of relief; four hours of courtly ceremony had wearied him, and he had been in constant terror lest something should call into use the ten thousand additional bamboos.

In another minute they were crossing a large quadrangle, paved with 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 theSouth 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 onlyprovisionallytransferred, 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 goonceround?" 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.

"Hemighthave waited to see us off!" said the old man, piteously.

"And he needn't have begun whistling the verymomenthe 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," hesaid. "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.

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

"It'll just take one more pebble."

"What everareyou 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 bodyis 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 saythatwater 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'sitsbusiness," said Lambert. "If Balbus says it's the same bulk, why, itisthe 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, brisklycutting away at the joint. "You know the old proverb 'Mutton first, mechanics afterwards'?"

The boys didnotknow 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 whatyouhave to find out!" Balbus gaily replied. "AllItell 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.

"Notjoining into itself, young man. Just before doingthat, 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 tohalfa pint of water, and plunge 4 inches of it into the water, so that the end of the cylindercomes 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 thereismore cylinder, occupying half the space between the 10-inch and the 12-inch marks, so that onlyonepint 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. "Whendoesthe 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 inbulk to the solid. Lardner says, precisely the same process occurs when a solid ispartiallyimmersed: 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 applyto 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 fallacysomewhere," 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.""HE REMAINS STEADFAST AND UNMOVED."

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

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

"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 woodenleg, and all the others watching? Ithinkit 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 whatdoesprove it!"

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

"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?"

"HowcanI tell?" gasped the terrified girl. She knew well what was coming.

"You can't, of course, withoutdata," her aunt replied: "but I'm just going to give you——"

"Give her a Chelsea bun, Miss! That's whatmost 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 melodiousrefrain:—

Chel-sea

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 relentlessold lady, who at once proceeded to count off thedataon 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 itdoeschange from Wednesday to Thursday, it does it at midnight—andonlyat midnight."

"Surely," said Balbus. Lambert was silent.

"Well, now, suppose it's midnight here in Chelsea. Then it's Wednesdaywestof Chelsea (say in Ireland or America) where midnight hasn't arrived yet: and it's Thursdayeastof 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, theremustbe some place—not Chelsea—that has different days on the two sides of it. And the worst of it is, the peoplethereget their days in the wrong order: they've got Wednesdayeastof them, and Thursdaywest—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 shipgoes 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 thereissuch 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 dayeastof them, and the new onewest: 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, Masterisat home, m'm," said thestately 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 theoleparty is a-waiting for you in the libery."

"I don't like his calling your father anoldparty," Mad Mathesis whispered to her niece, as they crossed the hall. And Clara had only just time to whisper in reply "he meant thewholeparty," 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. Thepièce de resistancewas a large green baize bag, which gave forth, as the old man restlessly lifted it from sideto 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—soon 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 wasanothercritical occasion. The ages of two of you were togetherdoublethat 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 thedata, and you shall have the money!"

"But weknowour 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 thedataonly! You mustn't even assumewhichit 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.

"Andyoushall have a similarcadeau," 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 agrinwith which the father turned away from his unhappy sons? Could it be—could it be awinkwith which the aunt abandoned her despairing niece? And were those—were those sounds of suppressedchucklingwhich 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!

"A knot!" said Alice. "Oh, do let me help to undo it!"

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 ¼ of an hour, up hill1⁄3, down hill1⁄6. Hence to go and return over the same mile, whether on the level or on the hill-side, takes ½ 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 3½ hours must be within ½ an hour of the time taken in reaching the peak; thus, as they started at 3, they got there within ½ an hour of ½ past 6.

Twenty-seven answers have come in. Of these, 9 are right, 16 partially right, and 2 wrong. The 16 give thedistancecorrectly, but they have failed to grasp the fact that the top of the hill might have been reached atanymoment between 6 o'clock and 7.

The two wrong answers are fromGerty VernonandA Nihilist. The former makes the distance "23 miles," while her revolutionary companion puts it at "27."Gerty Vernonsays "they had to go 4 miles along the plain, and got to the foot of the hill at 4 o'clock." Theymighthave done so, I grant; but you have no ground for saying theydidso. "It was 7½ miles to the top of the hill, and they reached that at ¼ before 7 o'clock." Here you go wrong in your arithmetic, and I must, however reluctantly, bid you farewell. 7½ miles, at 3 miles an hour, wouldnotrequire 2¾ hours.A Nihilistsays "Letxdenote the whole number of miles;ythe number of hours to hill-top; ∴ 3y= number of miles to hill-top, andx-3y= number of miles on the other side." You bewilder me. The other side ofwhat? "Of the hill," you say. But then, how did they get home again? However, to accommodate your views we will build a new hostelry at the foot of the hill on the opposite side, and also assume (what I grant you ispossible, though it is notnecessarilytrue) that there was no level road at all. Even then you go wrong.

You say

"y= 6 -(x- 3y)⁄6, ..... (i);x⁄4½= 6 ..... (ii)."

I grant you (i), but I deny (ii): it rests on the assumption that to gopartof the time at 3 miles an hour, and the rest at 6 miles an hour, comes to the same result as going thewholetime at 4½ miles an hour. But this would only be true if the "part" were an exacthalf, i.e., if they went up hill for 3 hours, and down hill for the other 3: which they certainly didnotdo.

The sixteen, who are partially right, areAgnes Bailey,F. K., Fifee, G. E. B., H. P.,Kit, M. E. T.,Mysie,A Mother's Son,Nairam,A Redruthian,A Socialist,Spear Maiden, T. B. C.,Vis Inertiæ, andYak. Of these,F. K., Fifee, T. B. C., andVis Inertiædo not attempt the second part at all. F. K. and H. P. give no working. The rest make particular assumptions, such as that there was no level road—that there were 6 miles of level road—and so on, all leading toparticulartimes being fixed for reaching the hill-top. The most curious assumption is that ofAgnes Bailey, who says "Letx= number of hours occupied in ascent; thenx⁄2= hours occupied in descent; and4x⁄3= hours occupied on thelevel." I suppose you were thinking of the relativerates, up hill and on the level; which we might express by saying that, if they wentxmiles up hill in a certain time, they would go4x⁄3miles on the levelin the same time. You have, in fact, assumed that they tookthe same timeon the level that they took in ascending the hill.Fifeeassumes that, when the aged knight said they had gone "four miles in the hour" on the level, he meant that four miles was thedistancegone, not merely the rate. This would have been—ifFifeewill excuse the slang expression—a "sell," ill-suited to the dignity of the hero.

And now "descend, ye classic Nine!" who have solved the whole problem, and let me sing your praises. Your names areBlithe, E. W., L. B.,A Marlborough Boy, O. V. L.,Putney Walker,Rose,Sea Breeze,Simple Susan, andMoney Spinner. (These last two I count as one, as they send a joint answer.)RoseandSimple SusanandCo. do not actually state that the hill-top was reached some time between 6 and 7, but, as they have clearly grasped the fact that a mile, ascended and descended, took the same time as two level miles, I mark them as "right."A Marlborough BoyandPutney Walkerdeserve honourable mention for their algebraical solutions being the only two who have perceivedthat the question leads toan indeterminate equation. E. W. brings a charge of untruthfulness against the aged knight—a serious charge, for he was the very pink of chivalry! She says "According to the data given, the time at the summit affords no clue to the total distance. It does not enable us to state precisely to an inch how much level and how much hill there was on the road." "Fair damsel," the aged knight replies, "—if, as I surmise, thy initials denote Early Womanhood—bethink thee that the word 'enable' is thine, not mine. I did but ask the time of reaching the hill-top as myconditionfor further parley. Ifnowthou wilt not grant that I am a truth-loving man, then will I affirm that those same initials denote Envenomed Wickedness!"

CLASS LIST.

A Marlborough Boy.Putney Walker.

II.

Blithe.E. W.L. B.O. V. L.Rose.Sea Breeze.{Simple Susan.{Money-Spinner.

Blithehas made so ingenious an addition to the problem, andSimple SusanandCo. have solved it in such tuneful verse, that I record both their answers in full. I have altered a word or two inBlithe's—which I trust she will excuse; it did not seem quite clear as it stood.

"Yet stay," said the youth, as a gleam of inspiration lighted up the relaxing muscles of his quiescent features. "Stay. Methinks it matters littlewhenwe reached that summit, the crown of our toil. For in the space of time wherein we clambered up one mile and bounded down the same on our return, we could have trudged thetwainon the level. We have plodded, then, four-and-twenty miles in these six mortal hours; for never a moment did we stop for catching of fleeting breath or for gazing on the scene around!"

"Very good," said the old man. "Twelve miles out and twelve miles in. And we reached the top some time between six and seven of the clock. Now mark me! For every five minutes that had fled since six of the clock when we stood on yonder peak, so many miles had we toiled upwards on the dreary mountainside!"

The youth moaned and rushed into the hostel.

Blithe.

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