Transcriber's note Minor punctuation errors have been changed without notice. Printererrors have been changed and are listed at the end. All otherinconsistencies are as in the original. In this version the square root symbol is indicated by [** sqrt], thesuperscript by ^, and the therefore symbol by [** therefore]. A TANGLED TALE [Decoration] [Illustration: "AT A PACE OF SIX MILES IN THE HOUR. " _Frontispiece. _] A TANGLED TALE BY LEWIS CARROLL _WITH SIX ILLUSTRATIONS_ BY ARTHUR B. FROST Hoc meum tale quale est accipe. _SECOND THOUSAND. _ London MACMILLAN AND CO. 1885 [_All Rights Reserved_] RICHARD CLAY & SONS, BREAD STREET HILL, LONDON, E. C. _And Bungay, Suffolk_. To My Pupil. Beloved pupil! Tamed by thee, Addish-, Subtrac-, Multiplica-tion, Division, Fractions, Rule of Three, Attest thy deft manipulation! Then onward! Let the voice of Fame From Age to Age repeat thy story, Till thou hast won thyself a name Exceeding even Euclid's glory! PREFACE. This Tale originally appeared as a serial in _The Monthly Packet_, beginning in April, 1880. The writer's intention was to embody in eachKnot (like the medicine so dexterously, but ineffectually, concealed inthe jam of our early childhood) one or more mathematical questions--inArithmetic, Algebra, or Geometry, as the case might be--for theamusement, and possible edification, of the fair readers of thatMagazine. L. C. _October, 1885. _ CONTENTS. KNOT PAGE I. EXCELSIOR 1 II. ELIGIBLE APARTMENTS 4 III. MAD MATHESIS 13 IV. THE DEAD RECKONING 19 V. OUGHTS AND CROSSES 27 VI. HER RADIANCY 34 VII. PETTY CASH 43 VIII. DE OMNIBUS REBUS 52 IX. A SERPENT WITH CORNERS 58 X. CHELSEA BUNS 66 ANSWERS TO KNOT I. 77 " " II. 84 " " III. 90 " " IV. 96 " " V. 102 " " VI. 106 " " VII. 112 " " VIII. 132 " " IX. 135 " " X. 142 A TANGLED TALE. KNOT I. EXCELSIOR. "Goblin, lead them up and down. " The ruddy glow of sunset was already fading into the sombre shadows ofnight, when two travellers might have been observed swiftly--at a paceof six miles in the hour--descending the rugged side of a mountain; theyounger bounding from crag to crag with the agility of a fawn, while hiscompanion, whose aged limbs seemed ill at ease in the heavy chain armourhabitually worn by tourists in that district, toiled on painfully at hisside. As is always the case under such circumstances, the younger knight wasthe first to break the silence. "A goodly pace, I trow!" he exclaimed. "We sped not thus in the ascent!" "Goodly, indeed!" the other echoed with a groan. "We clomb it but atthree miles in the hour. " "And on the dead level our pace is----?" the younger suggested; for hewas weak in statistics, and left all such details to his aged companion. "Four miles in the hour, " the other wearily replied. "Not an ouncemore, " he added, with that love of metaphor so common in old age, "andnot a farthing less!" "'Twas three hours past high noon when we left our hostelry, " the youngman said, musingly. "We shall scarce be back by supper-time. Perchancemine host will roundly deny us all food!" "He will chide our tardy return, " was the grave reply, "and such arebuke will be meet. " "A brave conceit!" cried the other, with a merry laugh. "And should webid him bring us yet another course, I trow his answer will be tart!" "We shall but get our deserts, " sighed the elder knight, who had neverseen a joke in his life, and was somewhat displeased at his companion'suntimely levity. "'Twill be nine of the clock, " he added in anundertone, "by the time we regain our hostelry. Full many a mile shallwe have plodded this day!" "How many? How many?" cried the eager youth, ever athirst for knowledge. The old man was silent. "Tell me, " he answered, after a moment's thought, "what time it was whenwe stood together on yonder peak. Not exact to the minute!" he addedhastily, reading a protest in the young man's face. "An' thy guess bewithin one poor half-hour of the mark, 'tis all I ask of thy mother'sson! Then will I tell thee, true to the last inch, how far we shall havetrudged betwixt three and nine of the clock. " A groan was the young man's only reply; while his convulsed features andthe deep wrinkles that chased each other across his manly brow, revealedthe abyss of arithmetical agony into which one chance question hadplunged him. KNOT II. ELIGIBLE APARTMENTS. "Straight down the crooked lane, And all round the square. " "Let's ask Balbus about it, " said Hugh. "All right, " said Lambert. "_He_ can guess it, " said Hugh. "Rather, " said Lambert. No more words were needed: the two brothers understood each otherperfectly. [Illustration: "BALBUS WAS ASSISTING HIS MOTHER-IN-LAW TO CONVINCE THEDRAGON. "] Balbus was waiting for them at the hotel: the journey down had tiredhim, he said: so his two pupils had been the round of the place, insearch of lodgings, without the old tutor who had been their inseparablecompanion from their childhood. They had named him after the hero oftheir Latin exercise-book, which overflowed with anecdotes of thatversatile genius--anecdotes whose vagueness in detail was more thancompensated by their sensational brilliance. "Balbus has overcome allhis enemies" had been marked by their tutor, in the margin of the book, "Successful Bravery. " In this way he had tried to extract a moral fromevery anecdote about Balbus--sometimes one of warning, as in "Balbus hadborrowed a healthy dragon, " against which he had written "Rashness inSpeculation"--sometimes of encouragement, as in the words "Influence ofSympathy in United Action, " which stood opposite to the anecdote "Balbuswas assisting his mother-in-law to convince the dragon"--and sometimesit dwindled down to a single word, such as "Prudence, " which was all hecould extract from the touching record that "Balbus, having scorched thetail of the dragon, went away. " His pupils liked the short morals best, as it left them more room for marginal illustrations, and in thisinstance they required all the space they could get to exhibit therapidity of the hero's departure. Their report of the state of things was discouraging. That mostfashionable of watering-places, Little Mendip, was "chockfull" (as theboys expressed it) from end to end. But in one Square they had seen noless than four cards, in different houses, all announcing in flamingcapitals "ELIGIBLE APARTMENTS. " "So there's plenty of choice, after all, you see, " said spokesman Hugh in conclusion. "That doesn't follow from the data, " said Balbus, as he rose from theeasy chair, where he had been dozing over _The Little Mendip Gazette_. "They may be all single rooms. However, we may as well see them. I shallbe glad to stretch my legs a bit. " An unprejudiced bystander might have objected that the operation wasneedless, and that this long, lank creature would have been all thebetter with even shorter legs: but no such thought occurred to hisloving pupils. One on each side, they did their best to keep up with hisgigantic strides, while Hugh repeated the sentence in their father'sletter, just received from abroad, over which he and Lambert had beenpuzzling. "He says a friend of his, the Governor of----_what_ was thatname again, Lambert?" ("Kgovjni, " said Lambert. ) "Well, yes. TheGovernor of----what-you-may-call-it----wants to give a _very_ smalldinner-party, and he means to ask his father's brother-in-law, hisbrother's father-in-law, his father-in-law's brother, and hisbrother-in-law's father: and we're to guess how many guests there willbe. " There was an anxious pause. "_How_ large did he say the pudding was tobe?" Balbus said at last. "Take its cubical contents, divide by thecubical contents of what each man can eat, and the quotient----" "He didn't say anything about pudding, " said Hugh, "--and here's theSquare, " as they turned a corner and came into sight of the "eligibleapartments. " "It _is_ a Square!" was Balbus' first cry of delight, as he gazed aroundhim. "Beautiful! Beau-ti-ful! Equilateral! _And_ rectangular!" The boys looked round with less enthusiasm. "Number nine is the firstwith a card, " said prosaic Lambert; but Balbus would not so soon awakefrom his dream of beauty. "See, boys!" he cried. "Twenty doors on a side! What symmetry! Each sidedivided into twenty-one equal parts! It's delicious!" "Shall I knock, or ring?" said Hugh, looking in some perplexity at asquare brass plate which bore the simple inscription "RING ALSO. " "Both, " said Balbus. "That's an Ellipsis, my boy. Did you never see anEllipsis before?" "I couldn't hardly read it, " said Hugh, evasively. "It's no good havingan Ellipsis, if they don't keep it clean. " "Which there is _one_ room, gentlemen, " said the smiling landlady. "Anda sweet room too! As snug a little back-room----" "We will see it, " said Balbus gloomily, as they followed her in. "I knewhow it would be! One room in each house! No view, I suppose?" "Which indeed there _is_, gentlemen!" the landlady indignantlyprotested, as she drew up the blind, and indicated the back garden. "Cabbages, I perceive, " said Balbus. "Well, they're green, at any rate. " "Which the greens at the shops, " their hostess explained, "are by nomeans dependable upon. Here you has them on the premises, _and_ of thebest. " "Does the window open?" was always Balbus' first question in testing alodging: and "Does the chimney smoke?" his second. Satisfied on allpoints, he secured the refusal of the room, and they moved on to NumberTwenty-five. This landlady was grave and stern. "I've nobbut one room left, " shetold them: "and it gives on the back-gyardin. " "But there are cabbages?" Balbus suggested. The landlady visibly relented. "There is, sir, " she said: "and goodones, though I say it as shouldn't. We can't rely on the shops forgreens. So we grows them ourselves. " "A singular advantage, " said Balbus: and, after the usual questions, they went on to Fifty-two. "And I'd gladly accommodate you all, if I could, " was the greeting thatmet them. "We are but mortal, " ("Irrelevant!" muttered Balbus) "and I'velet all my rooms but one. " "Which one is a back-room, I perceive, " said Balbus: "and looking outon--on cabbages, I presume?" "Yes, indeed, sir!" said their hostess. "Whatever _other_ folks may do, _we_ grows our own. For the shops----" "An excellent arrangement!" Balbus interrupted. "Then one can reallydepend on their being good. Does the window open?" The usual questions were answered satisfactorily: but this time Hughadded one of his own invention--"Does the cat scratch?" The landlady looked round suspiciously, as if to make sure the cat wasnot listening, "I will not deceive you, gentlemen, " she said. "It _do_scratch, but not without you pulls its whiskers! It'll never do it, " sherepeated slowly, with a visible effort to recall the exact words of somewritten agreement between herself and the cat, "without you pulls itswhiskers!" "Much may be excused in a cat so treated, " said Balbus, as they left thehouse and crossed to Number Seventy-three, leaving the landladycurtseying on the doorstep, and still murmuring to herself her partingwords, as if they were a form of blessing, "---- not without you pullsits whiskers!" At Number Seventy-three they found only a small shy girl to show thehouse, who said "yes'm" in answer to all questions. "The usual room, " said Balbus, as they marched in: "the usualback-garden, the usual cabbages. I suppose you can't get them good atthe shops?" "Yes'm, " said the girl. "Well, you may tell your mistress we will take the room, and that herplan of growing her own cabbages is simply _admirable_!" "Yes'm, " said the girl, as she showed them out. "One day-room and three bed-rooms, " said Balbus, as they returned to thehotel. "We will take as our day-room the one that gives us the leastwalking to do to get to it. " "Must we walk from door to door, and count the steps?" said Lambert. "No, no! Figure it out, my boys, figure it out!" Balbus gaily exclaimed, as he put pens, ink, and paper before his hapless pupils, and left theroom. "I say! It'll be a job!" said Hugh. "Rather!" said Lambert. KNOT III. MAD MATHESIS. "I waited for the train. " "Well, they call me so because I _am_ a little mad, I suppose, " shesaid, good-humouredly, in answer to Clara's cautiously-worded questionas to how she came by so strange a nick-name. "You see, I never do whatsane people are expected to do now-a-days. I never wear long trains, (talking of trains, that's the Charing Cross Metropolitan Station--I'vesomething to tell you about _that_), and I never play lawn-tennis. Ican't cook an omelette. I can't even set a broken limb! _There's_ anignoramus for you!" Clara was her niece, and full twenty years her junior; in fact, she wasstill attending a High School--an institution of which Mad Mathesisspoke with undisguised aversion. "Let a woman be meek and lowly!" shewould say. "None of your High Schools for me!" But it was vacation-timejust now, and Clara was her guest, and Mad Mathesis was showing her thesights of that Eighth Wonder of the world--London. "The Charing Cross Metropolitan Station!" she resumed, waving her handtowards the entrance as if she were introducing her niece to a friend. "The Bayswater and Birmingham Extension is just completed, and thetrains now run round and round continuously--skirting the border ofWales, just touching at York, and so round by the east coast back toLondon. The way the trains run is _most_ peculiar. The westerly ones goround in two hours; the easterly ones take three; but they always manageto start two trains from here, opposite ways, punctually everyquarter-of-an-hour. " "They part to meet again, " said Clara, her eyes filling with tears atthe romantic thought. "No need to cry about it!" her aunt grimly remarked. "They don't meet onthe same line of rails, you know. Talking of meeting, an idea strikesme!" she added, changing the subject with her usual abruptness. "Let'sgo opposite ways round, and see which can meet most trains. No need fora chaperon--ladies' saloon, you know. You shall go whichever way youlike, and we'll have a bet about it!" "I never make bets, " Clara said very gravely. "Our excellent preceptresshas often warned us----" "You'd be none the worse if you did!" Mad Mathesis interrupted. "Infact, you'd be the better, I'm certain!" "Neither does our excellent preceptress approve of puns, " said Clara. "But we'll have a match, if you like. Let me choose my train, " she addedafter a brief mental calculation, "and I'll engage to meet exactly halfas many again as you do. " "Not if you count fair, " Mad Mathesis bluntly interrupted. "Remember, weonly count the trains we meet _on the way_. You mustn't count the onethat starts as you start, nor the one that arrives as you arrive. " "That will only make the difference of _one_ train, " said Clara, as theyturned and entered the station. "But I never travelled alone before. There'll be no one to help me to alight. However, I don't mind. Let'shave a match. " A ragged little boy overheard her remark, and came running after her. "Buy a box of cigar-lights, Miss!" he pleaded, pulling her shawl toattract her attention. Clara stopped to explain. "I never smoke cigars, " she said in a meekly apologetic tone. "Ourexcellent preceptress----, " but Mad Mathesis impatiently hurried her on, and the little boy was left gazing after her with round eyes ofamazement. The two ladies bought their tickets and moved slowly down the centralplatform, Mad Mathesis prattling on as usual--Clara silent, anxiouslyreconsidering the calculation on which she rested her hopes of winningthe match. "Mind where you go, dear!" cried her aunt, checking her just in time. "One step more, and you'd have been in that pail of cold water!" "I know, I know, " Clara said, dreamily. "The pale, the cold, and themoony----" "Take your places on the spring-boards!" shouted a porter. "What are _they_ for!" Clara asked in a terrified whisper. "Merely to help us into the trains. " The elder lady spoke with thenonchalance of one quite used to the process. "Very few people can getinto a carriage without help in less than three seconds, and the trainsonly stop for one second. " At this moment the whistle was heard, and twotrains rushed into the station. A moment's pause, and they were goneagain; but in that brief interval several hundred passengers had beenshot into them, each flying straight to his place with the accuracy of aMinie bullet--while an equal number were showered out upon theside-platforms. Three hours had passed away, and the two friends met again on theCharing Cross platform, and eagerly compared notes. Then Clara turnedaway with a sigh. To young impulsive hearts, like hers, disappointmentis always a bitter pill. Mad Mathesis followed her, full of kindlysympathy. "Try again, my love!" she said, cheerily. "Let us vary the experiment. We will start as we did before, but not to begin counting till ourtrains meet. When we see each other, we will say 'One!' and so count ontill we come here again. " Clara brightened up. "I shall win _that_, " she exclaimed eagerly, "if Imay choose my train!" Another shriek of engine whistles, another upheaving of spring-boards, another living avalanche plunging into two trains as they flashed by:and the travellers were off again. Each gazed eagerly from her carriage window, holding up her handkerchiefas a signal to her friend. A rush and a roar. Two trains shot past eachother in a tunnel, and two travellers leaned back in their corners witha sigh--or rather with _two_ sighs--of relief. "One!" Clara murmured toherself. "Won! It's a word of good omen. _This_ time, at any rate, thevictory will be mine!" But _was_ it? KNOT IV. THE DEAD RECKONING. "I did dream of money-bags to-night. " Noonday on the open sea within a few degrees of the Equator is apt to beoppressively warm; and our two travellers were now airily clad in suitsof dazzling white linen, having laid aside the chain-armour which theyhad found not only endurable in the cold mountain air they had latelybeen breathing, but a necessary precaution against the daggers of thebanditti who infested the heights. Their holiday-trip was over, and theywere now on their way home, in the monthly packet which plied betweenthe two great ports of the island they had been exploring. Along with their armour, the tourists had laid aside the antiquatedspeech it had pleased them to affect while in knightly disguise, andhad returned to the ordinary style of two country gentlemen of theTwentieth Century. Stretched on a pile of cushions, under the shade of a huge umbrella, they were lazily watching some native fishermen, who had come on boardat the last landing-place, each carrying over his shoulder a small butheavy sack. A large weighing-machine, that had been used for cargo atthe last port, stood on the deck; and round this the fishermen hadgathered, and, with much unintelligible jabber, seemed to be weighingtheir sacks. "More like sparrows in a tree than human talk, isn't it?" the eldertourist remarked to his son, who smiled feebly, but would not exerthimself so far as to speak. The old man tried another listener. "What have they got in those sacks, Captain?" he inquired, as that greatbeing passed them in his never ending parade to and fro on the deck. The Captain paused in his march, and towered over the travellers--tall, grave, and serenely self-satisfied. "Fishermen, " he explained, "are often passengers in My ship. These fiveare from Mhruxi--the place we last touched at--and that's the way theycarry their money. The money of this island is heavy, gentlemen, but itcosts little, as you may guess. We buy it from them by weight--aboutfive shillings a pound. I fancy a ten pound-note would buy all thosesacks. " By this time the old man had closed his eyes--in order, no doubt, toconcentrate his thoughts on these interesting facts; but the Captainfailed to realise his motive, and with a grunt resumed his monotonousmarch. Meanwhile the fishermen were getting so noisy over the weighing-machinethat one of the sailors took the precaution of carrying off all theweights, leaving them to amuse themselves with such substitutes in theform of winch-handles, belaying-pins, &c. , as they could find. Thisbrought their excitement to a speedy end: they carefully hid their sacksin the folds of the jib that lay on the deck near the tourists, andstrolled away. When next the Captain's heavy footfall passed, the younger man rousedhimself to speak. "_What_ did you call the place those fellows came from, Captain?" heasked. "Mhruxi, sir. " "And the one we are bound for?" The Captain took a long breath, plunged into the word, and came out ofit nobly. "They call it Kgovjni, sir. " "K--I give it up!" the young man faintly said. He stretched out his hand for a glass of iced water which thecompassionate steward had brought him a minute ago, and had set down, unluckily, just outside the shadow of the umbrella. It was scalding hot, and he decided not to drink it. The effort of making this resolution, coming close on the fatiguing conversation he had just gone through, wastoo much for him: he sank back among the cushions in silence. His father courteously tried to make amends for his _nonchalance_. "Whereabouts are we now, Captain?" said he, "Have you any idea?" The Captain cast a pitying look on the ignorant landsman. "I could tellyou _that_, sir, " he said, in a tone of lofty condescension, "to aninch!" "You don't say so!" the old man remarked, in a tone of languid surprise. "And mean so, " persisted the Captain. "Why, what do you suppose wouldbecome of My ship, if I were to lose My Longitude and My Latitude?Could _you_ make anything of My Dead Reckoning?" "Nobody could, I'm sure!" the other heartily rejoined. But he had overdone it. "It's _perfectly_ intelligible, " the Captain said, in an offended tone, "to any one that understands such things. " With these words he movedaway, and began giving orders to the men, who were preparing to hoistthe jib. Our tourists watched the operation with such interest that neither ofthem remembered the five money-bags, which in another moment, as thewind filled out the jib, were whirled overboard and fell heavily intothe sea. But the poor fishermen had not so easily forgotten their property. In amoment they had rushed to the spot, and stood uttering cries of fury, and pointing, now to the sea, and now to the sailors who had caused thedisaster. The old man explained it to the Captain. "Let us make it up among us, " he added in conclusion. "Ten pounds willdo it, I think you said?" [Illustration] But the Captain put aside the suggestion with a wave of the hand. "No, sir!" he said, in his grandest manner. "You will excuse Me, I amsure; but these are My passengers. The accident has happened on board Myship, and under My orders. It is for Me to make compensation. " Heturned to the angry fishermen. "Come here, my men!" he said, in theMhruxian dialect. "Tell me the weight of each sack. I saw you weighingthem just now. " Then ensued a perfect Babel of noise, as the five natives explained, allscreaming together, how the sailors had carried off the weights, andthey had done what they could with whatever came handy. Two iron belaying-pins, three blocks, six holystones, fourwinch-handles, and a large hammer, were now carefully weighed, theCaptain superintending and noting the results. But the matter did notseem to be settled, even then: an angry discussion followed, in whichthe sailors and the five natives all joined: and at last the Captainapproached our tourists with a disconcerted look, which he tried toconceal under a laugh. "It's an absurd difficulty, " he said. "Perhaps one of you gentlemen cansuggest something. It seems they weighed the sacks two at a time!" "If they didn't have five separate weighings, of course you can't valuethem separately, " the youth hastily decided. "Let's hear all about it, " was the old man's more cautious remark. "They _did_ have five separate weighings, " the Captain said, "but--Well, it beats _me_ entirely!" he added, in a sudden burst of candour. "Here'sthe result. First and second sack weighed twelve pounds; second andthird, thirteen and a half; third and fourth, eleven and a half; fourthand fifth, eight: and then they say they had only the large hammer left, and it took _three_ sacks to weigh it down--that's the first, third andfifth--and _they_ weighed sixteen pounds. There, gentlemen! Did you everhear anything like _that_?" The old man muttered under his breath "If only my sister were here!" andlooked helplessly at his son. His son looked at the five natives. Thefive natives looked at the Captain. The Captain looked at nobody: hiseyes were cast down, and he seemed to be saying softly to himself"Contemplate one another, gentlemen, if such be your good pleasure. _I_contemplate _Myself_!" KNOT V. OUGHTS AND CROSSES. "Look here, upon this picture, and on this. " "And what made you choose the first train, Goosey?" said Mad Mathesis, as they got into the cab. "Couldn't you count better than _that_?" "I took an extreme case, " was the tearful reply. "Our excellentpreceptress always says 'When in doubt, my dears, take an extreme case. 'And I _was_ in doubt. " "Does it always succeed?" her aunt enquired. Clara sighed. "Not _always_, " she reluctantly admitted. "And I can'tmake out why. One day she was telling the little girls--they make such anoise at tea, you know--'The more noise you make, the less jam you willhave, and _vice versā_. ' And I thought they wouldn't know what '_viceversā_' meant: so I explained it to them. I said 'If you make aninfinite noise, you'll get no jam: and if you make no noise, you'll getan infinite lot of jam. ' But our excellent preceptress said that wasn'ta good instance. _Why_ wasn't it?" she added plaintively. Her aunt evaded the question. "One sees certain objections to it, " shesaid. "But how did you work it with the Metropolitan trains? None ofthem go infinitely fast, I believe. " "I called them hares and tortoises, " Clara said--a little timidly, forshe dreaded being laughed at. "And I thought there couldn't be so manyhares as tortoises on the Line: so I took an extreme case--one hare andan infinite number of tortoises. " "An extreme case, indeed, " her aunt remarked with admirable gravity:"and a most dangerous state of things!" "And I thought, if I went with a tortoise, there would be only _one_hare to meet: but if I went with the hare--you know there were _crowds_of tortoises!" "It wasn't a bad idea, " said the elder lady, as they left the cab, atthe entrance of Burlington House. "You shall have another chance to-day. We'll have a match in marking pictures. " Clara brightened up. "I should like to try again, very much, " she said. "I'll take more care this time. How are we to play?" To this question Mad Mathesis made no reply: she was busy drawing linesdown the margins of the catalogue. "See, " she said after a minute, "I'vedrawn three columns against the names of the pictures in the long room, and I want you to fill them with oughts and crosses--crosses for goodmarks and oughts for bad. The first column is for choice of subject, thesecond for arrangement, the third for colouring. And these are theconditions of the match. You must give three crosses to two or threepictures. You must give two crosses to four or five----" "Do you mean _only_ two crosses?" said Clara. "Or may I count thethree-cross pictures among the two-cross pictures?" "Of course you may, " said her aunt. "Any one, that has _three_ eyes, maybe said to have _two_ eyes, I suppose?" Clara followed her aunt's dreamy gaze across the crowded gallery, half-dreading to find that there was a three-eyed person in sight. "And you must give one cross to nine or ten. " "And which wins the match?" Clara asked, as she carefully entered theseconditions on a blank leaf in her catalogue. "Whichever marks fewest pictures. " "But suppose we marked the same number?" "Then whichever uses most marks. " Clara considered. "I don't think it's much of a match, " she said. "Ishall mark nine pictures, and give three crosses to three of them, twocrosses to two more, and one cross each to all the rest. " "Will you, indeed?" said her aunt. "Wait till you've heard all theconditions, my impetuous child. You must give three oughts to one or twopictures, two oughts to three or four, and one ought to eight or nine. Idon't want you to be _too_ hard on the R. A. 's. " Clara quite gasped as she wrote down all these fresh conditions. "It's agreat deal worse than Circulating Decimals!" she said. "But I'mdetermined to win, all the same!" Her aunt smiled grimly. "We can begin _here_, " she said, as they pausedbefore a gigantic picture, which the catalogue informed them was the"Portrait of Lieutenant Brown, mounted on his favorite elephant. " "He looks awfully conceited!" said Clara. "I don't think he was theelephant's favorite Lieutenant. What a hideous picture it is! And ittakes up room enough for twenty!" "Mind what you say, my dear!" her aunt interposed. "It's by an R. A. !" But Clara was quite reckless. "I don't care who it's by!" she cried. "And I shall give it three bad marks!" Aunt and niece soon drifted away from each other in the crowd, and forthe next half-hour Clara was hard at work, putting in marks and rubbingthem out again, and hunting up and down for suitable pictures. This shefound the hardest part of all. "I _can't_ find the one I want!" sheexclaimed at last, almost crying with vexation. "What is it you want to find, my dear?" The voice was strange to Clara, but so sweet and gentle that she felt attracted to the owner of it, evenbefore she had seen her; and when she turned, and met the smiling looksof two little old ladies, whose round dimpled faces, exactly alike, seemed never to have known a care, it was as much as she could do--asshe confessed to Aunt Mattie afterwards--to keep herself from huggingthem both. "I was looking for a picture, " she said, "that has a good subject--andthat's well arranged--but badly coloured. " The little old ladies glanced at each other in some alarm. "Calmyourself, my dear, " said the one who had spoken first, "and try toremember which it was. What _was_ the subject?" "Was it an elephant, for instance?" the other sister suggested. Theywere still in sight of Lieutenant Brown. "I don't know, indeed!" Clara impetuously replied. "You know it doesn'tmatter a bit what the subject _is_, so long as it's a good one!" Once more the sisters exchanged looks of alarm, and one of themwhispered something to the other, of which Clara caught only the oneword "mad. " "They mean Aunt Mattie, of course, " she said to herself--fancying, inher innocence, that London was like her native town, where everybodyknew everybody else. "If you mean my aunt, " she added aloud, "she's_there_--just three pictures beyond Lieutenant Brown. " "Ah, well! Then you'd better go to her, my dear!" her new friend said, soothingly. "_She'll_ find you the picture you want. Good-bye, dear!" "Good-bye, dear!" echoed the other sister, "Mind you don't lose sight ofyour aunt!" And the pair trotted off into another room, leaving Clararather perplexed at their manner. "They're real darlings!" she soliloquised. "I wonder why they pity meso!" And she wandered on, murmuring to herself "It must have two goodmarks, and----" KNOT VI. HER RADIANCY. "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 waythey were met by the Governor, who welcomed them in English--a greatrelief 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 manwhispered to his son. "And why do they say 'Bamboo!' so often?" "It alludes to a local custom, " replied the Governor, who had overheardthe question. "Such persons as happen in any way to displease HerRadiancy are usually beaten with rods. " [Illustration: "WHY DO THEY SAY 'BAMBOO!' SO OFTEN?"] The old man shuddered. "A most objectional local custom!" he remarkedwith strong emphasis. "I wish we had never landed! Did you notice thatblack fellow, Norman, opening his great mouth at us? I verily believe hewould like to eat us!" Norman appealed to the Governor, who was walking at his other side. "Dothey often eat distinguished strangers here?" he said, in as indifferenta 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 shallbe without a doubt. It's a comfort to know it won't be Beaten withoutthe B! My dear boy, just look at the peacocks!" They were now walking between two unbroken lines of those gorgeousbirds, each held in check, by means of a golden collar and chain, by ablack slave, who stood well behind, so as not to interrupt the view ofthe glittering tail, with its network of rustling feathers and itshundred eyes. The Governor smiled proudly. "In your honour, " he said, "Her Radiancyhas 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 ofcharm 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 Ihaven't, Norman!" "He has not, as yet, " Norman gently explained. And the Governorrepeated, in a satisfied tone, "Not as yet. " "Yours is a wondrous country!" the Governor resumed, after a pause. "Nowhere is a letter from a friend of mine, a merchant, in London. He andhis brother went there a year ago, with a thousand pounds apiece; and onNew-Year's-day they had sixty thousand pounds between them!" "How did they do it?" Norman eagerly exclaimed. Even the elder travellerlooked excited. The Governor handed him the open letter. "Anybody can do it, when oncethey know how, " so ran this oracular document. "We borrowed nought: westole nought. We began the year with only a thousand pounds apiece: andlast New-Year's-day we had sixty thousand pounds between us--sixtythousand golden sovereigns!" Norman looked grave and thoughtful as he handed back the letter. Hisfather hazarded one guess. "Was it by gambling?" "A Kgovjnian never gambles, " said the Governor gravely, as he usheredthem through the palace gates. They followed him in silence down a longpassage, and soon found themselves in a lofty hall, lined entirely withpeacocks' feathers. In the centre was a pile of crimson cushions, whichalmost concealed the figure of Her Radiancy--a plump little damsel, in arobe of green satin dotted with silver stars, whose pale round face litup 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 shelanguidly murmured a word or two in the Kgovjnian dialect. The Governor interpreted. "Her Radiancy welcomes you. She notes theImpenetrable Placidity of the old one, and the Imperceptible Acutenessof the youth. " Here the little potentate clapped her hands, and a troop of slavesinstantly appeared, carrying trays of coffee and sweetmeats, which theyoffered to the guests, who had, at a signal from the Governor, seatedthemselves on the carpet. "Sugar-plums!" muttered the old man. "One might as well be at aconfectioner's! Ask for a penny bun, Norman!" "Not so loud!" his son whispered. "Say something complimentary!" For theGovernor was evidently expecting a speech. "We thank Her Exalted Potency, " the old man timidly began. "We bask inthe light of her smile, which----" "The words of old men are weak!" the Governor interrupted angrily. "Letthe youth speak!" "Tell her, " cried Norman, in a wild burst of eloquence, "that, like twograsshoppers in a volcano, we are shrivelled up in the presence of HerSpangled Vehemence!" "It is well, " said the Governor, and translated this into Kgovjnian. "Iam now to tell you, " he proceeded, "what Her Radiancy requires of youbefore you go. The yearly competition for the post of ImperialScarf-maker is just ended; you are the judges. You will take account ofthe rate of work, the lightness of the scarves, and their warmth. Usually the competitors differ in one point only. Thus, last year, Fifiand Gogo made the same number of scarves in the trial-week, and theywere equally light; but Fifi's were twice as warm as Gogo's and she waspronounced twice as good. But this year, woe is me, who can judge it?Three competitors are here, and they differ in all points! While yousettle their claims, you shall be lodged, Her Radiancy bids me say, freeof expense--in the best dungeon, and abundantly fed on the best breadand water. " The old man groaned. "All is lost!" he wildly exclaimed. But Normanheeded him not: he had taken out his note-book, and was calmly jottingdown the particulars. "Three they be, " the Governor proceeded, "Lolo, Mimi, and Zuzu. Lolomakes 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 morethan one of Lolo's; yet Mimi's is lighter still--5 of hers will butbalance 3 of Zuzu's! And for warmth one of Mimi's is equal to 4 ofZuzu'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 HerRadiancy your farewell compliments--and walk out backwards. " The walking part was all the elder tourist could manage. Norman simplysaid "Tell Her Radiancy we are transfixed by the spectacle of Her SereneBrilliance, and bid an agonized farewell to her Condensed Milkiness!" "Her Radiancy is pleased, " the Governor reported, after duly translatingthis. "She casts on you a glance from Her Imperial Eyes, and isconfident that you will catch it!" "That I warrant we shall!" the elder traveller moaned to himselfdistractedly. Once more they bowed low, and then followed the Governor down a windingstaircase to the Imperial Dungeon, which they found to be lined withcoloured marble, lighted from the roof, and splendidly though notluxuriously furnished with a bench of polished malachite. "I trust youwill not delay the calculation, " the Governor said, ushering them inwith much ceremony. "I have known great inconvenience--great and seriousinconvenience--result to those unhappy ones who have delayed to executethe commands of Her Radiancy! And on this occasion she is resolute: shesays the thing must and shall be done: and she has ordered up tenthousand additional bamboos!" With these words he left them, and theyheard him lock and bar the door on the outside. "I told you how it would end!" moaned the elder traveller, wringing hishands, and quite forgetting in his anguish that he had himself proposedthe expedition, and had never predicted anything of the sort. "Oh thatwe were well out of this miserable business!" "Courage!" cried the younger cheerily. "_Hęc 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 rockedhimself to and fro on the malachite bench. "Glory without the L!" FOOTNOTE: [Footnote A: "_Maskee_, " in Pigeon-English, means "_without_. "] KNOT VII. PETTY CASH. "Base is the slave that pays. " "Aunt Mattie!" "My child?" "_Would_ you mind writing it down at once? I shall be quite _certain_ toforget it if you don't!" "My dear, we really must wait till the cab stops. How can I possiblywrite anything in the midst of all this jolting?" "But _really_ I shall be forgetting it!" Clara's voice took the plaintive tone that her aunt never knew how toresist, and with a sigh the old lady drew forth her ivory tablets andprepared to record the amount that Clara had just spent at theconfectioner's shop. Her expenditure was always made out of her aunt'spurse, but the poor girl knew, by bitter experience, that sooner orlater "Mad Mathesis" would expect an exact account of every penny thathad gone, and she waited, with ill-concealed impatience, while the oldlady 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'sluncheon duly entered. _One glass lemonade_ (Why can't you drink water, like me?) _three sandwiches_ (They never put in half mustard enough. Itold the young woman so, to her face; and she tossed her head--like herimpudence!) _and seven biscuits_. _Total one-and-two-pence. _ Well, nowfor to-day's?" "One glass of lemonade----" Clara was beginning to say, when suddenlythe cab drew up, and a courteous railway-porter was handing out thebewildered 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, whatever _you_ maythink--afterwards. " And she proceeded to pay the driver, and to givevoluminous orders about the luggage, quite deaf to the entreaties of herunhappy niece that she would enter the rest of the luncheon account. "My dear, you really must cultivate a more capacious mind!" was all theconsolation she vouchsafed to the poor girl. "Are not the tablets ofyour 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, andboth ladies turned in some surprise to see who it was that had sosuddenly struck into their conversation. A fat little old lady wasstanding at the door of a cab, helping the driver to extricate whatseemed an exact duplicate of herself: it would have been no easy task todecide which was the fatter, or which looked the more good-humoured ofthe two sisters. "I tell you the cab-door isn't half wide enough!" she repeated, as hersister finally emerged, somewhat after the fashion of a pellet from apop-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 withsmiles. "Some folks is too wide for 'em, " growled the cab-driver. [Illustration: "I TELL YOU THE CAB-DOOR ISN'T HALF WIDE ENOUGH!"] "Don't provoke me, man!" cried the little old lady, in what she meantfor a tempest of fury. "Say another word and I'll put you into theCounty Court, and sue you for a _Habeas Corpus_!" The cabman touched hishat, and marched off, grinning. "Nothing like a little Law to cow the ruffians, my dear!" she remarkedconfidentially to Clara. "You saw how he quailed when I mentioned the_Habeas Corpus_? Not that I've any idea what it means, but it soundsvery grand, doesn't it?" "It's very provoking, " Clara replied, a little vaguely. "Very!" the little old lady eagerly repeated. "And we're very muchprovoked 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. "Imet them first in the Royal Academy--and they were very kind to me--andthey were lunching at the next table to us, just now, you know--and theytried to help me to find the picture I wanted--and I'm sure they're dearold things!" "Friends of yours, are they?" said Mad Mathesis. "Well, I like theirlooks. You can be civil to them, while I get the tickets. But do try andarrange your ideas a little more chronologically!" And so it came to pass that the four ladies found themselves seated sideby side on the same bench waiting for the train, and chatting as if theyhad known one another for years. "Now this I call quite a remarkable coincidence!" exclaimed the smallerand more talkative of the two sisters--the one whose legal knowledge hadannihilated the cab-driver. "Not only that we should be waiting for thesame train, and at the same station--_that_ would be curious enough--butactually on the same day, and the same hour of the day! That's whatstrikes _me_ so forcibly!" She glanced at the fatter and more silentsister, whose chief function in life seemed to be to support the familyopinion, and who meekly responded-- "And me too, sister!" "Those are not _independent_ coincidences----" Mad Mathesis was justbeginning, when Clara ventured to interpose. "There's no jolting here, " she pleaded meekly. "_Would_ you mind writingit down now?" Out came the ivory tablets once more. "What was it, then?" said heraunt. "One glass of lemonade, one sandwich, one biscuit--Oh dear me!" criedpoor Clara, the historical tone suddenly changing to a wail of agony. "Toothache?" said her aunt calmly, as she wrote down the items. The twosisters instantly opened their reticules and produced two differentremedies for neuralgia, each marked "unequalled. " "It isn't that!" said poor Clara. "Thank you very much. It's only that I_can't_ remember how much I paid!" "Well, try and make it out, then, " said her aunt. "You've gotyesterday's luncheon to help you, you know. And here's the luncheon wehad the day before--the first day we went to that shop--_oneglass lemonade_, _four sandwiches_, _ten biscuits_. _Total, one-and-fivepence. _" She handed the tablets to Clara, who gazed at themwith eyes so dim with tears that she did not at first notice that shewas holding them upside down. The two sisters had been listening to all this with the deepestinterest, and at this juncture the smaller one softly laid her hand onClara's arm. "Do you know, my dear, " she said coaxingly, "my sister and I are in thevery 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 thelittle one would not wait for her to finish it. "Yes, my dear, " she resumed; "we were lunching at the very same shop asyou were--and we had two glasses of lemonade and three sandwiches andfive biscuits--and neither of us has the least idea what we paid. Havewe, sister?" "Quite identically and absolutely----" murmured the other, who evidentlyconsidered that she was now a whole sentence in arrears, and that sheought to discharge one obligation before contracting any freshliabilities; but the little lady broke in again, and she retired fromthe conversation a bankrupt. "_Would_ you make it out for us, my dear?" pleaded the little old lady. "You can do Arithmetic, I trust?" her aunt said, a little anxiously, asClara turned from one tablet to another, vainly trying to collect herthoughts. Her mind was a blank, and all human expression was rapidlyfading out of her face. A gloomy silence ensued. KNOT VIII. DE OMNIBUS REBUS. "This little pig went to market: This little pig staid at home. " "By Her Radiancy's express command, " said the Governor, as he conductedthe travellers, for the last time, from the Imperial presence, "I shallnow have the ecstasy of escorting you as far as the outer gate of theMilitary Quarter, where the agony of parting--if indeed Nature cansurvive the shock--must be endured! From that gate grurmstipths startevery quarter of an hour, both ways----" "Would you mind repeating that word?" said Norman. "Grurm----?" "Grurmstipths, " the Governor repeated. "You call them omnibuses inEngland. They run both ways, and you can travel by one of them all theway down to the harbour. " The old man breathed a sigh of relief; four hours of courtly ceremonyhad wearied him, and he had been in constant terror lest somethingshould call into use the ten thousand additional bamboos. In another minute they were crossing a large quadrangle, paved withmarble, and tastefully decorated with a pigsty in each corner. Soldiers, carrying pigs, were marching in all directions: and in the middle stooda gigantic officer giving orders in a voice of thunder, which madeitself heard above all the uproar of the pigs. "It is the Commander-in-Chief!" the Governor hurriedly whispered to hiscompanions, who at once followed his example in prostrating themselvesbefore the great man. The Commander gravely bowed in return. He wascovered with gold lace from head to foot: his face wore an expression ofdeep misery: and he had a little black pig under each arm. Still thegallant fellow did his best, in the midst of the orders he was everymoment issuing to his men, to bid a courteous farewell to the departingguests. "Farewell, oh old one--carry these three to the South corner--andfarewell to thee, thou young one--put this fat one on the top of theothers in the Western sty--may your shadows never be less--woe is me, itis wrongly done! Empty out all the sties, and begin again!" And thesoldier leant upon his sword, and wiped away a tear. "He is in distress, " the Governor explained as they left the court. "HerRadiancy has commanded him to place twenty-four pigs in those foursties, so that, as she goes round the court, she may always find thenumber 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 isnearer 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 themin vain for four months, " he said. "What hope remains? And Her Radiancyhas ordered up ten thousand additional----" "The pigs don't seem to enjoy being transferred, " the old man hastilyinterrupted. He did not like the subject of bamboos. "They are only _provisionally_ transferred, you know, " said theGovernor. "In most cases they are immediately carried back again: sothey need not mind it. And all is done with the greatest care, under thepersonal 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 outergate, and we must part!" He sobbed as he shook hands with them, and thenext 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!" saidthe young one, severely. "But look sharp--here are two what's-his-namesin 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 tookout his watch. "Just twelve minutes and a half since we started, " heremarked in an absent manner. Suddenly the vacant face brightened; theold man had an idea. "My boy!" he shouted, bringing his hand down uponNorman's shoulder so suddenly as for a moment to transfer his centre ofgravity beyond the base of support. Thus taken off his guard, the young man wildly staggered forwards, andseemed about to plunge into space: but in another moment he hadgracefully recovered himself. "Problem in Precession and Nutation, " heremarked--in tones where filial respect only just managed to conceal ashade of annoyance. "What is it?" he hastily added, fearing his fathermight 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 brokenby the distant squeals of the unfortunate little pigs, who were stillbeing provisionally transferred from sty to sty, under the personalsuperintendence 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 waslying 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, " saidLambert. "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 quiteimmersed: 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, andhanded the big one to Lambert. "Why, there's hardly a teacupful! Do youmean 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 thewater from the big bucket into the little one. "Why, it doesn't halffill it!" "That's _its_ business, " said Lambert. "If Balbus says it's the samebulk, 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 oncepropounded his difficulty. "Let's get you helped first, " said Balbus, briskly cutting away at thejoint. "You know the old proverb 'Mutton first, mechanics afterwards'?" The boys did _not_ know the proverb, but they accepted it in perfectgood faith, as they did every piece of information, however startling, that came from so infallible an authority as their tutor. They ate onsteadily in silence, and, when dinner was over, Hugh set out the usualarray of pens, ink, and paper, while Balbus repeated to them the problemhe had prepared for their afternoon's task. "A friend of mine has a flower-garden--a very pretty one, though nogreat 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 itswidth--and that a gravel-walk, one yard wide, begins at one corner andruns all round it. " "Joining into itself?" said Hugh. "_Not_ joining into itself, young man. Just before doing _that_, itturns a corner, and runs round the garden again, alongside of the firstportion, and then inside that again, winding in and in, and each laptouching 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 afurlong. 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 leavingthe room. "I did, " said Balbus. "Where do the flowers grow?" said Hugh. But Balbus thought it best notto hear the question. He left the boys to their problem, and, in thesilence of his own room, set himself to unravel Hugh's mechanicalparadox. "To fix our thoughts, " he murmured to himself, as, with handsdeep-buried in his pockets, he paced up and down the room, "we will takea cylindrical glass jar, with a scale of inches marked up the side, andfill it with water up to the 10-inch mark: and we will assume that everyinch depth of jar contains a pint of water. We will now take a solidcylinder, such that every inch of it is equal in bulk to _half_ a pintof water, and plunge 4 inches of it into the water, so that the end ofthe cylinder comes down to the 6-inch mark. Well, that displaces 2pints of water. What becomes of them? Why, if there were no morecylinder, they would lie comfortably on the top, and fill the jar up tothe 12-inch mark. But unfortunately there _is_ more cylinder, occupyinghalf 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 otherpint? Why, if there were no more cylinder, it would lie on the top, andfill the jar up to the 13-inch mark. But unfortunately----Shade ofNewton!" he exclaimed, in sudden accents of terror. "When _does_ thewater 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 itdisplaces a portion of the liquid equal to itself in bulk, and that thelevel of the liquid rises just so much as it would rise if a quantity ofliquid 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 ofthe solid which is immersed, and the rise of the level being inproportion. "Suppose a solid held above the surface of a liquid and partiallyimmersed: a portion of the liquid is displaced, and the level of theliquid rises. But, by this rise of level, a little bit more of the solidis of course immersed, and so there is a new displacement of a secondportion of the liquid, and a consequent rise of level. Again, thissecond rise of level causes a yet further immersion, and by consequenceanother displacement of liquid and another rise. It is self-evident thatthis process must continue till the entire solid is immersed, and thatthe 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 asto the source from which the water is supplied--which belongs to a highbranch of mathematics, and is therefore beyond our present scope--doesnot apply to the sea. Let us therefore take the familiar instance of aman standing at the edge of the sea, at ebb-tide, with a solid in hishand, which he partially immerses: he remains steadfast and unmoved, andwe all know that he must be drowned. The multitudes who daily perish inthis manner to attest a philosophical truth, and whose bodies theunreasoning wave casts sullenly upon our thankless shores, have a truerclaim to be called the martyrs of science than a Galileo or a Kepler. Touse Kossuth's eloquent phrase, they are the unnamed demigods of thenineteenth century. "[B] * * * * * "There's a fallacy _somewhere_, " he murmured drowsily, as he stretchedhis long legs upon the sofa. "I must think it over again. " He closed hiseyes, in order to concentrate his attention more perfectly, and for thenext hour or so his slow and regular breathing bore witness to thecareful deliberation with which he was investigating this new andperplexing view of the subject. [Illustration: "HE REMAINS STEADFAST AND UNMOVED. "] FOOTNOTE: [Footnote B: _Note by the writer. _--For the above Essay I am indebted toa 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 girlfilled with tears as she spoke. "Sad--but very curious when you come to look at it arithmetically, " washer aunt's less romantic reply. "Some of them have lost an arm in theircountry's service, some a leg, some an ear, some an eye----" "And some, perhaps, _all_!" Clara murmured dreamily, as they passed thelong rows of weather-beaten heroes basking in the sun. "Did you noticethat very old one, with a red face, who was drawing a map in the dustwith his wooden leg, and all the others watching? I _think_ it was aplan 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 aslively as you and me put together! Why, if drawing a map in thedust--with one's wooden leg--doesn't prove one to be alive, perhapsyou'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 oldlady never let slip an opportunity of driving her niece into acalculation--"what percentage do you suppose must have lost all four--aleg, an arm, an eye, and an ear?" "How _can_ I tell?" gasped the terrified girl. She knew well what wascoming. "You can't, of course, without _data_, " her aunt replied: "but I'm justgoing to give you----" "Give her a Chelsea bun, Miss! That's what most young ladies likesbest!" The voice was rich and musical, and the speaker dexterouslywhipped back the snowy cloth that covered his basket, and disclosed atempting 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 oldlady waved her parasol threateningly: but nothing seemed to disturb thegood-humour of the jolly old man, who marched on, chanting his melodiousrefrain:-- [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 willagree with you ever so much better!" Clara sighed, and there was a hungry look in her eyes as she watched thebasket lessening in the distance: but she meekly listened to therelentless old lady, who at once proceeded to count off the _data_ onher fingers. "Say that 70 per cent. Have lost an eye--75 per cent. An ear--80 percent. An arm--85 per cent. A leg--that'll do it beautifully. Now, mydear, what percentage, _at least_, must have lost all four?" No more conversation occurred--unless a smothered exclamation of "Pipinghot!" which escaped from Clara's lips as the basket vanished round acorner could be counted as such--until they reached the old Chelseamansion, where Clara's father was then staying, with his three sons andtheir old tutor. Balbus, Lambert, and Hugh had entered the house only a few minutesbefore them. They had been out walking, and Hugh had been propounding adifficulty which had reduced Lambert to the depths of gloom, and hadeven puzzled Balbus. "It changes from Wednesday to Thursday at midnight, doesn't it?" Hughhad begun. "Sometimes, " said Balbus, cautiously. "Always, " said Lambert, decisively. "_Sometimes_, " Balbus gently insisted. "Six midnights out of seven, itchanges to some other name. " "I meant, of course, " Hugh corrected himself, "when it _does_ changefrom Wednesday to Thursday, it does it at midnight--and _only_ atmidnight. " "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'tarrived yet: and it's Thursday _east_ of Chelsea (say in Germany orRussia) 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 oneday to another anywhere else. And yet, if Ireland and America and so oncall it Wednesday, and Germany and Russia and so on call it Thursday, there _must_ be some place--not Chelsea--that has different days on thetwo sides of it. And the worst of it is, the people _there_ get theirdays in the wrong order: they've got Wednesday _east_ of them, andThursday _west_--just as if their day had changed from Thursday toWednesday!" "I've heard that puzzle before!" cried Lambert. "And I'll tell you theexplanation. When a ship goes round the world from east to west, weknow 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 yougo the other way round you gain a day. " "I know all that, " said Hugh, in reply to this not very lucidexplanation: "but it doesn't help me, because the ship hasn't properdays. One way round, you get more than twenty-four hours to the day, andthe other way you get less: so of course the names get wrong: but peoplethat live on in one place always get twenty-four hours to the day. " "I suppose there _is_ such a place, " Balbus said, meditatively, "thoughI never heard of it. And the people must find it very queer, as Hughsays, 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 ofit and the old one behind it, one doesn't see exactly what happens. Imust think it over. " So they had entered the house in the state I have described--Balbuspuzzled, 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 ofthree 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 Mathesiswhispered to her niece, as they crossed the hall. And Clara had onlyjust time to whisper in reply "he meant the _whole_ party, " before theywere ushered into the library, and the sight of the five solemn facesthere assembled chilled her into silence. Her father sat at the head of the table, and mutely signed to the ladiesto take the two vacant chairs, one on each side of him. His three sonsand Balbus completed the party. Writing materials had been arrangedround the table, after the fashion of a ghostly banquet: the butler hadevidently bestowed much thought on the grim device. Sheets of quartopaper, each flanked by a pen on one side and a pencil on the other, represented the plates--penwipers did duty for rolls of bread--whileink-bottles stood in the places usually occupied by wine-glasses. The_pičce de resistance_ was a large green baize bag, which gave forth, asthe 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 thetable with his fists. This disconcerted the unpractised orator. "Sister--" he began again, then paused a moment, moved the bag to theother side, and went on with a rush, "I mean--this being--a criticaloccasion--more or less--being the year when one of my sons comes ofage--" he paused again in some confusion, having evidently got into themiddle of his speech sooner than he intended: but it was too late to goback. "Hear, hear!" cried Balbus. "Quite so, " said the old gentleman, recovering his self-possession a little: "when first I began this annualcustom--my friend Balbus will correct me if I am wrong--" (Hughwhispered "with a strap!" but nobody heard him except Lambert, who onlyfrowned and shook his head at him) "--this annual custom of giving eachof my sons as many guineas as would represent his age--it was a criticaltime--so Balbus informed me--as the ages of two of you were togetherequal to that of the third--so on that occasion I made a speech----" Hepaused so long that Balbus thought it well to come to the rescue withthe words "It was a most----" but the old man checked him with a warninglook: "yes, made a speech, " he repeated. "A few years after that, Balbuspointed out--I say pointed out--" ("Hear, hear"! cried Balbus. "Quiteso, " said the grateful old man. ) "--that it was _another_ criticaloccasion. The ages of two of you were together _double_ that of thethird. So I made another speech--another speech. And now again it's acritical occasion--so Balbus says--and I am making----" (Here MadMathesis pointedly referred to her watch) "all the haste I can!" the oldman cried, with wonderful presence of mind. "Indeed, sister, I'm comingto the point now! The number of years that have passed since that firstoccasion 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 havethe money!" "But we _know_ our ages!" cried Hugh. "Silence, sir!" thundered the old man, rising to his full height (he wasexactly 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 aboutas much as he could do to carry it) he left the room. "And _you_ shall have a similar _cadeau_, " the old lady whispered to herniece, "when you've calculated that percentage!" And she followed herbrother. Nothing could exceed the solemnity with which the old couple had risenfrom the table, and yet was it--was it a _grin_ with which the fatherturned away from his unhappy sons? Could it be--could it be a _wink_with which the aunt abandoned her despairing niece? And were those--werethose sounds of suppressed _chuckling_ which floated into the room, justbefore 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 noto'er" them (for the butler brought in the lamp): the same obligingshades 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 otherepoch) seemed likely to "restore" them--to that peace of mind which hadonce been theirs ere ever these problems had swooped upon them, andcrushed them with a load of unfathomable mystery! "It's hardly fair, " muttered Hugh, "to give us such a jumble as this towork 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 alonga level road, up a hill, and home again: their pace on the level being 4miles 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 hill1/6. Hence to go and return over the same mile, whether on the level oron the hill-side, takes 1/2 an hour. Hence in 6 hours they went 12 milesout and 12 back. If the 12 miles out had been nearly all level, theywould have taken a little over 3 hours; if nearly all up hill, a littleunder 4. Hence 3-1/2 hours must be within 1/2 an hour of the time takenin reaching the peak; thus, as they started at 3, they got there within1/2 an hour of 1/2 past 6. * * * * * Twenty-seven answers have come in. Of these, 9 are right, 16 partiallyright, and 2 wrong. The 16 give the _distance_ correctly, but they havefailed to grasp the fact that the top of the hill might have beenreached at _any_ moment between 6 o'clock and 7. The two wrong answers are from GERTY VERNON and A NIHILIST. The formermakes the distance "23 miles, " while her revolutionary companion puts itat "27. " GERTY VERNON says "they had to go 4 miles along the plain, andgot to the foot of the hill at 4 o'clock. " They _might_ have done so, Igrant; but you have no ground for saying they _did_ so. "It was 7-1/2miles to the top of the hill, and they reached that at 1/4 before 7o'clock. " Here you go wrong in your arithmetic, and I must, howeverreluctantly, bid you farewell. 7-1/2 miles, at 3 miles an hour, would_not_ require 2-3/4 hours. A NIHILIST says "Let _x_ denote the wholenumber of miles; _y_ the number of hours to hill-top; [** therefore] 3_y_ =number of miles to hill-top, and _x_-3_y_ = number of miles on the otherside. " You bewilder me. The other side of _what_? "Of the hill, " yousay. But then, how did they get home again? However, to accommodate yourviews we will build a new hostelry at the foot of the hill on theopposite side, and also assume (what I grant you is _possible_, thoughit is not _necessarily_ true) that there was no level road at all. Eventhen you go wrong. You say "_y_ = 6 - (_x_ - 3_y_)/6, . . . . . (i); _x_/4-1/2 = 6 . . . . . (ii). " I grant you (i), but I deny (ii): it rests on the assumption that to go_part_ of the time at 3 miles an hour, and the rest at 6 miles an hour, comes to the same result as going the _whole_ time at 4-1/2 miles anhour. But this would only be true if the "_part_" were an exact _half_, i. E. , if they went up hill for 3 hours, and down hill for the other 3:which they certainly did _not_ do. The sixteen, who are partially right, are AGNES BAILEY, F. K. , FIFEE, G. E. B. , H. P. , KIT, M. E. T. , MYSIE, A MOTHER'S SON, NAIRAM, AREDRUTHIAN, A SOCIALIST, SPEAR MAIDEN, T. B. C, VIS INERTIĘ, and YAK. Ofthese, F. K. , FIFEE, T. B. C, and VIS INERTIĘ do not attempt the secondpart at all. F. K. And H. P. Give no working. The rest make particularassumptions, such as that there was no level road--that there were 6miles of level road--and so on, all leading to _particular_ times beingfixed for reaching the hill-top. The most curious assumption is that ofAGNES BAILEY, who says "Let _x_ = number of hours occupied in ascent;then _x_/2 = hours occupied in descent; and 4_x_/3 = hours occupied onthe level. " I suppose you were thinking of the relative _rates_, uphill and on the level; which we might express by saying that, if theywent x miles up hill in a certain time, they would go 4_x_/3 miles onthe level _in the same time_. You have, in fact, assumed that they took_the same time_ on the level that they took in ascending the hill. FIFEEassumes that, when the aged knight said they had gone "four miles in thehour" on the level, he meant that four miles was the _distance_ gone, not merely the rate. This would have been--if FIFEE will excuse theslang 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 are BLITHE, E. W. , L. B. , AMARLBOROUGH BOY, O. V. L. , PUTNEY WALKER, ROSE, SEA BREEZE, SIMPLESUSAN, and MONEY SPINNER. (These last two I count as one, as they send ajoint answer. ) ROSE and SIMPLE SUSAN and CO. Do not actually state thatthe hill-top was reached some time between 6 and 7, but, as they haveclearly grasped the fact that a mile, ascended and descended, took thesame time as two level miles, I mark them as "right. " A MARLBOROUGH BOYand PUTNEY WALKER deserve honourable mention for their algebraicalsolutions being the only two who have perceived that the question leadsto _an indeterminate equation_. E. W. Brings a charge of untruthfulnessagainst the aged knight--a serious charge, for he was the very pink ofchivalry! She says "According to the data given, the time at the summitaffords no clue to the total distance. It does not enable us to stateprecisely to an inch how much level and how much hill there was on theroad. " "Fair damsel, " the aged knight replies, "--if, as I surmise, thyinitials denote Early Womanhood--bethink thee that the word 'enable' isthine, not mine. I did but ask the time of reaching the hill-top as my_condition_ for further parley. If _now_ thou wilt not grant that I am atruth-loving man, then will I affirm that those same initials denoteEnvenomed Wickedness!" CLASS LIST. I. A MARLBOROUGH BOY. PUTNEY WALKER. II. BLITHE. E. W. L. B. O. V. L. ROSE. SEA BREEZE. {SIMPLE SUSAN. {MONEY-SPINNER. BLITHE has made so ingenious an addition to the problem, and SIMPLESUSAN and CO. Have solved it in such tuneful verse, that I record boththeir answers in full. I have altered a word or two in BLITHE'S--which Itrust she will excuse; it did not seem quite clear as it stood. * * * * * "Yet stay, " said the youth, as a gleam of inspiration lighted up therelaxing muscles of his quiescent features. "Stay. Methinks it matterslittle _when_ we reached that summit, the crown of our toil. For in thespace of time wherein we clambered up one mile and bounded down the sameon our return, we could have trudged the _twain_ on the level. We haveplodded, then, four-and-twenty miles in these six mortal hours; fornever a moment did we stop for catching of fleeting breath or for gazingon 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. Nowmark me! For every five minutes that had fled since six of the clockwhen we stood on yonder peak, so many miles had we toiled upwards on thedreary mountainside!" The youth moaned and rushed into the hostel. BLITHE. The elder and the younger knight, They sallied forth at three; How far they went on level ground It matters not to me; What time they reached the foot of hill, When they began to mount, Are problems which I hold to be Of very small account. The moment that each waved his hat Upon the topmost peak-- To trivial query such as this No answer will I seek. Yet can I tell the distance well They must have travelled o'er: On hill and plain, 'twixt three and nine, The miles were twenty-four. Four miles an hour their steady pace Along the level track, Three when they climbed--but six when they Came swiftly striding back Adown the hill; and little skill It needs, methinks, to show, Up hill and down together told, Four miles an hour they go. For whether long or short the time Upon the hill they spent, Two thirds were passed in going up, One third in the descent. Two thirds at three, one third at six, If rightly reckoned o'er, Will make one whole at four--the tale Is tangled now no more. SIMPLE SUSAN. MONEY SPINNER. ANSWERS TO KNOT II. § 1. THE DINNER PARTY. _Problem. _--"The Governor of Kgovjni wants to give a very small dinnerparty, and invites his father's brother-in-law, his brother'sfather-in-law, his father-in-law's brother, and his brother-in-law'sfather. Find the number of guests. " _Answer. _--"One. " * * * * * In this genealogy, males are denoted by capitals, and females by smallletters. The Governor is E and his guest is C. A = a | +------+-+----+ | | | b = B D = d C = c | | | | +---++--+ +-+-+ | | | | | | e = E | g = G | F ========= f Ten answers have been received. Of these, one is wrong, GALANTHUSNIVALIS MAJOR, who insists on inviting _two_ guests, one being theGovernor's _wife's brother's father_. If she had taken his _sister'shusband's father_ instead, she would have found it possible to reducethe guests to _one_. Of the nine who send right answers, SEA-BREEZE is the very faintestbreath that ever bore the name! She simply states that the Governor'suncle might fulfill all the conditions "by intermarriages"! "Wind of thewestern sea, " you have had a very narrow escape! Be thankful to appearin the Class-list at all! BOG-OAK and BRADSHAW OF THE FUTURE usegenealogies which require 16 people instead of 14, by inviting theGovernor's _father's sister's husband_ instead of his _father's wife'sbrother_. I cannot think this so good a solution as one that requiresonly 14. CAIUS and VALENTINE deserve special mention as the only two whohave supplied genealogies. CLASS LIST. I. BEE. CAIUS. M. M. MATTHEW MATTICKS. OLD CAT. VALENTINE. II. BOG-OAK. BRADSHAW OF THE FUTURE. III. SEA-BREEZE. § 2. THE LODGINGS. _Problem. _--"A Square has 20 doors on each side, which contains 21 equalparts. They are numbered all round, beginning at one corner. From whichof the four, Nos. 9, 25, 52, 73, is the sum of the distances, to theother three, least?" _Answer. _--"From No. 9. " * * * * * [Illustration] Let A be No. 9, B No. 25, C No. 52, and D No. 73. Then AB = [** sqrt](12^{2} + 5^{2}) = [** sqrt]169 = 13; AC = 21; AD = [** sqrt](9^{2} + 8^{2}) = [** sqrt]145 = 12 + (N. B. _i. E. _ "between 12 and 13. ") BC = [** sqrt](16^{2} + 12^{2}) = [** sqrt]400 = 20; BD = [** sqrt](3^{2} + 21^{2}) = [** sqrt]450 = 21+; CD = [** sqrt](9^{2} + 13^{2}) = [** sqrt]250 = 15+; Hence sum of distances from A is between 46 and 47; from B, between 54and 55; from C, between 56 and 57; from D, between 48 and 51. (Why not"between 48 and 49"? Make this out for yourselves. ) Hence the sum isleast for A. * * * * * Twenty-five solutions have been received. Of these, 15 must be marked"0, " 5 are partly right, and 5 right. Of the 15, I may dismissALPHABETICAL PHANTOM, BOG-OAK, DINAH MITE, FIFEE, GALANTHUS NIVALISMAJOR (I fear the cold spring has blighted our SNOWDROP), GUY, H. M. S. PINAFORE, JANET, and VALENTINE with the simple remark that they insiston the unfortunate lodgers _keeping to the pavement_. (I used the words"crossed to Number Seventy-three" for the special purpose of showingthat _short cuts_ were possible. ) SEA-BREEZE does the same, and addsthat "the result would be the same" even if they crossed the Square, butgives no proof of this. M. M. Draws a diagram, and says that No. 9 isthe house, "as the diagram shows. " I cannot see _how_ it does so. OLDCAT assumes that the house _must_ be No. 9 or No. 73. She does notexplain how she estimates the distances. Bee's Arithmetic is faulty: shemakes [** sqrt]169 + [** sqrt]442 + [** sqrt]130 = 741. (I suppose youmean [** sqrt]741, which would be a little nearer the truth. But rootscannot be added in this manner. Do you think [** sqrt]9 + [** sqrt]16 is25, or even [** sqrt]25?) But AYR'S state is more perilous still: shedraws illogical conclusions with a frightful calmness. After pointingout (rightly) that AC is less than BD she says, "therefore the nearesthouse to the other three must be A or C. " And again, after pointing out(rightly) that B and D are both within the half-square containing A, she says "therefore" AB + AD must be less than BC + CD. (There is nological force in either "therefore. " For the first, try Nos. 1, 21, 60, 70: this will make your premiss true, and your conclusion false. Similarly, for the second, try Nos. 1, 30, 51, 71. ) Of the five partly-right solutions, RAGS AND TATTERS and MAD HATTER (whosend one answer between them) make No. 25 6 units from the cornerinstead of 5. CHEAM, E. R. D. L. , and MEGGY POTTS leave openings at thecorners of the Square, which are not in the _data_: moreover CHEAM givesvalues for the distances without any hint that they are only_approximations_. CROPHI AND MOPHI make the bold and unfoundedassumption that there were really 21 houses on each side, instead of 20as stated by Balbus. "We may assume, " they add, "that the doors of Nos. 21, 42, 63, 84, are invisible from the centre of the Square"! What isthere, I wonder, that CROPHI AND MOPHI would _not_ assume? Of the five who are wholly right, I think BRADSHAW OF THE FUTURE, CAIUS, CLIFTON C. , and MARTREB deserve special praise for their full_analytical_ solutions. MATTHEW MATTICKS picks out No. 9, and proves itto be the right house in two ways, very neatly and ingeniously, but_why_ he picks it out does not appear. It is an excellent _synthetical_proof, but lacks the analysis which the other four supply. CLASS LIST. I. BRADSHAW OF THE FUTURE CAIUS. CLIFTON C. MARTREB. II. MATTHEW MATTICKS. III. CHEAM. CROPHI AND MOPHI. E. R. D. L. MEGGY POTTS. {RAGS AND TATTERS. {MAD HATTER. A remonstrance has reached me from SCRUTATOR on the subject of KNOT I. , which he declares was "no problem at all. " "Two questions, " he says, "are put. To solve one there is no data: the other answers itself. " Asto the first point, SCRUTATOR is mistaken; there _are_ (not "is") datasufficient to answer the question. As to the other, it is interesting toknow that the question "answers itself, " and I am sure it does thequestion great credit: still I fear I cannot enter it on the list ofwinners, as this competition is only open to human beings. ANSWERS TO KNOT III. _Problem. _--(1) "Two travellers, starting at the same time, wentopposite ways round a circular railway. Trains start each way every 15minutes, the easterly ones going round in 3 hours, the westerly in 2. How many trains did each meet on the way, not counting trains met at theterminus itself?" (2) "They went round, as before, each travellercounting as 'one' the train containing the other traveller. How many dideach meet?" _Answers. _--(1) 19. (2) The easterly traveller met 12; the other 8. * * * * * The trains one way took 180 minutes, the other way 120. Let us take theL. C. M. , 360, and divide the railway into 360 units. Then one set oftrains went at the rate of 2 units a minute and at intervals of 30units; the other at the rate of 3 units a minute and at intervals of 45units. An easterly train starting has 45 units between it and the firsttrain it will meet: it does 2-5ths of this while the other does 3-5ths, and thus meets it at the end of 18 units, and so all the way round. Awesterly train starting has 30 units between it and the first train itwill meet: it does 3-5ths of this while the other does 2-5ths, and thusmeets it at the end of 18 units, and so all the way round. Hence if therailway be divided, by 19 posts, into 20 parts, each containing 18units, trains meet at every post, and, in (1), each traveller passes 19posts in going round, and so meets 19 trains. But, in (2), the easterlytraveller only begins to count after traversing 2-5ths of the journey, _i. E. _, on reaching the 8th post, and so counts 12 posts: similarly theother counts 8. They meet at the end of 2-5ths of 3 hours, or 3-5ths of2 hours, _i. E. _, 72 minutes. * * * * * Forty-five answers have been received. Of these 12 are beyond the reachof discussion, as they give no working. I can but enumerate their names. ARDMORE, E. A. , F. A. D. , L. D. , MATTHEW MATTICKS, M. E. T. , POO-POO, and THE RED QUEEN are all wrong. BETA and ROWENA have got (1) right and(2) wrong. CHEEKY BOB and NAIRAM give the right answers, but it mayperhaps make the one less cheeky, and induce the other to take a lessinverted view of things, to be informed that, if this had been acompetition for a prize, they would have got no marks. [N. B. --I havenot ventured to put E. A. 's name in full, as she only gave itprovisionally, in case her answer should prove right. ] Of the 33 answers for which the working is given, 10 are wrong; 11half-wrong and half-right; 3 right, except that they cherish thedelusion that it was _Clara_ who travelled in the easterly train--apoint which the data do not enable us to settle; and 9 wholly right. The 10 wrong answers are from BO-PEEP, FINANCIER, I. W. T. , KATE B. , M. A. H. , Q. Y. Z. , SEA-GULL, THISTLEDOWN, TOM-QUAD, and an unsigned one. BO-PEEP rightly says that the easterly traveller met all trains whichstarted during the 3 hours of her trip, as well as all which startedduring the previous 2 hours, _i. E. _, all which started at thecommencements of 20 periods of 15 minutes each; and she is right instriking out the one she met at the moment of starting; but wrong instriking out the _last_ train, for she did not meet this at theterminus, but 15 minutes before she got there. She makes the samemistake in (2). FINANCIER thinks that any train, met for the secondtime, is not to be counted. I. W. T. Finds, by a process which is notstated, that the travellers met at the end of 71 minutes and 26-1/2seconds. KATE B. Thinks the trains which are met on starting and onarriving are _never_ to be counted, even when met elsewhere. Q. Y. Z. Tries a rather complex algebraical solution, and succeeds in finding thetime of meeting correctly: all else is wrong. SEA-GULL seems to thinkthat, in (1), the easterly train _stood still_ for 3 hours; and saysthat, in (2), the travellers met at the end of 71 minutes 40 seconds. THISTLEDOWN nobly confesses to having tried no calculation, but merelyhaving drawn a picture of the railway and counted the trains; in (1), she counts wrong; in (2) she makes them meet in 75 minutes. TOM-QUADomits (1): in (2) he makes Clara count the train she met on her arrival. The unsigned one is also unintelligible; it states that the travellersgo "1-24th more than the total distance to be traversed"! The "Clara"theory, already referred to, is adopted by 5 of these, viz. , BO-PEEP, FINANCIER, KATE B. , TOM-QUAD, and the nameless writer. The 11 half-right answers are from BOG-OAK, BRIDGET, CASTOR, CHESHIRECAT, G. E. B. , GUY, MARY, M. A. H. , OLD MAID, R. W. , and VENDREDI. Allthese adopt the "Clara" theory. CASTOR omits (1). VENDREDI gets (1)right, but in (2) makes the same mistake as BO-PEEP. I notice in yoursolution a marvellous proportion-sum:--"300 miles: 2 hours :: one mile:24 seconds. " May I venture to advise your acquiring, as soon aspossible, an utter disbelief in the possibility of a ratio existingbetween _miles_ and _hours_? Do not be disheartened by your two friends'sarcastic remarks on your "roundabout ways. " Their short method, ofadding 12 and 8, has the slight disadvantage of bringing the answerwrong: even a "roundabout" method is better than _that_! M. A. H. , in(2), makes the travellers count "one" _after_ they met, not _when_ theymet. CHESHIRE CAT and OLD MAID get "20" as answer for (1), by forgettingto strike out the train met on arrival. The others all get "18" invarious ways. BOG-OAK, GUY, and R. W. Divide the trains which thewesterly traveller has to meet into 2 sets, viz. , those already on theline, which they (rightly) make "11, " and those which started during her2 hours' journey (exclusive of train met on arrival), which they(wrongly) make "7"; and they make a similar mistake with the easterlytrain. BRIDGET (rightly) says that the westerly traveller met a trainevery 6 minutes for 2 hours, but (wrongly) makes the number "20"; itshould be "21. " G. E. B. Adopts BO-PEEP'S method, but (wrongly) strikesout (for the easterly traveller) the train which started at the_commencement_ of the previous 2 hours. MARY thinks a train, met onarrival, must not be counted, even when met on a _previous_ occasion. The 3, who are wholly right but for the unfortunate "Clara" theory, areF. LEE, G. S. C. , and X. A. B. And now "descend, ye classic Ten!" who have solved the whole problem. Your names are AIX-LES-BAINS, ALGERNON BRAY (thanks for a friendlyremark, which comes with a heart-warmth that not even the Atlantic couldchill), ARVON, BRADSHAW OF THE FUTURE, FIFEE, H. L. R. , J. L. O. , OMEGA, S. S. G. , and WAITING FOR THE TRAIN. Several of these have put Clara, provisionally, into the easterly train: but they seem to have understoodthat the data do not decide that point. CLASS LIST. I. AIX-LES-BAINS. ALGERNON BRAY. BRADSHAW OF THE FUTURE. FIFEE. H. L. R. OMEGA. S. S. G. WAITING FOR THE TRAIN. II. ARVON. J. L. O. III. F. LEE. G. S. C. X. A. B. ANSWERS TO KNOT IV. _Problem. _--"There are 5 sacks, of which Nos. 1, 2, weigh 12 lbs. ; Nos. 2, 3, 13-1/2 lbs. ; Nos. 3, 4, 11-1/2 lbs. ; Nos. 4, 5, 8 lbs. ; Nos. 1, 3, 5, 16 lbs. Required the weight of each sack. " _Answer. _--"5-1/2, 6-1/2, 7, 4-1/2, 3-1/2. " * * * * * The sum of all the weighings, 61 lbs. , includes sack No. 3 _thrice_ andeach other _twice_. Deducting twice the sum of the 1st and 4thweighings, we get 21 lbs. For _thrice_ No. 3, _i. E. _, 7 lbs. For No. 3. Hence, the 2nd and 3rd weighings give 6-1/2 lbs. , 4-1/2 lbs. For Nos. 2, 4; and hence again, the 1st and 4th weighings give 5-1/2 lbs. , 3-1/2lbs. , for Nos. 1, 5. * * * * * Ninety-seven answers have been received. Of these, 15 are beyond thereach of discussion, as they give no working. I can but enumerate theirnames, and I take this opportunity of saying that this is the last timeI shall put on record the names of competitors who give no sort of clueto the process by which their answers were obtained. In guessing aconundrum, or in catching a flea, we do not expect the breathless victorto give us afterwards, in cold blood, a history of the mental ormuscular efforts by which he achieved success; but a mathematicalcalculation is another thing. The names of this "mute inglorious" bandare COMMON SENSE, D. E. R. , DOUGLAS, E. L. , ELLEN, I. M. T. , J. M. C. , JOSEPH, KNOT I, LUCY, MEEK, M. F. C. , PYRAMUS, SHAH, VERITAS. Of the eighty-two answers with which the working, or some approach toit, is supplied, one is wrong: seventeen have given solutions which are(from one cause or another) practically valueless: the remainingsixty-four I shall try to arrange in a Class-list, according to thevarying degrees of shortness and neatness to which they seem to haveattained. The solitary wrong answer is from NELL. To be thus "alone in the crowd"is a distinction--a painful one, no doubt, but still a distinction. I amsorry for you, my dear young lady, and I seem to hear your tearfulexclamation, when you read these lines, "Ah! This is the knell of all myhopes!" Why, oh why, did you assume that the 4th and 5th bags weighed 4lbs. Each? And why did you not test your answers? However, please tryagain: and please don't change your _nom-de-plume_: let us have NELL inthe First Class next time! The seventeen whose solutions are practically valueless are ARDMORE, AREADY RECKONER, ARTHUR, BOG-LARK, BOG-OAK, BRIDGET, FIRST ATTEMPT, J. L. C. , M. E. T. , ROSE, ROWENA, SEA-BREEZE, SYLVIA, THISTLEDOWN, THREE-FIFTHS ASLEEP, VENDREDI, and WINIFRED. BOG-LARK tries it by a sortof "rule of false, " assuming experimentally that Nos. 1, 2, weigh 6 lbs. Each, and having thus produced 17-1/2, instead of 16, as the weight of1, 3, and 5, she removes "the superfluous pound and a half, " but doesnot explain how she knows from which to take it. THREE-FIFTHS ASLEEPsays that (when in that peculiar state) "it seemed perfectly clear" toher that, "3 out of the 5 sacks being weighed twice over, 3/5 of 45 =27, must be the total weight of the 5 sacks. " As to which I can onlysay, with the Captain, "it beats me entirely!" WINIFRED, on the pleathat "one must have a starting-point, " assumes (what I fear is a mereguess) that No. 1 weighed 5-1/2 lbs. The rest all do it, wholly orpartly, by guess-work. The problem is of course (as any Algebraist sees at once) a case of"simultaneous simple equations. " It is, however, easily soluble byArithmetic only; and, when this is the case, I hold that it is badworkmanship to use the more complex method. I have not, this time, givenmore credit to arithmetical solutions; but in future problems I shall(other things being equal) give the highest marks to those who use thesimplest machinery. I have put into Class I. Those whose answers seemedspecially short and neat, and into Class III. Those that seemedspecially long or clumsy. Of this last set, A. C. M. , FURZE-BUSH, JAMES, PARTRIDGE, R. W. , and WAITING FOR THE TRAIN, have sent long wanderingsolutions, the substitutions having no definite method, but seeming tohave been made to see what would come of it. CHILPOME and DUBLIN BOYomit some of the working. ARVON MARLBOROUGH BOY only finds the weight of_one_ sack. CLASS LIST I. B. E. D. C. H. CONSTANCE JOHNSON. GREYSTEAD. GUY. HOOPOE. J. F. A. M. A. H. NUMBER FIVE. PEDRO. R. E. X. SEVEN OLD MEN. VIS INERTIĘ. WILLY B. YAHOO. II. AMERICAN SUBSCRIBER. AN APPRECIATIVE SCHOOLMA'AM. AYR. BRADSHAW OF THE FUTURE. CHEAM. C. M. G. DINAH MITE. DUCKWING. E. C. M. E. N. Lowry. ERA. EUROCLYDON. F. H. W. FIFEE. G. E. B. HARLEQUIN. HAWTHORN. HOUGH GREEN. J. A. B. JACK TAR. J. B. B. KGOVJNI. LAND LUBBER. L. D. MAGPIE. MARY. MHRUXI. MINNIE. MONEY-SPINNER. NAIRAM. OLD CAT. POLICHINELLE. SIMPLE SUSAN. S. S. G. THISBE. VERENA. WAMBA. WOLFE. WYKEHAMICUS. Y. M. A. H. III. A. C. M. ARVON MARLBOROUGH BOY. CHILPOME. DUBLIN BOY. FURZE-BUSH. JAMES. PARTRIDGE. R. W. WAITING FOR THE TRAIN. ANSWERS TO KNOT V. _Problem. _--To mark pictures, giving 3 x's to 2 or 3, 2 to 4 or 5, and 1to 9 or 10; also giving 3 o's to 1 or 2, 2 to 3 or 4 and 1 to 8 or 9; soas to mark the smallest possible number of pictures, and to give themthe largest possible number of marks. _Answer. _--10 pictures; 29 marks; arranged thus:-- x x x x x x x x x o x x x x x o o o o x x o o o o o o o o _Solution. _--By giving all the x's possible, putting into brackets theoptional ones, we get 10 pictures marked thus:-- x x x x x x x x x (x) x x x x (x) x x (x) By then assigning o's in the same way, beginning at the other end, weget 9 pictures marked thus:-- (o) o (o) o o o (o) o o o o o o o o All we have now to do is to run these two wedges as close together asthey will go, so as to get the minimum number of pictures----erasingoptional marks where by so doing we can run them closer, but otherwiseletting them stand. There are 10 necessary marks in the 1st row, and inthe 3rd; but only 7 in the 2nd. Hence we erase all optional marks in the1st and 3rd rows, but let them stand in the 2nd. * * * * * Twenty-two answers have been received. Of these 11 give no working; so, in accordance with what I announced in my last review of answers, Ileave them unnamed, merely mentioning that 5 are right and 6 wrong. Of the eleven answers with which some working is supplied, 3 are wrong. C. H. Begins with the rash assertion that under the given conditions"the sum is impossible. For, " he or she adds (these initialedcorrespondents are dismally vague beings to deal with: perhaps "it"would be a better pronoun), "10 is the least possible number ofpictures" (granted): "therefore we must either give 2 x's to 6, or 2 o'sto 5. " Why "must, " oh alphabetical phantom? It is nowhere ordained thatevery picture "must" have 3 marks! FIFEE sends a folio page of solution, which deserved a better fate: she offers 3 answers, in each of which 10pictures are marked, with 30 marks; in one she gives 2 x's to 6pictures; in another to 7; in the 3rd she gives 2 o's to 5; thus inevery case ignoring the conditions. (I pause to remark that thecondition "2 x's to 4 or 5 pictures" can only mean "_either_ to 4 _orelse_ to 5": if, as one competitor holds, it might mean _any_ number notless than 4, the words "_or_ 5" would be superfluous. ) I. E. A. (I amhappy to say that none of these bloodless phantoms appear this time inthe class-list. Is it IDEA with the "D" left out?) gives 2 x's to 6pictures. She then takes me to task for using the word "ought" insteadof "nought. " No doubt, to one who thus rebels against the rules laiddown for her guidance, the word must be distasteful. But does not I. E. A. Remember the parallel case of "adder"? That creature was originally"a nadder": then the two words took to bandying the poor "n" backwardsand forwards like a shuttlecock, the final state of the game being "anadder. " May not "a nought" have similarly become "an ought"? Anyhow, "oughts and crosses" is a very old game. I don't think I ever heard itcalled "noughts and crosses. " In the following Class-list, I hope the solitary occupant of III. Willsheathe her claws when she hears how narrow an escape she has had of notbeing named at all. Her account of the process by which she got theanswer is so meagre that, like the nursery tale of "Jack-a-Minory" (Itrust I. E. A. Will be merciful to the spelling), it is scarcely to bedistinguished from "zero. " CLASS LIST. I. GUY. OLD CAT. SEA-BREEZE. II. AYR. BRADSHAW OF THE FUTURE. F. LEE. H. VERNON. III. CAT. ANSWERS TO KNOT VI. _Problem 1. _--_A_ and _B_ began the year with only 1, 000_l. _ a-piece. They borrowed nought; they stole nought. On the next New-Year's Day theyhad 60, 000_l. _ between them. How did they do it? _Solution. _--They went that day to the Bank of England. _A_ stood infront of it, while _B_ went round and stood behind it. * * * * * Two answers have been received, both worthy of much honour. ADDLEPATEmakes them borrow "0" and steal "0, " and uses both cyphers by puttingthem at the right-hand end of the 1, 000_l. _, thus producing 100, 000_l. _, which is well over the mark. But (or to express it in Latin) AT SPESINFRACTA has solved it even more ingeniously: with the first cypher sheturns the "1" of the 1, 000_l. _ into a "9, " and adds the result to theoriginal sum, thus getting 10, 000_l. _: and in this, by means of theother "0, " she turns the "1" into a "6, " thus hitting the exact60, 000_l. _ CLASS LIST I. AT SPES INFRACTA. II. ADDLEPATE. * * * * * _Problem 2. _--_L_ makes 5 scarves, while _M_ makes 2: _Z_ makes 4 while_L_ makes 3. Five scarves of _Z_'s weigh one of _L_'s; 5 of _M_'s weigh3 of _Z_'s. One of _M_'s is as warm as 4 of _Z_'s: and one of _L_'s aswarm as 3 of _M_'s. Which is best, giving equal weight in the result torapidity of work, lightness, and warmth? _Answer. _--The order is _M_, _L_, _Z_. * * * * * _Solution. _--As to rapidity (other things being constant) _L_'s merit isto _M_'s in the ratio of 5 to 2: _Z_'s to _L_'s in the ratio of 4 to 3. In order to get one set of 3 numbers fulfilling these conditions, it isperhaps simplest to take the one that occurs _twice_ as unity, andreduce the others to fractions: this gives, for _L_, _M_, and _Z_, themarks 1, 2/5, 4/3. In estimating for _lightness_, we observe that thegreater the weight, the less the merit, so that _Z_'s merit is to _L_'sas 5 to 1. Thus the marks for _lightness_ are 1/5, 5/3, 1. Andsimilarly, the marks for warmth are 3, 1, 1/4. To get the total result, we must _multiply_ _L_'s 3 marks together, and do the same for _M_ andfor _Z_. The final numbers are 1 × 1/5 × 3, 2/5 × 5/3 × 1, 4/3 × 1 ×1/4; _i. E. _ 3/5, 2/3, 1/3; _i. E. _ multiplying throughout by 15 (whichwill not alter the proportion), 9, 10, 5; showing the order of merit tobe _M_, _L_, _Z_. * * * * * Twenty-nine answers have been received, of which five are right, andtwenty-four wrong. These hapless ones have all (with three exceptions)fallen into the error of _adding_ the proportional numbers together, foreach candidate, instead of _multiplying_. _Why_ the latter is right, rather than the former, is fully proved in text-books, so I will notoccupy space by stating it here: but it can be _illustrated_ very easilyby the case of length, breadth, and depth. Suppose _A_ and _B_ are rivaldiggers of rectangular tanks: the amount of work done is evidentlymeasured by the number of _cubical feet_ dug out. Let _A_ dig a tank 10feet long, 10 wide, 2 deep: let _B_ dig one 6 feet long, 5 wide, 10deep. The cubical contents are 200, 300; _i. E. _ _B_ is best digger inthe ratio of 3 to 2. Now try marking for length, width, and depth, separately; giving a maximum mark of 10 to the best in each contest, andthen _adding_ the results! Of the twenty-four malefactors, one gives no working, and so has no realclaim to be named; but I break the rule for once, in deference to itssuccess in Problem 1: he, she, or it, is ADDLEPATE. The othertwenty-three may be divided into five groups. First and worst are, I take it, those who put the rightful winner_last_; arranging them as "Lolo, Zuzu, Mimi. " The names of thesedesperate wrong-doers are AYR, BRADSHAW OF THE FUTURE, FURZE-BUSH andPOLLUX (who send a joint answer), GREYSTEAD, GUY, OLD HEN, and SIMPLESUSAN. The latter was _once_ best of all; the Old Hen has takenadvantage of her simplicity, and beguiled her with the chaff which wasthe bane of her own chickenhood. Secondly, I point the finger of scorn at those who have put the worstcandidate at the top; arranging them as "Zuzu, Mimi, Lolo. " They areGRAECIA, M. M. , OLD CAT, and R. E. X. "'Tis Greece, but----. " The third set have avoided both these enormities, and have evensucceeded in putting the worst last, their answer being "Lolo, Mimi, Zuzu. " Their names are AYR (who also appears among the "quite too too"), CLIFTON C. , F. B. , FIFEE, GRIG, JANET, and MRS. SAIREY GAMP. F. B. Hasnot fallen into the common error; she _multiplies_ together theproportionate numbers she gets, but in getting them she goes wrong, byreckoning warmth as a _de_-merit. Possibly she is "Freshly Burnt, " orcomes "From Bombay. " JANET and MRS. SAIREY GAMP have also avoided thiserror: the method they have adopted is shrouded in mystery--I scarcelyfeel competent to criticize it. MRS. GAMP says "if Zuzu makes 4 whileLolo makes 3, Zuzu makes 6 while Lolo makes 5 (bad reasoning), whileMimi makes 2. " From this she concludes "therefore Zuzu excels in speedby 1" (_i. E. _ when compared with Lolo; but what about Mimi?). She thencompares the 3 kinds of excellence, measured on this mystic scale. JANETtakes the statement, that "Lolo makes 5 while Mimi makes 2, " to provethat "Lolo makes 3 while Mimi makes 1 and Zuzu 4" (worse reasoning thanMRS. GAMP'S), and thence concludes that "Zuzu excels in speed by 1/8"!JANET should have been ADELINE, "mystery of mysteries!" The fourth set actually put Mimi at the top, arranging them as "Mimi, Zuzu, Lolo. " They are MARQUIS AND CO. , MARTREB, S. B. B. (first initialscarcely legible: _may_ be meant for "J"), and STANZA. The fifth set consist of AN ANCIENT FISH and CAMEL. These ill-assortedcomrades, by dint of foot and fin, have scrambled into the right answer, but, as their method is wrong, of course it counts for nothing. Also ANANCIENT FISH has very ancient and fishlike ideas as to _how_ numbersrepresent merit: she says "Lolo gains 2-1/2 on Mimi. " Two and a half_what_? Fish, fish, art thou in thy duty? Of the five winners I put BALBUS and THE ELDER TRAVELLER slightly belowthe other three--BALBUS for defective reasoning, the other for scantyworking. BALBUS gives two reasons for saying that _addition_ of marks is_not_ the right method, and then adds "it follows that the decision mustbe made by _multiplying_ the marks together. " This is hardly morelogical than to say "This is not Spring: _therefore_ it must be Autumn. " CLASS LIST. I. DINAH MITE. E. B. D. L. JORAM. II. BALBUS. THE ELDER TRAVELLER. * * * * * With regard to Knot V. , I beg to express to VIS INERTIĘ and to anyothers who, like her, understood the condition to be that _every_ markedpicture must have _three_ marks, my sincere regret that the unfortunatephrase "_fill_ the columns with oughts and crosses" should have causedthem to waste so much time and trouble. I can only repeat that a_literal_ interpretation of "fill" would seem to _me_ to require that_every_ picture in the gallery should be marked. VIS INERTIĘ would havebeen in the First Class if she had sent in the solution she now offers. ANSWERS TO KNOT VII. _Problem. _--Given that one glass of lemonade, 3 sandwiches, and 7biscuits, cost 1_s. _ 2_d. _; and that one glass of lemonade, 4sandwiches, and 10 biscuits, cost 1_s. _ 5_d. _: find the cost of (1) aglass of lemonade, a sandwich, and a biscuit; and (2) 2 glasses oflemonade, 3 sandwiches, and 5 biscuits. _Answer. _--(1) 8_d. _; (2) 1_s. _ 7_d. _ _Solution. _--This is best treated algebraically. Let _x_ = the cost (inpence) of a glass of lemonade, _y_ of a sandwich, and _z_ of a biscuit. Then we have _x_ + 3_y_ + 7_z_ = 14, and _x_ + 4_y_ + 10_z_ = 17. And werequire the values of _x_ + _y_ + _z_, and of 2_x_ + 3_y_ + 5_z_. Now, from _two_ equations only, we cannot find, _separately_, the values of_three_ unknowns: certain _combinations_ of them may, however, be found. Also we know that we can, by the help of the given equations, eliminate2 of the 3 unknowns from the quantity whose value is required, whichwill then contain one only. If, then, the required value isascertainable at all, it can only be by the 3rd unknown vanishing ofitself: otherwise the problem is impossible. Let us then eliminate lemonade and sandwiches, and reduce everything tobiscuits--a state of things even more depressing than "if all the worldwere apple-pie"--by subtracting the 1st equation from the 2nd, whicheliminates lemonade, and gives _y_ + 3_z_ = 3, or _y_ = 3-3_z_; and thensubstituting this value of _y_ in the 1st, which gives _x_-2_z_ = 5, _i. E. _ _x_ = 5 + 2_z_. Now if we substitute these values of _x_, _y_, inthe quantities whose values are required, the first becomes (5 + 2_z_) +(3-3_z_) + _z_, _i. E. _ 8: and the second becomes 2(5 + 2_z_) + 3(3-3_z_)+ 5_z_, _i. E. _ 19. Hence the answers are (1) 8_d. _, (2) 1_s. _ 7_d. _ * * * * * The above is a _universal_ method: that is, it is absolutely certaineither to produce the answer, or to prove that no answer is possible. The question may also be solved by combining the quantities whose valuesare given, so as to form those whose values are required. This is merelya matter of ingenuity and good luck: and as it _may_ fail, even when thething is possible, and is of no use in proving it _im_possible, I cannotrank this method as equal in value with the other. Even when itsucceeds, it may prove a very tedious process. Suppose the 26competitors, who have sent in what I may call _accidental_ solutions, had had a question to deal with where every number contained 8 or 10digits! I suspect it would have been a case of "silvered is the ravenhair" (see "Patience") before any solution would have been hit on bythe most ingenious of them. Forty-five answers have come in, of which 44 give, I am happy to say, some sort of _working_, and therefore deserve to be mentioned by name, and to have their virtues, or vices as the case may be, discussed. Thirteen have made assumptions to which they have no right, and socannot figure in the Class-list, even though, in 10 of the 13 cases, theanswer is right. Of the remaining 28, no less than 26 have sent in_accidental_ solutions, and therefore fall short of the highest honours. I will now discuss individual cases, taking the worst first, as mycustom is. FROGGY gives no working--at least this is all he gives: after statingthe given equations, he says "therefore the difference, 1 sandwich + 3biscuits, = 3_d. _": then follow the amounts of the unknown bills, withno further hint as to how he got them. FROGGY has had a _very_ narrowescape of not being named at all! Of those who are wrong, VIS INERTIĘ has sent in a piece of incorrectworking. Peruse the horrid details, and shudder! She takes _x_ (call it"_y_") as the cost of a sandwich, and concludes (rightly enough) that abiscuit will cost (3-_y_)/3. She then subtracts the second equation fromthe first, and deduces 3_y_ + 7 × (3-_y_)/3-4_y_ + 10 × (3-_y_)/3 = 3. By making two mistakes in this line, she brings out _y_ = 3/2. Try itagain, oh VIS INERTIĘ! Away with INERTIĘ: infuse a little more VIS: andyou will bring out the correct (though uninteresting) result, 0 = 0!This will show you that it is hopeless to try to coax any one of these 3unknowns to reveal its _separate_ value. The other competitor, who iswrong throughout, is either J. M. C. Or T. M. C. : but, whether he be aJuvenile Mis-Calculator or a True Mathematician Confused, he makes theanswers 7_d. _ and 1_s. _ 5_d. _ He assumes, with Too Much Confidence, thatbiscuits were 1/2_d. _ each, and that Clara paid for 8, though she onlyate 7! We will now consider the 13 whose working is wrong, though the answer isright: and, not to measure their demerits too exactly, I will take themin alphabetical order. ANITA finds (rightly) that "1 sandwich and 3biscuits cost 3_d. _, " and proceeds "therefore 1 sandwich = 1-1/2_d. _, 3biscuits = 1-1/2_d. _, 1 lemonade = 6_d. _" DINAH MITE begins like ANITA:and thence proves (rightly) that a biscuit costs less than a 1_d. _:whence she concludes (wrongly) that it _must_ cost 1/2_d. _ F. C. W. Isso beautifully resigned to the certainty of a verdict of "guilty, " thatI have hardly the heart to utter the word, without adding a "recommendedto mercy owing to extenuating circumstances. " But really, you know, where _are_ the extenuating circumstances? She begins by assuming thatlemonade is 4_d. _ a glass, and sandwiches 3_d. _ each, (making with the 2given equations, _four_ conditions to be fulfilled by _three_ miserableunknowns!). And, having (naturally) developed this into a contradiction, she then tries 5_d. _ and 2_d. _ with a similar result. (N. B. _This_process might have been carried on through the whole of the TertiaryPeriod, without gratifying one single Megatherium. ) She then, by a"happy thought, " tries half-penny biscuits, and so obtains a consistentresult. This may be a good solution, viewing the problem as a conundrum:but it is _not_ scientific. JANET identifies sandwiches with biscuits!"One sandwich + 3 biscuits" she makes equal to "4. " Four _what_? MAYFAIRmakes the astounding assertion that the equation, _s_ + 3_b_ = 3, "isevidently only satisfied by _s_ = 3/2, _b_ = 1/2"! OLD CAT believes thatthe assumption that a sandwich costs 1-1/2_d. _ is "the only way to avoidunmanageable fractions. " But _why_ avoid them? Is there not a certainglow of triumph in taming such a fraction? "Ladies and gentlemen, thefraction now before you is one that for years defied all efforts of arefining nature: it was, in a word, hopelessly vulgar. Treating it as acirculating decimal (the treadmill of fractions) only made mattersworse. As a last resource, I reduced it to its lowest terms, andextracted its square root!" Joking apart, let me thank OLD CAT for somevery kind words of sympathy, in reference to a correspondent (whose nameI am happy to say I have now forgotten) who had found fault with me as adiscourteous critic. O. V. L. Is beyond my comprehension. He takes thegiven equations as (1) and (2): thence, by the process [(2)-(1)] deduces(rightly) equation (3) viz. _s_ + 3_b_ = 3: and thence again, by theprocess [x3] (a hopeless mystery), deduces 3_s_ + 4_b_ = 4. I havenothing to say about it: I give it up. SEA-BREEZE says "it is immaterialto the answer" (why?) "in what proportion 3_d. _ is divided between thesandwich and the 3 biscuits": so she assumes _s_ = l-1/2_d. _, _b_ =1/2_d. _ STANZA is one of a very irregular metre. At first she (likeJANET) identifies sandwiches with biscuits. She then tries twoassumptions (_s_ = 1, _b_ = 2/3, and _s_ = 1/2 _b_ = 5/6), and(naturally) ends in contradictions. Then she returns to the firstassumption, and finds the 3 unknowns separately: _quod est absurdum_. STILETTO identifies sandwiches and biscuits, as "articles. " Is the wordever used by confectioners? I fancied "What is the next article, Ma'am?"was limited to linendrapers. TWO SISTERS first assume that biscuits are4 a penny, and then that they are 2 a penny, adding that "the answerwill of course be the same in both cases. " It is a dreamy remark, making one feel something like Macbeth grasping at the spectral dagger. "Is this a statement that I see before me?" If you were to say "we bothwalked the same way this morning, " and _I_ were to say "_one_ of youwalked the same way, but the other didn't, " which of the three would bethe most hopelessly confused? TURTLE PYATE (what _is_ a Turtle Pyate, please?) and OLD CROW, who send a joint answer, and Y. Y. , adopt thesame method. Y. Y. Gets the equation _s_ + 3_b_ = 3: and then says "thissum must be apportioned in one of the three following ways. " It _may_be, I grant you: but Y. Y. Do you say "must"? I fear it is _possible_for Y. Y. To be _two_ Y's. The other two conspirators are less positive:they say it "can" be so divided: but they add "either of the threeprices being right"! This is bad grammar and bad arithmetic at once, ohmysterious birds! Of those who win honours, THE SHETLAND SNARK must have the 3rd class allto himself. He has only answered half the question, viz. The amount ofClara's luncheon: the two little old ladies he pitilessly leaves in themidst of their "difficulty. " I beg to assure him (with thanks for hisfriendly remarks) that entrance-fees and subscriptions are thingsunknown in that most economical of clubs, "The Knot-Untiers. " The authors of the 26 "accidental" solutions differ only in the numberof steps they have taken between the _data_ and the answers. In orderto do them full justice I have arranged the 2nd class in sections, according to the number of steps. The two Kings are fearfullydeliberate! I suppose walking quick, or taking short cuts, isinconsistent with kingly dignity: but really, in reading THESEUS'solution, one almost fancied he was "marking time, " and making noadvance at all! The other King will, I hope, pardon me for havingaltered "Coal" into "Cole. " King Coilus, or Coil, seems to have reignedsoon after Arthur's time. Henry of Huntingdon identifies him with theKing Coėl who first built walls round Colchester, which was named afterhim. In the Chronicle of Robert of Gloucester we read:-- "Aftur Kyng Aruirag, of wam we habbeth y told, Marius ys sone was kyng, quoynte mon & bold. And ys sone was aftur hym, _Coil_ was ys name, Bothe it were quoynte men, & of noble fame. " BALBUS lays it down as a general principle that "in order to ascertainthe cost of any one luncheon, it must come to the same amount upon twodifferent assumptions. " (_Query. _ Should not "it" be "we"? Otherwise the_luncheon_ is represented as wishing to ascertain its own cost!) He thenmakes two assumptions--one, that sandwiches cost nothing; the other, that biscuits cost nothing, (either arrangement would lead to the shopbeing inconveniently crowded!)--and brings out the unknown luncheons as8_d. _ and 19_d. _, on each assumption. He then concludes that thisagreement of results "shows that the answers are correct. " Now I proposeto disprove his general law by simply giving _one_ instance of itsfailing. One instance is quite enough. In logical language, in order todisprove a "universal affirmative, " it is enough to prove itscontradictory, which is a "particular negative. " (I must pause for adigression on Logic, and especially on Ladies' Logic. The universalaffirmative "everybody says he's a duck" is crushed instantly by provingthe particular negative "Peter says he's a goose, " which is equivalentto "Peter does _not_ say he's a duck. " And the universal negative"nobody calls on her" is well met by the particular affirmative "_I_called yesterday. " In short, either of two contradictories disproves theother: and the moral is that, since a particular proposition is muchmore easily proved than a universal one, it is the wisest course, inarguing with a Lady, to limit one's _own_ assertions to "particulars, "and leave _her_ to prove the "universal" contradictory, if she can. Youwill thus generally secure a _logical_ victory: a _practical_ victory isnot to be hoped for, since she can always fall back upon the crushingremark "_that_ has nothing to do with it!"--a move for which Man has notyet discovered any satisfactory answer. Now let us return to BALBUS. )Here is my "particular negative, " on which to test his rule. Suppose thetwo recorded luncheons to have been "2 buns, one queen-cake, 2sausage-rolls, and a bottle of Zoėdone: total, one-and-ninepence, " and"one bun, 2 queen-cakes, a sausage-roll, and a bottle of Zoėdone: total, one-and-fourpence. " And suppose Clara's unknown luncheon to have been "3buns, one queen-cake, one sausage-roll, and 2 bottles of Zoėdone:" whilethe two little sisters had been indulging in "8 buns, 4 queen-cakes, 2sausage-rolls, and 6 bottles of Zoėdone. " (Poor souls, how thirsty theymust have been!) If BALBUS will kindly try this by his principle of "twoassumptions, " first assuming that a bun is 1_d. _ and a queen-cake 2_d. _, and then that a bun is 3_d. _ and a queen-cake 3_d. _, he will bring outthe other two luncheons, on each assumption, as "one-and-nine-pence" and"four-and-ten-pence" respectively, which harmony of results, he willsay, "shows that the answers are correct. " And yet, as a matter of fact, the buns were 2_d. _ each, the queen-cakes 3_d. _, the sausage-rolls6_d. _, and the Zoėdone 2_d. _ a bottle: so that Clara's third luncheonhad cost one-and-sevenpence, and her thirsty friends had spentfour-and-fourpence! Another remark of BALBUS I will quote and discuss: for I think that italso may yield a moral for some of my readers. He says "it is the samething in substance whether in solving this problem we use words and callit Arithmetic, or use letters and signs and call it Algebra. " Now thisdoes not appear to me a correct description of the two methods: theArithmetical method is that of "synthesis" only; it goes from one knownfact to another, till it reaches its goal: whereas the Algebraicalmethod is that of "analysis": it begins with the goal, symbolicallyrepresented, and so goes backwards, dragging its veiled victim with it, till it has reached the full daylight of known facts, in which it cantear off the veil and say "I know you!" Take an illustration. Your house has been broken into and robbed, andyou appeal to the policeman who was on duty that night. "Well, Mum, Idid see a chap getting out over your garden-wall: but I was a good bitoff, so I didn't chase him, like. I just cut down the short way to theChequers, and who should I meet but Bill Sykes, coming full split roundthe corner. So I just ups and says 'My lad, you're wanted. ' That's all Isays. And he says 'I'll go along quiet, Bobby, ' he says, 'without thedarbies, ' he says. " There's your _Arithmetical_ policeman. Now try theother method. "I seed somebody a running, but he was well gone or ever_I_ got nigh the place. So I just took a look round in the garden. And Inoticed the foot-marks, where the chap had come right across yourflower-beds. They was good big foot-marks sure-ly. And I noticed as theleft foot went down at the heel, ever so much deeper than the other. AndI says to myself 'The chap's been a big hulking chap: and he goes lameon his left foot. ' And I rubs my hand on the wall where he got over, andthere was soot on it, and no mistake. So I says to myself 'Now where canI light on a big man, in the chimbley-sweep line, what's lame of onefoot?' And I flashes up permiscuous: and I says 'It's Bill Sykes!' saysI. " There is your _Algebraical_ policeman--a higher intellectual type, to my thinking, than the other. LITTLE JACK'S solution calls for a word of praise, as he has written outwhat really is an algebraical proof _in words_, without representing anyof his facts as equations. If it is all his own, he will make a goodalgebraist in the time to come. I beg to thank SIMPLE SUSAN for somekind words of sympathy, to the same effect as those received from OLDCAT. HECLA and MARTREB are the only two who have used a method _certain_either to produce the answer, or else to prove it impossible: so theymust share between them the highest honours. CLASS LIST. I. HECLA. MARTREB. II. § 1 (2 _steps_). ADELAIDE. CLIFTON C. . . . E. K. C. GUY. L'INCONNU. LITTLE JACK. NIL DESPERANDUM. SIMPLE SUSAN. YELLOW-HAMMER. WOOLLY ONE. § 2 (3 _steps_). A. A. A CHRISTMAS CAROL. AFTERNOON TEA. AN APPRECIATIVE SCHOOLMA'AM. BABY. BALBUS. BOG-OAK. THE RED QUEEN. WALL-FLOWER. § 3 (4 _steps_). HAWTHORN. JORAM. S. S. G. § 4 (5 _steps_). A STEPNEY COACH. § 5 (6 _steps_). BAY LAUREL. BRADSHAW OF THE FUTURE. § 6 (9 _steps_). OLD KING COLE. § 7 (14 _steps_). THESEUS. ANSWERS TO CORRESPONDENTS. I have received several letters on the subjects of Knots II. And VI. , which lead me to think some further explanation desirable. In Knot II. , I had intended the numbering of the houses to begin at onecorner of the Square, and this was assumed by most, if not all, of thecompetitors. TROJANUS however says "assuming, in default of anyinformation, that the street enters the square in the middle of eachside, it may be supposed that the numbering begins at a street. " Butsurely the other is the more natural assumption? In Knot VI. , the first Problem was of course a mere _jeu de mots_, whosepresence I thought excusable in a series of Problems whose aim is toentertain rather than to instruct: but it has not escaped thecontemptuous criticisms of two of my correspondents, who seem to thinkthat Apollo is in duty bound to keep his bow always on the stretch. Neither of them has guessed it: and this is true human nature. Only theother day--the 31st of September, to be quite exact--I met my old friendBrown, and gave him a riddle I had just heard. With one great effort ofhis colossal mind, Brown guessed it. "Right!" said I. "Ah, " said he, "it's very neat--very neat. And it isn't an answer that would occur toeverybody. Very neat indeed. " A few yards further on, I fell in withSmith and to him I propounded the same riddle. He frowned over it for aminute, and then gave it up. Meekly I faltered out the answer. "A poorthing, sir!" Smith growled, as he turned away. "A very poor thing! Iwonder you care to repeat such rubbish!" Yet Smith's mind is, ifpossible, even more colossal than Brown's. The second Problem of Knot VI. Is an example in ordinary Double Rule ofThree, whose essential feature is that the result depends on thevariation of several elements, which are so related to it that, if allbut one be constant, it varies as that one: hence, if none be constant, it varies as their product. Thus, for example, the cubical contents of arectangular tank vary as its length, if breadth and depth be constant, and so on; hence, if none be constant, it varies as the product of thelength, breadth, and depth. When the result is not thus connected with the varying elements, theProblem ceases to be Double Rule of Three and often becomes one of greatcomplexity. To illustrate this, let us take two candidates for a prize, _A_ and _B_, who are to compete in French, German, and Italian: (_a_) Let it be laid down that the result is to depend on their_relative_ knowledge of each subject, so that, whether their marks, forFrench, be "1, 2" or "100, 200, " the result will be the same: and let italso be laid down that, if they get equal marks on 2 papers, the finalmarks are to have the same ratio as those of the 3rd paper. This is acase of ordinary Double Rule of Three. We multiply _A_'s 3 markstogether, and do the same for _B_. Note that, if _A_ gets a single "0, "his final mark is "0, " even if he gets full marks for 2 papers while _B_gets only one mark for each paper. This of course would be very unfairon _A_, though a correct solution under the given conditions. (_b_) The result is to depend, as before, on _relative_ knowledge; butFrench is to have twice as much weight as German or Italian. This is anunusual form of question. I should be inclined to say "the resultingratio is to be nearer to the French ratio than if we multiplied as in(_a_), and so much nearer that it would be necessary to use the othermultipliers _twice_ to produce the same result as in (_a_):" _e. G. _ ifthe French Ratio were 9/10, and the others 4/9, 1/9 so that the ultimateratio, by method (_a_), would be 2/45, I should multiply instead by 2/3, 1/3, giving the result, 1/3 which is nearer to 9/10 than if he had usedmethod (_a_). (_c_) The result is to depend on _actual_ amount of knowledge of the 3subjects collectively. Here we have to ask two questions. (1) What isto be the "unit" (_i. E. _ "standard to measure by") in each subject? (2)Are these units to be of equal, or unequal value? The usual "unit" isthe knowledge shown by answering the whole paper correctly; calling this"100, " all lower amounts are represented by numbers between "0" and"100. " Then, if these units are to be of equal value, we simply add_A_'s 3 marks together, and do the same for _B_. (_d_) The conditions are the same as (_c_), but French is to have doubleweight. Here we simply double the French marks, and add as before. (_e_) French is to have such weight, that, if other marks be equal, theultimate ratio is to be that of the French paper, so that a "0" in thiswould swamp the candidate: but the other two subjects are only to affectthe result collectively, by the amount of knowledge shown, the two beingreckoned of equal value. Here I should add _A_'s German and Italianmarks together, and multiply by his French mark. But I need not go on: the problem may evidently be set with many varyingconditions, each requiring its own method of solution. The Problem inKnot VI. Was meant to belong to variety (_a_), and to make this clear, Iinserted the following passage: "Usually the competitors differ in one point only. Thus, last year, Fifiand Gogo made the same number of scarves in the trial week, and theywere equally light; but Fifi's were twice as warm as Gogo's, and she waspronounced twice as good. " What I have said will suffice, I hope, as an answer to BALBUS, who holdsthat (_a_) and (_c_) are the only possible varieties of the problem, andthat to say "We cannot use addition, therefore we must be intended touse multiplication, " is "no more illogical than, from knowledge that onewas not born in the night, to infer that he was born in the daytime";and also to FIFEE, who says "I think a little more consideration willshow you that our 'error of _adding_ the proportional numbers togetherfor each candidate instead of _multiplying_' is no error at all. " Why, even if addition _had_ been the right method to use, not one of thewriters (I speak from memory) showed any consciousness of the necessityof fixing a "unit" for each subject. "No error at all!" They werepositively steeped in error! One correspondent (I do not name him, as the communication is not quitefriendly in tone) writes thus:--"I wish to add, very respectfully, thatI think it would be in better taste if you were to abstain from the verytrenchant expressions which you are accustomed to indulge in whencriticising the answer. That such a tone must not be" ("be not"?)"agreeable to the persons concerned who have made mistakes may possiblyhave no great weight with you, but I hope you will feel that it would beas well not to employ it, _unless you are quite certain of being correctyourself_. " The only instances the writer gives of the "trenchantexpressions" are "hapless" and "malefactors. " I beg to assure him (andany others who may need the assurance: I trust there are none) that allsuch words have been used in jest, and with no idea that they couldpossibly annoy any one, and that I sincerely regret any annoyance I mayhave thus inadvertently given. May I hope that in future they willrecognise the distinction between severe language used in sober earnest, and the "words of unmeant bitterness, " which Coleridge has alluded to inthat lovely passage beginning "A little child, a limber elf"? If thewriter will refer to that passage, or to the preface to "Fire, Famine, and Slaughter, " he will find the distinction, for which I plead, farbetter drawn out than I could hope to do in any words of mine. The writer's insinuation that I care not how much annoyance I give to myreaders I think it best to pass over in silence; but to his concludingremark I must entirely demur. I hold that to use language likely toannoy any of my correspondents would not be in the least justified bythe plea that I was "quite certain of being correct. " I trust that theknot-untiers and I are not on such terms as those! I beg to thank _G. B. _ for the offer of a puzzle--which, however, is toolike the old one "Make four 9's into 100. " ANSWERS TO KNOT VIII. § 1. THE PIGS. _Problem. _--Place twenty-four pigs in four sties so that, as you goround and round, you may always find the number in each sty nearer toten than the number in the last. _Answer. _--Place 8 pigs in the first sty, 10 in the second, nothing inthe third, and 6 in the fourth: 10 is nearer ten than 8; nothing isnearer ten than 10; 6 is nearer ten than nothing; and 8 is nearer tenthan 6. * * * * * This problem is noticed by only two correspondents. BALBUS says "itcertainly cannot be solved mathematically, nor do I see how to solve itby any verbal quibble. " NOLENS VOLENS makes Her Radiancy change thedirection of going round; and even then is obliged to add "the pigs mustbe carried in front of her"! § 2. THE GRURMSTIPTHS. _Problem. _--Omnibuses start from a certain point, both ways, every 15minutes. A traveller, starting on foot along with one of them, meetsone in 12-1/2 minutes: when will he be overtaken by one? _Answer. _--In 6-1/4 minutes. * * * * * _Solution. _--Let "_a_" be the distance an omnibus goes in 15 minutes, and "_x_" the distance from the starting-point to where the traveller isovertaken. Since the omnibus met is due at the starting-point in 2-1/2minutes, it goes in that time as far as the traveller walks in 12-1/2;_i. E. _ it goes 5 times as fast. Now the overtaking omnibus is "_a_"behind the traveller when he starts, and therefore goes "_a_ + _x_"while he goes "_x_. " Hence _a_ + _x_ = 5_x_; _i. E. _ 4_x_ = _a_, and _x_= _a_/4. This distance would be traversed by an omnibus in 15/4 minutes, and therefore by the traveller in 5 × 15/4. Hence he is overtaken in18-3/4 minutes after starting, _i. E. _ in 6-1/4 minutes after meeting theomnibus. * * * * * Four answers have been received, of which two are wrong. DINAH MITErightly states that the overtaking omnibus reached the point where theymet the other omnibus 5 minutes after they left, but wrongly concludesthat, going 5 times as fast, it would overtake them in another minute. The travellers are 5-minutes-walk ahead of the omnibus, and must walk1-4th of this distance farther before the omnibus overtakes them, whichwill be 1-5th of the distance traversed by the omnibus in the same time:this will require 1-1/4 minutes more. NOLENS VOLENS tries it by aprocess like "Achilles and the Tortoise. " He rightly states that, whenthe overtaking omnibus leaves the gate, the travellers are 1-5th of"_a_" ahead, and that it will take the omnibus 3 minutes to traversethis distance; "during which time" the travellers, he tells us, go1-15th of "_a_" (this should be 1-25th). The travellers being now 1-15thof "_a_" ahead, he concludes that the work remaining to be done is forthe travellers to go 1-60th of "_a_, " while the omnibus goes l-12th. The_principle_ is correct, and might have been applied earlier. CLASS LIST. I. BALBUS. DELTA. ANSWERS TO KNOT IX. § 1. THE BUCKETS. _Problem. _--Lardner states that a solid, immersed in a fluid, displacesan amount equal to itself in bulk. How can this be true of a smallbucket floating in a larger one? _Solution. _--Lardner means, by "displaces, " "occupies a space whichmight be filled with water without any change in the surroundings. " Ifthe portion of the floating bucket, which is above the water, could beannihilated, and the rest of it transformed into water, the surroundingwater would not change its position: which agrees with Lardner'sstatement. * * * * * Five answers have been received, none of which explains the difficultyarising from the well-known fact that a floating body is the same weightas the displaced fluid. HECLA says that "only that portion of thesmaller bucket which descends below the original level of the water canbe properly said to be immersed, and only an equal bulk of water isdisplaced. " Hence, according to HECLA, a solid, whose weight was equalto that of an equal bulk of water, would not float till the whole of itwas below "the original level" of the water: but, as a matter of fact, it would float as soon as it was all under water. MAGPIE says thefallacy is "the assumption that one body can displace another from aplace where it isn't, " and that Lardner's assertion is incorrect, exceptwhen the containing vessel "was originally full to the brim. " But thequestion of floating depends on the present state of things, not on pasthistory. OLD KING COLE takes the same view as HECLA. TYMPANUM and VINDEXassume that "displaced" means "raised above its original level, " andmerely explain how it comes to pass that the water, so raised, is lessin bulk than the immersed portion of bucket, and thus landthemselves--or rather set themselves floating--in the same boat asHECLA. I regret that there is no Class-list to publish for this Problem. * * * * * § 2. BALBUS' ESSAY. _Problem. _--Balbus states that if a certain solid be immersed in acertain vessel of water, the water will rise through a series ofdistances, two inches, one inch, half an inch, &c. , which series has noend. He concludes that the water will rise without limit. Is this true? _Solution. _--No. This series can never reach 4 inches, since, howevermany terms we take, we are always short of 4 inches by an amount equalto the last term taken. * * * * * Three answers have been received--but only two seem to me worthy ofhonours. TYMPANUM says that the statement about the stick "is merely a blind, towhich the old answer may well be applied, _solvitur ambulando_, orrather _mergendo_. " I trust TYMPANUM will not test this in his ownperson, by taking the place of the man in Balbus' Essay! He wouldinfallibly be drowned. OLD KING COLE rightly points out that the series, 2, 1, &c. , is adecreasing Geometrical Progression: while VINDEX rightly identifies thefallacy as that of "Achilles and the Tortoise. " CLASS LIST. I. OLD KING COLE. VINDEX. * * * * * § 3. THE GARDEN. _Problem. _--An oblong garden, half a yard longer than wide, consistsentirely of a gravel-walk, spirally arranged, a yard wide and 3, 630yards long. Find the dimensions of the garden. _Answer. _--60, 60-1/2. _Solution. _--The number of yards and fractions of a yard traversed inwalking along a straight piece of walk, is evidently the same as thenumber of square-yards and fractions of a square-yard, contained in thatpiece of walk: and the distance, traversed in passing through asquare-yard at a corner, is evidently a yard. Hence the area of thegarden is 3, 630 square-yards: _i. E. _, if _x_ be the width, _x_ (_x_ +1/2) = 3, 630. Solving this Quadratic, we find _x_ = 60. Hence thedimensions are 60, 60-1/2. * * * * * Twelve answers have been received--seven right and five wrong. C. G. L. , NABOB, OLD CROW, and TYMPANUM assume that the number of yardsin the length of the path is equal to the number of square-yards in thegarden. This is true, but should have been proved. But each is guilty ofdarker deeds. C. G. L. 's "working" consists of dividing 3, 630 by 60. Whence came this divisor, oh Segiel? Divination? Or was it a dream? Ifear this solution is worth nothing. OLD CROW'S is shorter, and so (ifpossible) worth rather less. He says the answer "is at once seen to be60 × 60-1/2"! NABOB'S calculation is short, but "as rich as a Nabob" inerror. He says that the square root of 3, 630, multiplied by 2, equalsthe length plus the breadth. That is 60. 25 × 2 = 120-1/2. His firstassertion is only true of a _square_ garden. His second is irrelevant, since 60. 25 is _not_ the square-root of 3, 630! Nay, Bob, this will _not_do! TYMPANUM says that, by extracting the square-root of 3, 630, we get60 yards with a remainder of 30/60, or half-a-yard, which we add so asto make the oblong 60 × 60-1/2. This is very terrible: but worse remainsbehind. TYMPANUM proceeds thus:--"But why should there be the half-yardat all? Because without it there would be no space at all for flowers. By means of it, we find reserved in the very centre a small plot ofground, two yards long by half-a-yard wide, the only space not occupiedby walk. " But Balbus expressly said that the walk "used up the whole ofthe area. " Oh, TYMPANUM! My tympa is exhausted: my brain is num! I cansay no more. HECLA indulges, again and again, in that most fatal of all habits incomputation--the making _two_ mistakes which cancel each other. Shetakes _x_ as the width of the garden, in yards, and _x_ + 1/2 as itslength, and makes her first "coil" the sum of _x_-1/2, _x_-1/2, _x_-1, _x_-1, _i. E. _ 4_x_-3: but the fourth term should be _x_-1-1/2, so thather first coil is 1/2 a yard too long. Her second coil is the sum of_x_-2-1/2, _x_-2-1/2, _x_-3, _x_-3: here the first term should be _x_-2and the last _x_-3-1/2: these two mistakes cancel, and this coil istherefore right. And the same thing is true of every other coil but thelast, which needs an extra half-yard to reach the _end_ of the path: andthis exactly balances the mistake in the first coil. Thus the sum totalof the coils comes right though the working is all wrong. Of the seven who are right, DINAH MITE, JANET, MAGPIE, and TAFFY makethe same assumption as C. G. L. And Co. They then solve by a Quadratic. MAGPIE also tries it by Arithmetical Progression, but fails to noticethat the first and last "coils" have special values. ALUMNUS ETONĘ attempts to prove what C. G. L. Assumes by a particularinstance, taking a garden 6 by 5-1/2. He ought to have proved itgenerally: what is true of one number is not always true of others. OLDKING COLE solves it by an Arithmetical Progression. It is right, but toolengthy to be worth as much as a Quadratic. VINDEX proves it very neatly, by pointing out that a yard of walkmeasured along the middle represents a square yard of garden, "whetherwe consider the straight stretches of walk or the square yards at theangles, in which the middle line goes half a yard in one direction andthen turns a right angle and goes half a yard in another direction. " CLASS LIST. I. VINDEX. II. ALUMNUS ETONĘ. OLD KING COLE. III. DINAH MITE. JANET. MAGPIE. TAFFY. ANSWERS TO KNOT X. § 1. THE CHELSEA PENSIONERS. _Problem. _--If 70 per cent. Have lost an eye, 75 per cent. An ear, 80per cent. An arm, 85 per cent. A leg: what percentage, _at least_, musthave lost all four? _Answer. _--Ten. _Solution. _--(I adopt that of POLAR STAR, as being better than my own). Adding the wounds together, we get 70 + 75 + 80 + 85 = 310, among 100men; which gives 3 to each, and 4 to 10 men. Therefore the leastpercentage is 10. * * * * * Nineteen answers have been received. One is "5, " but, as no working isgiven with it, it must, in accordance with the rule, remain "a deedwithout a name. " JANET makes it "35 and 7/10ths. " I am sorry she hasmisunderstood the question, and has supposed that those who had lost anear were 75 per cent. _of those who had lost an eye_; and so on. Ofcourse, on this supposition, the percentages must all be multipliedtogether. This she has done correctly, but I can give her no honours, as I do not think the question will fairly bear her interpretation, THREE SCORE AND TEN makes it "19 and 3/8ths. " Her solution has givenme--I will not say "many anxious days and sleepless nights, " for I wishto be strictly truthful, but--some trouble in making any sense at all ofit. She makes the number of "pensioners wounded once" to be 310 ("percent. , " I suppose!): dividing by 4, she gets 77 and a half as "averagepercentage:" again dividing by 4, she gets 19 and 3/8ths as "percentagewounded four times. " Does she suppose wounds of different kinds to"absorb" each other, so to speak? Then, no doubt, the _data_ areequivalent to 77 pensioners with one wound each, and a half-pensionerwith a half-wound. And does she then suppose these concentrated woundsto be _transferable_, so that 3/4ths of these unfortunates can obtainperfect health by handing over their wounds to the remaining 1/4th?Granting these suppositions, her answer is right; or rather, _if_ thequestion had been "A road is covered with one inch of gravel, along 77and a half per cent. Of it. How much of it could be covered 4 inchesdeep with the same material?" her answer _would_ have been right. Butalas, that _wasn't_ the question! DELTA makes some most amazingassumptions: "let every one who has not lost an eye have lost an ear, ""let every one who has not lost both eyes and ears have lost an arm. "Her ideas of a battle-field are grim indeed. Fancy a warrior who wouldcontinue fighting after losing both eyes, both ears, and both arms! Thisis a case which she (or "it?") evidently considers _possible_. Next come eight writers who have made the unwarrantable assumption that, because 70 per cent. Have lost an eye, _therefore_ 30 per cent. Have_not_ lost one, so that they have _both_ eyes. This is illogical. If yougive me a bag containing 100 sovereigns, and if in an hour I come to you(my face _not_ beaming with gratitude nearly so much as when I receivedthe bag) to say "I am sorry to tell you that 70 of these sovereigns arebad, " do I thereby guarantee the other 30 to be good? Perhaps I have nottested them yet. The sides of this illogical octagon are as follows, inalphabetical order:--ALGERNON BRAY, DINAH MITE, G. S. C. , JANE E. , J. D. W. , MAGPIE (who makes the delightful remark "therefore 90 per cent. Havetwo of something, " recalling to one's memory that fortunate monarch, with whom Xerxes was so much pleased that "he gave him ten ofeverything!"), S. S. G. , and TOKIO. BRADSHAW OF THE FUTURE and T. R. Do the question in a piecemealfashion--on the principle that the 70 per cent. And the 75 per cent. , though commenced at opposite ends of the 100, must overlap by _at least_45 per cent. ; and so on. This is quite correct working, but not, Ithink, quite the best way of doing it. The other five competitors will, I hope, feel themselves sufficientlyglorified by being placed in the first class, without my composing aTriumphal Ode for each! CLASS LIST. I. OLD CAT. OLD HEN. POLAR STAR. SIMPLE SUSAN. WHITE SUGAR. II. BRADSHAW OF THE FUTURE. T. R. III. ALGERNON BRAY. DINAH MITE. G. S. C. JANE E. J. D. W. MAGPIE. S. S. G. TOKIO. § 2. CHANGE OF DAY. I must postpone, _sine die_, the geographical problem--partly because Ihave not yet received the statistics I am hoping for, and partly becauseI am myself so entirely puzzled by it; and when an examiner is himselfdimly hovering between a second class and a third how is he to decidethe position of others? § 3. THE SONS' AGES. _Problem. _--"At first, two of the ages are together equal to the third. A few years afterwards, two of them are together double of the third. When the number of years since the first occasion is two-thirds of thesum of the ages on that occasion, one age is 21. What are the other two? _Answer. _--"15 and 18. " * * * * * _Solution. _--Let the ages at first be _x_, _y_, (_x_ + _y_). Now, if _a_+ _b_ = 2_c_, then (_a_-_n_) + (_b_-_n_) = 2(_c_-_n_), whatever be thevalue of _n_. Hence the second relationship, if _ever_ true, was_always_ true. Hence it was true at first. But it cannot be true that_x_ and _y_ are together double of (_x_ + _y_). Hence it must be true of(_x_ + _y_), together with _x_ or _y_; and it does not matter which wetake. We assume, then, (_x_ + _y_) + _x_ = 2_y_; _i. E. _ _y_ = 2_x_. Hence the three ages were, at first, _x_, 2_x_, 3_x_; and the number ofyears, since that time is two-thirds of 6_x_, _i. E. _ is 4_x_. Hence thepresent ages are 5_x_, 6_x_, 7_x_. The ages are clearly _integers_, since this is only "the year when one of my sons comes of age. " Hence7_x_ = 21, _x_ = 3, and the other ages are 15, 18. * * * * * Eighteen answers have been received. One of the writers merely assertsthat the first occasion was 12 years ago, that the ages were then 9, 6, and 3; and that on the second occasion they were 14, 11, and 8! As aRoman father, I _ought_ to withhold the name of the rash writer; butrespect for age makes me break the rule: it is THREE SCORE AND TEN. JANEE. Also asserts that the ages at first were 9, 6, 3: then she calculatesthe present ages, leaving the _second_ occasion unnoticed. OLD HEN isnearly as bad; she "tried various numbers till I found one that fitted_all_ the conditions"; but merely scratching up the earth, and peckingabout, is _not_ the way to solve a problem, oh venerable bird! And closeafter OLD HEN prowls, with hungry eyes, OLD CAT, who calmly assumes, tobegin with, that the son who comes of age is the _eldest_. Eat yourbird, Puss, for you will get nothing from me! There are yet two zeroes to dispose of. MINERVA assumes that, on _every_occasion, a son comes of age; and that it is only such a son who is"tipped with gold. " Is it wise thus to interpret "now, my boys, calculate your ages, and you shall have the money"? BRADSHAW OF THEFUTURE says "let" the ages at first be 9, 6, 3, then assumes that thesecond occasion was 6 years afterwards, and on these baselessassumptions brings out the right answers. Guide _future_ travellers, anthou wilt: thou art no Bradshaw for _this_ Age! Of those who win honours, the merely "honourable" are two. DINAH MITEascertains (rightly) the relationship between the three ages at first, but then _assumes_ one of them to be "6, " thus making the rest of hersolution tentative. M. F. C. Does the algebra all right up to theconclusion that the present ages are 5_z_, 6_z_, and 7_z_; it thenassumes, without giving any reason, that 7_z_ = 21. Of the more honourable, DELTA attempts a novelty--to discover _which_son comes of age by elimination: it assumes, successively, that it isthe middle one, and that it is the youngest; and in each case it_apparently_ brings out an absurdity. Still, as the proof contains thefollowing bit of algebra, "63 = 7_x_ + 4_y_; [** therefore] 21 = _x_ + 4sevenths of _y_, " I trust it will admit that its proof is not _quite_conclusive. The rest of its work is good. MAGPIE betrays the deplorabletendency of her tribe--to appropriate any stray conclusion she comesacross, without having any _strict_ logical right to it. Assuming _A_, _B_, _C_, as the ages at first, and _D_ as the number of the years thathave elapsed since then, she finds (rightly) the 3 equations, 2_A_ =_B_, _C_ = _B_ + _A_, _D_ = 2_B_. She then says "supposing that _A_ = 1, then _B_ = 2, _C_ = 3, and _D_ = 4. Therefore for _A_, _B_, _C_, _D_, four numbers are wanted which shall be to each other as 1:2:3:4. " It isin the "therefore" that I detect the unconscientiousness of this bird. The conclusion _is_ true, but this is only because the equations are"homogeneous" (_i. E. _ having one "unknown" in each term), a fact which Istrongly suspect had not been grasped--I beg pardon, clawed--by her. Were I to lay this little pitfall, "_A_ + 1 = _B_, _B_ + 1 = _C_;supposing _A_ = 1, then _B_ = 2 and _C_ = 3. _Therefore_ for _A_, _B_, _C_, three numbers are wanted which shall be to one another as 1:2:3, "would you not flutter down into it, oh MAGPIE, as amiably as a Dove?SIMPLE SUSAN is anything but simple to _me_. After ascertaining that the3 ages at first are as 3:2:1, she says "then, as two-thirds of theirsum, added to one of them, = 21, the sum cannot exceed 30, andconsequently the highest cannot exceed 15. " I suppose her (mental)argument is something like this:--"two-thirds of sum, + one age, = 21;[** therefore] sum, + 3 halves of one age, = 31 and a half. But 3 halves ofone age cannot be less than 1 and-a-half (here I perceive that SIMPLESUSAN would on no account present a guinea to a new-born baby!) hencethe sum cannot exceed 30. " This is ingenious, but her proof, after that, is (as she candidly admits) "clumsy and roundabout. " She finds thatthere are 5 possible sets of ages, and eliminates four of them. Supposethat, instead of 5, there had been 5 million possible sets? Would SIMPLESUSAN have courageously ordered in the necessary gallon of ink and reamof paper? The solution sent in by C. R. Is, like that of SIMPLE SUSAN, partlytentative, and so does not rise higher than being Clumsily Right. Among those who have earned the highest honours, ALGERNON BRAY solvesthe problem quite correctly, but adds that there is nothing to excludethe supposition that all the ages were _fractional_. This would make thenumber of answers infinite. Let me meekly protest that I _never_intended my readers to devote the rest of their lives to writing outanswers! E. M. RIX points out that, if fractional ages be admissible, any one of the three sons might be the one "come of age"; but sherightly rejects this supposition on the ground that it would make theproblem indeterminate. WHITE SUGAR is the only one who has detected anoversight of mine: I had forgotten the possibility (which of courseought to be allowed for) that the son, who came of age that _year_, neednot have done so by that _day_, so that he _might_ be only 20. Thisgives a second solution, viz. , 20, 24, 28. Well said, pure Crystal!Verily, thy "fair discourse hath been as sugar"! CLASS LIST. I. ALGERNON BRAY. AN OLD FOGEY. E. M. RIX. G. S. C. S. S. G. TOKIO. T. R. WHITE SUGAR. II. C. R. DELTA. MAGPIE. SIMPLE SUSAN. III. DINAH MITE. M. F. C. * * * * * I have received more than one remonstrance on my assertion, in theChelsea Pensioners' problem, that it was illogical to assume, from the_datum_ "70 p. C. Have lost an eye, " that 30 p. C. Have _not_. ALGERNONBRAY states, as a parallel case, "suppose Tommy's father gives him 4apples, and he eats one of them, how many has he left?" and says "Ithink we are justified in answering, 3. " I think so too. There is no"must" here, and the _data_ are evidently meant to fix the answer_exactly_: but, if the question were set me "how many _must_ he haveleft?", I should understand the _data_ to be that his father gave him 4_at least_, but _may_ have given him more. I take this opportunity of thanking those who have sent, along withtheir answers to the Tenth Knot, regrets that there are no more Knots tocome, or petitions that I should recall my resolution to bring them toan end. I am most grateful for their kind words; but I think it wisestto end what, at best, was but a lame attempt. "The stretched metre of anantique song" is beyond my compass; and my puppets were neitherdistinctly _in_ my life (like those I now address), nor yet (like Aliceand the Mock Turtle) distinctly _out_ of it. Yet let me at least fancy, as I lay down the pen, that I carry with me into my silent life, dearreader, a farewell smile from your unseen face, and a kindly farewellpressure from your unfelt hand! And so, good night! Parting is suchsweet sorrow, that I shall say "good night!" till it be morrow. THE END LONDON: RICHARD CLAY AND SONS, PRINTERS. [TURN OVER. WORKS BY LEWIS CARROLL. ALICE'S ADVENTURES IN WONDERLAND. With Forty-two Illustrations by TENNIEL. Crown 8vo, cloth, gilt edges, price 6_s. _ Seventy-fifth Thousand. TRANSLATIONS OF THE SAME--into French, by HENRI BUÉ--into German, byANTONIE ZIMMERMANN--and into Italian, by T. PIETROCŅLA ROSSETTI--withTENNIEL'S Illustrations. Crown 8vo, cloth, gilt edges, price 6_s. _ each. THROUGH THE LOOKING-GLASS, AND WHAT ALICE FOUND THERE. With FiftyIllustrations by TENNIEL. Crown 8vo, cloth, gilt edges, price 6_s. _Fifty-sixth Thousand. RHYME? AND REASON? With Sixty-five Illustrations by ARTHUR B. FROST, andNine by HENRY HOLIDAY. (This book is a reprint, with a few additions, ofthe comic portion of "Phantasmagoria and other Poems, " and of "TheHunting of the Snark. " Mr. Frost's pictures are new. ) Crown 8vo, cloth, coloured edges, price 7_s. _ Fifty Thousand. A TANGLED TALE. Reprinted from _The Monthly Packet_. With SixIllustrations by Arthur B. Frost. Crown 8vo, 4_s. _ 6_d. _ * * * * * N. B. In selling the above-mentioned books to the Trade, Messrs. Macmillan and Co. Will abate 2_d. _ in the shilling (no odd copies), andallow 5 per cent. Discount for payment within six months, and 10 percent. For cash. In selling them to the Public (for cash only) they willallow 10 per cent. Discount. * * * * * MR. LEWIS CARROLL, having been requested to allow "AN EASTER GREETING"(a leaflet, addressed to children, and frequently given with his books)to be sold separately, has arranged with Messrs. Harrison, of 59, PallMall, who will supply a single copy for 1_d. _, or 12 for 9_d. _, or 100for 5_s. _ MACMILLAN AND CO. , LONDON. LONDON: RICHARD CLAY AND SONS, PRINTERS. * * * * * Transcriber's note The following changes have been made to the text: Page 88: "he corners of the" changed to "the corners of the". Page 95: "Aix-le-Bains" changed to "Aix-les-Bains". Page 108: "3/5, 2, 1/3" changed to "3/5, 2/3, 1/3". Page 114: "10 of the 12 cases" changed to "10 of the 13 cases". Page 121: "four-and fourpence" changed to "four-and-fourpence". Last page: "Fifth Thousand" changed to "Fifty Thousand".