,music transcribed by ,linda ,cantoni4 ,transcriber's note ,minor punctuation errors have been changed without notice4 ,printer errors have been changed and are listed at the end4 ,all other inconsistencies are as in the original4 ,in this version the square root symbol is indicated by 9999 sqrt1 the superscript by 1 and the therefore symbol by 9999 therefore4 ,a ,,tangled ,,tale ,decoration ,illustration3 ,,at ,a ,,pace ,,of ,,six ,,miles ,,in ,,the ,,hour4 ,frontispiece4 ,a ,,tangled ,,tale ,,by ,,lewis ,,carroll ,,with ,,six ,,illustrations ,,by ,,arthur ,b4 ,,frost ,hoc meum tale quale est accipe4 ,,second ,,thousand4 ,london ,,macmillan ,,and ,,co4 #ahhe ,all ,rights ,reserved ,,richard ,,clay ,,sons1 ,,bread ,,street ,,hill1 ,,london1 ,e4,c4 ,and ,bungay1 ,suffolk4 ,to ,my ,pupil4 ,beloved pupil6 ,tamed by thee1 ,addish-1 ,subtrac-1 ,multiplica-tion1 ,division1 ,fractions1 ,rule of ,three1 ,attest thy deft manipulation6 ,then onward6 ,let the voice of ,fame ,from ,age to ,age repeat thy story1 ,till thou hast won thyself a name ,exceeding even ,euclid's glory6 ,,preface4 ,this ,tale originally appeared as a serial in ,the ,monthly ,packet1 beginning in ,april1 #ahhj4 ,the writer's intention was to embody in each ,knot 7like the medicine so dexterously1 but ineffectually1 concealed in the jam of our early childhood7 one or more mathematical questions--in ,arithmetic1 ,algebra1 or ,geometry1 as the case might be--for the amusement1 and possible edification1 of the fair readers of that ,magazine4 ,l4 ,c4 ,october1 #ahhe4 ,,contents4 ,,knot ,,page ,i4 ,,excelsior #a ,,ii4 ,,eligible ,,apartments #d ,,iii4 ,,mad ,,mathesis #ac ,,iv4 ,,the ,,dead ,,reckoning #ai ,v4 ,,oughts ,,and ,,crosses #bg ,,vi4 ,,her ,,radiancy #cd ,,vii4 ,,petty ,,cash #dc ,,viii4 ,,de ,,omnibus ,,rebus #eb ,,ix4 ,a ,,serpent ,,with ,,corners #eh ,x4 ,,chelsea ,,buns #ff ,,answers ,,to ,,knot ,i4 #gg ,,ii4 #hd ,,iii4 #ij ,,iv4 #if ,v4 #ajb ,,vi4 #ajf ,,vii4 #aab ,,viii4 #acb ,,ix4 #ace ,x4 #adb ,a ,,tangled ,,tale4 ,,knot ,i4 ,,excelsior4 ,goblin1 lead them up and down4 ,the ruddy glow of sunset was already fading into the sombre shadows of night1 when two travellers might have been observed swiftly--at a pace of six miles in the hour--descending the rugged side of a mountain2 the younger bounding from crag to crag with the agility of a fawn1 while his companion1 whose aged limbs seemed ill at ease in the heavy chain armour habitually worn by tourists in that district1 toiled on painfully at his side4 ,as is always the case under such circumstances1 the younger knight was the first to break the silence4 ,a goodly pace1 ,i trow6 he exclaimed4 ,we sped not thus in the ascent6 ,goodly1 indeed6 the other echoed with a groan4 ,we clomb it but at three miles in the hour4 ,and on the dead level our pace is----8 the younger suggested2 for he was weak in statistics1 and left all such details to his aged companion4 ,four miles in the hour1 the other wearily replied4 ,not an ounce more1 he added1 with that love of metaphor so common in old age1 and not a farthing less6 ',twas three hours past high noon when we left our hostelry1 the young man said1 musingly4 ,we shall scarce be back by supper-time4 ,perchance mine host will roundly deny us all food6 ,he will chide our tardy return1 was the grave reply1 and such a rebuke will be meet4 ,a brave conceit6 cried the other1 with a merry laugh4 ,and should we bid him bring us yet another course1 ,i trow his answer will be tart6 ,we shall but get our deserts1 sighed the elder knight1 who had never seen a joke in his life1 and was somewhat displeased at his companion's untimely levity4 ',twill be nine of the clock1 he added in an undertone1 by the time we regain our hostelry4 ,full many a mile shall we have plodded this day6 ,how many8 ,how many8 cried the eager youth1 ever athirst for knowledge4 ,the old man was silent4 ,tell me1 he answered1 after a moment's thought1 what time it was when we stood together on yonder peak4 ,not exact to the minute6 he added hastily1 reading a protest in the young man's face4 ,an' thy guess be within one poor half-hour of the mark1 'tis all ,i ask of thy mother's son6 ,then will ,i tell thee1 true to the last inch1 how far we shall have trudged betwixt three and nine of the clock4 ,a groan was the young man's only reply2 while his convulsed features and the deep wrinkles that chased each other across his manly brow1 revealed the abyss of arithmetical agony into which one chance question had plunged him4 ,,knot ,,ii4 ,,eligible ,,apartments4 ,straight down the crooked lane1 ,and all round the square4 ,let's ask ,balbus about it1 said ,hugh4 ,all right1 said ,lambert4 ,he can guess it1 said ,hugh4 ,rather1 said ,lambert4 ,no more words were needed3 the two brothers understood each other perfectly4 ,illustration3 ,,balbus ,,was ,,assisting ,,his ,,mother-,,in-,,law ,,to ,,convince ,,the ,,dragon4 ,balbus was waiting for them at the hotel3 the journey down had tired him1 he said3 so his two pupils had been the round of the place1 in search of lodgings1 without the old tutor who had been their inseparable companion from their childhood4 ,they had named him after the hero of their ,latin exercise-book1 which overflowed with anecdotes of that versatile genius--anecdotes whose vagueness in detail was more than compensated by their sensational brilliance4 ,balbus has overcome all his enemies had been marked by their tutor1 in the margin of the book1 ,successful ,bravery4 ,in this way he had tried to extract a moral from every anecdote about ,balbus--sometimes one of warning1 as in ,balbus had borrowed a healthy dragon1 against which he had written ,rashness in ,speculation--sometimes of encouragement1 as in the words ,influence of ,sympathy in ,united ,action1 which stood opposite to the anecdote ,balbus was assisting his mother-in-law to convince the dragon--and sometimes it dwindled down to a single word1 such as ,prudence1 which was all he could extract from the touching record that ,balbus1 having scorched the tail of the dragon1 went away4 ,his pupils liked the short morals best1 as it left them more room for marginal illustrations1 and in this instance they required all the space they could get to exhibit the rapidity of the hero's departure4 ,their report of the state of things was discouraging4 ,that most fashionable of watering-places1 ,little ,mendip1 was chockfull 7as the boys expressed it7 from end to end4 ,but in one ,square they had seen no less than four cards1 in different houses1 all announcing in flaming capitals ,,eligible ,,apartments4 ,so there's plenty of choice1 after all1 you see1 said spokesman ,hugh in conclusion4 ,that doesn't follow from the data1 said ,balbus1 as he rose from the easy chair1 where he had been dozing over ,the ,little ,mendip ,gazette4 ,they may be all single rooms4 ,however1 we may as well see them4 ,i shall be glad to stretch my legs a bit4 ,an unprejudiced bystander might have objected that the operation was needless1 and that this long1 lank creature would have been all the better with even shorter legs3 but no such thought occurred to his loving pupils4 ,one on each side1 they did their best to keep up with his gigantic strides1 while ,hugh repeated the sentence in their father's letter1 just received from abroad1 over which he and ,lambert had been puzzling4 ,he says a friend of his1 the ,governor of----what was that name again1 ,lambert8 7,kgovjni1 said ,lambert47 ,well1 yes4 ,the ,governor of----what-you-may-call-it----wants to give a very small dinner-party1 and he means to ask his father's brother-in-law1 his brother's father-in-law1 his father-in-law's brother1 and his brother-in-law's father3 and we're to guess how many guests there will be4 ,there was an anxious pause4 ,how large did he say the pudding was to be8 ,balbus said at last4 ,take its cubical contents1 divide by the cubical contents of what each man can eat1 and the quotient---- ,he didn't say anything about pudding1 said ,hugh1 --and here's the ,square1 as they turned a corner and came into sight of the eligible apartments4 ,it is a ,square6 was ,balbus' first cry of delight1 as he gazed around him4 ,beautiful6 ,beau-ti-ful6 ,equilateral6 ,and rectangular6 ,the boys looked round with less enthusiasm4 ,number nine is the first with a card1 said prosaic ,lambert2 but ,balbus would not so soon awake from his dream of beauty4 ,see1 boys6 he cried4 ,twenty doors on a side6 ,what symmetry6 ,each side divided into twenty-one equal parts6 ,it's delicious6 ,shall ,i knock1 or ring8 said ,hugh1 looking in some perplexity at a square brass plate which bore the simple inscription ,,ring ,,also4 ,both1 said ,balbus4 ,that's an ,ellipsis1 my boy4 ,did you never see an ,ellipsis before8 ,i couldn't hardly read it1 said ,hugh1 evasively4 ,it's no good having an ,ellipsis1 if they don't keep it clean4 ,which there is one room1 gentlemen1 said the smiling landlady4 ,and a sweet room too6 ,as snug a little back-room---- ,we will see it1 said ,balbus gloomily1 as they followed her in4 ,i knew how it would be6 ,one room in each house6 ,no view1 ,i suppose8 ,which indeed there is1 gentlemen6 the landlady indignantly protested1 as she drew up the blind1 and indicated the back garden4 ,cabbages1 ,i perceive1 said ,balbus4 ,well1 they're green1 at any rate4 ,which the greens at the shops1 their hostess explained1 are by no means dependable upon4 ,here you has them on the premises1 and of the best4 ,does the window open8 was always ,balbus' first question in testing a lodging3 and ,does the chimney smoke8 his second4 ,satisfied on all points1 he secured the refusal of the room1 and they moved on to ,number ,twenty-five4 ,this landlady was grave and stern4 ,i've nobbut one room left1 she told them3 and it gives on the back-gyardin4 ,but there are cabbages8 ,balbus suggested4 ,the landlady visibly relented4 ,there is1 sir1 she said3 and good ones1 though ,i say it as shouldn't4 ,we can't rely on the shops for greens4 ,so we grows them ourselves4 ,a singular advantage1 said ,balbus3 and1 after the usual questions1 they went on to ,fifty-two4 ,and ,i'd gladly accommodate you all1 if ,i could1 was the greeting that met them4 ,we are but mortal1 7,irrelevant6 muttered ,balbus7 and ,i've let all my rooms but one4 ,which one is a back-room1 ,i perceive1 said ,balbus3 and looking out on--on cabbages1 ,i presume8 ,yes1 indeed1 sir6 said their hostess4 ,whatever other folks may do1 we grows our own4 ,for the shops---- ,an excellent arrangement6 ,balbus interrupted4 ,then one can really depend on their being good4 ,does the window open8 ,the usual questions were answered satisfactorily3 but this time ,hugh added one of his own invention--,does the cat scratch8 ,the landlady looked round suspiciously1 as if to make sure the cat was not listening1 ,i will not deceive you1 gentlemen1 she said4 ,it do scratch1 but not without you pulls its whiskers6 ,it'll never do it1 she repeated slowly1 with a visible effort to recall the exact words of some written agreement between herself and the cat1 without you pulls its whiskers6 ,much may be excused in a cat so treated1 said ,balbus1 as they left the house and crossed to ,number ,seventy-three1 leaving the landlady curtseying on the doorstep1 and still murmuring to herself her parting words1 as if they were a form of blessing1 ---- not without you pulls its whiskers6 ,at ,number ,seventy-three they found only a small shy girl to show the house1 who said yes'm in answer to all questions4 ,the usual room1 said ,balbus1 as they marched in3 the usual back-garden1 the usual cabbages4 ,i suppose you can't get them good at the shops8 ,yes'm1 said the girl4 ,well1 you may tell your mistress we will take the room1 and that her plan of growing her own cabbages is simply admirable6 ,yes'm1 said the girl1 as she showed them out4 ,one day-room and three bed-rooms1 said ,balbus1 as they returned to the hotel4 ,we will take as our day-room the one that gives us the least walking to do to get to it4 ,must we walk from door to door1 and count the steps8 said ,lambert4 ,no1 no6 ,figure it out1 my boys1 figure it out6 ,balbus gaily exclaimed1 as he put pens1 ink1 and paper before his hapless pupils1 and left the room4 ,i say6 ,it'll be a job6 said ,hugh4 ,rather6 said ,lambert4 ,,knot ,,iii4 ,,mad ,,mathesis4 ,i waited for the train4 ,well1 they call me so because ,i am a little mad1 ,i suppose1 she said1 good-humouredly1 in answer to ,clara's cautiously-worded question as to how she came by so strange a nick-name4 ,you see1 ,i never do what sane people are expected to do now-a-days4 ,i never wear long trains1 7talking of trains1 that's the ,charing ,cross ,metropolitan ,station--,i've something to tell you about that71 and ,i never play lawn-tennis4 ,i can't cook an omelette4 ,i can't even set a broken limb6 ,there's an ignoramus for you6 ,clara was her niece1 and full twenty years her junior2 in fact1 she was still attending a ,high ,school--an institution of which ,mad ,mathesis spoke with undisguised aversion4 ,let a woman be meek and lowly6 she would say4 ,none of your ,high ,schools for me6 ,but it was vacation-time just now1 and ,clara was her guest1 and ,mad ,mathesis was showing her the sights of that ,eighth ,wonder of the world--,london4 ,the ,charing ,cross ,metropolitan ,station6 she resumed1 waving her hand towards the entrance as if she were introducing her niece to a friend4 ,the ,bayswater and ,birmingham ,extension is just completed1 and the trains now run round and round continuously--skirting the border of ,wales1 just touching at ,york1 and so round by the east coast back to ,london4 ,the way the trains run is most peculiar4 ,the westerly ones go round in two hours2 the easterly ones take three2 but they always manage to start two trains from here1 opposite ways1 punctually every quarter-of-an-hour4 ,they part to meet again1 said ,clara1 her eyes filling with tears at the romantic thought4 ,no need to cry about it6 her aunt grimly remarked4 ,they don't meet on the same line of rails1 you know4 ,talking of meeting1 an idea strikes me6 she added1 changing the subject with her usual abruptness4 ,let's go opposite ways round1 and see which can meet most trains4 ,no need for a chaperon--ladies' saloon1 you know4 ,you shall go whichever way you like1 and we'll have a bet about it6 ,i never make bets1 ,clara said very gravely4 ,our excellent preceptress has often warned us---- ,you'd be none the worse if you did6 ,mad ,mathesis interrupted4 ,in fact1 you'd be the better1 ,i'm certain6 ,neither does our excellent preceptress approve of puns1 said ,clara4 ,but we'll have a match1 if you like4 ,let me choose my train1 she added after a brief mental calculation1 and ,i'll engage to meet exactly half as many again as you do4 ,not if you count fair1 ,mad ,mathesis bluntly interrupted4 ,remember1 we only count the trains we meet on the way4 ,you mustn't count the one that starts as you start1 nor the one that arrives as you arrive4 ,that will only make the difference of one train1 said ,clara1 as they turned and entered the station4 ,but ,i never travelled alone before4 ,there'll be no one to help me to alight4 ,however1 ,i don't mind4 ,let's have a match4 ,a ragged little boy overheard her remark1 and came running after her4 ,buy a box of cigar-lights1 ,miss6 he pleaded1 pulling her shawl to attract her attention4 ,clara stopped to explain4 ,i never smoke cigars1 she said in a meekly apologetic tone4 ,our excellent preceptress----1 but ,mad ,mathesis impatiently hurried her on1 and the little boy was left gazing after her with round eyes of amazement4 ,the two ladies bought their tickets and moved slowly down the central platform1 ,mad ,mathesis prattling on as usual--,clara silent1 anxiously reconsidering the calculation on which she rested her hopes of winning the match4 ,mind where you go1 dear6 cried her aunt1 checking her just in time4 ,one step more1 and you'd have been in that pail of cold water6 ,i know1 ,i know1 ,clara said1 dreamily4 ,the pale1 the cold1 and the moony---- ,take your places on the spring-boards6 shouted a porter4 ,what are they for6 ,clara asked in a terrified whisper4 ,merely to help us into the trains4 ,the elder lady spoke with the nonchalance of one quite used to the process4 ,very few people can get into a carriage without help in less than three seconds1 and the trains only stop for one second4 ,at this moment the whistle was heard1 and two trains rushed into the station4 ,a moment's pause1 and they were gone again2 but in that brief interval several hundred passengers had been shot into them1 each flying straight to his place with the accuracy of a ,minie bullet--while an equal number were showered out upon the side-platforms4 ,three hours had passed away1 and the two friends met again on the ,charing ,cross platform1 and eagerly compared notes4 ,then ,clara turned away with a sigh4 ,to young impulsive hearts1 like hers1 disappointment is always a bitter pill4 ,mad ,mathesis followed her1 full of kindly sympathy4 ,try again1 my love6 she said1 cheerily4 ,let us vary the experiment4 ,we will start as we did before1 but not to begin counting till our trains meet4 ,when we see each other1 we will say ',one6' and so count on till we come here again4 ,clara brightened up4 ,i shall win that1 she exclaimed eagerly1 if ,i may choose my train6 ,another shriek of engine whistles1 another upheaving of spring-boards1 another living avalanche plunging into two trains as they flashed by3 and the travellers were off again4 ,each gazed eagerly from her carriage window1 holding up her handkerchief as a signal to her friend4 ,a rush and a roar4 ,two trains shot past each other in a tunnel1 and two travellers leaned back in their corners with a sigh--or rather with two sighs--of relief4 ,one6 ,clara murmured to herself4 ,won6 ,it's a word of good omen4 ,this time1 at any rate1 the victory will be mine6 ,but was it8 ,,knot ,,iv4 ,,the ,,dead ,,reckoning4 ,i did dream of money-bags to-night4 ,noonday on the open sea within a few degrees of the ,equator is apt to be oppressively warm2 and our two travellers were now airily clad in suits of dazzling white linen1 having laid aside the chain-armour which they had found not only endurable in the cold mountain air they had lately been breathing1 but a necessary precaution against the daggers of the banditti who infested the heights4 ,their holiday-trip was over1 and they were now on their way home1 in the monthly packet which plied between the two great ports of the island they had been exploring4 ,along with their armour1 the tourists had laid aside the antiquated speech it had pleased them to affect while in knightly disguise1 and had returned to the ordinary style of two country gentlemen of the ,twentieth ,century4 ,stretched on a pile of cushions1 under the shade of a huge umbrella1 they were lazily watching some native fishermen1 who had come on board at the last landing-place1 each carrying over his shoulder a small but heavy sack4 ,a large weighing-machine1 that had been used for cargo at the last port1 stood on the deck2 and round this the fishermen had gathered1 and1 with much unintelligible jabber1 seemed to be weighing their sacks4 ,more like sparrows in a tree than human talk1 isn't it8 the elder tourist remarked to his son1 who smiled feebly1 but would not exert himself so far as to speak4 ,the old man tried another listener4 ,what have they got in those sacks1 ,captain8 he inquired1 as that great being passed them in his never ending parade to and fro on the deck4 ,the ,captain paused in his march1 and towered over the travellers--tall1 grave1 and serenely self-satisfied4 ,fishermen1 he explained1 are often passengers in ,my ship4 ,these five are from ,mhruxi--the place we last touched at--and that's the way they carry their money4 ,the money of this island is heavy1 gentlemen1 but it costs little1 as you may guess4 ,we buy it from them by weight--about five shillings a pound4 ,i fancy a ten pound-note would buy all those sacks4 ,by this time the old man had closed his eyes--in order1 no doubt1 to concentrate his thoughts on these interesting facts2 but the ,captain failed to realise his motive1 and with a grunt resumed his monotonous march4 ,meanwhile the fishermen were getting so noisy over the weighing-machine that one of the sailors took the precaution of carrying off all the weights1 leaving them to amuse themselves with such substitutes in the form of winch-handles1 belaying-pins1 c41 as they could find4 ,this brought their excitement to a speedy end3 they carefully hid their sacks in the folds of the jib that lay on the deck near the tourists1 and strolled away4 ,when next the ,captain's heavy footfall passed1 the younger man roused himself to speak4 ,what did you call the place those fellows came from1 ,captain8 he asked4 ,mhruxi1 sir4 ,and the one we are bound for8 ,the ,captain took a long breath1 plunged into the word1 and came out of it nobly4 ,they call it ,kgovjni1 sir4 ,k--,i give it up6 the young man faintly said4 ,he stretched out his hand for a glass of iced water which the compassionate steward had brought him a minute ago1 and had set down1 unluckily1 just outside the shadow of the umbrella4 ,it was scalding hot1 and he decided not to drink it4 ,the effort of making this resolution1 coming close on the fatiguing conversation he had just gone through1 was too much for him3 he sank back among the cushions in silence4 ,his father courteously tried to make amends for his nonchalance4 ,whereabouts are we now1 ,captain8 said he1 ,have you any idea8 ,the ,captain cast a pitying look on the ignorant landsman4 ,i could tell you that1 sir1 he said1 in a tone of lofty condescension1 to an inch6 ,you don't say so6 the old man remarked1 in a tone of languid surprise4 ,and mean so1 persisted the ,captain4 ,why1 what do you suppose would become of ,my ship1 if ,i were to lose ,my ,longitude and ,my ,latitude8 ,could you make anything of ,my ,dead ,reckoning8 ,nobody could1 ,i'm sure6 the other heartily rejoined4 ,but he had overdone it4 ,it's perfectly intelligible1 the ,captain said1 in an offended tone1 to any one that understands such things4 ,with these words he moved away1 and began giving orders to the men1 who were preparing to hoist the jib4 ,our tourists watched the operation with such interest that neither of them remembered the five money-bags1 which in another moment1 as the wind filled out the jib1 were whirled overboard and fell heavily into the sea4 ,but the poor fishermen had not so easily forgotten their property4 ,in a moment they had rushed to the spot1 and stood uttering cries of fury1 and pointing1 now to the sea1 and now to the sailors who had caused the disaster4 ,the old man explained it to the ,captain4 ,let us make it up among us1 he added in conclusion4 ,ten pounds will do it1 ,i think you said8 ,illustration ,but the ,captain put aside the suggestion with a wave of the hand4 ,no1 sir6 he said1 in his grandest manner4 ,you will excuse ,me1 ,i am sure2 but these are ,my passengers4 ,the accident has happened on board ,my ship1 and under ,my orders4 ,it is for ,me to make compensation4 ,he turned to the angry fishermen4 ,come here1 my men6 he said1 in the ,mhruxian dialect4 ,tell me the weight of each sack4 ,i saw you weighing them just now4 ,then ensued a perfect ,babel of noise1 as the five natives explained1 all screaming together1 how the sailors had carried off the weights1 and they had done what they could with whatever came handy4 ,two iron belaying-pins1 three blocks1 six holystones1 four winch-handles1 and a large hammer1 were now carefully weighed1 the ,captain superintending and noting the results4 ,but the matter did not seem to be settled1 even then3 an angry discussion followed1 in which the sailors and the five natives all joined3 and at last the ,captain approached our tourists with a disconcerted look1 which he tried to conceal under a laugh4 ,it's an absurd difficulty1 he said4 ,perhaps one of you gentlemen can suggest something4 ,it seems they weighed the sacks two at a time6 ,if they didn't have five separate weighings1 of course you can't value them separately1 the youth hastily decided4 ,let's hear all about it1 was the old man's more cautious remark4 ,they did have five separate weighings1 the ,captain said1 but--,well1 it beats me entirely6 he added1 in a sudden burst of candour4 ,here's the result4 ,first and second sack weighed twelve pounds2 second and third1 thirteen and a half2 third and fourth1 eleven and a half2 fourth and fifth1 eight3 and then they say they had only the large hammer left1 and it took three sacks to weigh it down--that's the first1 third and fifth--and they weighed sixteen pounds4 ,there1 gentlemen6 ,did you ever hear anything like that8 ,the old man muttered under his breath ,if only my sister were here6 and looked helplessly at his son4 ,his son looked at the five natives4 ,the five natives looked at the ,captain4 ,the ,captain looked at nobody3 his eyes were cast down1 and he seemed to be saying softly to himself ,contemplate one another1 gentlemen1 if such be your good pleasure4 ,i contemplate ,myself6 ,,knot ,v4 ,,oughts ,,and ,,crosses4 ,look here1 upon this picture1 and on this4 ,and what made you choose the first train1 ,goosey8 said ,mad ,mathesis1 as they got into the cab4 ,couldn't you count better than that8 ,i took an extreme case1 was the tearful reply4 ,our excellent preceptress always says ',when in doubt1 my dears1 take an extreme case4' ,and ,i was in doubt4 ,does it always succeed8 her aunt enquired4 ,clara sighed4 ,not always1 she reluctantly admitted4 ,and ,i can't make out why4 ,one day she was telling the little girls--they make such a noise at tea1 you know--',the more noise you make1 the less jam you will have1 and vice vers4' ,and ,i thought they wouldn't know what 'vice vers' meant3 so ,i explained it to them4 ,i said ',if you make an infinite noise1 you'll get no jam3 and if you make no noise1 you'll get an infinite lot of jam4' ,but our excellent preceptress said that wasn't a good instance4 ,why wasn't it8 she added plaintively4 ,her aunt evaded the question4 ,one sees certain objections to it1 she said4 ,but how did you work it with the ,metropolitan trains8 ,none of them go infinitely fast1 ,i believe4 ,i called them hares and tortoises1 ,clara said--a little timidly1 for she dreaded being laughed at4 ,and ,i thought there couldn't be so many hares as tortoises on the ,line3 so ,i took an extreme case--one hare and an infinite number of tortoises4 ,an extreme case1 indeed1 her aunt remarked with admirable gravity3 and a most dangerous state of things6 ,and ,i thought1 if ,i went with a tortoise1 there would be only one hare to meet3 but if ,i went with the hare--you know there were crowds of tortoises6 ,it wasn't a bad idea1 said the elder lady1 as they left the cab1 at the entrance of ,burlington ,house4 ,you shall have another chance to-day4 ,we'll have a match in marking pictures4 ,clara brightened up4 ,i should like to try again1 very much1 she said4 ,i'll take more care this time4 ,how are we to play8 ,to this question ,mad ,mathesis made no reply3 she was busy drawing lines down the margins of the catalogue4 ,see1 she said after a minute1 ,i've drawn three columns against the names of the pictures in the long room1 and ,i want you to fill them with oughts and crosses--crosses for good marks and oughts for bad4 ,the first column is for choice of subject1 the second for arrangement1 the third for colouring4 ,and these are the conditions of the match4 ,you must give three crosses to two or three pictures4 ,you must give two crosses to four or five---- ,do you mean only two crosses8 said ,clara4 ,or may ,i count the three-cross pictures among the two-cross pictures8 ,of course you may1 said her aunt4 ,any one1 that has three eyes1 may be said to have two eyes1 ,i suppose8 ,clara followed her aunt's dreamy gaze across the crowded gallery1 half-dreading to find that there was a three-eyed person in sight4 ,and you must give one cross to nine or ten4 ,and which wins the match8 ,clara asked1 as she carefully entered these conditions on a blank leaf in her catalogue4 ,whichever marks fewest pictures4 ,but suppose we marked the same number8 ,then whichever uses most marks4 ,clara considered4 ,i don't think it's much of a match1 she said4 ,i shall mark nine pictures1 and give three crosses to three of them1 two crosses to two more1 and one cross each to all the rest4 ,will you1 indeed8 said her aunt4 ,wait till you've heard all the conditions1 my impetuous child4 ,you must give three oughts to one or two pictures1 two oughts to three or four1 and one ought to eight or nine4 ,i don't want you to be too hard on the ,r4,a4's4 ,clara quite gasped as she wrote down all these fresh conditions4 ,it's a great deal worse than ,circulating ,decimals6 she said4 ,but ,i'm determined to win1 all the same6 ,her aunt smiled grimly4 ,we can begin here1 she said1 as they paused before a gigantic picture1 which the catalogue informed them was the ,portrait of ,lieutenant ,brown1 mounted on his favorite elephant4 ,he looks awfully conceited6 said ,clara4 ,i don't think he was the elephant's favorite ,lieutenant4 ,what a hideous picture it is6 ,and it takes up room enough for twenty6 ,mind what you say1 my dear6 her aunt interposed4 ,it's by an ,r4,a46 ,but ,clara was quite reckless4 ,i don't care who it's by6 she cried4 ,and ,i shall give it three bad marks6 ,aunt and niece soon drifted away from each other in the crowd1 and for the next half-hour ,clara was hard at work1 putting in marks and rubbing them out again1 and hunting up and down for suitable pictures4 ,this she found the hardest part of all4 ,i can't find the one ,i want6 she exclaimed at last1 almost crying with vexation4 ,what is it you want to find1 my dear8 ,the voice was strange to ,clara1 but so sweet and gentle that she felt attracted to the owner of it1 even before she had seen her2 and when she turned1 and met the smiling looks of two little old ladies1 whose round dimpled faces1 exactly alike1 seemed never to have known a care1 it was as much as she could do--as she confessed to ,aunt ,mattie afterwards--to keep herself from hugging them both4 ,i was looking for a picture1 she said1 that has a good subject--and that's well arranged--but badly coloured4 ,the little old ladies glanced at each other in some alarm4 ,calm yourself1 my dear1 said the one who had spoken first1 and try to remember which it was4 ,what was the subject8 ,was it an elephant1 for instance8 the other sister suggested4 ,they were still in sight of ,lieutenant ,brown4 ,i don't know1 indeed6 ,clara impetuously replied4 ,you know it doesn't matter a bit what the subject is1 so long as it's a good one6 ,once more the sisters exchanged looks of alarm1 and one of them whispered something to the other1 of which ,clara caught only the one word mad4 ,they mean ,aunt ,mattie1 of course1 she said to herself--fancying1 in her innocence1 that ,london was like her native town1 where everybody knew everybody else4 ,if you mean my aunt1 she added aloud1 she's there--just three pictures beyond ,lieutenant ,brown4 ,ah1 well6 ,then you'd better go to her1 my dear6 her new friend said1 soothingly4 ,she'll find you the picture you want4 ,good-bye1 dear6 ,good-bye1 dear6 echoed the other sister1 ,mind you don't lose sight of your aunt6 ,and the pair trotted off into another room1 leaving ,clara rather perplexed at their manner4 ,they're real darlings6 she soliloquised4 ,i wonder why they pity me so6 ,and she wandered on1 murmuring to herself ,it must have two good marks1 and---- ,,knot ,,vi4 ,,her ,,radiancy4 ,one piecee thing that my have got1 ,maskee,a that thing my no can do4 ,you talkee you no sabey what8 ,bamboo4 ,they landed1 and were at once conducted to the ,palace4 ,about half way they were met by the ,governor1 who welcomed them in ,english--a great relief to our travellers1 whose guide could speak nothing but ,kgovjnian4 ,i don't half like the way they grin at us as we go by6 the old man whispered to his son4 ,and why do they say ',bamboo6' so often8 ,it alludes to a local custom1 replied the ,governor1 who had overheard the question4 ,such persons as happen in any way to displease ,her ,radiancy are usually beaten with rods4 ,illustration3 ,,why ,,do ,,they ,,say ',,bamboo6' ,,so ,,often8 ,the old man shuddered4 ,a most objectional local custom6 he remarked with strong emphasis4 ,i wish we had never landed6 ,did you notice that black fellow1 ,norman1 opening his great mouth at us8 ,i verily believe he would like to eat us6 ,norman appealed to the ,governor1 who was walking at his other side4 ,do they often eat distinguished strangers here8 he said1 in as indifferent a tone as he could assume4 ,not often--not ever6 was the welcome reply4 ,they are not good for it4 ,pigs we eat1 for they are fat4 ,this old man is thin4 ,and thankful to be so6 muttered the elder traveller4 ,beaten we shall be without a doubt4 ,it's a comfort to know it won't be ,beaten without the ,b6 ,my dear boy1 just look at the peacocks6 ,they were now walking between two unbroken lines of those gorgeous birds1 each held in check1 by means of a golden collar and chain1 by a black slave1 who stood well behind1 so as not to interrupt the view of the glittering tail1 with its network of rustling feathers and its hundred eyes4 ,the ,governor smiled proudly4 ,in your honour1 he said1 ,her ,radiancy has ordered up ten thousand additional peacocks4 ,she will1 no doubt1 decorate you1 before you go1 with the usual ,star and ,feathers4 ,it'll be ,star without the ,s6 faltered one of his hearers4 ,come1 come6 ,don't lose heart6 said the other4 ,all this is full of charm for me4 ,you are young1 ,norman1 sighed his father2 young and light-hearted4 ,for me1 it is ,charm without the ,c4 ,the old one is sad1 the ,governor remarked with some anxiety4 ,he has1 without doubt1 effected some fearful crime8 ,but ,i haven't6 the poor old gentleman hastily exclaimed4 ,tell him ,i haven't1 ,norman6 ,he has not1 as yet1 ,norman gently explained4 ,and the ,governor repeated1 in a satisfied tone1 ,not as yet4 ,yours is a wondrous country6 the ,governor resumed1 after a pause4 ,now here is a letter from a friend of mine1 a merchant1 in ,london4 ,he and his brother went there a year ago1 with a thousand pounds apiece2 and on ,new-,year's-day they had sixty thousand pounds between them6 ,how did they do it8 ,norman eagerly exclaimed4 ,even the elder traveller looked excited4 ,the ,governor handed him the open letter4 ,anybody can do it1 when once they know how1 so ran this oracular document4 ,we borrowed nought3 we stole nought4 ,we began the year with only a thousand pounds apiece3 and last ,new-,year's-day we had sixty thousand pounds between us--sixty thousand golden sovereigns6 ,norman looked grave and thoughtful as he handed back the letter4 ,his father hazarded one guess4 ,was it by gambling8 ,a ,kgovjnian never gambles1 said the ,governor gravely1 as he ushered them through the palace gates4 ,they followed him in silence down a long passage1 and soon found themselves in a lofty hall1 lined entirely with peacocks' feathers4 ,in the centre was a pile of crimson cushions1 which almost concealed the figure of ,her ,radiancy--a plump little damsel1 in a robe of green satin dotted with silver stars1 whose pale round face lit up for a moment with a half-smile as the travellers bowed before her1 and then relapsed into the exact expression of a wax doll1 while she languidly murmured a word or two in the ,kgovjnian dialect4 ,the ,governor interpreted4 ,her ,radiancy welcomes you4 ,she notes the ,impenetrable ,placidity of the old one1 and the ,imperceptible ,acuteness of the youth4 ,here the little potentate clapped her hands1 and a troop of slaves instantly appeared1 carrying trays of coffee and sweetmeats1 which they offered to the guests1 who had1 at a signal from the ,governor1 seated themselves on the carpet4 ,sugar-plums6 muttered the old man4 ,one might as well be at a confectioner's6 ,ask for a penny bun1 ,norman6 ,not so loud6 his son whispered4 ,say something complimentary6 ,for the ,governor was evidently expecting a speech4 ,we thank ,her ,exalted ,potency1 the old man timidly began4 ,we bask in the light of her smile1 which---- ,the words of old men are weak6 the ,governor interrupted angrily4 ,let the youth speak6 ,tell her1 cried ,norman1 in a wild burst of eloquence1 that1 like two grasshoppers in a volcano1 we are shrivelled up in the presence of ,her ,spangled ,vehemence6 ,it is well1 said the ,governor1 and translated this into ,kgovjnian4 ,i am now to tell you1 he proceeded1 what ,her ,radiancy requires of you before you go4 ,the yearly competition for the post of ,imperial ,scarf-maker is just ended2 you are the judges4 ,you will take account of the rate of work1 the lightness of the scarves1 and their warmth4 ,usually the competitors differ in one point only4 ,thus1 last year1 ,fifi and ,gogo made the same number of scarves in the trial-week1 and they were equally light2 but ,fifi's were twice as warm as ,gogo's and she was pronounced twice as good4 ,but this year1 woe is me1 who can judge it8 ,three competitors are here1 and they differ in all points6 ,while you settle their claims1 you shall be lodged1 ,her ,radiancy bids me say1 free of expense--in the best dungeon1 and abundantly fed on the best bread and water4 ,the old man groaned4 ,all is lost6 he wildly exclaimed4 ,but ,norman heeded him not3 he had taken out his note-book1 and was calmly jotting down the particulars4 ,three they be1 the ,governor proceeded1 ,lolo1 ,mimi1 and ,zuzu4 ,lolo makes #e scarves while ,mimi makes #b2 but ,zuzu makes #d while ,lolo makes #c6 ,again1 so fairylike is ,zuzu's handiwork1 #e of her scarves weigh no more than one of ,lolo's2 yet ,mimi's is lighter still--#e of hers will but balance #c of ,zuzu's6 ,and for warmth one of ,mimi's is equal to #d of ,zuzu's2 yet one of ,lolo's is as warm as #c of ,mimi's6 ,here the little lady once more clapped her hands4 ,it is our signal of dismissal6 the ,governor hastily said4 ,pay ,her ,radiancy your farewell compliments--and walk out backwards4 ,the walking part was all the elder tourist could manage4 ,norman simply said ,tell ,her ,radiancy we are transfixed by the spectacle of ,her ,serene ,brilliance1 and bid an agonized farewell to her ,condensed ,milkiness6 ,her ,radiancy is pleased1 the ,governor reported1 after duly translating this4 ,she casts on you a glance from ,her ,imperial ,eyes1 and is confident that you will catch it6 ,that ,i warrant we shall6 the elder traveller moaned to himself distractedly4 ,once more they bowed low1 and then followed the ,governor down a winding staircase to the ,imperial ,dungeon1 which they found to be lined with coloured marble1 lighted from the roof1 and splendidly though not luxuriously furnished with a bench of polished malachite4 ,i trust you will not delay the calculation1 the ,governor said1 ushering them in with much ceremony4 ,i have known great inconvenience--great and serious inconvenience--result to those unhappy ones who have delayed to execute the commands of ,her ,radiancy6 ,and on this occasion she is resolute3 she says the thing must and shall be done3 and she has ordered up ten thousand additional bamboos6 ,with these words he left them1 and they heard him lock and bar the door on the outside4 ,i told you how it would end6 moaned the elder traveller1 wringing his hands1 and quite forgetting in his anguish that he had himself proposed the expedition1 and had never predicted anything of the sort4 ,oh that we were well out of this miserable business6 ,courage6 cried the younger cheerily4 ,hc olim meminisse juvabit6 ,the end of all this will be glory6 ,glory without the ,l6 was all the poor old man could say1 as he rocked himself to and fro on the malachite bench4 ,glory without the ,l6 ,,footnote3 ,footnote ,a3 ,maskee1 in ,pigeon-,english1 means without4 ,,knot ,,vii4 ,,petty ,,cash4 ,base is the slave that pays4 ,aunt ,mattie6 ,my child8 ,would you mind writing it down at once8 ,i shall be quite certain to forget it if you don't6 ,my dear1 we really must wait till the cab stops4 ,how can ,i possibly write anything in the midst of all this jolting8 ,but really ,i shall be forgetting it6 ,clara's voice took the plaintive tone that her aunt never knew how to resist1 and with a sigh the old lady drew forth her ivory tablets and prepared to record the amount that ,clara had just spent at the confectioner's shop4 ,her expenditure was always made out of her aunt's purse1 but the poor girl knew1 by bitter experience1 that sooner or later ,mad ,mathesis would expect an exact account of every penny that had gone1 and she waited1 with ill-concealed impatience1 while the old lady turned the tablets over and over1 till she had found the one headed ,,petty ,,cash4 ,here's the place1 she said at last1 and here we have yesterday's luncheon duly entered4 ,one glass lemonade 7,why can't you drink water1 like me87 three sandwiches 7,they never put in half mustard enough4 ,i told the young woman so1 to her face2 and she tossed her head--like her impudence67 and seven biscuits4 ,total one-and-two-pence4 ,well1 now for to-day's8 ,one glass of lemonade---- ,clara was beginning to say1 when suddenly the cab drew up1 and a courteous railway-porter was handing out the bewildered girl before she had had time to finish her sentence4 ,her aunt pocketed the tablets instantly4 ,business first1 she said3 petty cash--which is a form of pleasure1 whatever you may think--afterwards4 ,and she proceeded to pay the driver1 and to give voluminous orders about the luggage1 quite deaf to the entreaties of her unhappy niece that she would enter the rest of the luncheon account4 ,my dear1 you really must cultivate a more capacious mind6 was all the consolation she vouchsafed to the poor girl4 ,are not the tablets of your memory wide enough to contain the record of one single luncheon8 ,not wide enough6 ,not half wide enough6 was the passionate reply4 ,the words came in aptly enough1 but the voice was not that of ,clara1 and both ladies turned in some surprise to see who it was that had so suddenly struck into their conversation4 ,a fat little old lady was standing at the door of a cab1 helping the driver to extricate what seemed an exact duplicate of herself3 it would have been no easy task to decide which was the fatter1 or which looked the more good-humoured of the two sisters4 ,i tell you the cab-door isn't half wide enough6 she repeated1 as her sister finally emerged1 somewhat after the fashion of a pellet from a pop-gun1 and she turned to appeal to ,clara4 ,is it1 dear8 she said1 trying hard to bring a frown into a face that dimpled all over with smiles4 ,some folks is too wide for 'em1 growled the cab-driver4 ,illustration3 ,i ,,tell ,,you ,,the ,,cab-,,door ,,isn',t ,,half ,,wide ,,enough6 ,don't provoke me1 man6 cried the little old lady1 in what she meant for a tempest of fury4 ,say another word and ,i'll put you into the ,county ,court1 and sue you for a ,habeas ,corpus6 ,the cabman touched his hat1 and marched off1 grinning4 ,nothing like a little ,law to cow the ruffians1 my dear6 she remarked confidentially to ,clara4 ,you saw how he quailed when ,i mentioned the ,habeas ,corpus8 ,not that ,i've any idea what it means1 but it sounds very grand1 doesn't it8 ,it's very provoking1 ,clara replied1 a little vaguely4 ,very6 the little old lady eagerly repeated4 ,and we're very much provoked indeed4 ,aren't we1 sister8 ,i never was so provoked in all my life6 the fatter sister assented1 radiantly4 ,by this time ,clara had recognised her picture-gallery acquaintances1 and1 drawing her aunt aside1 she hastily whispered her reminiscences4 ,i met them first in the ,royal ,academy--and they were very kind to me--and they were lunching at the next table to us1 just now1 you know--and they tried to help me to find the picture ,i wanted--and ,i'm sure they're dear old things6 ,friends of yours1 are they8 said ,mad ,mathesis4 ,well1 ,i like their looks4 ,you can be civil to them1 while ,i get the tickets4 ,but do try and arrange your ideas a little more chronologically6 ,and so it came to pass that the four ladies found themselves seated side by side on the same bench waiting for the train1 and chatting as if they had known one another for years4 ,now this ,i call quite a remarkable coincidence6 exclaimed the smaller and more talkative of the two sisters--the one whose legal knowledge had annihilated the cab-driver4 ,not only that we should be waiting for the same train1 and at the same station--that would be curious enough--but actually on the same day1 and the same hour of the day6 ,that's what strikes me so forcibly6 ,she glanced at the fatter and more silent sister1 whose chief function in life seemed to be to support the family opinion1 and who meekly responded-- ,and me too1 sister6 ,those are not independent coincidences---- ,mad ,mathesis was just beginning1 when ,clara ventured to interpose4 ,there's no jolting here1 she pleaded meekly4 ,would you mind writing it down now8 ,out came the ivory tablets once more4 ,what was it1 then8 said her aunt4 ,one glass of lemonade1 one sandwich1 one biscuit--,oh dear me6 cried poor ,clara1 the historical tone suddenly changing to a wail of agony4 ,toothache8 said her aunt calmly1 as she wrote down the items4 ,the two sisters instantly opened their reticules and produced two different remedies for neuralgia1 each marked unequalled4 ,it isn't that6 said poor ,clara4 ,thank you very much4 ,it's only that ,i can't remember how much ,i paid6 ,well1 try and make it out1 then1 said her aunt4 ,you've got yesterday's luncheon to help you1 you know4 ,and here's the luncheon we had the day before--the first day we went to that shop--one glass lemonade1 four sandwiches1 ten biscuits4 ,total1 one-and-fivepence4 ,she handed the tablets to ,clara1 who gazed at them with eyes so dim with tears that she did not at first notice that she was holding them upside down4 ,the two sisters had been listening to all this with the deepest interest1 and at this juncture the smaller one softly laid her hand on ,clara's arm4 ,do you know1 my dear1 she said coaxingly1 my sister and ,i are in the very same predicament6 ,quite identically the very same predicament6 ,aren't we1 sister8 ,quite identically and absolutely the very---- began the fatter sister1 but she was constructing her sentence on too large a scale1 and the little one would not wait for her to finish it4 ,yes1 my dear1 she resumed2 we were lunching at the very same shop as you were--and we had two glasses of lemonade and three sandwiches and five biscuits--and neither of us has the least idea what we paid4 ,have we1 sister8 ,quite identically and absolutely---- murmured the other1 who evidently considered that she was now a whole sentence in arrears1 and that she ought to discharge one obligation before contracting any fresh liabilities2 but the little lady broke in again1 and she retired from the conversation a bankrupt4 ,would you make it out for us1 my dear8 pleaded the little old lady4 ,you can do ,arithmetic1 ,i trust8 her aunt said1 a little anxiously1 as ,clara turned from one tablet to another1 vainly trying to collect her thoughts4 ,her mind was a blank1 and all human expression was rapidly fading out of her face4 ,a gloomy silence ensued4 ,,knot ,,viii4 ,,de ,,omnibus ,,rebus4 ,this little pig went to market3 ,this little pig staid at home4 ,by ,her ,radiancy's express command1 said the ,governor1 as he conducted the travellers1 for the last time1 from the ,imperial presence1 ,i shall now have the ecstasy of escorting you as far as the outer gate of the ,military ,quarter1 where the agony of parting--if indeed ,nature can survive the shock--must be endured6 ,from that gate grurmstipths start every quarter of an hour1 both ways---- ,would you mind repeating that word8 said ,norman4 ,grurm----8 ,grurmstipths1 the ,governor repeated4 ,you call them omnibuses in ,england4 ,they run both ways1 and you can travel by one of them all the way down to the harbour4 ,the old man breathed a sigh of relief2 four hours of courtly ceremony had wearied him1 and he had been in constant terror lest something should call into use the ten thousand additional bamboos4 ,in another minute they were crossing a large quadrangle1 paved with marble1 and tastefully decorated with a pigsty in each corner4 ,soldiers1 carrying pigs1 were marching in all directions3 and in the middle stood a gigantic officer giving orders in a voice of thunder1 which made itself heard above all the uproar of the pigs4 ,it is the ,commander-in-,chief6 the ,governor hurriedly whispered to his companions1 who at once followed his example in prostrating themselves before the great man4 ,the ,commander gravely bowed in return4 ,he was covered with gold lace from head to foot3 his face wore an expression of deep misery3 and he had a little black pig under each arm4 ,still the gallant fellow did his best1 in the midst of the orders he was every moment issuing to his men1 to bid a courteous farewell to the departing guests4 ,farewell1 oh old one--carry these three to the ,south corner--and farewell to thee1 thou young one--put this fat one on the top of the others in the ,western sty--may your shadows never be less--woe is me1 it is wrongly done6 ,empty out all the sties1 and begin again6 ,and the soldier leant upon his sword1 and wiped away a tear4 ,he is in distress1 the ,governor explained as they left the court4 ,her ,radiancy has commanded him to place twenty-four pigs in those four sties1 so that1 as she goes round the court1 she may always find the number in each sty nearer to ten than the number in the last4 ,does she call ten nearer to ten than nine is8 said ,norman4 ,surely1 said the ,governor4 ,her ,radiancy would admit that ten is nearer to ten than nine is--and also nearer than eleven is4 ,then ,i think it can be done1 said ,norman4 ,the ,governor shook his head4 ,the ,commander has been transferring them in vain for four months1 he said4 ,what hope remains8 ,and ,her ,radiancy has ordered up ten thousand additional---- ,the pigs don't seem to enjoy being transferred1 the old man hastily interrupted4 ,he did not like the subject of bamboos4 ,they are only provisionally transferred1 you know1 said the ,governor4 ,in most cases they are immediately carried back again3 so they need not mind it4 ,and all is done with the greatest care1 under the personal superintendence of the ,commander-in-,chief4 ,of course she would only go once round8 said ,norman4 ,alas1 no6 sighed their conductor4 ,round and round4 ,round and round4 ,these are ,her ,radiancy's own words4 ,but oh1 agony6 ,here is the outer gate1 and we must part6 ,he sobbed as he shook hands with them1 and the next moment was briskly walking away4 ,he might have waited to see us off6 said the old man1 piteously4 ,and he needn't have begun whistling the very moment he left us6 said the young one1 severely4 ,but look sharp--here are two what's-his-names in the act of starting6 ,unluckily1 the sea-bound omnibus was full4 ,never mind6 said ,norman1 cheerily4 ,we'll walk on till the next one overtakes us4 ,they trudged on in silence1 both thinking over the military problem1 till they met an omnibus coming from the sea4 ,the elder traveller took out his watch4 ,just twelve minutes and a half since we started1 he remarked in an absent manner4 ,suddenly the vacant face brightened2 the old man had an idea4 ,my boy6 he shouted1 bringing his hand down upon ,norman's shoulder so suddenly as for a moment to transfer his centre of gravity beyond the base of support4 ,thus taken off his guard1 the young man wildly staggered forwards1 and seemed about to plunge into space3 but in another moment he had gracefully recovered himself4 ,problem in ,precession and ,nutation1 he remarked--in tones where filial respect only just managed to conceal a shade of annoyance4 ,what is it8 he hastily added1 fearing his father might have been taken ill4 ,will you have some brandy8 ,when will the next omnibus overtake us8 ,when8 ,when8 the old man cried1 growing more excited every moment4 ,norman looked gloomy4 ,give me time1 he said4 ,i must think it over4 ,and once more the travellers passed on in silence--a silence only broken by the distant squeals of the unfortunate little pigs1 who were still being provisionally transferred from sty to sty1 under the personal superintendence of the ,commander-in-,chief4 ,,knot ,,ix4 ,a ,,serpent ,,with ,,corners4 ,water1 water1 every where1 ,nor any drop to drink4 ,it'll just take one more pebble4 ,what ever are you doing with those buckets8 ,the speakers were ,hugh and ,lambert4 ,place1 the beach of ,little ,mendip4 ,time1 #a4cj1 ,p4,m4 ,hugh was floating a bucket in another a size larger1 and trying how many pebbles it would carry without sinking4 ,lambert was lying on his back1 doing nothing4 ,for the next minute or two ,hugh was silent1 evidently deep in thought4 ,suddenly he started4 ,i say1 look here1 ,lambert6 he cried4 ,if it's alive1 and slimy1 and with legs1 ,i don't care to1 said ,lambert4 ,didn't ,balbus say this morning that1 if a body is immersed in liquid1 it displaces as much liquid as is equal to its own bulk8 said ,hugh4 ,he said things of that sort1 ,lambert vaguely replied4 ,well1 just look here a minute4 ,here's the little bucket almost quite immersed3 so the water displaced ought to be just about the same bulk4 ,and now just look at it6 ,he took out the little bucket as he spoke1 and handed the big one to ,lambert4 ,why1 there's hardly a teacupful6 ,do you mean to say that water is the same bulk as the little bucket8 ,course it is1 said ,lambert4 ,well1 look here again6 cried ,hugh1 triumphantly1 as he poured the water from the big bucket into the little one4 ,why1 it doesn't half fill it6 ,that's its business1 said ,lambert4 ,if ,balbus says it's the same bulk1 why1 it is the same bulk1 you know4 ,well1 ,i don't believe it1 said ,hugh4 ,you needn't1 said ,lambert4 ,besides1 it's dinner-time4 ,come along4 ,they found ,balbus waiting dinner for them1 and to him ,hugh at once propounded his difficulty4 ,let's get you helped first1 said ,balbus1 briskly cutting away at the joint4 ,you know the old proverb ',mutton first1 mechanics afterwards'8 ,the boys did not know the proverb1 but they accepted it in perfect good faith1 as they did every piece of information1 however startling1 that came from so infallible an authority as their tutor4 ,they ate on steadily in silence1 and1 when dinner was over1 ,hugh set out the usual array of pens1 ink1 and paper1 while ,balbus repeated to them the problem he had prepared for their afternoon's task4 ,a friend of mine has a flower-garden--a very pretty one1 though no great size-- ,how big is it8 said ,hugh4 ,that's what you have to find out6 ,balbus gaily replied4 ,all ,i tell you is that it is oblong in shape--just half a yard longer than its width--and that a gravel-walk1 one yard wide1 begins at one corner and runs all round it4 ,joining into itself8 said ,hugh4 ,not joining into itself1 young man4 ,just before doing that1 it turns a corner1 and runs round the garden again1 alongside of the first portion1 and then inside that again1 winding in and in1 and each lap touching the last one1 till it has used up the whole of the area4 ,like a serpent with corners8 said ,lambert4 ,exactly so4 ,and if you walk the whole length of it1 to the last inch1 keeping in the centre of the path1 it's exactly two miles and half a furlong4 ,now1 while you find out the length and breadth of the garden1 ,i'll see if ,i can think out that sea-water puzzle4 ,you said it was a flower-garden8 ,hugh inquired1 as ,balbus was leaving the room4 ,i did1 said ,balbus4 ,where do the flowers grow8 said ,hugh4 ,but ,balbus thought it best not to hear the question4 ,he left the boys to their problem1 and1 in the silence of his own room1 set himself to unravel ,hugh's mechanical paradox4 ,to fix our thoughts1 he murmured to himself1 as1 with hands deep-buried in his pockets1 he paced up and down the room1 we will take a cylindrical glass jar1 with a scale of inches marked up the side1 and fill it with water up to the #aj-;inch mark3 and we will assume that every inch depth of jar contains a pint of water4 ,we will now take a solid cylinder1 such that every inch of it is equal in bulk to half a pint of water1 and plunge #d inches of it into the water1 so that the end of the cylinder comes down to the #f-;inch mark4 ,well1 that displaces #b pints of water4 ,what becomes of them8 ,why1 if there were no more cylinder1 they would lie comfortably on the top1 and fill the jar up to the #ab-;inch mark4 ,but unfortunately there is more cylinder1 occupying half the space between the #aj-;inch and the #ab-;inch marks1 so that only one pint of water can be accommodated there4 ,what becomes of the other pint8 ,why1 if there were no more cylinder1 it would lie on the top1 and fill the jar up to the #ac-;inch mark4 ,but unfortunately----,shade of ,newton6 he exclaimed1 in sudden accents of terror4 ,when does the water stop rising8 ,a bright idea struck him4 ,i'll write a little essay on it1 he said4 99 99 99 99 99 ,balbus's ,essay4 ,when a solid is immersed in a liquid1 it is well known that it displaces a portion of the liquid equal to itself in bulk1 and that the level of the liquid rises just so much as it would rise if a quantity of liquid had been added to it1 equal in bulk to the solid4 ,lardner says1 precisely the same process occurs when a solid is partially immersed3 the quantity of liquid displaced1 in this case1 equalling the portion of the solid which is immersed1 and the rise of the level being in proportion4 ,suppose a solid held above the surface of a liquid and partially immersed3 a portion of the liquid is displaced1 and the level of the liquid rises4 ,but1 by this rise of level1 a little bit more of the solid is of course immersed1 and so there is a new displacement of a second portion of the liquid1 and a consequent rise of level4 ,again1 this second rise of level causes a yet further immersion1 and by consequence another displacement of liquid and another rise4 ,it is self-evident that this process must continue till the entire solid is immersed1 and that the liquid will then begin to immerse whatever holds the solid1 which1 being connected with it1 must for the time be considered a part of it4 ,if you hold a stick1 six feet long1 with its end in a tumbler of water1 and wait long enough1 you must eventually be immersed4 ,the question as to the source from which the water is supplied--which belongs to a high branch of mathematics1 and is therefore beyond our present scope--does not apply to the sea4 ,let us therefore take the familiar instance of a man standing at the edge of the sea1 at ebb-tide1 with a solid in his hand1 which he partially immerses3 he remains steadfast and unmoved1 and we all know that he must be drowned4 ,the multitudes who daily perish in this manner to attest a philosophical truth1 and whose bodies the unreasoning wave casts sullenly upon our thankless shores1 have a truer claim to be called the martyrs of science than a ,galileo or a ,kepler4 ,to use ,kossuth's eloquent phrase1 they are the unnamed demigods of the nineteenth century4,b 99 99 99 99 99 ,there's a fallacy somewhere1 he murmured drowsily1 as he stretched his long legs upon the sofa4 ,i must think it over again4 ,he closed his eyes1 in order to concentrate his attention more perfectly1 and for the next hour or so his slow and regular breathing bore witness to the careful deliberation with which he was investigating this new and perplexing view of the subject4 ,illustration3 ,,he ,,remains ,,steadfast ,,and ,,unmoved4 ,,footnote3 ,footnote ,b3 ,note by the writer4--,for the above ,essay ,i am indebted to a dear friend1 now deceased4 ,,knot ,x4 ,,chelsea ,,buns4 ,yea1 buns1 and buns1 and buns6 ,,old ,,song4 ,how very1 very sad6 exclaimed ,clara2 and the eyes of the gentle girl filled with tears as she spoke4 ,sad--but very curious when you come to look at it arithmetically1 was her aunt's less romantic reply4 ,some of them have lost an arm in their country's service1 some a leg1 some an ear1 some an eye---- ,and some1 perhaps1 all6 ,clara murmured dreamily1 as they passed the long rows of weather-beaten heroes basking in the sun4 ,did you notice that very old one1 with a red face1 who was drawing a map in the dust with his wooden leg1 and all the others watching8 ,i think it was a plan of a battle---- ,the battle of ,trafalgar1 no doubt1 her aunt interrupted1 briskly4 ,hardly that1 ,i think1 ,clara ventured to say4 ,you see1 in that case1 he couldn't well be alive---- ,couldn't well be alive6 the old lady contemptuously repeated4 ,he's as lively as you and me put together6 ,why1 if drawing a map in the dust--with one's wooden leg--doesn't prove one to be alive1 perhaps you'll kindly mention what does prove it6 ,clara did not see her way out of it4 ,logic had never been her forte4 ,to return to the arithmetic1 ,mad ,mathesis resumed--the eccentric old lady never let slip an opportunity of driving her niece into a calculation--what percentage do you suppose must have lost all four--a leg1 an arm1 an eye1 and an ear8 ,how can ,i tell8 gasped the terrified girl4 ,she knew well what was coming4 ,you can't1 of course1 without data1 her aunt replied3 but ,i'm just going to give you---- ,give her a ,chelsea bun1 ,miss6 ,that's what most young ladies likes best6 ,the voice was rich and musical1 and the speaker dexterously whipped back the snowy cloth that covered his basket1 and disclosed a tempting array of the familiar square buns1 joined together in rows1 richly egged and browned1 and glistening in the sun4 ,no1 sir6 ,i shall give her nothing so indigestible6 ,be off6 ,the old lady waved her parasol threateningly3 but nothing seemed to disturb the good-humour of the jolly old man1 who marched on1 chanting his melodious refrain3-- ,music3 ,chel-sea buns6 ,chel-sea buns hot6 ,chel-sea buns6 ,pi-ping hot6 ,chel-sea buns hot6 ,chel-sea buns6 ,far too indigestible1 my love6 said the old lady4 ,percentages will agree with you ever so much better6 ,clara sighed1 and there was a hungry look in her eyes as she watched the basket lessening in the distance3 but she meekly listened to the relentless old lady1 who at once proceeded to count off the data on her fingers4 ,say that #gj per cent4 have lost an eye--#ge per cent4 an ear--#hj per cent4 an arm--#he per cent4 a leg--that'll do it beautifully4 ,now1 my dear1 what percentage1 at least1 must have lost all four8 ,no more conversation occurred--unless a smothered exclamation of ,piping hot6 which escaped from ,clara's lips as the basket vanished round a corner could be counted as such--until they reached the old ,chelsea mansion1 where ,clara's father was then staying1 with his three sons and their old tutor4 ,balbus1 ,lambert1 and ,hugh had entered the house only a few minutes before them4 ,they had been out walking1 and ,hugh had been propounding a difficulty which had reduced ,lambert to the depths of gloom1 and had even puzzled ,balbus4 ,it changes from ,wednesday to ,thursday at midnight1 doesn't it8 ,hugh had begun4 ,sometimes1 said ,balbus1 cautiously4 ,always1 said ,lambert1 decisively4 ,sometimes1 ,balbus gently insisted4 ,six midnights out of seven1 it changes to some other name4 ,i meant1 of course1 ,hugh corrected himself1 when it does change from ,wednesday to ,thursday1 it does it at midnight--and only at midnight4 ,surely1 said ,balbus4 ,lambert was silent4 ,well1 now1 suppose it's midnight here in ,chelsea4 ,then it's ,wednesday west of ,chelsea 7say in ,ireland or ,america7 where midnight hasn't arrived yet3 and it's ,thursday east of ,chelsea 7say in ,germany or ,russia7 where midnight has just passed by8 ,surely1 ,balbus said again4 ,even ,lambert nodded this time4 ,but it isn't midnight1 anywhere else2 so it can't be changing from one day to another anywhere else4 ,and yet1 if ,ireland and ,america and so on call it ,wednesday1 and ,germany and ,russia and so on call it ,thursday1 there must be some place--not ,chelsea--that has different days on the two sides of it4 ,and the worst of it is1 the people there get their days in the wrong order3 they've got ,wednesday east of them1 and ,thursday west--just as if their day had changed from ,thursday to ,wednesday6 ,i've heard that puzzle before6 cried ,lambert4 ,and ,i'll tell you the explanation4 ,when a ship goes round the world from east to west1 we know that it loses a day in its reckoning3 so that when it gets home1 and calls its day ,wednesday1 it finds people here calling it ,thursday1 because we've had one more midnight than the ship has had4 ,and when you go the other way round you gain a day4 ,i know all that1 said ,hugh1 in reply to this not very lucid explanation3 but it doesn't help me1 because the ship hasn't proper days4 ,one way round1 you get more than twenty-four hours to the day1 and the other way you get less3 so of course the names get wrong3 but people that live on in one place always get twenty-four hours to the day4 ,i suppose there is such a place1 ,balbus said1 meditatively1 though ,i never heard of it4 ,and the people must find it very queer1 as ,hugh says1 to have the old day east of them1 and the new one west3 because1 when midnight comes round to them1 with the new day in front of it and the old one behind it1 one doesn't see exactly what happens4 ,i must think it over4 ,so they had entered the house in the state ,i have described--,balbus puzzled1 and ,lambert buried in gloomy thought4 ,yes1 m'm1 ,master is at home1 m'm1 said the stately old butler4 7,n4,b4--,it is only a butler of experience who can manage a series of three ,m's together1 without any interjacent vowels47 ,and the ole party is a-waiting for you in the libery4 ,i don't like his calling your father an old party1 ,mad ,mathesis whispered to her niece1 as they crossed the hall4 ,and ,clara had only just time to whisper in reply he meant the whole party1 before they were ushered into the library1 and the sight of the five solemn faces there assembled chilled her into silence4 ,her father sat at the head of the table1 and mutely signed to the ladies to take the two vacant chairs1 one on each side of him4 ,his three sons and ,balbus completed the party4 ,writing materials had been arranged round the table1 after the fashion of a ghostly banquet3 the butler had evidently bestowed much thought on the grim device4 ,sheets of quarto paper1 each flanked by a pen on one side and a pencil on the other1 represented the plates--penwipers did duty for rolls of bread--while ink-bottles stood in the places usually occupied by wine-glasses4 ,the pice de resistance was a large green baize bag1 which gave forth1 as the old man restlessly lifted it from side to side1 a charming jingle1 as of innumerable golden guineas4 ,sister1 daughter1 sons--and ,balbus--1 the old man began1 so nervously1 that ,balbus put in a gentle ,hear1 hear6 while ,hugh drummed on the table with his fists4 ,this disconcerted the unpractised orator4 ,sister-- he began again1 then paused a moment1 moved the bag to the other side1 and went on with a rush1 ,i mean--this being--a critical occasion--more or less--being the year when one of my sons comes of age-- he paused again in some confusion1 having evidently got into the middle of his speech sooner than he intended3 but it was too late to go back4 ,hear1 hear6 cried ,balbus4 ,quite so1 said the old gentleman1 recovering his self-possession a little3 when first ,i began this annual custom--my friend ,balbus will correct me if ,i am wrong-- 7,hugh whispered with a strap6 but nobody heard him except ,lambert1 who only frowned and shook his head at him7 --this annual custom of giving each of my sons as many guineas as would represent his age--it was a critical time--so ,balbus informed me--as the ages of two of you were together equal to that of the third--so on that occasion ,i made a speech---- ,he paused so long that ,balbus thought it well to come to the rescue with the words ,it was a most---- but the old man checked him with a warning look3 yes1 made a speech1 he repeated4 ,a few years after that1 ,balbus pointed out--,i say pointed out-- 7,hear1 hear6 cried ,balbus4 ,quite so1 said the grateful old man47 --that it was another critical occasion4 ,the ages of two of you were together double that of the third4 ,so ,i made another speech--another speech4 ,and now again it's a critical occasion--so ,balbus says--and ,i am making---- 7,here ,mad ,mathesis pointedly referred to her watch7 all the haste ,i can6 the old man cried1 with wonderful presence of mind4 ,indeed1 sister1 ,i'm coming to the point now6 ,the number of years that have passed since that first occasion is just two-thirds of the number of guineas ,i then gave you4 ,now1 my boys1 calculate your ages from the data1 and you shall have the money6 ,but we know our ages6 cried ,hugh4 ,silence1 sir6 thundered the old man1 rising to his full height 7he was exactly five-foot five7 in his indignation4 ,i say you must use the data only6 ,you mustn't even assume which it is that comes of age6 ,he clutched the bag as he spoke1 and with tottering steps 7it was about as much as he could do to carry it7 he left the room4 ,and you shall have a similar cadeau1 the old lady whispered to her niece1 when you've calculated that percentage6 ,and she followed her brother4 ,nothing could exceed the solemnity with which the old couple had risen from the table1 and yet was it--was it a grin with which the father turned away from his unhappy sons8 ,could it be--could it be a wink with which the aunt abandoned her despairing niece8 ,and were those--were those sounds of suppressed chuckling which floated into the room1 just before ,balbus 7who had followed them out7 closed the door8 ,surely not3 and yet the butler told the cook--but no1 that was merely idle gossip1 and ,i will not repeat it4 ,the shades of evening granted their unuttered petition1 and closed not o'er them 7for the butler brought in the lamp73 the same obliging shades left them a lonely bark 7the wail of a dog1 in the back-yard1 baying the moon7 for awhile3 but neither morn1 alas1 7nor any other epoch7 seemed likely to restore them--to that peace of mind which had once been theirs ere ever these problems had swooped upon them1 and crushed them with a load of unfathomable mystery6 ,it's hardly fair1 muttered ,hugh1 to give us such a jumble as this to work out6 ,fair8 ,clara echoed1 bitterly4 ,well6 ,and to all my readers ,i can but repeat the last words of gentle ,clara-- ,,fare-,,well6 ,,appendix4 ,a knot6 said ,alice4 ,oh1 do let me help to undo it6 ,,answers ,,to ,,knot ,i4 ,problem4--,two travellers spend from #c o'clock till #i in walking along a level road1 up a hill1 and home again3 their pace on the level being #d miles an hour1 up hill #c1 and down hill #f4 ,find distance walked3 also 7within half an hour7 time of reaching top of hill4 ,answer4--#bd miles3 half-past #f4 99 99 99 99 99 ,solution4--,a level mile takes #a_/d of an hour1 up hill #a_/c1 down hill #a_/f4 ,hence to go and return over the same mile1 whether on the level or on the hill-side1 takes #a_/b an hour4 ,hence in #f hours they went #ab miles out and #ab back4 ,if the #ab miles out had been nearly all level1 they would have taken a little over #c hours2 if nearly all up hill1 a little under #d4 ,hence #c-a_/b hours must be within #a_/b an hour of the time taken in reaching the peak2 thus1 as they started at #c1 they got there within #a_/b an hour of #a_/b past #f4 99 99 99 99 99 ,twenty-seven answers have come in4 ,of these1 #i are right1 #af partially right1 and #b wrong4 ,the #af give the distance correctly1 but they have failed to grasp the fact that the top of the hill might have been reached at any moment between #f o'clock and #g4 ,the two wrong answers are from ,,gerty ,,vernon and ,a ,,nihilist4 ,the former makes the distance #bc miles1 while her revolutionary companion puts it at #bg4 ,,gerty ,,vernon says they had to go #d miles along the plain1 and got to the foot of the hill at #d o'clock4 ,they might have done so1 ,i grant2 but you have no ground for saying they did so4 ,it was #g-a_/b miles to the top of the hill1 and they reached that at #a_/d before #g o'clock4 ,here you go wrong in your arithmetic1 and ,i must1 however reluctantly1 bid you farewell4 #g-a_/b miles1 at #c miles an hour1 would not require #b-c_/d hours4 ,a ,,nihilist says ,let x denote the whole number of miles2 y the number of hours to hill-top2 9999 therefore #cy number of miles to hill-top1 and x-#cy number of miles on the other side4 ,you bewilder me4 ,the other side of what8 ,of the hill1 you say4 ,but then1 how did they get home again8 ,however1 to accommodate your views we will build a new hostelry at the foot of the hill on the opposite side1 and also assume 7what ,i grant you is possible1 though it is not necessarily true7 that there was no level road at all4 ,even then you go wrong4 ,you say y #f - 7x - #cy7_/#f1 44444 7i72 x_/#d-a_/b #f 44444 7ii74 ,i grant you 7i71 but ,i deny 7ii73 it rests on the assumption that to go part of the time at #c miles an hour1 and the rest at #f miles an hour1 comes to the same result as going the whole time at #d-a_/b miles an hour4 ,but this would only be true if the part were an exact half1 i4e41 if they went up hill for #c hours1 and down hill for the other #c3 which they certainly did not do4 ,the sixteen1 who are partially right1 are ,,agnes ,,bailey1 ,f4 ,k41 ,,fifee1 ,g4 ,e4 ,b41 ,h4 ,p41 ,,kit1 ,m4 ,e4 ,t41 ,,mysie1 ,a ,,mother',s ,,son1 ,,nairam1 ,a ,,redruthian1 ,a ,,socialist1 ,,spear ,,maiden1 ,t4 ,b4 ,c41 ,,vis ,,inerti1 and ,,yak4 ,of these1 ,f4 ,k41 ,,fifee1 ,t4 ,b4 ,c41 and ,,vis ,,inerti do not attempt the second part at all4 ,f4 ,k4 and ,h4 ,p4 give no working4 ,the rest make particular assumptions1 such as that there was no level road--that there were #f miles of level road--and so on1 all leading to particular times being fixed for reaching the hill-top4 ,the most curious assumption is that of ,,agnes ,,bailey1 who says ,let x number of hours occupied in ascent2 then x_/#b hours occupied in descent2 and #dx_/#c hours occupied on the level4 ,i suppose you were thinking of the relative rates1 up hill and on the level2 which we might express by saying that1 if they went x miles up hill in a certain time1 they would go #dx_/#c miles on the level in the same time4 ,you have1 in fact1 assumed that they took the same time on the level that they took in ascending the hill4 ,,fifee assumes that1 when the aged knight said they had gone four miles in the hour on the level1 he meant that four miles was the distance gone1 not merely the rate4 ,this would have been--if ,,fifee will excuse the slang expression--a sell1 ill-suited to the dignity of the hero4 ,and now descend1 ye classic ,nine6 who have solved the whole problem1 and let me sing your praises4 ,your names are ,,blithe1 ,e4 ,w41 ,l4 ,b41 ,a ,,marlborough ,,boy1 ,o4 ,v4 ,l41 ,,putney ,,walker1 ,,rose1 ,,sea ,,breeze1 ,,simple ,,susan1 and ,,money ,,spinner4 7,these last two ,i count as one1 as they send a joint answer47 ,,rose and ,,simple ,,susan and ,,co4 do not actually state that the hill-top was reached some time between #f and #g1 but1 as they have clearly grasped the fact that a mile1 ascended and descended1 took the same time as two level miles1 ,i mark them as right4 ,a ,,marlborough ,,boy and ,,putney ,,walker deserve honourable mention for their algebraical solutions being the only two who have perceived that the question leads to an indeterminate equation4 ,e4 ,w4 brings a charge of untruthfulness against the aged knight--a serious charge1 for he was the very pink of chivalry6 ,she says ,according to the data given1 the time at the summit affords no clue to the total distance4 ,it does not enable us to state precisely to an inch how much level and how much hill there was on the road4 ,fair damsel1 the aged knight replies1 --if1 as ,i surmise1 thy initials denote ,early ,womanhood--bethink thee that the word 'enable' is thine1 not mine4 ,i did but ask the time of reaching the hill-top as my condition for further parley4 ,if now thou wilt not grant that ,i am a truth-loving man1 then will ,i affirm that those same initials denote ,envenomed ,wickedness6 ,,class ,,list4 ,i4 ,a ,,marlborough ,,boy4 ,,putney ,,walker4 ,,ii4 ,,blithe4 ,e4 ,w4 ,l4 ,b4 ,o4 ,v4 ,l4 ,,rose4 ,,sea ,,breeze4 ,,simple ,,susan4 ,,money-,,spinner4 ,,blithe has made so ingenious an addition to the problem1 and ,,simple ,,susan and ,,co4 have solved it in such tuneful verse1 that ,i record both their answers in full4 ,i have altered a word or two in ,,blithe',s--which ,i trust she will excuse2 it did not seem quite clear as it stood4 99 99 99 99 99 ,yet stay1 said the youth1 as a gleam of inspiration lighted up the relaxing muscles of his quiescent features4 ,stay4 ,methinks it matters little when we reached that summit1 the crown of our toil4 ,for in the space of time wherein we clambered up one mile and bounded down the same on our return1 we could have trudged the twain on the level4 ,we have plodded1 then1 four-and-twenty miles in these six mortal hours2 for never a moment did we stop for catching of fleeting breath or for gazing on the scene around6 ,very good1 said the old man4 ,twelve miles out and twelve miles in4 ,and we reached the top some time between six and seven of the clock4 ,now mark me6 ,for every five minutes that had fled since six of the clock when we stood on yonder peak1 so many miles had we toiled upwards on the dreary mountainside6 ,the youth moaned and rushed into the hostel4 ,,blithe4 ,the elder and the younger knight1 ,they sallied forth at three2 ,how far they went on level ground ,it matters not to me2 ,what time they reached the foot of hill1 ,when they began to mount1 ,are problems which ,i hold to be ,of very small account4 ,the moment that each waved his hat ,upon the topmost peak-- ,to trivial query such as this ,no answer will ,i seek4 ,yet can ,i tell the distance well ,they must have travelled o'er3 ,on hill and plain1 'twixt three and nine1 ,the miles were twenty-four4 ,four miles an hour their steady pace ,along the level track1 ,three when they climbed--but six when they ,came swiftly striding back ,adown the hill2 and little skill ,it needs1 methinks1 to show1 ,up hill and down together told1 ,four miles an hour they go4 ,for whether long or short the time ,upon the hill they spent1 ,two thirds were passed in going up1 ,one third in the descent4 ,two thirds at three1 one third at six1 ,if rightly reckoned o'er1 ,will make one whole at four--the tale ,is tangled now no more4 ,,simple ,,susan4 ,,money ,,spinner4 ,,answers ,,to ,,knot ,,ii4 #a4 ,,the ,,dinner ,,party4 ,problem4--,the ,governor of ,kgovjni wants to give a very small dinner party1 and invites his father's brother-in-law1 his brother's father-in-law1 his father-in-law's brother1 and his brother-in-law's father4 ,find the number of guests4 ,answer4--,one4 99 99 99 99 99 ,in this genealogy1 males are denoted by capitals1 and females by small letters4 ,the ,governor is ,e and his guest is ,c4 ,a a ----------- b ,b ,d d ,c c ----- -- e ,e g ,g ,f f ,ten answers have been received4 ,of these1 one is wrong1 ,,galanthus ,,nivalis ,,major1 who insists on inviting two guests1 one being the ,governor's wife's brother's father4 ,if she had taken his sister's husband's father instead1 she would have found it possible to reduce the guests to one4 ,of the nine who send right answers1 ,,sea-,,breeze is the very faintest breath that ever bore the name6 ,she simply states that the ,governor's uncle might fulfill all the conditions by intermarriages6 ,wind of the western sea1 you have had a very narrow escape6 ,be thankful to appear in the ,class-list at all6 ,,bog-,,oak and ,,bradshaw ,,of ,,the ,,future use genealogies which require #af people instead of #ad1 by inviting the ,governor's father's sister's husband instead of his father's wife's brother4 ,i cannot think this so good a solution as one that requires only #ad4 ,,caius and ,,valentine deserve special mention as the only two who have supplied genealogies4 ,,class ,,list4 ,i4 ,,bee4 ,,caius4 ,m4 ,m4 ,,matthew ,,matticks4 ,,old ,,cat4 ,,valentine4 ,,ii4 ,,bog-,,oak4 ,,bradshaw ,,of ,,the ,,future4 ,,iii4 ,,sea-,,breeze4 #b4 ,,the ,,lodgings4 ,problem4--,a ,square has #bj doors on each side1 which contains #ba equal parts4 ,they are numbered all round1 beginning at one corner4 ,from which of the four1 ,nos4 #i1 #be1 #eb1 #gc1 is the sum of the distances1 to the other three1 least8 ,answer4--,from ,no4 #i4 99 99 99 99 99 ,illustration ,let ,a be ,no4 #i1 ,b ,no4 #be1 ,c ,no4 #eb1 and ,d ,no4 #gc4 ,then ,,ab 9999 sqrt7#abb #eb7 9999 sqrt#afi #ac2 ,,ac #ba2 ,,ad 9999 sqrt7#ib #hb7 9999 sqrt#ade #ab 7,n4,b4 i4e4 between #ab and #ac47 ,,bc 9999 sqrt7#afb #abb7 9999 sqrt#djj #bj2 ,,bd 9999 sqrt7#cb #bab7 9999 sqrt#dej #ba2 ,,cd 9999 sqrt7#ib #acb7 9999 sqrt#bej #ae2 ,hence sum of distances from ,a is between #df and #dg2 from ,b1 between #ed and #ee2 from ,c1 between #ef and #eg2 from ,d1 between #dh and #ea4 7,why not between #dh and #di8 ,make this out for yourselves47 ,hence the sum is least for ,a4 99 99 99 99 99 ,twenty-five solutions have been received4 ,of these1 #ae must be marked #j1 #e are partly right1 and #e right4 ,of the #ae1 ,i may dismiss ,,alphabetical ,,phantom1 ,,bog-,,oak1 ,,dinah ,,mite1 ,,fifee1 ,,galanthus ,,nivalis ,,major 7,i fear the cold spring has blighted our ,,snowdrop71 ,,guy1 ,h4,m4,s4 ,,pinafore1 ,,janet1 and ,,valentine with the simple remark that they insist on the unfortunate lodgers keeping to the pavement4 7,i used the words crossed to ,number ,seventy-three for the special purpose of showing that short cuts were possible47 ,,sea-,,breeze does the same1 and adds that the result would be the same even if they crossed the ,square1 but gives no proof of this4 ,m4 ,m4 draws a diagram1 and says that ,no4 #i is the house1 as the diagram shows4 ,i cannot see how it does so4 ,,old ,,cat assumes that the house must be ,no4 #i or ,no4 #gc4 ,she does not explain how she estimates the distances4 ,,bee's ,arithmetic is faulty3 she makes 9999 sqrt#afi 9999 sqrt#ddb 9999 sqrt#acj #gda4 7,i suppose you mean 9999 sqrt#gda1 which would be a little nearer the truth4 ,but roots cannot be added in this manner4 ,do you think 9999 sqrt#i 9999 sqrt#af is #be1 or even 9999 sqrt#be87 ,but ,,ayr',s state is more perilous still3 she draws illogical conclusions with a frightful calmness4 ,after pointing out 7rightly7 that ,,ac is less than ,,bd she says1 therefore the nearest house to the other three must be ,a or ,c4 ,and again1 after pointing out 7rightly7 that ,b and ,d are both within the half-square containing ,a1 she says therefore ,,ab ,,ad must be less than ,,bc ,,cd4 7,there is no logical force in either therefore4 ,for the first1 try ,nos4 #a1 #ba1 #fj1 #gj3 this will make your premiss true1 and your conclusion false4 ,similarly1 for the second1 try ,nos4 #a1 #cj1 #ea1 #ga47 ,of the five partly-right solutions1 ,,rags ,,and ,,tatters and ,,mad ,,hatter 7who send one answer between them7 make ,no4 #be #f units from the corner instead of #e4 ,,cheam1 ,e4 ,r4 ,d4 ,l41 and ,,meggy ,,potts leave openings at the corners of the ,square1 which are not in the data3 moreover ,,cheam gives values for the distances without any hint that they are only approximations4 ,,crophi ,,and ,,mophi make the bold and unfounded assumption that there were really #ba houses on each side1 instead of #bj as stated by ,balbus4 ,we may assume1 they add1 that the doors of ,nos4 #ba1 #db1 #fc1 #hd1 are invisible from the centre of the ,square6 ,what is there1 ,i wonder1 that ,,crophi ,,and ,,mophi would not assume8 ,of the five who are wholly right1 ,i think ,,bradshaw ,,of ,,the ,,future1 ,,caius1 ,,clifton ,c41 and ,,martreb deserve special praise for their full analytical solutions4 ,,matthew ,,matticks picks out ,no4 #i1 and proves it to be the right house in two ways1 very neatly and ingeniously1 but why he picks it out does not appear4 ,it is an excellent synthetical proof1 but lacks the analysis which the other four supply4 ,,class ,,list4 ,i4 ,,bradshaw ,,of ,,the ,,future ,,caius4 ,,clifton ,c4 ,,martreb4 ,,ii4 ,,matthew ,,matticks4 ,,iii4 ,,cheam4 ,,crophi ,,and ,,mophi4 ,e4 ,r4 ,d4 ,l4 ,,meggy ,,potts4 ,,rags ,,and ,,tatters4 ,,mad ,,hatter4 ,a remonstrance has reached me from ,,scrutator on the subject of ,,knot ,i41 which he declares was no problem at all4 ,two questions1 he says1 are put4 ,to solve one there is no data3 the other answers itself4 ,as to the first point1 ,,scrutator is mistaken2 there are 7not is7 data sufficient to answer the question4 ,as to the other1 it is interesting to know that the question answers itself1 and ,i am sure it does the question great credit3 still ,i fear ,i cannot enter it on the list of winners1 as this competition is only open to human beings4 ,,answers ,,to ,,knot ,,iii4 ,problem4--7#a7 ,two travellers1 starting at the same time1 went opposite ways round a circular railway4 ,trains start each way every #ae minutes1 the easterly ones going round in #c hours1 the westerly in #b4 ,how many trains did each meet on the way1 not counting trains met at the terminus itself8 7#b7 ,they went round1 as before1 each traveller counting as 'one' the train containing the other traveller4 ,how many did each meet8 ,answers4--7#a7 #ai4 7#b7 ,the easterly traveller met #ab2 the other #h4 99 99 99 99 99 ,the trains one way took #ahj minutes1 the other way #abj4 ,let us take the ,l4 ,c4 ,m41 #cfj1 and divide the railway into #cfj units4 ,then one set of trains went at the rate of #b units a minute and at intervals of #cj units2 the other at the rate of #c units a minute and at intervals of #de units4 ,an easterly train starting has #de units between it and the first train it will meet3 it does #b-eths of this while the other does #c-eths1 and thus meets it at the end of #ah units1 and so all the way round4 ,a westerly train starting has #cj units between it and the first train it will meet3 it does #c-eths of this while the other does #b-eths1 and thus meets it at the end of #ah units1 and so all the way round4 ,hence if the railway be divided1 by #ai posts1 into #bj parts1 each containing #ah units1 trains meet at every post1 and1 in 7#a71 each traveller passes #ai posts in going round1 and so meets #ai trains4 ,but1 in 7#b71 the easterly traveller only begins to count after traversing #b-eths of the journey1 i4e41 on reaching the #hth post1 and so counts #ab posts3 similarly the other counts #h4 ,they meet at the end of #b-eths of #c hours1 or #c-eths of #b hours1 i4e41 #gb minutes4 99 99 99 99 99 ,forty-five answers have been received4 ,of these #ab are beyond the reach of discussion1 as they give no working4 ,i can but enumerate their names4 ,,ardmore1 ,e4 ,a41 ,f4 ,a4 ,d41 ,l4 ,d41 ,,matthew ,,matticks1 ,m4 ,e4 ,t41 ,,poo-,,poo1 and ,,the ,,red ,,queen are all wrong4 ,,beta and ,,rowena have got 7#a7 right and 7#b7 wrong4 ,,cheeky ,,bob and ,,nairam give the right answers1 but it may perhaps make the one less cheeky1 and induce the other to take a less inverted view of things1 to be informed that1 if this had been a competition for a prize1 they would have got no marks4 ,n4,b4--,i have not ventured to put ,e4 ,a4's name in full1 as she only gave it provisionally1 in case her answer should prove right4 ,of the #cc answers for which the working is given1 #aj are wrong2 #aa half-wrong and half-right2 #c right1 except that they cherish the delusion that it was ,clara who travelled in the easterly train--a point which the data do not enable us to settle2 and #i wholly right4 ,the #aj wrong answers are from ,,bo-,,peep1 ,,financier1 ,i4 ,w4 ,t41 ,,kate ,b41 ,m4 ,a4 ,h41 ,q4 ,y4 ,z41 ,,sea-,,gull1 ,,thistledown1 ,,tom-,,quad1 and an unsigned one4 ,,bo-,,peep rightly says that the easterly traveller met all trains which started during the #c hours of her trip1 as well as all which started during the previous #b hours1 i4e41 all which started at the commencements of #bj periods of #ae minutes each2 and she is right in striking out the one she met at the moment of starting2 but wrong in striking out the last train1 for she did not meet this at the terminus1 but #ae minutes before she got there4 ,she makes the same mistake in 7#b74 ,,financier thinks that any train1 met for the second time1 is not to be counted4 ,i4 ,w4 ,t4 finds1 by a process which is not stated1 that the travellers met at the end of #ga minutes and #bf-a_/b seconds4 ,,kate ,b4 thinks the trains which are met on starting and on arriving are never to be counted1 even when met elsewhere4 ,q4 ,y4 ,z4 tries a rather complex algebraical solution1 and succeeds in finding the time of meeting correctly3 all else is wrong4 ,,sea-,,gull seems to think that1 in 7#a71 the easterly train stood still for #c hours2 and says that1 in 7#b71 the travellers met at the end of #ga minutes #dj seconds4 ,,thistledown nobly confesses to having tried no calculation1 but merely having drawn a picture of the railway and counted the trains2 in 7#a71 she counts wrong2 in 7#b7 she makes them meet in #ge minutes4 ,,tom-,,quad omits 7#a73 in 7#b7 he makes ,clara count the train she met on her arrival4 ,the unsigned one is also unintelligible2 it states that the travellers go #a-bdth more than the total distance to be traversed6 ,the ,clara theory1 already referred to1 is adopted by #e of these1 viz41 ,,bo-,,peep1 ,,financier1 ,,kate ,b41 ,,tom-,,quad1 and the nameless writer4 ,the #aa half-right answers are from ,,bog-,,oak1 ,,bridget1 ,,castor1 ,,cheshire ,,cat1 ,g4 ,e4 ,b41 ,,guy1 ,,mary1 ,m4 ,a4 ,h41 ,,old ,,maid1 ,r4 ,w41 and ,,vendredi4 ,all these adopt the ,clara theory4 ,,castor omits 7#a74 ,,vendredi gets 7#a7 right1 but in 7#b7 makes the same mistake as ,,bo-,,peep4 ,i notice in your solution a marvellous proportion-sum3--#cjj miles3 #b hours 33 one mile3 #bd seconds4 ,may ,i venture to advise your acquiring1 as soon as possible1 an utter disbelief in the possibility of a ratio existing between miles and hours8 ,do not be disheartened by your two friends' sarcastic remarks on your roundabout ways4 ,their short method1 of adding #ab and #h1 has the slight disadvantage of bringing the answer wrong3 even a roundabout method is better than that6 ,m4 ,a4 ,h41 in 7#b71 makes the travellers count one after they met1 not when they met4 ,,cheshire ,,cat and ,,old ,,maid get #bj as answer for 7#a71 by forgetting to strike out the train met on arrival4 ,the others all get #ah in various ways4 ,,bog-,,oak1 ,,guy1 and ,r4 ,w4 divide the trains which the westerly traveller has to meet into #b sets1 viz41 those already on the line1 which they 7rightly7 make #aa1 and those which started during her #b hours' journey 7exclusive of train met on arrival71 which they 7wrongly7 make #g2 and they make a similar mistake with the easterly train4 ,,bridget 7rightly7 says that the westerly traveller met a train every #f minutes for #b hours1 but 7wrongly7 makes the number #bj2 it should be #ba4 ,g4 ,e4 ,b4 adopts ,,bo-,,peep',s method1 but 7wrongly7 strikes out 7for the easterly traveller7 the train which started at the commencement of the previous #b hours4 ,,mary thinks a train1 met on arrival1 must not be counted1 even when met on a previous occasion4 ,the #c1 who are wholly right but for the unfortunate ,clara theory1 are ,f4 ,,lee1 ,g4 ,s4 ,c41 and ,x4 ,a4 ,b4 ,and now descend1 ye classic ,ten6 who have solved the whole problem4 ,your names are ,,aix-,,les-,,bains1 ,,algernon ,,bray 7thanks for a friendly remark1 which comes with a heart-warmth that not even the ,atlantic could chill71 ,,arvon1 ,,bradshaw ,,of ,,the ,,future1 ,,fifee1 ,h4 ,l4 ,r41 ,j4 ,l4 ,o41 ,,omega1 ,s4 ,s4 ,g41 and ,,waiting ,,for ,,the ,,train4 ,several of these have put ,clara1 provisionally1 into the easterly train3 but they seem to have understood that the data do not decide that point4 ,,class ,,list4 ,i4 ,,aix-,,les-,,bains4 ,,algernon ,,bray4 ,,bradshaw ,,of ,,the ,,future4 ,,fifee4 ,h4 ,l4 ,r4 ,,omega4 ,s4 ,s4 ,g4 ,,waiting ,,for ,,the ,,train4 ,,ii4 ,,arvon4 ,j4 ,l4 ,o4 ,,iii4 ,f4 ,,lee4 ,g4 ,s4 ,c4 ,x4 ,a4 ,b4 ,,answers ,,to ,,knot ,,iv4 ,problem4--,there are #e sacks1 of which ,nos4 #a1 #b1 weigh #ab lbs42 ,nos4 #b1 #c1 #ac-a_/b lbs42 ,nos4 #c1 #d1 #aa-a_/b lbs42 ,nos4 #d1 #e1 #h lbs42 ,nos4 #a1 #c1 #e1 #af lbs4 ,required the weight of each sack4 ,answer4--#e-a_/b1 #f-a_/b1 #g1 #d-a_/b1 #c-a_/b4 99 99 99 99 99 ,the sum of all the weighings1 #fa lbs41 includes sack ,no4 #c thrice and each other twice4 ,deducting twice the sum of the #ast and #dth weighings1 we get #ba lbs4 for thrice ,no4 #c1 i4e41 #g lbs4 for ,no4 #c4 ,hence1 the #bnd and #crd weighings give #f-a_/b lbs41 #d-a_/b lbs4 for ,nos4 #b1 #d2 and hence again1 the #ast and #dth weighings give #e-a_/b lbs41 #c-a_/b lbs41 for ,nos4 #a1 #e4 99 99 99 99 99 ,ninety-seven answers have been received4 ,of these1 #ae are beyond the reach of discussion1 as they give no working4 ,i can but enumerate their names1 and ,i take this opportunity of saying that this is the last time ,i shall put on record the names of competitors who give no sort of clue to the process by which their answers were obtained4 ,in guessing a conundrum1 or in catching a flea1 we do not expect the breathless victor to give us afterwards1 in cold blood1 a history of the mental or muscular efforts by which he achieved success2 but a mathematical calculation is another thing4 ,the names of this mute inglorious band are ,,common ,,sense1 ,d4 ,e4 ,r41 ,,douglas1 ,e4 ,l41 ,,ellen1 ,i4 ,m4 ,t41 ,j4 ,m4 ,c41 ,,joseph1 ,,knot ,i1 ,,lucy1 ,,meek1 ,m4 ,f4 ,c41 ,,pyramus1 ,,shah1 ,,veritas4 ,of the eighty-two answers with which the working1 or some approach to it1 is supplied1 one is wrong3 seventeen have given solutions which are 7from one cause or another7 practically valueless3 the remaining sixty-four ,i shall try to arrange in a ,class-list1 according to the varying degrees of shortness and neatness to which they seem to have attained4 ,the solitary wrong answer is from ,,nell4 ,to be thus alone in the crowd is a distinction--a painful one1 no doubt1 but still a distinction4 ,i am sorry for you1 my dear young lady1 and ,i seem to hear your tearful exclamation1 when you read these lines1 ,ah6 ,this is the knell of all my hopes6 ,why1 oh why1 did you assume that the #dth and #eth bags weighed #d lbs4 each8 ,and why did you not test your answers8 ,however1 please try again3 and please don't change your nom-de-plume3 let us have ,,nell in the ,first ,class next time6 ,the seventeen whose solutions are practically valueless are ,,ardmore1 ,a ,,ready ,,reckoner1 ,,arthur1 ,,bog-,,lark1 ,,bog-,,oak1 ,,bridget1 ,,first ,,attempt1 ,j4 ,l4 ,c41 ,m4 ,e4 ,t41 ,,rose1 ,,rowena1 ,,sea-,,breeze1 ,,sylvia1 ,,thistledown1 ,,three-,,fifths ,,asleep1 ,,vendredi1 and ,,winifred4 ,,bog-,,lark tries it by a sort of rule of false1 assuming experimentally that ,nos4 #a1 #b1 weigh #f lbs4 each1 and having thus produced #ag-a_/b1 instead of #af1 as the weight of #a1 #c1 and #e1 she removes the superfluous pound and a half1 but does not explain how she knows from which to take it4 ,,three-,,fifths ,,asleep says that 7when in that peculiar state7 it seemed perfectly clear to her that1 #c out of the #e sacks being weighed twice over1 #c_/e of #de #bg1 must be the total weight of the #e sacks4 ,as to which ,i can only say1 with the ,captain1 it beats me entirely6 ,,winifred1 on the plea that one must have a starting-point1 assumes 7what ,i fear is a mere guess7 that ,no4 #a weighed #e-a_/b lbs4 ,the rest all do it1 wholly or partly1 by guess-work4 ,the problem is of course 7as any ,algebraist sees at once7 a case of simultaneous simple equations4 ,it is1 however1 easily soluble by ,arithmetic only2 and1 when this is the case1 ,i hold that it is bad workmanship to use the more complex method4 ,i have not1 this time1 given more credit to arithmetical solutions2 but in future problems ,i shall 7other things being equal7 give the highest marks to those who use the simplest machinery4 ,i have put into ,class ,i4 those whose answers seemed specially short and neat1 and into ,class ,,iii4 those that seemed specially long or clumsy4 ,of this last set1 ,a4 ,c4 ,m41 ,,furze-,,bush1 ,,james1 ,,partridge1 ,r4 ,w41 and ,,waiting ,,for ,,the ,,train1 have sent long wandering solutions1 the substitutions having no definite method1 but seeming to have been made to see what would come of it4 ,,chilpome and ,,dublin ,,boy omit some of the working4 ,,arvon ,,marlborough ,,boy only finds the weight of one sack4 ,,class ,,list ,i4 ,b4 ,e4 ,d4 ,c4 ,h4 ,,constance ,,johnson4 ,,greystead4 ,,guy4 ,,hoopoe4 ,j4 ,f4 ,a4 ,m4 ,a4 ,h4 ,,number ,,five4 ,,pedro4 ,r4 ,e4 ,x4 ,,seven ,,old ,,men4 ,,vis ,,inerti4 ,,willy ,b4 ,,yahoo4 ,,ii4 ,,american ,,subscriber4 ,,an ,,appreciative ,,schoolma',,am4 ,,ayr4 ,,bradshaw ,,of ,,the ,,future4 ,,cheam4 ,c4 ,m4 ,g4 ,,dinah ,,mite4 ,,duckwing4 ,e4 ,c4 ,m4 ,e4 ,n4 ,lowry4 ,,era4 ,,euroclydon4 ,f4 ,h4 ,w4 ,,fifee4 ,g4 ,e4 ,b4 ,,harlequin4 ,,hawthorn4 ,,hough ,,green4 ,j4 ,a4 ,b4 ,,jack ,,tar4 ,j4 ,b4 ,b4 ,,kgovjni4 ,,land ,,lubber4 ,l4 ,d4 ,,magpie4 ,,mary4 ,,mhruxi4 ,,minnie4 ,,money-,,spinner4 ,,nairam4 ,,old ,,cat4 ,,polichinelle4 ,,simple ,,susan4 ,s4 ,s4 ,g4 ,,thisbe4 ,,verena4 ,,wamba4 ,,wolfe4 ,,wykehamicus4 ,y4 ,m4 ,a4 ,h4 ,,iii4 ,a4 ,c4 ,m4 ,,arvon ,,marlborough ,,boy4 ,,chilpome4 ,,dublin ,,boy4 ,,furze-,,bush4 ,,james4 ,,partridge4 ,r4 ,w4 ,,waiting ,,for ,,the ,,train4 ,,answers ,,to ,,knot ,v4 ,problem4--,to mark pictures1 giving #c x's to #b or #c1 #b to #d or #e1 and #a to #i or #aj2 also giving #c o's to #a or #b1 #b to #c or #d and #a to #h or #i2 so as to mark the smallest possible number of pictures1 and to give them the largest possible number of marks4 ,answer4--#aj pictures2 #bi marks2 arranged thus3-- 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 ,solution4--,by giving all the x's possible1 putting into brackets the optional ones1 we get #aj pictures marked thus3-- x x x x x x x x x 7x7 x x x x 7x7 x x 7x7 ,by then assigning o's in the same way1 beginning at the other end1 we get #i pictures marked thus3-- 7o7 o 7o7 o o o 7o7 o o o o o o o o ,all we have now to do is to run these two wedges as close together as they will go1 so as to get the minimum number of pictures----erasing optional marks where by so doing we can run them closer1 but otherwise letting them stand4 ,there are #aj necessary marks in the #ast row1 and in the #crd2 but only #g in the #bnd4 ,hence we erase all optional marks in the #ast and #crd rows1 but let them stand in the #bnd4 99 99 99 99 99 ,twenty-two answers have been received4 ,of these #aa give no working2 so1 in accordance with what ,i announced in my last review of answers1 ,i leave them unnamed1 merely mentioning that #e are right and #f wrong4 ,of the eleven answers with which some working is supplied1 #c are wrong4 ,c4 ,h4 begins with the rash assertion that under the given conditions the sum is impossible4 ,for1 he or she adds 7these initialed correspondents are dismally vague beings to deal with3 perhaps it would be a better pronoun71 #aj is the least possible number of pictures 7granted73 therefore we must either give #b x's to #f1 or #b o's to #e4 ,why must1 oh alphabetical phantom8 ,it is nowhere ordained that every picture must have #c marks6 ,,fifee sends a folio page of solution1 which deserved a better fate3 she offers #c answers1 in each of which #aj pictures are marked1 with #cj marks2 in one she gives #b x's to #f pictures2 in another to #g2 in the #crd she gives #b o's to #e2 thus in every case ignoring the conditions4 7,i pause to remark that the condition #b x's to #d or #e pictures can only mean either to #d or else to #e3 if1 as one competitor holds1 it might mean any number not less than #d1 the words or #e would be superfluous47 ,i4 ,e4 ,a4 7,i am happy to say that none of these bloodless phantoms appear this time in the class-list4 ,is it ,,idea with the ,d left out87 gives #b x's to #f pictures4 ,she then takes me to task for using the word ought instead of nought4 ,no doubt1 to one who thus rebels against the rules laid down for her guidance1 the word must be distasteful4 ,but does not ,i4 ,e4 ,a4 remember the parallel case of adder8 ,that creature was originally a nadder3 then the two words took to bandying the poor n backwards and forwards like a shuttlecock1 the final state of the game being an adder4 ,may not a nought have similarly become an ought8 ,anyhow1 oughts and crosses is a very old game4 ,i don't think ,i ever heard it called noughts and crosses4 ,in the following ,class-list1 ,i hope the solitary occupant of ,,iii4 will sheathe her claws when she hears how narrow an escape she has had of not being named at all4 ,her account of the process by which she got the answer is so meagre that1 like the nursery tale of ,jack-a-,minory 7,i trust ,i4 ,e4 ,a4 will be merciful to the spelling71 it is scarcely to be distinguished from zero4 ,,class ,,list4 ,i4 ,,guy4 ,,old ,,cat4 ,,sea-,,breeze4 ,,ii4 ,,ayr4 ,,bradshaw ,,of ,,the ,,future4 ,f4 ,,lee4 ,h4 ,,vernon4 ,,iii4 ,,cat4 ,,answers ,,to ,,knot ,,vi4 ,problem #a4--;,a and ,b began the year with only #a1jjjl4 a-piece4 ,they borrowed nought2 they stole nought4 ,on the next ,new-,year's ,day they had #fj1jjjl4 between them4 ,how did they do it8 ,solution4--,they went that day to the ,bank of ,england4 ,a stood in front of it1 while ,b went round and stood behind it4 99 99 99 99 99 ,two answers have been received1 both worthy of much honour4 ,,addlepate makes them borrow #j and steal #j1 and uses both cyphers by putting them at the right-hand end of the #a1jjjl41 thus producing #ajj1jjjl41 which is well over the mark4 ,but 7or to express it in ,latin7 ,,at ,,spes ,,infracta has solved it even more ingeniously3 with the first cypher she turns the #a of the #a1jjjl4 into a #i1 and adds the result to the original sum1 thus getting #aj1jjjl43 and in this1 by means of the other #j1 she turns the #a into a #f1 thus hitting the exact #fj1jjjl4 ,,class ,,list ,i4 ,,at ,,spes ,,infracta4 ,,ii4 ,,addlepate4 99 99 99 99 99 ,problem #b4--,l makes #e scarves1 while ,m makes #b3 ,z makes #d while ,l makes #c4 ,five scarves of ,z's weigh one of ,l's2 #e of ,m's weigh #c of ,z's4 ,one of ,m's is as warm as #d of ,z's3 and one of ,l's as warm as #c of ,m's4 ,which is best1 giving equal weight in the result to rapidity of work1 lightness1 and warmth8 ,answer4--,the order is ,m1 ,l1 ,z4 99 99 99 99 99 ,solution4--,as to rapidity 7other things being constant7 ,l's merit is to ,m's in the ratio of #e to #b3 ,z's to ,l's in the ratio of #d to #c4 ,in order to get one set of #c numbers fulfilling these conditions1 it is perhaps simplest to take the one that occurs twice as unity1 and reduce the others to fractions3 this gives1 for ,l1 ,m1 and ,z1 the marks #a1 #b_/e1 #d_/c4 ,in estimating for lightness1 we observe that the greater the weight1 the less the merit1 so that ,z's merit is to ,l's as #e to #a4 ,thus the marks for lightness are #a_/e1 #e_/c1 #a4 ,and similarly1 the marks for warmth are #c1 #a1 #a_/d4 ,to get the total result1 we must multiply ,l's #c marks together1 and do the same for ,m and for ,z4 ,the final numbers are #a #a_/e #c1 #b_/e #e_/c #a1 #d_/c #a #a_/d2 i4e4 #c_/e1 #b_/c1 #a_/c2 i4e4 multiplying throughout by #ae 7which will not alter the proportion71 #i1 #aj1 #e2 showing the order of merit to be ,m1 ,l1 ,z4 99 99 99 99 99 ,twenty-nine answers have been received1 of which five are right1 and twenty-four wrong4 ,these hapless ones have all 7with three exceptions7 fallen into the error of adding the proportional numbers together1 for each candidate1 instead of multiplying4 ,why the latter is right1 rather than the former1 is fully proved in text-books1 so ,i will not occupy space by stating it here3 but it can be illustrated very easily by the case of length1 breadth1 and depth4 ,suppose ,a and ,b are rival diggers of rectangular tanks3 the amount of work done is evidently measured by the number of cubical feet dug out4 ,let ,a dig a tank #aj feet long1 #aj wide1 #b deep3 let ,b dig one #f feet long1 #e wide1 #aj deep4 ,the cubical contents are #bjj1 #cjj2 i4e4 ,b is best digger in the ratio of #c to #b4 ,now try marking for length1 width1 and depth1 separately2 giving a maximum mark of #aj to the best in each contest1 and then adding the results6 ,of the twenty-four malefactors1 one gives no working1 and so has no real claim to be named2 but ,i break the rule for once1 in deference to its success in ,problem #a3 he1 she1 or it1 is ,,addlepate4 ,the other twenty-three may be divided into five groups4 ,first and worst are1 ,i take it1 those who put the rightful winner last2 arranging them as ,lolo1 ,zuzu1 ,mimi4 ,the names of these desperate wrong-doers are ,,ayr1 ,,bradshaw ,,of ,,the ,,future1 ,,furze-,,bush and ,,pollux 7who send a joint answer71 ,,greystead1 ,,guy1 ,,old ,,hen1 and ,,simple ,,susan4 ,the latter was once best of all2 the ,old ,hen has taken advantage of her simplicity1 and beguiled her with the chaff which was the bane of her own chickenhood4 ,secondly1 ,i point the finger of scorn at those who have put the worst candidate at the top2 arranging them as ,zuzu1 ,mimi1 ,lolo4 ,they are ,,graecia1 ,m4 ,m41 ,,old ,,cat1 and ,r4 ,e4 ,x4 ',tis ,greece1 but----4 ,the third set have avoided both these enormities1 and have even succeeded in putting the worst last1 their answer being ,lolo1 ,mimi1 ,zuzu4 ,their names are ,,ayr 7who also appears among the quite too too71 ,,clifton ,c41 ,f4 ,b41 ,,fifee1 ,,grig1 ,,janet1 and ,,mrs4 ,,sairey ,,gamp4 ,f4 ,b4 has not fallen into the common error2 she multiplies together the proportionate numbers she gets1 but in getting them she goes wrong1 by reckoning warmth as a de-merit4 ,possibly she is ,freshly ,burnt1 or comes ,from ,bombay4 ,,janet and ,,mrs4 ,,sairey ,,gamp have also avoided this error3 the method they have adopted is shrouded in mystery--,i scarcely feel competent to criticize it4 ,,mrs4 ,,gamp says if ,zuzu makes #d while ,lolo makes #c1 ,zuzu makes #f while ,lolo makes #e 7bad reasoning71 while ,mimi makes #b4 ,from this she concludes therefore ,zuzu excels in speed by #a 7i4e4 when compared with ,lolo2 but what about ,mimi874 ,she then compares the #c kinds of excellence1 measured on this mystic scale4 ,,janet takes the statement1 that ,lolo makes #e while ,mimi makes #b1 to prove that ,lolo makes #c while ,mimi makes #a and ,zuzu #d 7worse reasoning than ,,mrs4 ,,gamp',s71 and thence concludes that ,zuzu excels in speed by #a_/h6 ,,janet should have been ,,adeline1 mystery of mysteries6 ,the fourth set actually put ,mimi at the top1 arranging them as ,mimi1 ,zuzu1 ,lolo4 ,they are ,,marquis ,,and ,,co41 ,,martreb1 ,s4 ,b4 ,b4 7first initial scarcely legible3 may be meant for ,j71 and ,,stanza4 ,the fifth set consist of ,,an ,,ancient ,,fish and ,,camel4 ,these ill-assorted comrades1 by dint of foot and fin1 have scrambled into the right answer1 but1 as their method is wrong1 of course it counts for nothing4 ,also ,,an ,,ancient ,,fish has very ancient and fishlike ideas as to how numbers represent merit3 she says ,lolo gains #b-a_/b on ,mimi4 ,two and a half what8 ,fish1 fish1 art thou in thy duty8 ,of the five winners ,i put ,,balbus and ,,the ,,elder ,,traveller slightly below the other three--,,balbus for defective reasoning1 the other for scanty working4 ,,balbus gives two reasons for saying that addition of marks is not the right method1 and then adds it follows that the decision must be made by multiplying the marks together4 ,this is hardly more logical than to say ,this is not ,spring3 therefore it must be ,autumn4 ,,class ,,list4 ,i4 ,,dinah ,,mite4 ,e4 ,b4 ,d4 ,l4 ,,joram4 ,,ii4 ,,balbus4 ,,the ,,elder ,,traveller4 99 99 99 99 99 ,with regard to ,knot ,v41 ,i beg to express to ,,vis ,,inerti and to any others who1 like her1 understood the condition to be that every marked picture must have three marks1 my sincere regret that the unfortunate phrase fill the columns with oughts and crosses should have caused them to waste so much time and trouble4 ,i can only repeat that a literal interpretation of fill would seem to me to require that every picture in the gallery should be marked4 ,,vis ,,inerti would have been in the ,first ,class if she had sent in the solution she now offers4 ,,answers ,,to ,,knot ,,vii4 ,problem4--,given that one glass of lemonade1 #c sandwiches1 and #g biscuits1 cost #as4 #b;d42 and that one glass of lemonade1 #d sandwiches1 and #aj biscuits1 cost #as4 #e;d43 find the cost of 7#a7 a glass of lemonade1 a sandwich1 and a biscuit2 and 7#b7 #b glasses of lemonade1 #c sandwiches1 and #e biscuits4 ,answer4--7#a7 #h;d42 7#b7 #as4 #g;d4 ,solution4--,this is best treated algebraically4 ,let x the cost 7in pence7 of a glass of lemonade1 y of a sandwich1 and z of a biscuit4 ,then we have x #cy #gz #ad1 and x #dy #ajz #ag4 ,and we require the values of x y z1 and of #bx #cy #ez4 ,now1 from two equations only1 we cannot find1 separately1 the values of three unknowns3 certain combinations of them may1 however1 be found4 ,also we know that we can1 by the help of the given equations1 eliminate #b of the #c unknowns from the quantity whose value is required1 which will then contain one only4 ,if1 then1 the required value is ascertainable at all1 it can only be by the #crd unknown vanishing of itself3 otherwise the problem is impossible4 ,let us then eliminate lemonade and sandwiches1 and reduce everything to biscuits--a state of things even more depressing than if all the world were apple-pie--by subtracting the #ast equation from the #bnd1 which eliminates lemonade1 and gives y #cz #c1 or y #c-cz2 and then substituting this value of y in the #ast1 which gives x-#bz #e1 i4e4 x #e #bz4 ,now if we substitute these values of x1 y1 in the quantities whose values are required1 the first becomes 7#e #bz7 7#c-cz7 z1 i4e4 #h3 and the second becomes #b7e #bz7 #c7c-cz7 #ez1 i4e4 #ai4 ,hence the answers are 7#a7 #h;d41 7#b7 #as4 #g;d4 99 99 99 99 99 ,the above is a universal method3 that is1 it is absolutely certain either to produce the answer1 or to prove that no answer is possible4 ,the question may also be solved by combining the quantities whose values are given1 so as to form those whose values are required4 ,this is merely a matter of ingenuity and good luck3 and as it may fail1 even when the thing is possible1 and is of no use in proving it impossible1 ,i cannot rank this method as equal in value with the other4 ,even when it succeeds1 it may prove a very tedious process4 ,suppose the #bf competitors1 who have sent in what ,i may call accidental solutions1 had had a question to deal with where every number contained #h or #aj digits6 ,i suspect it would have been a case of silvered is the raven hair 7see ,patience7 before any solution would have been hit on by the most ingenious of them4 ,forty-five answers have come in1 of which #dd give1 ,i am happy to say1 some sort of working1 and therefore deserve to be mentioned by name1 and to have their virtues1 or vices as the case may be1 discussed4 ,thirteen have made assumptions to which they have no right1 and so cannot figure in the ,class-list1 even though1 in #aj of the #ac cases1 the answer is right4 ,of the remaining #bh1 no less than #bf have sent in accidental solutions1 and therefore fall short of the highest honours4 ,i will now discuss individual cases1 taking the worst first1 as my custom is4 ,,froggy gives no working--at least this is all he gives3 after stating the given equations1 he says therefore the difference1 #a sandwich #c biscuits1 #c;d43 then follow the amounts of the unknown bills1 with no further hint as to how he got them4 ,,froggy has had a very narrow escape of not being named at all6 ,of those who are wrong1 ,,vis ,,inerti has sent in a piece of incorrect working4 ,peruse the horrid details1 and shudder6 ,she takes x 7call it y7 as the cost of a sandwich1 and concludes 7rightly enough7 that a biscuit will cost 7#c-y7_/#c4 ,she then subtracts the second equation from the first1 and deduces #cy #g 7#c-y7_/#c-dy #aj 7#c-y7_/#c #c4 ,by making two mistakes in this line1 she brings out y #c_/b4 ,try it again1 oh ,,vis ,,inerti6 ,away with ,,inerti3 infuse a little more ,,vis3 and you will bring out the correct 7though uninteresting7 result1 #j #j6 ,this will show you that it is hopeless to try to coax any one of these #c unknowns to reveal its separate value4 ,the other competitor1 who is wrong throughout1 is either ,j4 ,m4 ,c4 or ,t4 ,m4 ,c43 but1 whether he be a ,juvenile ,mis-,calculator or a ,true ,mathematician ,confused1 he makes the answers #g;d4 and #as4 #e;d4 ,he assumes1 with ,too ,much ,confidence1 that biscuits were #a_/b;d4 each1 and that ,clara paid for #h1 though she only ate #g6 ,we will now consider the #ac whose working is wrong1 though the answer is right3 and1 not to measure their demerits too exactly1 ,i will take them in alphabetical order4 ,,anita finds 7rightly7 that #a sandwich and #c biscuits cost #c;d41 and proceeds therefore #a sandwich #a-a_/b;d41 #c biscuits #a-a_/b;d41 #a lemonade #f;d4 ,,dinah ,,mite begins like ,,anita3 and thence proves 7rightly7 that a biscuit costs less than a #a;d43 whence she concludes 7wrongly7 that it must cost #a_/b;d4 ,f4 ,c4 ,w4 is so beautifully resigned to the certainty of a verdict of guilty1 that ,i have hardly the heart to utter the word1 without adding a recommended to mercy owing to extenuating circumstances4 ,but really1 you know1 where are the extenuating circumstances8 ,she begins by assuming that lemonade is #d;d4 a glass1 and sandwiches #c;d4 each1 7making with the #b given equations1 four conditions to be fulfilled by three miserable unknowns674 ,and1 having 7naturally7 developed this into a contradiction1 she then tries #e;d4 and #b;d4 with a similar result4 7,n4,b4 ,this process might have been carried on through the whole of the ,tertiary ,period1 without gratifying one single ,megatherium47 ,she then1 by a happy thought1 tries half-penny biscuits1 and so obtains a consistent result4 ,this may be a good solution1 viewing the problem as a conundrum3 but it is not scientific4 ,,janet identifies sandwiches with biscuits6 ,one sandwich #c biscuits she makes equal to #d4 ,four what8 ,,mayfair makes the astounding assertion that the equation1 s #c;b #c1 is evidently only satisfied by s #c_/b1 b #a_/b6 ,,old ,,cat believes that the assumption that a sandwich costs #a-a_/b;d4 is the only way to avoid unmanageable fractions4 ,but why avoid them8 ,is there not a certain glow of triumph in taming such a fraction8 ,ladies and gentlemen1 the fraction now before you is one that for years defied all efforts of a refining nature3 it was1 in a word1 hopelessly vulgar4 ,treating it as a circulating decimal 7the treadmill of fractions7 only made matters worse4 ,as a last resource1 ,i reduced it to its lowest terms1 and extracted its square root6 ,joking apart1 let me thank ,,old ,,cat for some very kind words of sympathy1 in reference to a correspondent 7whose name ,i am happy to say ,i have now forgotten7 who had found fault with me as a discourteous critic4 ,o4 ,v4 ,l4 is beyond my comprehension4 ,he takes the given equations as 7#a7 and 7#b73 thence1 by the process 7#b7-7a7 deduces 7rightly7 equation 7#c7 viz4 s #c;b #c3 and thence again1 by the process #cc 7a hopeless mystery71 deduces #cs #d;b #d4 ,i have nothing to say about it3 ,i give it up4 ,,sea-,,breeze says it is immaterial to the answer 7why87 in what proportion #c;d4 is divided between the sandwich and the #c biscuits3 so she assumes s l-#a_/b;d41 b #a_/b;d4 ,,stanza is one of a very irregular metre4 ,at first she 7like ,,janet7 identifies sandwiches with biscuits4 ,she then tries two assumptions 7s #a1 b #b_/c1 and s #a_/b b #e_/f71 and 7naturally7 ends in contradictions4 ,then she returns to the first assumption1 and finds the #c unknowns separately3 quod est absurdum4 ,,stiletto identifies sandwiches and biscuits1 as articles4 ,is the word ever used by confectioners8 ,i fancied ,what is the next article1 ,ma'am8 was limited to linendrapers4 ,,two ,,sisters first assume that biscuits are #d a penny1 and then that they are #b a penny1 adding that the answer will of course be the same in both cases4 ,it is a dreamy remark1 making one feel something like ,macbeth grasping at the spectral dagger4 ,is this a statement that ,i see before me8 ,if you were to say we both walked the same way this morning1 and ,i were to say one of you walked the same way1 but the other didn't1 which of the three would be the most hopelessly confused8 ,,turtle ,,pyate 7what is a ,turtle ,pyate1 please87 and ,,old ,,crow1 who send a joint answer1 and ,y4 ,y41 adopt the same method4 ,y4 ,y4 gets the equation s #c;b #c3 and then says this sum must be apportioned in one of the three following ways4 ,it may be1 ,i grant you3 but ,y4 ,y4 do you say must8 ,i fear it is possible for ,y4 ,y4 to be two ,y's4 ,the other two conspirators are less positive3 they say it can be so divided3 but they add either of the three prices being right6 ,this is bad grammar and bad arithmetic at once1 oh mysterious birds6 ,of those who win honours1 ,,the ,,shetland ,,snark must have the #crd class all to himself4 ,he has only answered half the question1 viz4 the amount of ,clara's luncheon3 the two little old ladies he pitilessly leaves in the midst of their difficulty4 ,i beg to assure him 7with thanks for his friendly remarks7 that entrance-fees and subscriptions are things unknown in that most economical of clubs1 ,the ,knot-,untiers4 ,the authors of the #bf accidental solutions differ only in the number of steps they have taken between the data and the answers4 ,in order to do them full justice ,i have arranged the #bnd class in sections1 according to the number of steps4 ,the two ,kings are fearfully deliberate6 ,i suppose walking quick1 or taking short cuts1 is inconsistent with kingly dignity3 but really1 in reading ,,theseus' solution1 one almost fancied he was marking time1 and making no advance at all6 ,the other ,king will1 ,i hope1 pardon me for having altered ,coal into ,cole4 ,king ,coilus1 or ,coil1 seems to have reigned soon after ,arthur's time4 ,henry of ,huntingdon identifies him with the ,king ,col who first built walls round ,colchester1 which was named after him4 ,in the ,chronicle of ,robert of ,gloucester we read3-- ,aftur ,kyng ,aruirag1 of wam we habbeth y told1 ,marius ys sone was kyng1 quoynte mon bold4 ,and ys sone was aftur hym1 ,coil was ys name1 ,bothe it were quoynte men1 of noble fame4 ,,balbus lays it down as a general principle that in order to ascertain the cost of any one luncheon1 it must come to the same amount upon two different assumptions4 7,query4 ,should not it be we8 ,otherwise the luncheon is represented as wishing to ascertain its own cost67 ,he then makes two assumptions--one1 that sandwiches cost nothing2 the other1 that biscuits cost nothing1 7either arrangement would lead to the shop being inconveniently crowded67--and brings out the unknown luncheons as #h;d4 and #ai;d41 on each assumption4 ,he then concludes that this agreement of results shows that the answers are correct4 ,now ,i propose to disprove his general law by simply giving one instance of its failing4 ,one instance is quite enough4 ,in logical language1 in order to disprove a universal affirmative1 it is enough to prove its contradictory1 which is a particular negative4 7,i must pause for a digression on ,logic1 and especially on ,ladies' ,logic4 ,the universal affirmative everybody says he's a duck is crushed instantly by proving the particular negative ,peter says he's a goose1 which is equivalent to ,peter does not say he's a duck4 ,and the universal negative nobody calls on her is well met by the particular affirmative ,i called yesterday4 ,in short1 either of two contradictories disproves the other3 and the moral is that1 since a particular proposition is much more easily proved than a universal one1 it is the wisest course1 in arguing with a ,lady1 to limit one's own assertions to particulars1 and leave her to prove the universal contradictory1 if she can4 ,you will thus generally secure a logical victory3 a practical victory is not to be hoped for1 since she can always fall back upon the crushing remark that has nothing to do with it6--a move for which ,man has not yet discovered any satisfactory answer4 ,now let us return to ,,balbus47 ,here is my particular negative1 on which to test his rule4 ,suppose the two recorded luncheons to have been #b buns1 one queen-cake1 #b sausage-rolls1 and a bottle of ,zodone3 total1 one-and-ninepence1 and one bun1 #b queen-cakes1 a sausage-roll1 and a bottle of ,zodone3 total1 one-and-fourpence4 ,and suppose ,clara's unknown luncheon to have been #c buns1 one queen-cake1 one sausage-roll1 and #b bottles of ,zodone3 while the two little sisters had been indulging in #h buns1 #d queen-cakes1 #b sausage-rolls1 and #f bottles of ,zodone4 7,poor souls1 how thirsty they must have been67 ,if ,,balbus will kindly try this by his principle of two assumptions1 first assuming that a bun is #a;d4 and a queen-cake #b;d41 and then that a bun is #c;d4 and a queen-cake #c;d41 he will bring out the other two luncheons1 on each assumption1 as one-and-nine-pence and four-and-ten-pence respectively1 which harmony of results1 he will say1 shows that the answers are correct4 ,and yet1 as a matter of fact1 the buns were #b;d4 each1 the queen-cakes #c;d41 the sausage-rolls #f;d41 and the ,zodone #b;d4 a bottle3 so that ,clara's third luncheon had cost one-and-sevenpence1 and her thirsty friends had spent four-and-fourpence6 ,another remark of ,,balbus ,i will quote and discuss3 for ,i think that it also may yield a moral for some of my readers4 ,he says it is the same thing in substance whether in solving this problem we use words and call it ,arithmetic1 or use letters and signs and call it ,algebra4 ,now this does not appear to me a correct description of the two methods3 the ,arithmetical method is that of synthesis only2 it goes from one known fact to another1 till it reaches its goal3 whereas the ,algebraical method is that of analysis3 it begins with the goal1 symbolically represented1 and so goes backwards1 dragging its veiled victim with it1 till it has reached the full daylight of known facts1 in which it can tear off the veil and say ,i know you6 ,take an illustration4 ,your house has been broken into and robbed1 and you appeal to the policeman who was on duty that night4 ,well1 ,mum1 ,i did see a chap getting out over your garden-wall3 but ,i was a good bit off1 so ,i didn't chase him1 like4 ,i just cut down the short way to the ,chequers1 and who should ,i meet but ,bill ,sykes1 coming full split round the corner4 ,so ,i just ups and says ',my lad1 you're wanted4' ,that's all ,i says4 ,and he says ',i'll go along quiet1 ,bobby1' he says1 'without the darbies1' he says4 ,there's your ,arithmetical policeman4 ,now try the other method4 ,i seed somebody a running1 but he was well gone or ever ,i got nigh the place4 ,so ,i just took a look round in the garden4 ,and ,i noticed the foot-marks1 where the chap had come right across your flower-beds4 ,they was good big foot-marks sure-ly4 ,and ,i noticed as the left foot went down at the heel1 ever so much deeper than the other4 ,and ,i says to myself ',the chap's been a big hulking chap3 and he goes lame on his left foot4' ,and ,i rubs my hand on the wall where he got over1 and there was soot on it1 and no mistake4 ,so ,i says to myself ',now where can ,i light on a big man1 in the chimbley-sweep line1 what's lame of one foot8' ,and ,i flashes up permiscuous3 and ,i says ',it's ,bill ,sykes6' says ,i4 ,there is your ,algebraical policeman--a higher intellectual type1 to my thinking1 than the other4 ,,little ,,jack',s solution calls for a word of praise1 as he has written out what really is an algebraical proof in words1 without representing any of his facts as equations4 ,if it is all his own1 he will make a good algebraist in the time to come4 ,i beg to thank ,,simple ,,susan for some kind words of sympathy1 to the same effect as those received from ,,old ,,cat4 ,,hecla and ,,martreb are the only two who have used a method certain either to produce the answer1 or else to prove it impossible3 so they must share between them the highest honours4 ,,class ,,list4 ,i4 ,,hecla4 ,,martreb4 ,,ii4 #a 7#b steps74 ,,adelaide4 ,,clifton ,c4444 ,e4 ,k4 ,c4 ,,guy4 ,l',,inconnu4 ,,little ,,jack4 ,,nil ,,desperandum4 ,,simple ,,susan4 ,,yellow-,,hammer4 ,,woolly ,,one4 #b 7#c steps74 ,a4 ,a4 ,a ,,christmas ,,carol4 ,,afternoon ,,tea4 ,,an ,,appreciative ,,schoolma',,am4 ,,baby4 ,,balbus4 ,,bog-,,oak4 ,,the ,,red ,,queen4 ,,wall-,,flower4 #c 7#d steps74 ,,hawthorn4 ,,joram4 ,s4 ,s4 ,g4 #d 7#e steps74 ,a ,,stepney ,,coach4 #e 7#f steps74 ,,bay ,,laurel4 ,,bradshaw ,,of ,,the ,,future4 #f 7#i steps74 ,,old ,,king ,,cole4 #g 7#ad steps74 ,,theseus4 ,,answers ,,to ,,correspondents4 ,i have received several letters on the subjects of ,knots ,,ii4 and ,,vi41 which lead me to think some further explanation desirable4 ,in ,knot ,,ii41 ,i had intended the numbering of the houses to begin at one corner of the ,square1 and this was assumed by most1 if not all1 of the competitors4 ,,trojanus however says assuming1 in default of any information1 that the street enters the square in the middle of each side1 it may be supposed that the numbering begins at a street4 ,but surely the other is the more natural assumption8 ,in ,knot ,,vi41 the first ,problem was of course a mere jeu de mots1 whose presence ,i thought excusable in a series of ,problems whose aim is to entertain rather than to instruct3 but it has not escaped the contemptuous criticisms of two of my correspondents1 who seem to think that ,apollo is in duty bound to keep his bow always on the stretch4 ,neither of them has guessed it3 and this is true human nature4 ,only the other day--the #cast of ,september1 to be quite exact--,i met my old friend ,brown1 and gave him a riddle ,i had just heard4 ,with one great effort of his colossal mind1 ,brown guessed it4 ,right6 said ,i4 ,ah1 said he1 it's very neat--very neat4 ,and it isn't an answer that would occur to everybody4 ,very neat indeed4 ,a few yards further on1 ,i fell in with ,smith and to him ,i propounded the same riddle4 ,he frowned over it for a minute1 and then gave it up4 ,meekly ,i faltered out the answer4 ,a poor thing1 sir6 ,smith growled1 as he turned away4 ,a very poor thing6 ,i wonder you care to repeat such rubbish6 ,yet ,smith's mind is1 if possible1 even more colossal than ,brown's4 ,the second ,problem of ,knot ,,vi4 is an example in ordinary ,double ,rule of ,three1 whose essential feature is that the result depends on the variation of several elements1 which are so related to it that1 if all but one be constant1 it varies as that one3 hence1 if none be constant1 it varies as their product4 ,thus1 for example1 the cubical contents of a rectangular tank vary as its length1 if breadth and depth be constant1 and so on2 hence1 if none be constant1 it varies as the product of the length1 breadth1 and depth4 ,when the result is not thus connected with the varying elements1 the ,problem ceases to be ,double ,rule of ,three and often becomes one of great complexity4 ,to illustrate this1 let us take two candidates for a prize1 ,a and ,b1 who are to compete in ,french1 ,german1 and ,italian3 7a7 ,let it be laid down that the result is to depend on their relative knowledge of each subject1 so that1 whether their marks1 for ,french1 be #a1 #b or #ajj1 #bjj1 the result will be the same3 and let it also be laid down that1 if they get equal marks on #b papers1 the final marks are to have the same ratio as those of the #crd paper4 ,this is a case of ordinary ,double ,rule of ,three4 ,we multiply ,a's #c marks together1 and do the same for ,b4 ,note that1 if ,a gets a single #j1 his final mark is #j1 even if he gets full marks for #b papers while ,b gets only one mark for each paper4 ,this of course would be very unfair on ,a1 though a correct solution under the given conditions4 7b7 ,the result is to depend1 as before1 on relative knowledge2 but ,french is to have twice as much weight as ,german or ,italian4 ,this is an unusual form of question4 ,i should be inclined to say the resulting ratio is to be nearer to the ,french ratio than if we multiplied as in 7a71 and so much nearer that it would be necessary to use the other multipliers twice to produce the same result as in 7a73 e4g4 if the ,french ,ratio were #i_/aj1 and the others #d_/i1 #a_/i so that the ultimate ratio1 by method 7a71 would be #b_/de1 ,i should multiply instead by #b_/c1 #a_/c1 giving the result1 #a_/c which is nearer to #i_/aj than if he had used method 7a74 7c7 ,the result is to depend on actual amount of knowledge of the #c subjects collectively4 ,here we have to ask two questions4 7#a7 ,what is to be the unit 7i4e4 standard to measure by7 in each subject8 7#b7 ,are these units to be of equal1 or unequal value8 ,the usual unit is the knowledge shown by answering the whole paper correctly2 calling this #ajj1 all lower amounts are represented by numbers between #j and #ajj4 ,then1 if these units are to be of equal value1 we simply add ,a's #c marks together1 and do the same for ,b4 7d7 ,the conditions are the same as 7c71 but ,french is to have double weight4 ,here we simply double the ,french marks1 and add as before4 7e7 ,french is to have such weight1 that1 if other marks be equal1 the ultimate ratio is to be that of the ,french paper1 so that a #j in this would swamp the candidate3 but the other two subjects are only to affect the result collectively1 by the amount of knowledge shown1 the two being reckoned of equal value4 ,here ,i should add ,a's ,german and ,italian marks together1 and multiply by his ,french mark4 ,but ,i need not go on3 the problem may evidently be set with many varying conditions1 each requiring its own method of solution4 ,the ,problem in ,knot ,,vi4 was meant to belong to variety 7a71 and to make this clear1 ,i inserted the following passage3 ,usually the competitors differ in one point only4 ,thus1 last year1 ,fifi and ,gogo made the same number of scarves in the trial week1 and they were equally light2 but ,fifi's were twice as warm as ,gogo's1 and she was pronounced twice as good4 ,what ,i have said will suffice1 ,i hope1 as an answer to ,,balbus1 who holds that 7a7 and 7c7 are the only possible varieties of the problem1 and that to say ,we cannot use addition1 therefore we must be intended to use multiplication1 is no more illogical than1 from knowledge that one was not born in the night1 to infer that he was born in the daytime2 and also to ,,fifee1 who says ,i think a little more consideration will show you that our 'error of adding the proportional numbers together for each candidate instead of multiplying' is no error at all4 ,why1 even if addition had been the right method to use1 not one of the writers 7,i speak from memory7 showed any consciousness of the necessity of fixing a unit for each subject4 ,no error at all6 ,they were positively steeped in error6 ,one correspondent 7,i do not name him1 as the communication is not quite friendly in tone7 writes thus3--,i wish to add1 very respectfully1 that ,i think it would be in better taste if you were to abstain from the very trenchant expressions which you are accustomed to indulge in when criticising the answer4 ,that such a tone must not be 7be not87 agreeable to the persons concerned who have made mistakes may possibly have no great weight with you1 but ,i hope you will feel that it would be as well not to employ it1 unless you are quite certain of being correct yourself4 ,the only instances the writer gives of the trenchant expressions are hapless and malefactors4 ,i beg to assure him 7and any others who may need the assurance3 ,i trust there are none7 that all such words have been used in jest1 and with no idea that they could possibly annoy any one1 and that ,i sincerely regret any annoyance ,i may have thus inadvertently given4 ,may ,i hope that in future they will recognise the distinction between severe language used in sober earnest1 and the words of unmeant bitterness1 which ,coleridge has alluded to in that lovely passage beginning ,a little child1 a limber elf8 ,if the writer will refer to that passage1 or to the preface to ,fire1 ,famine1 and ,slaughter1 he will find the distinction1 for which ,i plead1 far better drawn out than ,i could hope to do in any words of mine4 ,the writer's insinuation that ,i care not how much annoyance ,i give to my readers ,i think it best to pass over in silence2 but to his concluding remark ,i must entirely demur4 ,i hold that to use language likely to annoy any of my correspondents would not be in the least justified by the plea that ,i was quite certain of being correct4 ,i trust that the knot-untiers and ,i are not on such terms as those6 ,i beg to thank ,g4 ,b4 for the offer of a puzzle--which1 however1 is too like the old one ,make four #i's into #ajj4 ,,answers ,,to ,,knot ,,viii4 #a4 ,,the ,,pigs4 ,problem4--,place twenty-four pigs in four sties so that1 as you go round and round1 you may always find the number in each sty nearer to ten than the number in the last4 ,answer4--,place #h pigs in the first sty1 #aj in the second1 nothing in the third1 and #f in the fourth3 #aj is nearer ten than #h2 nothing is nearer ten than #aj2 #f is nearer ten than nothing2 and #h is nearer ten than #f4 99 99 99 99 99 ,this problem is noticed by only two correspondents4 ,,balbus says it certainly cannot be solved mathematically1 nor do ,i see how to solve it by any verbal quibble4 ,,nolens ,,volens makes ,her ,radiancy change the direction of going round2 and even then is obliged to add the pigs must be carried in front of her6 #b4 ,,the ,,grurmstipths4 ,problem4--,omnibuses start from a certain point1 both ways1 every #ae minutes4 ,a traveller1 starting on foot along with one of them1 meets one in #ab-a_/b minutes3 when will he be overtaken by one8 ,answer4--,in #f-a_/d minutes4 99 99 99 99 99 ,solution4--,let a be the distance an omnibus goes in #ae minutes1 and x the distance from the starting-point to where the traveller is overtaken4 ,since the omnibus met is due at the starting-point in #b-a_/b minutes1 it goes in that time as far as the traveller walks in #ab-a_/b2 i4e4 it goes #e times as fast4 ,now the overtaking omnibus is a behind the traveller when he starts1 and therefore goes a x while he goes x4 ,hence a x #ex2 i4e4 #dx a1 and x a_/#d4 ,this distance would be traversed by an omnibus in #ae_/d minutes1 and therefore by the traveller in #e #ae_/d4 ,hence he is overtaken in #ah-c_/d minutes after starting1 i4e4 in #f-a_/d minutes after meeting the omnibus4 ,four answers have been received1 of which two are wrong4 ,,dinah ,,mite rightly states that the overtaking omnibus reached the point where they met the other omnibus #e minutes after they left1 but wrongly concludes that1 going #e times as fast1 it would overtake them in another minute4 ,the travellers are #e-minutes-walk ahead of the omnibus1 and must walk #a-dth of this distance farther before the omnibus overtakes them1 which will be #a-eth of the distance traversed by the omnibus in the same time3 this will require #a-a_/d minutes more4 ,,nolens ,,volens tries it by a process like ,achilles and the ,tortoise4 ,he rightly states that1 when the overtaking omnibus leaves the gate1 the travellers are #a-eth of a ahead1 and that it will take the omnibus #c minutes to traverse this distance2 during which time the travellers1 he tells us1 go #a-aeth of a 7this should be #a-beth74 ,the travellers being now #a-aeth of a ahead1 he concludes that the work remaining to be done is for the travellers to go #a-fjth of a1 while the omnibus goes #a-abth4 ,the principle is correct1 and might have been applied earlier4 ,,class ,,list4 ,i4 ,,balbus4 ,,delta4 ,,answers ,,to ,,knot ,,ix4 #a4 ,,the ,,buckets4 ,problem4--,lardner states that a solid1 immersed in a fluid1 displaces an amount equal to itself in bulk4 ,how can this be true of a small bucket floating in a larger one8 ,solution4--,lardner means1 by displaces1 occupies a space which might be filled with water without any change in the surroundings4 ,if the portion of the floating bucket1 which is above the water1 could be annihilated1 and the rest of it transformed into water1 the surrounding water would not change its position3 which agrees with ,lardner's statement4 99 99 99 99 99 ,five answers have been received1 none of which explains the difficulty arising from the well-known fact that a floating body is the same weight as the displaced fluid4 ,,hecla says that only that portion of the smaller bucket which descends below the original level of the water can be properly said to be immersed1 and only an equal bulk of water is displaced4 ,hence1 according to ,,hecla1 a solid1 whose weight was equal to that of an equal bulk of water1 would not float till the whole of it was below the original level of the water3 but1 as a matter of fact1 it would float as soon as it was all under water4 ,,magpie says the fallacy is the assumption that one body can displace another from a place where it isn't1 and that ,lardner's assertion is incorrect1 except when the containing vessel was originally full to the brim4 ,but the question of floating depends on the present state of things1 not on past history4 ,,old ,,king ,,cole takes the same view as ,,hecla4 ,,tympanum and ,,vindex assume that displaced means raised above its original level1 and merely explain how it comes to pass that the water1 so raised1 is less in bulk than the immersed portion of bucket1 and thus land themselves--or rather set themselves floating--in the same boat as ,,hecla4 ,i regret that there is no ,class-list to publish for this ,problem4 99 99 99 99 99 #b4 ,,balbus' ,,essay4 ,problem4--,balbus states that if a certain solid be immersed in a certain vessel of water1 the water will rise through a series of distances1 two inches1 one inch1 half an inch1 c41 which series has no end4 ,he concludes that the water will rise without limit4 ,is this true8 ,solution4--,no4 ,this series can never reach #d inches1 since1 however many terms we take1 we are always short of #d inches by an amount equal to the last term taken4 99 99 99 99 99 ,three answers have been received--but only two seem to me worthy of honours4 ,,tympanum says that the statement about the stick is merely a blind1 to which the old answer may well be applied1 solvitur ambulando1 or rather mergendo4 ,i trust ,,tympanum will not test this in his own person1 by taking the place of the man in ,balbus' ,essay6 ,he would infallibly be drowned4 ,,old ,,king ,,cole rightly points out that the series1 #b1 #a1 c41 is a decreasing ,geometrical ,progression3 while ,,vindex rightly identifies the fallacy as that of ,achilles and the ,tortoise4 ,,class ,,list4 ,i4 ,,old ,,king ,,cole4 ,,vindex4 99 99 99 99 99 #c4 ,,the ,,garden4 ,problem4--,an oblong garden1 half a yard longer than wide1 consists entirely of a gravel-walk1 spirally arranged1 a yard wide and #c1fcj yards long4 ,find the dimensions of the garden4 ,answer4--#fj1 #fj-a_/b4 ,solution4--,the number of yards and fractions of a yard traversed in walking along a straight piece of walk1 is evidently the same as the number of square-yards and fractions of a square-yard1 contained in that piece of walk3 and the distance1 traversed in passing through a square-yard at a corner1 is evidently a yard4 ,hence the area of the garden is #c1fcj square-yards3 i4e41 if x be the width1 x 7x #a_/b7 #c1fcj4 ,solving this ,quadratic1 we find x #fj4 ,hence the dimensions are #fj1 #fj-a_/b4 99 99 99 99 99 ,twelve answers have been received--seven right and five wrong4 ,c4 ,g4 ,l41 ,,nabob1 ,,old ,,crow1 and ,,tympanum assume that the number of yards in the length of the path is equal to the number of square-yards in the garden4 ,this is true1 but should have been proved4 ,but each is guilty of darker deeds4 ,c4 ,g4 ,l4's working consists of dividing #c1fcj by #fj4 ,whence came this divisor1 oh ,segiel8 ,divination8 ,or was it a dream8 ,i fear this solution is worth nothing4 ,,old ,,crow',s is shorter1 and so 7if possible7 worth rather less4 ,he says the answer is at once seen to be #fj #fj-a_/b6 ,,nabob',s calculation is short1 but as rich as a ,nabob in error4 ,he says that the square root of #c1fcj1 multiplied by #b1 equals the length plus the breadth4 ,that is #fj4be #b #abj-a_/b4 ,his first assertion is only true of a square garden4 ,his second is irrelevant1 since #fj4be is not the square-root of #c1fcj6 ,nay1 ,bob1 this will not do6 ,,tympanum says that1 by extracting the square-root of #c1fcj1 we get #fj yards with a remainder of #cj_/fj1 or half-a-yard1 which we add so as to make the oblong #fj #fj-a_/b4 ,this is very terrible3 but worse remains behind4 ,,tympanum proceeds thus3--,but why should there be the half-yard at all8 ,because without it there would be no space at all for flowers4 ,by means of it1 we find reserved in the very centre a small plot of ground1 two yards long by half-a-yard wide1 the only space not occupied by walk4 ,but ,balbus expressly said that the walk used up the whole of the area4 ,oh1 ,,tympanum6 ,my tympa is exhausted3 my brain is num6 ,i can say no more4 ,,hecla indulges1 again and again1 in that most fatal of all habits in computation--the making two mistakes which cancel each other4 ,she takes x as the width of the garden1 in yards1 and x #a_/b as its length1 and makes her first coil the sum of x-#a_/b1 x-#a_/b1 x-#a1 x-#a1 i4e4 #dx-#c3 but the fourth term should be x-#a-a_/b1 so that her first coil is #a_/b a yard too long4 ,her second coil is the sum of x-#b-a_/b1 x-#b-a_/b1 x-#c1 x-#c3 here the first term should be x-#b and the last x-#c-a_/b3 these two mistakes cancel1 and this coil is therefore right4 ,and the same thing is true of every other coil but the last1 which needs an extra half-yard to reach the end of the path3 and this exactly balances the mistake in the first coil4 ,thus the sum total of the coils comes right though the working is all wrong4 ,of the seven who are right1 ,,dinah ,,mite1 ,,janet1 ,,magpie1 and ,,taffy make the same assumption as ,c4 ,g4 ,l4 and ,co4 ,they then solve by a ,quadratic4 ,,magpie also tries it by ,arithmetical ,progression1 but fails to notice that the first and last coils have special values4 ,,alumnus ,,eton attempts to prove what ,c4 ,g4 ,l4 assumes by a particular instance1 taking a garden #f by #e-a_/b4 ,he ought to have proved it generally3 what is true of one number is not always true of others4 ,,old ,,king ,,cole solves it by an ,arithmetical ,progression4 ,it is right1 but too lengthy to be worth as much as a ,quadratic4 ,,vindex proves it very neatly1 by pointing out that a yard of walk measured along the middle represents a square yard of garden1 whether we consider the straight stretches of walk or the square yards at the angles1 in which the middle line goes half a yard in one direction and then turns a right angle and goes half a yard in another direction4 ,,class ,,list4 ,i4 ,,vindex4 ,,ii4 ,,alumnus ,,eton4 ,,old ,,king ,,cole4 ,,iii4 ,,dinah ,,mite4 ,,janet4 ,,magpie4 ,,taffy4 ,,answers ,,to ,,knot ,x4 #a4 ,,the ,,chelsea ,,pensioners4 ,problem4--,if #gj per cent4 have lost an eye1 #ge per cent4 an ear1 #hj per cent4 an arm1 #he per cent4 a leg3 what percentage1 at least1 must have lost all four8 ,answer4--,ten4 99 99 99 99 99 ,solution4--7,i adopt that of ,,polar ,,star1 as being better than my own74 ,adding the wounds together1 we get #gj #ge #hj #he #caj1 among #ajj men2 which gives #c to each1 and #d to #aj men4 ,therefore the least percentage is #aj4 99 99 99 99 99 ,nineteen answers have been received4 ,one is #e1 but1 as no working is given with it1 it must1 in accordance with the rule1 remain a deed without a name4 ,,janet makes it #ce and #g_/ajths4 ,i am sorry she has misunderstood the question1 and has supposed that those who had lost an ear were #ge per cent4 of those who had lost an eye2 and so on4 ,of course1 on this supposition1 the percentages must all be multiplied together4 ,this she has done correctly1 but ,i can give her no honours1 as ,i do not think the question will fairly bear her interpretation1 ,,three ,,score ,,and ,,ten makes it #ai and #c_/hths4 ,her solution has given me--,i will not say many anxious days and sleepless nights1 for ,i wish to be strictly truthful1 but--some trouble in making any sense at all of it4 ,she makes the number of pensioners wounded once to be #caj 7per cent41 ,i suppose673 dividing by #d1 she gets #gg and a half as average percentage3 again dividing by #d1 she gets #ai and #c_/hths as percentage wounded four times4 ,does she suppose wounds of different kinds to absorb each other1 so to speak8 ,then1 no doubt1 the data are equivalent to #gg pensioners with one wound each1 and a half-pensioner with a half-wound4 ,and does she then suppose these concentrated wounds to be transferable1 so that #c_/dths of these unfortunates can obtain perfect health by handing over their wounds to the remaining #a_/dth8 ,granting these suppositions1 her answer is right2 or rather1 if the question had been ,a road is covered with one inch of gravel1 along #gg and a half per cent4 of it4 ,how much of it could be covered #d inches deep with the same material8 her answer would have been right4 ,but alas1 that wasn't the question6 ,,delta makes some most amazing assumptions3 let every one who has not lost an eye have lost an ear1 let every one who has not lost both eyes and ears have lost an arm4 ,her ideas of a battle-field are grim indeed4 ,fancy a warrior who would continue fighting after losing both eyes1 both ears1 and both arms6 ,this is a case which she 7or it87 evidently considers possible4 ,next come eight writers who have made the unwarrantable assumption that1 because #gj per cent4 have lost an eye1 therefore #cj per cent4 have not lost one1 so that they have both eyes4 ,this is illogical4 ,if you give me a bag containing #ajj sovereigns1 and if in an hour ,i come to you 7my face not beaming with gratitude nearly so much as when ,i received the bag7 to say ,i am sorry to tell you that #gj of these sovereigns are bad1 do ,i thereby guarantee the other #cj to be good8 ,perhaps ,i have not tested them yet4 ,the sides of this illogical octagon are as follows1 in alphabetical order3--,,algernon ,,bray1 ,,dinah ,,mite1 ,g4 ,s4 ,c41 ,,jane ,e41 ,j4 ,d4 ,w41 ,,magpie 7who makes the delightful remark therefore #ij per cent4 have two of something1 recalling to one's memory that fortunate monarch1 with whom ,xerxes was so much pleased that he gave him ten of everything671 ,s4 ,s4 ,g41 and ,,tokio4 ,,bradshaw ,,of ,,the ,,future and ,t4 ,r4 do the question in a piecemeal fashion--on the principle that the #gj per cent4 and the #ge per cent41 though commenced at opposite ends of the #ajj1 must overlap by at least #de per cent42 and so on4 ,this is quite correct working1 but not1 ,i think1 quite the best way of doing it4 ,the other five competitors will1 ,i hope1 feel themselves sufficiently glorified by being placed in the first class1 without my composing a ,triumphal ,ode for each6 ,,class ,,list4 ,i4 ,,old ,,cat4 ,,old ,,hen4 ,,polar ,,star4 ,,simple ,,susan4 ,,white ,,sugar4 ,,ii4 ,,bradshaw ,,of ,,the ,,future4 ,t4 ,r4 ,,iii4 ,,algernon ,,bray4 ,,dinah ,,mite4 ,g4 ,s4 ,c4 ,,jane ,e4 ,j4 ,d4 ,w4 ,,magpie4 ,s4 ,s4 ,g4 ,,tokio4 #b4 ,,change ,,of ,,day4 ,i must postpone1 sine die1 the geographical problem--partly because ,i have not yet received the statistics ,i am hoping for1 and partly because ,i am myself so entirely puzzled by it2 and when an examiner is himself dimly hovering between a second class and a third how is he to decide the position of others8 #c4 ,,the ,,sons' ,,ages4 ,problem4--,at first1 two of the ages are together equal to the third4 ,a few years afterwards1 two of them are together double of the third4 ,when the number of years since the first occasion is two-thirds of the sum of the ages on that occasion1 one age is #ba4 ,what are the other two8 ,answer4--#ae and #ah4 99 99 99 99 99 ,solution4--,let the ages at first be x1 y1 7x y74 ,now1 if a b #b;c1 then 7a-n7 7b-n7 #b7;c-n71 whatever be the value of n4 ,hence the second relationship1 if ever true1 was always true4 ,hence it was true at first4 ,but it cannot be true that x and y are together double of 7x y74 ,hence it must be true of 7x y71 together with x or y2 and it does not matter which we take4 ,we assume1 then1 7x y7 x #by2 i4e4 y #bx4 ,hence the three ages were1 at first1 x1 #bx1 #cx2 and the number of years1 since that time is two-thirds of #fx1 i4e4 is #dx4 ,hence the present ages are #ex1 #fx1 #gx4 ,the ages are clearly integers1 since this is only the year when one of my sons comes of age4 ,hence #gx #ba1 x #c1 and the other ages are #ae1 #ah4 99 99 99 99 99 ,eighteen answers have been received4 ,one of the writers merely asserts that the first occasion was #ab years ago1 that the ages were then #i1 #f1 and #c2 and that on the second occasion they were #ad1 #aa1 and #h6 ,as a ,roman father1 ,i ought to withhold the name of the rash writer2 but respect for age makes me break the rule3 it is ,,three ,,score ,,and ,,ten4 ,,jane ,e4 also asserts that the ages at first were #i1 #f1 #c3 then she calculates the present ages1 leaving the second occasion unnoticed4 ,,old ,,hen is nearly as bad2 she tried various numbers till ,i found one that fitted all the conditions2 but merely scratching up the earth1 and pecking about1 is not the way to solve a problem1 oh venerable bird6 ,and close after ,,old ,,hen prowls1 with hungry eyes1 ,,old ,,cat1 who calmly assumes1 to begin with1 that the son who comes of age is the eldest4 ,eat your bird1 ,puss1 for you will get nothing from me6 ,there are yet two zeroes to dispose of4 ,,minerva assumes that1 on every occasion1 a son comes of age2 and that it is only such a son who is tipped with gold4 ,is it wise thus to interpret now1 my boys1 calculate your ages1 and you shall have the money8 ,,bradshaw ,,of ,,the ,,future says let the ages at first be #i1 #f1 #c1 then assumes that the second occasion was #f years afterwards1 and on these baseless assumptions brings out the right answers4 ,guide future travellers1 an thou wilt3 thou art no ,bradshaw for this ,age6 ,of those who win honours1 the merely honourable are two4 ,,dinah ,,mite ascertains 7rightly7 the relationship between the three ages at first1 but then assumes one of them to be #f1 thus making the rest of her solution tentative4 ,m4 ,f4 ,c4 does the algebra all right up to the conclusion that the present ages are #ez1 #fz1 and #gz2 it then assumes1 without giving any reason1 that #gz #ba4 ,of the more honourable1 ,,delta attempts a novelty--to discover which son comes of age by elimination3 it assumes1 successively1 that it is the middle one1 and that it is the youngest2 and in each case it apparently brings out an absurdity4 ,still1 as the proof contains the following bit of algebra1 #fc #gx #dy2 9999 therefore #ba x #d sevenths of y1 ,i trust it will admit that its proof is not quite conclusive4 ,the rest of its work is good4 ,,magpie betrays the deplorable tendency of her tribe--to appropriate any stray conclusion she comes across1 without having any strict logical right to it4 ,assuming ,a1 ,b1 ,c1 as the ages at first1 and ,d as the number of the years that have elapsed since then1 she finds 7rightly7 the #c equations1 #b;,a ,b1 ,c ,b ,a1 ,d #b;,b4 ,she then says supposing that ,a #a1 then ,b #b1 ,c #c1 and ,d #d4 ,therefore for ,a1 ,b1 ,c1 ,d1 four numbers are wanted which shall be to each other as #a3b3c3d4 ,it is in the therefore that ,i detect the unconscientiousness of this bird4 ,the conclusion is true1 but this is only because the equations are homogeneous 7i4e4 having one unknown in each term71 a fact which ,i strongly suspect had not been grasped--,i beg pardon1 clawed--by her4 ,were ,i to lay this little pitfall1 ,a #a ,b1 ,b #a ,c2 supposing ,a #a1 then ,b #b and ,c #c4 ,therefore for ,a1 ,b1 ,c1 three numbers are wanted which shall be to one another as #a3b3c1 would you not flutter down into it1 oh ,,magpie1 as amiably as a ,dove8 ,,simple ,,susan is anything but simple to me4 ,after ascertaining that the #c ages at first are as #c3b3a1 she says then1 as two-thirds of their sum1 added to one of them1 #ba1 the sum cannot exceed #cj1 and consequently the highest cannot exceed #ae4 ,i suppose her 7mental7 argument is something like this3--two-thirds of sum1 one age1 #ba2 9999 therefore sum1 #c halves of one age1 #ca and a half4 ,but #c halves of one age cannot be less than #a and-a-half 7here ,i perceive that ,,simple ,,susan would on no account present a guinea to a new-born baby67 hence the sum cannot exceed #cj4 ,this is ingenious1 but her proof1 after that1 is 7as she candidly admits7 clumsy and roundabout4 ,she finds that there are #e possible sets of ages1 and eliminates four of them4 ,suppose that1 instead of #e1 there had been #e million possible sets8 ,would ,,simple ,,susan have courageously ordered in the necessary gallon of ink and ream of paper8 ,the solution sent in by ,c4 ,r4 is1 like that of ,,simple ,,susan1 partly tentative1 and so does not rise higher than being ,clumsily ,right4 ,among those who have earned the highest honours1 ,,algernon ,,bray solves the problem quite correctly1 but adds that there is nothing to exclude the supposition that all the ages were fractional4 ,this would make the number of answers infinite4 ,let me meekly protest that ,i never intended my readers to devote the rest of their lives to writing out answers6 ,e4 ,m4 ,,rix points out that1 if fractional ages be admissible1 any one of the three sons might be the one come of age2 but she rightly rejects this supposition on the ground that it would make the problem indeterminate4 ,,white ,,sugar is the only one who has detected an oversight of mine3 ,i had forgotten the possibility 7which of course ought to be allowed for7 that the son1 who came of age that year1 need not have done so by that day1 so that he might be only #bj4 ,this gives a second solution1 viz41 #bj1 #bd1 #bh4 ,well said1 pure ,crystal6 ,verily1 thy fair discourse hath been as sugar6 ,,class ,,list4 ,i4 ,,algernon ,,bray4 ,,an ,,old ,,fogey4 ,e4 ,m4 ,,rix4 ,g4 ,s4 ,c4 ,s4 ,s4 ,g4 ,,tokio4 ,t4 ,r4 ,,white ,,sugar4 ,,ii4 ,c4 ,r4 ,,delta4 ,,magpie4 ,,simple ,,susan4 ,,iii4 ,,dinah ,,mite4 ,m4 ,f4 ,c4 99 99 99 99 99 ,i have received more than one remonstrance on my assertion1 in the ,chelsea ,pensioners' problem1 that it was illogical to assume1 from the datum #gj p4 c4 have lost an eye1 that #cj p4 c4 have not4 ,,algernon ,,bray states1 as a parallel case1 suppose ,tommy's father gives him #d apples1 and he eats one of them1 how many has he left8 and says ,i think we are justified in answering1 #c4 ,i think so too4 ,there is no must here1 and the data are evidently meant to fix the answer exactly3 but1 if the question were set me how many must he have left81 ,i should understand the data to be that his father gave him #d at least1 but may have given him more4 ,i take this opportunity of thanking those who have sent1 along with their answers to the ,tenth ,knot1 regrets that there are no more ,knots to come1 or petitions that ,i should recall my resolution to bring them to an end4 ,i am most grateful for their kind words2 but ,i think it wisest to end what1 at best1 was but a lame attempt4 ,the stretched metre of an antique song is beyond my compass2 and my puppets were neither distinctly in my life 7like those ,i now address71 nor yet 7like ,alice and the ,mock ,turtle7 distinctly out of it4 ,yet let me at least fancy1 as ,i lay down the pen1 that ,i carry with me into my silent life1 dear reader1 a farewell smile from your unseen face1 and a kindly farewell pressure from your unfelt hand6 ,and so1 good night6 ,parting is such sweet sorrow1 that ,i shall say good night6 till it be morrow4 ,,the ,,end ,,london3 ,,richard ,,clay ,,and ,,sons1 ,,printers4 ,,turn ,,over4 ,,works ,,by ,,lewis ,,carroll4 ,,alice',s ,,adventures ,,in ,,wonderland4 ,with ,forty-two ,illustrations by ,,tenniel4 ,crown #hvo1 cloth1 gilt edges1 price #fs4 ,seventy-fifth ,thousand4 ,,translations ,,of ,,the ,,same--into ,french1 by ,,henri ,,bu--into ,german1 by ,,antonie ,,zimmermann--and into ,italian1 by ,t4 ,,pietroc,,la ,,rossetti--with ,,tenniel',s ,illustrations4 ,crown #hvo1 cloth1 gilt edges1 price #fs4 each4 ,,through ,,the ,,looking-,,glass1 ,,and ,,what ,,alice ,,found ,,there4 ,with ,fifty ,illustrations by ,,tenniel4 ,crown #hvo1 cloth1 gilt edges1 price #fs4 ,fifty-sixth ,thousand4 ,,rhyme8 ,,and ,,reason8 ,with ,sixty-five ,illustrations by ,,arthur ,b4 ,,frost1 and ,nine by ,,henry ,,holiday4 7,this book is a reprint1 with a few additions1 of the comic portion of ,phantasmagoria and other ,poems1 and of ,the ,hunting of the ,snark4 ,mr4 ,frost's pictures are new47 ,crown #hvo1 cloth1 coloured edges1 price #gs4 ,fifty ,thousand4 ,a ,,tangled ,,tale4 ,reprinted from ,the ,monthly ,packet4 ,with ,six ,illustrations by ,arthur ,b4 ,frost4 ,crown #hvo1 #ds4 #f;d4 99 99 99 99 99 ,n4,b4 ,in selling the above-mentioned books to the ,trade1 ,messrs4 ,macmillan and ,co4 will abate #b;d4 in the shilling 7no odd copies71 and allow #e per cent4 discount for payment within six months1 and #aj per cent4 for cash4 ,in selling them to the ,public 7for cash only7 they will allow #aj per cent4 discount4 99 99 99 99 99 ,,mr4 ,,lewis ,,carroll1 having been requested to allow ,,an ,,easter ,,greeting 7a leaflet1 addressed to children1 and frequently given with his books7 to be sold separately1 has arranged with ,messrs4 ,harrison1 of #ei1 ,pall ,mall1 who will supply a single copy for #a;d41 or #ab for #i;d41 or #ajj for #es4 ,,macmillan ,,and ,,co41 ,,london4 ,,london3 ,,richard ,,clay ,,and ,,sons1 ,,printers4 99 99 99 99 99 ,transcriber's note ,the following changes have been made to the text3 ,page #hh3 he corners of the changed to the corners of the4 ,page #ie3 ,aix-le-,bains changed to ,aix-les-,bains4 ,page #ajh3 #c_/e1 #b1 #a_/c changed to #c_/e1 #b_/c1 #a_/c4 ,page #aad3 #aj of the #ab cases changed to #aj of the #ac cases4 ,page #aba3 four-and fourpence changed to four-and-fourpence4 ,last page3 ,fifth ,thousand changed to ,fifty ,thousand4