"There's a fallacy somewhere," he murmured drowsily, as he stretched his long legs upon the sofa. "I must think it over again." He closed his eyes, in order to concentrate his attention more perfectly, and for the next hour or so his slow and regular breathing bore witness to the careful deliberation with which he was investigating this new and perplexing view of the subject.
"HE REMAINS STEADFAST AND UNMOVED."
FOOTNOTE:
[B] Note by the writer.—For the above Essay I am indebted to a dear friend, now deceased.
KNOT X.
CHELSEA BUNS.
"Yea, buns, and buns, and buns!"
Old Song.
"How very, very sad!" exclaimed Clara; and the eyes of the gentle girl filled with tears as she spoke.
"Sad—but very curious when you come to look at it arithmetically," was her aunt's less romantic reply. "Some of them have lost an arm in their country's service, some a leg, some an ear, some an eye——"
"And some, perhaps, all!" Clara murmured dreamily, as they passed the long rows of weather-beaten heroes basking in the sun. "Did you notice that very old one, with a red face, who was drawing a map in the dust with his wooden leg, and all the others watching? I think it was a plan of a battle——"
"The battle of Trafalgar, no doubt," her aunt interrupted, briskly.
"Hardly that, I think," Clara ventured to say. "You see, in that case, he couldn't well be alive——"
"Couldn't well be alive!" the old lady contemptuously repeated. "He's as lively as you and me put together! Why, if drawing a map in the dust—with one's wooden leg—doesn't prove one to be alive, perhaps you'll kindly mention what does prove it!"
Clara did not see her way out of it. Logic had never been her forte.
"To return to the arithmetic," Mad Mathesis resumed—the eccentric old lady never let slip an opportunity of driving her niece into a calculation—"what percentage do you suppose must have lost all four—a leg, an arm, an eye, and an ear?"
"How can I tell?" gasped the terrified girl. She knew well what was coming.
"You can't, of course, without data," her aunt replied: "but I'm just going to give you——"
"Give her a Chelsea bun, Miss! That's what most young ladies likes best!" The voice was rich and musical, and the speaker dexterously whipped back the snowy cloth that covered his basket, and disclosed a tempting array of the familiar square buns, joined together in rows, richly egged and browned, and glistening in the sun.
"No, sir! I shall give her nothing so indigestible! Be off!" The old lady waved her parasol threateningly: but nothing seemed to disturb the good-humour of the jolly old man, who marched on, chanting his melodious refrain:—
Listen
"Far too indigestible, my love!" said the old lady. "Percentages will agree with you ever so much better!"
Clara sighed, and there was a hungry look in her eyes as she watched the basket lessening in the distance: but she meekly listened to the relentless old lady, who at once proceeded to count off the data on her fingers.
"Say that 70 per cent. have lost an eye—75 per cent. an ear—80 per cent. an arm—85 per cent. a leg—that'll do it beautifully. Now, my dear, what percentage, at least, must have lost all four?"
No more conversation occurred—unless a smothered exclamation of "Piping hot!" which escaped from Clara's lips as the basket vanished round a corner could be counted as such—until they reached the old Chelsea mansion, where Clara's father was then staying, with his three sons and their old tutor.
Balbus, Lambert, and Hugh had entered the house only a few minutes before them. They had been out walking, and Hugh had been propounding a difficulty which had reduced Lambert to the depths of gloom, and had even puzzled Balbus.
"It changes from Wednesday to Thursday at midnight, doesn't it?" Hugh had begun.
"Sometimes," said Balbus, cautiously.
"Always," said Lambert, decisively.
"Sometimes," Balbus gently insisted. "Six midnights out of seven, it changes to some other name."
"I meant, of course," Hugh corrected himself, "when it does change from Wednesday to Thursday, it does it at midnight—and only at midnight."
"Surely," said Balbus. Lambert was silent.
"Well, now, suppose it's midnight here in Chelsea. Then it's Wednesday west of Chelsea (say in Ireland or America) where midnight hasn't arrived yet: and it's Thursday east of Chelsea (say in Germany or Russia) where midnight has just passed by?"
"Surely," Balbus said again. Even Lambert nodded this time.
"But it isn't midnight, anywhere else; so it can't be changing from one day to another anywhere else. And yet, if Ireland and America and so on call it Wednesday, and Germany and Russia and so on call it Thursday, there must be some place—not Chelsea—that has different days on the two sides of it. And the worst of it is, the people there get their days in the wrong order: they've got Wednesday east of them, and Thursday west—just as if their day had changed from Thursday to Wednesday!"
"I've heard that puzzle before!" cried Lambert. "And I'll tell you the explanation. When a ship goes round the world from east to west, we know that it loses a day in its reckoning: so that when it gets home, and calls its day Wednesday, it finds people here calling it Thursday, because we've had one more midnight than the ship has had. And when you go the other way round you gain a day."
"I know all that," said Hugh, in reply to this not very lucid explanation: "but it doesn't help me, because the ship hasn't proper days. One way round, you get more than twenty-four hours to the day, and the other way you get less: so of course the names get wrong: but people that live on in one place always get twenty-four hours to the day."
"I suppose there is such a place," Balbus said, meditatively, "though I never heard of it. And the people must find it very queer, as Hugh says, to have the old day east of them, and the new one west: because, when midnight comes round to them, with the new day in front of it and the old one behind it, one doesn't see exactly what happens. I must think it over."
So they had entered the house in the state I have described—Balbus puzzled, and Lambert buried in gloomy thought.
"Yes, m'm, Master is at home, m'm," said the stately old butler. (N.B.—It is only a butler of experience who can manage a series of three M's together, without any interjacent vowels.) "And the ole party is a-waiting for you in the libery."
"I don't like his calling your father an old party," Mad Mathesis whispered to her niece, as they crossed the hall. And Clara had only just time to whisper in reply "he meant the whole party," before they were ushered into the library, and the sight of the five solemn faces there assembled chilled her into silence.
Her father sat at the head of the table, and mutely signed to the ladies to take the two vacant chairs, one on each side of him. His three sons and Balbus completed the party. Writing materials had been arranged round the table, after the fashion of a ghostly banquet: the butler had evidently bestowed much thought on the grim device. Sheets of quarto paper, each flanked by a pen on one side and a pencil on the other, represented the plates—penwipers did duty for rolls of bread—while ink-bottles stood in the places usually occupied by wine-glasses. The pièce de resistance was a large green baize bag, which gave forth, as the old man restlessly lifted it from side to side, a charming jingle, as of innumerable golden guineas.
"Sister, daughter, sons—and Balbus—," the old man began, so nervously, that Balbus put in a gentle "Hear, hear!" while Hugh drummed on the table with his fists. This disconcerted the unpractised orator. "Sister—" he began again, then paused a moment, moved the bag to the other side, and went on with a rush, "I mean—this being—a critical occasion—more or less—being the year when one of my sons comes of age—" he paused again in some confusion, having evidently got into the middle of his speech sooner than he intended: but it was too late to go back. "Hear, hear!" cried Balbus. "Quite so," said the old gentleman, recovering his self-possession a little: "when first I began this annual
custom—my friend Balbus will correct me if I am wrong—" (Hugh whispered "with a strap!" but nobody heard him except Lambert, who only frowned and shook his head at him) "—this annual custom of giving each of my sons as many guineas as would represent his age—it was a critical time—so Balbus informed me—as the ages of two of you were together equal to that of the third—so on that occasion I made a speech——" He paused so long that Balbus thought it well to come to the rescue with the words "It was a most——" but the old man checked him with a warning look: "yes, made a speech," he repeated. "A few years after that, Balbus pointed out—I say pointed out—" ("Hear, hear"! cried Balbus. "Quite so," said the grateful old man.) "—that it was another critical occasion. The ages of two of you were together double that of the third. So I made another speech—another speech. And now again it's a critical occasion—so Balbus says—and I am making——" (Here Mad Mathesis pointedly referred to her watch) "all the haste I can!" the old man cried, with wonderful presence of mind. "Indeed, sister, I'm coming to the point now! The number of years that have passed since that first occasion is just two-thirds of the number of guineas I then gave you. Now, my boys, calculate your ages from the data, and you shall have the money!"
"But we know our ages!" cried Hugh.
"Silence, sir!" thundered the old man, rising to his full height (he was exactly five-foot five) in his indignation. "I say you must use the data only! You mustn't even assume which it is that comes of age!" He clutched the bag as he spoke, and with tottering steps (it was about as much as he could do to carry it) he left the room.
"And you shall have a similar cadeau," the old lady whispered to her niece, "when you've calculated that percentage!" And she followed her brother.
Nothing could exceed the solemnity with which the old couple had risen from the table, and yet was it—was it a grin with which the father turned away from his unhappy sons? Could it be—could it be a wink with which the aunt abandoned her despairing niece? And were those—were those sounds of suppressed chuckling which floated into the room, just before Balbus (who had followed them out) closed the door? Surely not: and yet the butler told the cook—but no, that was merely idle gossip, and I will not repeat it.
The shades of evening granted their unuttered petition, and "closed not o'er" them (for the butler brought in the lamp): the same obliging shades left them a "lonely bark" (the wail of a dog, in the back-yard, baying the moon) for "awhile": but neither "morn, alas," (nor any other epoch) seemed likely to "restore" them—to that peace of mind which had once been theirs ere ever these problems had swooped upon them, and crushed them with a load of unfathomable mystery!
"It's hardly fair," muttered Hugh, "to give us such a jumble as this to work out!"
"Fair?" Clara echoed, bitterly. "Well!"
And to all my readers I can but repeat the last words of gentle Clara—
Fare-well!
APPENDIX.
"A knot!" said Alice. "Oh, do let me help to undo it!"
ANSWERS TO KNOT I.
Problem.—"Two travellers spend from 3 o'clock till 9 in walking along a level road, up a hill, and home again: their pace on the level being 4 miles an hour, up hill 3, and down hill 6. Find distance walked: also (within half an hour) time of reaching top of hill."
Answer.—"24 miles: half-past 6."
Solution.—A level mile takes ¼ of an hour, up hill 1⁄3, down hill 1⁄6. Hence to go and return over the same mile, whether on the level or on the hill-side, takes ½ an hour. Hence in 6 hours they went 12 miles out and 12 back. If the 12 miles out had been nearly all level, they would have taken a little over 3 hours; if nearly all up hill, a little under 4. Hence 3½ hours must be within ½ an hour of the time taken in reaching the peak; thus, as they started at 3, they got there within ½ an hour of ½ past 6.
Twenty-seven answers have come in. Of these, 9 are right, 16 partially right, and 2 wrong. The 16 give the distance correctly, but they have failed to grasp the fact that the top of the hill might have been reached at any moment between 6 o'clock and 7.
The two wrong answers are from Gerty Vernon and A Nihilist. The former makes the distance "23 miles," while her revolutionary companion puts it at "27." Gerty Vernon says "they had to go 4 miles along the plain, and got to the foot of the hill at 4 o'clock." They might have done so, I grant; but you have no ground for saying they did so. "It was 7½ miles to the top of the hill, and they reached that at ¼ before 7 o'clock." Here you go wrong in your arithmetic, and I must, however reluctantly, bid you farewell. 7½ miles, at 3 miles an hour, would not require 2¾ hours. A Nihilist says "Let x denote the whole number of miles; y the number of hours to hill-top; ∴ 3y = number of miles to hill-top, and x-3y = number of miles on the other side." You bewilder me. The other side of what? "Of the hill," you say. But then, how did they get home again? However, to accommodate your views we will build a new hostelry at the foot of the hill on the opposite side, and also assume (what I grant you is possible, though it is not necessarily true) that there was no level road at all. Even then you go wrong.
You say
"y = 6 - (x - 3y)⁄6, ..... (i);
x⁄4½ = 6 ..... (ii)."
I grant you (i), but I deny (ii): it rests on the assumption that to go part of the time at 3 miles an hour, and the rest at 6 miles an hour, comes to the same result as going the whole time at 4½ miles an hour. But this would only be true if the "part" were an exact half, i.e., if they went up hill for 3 hours, and down hill for the other 3: which they certainly did not do.
The sixteen, who are partially right, are Agnes Bailey, F. K., Fifee, G. E. B., H. P., Kit, M. E. T., Mysie, A Mother's Son, Nairam, A Redruthian, A Socialist, Spear Maiden, T. B. C, Vis Inertiæ, and Yak. Of these, F. K., Fifee, T. B. C, and Vis Inertiæ do not attempt the second part at all. F. K. and H. P. give no working. The rest make particular assumptions, such as that there was no level road—that there were 6 miles of level road—and so on, all leading to particular times being fixed for reaching the hill-top. The most curious assumption is that of Agnes Bailey, who says "Let x = number of hours occupied in ascent; then x⁄2 = hours occupied in descent; and 4x⁄3 = hours occupied on the level." I suppose you were thinking of the relative rates, up hill and on the level; which we might express by saying that, if they went x miles up hill in a certain time, they would go 4x⁄3 miles on the level in the same time. You have, in fact, assumed that they took the same time on the level that they took in ascending the hill. Fifee assumes that, when the aged knight said they had gone "four miles in the hour" on the level, he meant that four miles was the distance gone, not merely the rate. This would have been—if Fifee will excuse the slang expression—a "sell," ill-suited to the dignity of the hero.
And now "descend, ye classic Nine!" who have solved the whole problem, and let me sing your praises. Your names are Blithe, E. W., L. B., A Marlborough Boy, O. V. L., Putney Walker, Rose, Sea Breeze, Simple Susan, and Money Spinner. (These last two I count as one, as they send a joint answer.) Rose and Simple Susan and Co. do not actually state that the hill-top was reached some time between 6 and 7, but, as they have clearly grasped the fact that a mile, ascended and descended, took the same time as two level miles, I mark them as "right." A Marlborough 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 equation. E. W. brings a charge of untruthfulness against the aged knight—a serious charge, for he was the very pink of chivalry! She says "According to the data given, the time at the summit affords no clue to the total distance. It does not enable us to state precisely to an inch how much level and how much hill there was on the road." "Fair damsel," the aged knight replies, "—if, as I surmise, thy initials denote Early Womanhood—bethink thee that the word 'enable' is thine, not mine. I did but ask the time of reaching the hill-top as my condition for further parley. If now thou wilt not grant that I am a truth-loving man, then will I affirm that those same initials denote Envenomed Wicke
dness!"
CLASS LIST.
I.
A Marlborough Boy.
Putney Walker.
II.
Blithe.
E. W.
L. B.
O. V. L.
Rose.
Sea Breeze.
{Simple Susan.
{Money-Spinner.
Blithe has made so ingenious an addition to the problem, and Simple Susan and Co. have solved it in such tuneful verse, that I record both their answers in full. I have altered a word or two in Blithe's—which I trust she will excuse; it did not seem quite clear as it stood.
"Yet stay," said the youth, as a gleam of inspiration lighted up the relaxing muscles of his quiescent features. "Stay. Methinks it matters little when we reached that summit, the crown of our toil. For in the space of time wherein we clambered up one mile and bounded down the same on our return, we could have trudged the twain on the level. We have plodded, then, four-and-twenty miles in these six mortal hours; for never a moment did we stop for catching of fleeting breath or for gazing on the scene around!"
Complete Works of Lewis Carroll Page 68