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diff --git a/29042-h/29042-h.htm b/29042-h/29042-h.htm new file mode 100644 index 0000000..b71075e --- /dev/null +++ b/29042-h/29042-h.htm @@ -0,0 +1,4610 @@ +<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" + "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> + +<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en" lang="en"> + <head> + <meta http-equiv="Content-Type" content="text/html;charset=iso-8859-1" /> + <meta http-equiv="Content-Style-Type" content="text/css" /> + <title> + The Project Gutenberg eBook of A Tangled Tale, by Lewis Carroll. + </title> + <style type="text/css"> + +body { + margin-left: 10%; + margin-right: 10%; +} + + h1,h2,h3,h4,h5,h6 { + text-align: center; /* all headings centered */ + clear: both; +} + +p { + margin-top: .75em; + text-align: justify; + margin-bottom: .75em; +} + +.fm2 {font-size: 125%; + text-align: center; + font-weight: bold; +} + +.fm3 {font-size: 100%; + text-align: center; + font-weight: bold; +} + +.fm4 {font-size: 90%; + text-align: center; + font-weight: bold; +} + +hr { + width: 33%; + margin-top: 2em; + margin-bottom: 2em; + margin-left: auto; + margin-right: auto; + clear: both; +} + +table {margin: auto; text-align: center; width: 35em;} +td.tdl {text-align: left; padding-right: .5em;} +td.tdr {text-align: right; padding-left: .5em;} +td.page {font-size: 90%;} + +.author {text-align: right; margin-right: 20%;} + +.pagenum { /* uncomment the next line for invisible page numbers */ + /* visibility: hidden; */ + position: absolute; + left: 92%; + font-size: smaller; + text-align: right; +} /* page numbers */ + + +.center {text-align: center;} + +.smcap {font-variant: small-caps;} + + +.transnote { background-color: #ADD8E6; color: inherit; margin: 2em 10% 1em 10%; font-size: 80%; padding: 0.5em 1em 0.5em 1em;} +.transnote p { text-align: left;} +a.correction {text-decoration: none; border-bottom: thin dotted red; color: inherit; background-color: inherit;} +a.correction:hover {text-decoration: none;} + +/* Poetry */ +.poem { + margin-left:10%; + margin-right:10%; + text-align: left; +} + +.poem br {display: none;} + +.poem .stanza {margin: 1em 0em 1em 0em;} + +.poem span.i0 { + display: block; + margin-left: 10em; + padding-left: 3em; + text-indent: -3em; +} + + .poem span.i1 {display: block; margin-left: 11em; padding-left: 3em; text-indent: -3em;} + .poem span.i11 {display: block; margin-left: 21em; padding-left: 3em; text-indent: -3em;} + + +.caption {font-weight: bold;} + +/* Images */ +.figcenter { + margin: auto; + text-align: center; +} + +.figright { + float: right; + clear: right; + margin-left: 1em; + margin-bottom: + 1em; + margin-top: 1em; + margin-right: 0; + padding: 0; + text-align: center; +} + +/* Footnotes */ +.footnotes {border: dashed 1px;} + +.footnote {margin-left: 10%; margin-right: 10%; font-size: 0.9em;} + +.footnote .label {position: absolute; right: 84%; text-align: right;} + +.fnanchor { + vertical-align: super; + font-size: .8em; + text-decoration: + none; +} + + </style> + </head> +<body> + + +<pre> + +The Project Gutenberg EBook of A Tangled Tale, by Lewis Carroll + +This eBook is for the use of anyone anywhere at no cost and with +almost no restrictions whatsoever. You may copy it, give it away or +re-use it under the terms of the Project Gutenberg License included +with this eBook or online at www.gutenberg.org + + +Title: A Tangled Tale + +Author: Lewis Carroll + +Illustrator: Arthur B. Frost + +Release Date: June 5, 2009 [EBook #29042] + +Language: English + +Character set encoding: ISO-8859-1 + +*** START OF THIS PROJECT GUTENBERG EBOOK A TANGLED TALE *** + + + + +Produced by Chris Curnow, Carla Foust, Lindy Walsh and the +Online Distributed Proofreading Team at https://www.pgdp.net. +Music transcribed by Linda Cantoni. + + + + + + +</pre> + + +<div class="transnote"> +<h3>Transcriber's note</h3> +<p>Minor punctuation errors have been changed without notice. Printer +errors have been changed, and they are indicated with +a <a class="correction" title="like this" href="#tnotes">mouse-hover</a> +and listed at the +<a href="#tnotes">end of this book</a>. All other +inconsistencies are as in the original. +</p> + +<p>On page <a href="#Page_68">68</a> a Midi file has been provided for the +song. To hear it, click on the [Listen] link.</p> +</div> + +<hr style="width: 65%;" /> +<p><span class='pagenum'><a name="Page_i" id="Page_i"> </a></span></p> +<div class="figcenter" style="width: 197px;"> +<img src="images/icover.jpg" width="197" height="300" alt="" title="" /> +</div> + +<p class="fm2">A TANGLED TALE</p> + +<hr style="width: 65%;" /> +<div class="figcenter" style="width: 260px;"> +<img src="images/i003.png" width="260" height="251" alt="" title="" /> +</div> + +<hr style="width: 65%;" /> +<p><span class='pagenum'><a name="Page_iii" id="Page_iii"> </a></span></p> +<div class="figcenter" style="width: 339px;"> +<img src="images/i005.png" width="339" height="500" alt=""AT A PACE OF SIX MILES IN THE HOUR." + +Frontispiece." title="" /> +<span class="caption">"AT A PACE OF SIX MILES IN THE HOUR." + +Frontispiece.</span> +</div> + + + +<hr style="width: 65%;" /> +<h1>A TANGLED TALE</h1> + +<p><br /><br /></p> +<p class="fm4">BY</p> +<p class="fm2">LEWIS CARROLL</p> + +<p><br /><br /></p> +<p class="fm3"><i>WITH SIX ILLUSTRATIONS</i></p> +<p class="fm4">BY</p> +<p class="fm2">ARTHUR B. FROST</p> + +<p><br /><br /></p> +<p class="fm4">Hoc meum tale quale est accipe.</p> + +<p><br /><br /></p> +<p class="fm3"><i>SECOND THOUSAND.</i></p> + +<p><br /><br /></p> +<p class="fm3">London</p> +<p class="fm2">MACMILLAN AND CO.<br /> +1885</p> + +<p><br /><br /></p> +<p class="fm4">[<i>All Rights Reserved</i>]</p> + + + +<hr style="width: 65%;" /> +<p class="fm3"><span class="smcap">Richard Clay & Sons</span>,<br /> +BREAD STREET HILL, LONDON, E.C.</p> +<p class="fm4"><i>And Bungay, Suffolk</i>.</p> + + + +<hr style="width: 65%;" /> +<h2><a name="To_My_Pupil" id="To_My_Pupil"></a>To My Pupil.</h2> + +<div class="poem"><div class="stanza"> +<span class="i0">Beloved pupil! Tamed by thee,<br /></span> +<span class="i1">Addish-, Subtrac-, Multiplica-tion,<br /></span> +<span class="i0">Division, Fractions, Rule of Three,<br /></span> +<span class="i1">Attest thy deft manipulation!<br /></span> +</div><div class="stanza"> +<span class="i0">Then onward! Let the voice of Fame<br /></span> +<span class="i1">From Age to Age repeat thy story,<br /></span> +<span class="i0">Till thou hast won thyself a name<br /></span> +<span class="i1">Exceeding even Euclid's glory!<br /></span> +</div></div> + +<hr style="width: 65%;" /> +<h2>PREFACE.<br /></h2> + +<p>This Tale originally appeared as a serial in <i>The Monthly Packet</i>, +beginning in April, 1880. The writer's intention was to embody in each +Knot (like the medicine so dexterously, but ineffectually, concealed in +the jam of our early childhood) one or more mathematical questions—in +Arithmetic, Algebra, or Geometry, as the case might be—for the +amusement, and possible edification, of the fair readers of that +Magazine.</p> + +<p class="author">L. C.</p> + +<p><span style="margin-left: 1.5em;"><i>October, 1885.</i></span></p> + + +<hr style="width: 65%;" /> +<h2>CONTENTS.</h2> + +<table summary="CONTENTS"> +<tr> +<td class="tdr">KNOT</td> +<td class="tdl"> </td> +<td class="tdr">PAGE</td> +</tr> +<tr> +<td class="tdr">I.</td> +<td class="tdl"><span class="smcap">Excelsior</span></td> +<td class="tdr"><a href="#Page_1">1</a></td> +</tr> +<tr> +<td class="tdr">II.</td> +<td class="tdl"><span class="smcap">Eligible Apartments</span></td> +<td class="tdr"><a href="#Page_4">4</a></td> +</tr> +<tr> +<td class="tdr">III.</td> +<td class="tdl"><span class="smcap">Mad Mathesis</span></td> +<td class="tdr"><a href="#Page_13">13</a></td> +</tr> +<tr> +<td class="tdr">IV.</td> +<td class="tdl"><span class="smcap">The Dead Reckoning</span></td> +<td class="tdr"><a href="#Page_19">19</a></td> +</tr> +<tr> +<td class="tdr">V.</td> +<td class="tdl"><span class="smcap">Oughts and Crosses</span></td> +<td class="tdr"><a href="#Page_27">27</a></td> +</tr> +<tr> +<td class="tdr">VI.</td> +<td class="tdl"><span class="smcap">Her Radiancy</span></td> +<td class="tdr"><a href="#Page_34">34</a></td> +</tr> +<tr> +<td class="tdr">VII.</td> +<td class="tdl"><span class="smcap">Petty Cash</span></td> +<td class="tdr"><a href="#Page_43">43</a></td> +</tr> +<tr> +<td class="tdr">VIII.</td> +<td class="tdl"><span class="smcap">De Omnibus Rebus</span></td> +<td class="tdr"><a href="#Page_52">52</a></td> +</tr> +<tr> +<td class="tdr">IX.</td> +<td class="tdl"><span class="smcap">A Serpent with Corners</span></td> +<td class="tdr"><a href="#Page_58">58</a></td> +</tr> +<tr> +<td class="tdr">X.</td> +<td class="tdl"><span class="smcap">Chelsea Buns</span></td> +<td class="tdr"><a href="#Page_66">66</a></td> +</tr> +</table> + +<table summary="CONTENTS"> +<tr> +<td class="tdl"><span class="smcap"> Answers to Knot I.</span></td> +<td class="tdl"> </td> +<td class="tdr"><a href="#Page_77">77</a></td> +</tr> +<tr> +<td class="tdl"><span class="smcap"> + " + " + II.</span></td> +<td class="tdr"> </td> +<td class="tdr"><a href="#Page_84">84</a></td> +</tr> +<tr> +<td class="tdl"><span class="smcap"> + " + " + III.</span></td> +<td class="tdr"> </td> +<td class="tdr"><a href="#Page_90">90</a></td> +</tr> +<tr> +<td class="tdl"><span class="smcap"> + " + " + IV.</span></td> +<td class="tdr"> </td> +<td class="tdr"><a href="#Page_96">96</a></td> +</tr> +<tr> +<td class="tdl"><span class="smcap"> + " + " + V.</span></td> +<td class="tdr"> </td> +<td class="tdr"><a href="#Page_102">102</a></td> +</tr> +<tr> +<td class="tdl"><span class="smcap"> + " + " + VI.</span></td> +<td class="tdr"> </td> +<td class="tdr"><a href="#Page_106">106</a></td> +</tr> +<tr> +<td class="tdl"><span class="smcap"> + " + " + VII.</span></td> +<td class="tdr"> </td> +<td class="tdr"><a href="#Page_112">112</a></td> +</tr> +<tr> +<td class="tdl"><span class="smcap"> + " + " + VIII.</span></td> +<td class="tdr"> </td> +<td class="tdr"><a href="#Page_132">132</a></td> +</tr> +<tr> +<td class="tdl"><span class="smcap"> + " + " + IX.</span></td> +<td class="tdr"> </td> +<td class="tdr"><a href="#Page_135">135</a></td> +</tr> +<tr> +<td class="tdl"><span class="smcap"> + " + " + X.</span></td> +<td class="tdr"> </td> +<td class="tdr"><a href="#Page_142">142</a></td> +</tr> +</table> + + +<hr style="width: 65%;" /> +<p><span class='pagenum'><a name="Page_1" id="Page_1">[1]</a></span></p> + + + + +<p class="fm2">A TANGLED TALE.</p> + + + + +<h2>KNOT I.</h2> + +<h3>EXCELSIOR.</h3> + +<p class="center">"Goblin, lead them up and down."</p> + +<p>The ruddy glow of sunset was already fading into the sombre shadows of +night, when two travellers might have been observed swiftly—at a pace +of six miles in the hour—descending the rugged side of a mountain; the +younger bounding from crag to crag with the agility of a fawn, while his +companion, whose aged limbs seemed ill at ease in the heavy chain armour +habitually worn by tourists in that district, toiled on painfully at his +side.</p> + +<p>As is always the case under such circumstances, the younger knight was +the first to break the silence.<span class='pagenum'><a name="Page_2" id="Page_2">[2]</a></span></p> + +<p>"A goodly pace, I trow!" he exclaimed. "We sped not thus in the ascent!"</p> + +<p>"Goodly, indeed!" the other echoed with a groan. "We clomb it but at +three miles in the hour."</p> + +<p>"And on the dead level our pace is——?" the younger suggested; for he +was weak in statistics, and left all such details to his aged companion.</p> + +<p>"Four miles in the hour," the other wearily replied. "Not an ounce +more," he added, with that love of metaphor so common in old age, "and +not a farthing less!"</p> + +<p>"'Twas three hours past high noon when we left our hostelry," the young +man said, musingly. "We shall scarce be back by supper-time. Perchance +mine host will roundly deny us all food!"</p> + +<p>"He will chide our tardy return," was the grave reply, "and such a +rebuke will be meet."</p> + +<p>"A brave conceit!" cried the other, with a merry laugh. "And should we +bid him bring us yet another course, I trow his answer will be tart!"</p> + +<p>"We shall but get our deserts," sighed the elder knight, who had never +seen a joke in his life, and was somewhat displeased at his companion's +untimely levity. "'Twill be nine of the clock," he<span class='pagenum'><a name="Page_3" id="Page_3">[3]</a></span> added in an +undertone, "by the time we regain our hostelry. Full many a mile shall +we have plodded this day!" +</p> +<p>"How many? How many?" cried the eager youth, ever athirst for knowledge.</p> + +<p>The old man was silent.</p> + +<p>"Tell me," he answered, after a moment's thought, "what time it was when +we stood together on yonder peak. Not exact to the minute!" he added +hastily, reading a protest in the young man's face. "An' thy guess be +within one poor half-hour of the mark, 'tis all I ask of thy mother's +son! Then will I tell thee, true to the last inch, how far we shall have +trudged betwixt three and nine of the clock."</p> + +<p>A groan was the young man's only reply; while his convulsed features and +the deep wrinkles that chased each other across his manly brow, revealed +the abyss of arithmetical agony into which one chance question had +plunged him.</p> + + +<hr style="width: 65%;" /> +<p><span class='pagenum'><a name="Page_4" id="Page_4">[4]</a></span></p> +<h2>KNOT II.</h2> + +<h3>ELIGIBLE APARTMENTS.</h3> + +<div class="poem"><div class="stanza"> +<span class="i0">"Straight down the crooked lane,<br /></span> +<span class="i0">And all round the square."<br /></span> +</div></div> + + +<p>"Let's ask Balbus about it," said Hugh.</p> + +<p>"All right," said Lambert.</p> + +<p>"<i>He</i> can guess it," said Hugh.</p> + +<p>"Rather," said Lambert.</p> + +<p>No more words were needed: the two brothers understood each other +perfectly.</p> + +<p><span class='pagenum'><a name="Page_5" id="Page_5">[5]</a></span></p> +<div class="figcenter" style="width: 290px;"> +<img src="images/i018.png" width="290" height="425" alt=""BALBUS WAS ASSISTING HIS MOTHER-IN-LAW TO CONVINCE THE +DRAGON."" title="" /> +<span class="caption">"BALBUS WAS ASSISTING HIS MOTHER-IN-LAW TO CONVINCE THE +DRAGON."</span> +</div> + +<p>Balbus was waiting for them at the hotel: the journey down had tired +him, he said: so his two pupils had been the round of the place, in +search of lodgings, without the old tutor who had been their inseparable +companion from their childhood. They had named him after the hero of +their Latin exercise-book, which overflowed with anecdotes of that +versatile genius—anecdotes whose vagueness<span class='pagenum'><a name="Page_6" id="Page_6">[6]</a></span> in detail was more than +compensated by their sensational brilliance. "Balbus has overcome all +his enemies" had been marked by their tutor, in the margin of the book, +"Successful Bravery." In this way he had tried to extract a moral from +every anecdote about Balbus—sometimes one of warning, as in "Balbus had +borrowed a healthy dragon," against which he had written "Rashness in +Speculation"—sometimes of encouragement, as in the words "Influence of +Sympathy in United Action," 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 word, such as "Prudence," which was all he +could extract from the touching record that "Balbus, having scorched the +tail of the dragon, went away." His pupils liked the short morals best, +as it left them more room for marginal illustrations, and in this +instance they required all the space they could get to exhibit the +rapidity of the hero's departure.</p> + +<p>Their report of the state of things was discouraging. That most +fashionable of watering-places, Little Mendip, was "chockfull" (as the +boys expressed it) from end to end. But in one Square they had seen no +less than four cards, in<span class='pagenum'><a name="Page_7" id="Page_7">[7]</a></span> different houses, all announcing in flaming +capitals "<span class="smcap">ELIGIBLE APARTMENTS</span>." "So there's plenty of choice, after all, +you see," said spokesman Hugh in conclusion.</p> + +<p>"That doesn't follow from the data," said Balbus, as he rose from the +easy chair, where he had been dozing over <i>The Little Mendip Gazette</i>. +"They may be all single rooms. However, we may as well see them. I shall +be glad to stretch my legs a bit."</p> + +<p>An unprejudiced bystander might have objected that the operation was +needless, and that this long, lank creature would have been all the +better with even shorter legs: but no such thought occurred to his +loving pupils. One on each side, they did their best to keep up with his +gigantic strides, while Hugh repeated the sentence in their father's +letter, just received from abroad, over which he and Lambert had been +puzzling. "He says a friend of his, the Governor of——<i>what</i> was that +name again, Lambert?" ("Kgovjni," said Lambert.) "Well, yes. The +Governor of——what-you-may-call-it——wants to give a <i>very</i> small +dinner-party, and he means to ask his father's brother-in-law, his +brother's father-in-law, his father-in-law's brother,<span class='pagenum'><a name="Page_8" id="Page_8">[8]</a></span> and his +brother-in-law's father: and we're to guess how many guests there will +be."</p> + +<p>There was an anxious pause. "<i>How</i> large did he say the pudding was to +be?" Balbus said at last. "Take its cubical contents, divide by the +cubical contents of what each man can eat, and the quotient——"</p> + +<p>"He didn't say anything about pudding," said Hugh, "—and here's the +Square," as they turned a corner and came into sight of the "eligible +apartments."</p> + +<p>"It <i>is</i> a Square!" was Balbus' first cry of delight, as he gazed around +him. "Beautiful! Beau-ti-ful! Equilateral! <i>And</i> rectangular!"</p> + +<p>The boys looked round with less enthusiasm. "Number nine is the first +with a card," said prosaic Lambert; but Balbus would not so soon awake +from his dream of beauty.</p> + +<p>"See, boys!" he cried. "Twenty doors on a side! What symmetry! Each side +divided into twenty-one equal parts! It's delicious!"</p> + +<p>"Shall I knock, or ring?" said Hugh, looking in some perplexity at a +square brass plate which bore the simple inscription "<span class="smcap">RING ALSO</span>."</p> + +<p>"Both," said Balbus. "That's an Ellipsis,<span class='pagenum'><a name="Page_9" id="Page_9">[9]</a></span> my boy. Did you never see an +Ellipsis before?"</p> + +<p>"I couldn't hardly read it," said Hugh, evasively. "It's no good having +an Ellipsis, if they don't keep it clean."</p> + +<p>"Which there is <i>one</i> room, gentlemen," said the smiling landlady. "And +a sweet room too! As snug a little back-room——"</p> + +<p>"We will see it," said Balbus gloomily, as they followed her in. "I knew +how it would be! One room in each house! No view, I suppose?"</p> + +<p>"Which indeed there <i>is</i>, gentlemen!" the landlady indignantly +protested, as she drew up the blind, and indicated the back garden.</p> + +<p>"Cabbages, I perceive," said Balbus. "Well, they're green, at any rate."</p> + +<p>"Which the greens at the shops," their hostess explained, "are by no +means dependable upon. Here you has them on the premises, <i>and</i> of the +best."</p> + +<p>"Does the window open?" was always Balbus' first question in testing a +lodging: and "Does the chimney smoke?" his second. Satisfied on all +points, he secured the refusal of the room, and they moved on to Number +Twenty-five.</p> + +<p>This landlady was grave and stern. "I've<span class='pagenum'><a name="Page_10" id="Page_10">[10]</a></span> nobbut one room left," she +told them: "and it gives on the back-gyardin."</p> + +<p>"But there are cabbages?" Balbus suggested.</p> + +<p>The landlady visibly relented. "There is, sir," she said: "and good +ones, though I say it as shouldn't. We can't rely on the shops for +greens. So we grows them ourselves."</p> + +<p>"A singular advantage," said Balbus: and, after the usual questions, +they went on to Fifty-two.</p> + +<p>"And I'd gladly accommodate you all, if I could," was the greeting that +met them. "We are but mortal," ("Irrelevant!" muttered Balbus) "and I've +let all my rooms but one."</p> + +<p>"Which one is a back-room, I perceive," said Balbus: "and looking out +on—on cabbages, I presume?"</p> + +<p>"Yes, indeed, sir!" said their hostess. "Whatever <i>other</i> folks may do, +<i>we</i> grows our own. For the shops——"</p> + +<p>"An excellent arrangement!" Balbus interrupted. "Then one can really +depend on their being good. Does the window open?"</p> + +<p>The usual questions were answered satisfactorily: but this time Hugh +added one of his own invention—"Does the cat scratch?"<span class='pagenum'><a name="Page_11" id="Page_11">[11]</a></span></p> + +<p>The landlady looked round suspiciously, as if to make sure the cat was +not listening, "I will not deceive you, gentlemen," she said. "It <i>do</i> +scratch, but not without you pulls its whiskers! It'll never do it," she +repeated slowly, with a visible effort to recall the exact words of some +written agreement between herself and the cat, "without you pulls its +whiskers!"</p> + +<p>"Much may be excused in a cat so treated," said Balbus, as they left the +house and crossed to Number Seventy-three, leaving the landlady +curtseying on the doorstep, and still murmuring to herself her parting +words, as if they were a form of blessing, "—— not without you pulls +its whiskers!"</p> + +<p>At Number Seventy-three they found only a small shy girl to show the +house, who said "yes'm" in answer to all questions.</p> + +<p>"The usual room," said Balbus, as they marched in: "the usual +back-garden, the usual cabbages. I suppose you can't get them good at +the shops?"</p> + +<p>"Yes'm," said the girl.</p> + +<p>"Well, you may tell your mistress we will take the room, and that her +plan of growing her own cabbages is simply <i>admirable</i>!"<span class='pagenum'><a name="Page_12" id="Page_12">[12]</a></span></p> + +<p>"Yes'm," said the girl, as she showed them out.</p> + +<p>"One day-room and three bed-rooms," said Balbus, as they returned to the +hotel. "We will take as our day-room the one that gives us the least +walking to do to get to it."</p> + +<p>"Must we walk from door to door, and count the steps?" said Lambert.</p> + +<p>"No, no! Figure it out, my boys, figure it out!" Balbus gaily exclaimed, +as he put pens, ink, and paper before his hapless pupils, and left the +room.</p> + +<p>"I say! It'll be a job!" said Hugh.</p> + +<p>"Rather!" said Lambert.</p> + + + +<hr style="width: 65%;" /> +<p><span class='pagenum'><a name="Page_13" id="Page_13">[13]</a></span></p> +<h2><a name="KNOT_III" id="KNOT_III"></a>KNOT III.</h2> + +<h3>MAD MATHESIS.</h3> + +<p class="center">"I waited for the train."</p> + +<p>"Well, they call me so because I <i>am</i> a little mad, I suppose," she +said, good-humouredly, in answer to Clara's cautiously-worded question +as to how she came by so strange a nick-name. "You see, I never do what +sane people are expected to do now-a-days. I never wear long trains, +(talking of trains, that's the Charing Cross Metropolitan Station—I've +something to tell you about <i>that</i>), and I never play lawn-tennis. I +can't cook an omelette. I can't even set a broken limb! <i>There's</i> an +ignoramus for you!"</p> + +<p>Clara was her niece, and full twenty years her junior; in fact, she was +still attending a High School—an institution of which Mad Mathesis +spoke with undisguised aversion. "Let a woman<span class='pagenum'><a name="Page_14" id="Page_14">[14]</a></span> be meek and lowly!" she +would say. "None of your High Schools for me!" But it was vacation-time +just now, and Clara was her guest, and Mad Mathesis was showing her the +sights of that Eighth Wonder of the world—London.</p> + +<p>"The Charing Cross Metropolitan Station!" she resumed, waving her hand +towards the entrance as if she were introducing her niece to a friend. +"The Bayswater and Birmingham Extension is just completed, and the +trains now run round and round continuously—skirting the border of +Wales, just touching at York, and so round by the east coast back to +London. The way the trains run is <i>most</i> peculiar. The westerly ones go +round in two hours; the easterly ones take three; but they always manage +to start two trains from here, opposite ways, punctually every +quarter-of-an-hour."</p> + +<p>"They part to meet again," said Clara, her eyes filling with tears at +the romantic thought.</p> + +<p>"No need to cry about it!" her aunt grimly remarked. "They don't meet on +the same line of rails, you know. Talking of meeting, an idea strikes +me!" she added, changing the subject with her usual abruptness. "Let's +go opposite ways<span class='pagenum'><a name="Page_15" id="Page_15">[15]</a></span> round, and see which can meet most trains. No need for +a chaperon—ladies' saloon, you know. You shall go whichever way you +like, and we'll have a bet about it!"</p> + +<p>"I never make bets," Clara said very gravely. "Our excellent preceptress +has often warned us——"</p> + +<p>"You'd be none the worse if you did!" Mad Mathesis interrupted. "In +fact, you'd be the better, I'm certain!"</p> + +<p>"Neither does our excellent preceptress approve of puns," said Clara. +"But we'll have a match, if you like. Let me choose my train," she added +after a brief mental calculation, "and I'll engage to meet exactly half +as many again as you do."</p> + +<p>"Not if you count fair," Mad Mathesis bluntly interrupted. "Remember, we +only count the trains we meet <i>on the way</i>. You mustn't count the one +that starts as you start, nor the one that arrives as you arrive."</p> + +<p>"That will only make the difference of <i>one</i> train," said Clara, as they +turned and entered the station. "But I never travelled alone before. +There'll be no one to help me to alight. However, I don't mind. Let's +have a match."<span class='pagenum'><a name="Page_16" id="Page_16">[16]</a></span></p> + +<p>A ragged little boy overheard her remark, and came running after her. +"Buy a box of cigar-lights, Miss!" he pleaded, pulling her shawl to +attract her attention. Clara stopped to explain.</p> + +<p>"I never smoke cigars," she said in a meekly apologetic tone. "Our +excellent preceptress——," but Mad Mathesis impatiently hurried her on, +and the little boy was left gazing after her with round eyes of +amazement.</p> + +<p>The two ladies bought their tickets and moved slowly down the central +platform, Mad Mathesis prattling on as usual—Clara silent, anxiously +reconsidering the calculation on which she rested her hopes of winning +the match.</p> + +<p>"Mind where you go, dear!" cried her aunt, checking her just in time. +"One step more, and you'd have been in that pail of cold water!"</p> + +<p>"I know, I know," Clara said, dreamily. "The pale, the cold, and the +moony——"</p> + +<p>"Take your places on the spring-boards!" shouted a porter.</p> + +<p>"What are <i>they</i> for!" Clara asked in a terrified whisper.</p> + +<p>"Merely to help us into the trains." The elder lady spoke with the +nonchalance of one quite used<span class='pagenum'><a name="Page_17" id="Page_17">[17]</a></span> to the process. "Very few people can get +into a carriage without help in less than three seconds, and the trains +only stop for one second." At this moment the whistle was heard, and two +trains rushed into the station. A moment's pause, and they were gone +again; but in that brief interval several hundred passengers had been +shot into them, each flying straight to his place with the accuracy of a +Minie bullet—while an equal number were showered out upon the +side-platforms.</p> + +<p>Three hours had passed away, and the two friends met again on the +Charing Cross platform, and eagerly compared notes. Then Clara turned +away with a sigh. To young impulsive hearts, like hers, disappointment +is always a bitter pill. Mad Mathesis followed her, full of kindly +sympathy.</p> + +<p>"Try again, my love!" she said, cheerily. "Let us vary the experiment. +We will start as we did before, but not to begin counting till our +trains meet. When we see each other, we will say 'One!' and so count on +till we come here again."</p> + +<p>Clara brightened up. "I shall win <i>that</i>," she exclaimed eagerly, "if I +may choose my train!"</p> + +<p>Another shriek of engine whistles, another upheaving of spring-boards, +another living avalanche<span class='pagenum'><a name="Page_18" id="Page_18">[18]</a></span> plunging into two trains as they flashed by: +and the travellers were off again.</p> + +<p>Each gazed eagerly from her carriage window, holding up her handkerchief +as a signal to her friend. A rush and a roar. Two trains shot past each +other in a tunnel, and two travellers leaned back in their corners with +a sigh—or rather with <i>two</i> sighs—of relief. "One!" Clara murmured to +herself. "Won! It's a word of good omen. <i>This</i> time, at any rate, the +victory will be mine!"</p> + +<p>But <i>was</i> it?</p> + + + +<hr style="width: 65%;" /> +<p><span class='pagenum'><a name="Page_19" id="Page_19">[19]</a></span></p> +<h2><a name="KNOT_IV" id="KNOT_IV"></a>KNOT IV.</h2> + +<h3>THE DEAD RECKONING.</h3> + +<p class="center">"I did dream of money-bags to-night."</p> + + +<p>Noonday on the open sea within a few degrees of the Equator is apt to be +oppressively warm; and our two travellers were now airily clad in suits +of dazzling white linen, having laid aside the chain-armour which they +had found not only endurable in the cold mountain air they had lately +been breathing, but a necessary precaution against the daggers of the +banditti who infested the heights. Their holiday-trip was over, and they +were now on their way home, in the monthly packet which plied between +the two great ports of the island they had been exploring.</p> + +<p>Along with their armour, the tourists had laid aside the antiquated +speech it had pleased them to affect while in knightly disguise, and +had<span class='pagenum'><a name="Page_20" id="Page_20">[20]</a></span> returned to the ordinary style of two country gentlemen of the +Twentieth Century.</p> + +<p>Stretched on a pile of cushions, under the shade of a huge umbrella, +they were lazily watching some native fishermen, who had come on board +at the last landing-place, each carrying over his shoulder a small but +heavy sack. A large weighing-machine, that had been used for cargo at +the last port, stood on the deck; and round this the fishermen had +gathered, and, with much unintelligible jabber, seemed to be weighing +their sacks.</p> + +<p>"More like sparrows in a tree than human talk, isn't it?" the elder +tourist remarked to his son, who smiled feebly, but would not exert +himself so far as to speak. The old man tried another listener.</p> + +<p>"What have they got in those sacks, Captain?" he inquired, as that great +being passed them in his never ending parade to and fro on the deck.</p> + +<p>The Captain paused in his march, and towered over the travellers—tall, +grave, and serenely self-satisfied.</p> + +<p>"Fishermen," he explained, "are often passengers in My ship. These five +are from Mhruxi—the<span class='pagenum'><a name="Page_21" id="Page_21">[21]</a></span> place we last touched at—and that's the way they +carry their money. The money of this island is heavy, gentlemen, but it +costs little, as you may guess. We buy it from them by weight—about +five shillings a pound. I fancy a ten pound-note would buy all those +sacks."</p> + +<p>By this time the old man had closed his eyes—in order, no doubt, to +concentrate his thoughts on these interesting facts; but the Captain +failed to realise his motive, and with a grunt resumed his monotonous +march.</p> + +<p>Meanwhile the fishermen were getting so noisy over the weighing-machine +that one of the sailors took the precaution of carrying off all the +weights, leaving them to amuse themselves with such substitutes in the +form of winch-handles, belaying-pins, &c., as they could find. This +brought their excitement to a speedy end: they carefully hid their sacks +in the folds of the jib that lay on the deck near the tourists, and +strolled away.</p> + +<p>When next the Captain's heavy footfall passed, the younger man roused +himself to speak.</p> + +<p>"<i>What</i> did you call the place those fellows came from, Captain?" he +asked.</p> + +<p>"Mhruxi, sir."<span class='pagenum'><a name="Page_22" id="Page_22">[22]</a></span></p> + +<p>"And the one we are bound for?"</p> + +<p>The Captain took a long breath, plunged into the word, and came out of +it nobly. "They call it Kgovjni, sir."</p> + +<p>"K—I give it up!" the young man faintly said.</p> + +<p>He stretched out his hand for a glass of iced water which the +compassionate steward had brought him a minute ago, and had set down, +unluckily, just outside the shadow of the umbrella. It was scalding hot, +and he decided not to drink it. The effort of making this resolution, +coming close on the fatiguing conversation he had just gone through, was +too much for him: he sank back among the cushions in silence.</p> + +<p>His father courteously tried to make amends for his <i>nonchalance</i>.</p> + +<p>"Whereabouts are we now, Captain?" said he, "Have you any idea?"</p> + +<p>The Captain cast a pitying look on the ignorant landsman. "I could tell +you <i>that</i>, sir," he said, in a tone of lofty condescension, "to an +inch!"</p> + +<p>"You don't say so!" the old man remarked, in a tone of languid surprise.</p> + +<p>"And mean so," persisted the Captain. "Why, what do you suppose would +become of My<span class='pagenum'><a name="Page_23" id="Page_23">[23]</a></span> ship, if I were to lose My Longitude and My Latitude? +Could <i>you</i> make anything of My Dead Reckoning?"</p> + +<p>"Nobody could, I'm sure!" the other heartily rejoined.</p> + +<p>But he had overdone it.</p> + +<p>"It's <i>perfectly</i> intelligible," the Captain said, in an offended tone, +"to any one that understands such things." With these words he moved +away, and began giving orders to the men, who were preparing to hoist +the jib.</p> + +<p>Our tourists watched the operation with such interest that neither of +them remembered the five money-bags, which in another moment, as the +wind filled out the jib, were whirled overboard and fell heavily into +the sea.</p> + +<p>But the poor fishermen had not so easily forgotten their property. In a +moment they had rushed to the spot, and stood uttering cries of fury, +and pointing, now to the sea, and now to the sailors who had caused the +disaster.</p> + +<p>The old man explained it to the Captain.</p> + +<p>"Let us make it up among us," he added in conclusion. "Ten pounds will +do it, I think you said?"<span class='pagenum'><a name="Page_24" id="Page_24">[24]</a></span></p> + +<div class="figcenter" style="width: 336px;"> +<img src="images/i037.png" width="336" height="375" alt="" title="" /> +</div> + +<p>But the Captain put aside the suggestion with a wave of the hand.</p> + +<p>"No, sir!" he said, in his grandest manner. "You will excuse Me, I am +sure; but these are My passengers. The accident has happened on board My +ship, and under My orders. It is for Me to<span class='pagenum'><a name="Page_25" id="Page_25">[25]</a></span> make compensation." He +turned to the angry fishermen. "Come here, my men!" he said, in the +Mhruxian dialect. "Tell me the weight of each sack. I saw you weighing +them just now."</p> + +<p>Then ensued a perfect Babel of noise, as the five natives explained, all +screaming together, how the sailors had carried off the weights, and +they had done what they could with whatever came handy.</p> + +<p>Two iron belaying-pins, three blocks, six holystones, four +winch-handles, and a large hammer, were now carefully weighed, the +Captain superintending and noting the results. But the matter did not +seem to be settled, even then: an angry discussion followed, in which +the sailors and the five natives all joined: and at last the Captain +approached our tourists with a disconcerted look, which he tried to +conceal under a laugh.</p> + +<p>"It's an absurd difficulty," he said. "Perhaps one of you gentlemen can +suggest something. It seems they weighed the sacks two at a time!"</p> + +<p>"If they didn't have five separate weighings, of course you can't value +them separately," the youth hastily decided.<span class='pagenum'><a name="Page_26" id="Page_26">[26]</a></span></p> + +<p>"Let's hear all about it," was the old man's more cautious remark.</p> + +<p>"They <i>did</i> have five separate weighings," the Captain said, "but—Well, +it beats <i>me</i> entirely!" he added, in a sudden burst of candour. "Here's +the result. First and second sack weighed twelve pounds; second and +third, thirteen and a half; third and fourth, eleven and a half; fourth +and fifth, eight: and then they say they had only the large hammer left, +and it took <i>three</i> sacks to weigh it down—that's the first, third and +fifth—and <i>they</i> weighed sixteen pounds. There, gentlemen! Did you ever +hear anything like <i>that</i>?"</p> + +<p>The old man muttered under his breath "If only my sister were here!" and +looked helplessly at his son. His son looked at the five natives. The +five natives looked at the Captain. The Captain looked at nobody: his +eyes were cast down, and he seemed to be saying softly to himself +"Contemplate one another, gentlemen, if such be your good pleasure. <i>I</i> +contemplate <i>Myself</i>!"</p> + + + +<hr style="width: 65%;" /> +<p><span class='pagenum'><a name="Page_27" id="Page_27">[27]</a></span></p> +<h2><a name="KNOT_V" id="KNOT_V"></a>KNOT V.</h2> + +<h3>OUGHTS AND CROSSES.</h3> + +<p class="center">"Look here, upon this picture, and on this."</p> + + +<p>"And what made you choose the first train, Goosey?" said Mad Mathesis, +as they got into the cab. "Couldn't you count better than <i>that</i>?"</p> + +<p>"I took an extreme case," was the tearful reply. "Our excellent +preceptress always says 'When in doubt, my dears, take an extreme case.' +And I <i>was</i> in doubt."</p> + +<p>"Does it always succeed?" her aunt enquired.</p> + +<p>Clara sighed. "Not <i>always</i>," she reluctantly admitted. "And I can't +make out why. One day she was telling the little girls—they make such a +noise at tea, you know—'The more noise you make, the less jam you will +have, and <i>vice versâ</i>.' And I thought they wouldn't know what '<i>vice +versâ</i>' meant: so I explained it to them. I<span class='pagenum'><a name="Page_28" id="Page_28">[28]</a></span> said 'If you make an +infinite noise, you'll get no jam: and if you make no noise, you'll get +an infinite lot of jam.' But our excellent preceptress said that wasn't +a good instance. <i>Why</i> wasn't it?" she added plaintively.</p> + +<p>Her aunt evaded the question. "One sees certain objections to it," she +said. "But how did you work it with the Metropolitan trains? None of +them go infinitely fast, I believe."</p> + +<p>"I called them hares and tortoises," Clara said—a little timidly, for +she dreaded being laughed at. "And I thought there couldn't be so many +hares as tortoises on the Line: so I took an extreme case—one hare and +an infinite number of tortoises."</p> + +<p>"An extreme case, indeed," her aunt remarked with admirable gravity: +"and a most dangerous state of things!"</p> + +<p>"And I thought, if I went with a tortoise, there would be only <i>one</i> +hare to meet: but if I went with the hare—you know there were <i>crowds</i> +of tortoises!"</p> + +<p>"It wasn't a bad idea," said the elder lady, as they left the cab, at +the entrance of Burlington House. "You shall have another chance to-day. +We'll have a match in marking pictures."<span class='pagenum'><a name="Page_29" id="Page_29">[29]</a></span></p> + +<p>Clara brightened up. "I should like to try again, very much," she said. +"I'll take more care this time. How are we to play?"</p> + +<p>To this question Mad Mathesis made no reply: she was busy drawing lines +down the margins of the catalogue. "See," she said after a minute, "I've +drawn three columns against the names of the pictures in the long room, +and I want you to fill them with oughts and crosses—crosses for good +marks and oughts for bad. The first column is for choice of subject, the +second for arrangement, the third for colouring. And these are the +conditions of the match. You must give three crosses to two or three +pictures. You must give two crosses to four or five——"</p> + +<p>"Do you mean <i>only</i> two crosses?" said Clara. "Or may I count the +three-cross pictures among the two-cross pictures?"</p> + +<p>"Of course you may," said her aunt. "Any one, that has <i>three</i> eyes, may +be said to have <i>two</i> eyes, I suppose?"</p> + +<p>Clara followed her aunt's dreamy gaze across the crowded gallery, +half-dreading to find that there was a three-eyed person in sight.</p> + +<p>"And you must give one cross to nine or ten."<span class='pagenum'><a name="Page_30" id="Page_30">[30]</a></span></p> + +<p>"And which wins the match?" Clara asked, as she carefully entered these +conditions on a blank leaf in her catalogue.</p> + +<p>"Whichever marks fewest pictures."</p> + +<p>"But suppose we marked the same number?"</p> + +<p>"Then whichever uses most marks."</p> + +<p>Clara considered. "I don't think it's much of a match," she said. "I +shall mark nine pictures, and give three crosses to three of them, two +crosses to two more, and one cross each to all the rest."</p> + +<p>"Will you, indeed?" said her aunt. "Wait till you've heard all the +conditions, my impetuous child. You must give three oughts to one or two +pictures, two oughts to three or four, and one ought to eight or nine. I +don't want you to be <i>too</i> hard on the R.A.'s."</p> + +<p>Clara quite gasped as she wrote down all these fresh conditions. "It's a +great deal worse than Circulating Decimals!" she said. "But I'm +determined to win, all the same!"</p> + +<p>Her aunt smiled grimly. "We can begin <i>here</i>," she said, as they paused +before a gigantic picture, which the catalogue informed them was the +"Portrait of Lieutenant Brown, mounted on his favorite elephant."<span class='pagenum'><a name="Page_31" id="Page_31">[31]</a></span></p> + +<p>"He looks awfully conceited!" said Clara. "I don't think he was the +elephant's favorite Lieutenant. What a hideous picture it is! And it +takes up room enough for twenty!"</p> + +<p>"Mind what you say, my dear!" her aunt interposed. "It's by an R.A.!"</p> + +<p>But Clara was quite reckless. "I don't care who it's by!" she cried. +"And I shall give it three bad marks!"</p> + +<p>Aunt and niece soon drifted away from each other in the crowd, and for +the next half-hour Clara was hard at work, putting in marks and rubbing +them out again, and hunting up and down for suitable pictures. This she +found the hardest part of all. "I <i>can't</i> find the one I want!" she +exclaimed at last, almost crying with vexation.</p> + +<p>"What is it you want to find, my dear?" The voice was strange to Clara, +but so sweet and gentle that she felt attracted to the owner of it, even +before she had seen her; and when she turned, and met the smiling looks +of two little old ladies, whose round dimpled faces, exactly alike, +seemed never to have known a care, it was as much as she could do—as +she confessed to Aunt Mattie afterwards—to keep herself from hugging +them both.<span class='pagenum'><a name="Page_32" id="Page_32">[32]</a></span></p> + +<p>"I was looking for a picture," she said, "that has a good subject—and +that's well arranged—but badly coloured."</p> + +<p>The little old ladies glanced at each other in some alarm. "Calm +yourself, my dear," said the one who had spoken first, "and try to +remember which it was. What <i>was</i> the subject?"</p> + +<p>"Was it an elephant, for instance?" the other sister suggested. They +were still in sight of Lieutenant Brown.</p> + +<p>"I don't know, indeed!" Clara impetuously replied. "You know it doesn't +matter a bit what the subject <i>is</i>, so long as it's a good one!"</p> + +<p>Once more the sisters exchanged looks of alarm, and one of them +whispered something to the other, of which Clara caught only the one +word "mad."</p> + +<p>"They mean Aunt Mattie, of course," she said to herself—fancying, in +her innocence, that London was like her native town, where everybody +knew everybody else. "If you mean my aunt," she added aloud, "she's +<i>there</i>—just three pictures beyond Lieutenant Brown."</p> + +<p>"Ah, well! Then you'd better go to her, my dear!" her new friend said, +soothingly. "<i>She'll</i> find you the picture you want. Good-bye, dear!"<span class='pagenum'><a name="Page_33" id="Page_33">[33]</a></span></p> + +<p>"Good-bye, dear!" echoed the other sister, "Mind you don't lose sight of +your aunt!" And the pair trotted off into another room, leaving Clara +rather perplexed at their manner.</p> + +<p>"They're real darlings!" she soliloquised. "I wonder why they pity me +so!" And she wandered on, murmuring to herself "It must have two good +marks, and——"</p> + + + +<hr style="width: 65%;" /> +<p><span class='pagenum'><a name="Page_34" id="Page_34">[34]</a></span></p> +<h2><a name="KNOT_VI" id="KNOT_VI"></a>KNOT VI.</h2> + +<h3>HER RADIANCY.</h3> + +<div class="poem"><div class="stanza"> +<span class="i0">"One piecee thing that my have got,<br /></span> +<span class="i1">Maskee<a name="FNanchor_A_1" id="FNanchor_A_1"></a><a href="#Footnote_A_1" class="fnanchor">[A]</a> that thing my no can do.<br /></span> +<span class="i0">You talkee you no sabey what?<br /></span> +<span class="i11">Bamboo."<br /></span> +</div></div> + + +<p>They landed, and were at once conducted to the Palace. About half way +they were met by the Governor, who welcomed them in English—a great +relief to our travellers, whose guide could speak nothing but Kgovjnian.</p> + +<p>"I don't half like the way they grin at us as we go by!" the old man +whispered to his son. "And why do they say 'Bamboo!' so often?"</p> + +<p>"It alludes to a local custom," replied the Governor, who had overheard +the question. "Such persons as happen in any way to displease Her +Radiancy are usually beaten with rods."<span class='pagenum'><a name="Page_35" id="Page_35">[35]</a></span></p> + +<div class="figcenter" style="width: 266px;"> +<img src="images/i048.png" width="266" height="400" alt=""WHY DO THEY SAY 'BAMBOO!' SO OFTEN?"" title="" /> +<span class="caption">"WHY DO THEY SAY 'BAMBOO!' SO OFTEN?"</span> +</div><p><span class='pagenum'><a name="Page_36" id="Page_36">[36]</a></span></p> + +<p>The old man shuddered. "A most objectional local custom!" he remarked +with strong emphasis. "I wish we had never landed! Did you notice that +black fellow, Norman, opening his great mouth at us? I verily believe he +would like to eat us!"</p> + +<p>Norman appealed to the Governor, who was walking at his other side. "Do +they often eat distinguished strangers here?" he said, in as indifferent +a tone as he could assume.</p> + +<p>"Not often—not ever!" was the welcome reply. "They are not good for it. +Pigs we eat, for they are fat. This old man is thin."</p> + +<p>"And thankful to be so!" muttered the elder traveller. "Beaten we shall +be without a doubt. It's a comfort to know it won't be Beaten without +the B! My dear boy, just look at the peacocks!"</p> + +<p>They were now walking between two unbroken lines of those gorgeous +birds, each held in check, by means of a golden collar and chain, by a +black slave, who stood well behind, so as not to interrupt the view of +the glittering tail, with its network of rustling feathers and its +hundred eyes.</p> + +<p>The Governor smiled proudly. "In your honour," he said, "Her Radiancy +has ordered up ten thousand additional peacocks. She will, no doubt,<span class='pagenum'><a name="Page_37" id="Page_37">[37]</a></span> +decorate you, before you go, with the usual Star and Feathers."</p> + +<p>"It'll be Star without the S!" faltered one of his hearers.</p> + +<p>"Come, come! Don't lose heart!" said the other. "All this is full of +charm for me."</p> + +<p>"You are young, Norman," sighed his father; "young and light-hearted. +For me, it is Charm without the C."</p> + +<p>"The old one is sad," the Governor remarked with some anxiety. "He has, +without doubt, effected some fearful crime?"</p> + +<p>"But I haven't!" the poor old gentleman hastily exclaimed. "Tell him I +haven't, Norman!"</p> + +<p>"He has not, as yet," Norman gently explained. And the Governor +repeated, in a satisfied tone, "Not as yet."</p> + +<p>"Yours is a wondrous country!" the Governor resumed, after a pause. "Now +here is a letter from a friend of mine, a merchant, in London. He and +his brother went there a year ago, with a thousand pounds apiece; and on +New-Year's-day they had sixty thousand pounds between them!"</p> + +<p>"How did they do it?" Norman eagerly exclaimed. Even the elder traveller +looked excited.<span class='pagenum'><a name="Page_38" id="Page_38">[38]</a></span></p> + +<p>The Governor handed him the open letter. "Anybody can do it, when once +they know how," so ran this oracular document. "We borrowed nought: we +stole nought. We began the year with only a thousand pounds apiece: and +last New-Year's-day we had sixty thousand pounds between us—sixty +thousand golden sovereigns!"</p> + +<p>Norman looked grave and thoughtful as he handed back the letter. His +father hazarded one guess. "Was it by gambling?"</p> + +<p>"A Kgovjnian never gambles," said the Governor gravely, as he ushered +them through the palace gates. They followed him in silence down a long +passage, and soon found themselves in a lofty hall, lined entirely with +peacocks' feathers. In the centre was a pile of crimson cushions, which +almost concealed the figure of Her Radiancy—a plump little damsel, in a +robe of green satin dotted with silver stars, whose pale round face lit +up for a moment with a half-smile as the travellers bowed before her, +and then relapsed into the exact expression of a wax doll, while she +languidly murmured a word or two in the Kgovjnian dialect.</p> + +<p>The Governor interpreted. "Her Radiancy welcomes you. She notes the +Impenetrable Placidity<span class='pagenum'><a name="Page_39" id="Page_39">[39]</a></span> of the old one, and the Imperceptible Acuteness +of the youth."</p> + +<p>Here the little potentate clapped her hands, and a troop of slaves +instantly appeared, carrying trays of coffee and sweetmeats, which they +offered to the guests, who had, at a signal from the Governor, seated +themselves on the carpet.</p> + +<p>"Sugar-plums!" muttered the old man. "One might as well be at a +confectioner's! Ask for a penny bun, Norman!"</p> + +<p>"Not so loud!" his son whispered. "Say something complimentary!" For the +Governor was evidently expecting a speech.</p> + +<p>"We thank Her Exalted Potency," the old man timidly began. "We bask in +the light of her smile, which——"</p> + +<p>"The words of old men are weak!" the Governor interrupted angrily. "Let +the youth speak!"</p> + +<p>"Tell her," cried Norman, in a wild burst of eloquence, "that, like two +grasshoppers in a volcano, we are shrivelled up in the presence of Her +Spangled Vehemence!"</p> + +<p>"It is well," said the Governor, and translated this into Kgovjnian. "I +am now to tell you," he proceeded, "what Her Radiancy requires of you<span class='pagenum'><a name="Page_40" id="Page_40">[40]</a></span> +before you go. The yearly competition for the post of Imperial +Scarf-maker is just ended; you are the judges. You will take account of +the rate of work, the lightness of the scarves, and their warmth. +Usually the competitors differ in one point only. Thus, last year, Fifi +and Gogo made the same number of scarves in the trial-week, and they +were equally light; but Fifi's were twice as warm as Gogo's and she was +pronounced twice as good. But this year, woe is me, who can judge it? +Three competitors are here, and they differ in all points! While you +settle their claims, you shall be lodged, Her Radiancy bids me say, free +of expense—in the best dungeon, and abundantly fed on the best bread +and water."</p> + +<p>The old man groaned. "All is lost!" he wildly exclaimed. But Norman +heeded him not: he had taken out his note-book, and was calmly jotting +down the particulars.</p> + +<p>"Three they be," the Governor proceeded, "Lolo, Mimi, and Zuzu. Lolo +makes 5 scarves while Mimi makes 2; but Zuzu makes 4 while Lolo makes 3! +Again, so fairylike is Zuzu's handiwork, 5 of her scarves weigh no more +than one of Lolo's; yet Mimi's is lighter still—5 of hers will but +balance<span class='pagenum'><a name="Page_41" id="Page_41">[41]</a></span> 3 of Zuzu's! And for warmth one of Mimi's is equal to 4 of +Zuzu's; yet one of Lolo's is as warm as 3 of Mimi's!"</p> + +<p>Here the little lady once more clapped her hands.</p> + +<p>"It is our signal of dismissal!" the Governor hastily said. "Pay Her +Radiancy your farewell compliments—and walk out backwards."</p> + +<p>The walking part was all the elder tourist could manage. Norman simply +said "Tell Her Radiancy we are transfixed by the spectacle of Her Serene +Brilliance, and bid an agonized farewell to her Condensed Milkiness!"</p> + +<p>"Her Radiancy is pleased," the Governor reported, after duly translating +this. "She casts on you a glance from Her Imperial Eyes, and is +confident that you will catch it!"</p> + +<p>"That I warrant we shall!" the elder traveller moaned to himself +distractedly.</p> + +<p>Once more they bowed low, and then followed the Governor down a winding +staircase to the Imperial Dungeon, which they found to be lined with +coloured marble, lighted from the roof, and splendidly though not +luxuriously furnished with a bench of polished malachite. "I trust you +will<span class='pagenum'><a name="Page_42" id="Page_42">[42]</a></span> not delay the calculation," the Governor said, ushering them in +with much ceremony. "I have known great inconvenience—great and serious +inconvenience—result to those unhappy ones who have delayed to execute +the commands of Her Radiancy! And on this occasion she is resolute: she +says the thing must and shall be done: and she has ordered up ten +thousand additional bamboos!" With these words he left them, and they +heard him lock and bar the door on the outside.</p> + +<p>"I told you how it would end!" moaned the elder traveller, wringing his +hands, and quite forgetting in his anguish that he had himself proposed +the expedition, and had never predicted anything of the sort. "Oh that +we were well out of this miserable business!"</p> + +<p>"Courage!" cried the younger cheerily. "<i>Hæc olim meminisse juvabit!</i> +The end of all this will be glory!"</p> + +<p>"Glory without the L!" was all the poor old man could say, as he rocked +himself to and fro on the malachite bench. "Glory without the L!"</p> + +<div class="footnotes"><h3>FOOTNOTE:</h3> +<div class="footnote"><p><a name="Footnote_A_1" id="Footnote_A_1"></a><a href="#FNanchor_A_1"><span class="label">[A]</span></a> "<i>Maskee</i>," in Pigeon-English, means "<i>without</i>."</p></div> +</div> + + +<hr style="width: 65%;" /> +<p><span class='pagenum'><a name="Page_43" id="Page_43">[43]</a></span></p> +<h2><a name="KNOT_VII" id="KNOT_VII"></a>KNOT VII.</h2> + +<h3>PETTY CASH.</h3> + +<p class="center">"Base is the slave that pays."</p> + + +<p>"Aunt Mattie!"</p> + +<p>"My child?"</p> + +<p>"<i>Would</i> you mind writing it down at once? I shall be quite <i>certain</i> to +forget it if you don't!"</p> + +<p>"My dear, we really must wait till the cab stops. How can I possibly +write anything in the midst of all this jolting?"</p> + +<p>"But <i>really</i> I shall be forgetting it!"</p> + +<p>Clara's voice took the plaintive tone that her aunt never knew how to +resist, and with a sigh the old lady drew forth her ivory tablets and +prepared to record the amount that Clara had just spent at the +confectioner's shop. Her expenditure was always made out of her aunt's +purse, but the poor girl knew, by bitter experience, that sooner or +later<span class='pagenum'><a name="Page_44" id="Page_44">[44]</a></span> "Mad Mathesis" would expect an exact account of every penny that +had gone, and she waited, with ill-concealed impatience, while the old +lady turned the tablets over and over, till she had found the one headed +"<span class="smcap">PETTY CASH</span>."</p> + +<p>"Here's the place," she said at last, "and here we have yesterday's +luncheon duly entered. <i>One glass lemonade</i> (Why can't you drink water, +like me?) <i>three sandwiches</i> (They never put in half mustard enough. I +told the young woman so, to her face; and she tossed her head—like her +impudence!) <i>and seven biscuits</i>. <i>Total one-and-two-pence.</i> Well, now +for to-day's?"</p> + +<p>"One glass of lemonade——" Clara was beginning to say, when suddenly +the cab drew up, and a courteous railway-porter was handing out the +bewildered girl before she had had time to finish her sentence.</p> + +<p>Her aunt pocketed the tablets instantly. "Business first," she said: +"petty cash—which is a form of pleasure, whatever <i>you</i> may +think—afterwards." And she proceeded to pay the driver, and to give +voluminous orders about the luggage, quite deaf to the entreaties of her +unhappy niece that she would enter the rest of the luncheon account. +<span class='pagenum'><a name="Page_45" id="Page_45">[45]</a></span> +"My dear, you really must cultivate a more capacious mind!" was all the +consolation she vouchsafed to the poor girl. "Are not the tablets of +your memory wide enough to contain the record of one single luncheon?"</p> + +<p>"Not wide enough! Not half wide enough!" was the passionate reply.</p> + +<p>The words came in aptly enough, but the voice was not that of Clara, and +both ladies turned in some surprise to see who it was that had so +suddenly struck into their conversation. A fat little old lady was +standing at the door of a cab, helping the driver to extricate what +seemed an exact duplicate of herself: it would have been no easy task to +decide which was the fatter, or which looked the more good-humoured of +the two sisters.</p> + +<p>"I tell you the cab-door isn't half wide enough!" she repeated, as her +sister finally emerged, somewhat after the fashion of a pellet from a +pop-gun, and she turned to appeal to Clara. "Is it, dear?" she said, +trying hard to bring a frown into a face that dimpled all over with +smiles.</p> + +<p>"Some folks is too wide for 'em," growled the cab-driver.</p> + +<p><span class='pagenum'><a name="Page_46" id="Page_46">[46]</a></span></p> +<div class="figcenter" style="width: 266px;"> +<img src="images/i059.png" width="266" height="400" alt=""I TELL YOU THE CAB-DOOR ISN'T HALF WIDE ENOUGH!"" title="" /> +<span class="caption">"I TELL YOU THE CAB-DOOR ISN'T HALF WIDE ENOUGH!"</span> +</div> + +<p>"Don't provoke me, man!" cried the little old<span class='pagenum'><a name="Page_47" id="Page_47">[47]</a></span> lady, in what she meant +for a tempest of fury. "Say another word and I'll put you into the +County Court, and sue you for a <i>Habeas Corpus</i>!" The cabman touched his +hat, and marched off, grinning.</p> + +<p>"Nothing like a little Law to cow the ruffians, my dear!" she remarked +confidentially to Clara. "You saw how he quailed when I mentioned the +<i>Habeas Corpus</i>? Not that I've any idea what it means, but it sounds +very grand, doesn't it?"</p> + +<p>"It's very provoking," Clara replied, a little vaguely.</p> + +<p>"Very!" the little old lady eagerly repeated. "And we're very much +provoked indeed. Aren't we, sister?"</p> + +<p>"I never was so provoked in all my life!" the fatter sister assented, +radiantly.</p> + +<p>By this time Clara had recognised her picture-gallery acquaintances, +and, drawing her aunt aside, she hastily whispered her reminiscences. "I +met them first in the Royal Academy—and they were very kind to me—and +they were lunching at the next table to us, just now, you know—and they +tried to help me to find the picture I wanted—and I'm sure they're dear +old things!"<span class='pagenum'><a name="Page_48" id="Page_48">[48]</a></span></p> + +<p>"Friends of yours, are they?" said Mad Mathesis. "Well, I like their +looks. You can be civil to them, while I get the tickets. But do try and +arrange your ideas a little more chronologically!"</p> + +<p>And so it came to pass that the four ladies found themselves seated side +by side on the same bench waiting for the train, and chatting as if they +had known one another for years.</p> + +<p>"Now this I call quite a remarkable coincidence!" exclaimed the smaller +and more talkative of the two sisters—the one whose legal knowledge had +annihilated the cab-driver. "Not only that we should be waiting for the +same train, and at the same station—<i>that</i> would be curious enough—but +actually on the same day, and the same hour of the day! That's what +strikes <i>me</i> so forcibly!" She glanced at the fatter and more silent +sister, whose chief function in life seemed to be to support the family +opinion, and who meekly responded—</p> + +<p>"And me too, sister!"</p> + +<p>"Those are not <i>independent</i> coincidences——" Mad Mathesis was just +beginning, when Clara ventured to interpose.<span class='pagenum'><a name="Page_49" id="Page_49">[49]</a></span></p> + +<p>"There's no jolting here," she pleaded meekly. "<i>Would</i> you mind writing +it down now?"</p> + +<p>Out came the ivory tablets once more. "What was it, then?" said her +aunt.</p> + +<p>"One glass of lemonade, one sandwich, one biscuit—Oh dear me!" cried +poor Clara, the historical tone suddenly changing to a wail of agony.</p> + +<p>"Toothache?" said her aunt calmly, as she wrote down the items. The two +sisters instantly opened their reticules and produced two different +remedies for neuralgia, each marked "unequalled."</p> + +<p>"It isn't that!" said poor Clara. "Thank you very much. It's only that I +<i>can't</i> remember how much I paid!"</p> + +<p>"Well, try and make it out, then," said her aunt. "You've got +yesterday's luncheon to help you, you know. And here's the luncheon we +had the day before—the first day we went to that shop—<i>one glass +lemonade</i>, <i>four sandwiches</i>, <i>ten biscuits</i>. <i>Total, +one-and-fivepence.</i>" She handed the tablets to Clara, who gazed at them +with eyes so dim with tears that she did not at first notice that she +was holding them upside down.</p> + +<p>The two sisters had been listening to all this<span class='pagenum'><a name="Page_50" id="Page_50">[50]</a></span> with the deepest +interest, and at this juncture the smaller one softly laid her hand on +Clara's arm.</p> + +<p>"Do you know, my dear," she said coaxingly, "my sister and I are in the +very same predicament! Quite identically the very same predicament! +Aren't we, sister?"</p> + +<p>"Quite identically and absolutely the very——" began the fatter sister, +but she was constructing her sentence on too large a scale, and the +little one would not wait for her to finish it.</p> + +<p>"Yes, my dear," she resumed; "we were lunching at the very same shop as +you were—and we had two glasses of lemonade and three sandwiches and +five biscuits—and neither of us has the least idea what we paid. Have +we, sister?"</p> + +<p>"Quite identically and absolutely——" murmured the other, who evidently +considered that she was now a whole sentence in arrears, and that she +ought to discharge one obligation before contracting any fresh +liabilities; but the little lady broke in again, and she retired from +the conversation a bankrupt.</p> + +<p>"<i>Would</i> you make it out for us, my dear?" pleaded the little old lady.<span class='pagenum'><a name="Page_51" id="Page_51">[51]</a></span></p> + +<p>"You can do Arithmetic, I trust?" her aunt said, a little anxiously, as +Clara turned from one tablet to another, vainly trying to collect her +thoughts. Her mind was a blank, and all human expression was rapidly +fading out of her face.</p> + +<p>A gloomy silence ensued.</p> + + + +<hr style="width: 65%;" /> +<p><span class='pagenum'><a name="Page_52" id="Page_52">[52]</a></span></p> +<h2><a name="KNOT_VIII" id="KNOT_VIII"></a>KNOT VIII.</h2> + +<h3>DE OMNIBUS REBUS.</h3> + +<div class="poem"><div class="stanza"> +<span class="i0">"This little pig went to market:<br /></span> +<span class="i0">This little pig staid at home."<br /></span> +</div></div> + + +<p>"By Her Radiancy's express command," said the Governor, as he conducted +the travellers, for the last time, from the Imperial presence, "I shall +now have the ecstasy of escorting you as far as the outer gate of the +Military Quarter, where the agony of parting—if indeed Nature can +survive the shock—must be endured! From that gate grurmstipths start +every quarter of an hour, both ways——"</p> + +<p>"Would you mind repeating that word?" said Norman. "Grurm——?"</p> + +<p>"Grurmstipths," the Governor repeated. "You call them omnibuses in +England. They run both ways, and you can travel by one of them all the +way down to the harbour."<span class='pagenum'><a name="Page_53" id="Page_53">[53]</a></span></p> + +<p>The old man breathed a sigh of relief; four hours of courtly ceremony +had wearied him, and he had been in constant terror lest something +should call into use the ten thousand additional bamboos.</p> + +<p>In another minute they were crossing a large quadrangle, paved with +marble, and tastefully decorated with a pigsty in each corner. Soldiers, +carrying pigs, were marching in all directions: and in the middle stood +a gigantic officer giving orders in a voice of thunder, which made +itself heard above all the uproar of the pigs.</p> + +<p>"It is the Commander-in-Chief!" the Governor hurriedly whispered to his +companions, who at once followed his example in prostrating themselves +before the great man. The Commander gravely bowed in return. He was +covered with gold lace from head to foot: his face wore an expression of +deep misery: and he had a little black pig under each arm. Still the +gallant fellow did his best, in the midst of the orders he was every +moment issuing to his men, to bid a courteous farewell to the departing +guests.</p> + +<p>"Farewell, oh old one—carry these three to the<span class='pagenum'><a name="Page_54" id="Page_54">[54]</a></span> South corner—and +farewell to thee, thou young one—put this fat one on the top of the +others in the Western sty—may your shadows never be less—woe is me, it +is wrongly done! Empty out all the sties, and begin again!" And the +soldier leant upon his sword, and wiped away a tear.</p> + +<p>"He is in distress," the Governor explained as they left the court. "Her +Radiancy has commanded him to place twenty-four pigs in those four +sties, so that, as she goes round the court, she may always find the +number in each sty nearer to ten than the number in the last."</p> + +<p>"Does she call ten nearer to ten than nine is?" said Norman.</p> + +<p>"Surely," said the Governor. "Her Radiancy would admit that ten is +nearer to ten than nine is—and also nearer than eleven is."</p> + +<p>"Then I think it can be done," said Norman.</p> + +<p>The Governor shook his head. "The Commander has been transferring them +in vain for four months," he said. "What hope remains? And Her Radiancy +has ordered up ten thousand additional——"</p> + +<p>"The pigs don't seem to enjoy being transferred,"<span class='pagenum'><a name="Page_55" id="Page_55">[55]</a></span> the old man hastily +interrupted. He did not like the subject of bamboos.</p> + +<p>"They are only <i>provisionally</i> transferred, you know," said the +Governor. "In most cases they are immediately carried back again: so +they need not mind it. And all is done with the greatest care, under the +personal superintendence of the Commander-in-Chief."</p> + +<p>"Of course she would only go <i>once</i> round?" said Norman.</p> + +<p>"Alas, no!" sighed their conductor. "Round and round. Round and round. +These are Her Radiancy's own words. But oh, agony! Here is the outer +gate, and we must part!" He sobbed as he shook hands with them, and the +next moment was briskly walking away.</p> + +<p>"He <i>might</i> have waited to see us off!" said the old man, piteously.</p> + +<p>"And he needn't have begun whistling the very <i>moment</i> he left us!" said +the young one, severely. "But look sharp—here are two what's-his-names +in the act of starting!"</p> + +<p>Unluckily, the sea-bound omnibus was full. "Never mind!" said Norman, +cheerily. "We'll walk on till the next one overtakes us."<span class='pagenum'><a name="Page_56" id="Page_56">[56]</a></span></p> + +<p>They trudged on in silence, both thinking over the military problem, +till they met an omnibus coming from the sea. The elder traveller took +out his watch. "Just twelve minutes and a half since we started," he +remarked in an absent manner. Suddenly the vacant face brightened; the +old man had an idea. "My boy!" he shouted, bringing his hand down upon +Norman's shoulder so suddenly as for a moment to transfer his centre of +gravity beyond the base of support.</p> + +<p>Thus taken off his guard, the young man wildly staggered forwards, and +seemed about to plunge into space: but in another moment he had +gracefully recovered himself. "Problem in Precession and Nutation," he +remarked—in tones where filial respect only just managed to conceal a +shade of annoyance. "What is it?" he hastily added, fearing his father +might have been taken ill. "Will you have some brandy?"</p> + +<p>"When will the next omnibus overtake us? When? When?" the old man cried, +growing more excited every moment.</p> + +<p>Norman looked gloomy. "Give me time," he<span class='pagenum'><a name="Page_57" id="Page_57">[57]</a></span> said. "I must think it over." +And once more the travellers passed on in silence—a silence only broken +by the distant squeals of the unfortunate little pigs, who were still +being provisionally transferred from sty to sty, under the personal +superintendence of the Commander-in-Chief.</p> + + + +<hr style="width: 65%;" /> +<p><span class='pagenum'><a name="Page_58" id="Page_58">[58]</a></span></p> +<h2><a name="KNOT_IX" id="KNOT_IX"></a>KNOT IX.</h2> + +<h3>A SERPENT WITH CORNERS.</h3> + +<div class="poem"><div class="stanza"> +<span class="i0">"Water, water, every where,<br /></span> +<span class="i0">Nor any drop to drink."<br /></span> +</div></div> + + +<p>"It'll just take one more pebble."</p> + +<p>"What ever <i>are</i> you doing with those buckets?"</p> + +<p>The speakers were Hugh and Lambert. Place, the beach of Little Mendip. +Time, 1.30, <span class="smcap">P.M.</span> Hugh was floating a bucket in another a size larger, +and trying how many pebbles it would carry without sinking. Lambert was +lying on his back, doing nothing.</p> + +<p>For the next minute or two Hugh was silent, evidently deep in thought. +Suddenly he started. "I say, look here, Lambert!" he cried.</p> + +<p>"If it's alive, and slimy, and with legs, I don't care to," said +Lambert.</p> + +<p>"Didn't Balbus say this morning that, if a body<span class='pagenum'><a name="Page_59" id="Page_59">[59]</a></span> is immersed in liquid, +it displaces as much liquid as is equal to its own bulk?" said Hugh.</p> + +<p>"He said things of that sort," Lambert vaguely replied.</p> + +<p>"Well, just look here a minute. Here's the little bucket almost quite +immersed: so the water displaced ought to be just about the same bulk. +And now just look at it!" He took out the little bucket as he spoke, and +handed the big one to Lambert. "Why, there's hardly a teacupful! Do you +mean to say <i>that</i> water is the same bulk as the little bucket?"</p> + +<p>"Course it is," said Lambert.</p> + +<p>"Well, look here again!" cried Hugh, triumphantly, as he poured the +water from the big bucket into the little one. "Why, it doesn't half +fill it!"</p> + +<p>"That's <i>its</i> business," said Lambert. "If Balbus says it's the same +bulk, why, it <i>is</i> the same bulk, you know."</p> + +<p>"Well, I don't believe it," said Hugh.</p> + +<p>"You needn't," said Lambert. "Besides, it's dinner-time. Come along."</p> + +<p>They found Balbus waiting dinner for them, and to him Hugh at once +propounded his difficulty.</p> + +<p>"Let's get you helped first," said Balbus, briskly<span class='pagenum'><a name="Page_60" id="Page_60">[60]</a></span> cutting away at the +joint. "You know the old proverb 'Mutton first, mechanics afterwards'?"</p> + +<p>The boys did <i>not</i> know the proverb, but they accepted it in perfect +good faith, as they did every piece of information, however startling, +that came from so infallible an authority as their tutor. They ate on +steadily in silence, and, when dinner was over, Hugh set out the usual +array of pens, ink, and paper, while Balbus repeated to them the problem +he had prepared for their afternoon's task.</p> + +<p>"A friend of mine has a flower-garden—a very pretty one, though no +great size—"</p> + +<p>"How big is it?" said Hugh.</p> + +<p>"That's what <i>you</i> have to find out!" Balbus gaily replied. "All <i>I</i> +tell you is that it is oblong in shape—just half a yard longer than its +width—and that a gravel-walk, one yard wide, begins at one corner and +runs all round it."</p> + +<p>"Joining into itself?" said Hugh.</p> + +<p>"<i>Not</i> joining into itself, young man. Just before doing <i>that</i>, it +turns a corner, and runs round the garden again, alongside of the first +portion, and then inside that again, winding in and in, and each lap +touching the last one, till it has used up the whole of the area."<span class='pagenum'><a name="Page_61" id="Page_61">[61]</a></span></p> + +<p>"Like a serpent with corners?" said Lambert.</p> + +<p>"Exactly so. And if you walk the whole length of it, to the last inch, +keeping in the centre of the path, it's exactly two miles and half a +furlong. Now, while you find out the length and breadth of the garden, +I'll see if I can think out that sea-water puzzle."</p> + +<p>"You said it was a flower-garden?" Hugh inquired, as Balbus was leaving +the room.</p> + +<p>"I did," said Balbus.</p> + +<p>"Where do the flowers grow?" said Hugh. But Balbus thought it best not +to hear the question. He left the boys to their problem, and, in the +silence of his own room, set himself to unravel Hugh's mechanical +paradox.</p> + +<p>"To fix our thoughts," he murmured to himself, as, with hands +deep-buried in his pockets, he paced up and down the room, "we will take +a cylindrical glass jar, with a scale of inches marked up the side, and +fill it with water up to the 10-inch mark: and we will assume that every +inch depth of jar contains a pint of water. We will now take a solid +cylinder, such that every inch of it is equal in bulk to <i>half</i> a pint +of water, and plunge 4 inches of it into the water, so that the end of +the cylinder<span class='pagenum'><a name="Page_62" id="Page_62">[62]</a></span> comes down to the 6-inch mark. Well, that displaces 2 +pints of water. What becomes of them? Why, if there were no more +cylinder, they would lie comfortably on the top, and fill the jar up to +the 12-inch mark. But unfortunately there <i>is</i> more cylinder, occupying +half the space between the 10-inch and the 12-inch marks, so that only +<i>one</i> pint of water can be accommodated there. What becomes of the other +pint? Why, if there were no more cylinder, it would lie on the top, and +fill the jar up to the 13-inch mark. But unfortunately——Shade of +Newton!" he exclaimed, in sudden accents of terror. "When <i>does</i> the +water stop rising?"</p> + +<p>A bright idea struck him. "I'll write a little essay on it," he said.</p> + +<hr style="width: 45%;" /> + +<p class="center"><i>Balbus's Essay.</i></p> + +<p>"When a solid is immersed in a liquid, it is well known that it +displaces a portion of the liquid equal to itself in bulk, and that the +level of the liquid rises just so much as it would rise if a quantity of +liquid had been added to it, equal in<span class='pagenum'><a name="Page_63" id="Page_63">[63]</a></span> bulk to the solid. Lardner says, +precisely the same process occurs when a solid is <i>partially</i> immersed: +the quantity of liquid displaced, in this case, equalling the portion of +the solid which is immersed, and the rise of the level being in +proportion.</p> + +<p>"Suppose a solid held above the surface of a liquid and partially +immersed: a portion of the liquid is displaced, and the level of the +liquid rises. But, by this rise of level, a little bit more of the solid +is of course immersed, and so there is a new displacement of a second +portion of the liquid, and a consequent rise of level. Again, this +second rise of level causes a yet further immersion, and by consequence +another displacement of liquid and another rise. It is self-evident that +this process must continue till the entire solid is immersed, and that +the liquid will then begin to immerse whatever holds the solid, which, +being connected with it, must for the time be considered a part of it. +If you hold a stick, six feet long, with its end in a tumbler of water, +and wait long enough, you must eventually be immersed. The question as +to the source from which the water is supplied—which belongs to a high +branch of mathematics, and is therefore beyond our present scope—does +not apply<span class='pagenum'><a name="Page_64" id="Page_64">[64]</a></span> to the sea. Let us therefore take the familiar instance of a +man standing at the edge of the sea, at ebb-tide, with a solid in his +hand, which he partially immerses: he remains steadfast and unmoved, and +we all know that he must be drowned. The multitudes who daily perish in +this manner to attest a philosophical truth, and whose bodies the +unreasoning wave casts sullenly upon our thankless shores, have a truer +claim to be called the martyrs of science than a Galileo or a Kepler. To +use Kossuth's eloquent phrase, they are the unnamed demigods of the +nineteenth century."<a name="FNanchor_B_2" id="FNanchor_B_2"></a><a href="#Footnote_B_2" class="fnanchor">[B]</a></p> + +<hr style="width: 45%;" /> + +<p>"There's a fallacy <i>somewhere</i>," 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.<span class='pagenum'><a name="Page_65" id="Page_65">[65]</a></span></p> + +<div class="figcenter" style="width: 323px;"> +<img src="images/i078.png" width="323" height="475" alt=""HE REMAINS STEADFAST AND UNMOVED."" title="" /> +<span class="caption">"HE REMAINS STEADFAST AND UNMOVED."</span> +</div> + +<p><br /></p> + +<div class="footnotes"><h3>FOOTNOTE:</h3> +<div class="footnote"><p><a name="Footnote_B_2" id="Footnote_B_2"></a><a href="#FNanchor_B_2"><span class="label">[B]</span></a> <i>Note by the writer.</i>—For the above Essay I am indebted to +a dear friend, now deceased.</p></div> +</div> + + +<hr style="width: 65%;" /> +<p><span class='pagenum'><a name="Page_66" id="Page_66">[66]</a></span></p> +<h2><a name="KNOT_X" id="KNOT_X"></a>KNOT X.</h2> + +<h3>CHELSEA BUNS.</h3> + +<p class="center">"Yea, buns, and buns, and buns!"</p> + +<p class="author"><span class="smcap">Old Song.</span></p> + + +<p>"How very, very sad!" exclaimed Clara; and the eyes of the gentle girl +filled with tears as she spoke.</p> + +<p>"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——"</p> + +<p>"And some, perhaps, <i>all</i>!" 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<span class='pagenum'><a name="Page_67" id="Page_67">[67]</a></span> leg, and all the others watching? I <i>think</i> it was a +plan of a battle——"</p> + +<p>"The battle of Trafalgar, no doubt," her aunt interrupted, briskly.</p> + +<p>"Hardly that, I think," Clara ventured to say. "You see, in that case, +he couldn't well be alive——"</p> + +<p>"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 <i>does</i> prove it!"</p> + +<p>Clara did not see her way out of it. Logic had never been her <i>forte</i>.</p> + +<p>"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?"</p> + +<p>"How <i>can</i> I tell?" gasped the terrified girl. She knew well what was +coming.</p> + +<p>"You can't, of course, without <i>data</i>," her aunt replied: "but I'm just +going to give you——"</p> + +<p>"Give her a Chelsea bun, Miss! That's what<span class='pagenum'><a name="Page_68" id="Page_68">[68]</a></span> 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.</p> + +<p>"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 +<a name="corr7" id="corr7"></a><a class="correction" href="#cn7" title="See Music Transcriber's note.">refrain</a>:—</p> + +<div class="figcenter" style="width: 600px;"> +<img src="images/i081.png" width="600" height="208" alt="Chel-sea" title="" /> +</div> + +<p class="center"><a href="music/chelsea.mid">Listen</a></p> + +<p>"Far too indigestible, my love!" said the old lady. "Percentages will +agree with you ever so much better!"</p> + +<p>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<span class='pagenum'><a name="Page_69" id="Page_69">[69]</a></span> old lady, who at once proceeded to count off the <i>data</i> on +her fingers.</p> + +<p>"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, <i>at least</i>, must have lost all four?"</p> + +<p>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.</p> + +<p>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.</p> + +<p>"It changes from Wednesday to Thursday at midnight, doesn't it?" Hugh +had begun.</p> + +<p>"Sometimes," said Balbus, cautiously.</p> + +<p>"Always," said Lambert, decisively.</p> + +<p>"<i>Sometimes</i>," Balbus gently insisted. "Six midnights out of seven, it +changes to some other name."<span class='pagenum'><a name="Page_70" id="Page_70">[70]</a></span></p> + +<p>"I meant, of course," Hugh corrected himself, "when it <i>does</i> change +from Wednesday to Thursday, it does it at midnight—and <i>only</i> at +midnight."</p> + +<p>"Surely," said Balbus. Lambert was silent.</p> + +<p>"Well, now, suppose it's midnight here in Chelsea. Then it's Wednesday +<i>west</i> of Chelsea (say in Ireland or America) where midnight hasn't +arrived yet: and it's Thursday <i>east</i> of Chelsea (say in Germany or +Russia) where midnight has just passed by?"</p> + +<p>"Surely," Balbus said again. Even Lambert nodded this time.</p> + +<p>"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 <i>must</i> be some place—not Chelsea—that has different days on the +two sides of it. And the worst of it is, the people <i>there</i> get their +days in the wrong order: they've got Wednesday <i>east</i> of them, and +Thursday <i>west</i>—just as if their day had changed from Thursday to +Wednesday!"</p> + +<p>"I've heard that puzzle before!" cried Lambert. "And I'll tell you the +explanation. When a ship<span class='pagenum'><a name="Page_71" id="Page_71">[71]</a></span> 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."</p> + +<p>"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."</p> + +<p>"I suppose there <i>is</i> 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 <i>east</i> of them, and the new one <i>west</i>: +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."</p> + +<p>So they had entered the house in the state I have described—Balbus +puzzled, and Lambert buried in gloomy thought.</p> + +<p>"Yes, m'm, Master <i>is</i> at home, m'm," said the<span class='pagenum'><a name="Page_72" id="Page_72">[72]</a></span> 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 <i>ole</i> +party is a-waiting for you in the libery."</p> + +<p>"I don't like his calling your father an <i>old</i> 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 <i>whole</i> party," before they +were ushered into the library, and the sight of the five solemn faces +there assembled chilled her into silence.</p> + +<p>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 +<i>pièce de resistance</i> was a large green baize bag, which gave forth, as +the old man restlessly lifted it from side<span class='pagenum'><a name="Page_73" id="Page_73">[73]</a></span> to side, a charming jingle, +as of innumerable golden guineas.</p> + +<p>"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<span class='pagenum'><a name="Page_74" id="Page_74">[74]</a></span> 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 <i>another</i> critical +occasion. The ages of two of you were together <i>double</i> 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 <i>data</i>, and you shall have +the money!"</p> + +<p>"But we <i>know</i> our ages!" cried Hugh.</p> + +<p>"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 +<i>data</i><span class='pagenum'><a name="Page_75" id="Page_75">[75]</a></span> only! You mustn't even assume <i>which</i> 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.</p> + +<p>"And <i>you</i> shall have a similar <i>cadeau</i>," the old lady whispered to her +niece, "when you've calculated that percentage!" And she followed her +brother.</p> + +<p>Nothing could exceed the solemnity with which the old couple had risen +from the table, and yet was it—was it a <i>grin</i> with which the father +turned away from his unhappy sons? Could it be—could it be a <i>wink</i> +with which the aunt abandoned her despairing niece? And were those—were +those sounds of suppressed <i>chuckling</i> 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.</p> + +<p>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)<span class='pagenum'><a name="Page_76" id="Page_76">[76]</a></span> 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!</p> + +<p>"It's hardly fair," muttered Hugh, "to give us such a jumble as this to +work out!"</p> + +<p>"Fair?" Clara echoed, bitterly. "Well!"</p> + +<p>And to all my readers I can but repeat the last words of gentle Clara—</p> + +<p class="fm2">Fare-well!</p> + + +<hr style="width: 65%;" /> +<p><span class='pagenum'><a name="Page_77" id="Page_77">[77]</a></span></p> +<h2><a name="APPENDIX" id="APPENDIX"></a>APPENDIX.</h2> + +<p class="center">"A knot!" said Alice. "Oh, do let me help to undo it!"</p> + +<h3>ANSWERS TO KNOT I.</h3> + + +<p><i>Problem.</i>—"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."</p> + +<p><i>Answer.</i>—"24 miles: half-past 6."</p> + +<hr style="width: 45%;" /> + +<p><i>Solution.</i>—A level mile takes ¼ of an hour, up hill <sup>1</sup>⁄<sub>3</sub>, +down hill +<sup>1</sup>⁄<sub>6</sub>. 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.</p> + +<p><span class='pagenum'><a name="Page_78" id="Page_78">[78]</a></span></p><hr style="width: 45%;" /> + +<p>Twenty-seven answers have come in. Of these, 9 are right, 16 partially +right, and 2 wrong. The 16 give the <i>distance</i> correctly, but they have +failed to grasp the fact that the top of the hill might have been +reached at <i>any</i> moment between 6 o'clock and 7.</p> + +<p>The two wrong answers are from <span class="smcap">Gerty Vernon</span> and <span class="smcap">A Nihilist</span>. The former +makes the distance "23 miles," while her revolutionary companion puts it +at "27." <span class="smcap">Gerty Vernon</span> says "they had to go 4 miles along the plain, and +got to the foot of the hill at 4 o'clock." They <i>might</i> have done so, I +grant; but you have no ground for saying they <i>did</i> 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 +<i>not</i> require 2¾ hours. <span class="smcap">A Nihilist</span> says "Let <i>x</i> denote the whole +number of miles; <i>y</i> the number of hours to hill-top; ∴ 3<i>y</i> = +number of miles to hill-top, and <i>x</i>-3<i>y</i> = number of miles on the other +side." You bewilder me. The other side of <i>what</i>? "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 <i>possible</i>, though +it is not <i>necessarily</i> true) that there was no level road at all. Even +then you go wrong.<span class='pagenum'><a name="Page_79" id="Page_79">[79]</a></span></p> + +<p>You say</p> + +<p> +<span style="margin-left: 10.5em;">"<i>y</i> = 6 - <sup>(<i>x</i> - 3<i>y</i>)</sup>⁄<sub>6</sub>, ..... (i);</span><br /> +<br /> +<span style="margin-left: 10.5em;"><sup><i>x</i></sup>⁄<sub>4½</sub> = 6 ..... (ii)."</span><br /> +</p> + +<p>I grant you (i), but I deny (ii): it rests on the assumption that to go +<i>part</i> of the time at 3 miles an hour, and the rest at 6 miles an hour, +comes to the same result as going the <i>whole</i> time at 4½ miles an +hour. But this would only be true if the "<i>part</i>" were an exact <i>half</i>, +i.e., if they went up hill for 3 hours, and down hill for the other 3: +which they certainly did <i>not</i> do.</p> + +<p>The sixteen, who are partially right, are <span class="smcap">Agnes Bailey</span>, <span class="smcap">F. K., Fifee</span>, G. +E. B., H. P., <span class="smcap">Kit</span>, M. E. T., <span class="smcap">Mysie</span>, <span class="smcap">A Mother's Son</span>, <span class="smcap">Nairam</span>, <span class="smcap">A +Redruthian</span>, <span class="smcap">A Socialist</span>, <span class="smcap">Spear Maiden</span>, T. B. C., <span class="smcap">Vis Inertiæ</span>, and <span class="smcap">Yak</span>. Of +these, <span class="smcap">F. K., Fifee</span>, T. B. C., and <span class="smcap">Vis Inertiæ</span> 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 <i>particular</i> times being +fixed for reaching the hill-top. The most curious assumption is that of +<span class="smcap">Agnes Bailey</span>, who says "Let <i>x</i> = number of hours occupied in ascent; +then <sup><i>x</i></sup>⁄<sub>2</sub> = hours occupied in descent; and <sup>4<i>x</i></sup>⁄<sub>3</sub> = hours occupied on +the<span class='pagenum'><a name="Page_80" id="Page_80">[80]</a></span> level." I suppose you were thinking of the relative <i>rates</i>, up +hill and on the level; which we might express by saying that, if they +went <i>x</i> miles up hill in a certain time, they would go <sup>4<i>x</i></sup>⁄<sub>3</sub> miles on +the level <i>in the same time</i>. You have, in fact, assumed that they took +<i>the same time</i> on the level that they took in ascending the hill. <span class="smcap">Fifee</span> +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 <i>distance</i> gone, +not merely the rate. This would have been—if <span class="smcap">Fifee</span> will excuse the +slang expression—a "sell," ill-suited to the dignity of the hero.</p> + +<p>And now "descend, ye classic Nine!" who have solved the whole problem, +and let me sing your praises. Your names are <span class="smcap">Blithe</span>, E. W., L. B., <span class="smcap">A +Marlborough Boy</span>, O. V. L., <span class="smcap">Putney Walker</span>, <span class="smcap">Rose</span>, <span class="smcap">Sea Breeze</span>, <span class="smcap">Simple +Susan</span>, and <span class="smcap">Money Spinner</span>. (These last two I count as one, as they send a +joint answer.) <span class="smcap">Rose</span> and <span class="smcap">Simple Susan</span> and <span class="smcap">Co</span>. 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." <span class="smcap">A Marlborough Boy</span> +and <span class="smcap">Putney Walker</span> deserve honourable mention for their algebraical +solutions being the only two who have perceived<span class='pagenum'><a name="Page_81" id="Page_81">[81]</a></span> that the question leads +to <i>an indeterminate equation</i>. 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 +<i>condition</i> for further parley. If <i>now</i> thou wilt not grant that I am a +truth-loving man, then will I affirm that those same initials denote +Envenomed Wickedness!"</p> + +<hr style="width: 45%;" /> +<p class="center">CLASS LIST.</p> + +<p><span style="margin-left: 12.5em;">I.</span></p> + +<p> +<span style="margin-left: 10.5em;"><span class="smcap">A Marlborough Boy.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Putney Walker.</span></span> +</p> + +<p><span style="margin-left: 12.5em;">II.</span></p> + +<p> +<span style="margin-left: 10.5em;"><span class="smcap">Blithe.</span></span><br /> +<span style="margin-left: 10.5em;">E. W.</span><br /> +<span style="margin-left: 10.5em;">L. B.</span><br /> +<span style="margin-left: 10.5em;">O. V. L.</span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Rose.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Sea Breeze.</span></span><br /> +<span style="margin-left: 10.5em;">{<span class="smcap">Simple Susan.</span></span><br /> +<span style="margin-left: 10.5em;">{<span class="smcap">Money-Spinner.</span></span><br /> +<span class='pagenum'><a name="Page_82" id="Page_82">[82]</a></span></p> + +<p><span class="smcap">Blithe</span> has made so ingenious an addition to the problem, and <span class="smcap">Simple +Susan</span> and <span class="smcap">Co</span>. have solved it in such tuneful verse, that I record both +their answers in full. I have altered a word or two in <span class="smcap">Blithe's</span>—which I +trust she will excuse; it did not seem quite clear as it stood.</p> + +<hr style="width: 45%;" /> + +<p>"Yet stay," said the youth, as a gleam of inspiration lighted up the +relaxing muscles of his quiescent features. "Stay. Methinks it matters +little <i>when</i> 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 <i>twain</i> 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!"</p> + +<p>"Very good," said the old man. "Twelve miles out and twelve miles in. +And we reached the top some time between six and seven of the clock. Now +mark me! For every five minutes that had fled since six of the clock +when we stood on yonder peak, so many miles had we toiled upwards on the +dreary mountainside!"</p> + +<p>The youth moaned and rushed into the hostel.</p> + +<p class="author"><span class="smcap">Blithe.</span></p> +<hr style="width: 45%;" /> +<p><span class='pagenum'><a name="Page_83" id="Page_83">[83]</a></span></p> + +<div class="poem"><div class="stanza"> +<span class="i0">The elder and the younger knight,<br /></span> +<span class="i1">They sallied forth at three;<br /></span> +<span class="i0">How far they went on level ground<br /></span> +<span class="i1">It matters not to me;<br /></span> +<span class="i0">What time they reached the foot of hill,<br /></span> +<span class="i1">When they began to mount,<br /></span> +<span class="i0">Are problems which I hold to be<br /></span> +<span class="i1">Of very small account.<br /></span> +</div><div class="stanza"> +<span class="i0">The moment that each waved his hat<br /></span> +<span class="i1">Upon the topmost peak—<br /></span> +<span class="i0">To trivial query such as this<br /></span> +<span class="i1">No answer will I seek.<br /></span> +<span class="i0">Yet can I tell the distance well<br /></span> +<span class="i1">They must have travelled o'er:<br /></span> +<span class="i0">On hill and plain, 'twixt three and nine,<br /></span> +<span class="i1">The miles were twenty-four.<br /></span> +</div><div class="stanza"> +<span class="i0">Four miles an hour their steady pace<br /></span> +<span class="i1">Along the level track,<br /></span> +<span class="i0">Three when they climbed—but six when they<br /></span> +<span class="i1">Came swiftly striding back<br /></span> +<span class="i0">Adown the hill; and little skill<br /></span> +<span class="i1">It needs, methinks, to show,<br /></span> +<span class="i0">Up hill and down together told,<br /></span> +<span class="i1">Four miles an hour they go.<br /></span> +</div><div class="stanza"> +<span class="i0">For whether long or short the time<br /></span> +<span class="i1">Upon the hill they spent,<br /></span> +<span class="i0">Two thirds were passed in going up,<br /></span> +<span class="i1">One third in the descent.<br /></span> +<span class="i0">Two thirds at three, one third at six,<br /></span> +<span class="i1">If rightly reckoned o'er,<br /></span> +<span class="i0">Will make one whole at four—the tale<br /></span> +<span class="i1">Is tangled now no more.<br /></span> +</div></div> + +<p class="author"><span class="smcap">Simple Susan.</span><br /> +<span class="smcap">Money Spinner.</span></p> + +<hr style="width: 45%;" /> +<p><span class='pagenum'><a name="Page_84" id="Page_84">[84]</a></span></p> +<h3>ANSWERS TO KNOT II.</h3> + +<p class="fm3">§ 1. <span class="smcap">The Dinner Party.</span></p> + +<p><i>Problem.</i>—"The Governor of Kgovjni wants to give a very small dinner +party, and invites his father's brother-in-law, his brother's +father-in-law, his father-in-law's brother, and his brother-in-law's +father. Find the number of guests."</p> + +<p><i>Answer.</i>—"One."</p> + +<hr style="width: 45%;" /> + +<p>In this genealogy, males are denoted by capitals, and females by small +letters.</p> + +<p>The Governor is E and his guest is C.</p> + +<div class="figright" style="width: 400px;"> +<img src="images/i097.png" width="400" height="243" alt="" title="" /> +</div> + +<p>Ten answers have been received. Of these, one is wrong, <span class="smcap">Galanthus +Nivalis Major</span>, who insists on inviting <i>two</i> guests, one being the +Governor's <i>wife's brother's father</i>. If she had taken his <i>sister's +husband's father</i> instead, she would have found it possible to reduce +the guests to <i>one</i>.<span class='pagenum'><a name="Page_85" id="Page_85">[85]</a></span></p> + +<p>Of the nine who send right answers, <span class="smcap">Sea-Breeze</span> is the very faintest +breath that ever bore the name! She simply states that the Governor's +uncle might fulfill all the conditions "by intermarriages"! "Wind of the +western sea," you have had a very narrow escape! Be thankful to appear +in the Class-list at all! <span class="smcap">Bog-Oak</span> and <span class="smcap">Bradshaw of the Future</span> use +genealogies which require 16 people instead of 14, by inviting the +Governor's <i>father's sister's husband</i> instead of his <i>father's wife's +brother</i>. I cannot think this so good a solution as one that requires +only 14. <span class="smcap">Caius</span> and <span class="smcap">Valentine</span> deserve special mention as the only two who +have supplied genealogies.</p> + + +<p class="center">CLASS LIST.</p> + +<p><span style="margin-left: 12.5em;">I.</span></p> + +<p> +<span style="margin-left: 10.5em;"><span class="smcap">Bee.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Caius.</span></span><br /> +<span style="margin-left: 10.5em;">M. M.</span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Matthew Matticks.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Old Cat.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Valentine.</span></span><br /> +</p> + +<p><span style="margin-left: 12.5em;">II.</span></p> + +<p> +<span style="margin-left: 10.5em;"><span class="smcap">Bog-Oak.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Bradshaw of the Future.</span></span><br /> +</p> + +<p><span style="margin-left: 12.5em;">III.</span></p> + +<p> +<span style="margin-left: 10.5em;"><span class="smcap">Sea-Breeze.</span></span><br /> +<span class='pagenum'><a name="Page_86" id="Page_86">[86]</a></span></p> + + +<p class="fm3">§ 2. <span class="smcap">The Lodgings.</span></p> + +<p><i>Problem.</i>—"A Square has 20 doors on each side, which contains 21 equal +parts. They are numbered all round, beginning at one corner. From which +of the four, Nos. 9, 25, 52, 73, is the sum of the distances, to the +other three, least?"</p> + +<p><i>Answer.</i>—"From No. 9."</p> + +<hr style="width: 45%;" /> + +<div class="figright" style="width: 319px;"> +<img src="images/i099.png" width="319" height="372" alt="" title="" /> +</div> + +<p>Let A be No. 9, B No. 25, C No. 52, and D No. 73.</p> + +<p> +<span style="margin-left: 1.5em;">Then AB = √(12<sup>2</sup> + 5<sup>2</sup>) = √169 = 13;</span><br /> +<span style="margin-left: 1.5em;">AC = 21;</span><br /> +<span style="margin-left: 1.5em;">AD = √(9<sup>2</sup> + 8<sup>2</sup>) = √145 = 12+</span><br /> +<span style="margin-left: 1.5em;">(N.B. <i>i.e.</i> "between 12 and 13.")</span><br /> +<span style="margin-left: 1.5em;">BC = √(16<sup>2</sup> + 12<sup>2</sup>) = √400 = 20;</span><br /> +<span style="margin-left: 1.5em;">BD = √(3<sup>2</sup> + 21<sup>2</sup>) = √450 = 21+;</span><br /> +<span style="margin-left: 1.5em;">CD = √(9<sup>2</sup> + 13<sup>2</sup>) = √250 = 15+;</span><br /> +</p> + + +<p>Hence sum of distances from A is between 46 and 47; from B, between 54 +and 55; from C, between 56 and 57; from D, between 48 and 51. (Why not +"between 48 and 49"? Make this out for yourselves.) Hence the sum is +least for A.</p> + +<hr style="width: 45%;" /> +<p><span class='pagenum'><a name="Page_87" id="Page_87">[87]</a></span></p> +<p>Twenty-five solutions have been received. Of these, 15 must be marked +"0," 5 are partly right, and 5 right. Of the 15, I may dismiss +<span class="smcap">Alphabetical Phantom</span>, <span class="smcap">Bog-Oak</span>, <span class="smcap">Dinah Mite</span>, <span class="smcap">Fifee</span>, <span class="smcap">Galanthus Nivalis +Major</span> (I fear the cold spring has blighted our <span class="smcap">Snowdrop</span>), <span class="smcap">Guy</span>, <span class="smcap">H.M.S. +Pinafore</span>, <span class="smcap">Janet</span>, and <span class="smcap">Valentine</span> with the simple remark that they insist +on the unfortunate lodgers <i>keeping to the pavement</i>. (I used the words +"crossed to Number Seventy-three" for the special purpose of showing +that <i>short cuts</i> were possible.) <span class="smcap">Sea-Breeze</span> does the same, and adds +that "the result would be the same" even if they crossed the Square, but +gives no proof of this. M. M. draws a diagram, and says that No. 9 is +the house, "as the diagram shows." I cannot see <i>how</i> it does so. <span class="smcap">Old +Cat</span> assumes that the house <i>must</i> be No. 9 or No. 73. She does not +explain how she estimates the distances. BEE's Arithmetic is faulty: she +makes √169 + √442 + √130 = 741. (I suppose you +mean √741, which would be a little nearer the truth. But roots +cannot be added in this manner. Do you think √9 + √16 is +25, or even √25?) But <span class="smcap">Ayr's</span> state is more perilous still: she +draws illogical conclusions with a frightful calmness. After pointing +out (rightly) that AC is less than BD she says, "therefore the nearest +house to the other three must be A or C." And again, after pointing out +(rightly) that B and D are both within the half-square containing<span class='pagenum'><a name="Page_88" id="Page_88">[88]</a></span> A, +she says "therefore" AB + AD must be less than BC + CD. (There is no +logical force in either "therefore." For the first, try Nos. 1, 21, 60, +70: this will make your premiss true, and your conclusion false. +Similarly, for the second, try Nos. 1, 30, 51, 71.)</p> + +<p>Of the five partly-right solutions, <span class="smcap">Rags and Tatters</span> and <span class="smcap">Mad Hatter</span> (who +send one answer between them) make No. 25 6 units from the corner +instead of 5. <span class="smcap">Cheam</span>, E. R. D. L., and <span class="smcap">Meggy Potts</span> leave openings at +<a name="corr1" id="corr1"></a><a class="correction" href="#cn1" title="changed from 'he'">the</a> +corners of the Square, which are not in the <i>data</i>: moreover <span class="smcap">Cheam</span> gives +values for the distances without any hint that they are only +<i>approximations</i>. <span class="smcap">Crophi and Mophi</span> make the bold and unfounded +assumption that there were really 21 houses on each side, instead of 20 +as stated by Balbus. "We may assume," they add, "that the doors of Nos. +21, 42, 63, 84, are invisible from the centre of the Square"! What is +there, I wonder, that <span class="smcap">Crophi and Mophi</span> would <i>not</i> assume?</p> + +<p>Of the five who are wholly right, I think <span class="smcap">Bradshaw Of the Future</span>, <span class="smcap">Caius</span>, +<span class="smcap">Clifton C.</span>, and <span class="smcap">Martreb</span> deserve special praise for their full +<i>analytical</i> solutions. <span class="smcap">Matthew Matticks</span> picks out No. 9, and proves it +to be the right house in two ways, very neatly and ingeniously, but +<i>why</i> he picks it out does not appear. It is an excellent <i>synthetical</i> +proof, but lacks the analysis which the other four supply.</p> + +<p><span class='pagenum'><a name="Page_89" id="Page_89">[89]</a></span></p> +<p class="center">CLASS LIST.</p> + +<p><span style="margin-left: 12.5em;">I.</span></p> + +<p> +<span style="margin-left: 10.5em;"><span class="smcap">Bradshaw of the Future</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Caius.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Clifton C.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Martreb.</span></span><br /> +</p> + +<p><span style="margin-left: 12.5em;">II.</span></p> + +<p><span style="margin-left: 10.5em;"><span class="smcap">Matthew Matticks.</span></span></p> + +<p><span style="margin-left: 12.5em;">III.</span></p> + +<p> +<span style="margin-left: 10.5em;"><span class="smcap">Cheam.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Crophi and Mophi.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">E. R. D. L.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Meggy Potts.</span></span><br /> +<span style="margin-left: 10.5em;">{<span class="smcap">Rags and Tatters.</span></span><br /> +<span style="margin-left: 10.5em;">{<span class="smcap">Mad Hatter.</span></span><br /> +</p> + + +<p>A remonstrance has reached me from <span class="smcap">Scrutator</span> on the subject of <span class="smcap">Knot I.</span>, +which he declares was "no problem at all." "Two questions," he says, +"are put. To solve one there is no data: the other answers itself." As +to the first point, <span class="smcap">Scrutator</span> is mistaken; there <i>are</i> (not "is") data +sufficient to answer the question. As to the other, it is interesting to +know that the question "answers itself," and I am sure it does the +question great credit: still I fear I cannot enter it on the list of +winners, as this competition is only open to human beings.</p> + +<hr style="width: 45%;" /> +<p><span class='pagenum'><a name="Page_90" id="Page_90">[90]</a></span></p> +<h3>ANSWERS TO KNOT III.</h3> + +<p><i>Problem.</i>—(1) "Two travellers, starting at the same time, went +opposite ways round a circular railway. Trains start each way every 15 +minutes, the easterly ones going round in 3 hours, the westerly in 2. +How many trains did each meet on the way, not counting trains met at the +terminus itself?" (2) "They went round, as before, each traveller +counting as 'one' the train containing the other traveller. How many did +each meet?"</p> + +<p><i>Answers.</i>—(1) 19. (2) The easterly traveller met 12; the other 8.</p> + +<hr style="width: 45%;" /> + +<p>The trains one way took 180 minutes, the other way 120. Let us take the +L. C. M., 360, and divide the railway into 360 units. Then one set of +trains went at the rate of 2 units a minute and at intervals of 30 +units; the other at the rate of 3 units a minute and at intervals of 45 +units. An easterly train starting has 45 units between it and the first +train it will meet: it does 2-5ths of this while the other does 3-5ths, +and<span class='pagenum'><a name="Page_91" id="Page_91">[91]</a></span> thus meets it at the end of 18 units, and so all the way round. A +westerly train starting has 30 units between it and the first train it +will meet: it does 3-5ths of this while the other does 2-5ths, and thus +meets it at the end of 18 units, and so all the way round. Hence if the +railway be divided, by 19 posts, into 20 parts, each containing 18 +units, trains meet at every post, and, in (1), each traveller passes 19 +posts in going round, and so meets 19 trains. But, in (2), the easterly +traveller only begins to count after traversing 2-5ths of the journey, +<i>i.e.</i>, on reaching the 8th post, and so counts 12 posts: similarly the +other counts 8. They meet at the end of 2-5ths of 3 hours, or 3-5ths of +2 hours, <i>i.e.</i>, 72 minutes.</p> + +<hr style="width: 45%;" /> + +<p>Forty-five answers have been received. Of these 12 are beyond the reach +of discussion, as they give no working. I can but enumerate their names. +<span class="smcap">Ardmore</span>, E. A., F. A. D., L. D., <span class="smcap">Matthew Matticks</span>, M. E. T., <span class="smcap">Poo-Poo</span>, +and <span class="smcap">The Red Queen</span> are all wrong. <span class="smcap">Beta</span> and <span class="smcap">Rowena</span> have got (1) right and +(2) wrong. <span class="smcap">Cheeky Bob</span> and <span class="smcap">Nairam</span> give the right answers, but it may +perhaps make the one less cheeky, and induce the other to take a less +inverted view of things, to be informed that, if this had been a +competition for a<span class='pagenum'><a name="Page_92" id="Page_92">[92]</a></span> prize, they would have got no marks. [N.B.—I have +not ventured to put E. A.'s name in full, as she only gave it +provisionally, in case her answer should prove right.]</p> + +<p>Of the 33 answers for which the working is given, 10 are wrong; 11 +half-wrong and half-right; 3 right, except that they cherish the +delusion that it was <i>Clara</i> who travelled in the easterly train—a +point which the data do not enable us to settle; and 9 wholly right.</p> + +<p>The 10 wrong answers are from <span class="smcap">Bo-Peep</span>, <span class="smcap">Financier</span>, I. W. T., <span class="smcap">Kate B.</span>, M. +A. H., Q. Y. Z., <span class="smcap">Sea-Gull</span>, <span class="smcap">Thistledown</span>, <span class="smcap">Tom-Quad</span>, and an unsigned one. +<span class="smcap">Bo-Peep</span> rightly says that the easterly traveller met all trains which +started during the 3 hours of her trip, as well as all which started +during the previous 2 hours, <i>i.e.</i>, all which started at the +commencements of 20 periods of 15 minutes each; and she is right in +striking out the one she met at the moment of starting; but wrong in +striking out the <i>last</i> train, for she did not meet this at the +terminus, but 15 minutes before she got there. She makes the same +mistake in (2). <span class="smcap">Financier</span> thinks that any train, met for the second +time, is not to be counted. I. W. T. finds, by a process which is not +stated, that the travellers met at the end of 71 minutes and 26½ +seconds. <span class="smcap">Kate B.</span> thinks the trains which are met on starting and on +arriving<span class='pagenum'><a name="Page_93" id="Page_93">[93]</a></span> are <i>never</i> to be counted, even when met elsewhere. Q. Y. Z. +tries a rather complex algebraical solution, and succeeds in finding the +time of meeting correctly: all else is wrong. <span class="smcap">Sea-Gull</span> seems to think +that, in (1), the easterly train <i>stood still</i> for 3 hours; and says +that, in (2), the travellers met at the end of 71 minutes 40 seconds. +<span class="smcap">Thistledown</span> nobly confesses to having tried no calculation, but merely +having drawn a picture of the railway and counted the trains; in (1), +she counts wrong; in (2) she makes them meet in 75 minutes. <span class="smcap">Tom-Quad</span> +omits (1): in (2) he makes Clara count the train she met on her arrival. +The unsigned one is also unintelligible; it states that the travellers +go "1-24th more than the total distance to be traversed"! The "Clara" +theory, already referred to, is adopted by 5 of these, viz., <span class="smcap">Bo-Peep</span>, +<span class="smcap">Financier</span>, <span class="smcap">Kate B.</span>, <span class="smcap">Tom-Quad</span>, and the nameless writer.</p> + +<p>The 11 half-right answers are from <span class="smcap">Bog-Oak</span>, <span class="smcap">Bridget</span>, <span class="smcap">Castor</span>, <span class="smcap">Cheshire +Cat</span>, G. E. B., <span class="smcap">Guy</span>, <span class="smcap">Mary</span>, M. A. H., <span class="smcap">Old Maid</span>, R. W., and <span class="smcap">Vendredi</span>. All +these adopt the "Clara" theory. <span class="smcap">Castor</span> omits (1). <span class="smcap">Vendredi</span> gets (1) +right, but in (2) makes the same mistake as <span class="smcap">Bo-Peep</span>. I notice in your +solution a marvellous proportion-sum:—"300 miles: 2 hours :: one mile: +24 seconds." May I venture to advise your acquiring, as soon as +possible, an utter disbelief in the possibility of a ratio<span class='pagenum'><a name="Page_94" id="Page_94">[94]</a></span> existing +between <i>miles</i> and <i>hours</i>? Do not be disheartened by your two friends' +sarcastic remarks on your "roundabout ways." Their short method, of +adding 12 and 8, has the slight disadvantage of bringing the answer +wrong: even a "roundabout" method is better than <i>that</i>! M. A. H., in +(2), makes the travellers count "one" <i>after</i> they met, not <i>when</i> they +met. <span class="smcap">Cheshire Cat</span> and <span class="smcap">Old Maid</span> get "20" as answer for (1), by forgetting +to strike out the train met on arrival. The others all get "18" in +various ways. <span class="smcap">Bog-Oak</span>, <span class="smcap">Guy</span>, and R. W. divide the trains which the +westerly traveller has to meet into 2 sets, viz., those already on the +line, which they (rightly) make "11," and those which started during her +2 hours' journey (exclusive of train met on arrival), which they +(wrongly) make "7"; and they make a similar mistake with the easterly +train. <span class="smcap">Bridget</span> (rightly) says that the westerly traveller met a train +every 6 minutes for 2 hours, but (wrongly) makes the number "20"; it +should be "21." G. E. B. adopts <span class="smcap">Bo-Peep's</span> method, but (wrongly) strikes +out (for the easterly traveller) the train which started at the +<i>commencement</i> of the previous 2 hours. <span class="smcap">Mary</span> thinks a train, met on +arrival, must not be counted, even when met on a <i>previous</i> occasion.</p> + +<p>The 3, who are wholly right but for the unfortunate "Clara" theory, are +<span class="smcap">F. Lee</span>, G. S. C., and X. A. B.</p> + +<p>And now "descend, ye classic Ten!" who have<span class='pagenum'><a name="Page_95" id="Page_95">[95]</a></span> solved the whole problem. +Your names are +<span class="smcap">Aix-les-Bains</span>, +<span class="smcap">Algernon Bray</span> (thanks for a friendly +remark, which comes with a heart-warmth that not even the Atlantic could +chill), <span class="smcap">Arvon</span>, <span class="smcap">Bradshaw of the Future</span>, <span class="smcap">Fifee</span>, H. L. R., J. L. O., <span class="smcap">Omega</span>, +S. S. G., and <span class="smcap">Waiting for the Train</span>. Several of these have put Clara, +provisionally, into the easterly train: but they seem to have understood +that the data do not decide that point.</p> + + +<p class="center">CLASS LIST.</p> + +<p><span style="margin-left: 12.5em;">I.</span></p> + +<p> +<span style="margin-left: 10.5em;"><a name="corr2" id="corr2"></a><a class="correction" href="#cn2" title="changed from 'Aix-le-Bains'"><span class="smcap">Aix-les-Bains.</span></a></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Algernon Bray.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Bradshaw of the Future.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Fifee.</span></span><br /> +<span style="margin-left: 10.5em;">H. L. R.</span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Omega.</span></span><br /> +<span style="margin-left: 10.5em;">S. S. G.</span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Waiting for the train.</span></span><br /> +</p> + +<p><span style="margin-left: 12.5em;">II.</span></p> + +<p> +<span style="margin-left: 10.5em;"><span class="smcap">Arvon.</span></span><br /> +<span style="margin-left: 10.5em;">J. L. O.</span><br /> +</p> + +<p><span style="margin-left: 12.5em;">III.</span></p> + +<p> +<span style="margin-left: 10.5em;"><span class="smcap">F. Lee.</span></span><br /> +<span style="margin-left: 10.5em;">G. S. C.</span><br /> +<span style="margin-left: 10.5em;">X. A. B.</span><br /> +</p> + +<hr style="width: 45%;" /> +<p><span class='pagenum'><a name="Page_96" id="Page_96">[96]</a></span></p> +<h3>ANSWERS TO KNOT IV.</h3> + +<p><i>Problem.</i>—"There are 5 sacks, of which Nos. 1, 2, weigh 12 lbs.; Nos. +2, 3, 13½ lbs.; Nos. 3, 4, 11½ lbs.; Nos. 4, 5, 8 lbs.; Nos. 1, 3, +5, 16 lbs. Required the weight of each sack."</p> + +<p><i>Answer.</i>—"5½, 6½, 7, 4½, 3½."</p> + +<hr style="width: 45%;" /> + +<p>The sum of all the weighings, 61 lbs., includes sack No. 3 <i>thrice</i> and +each other <i>twice</i>. Deducting twice the sum of the 1st and 4th +weighings, we get 21 lbs. for <i>thrice</i> No. 3, <i>i.e.</i>, 7 lbs. for No. 3. +Hence, the 2nd and 3rd weighings give 6½ lbs., 4½ lbs. for Nos. 2, +4; and hence again, the 1st and 4th weighings give 5½ lbs., 3½ +lbs., for Nos. 1, 5.</p> + +<hr style="width: 45%;" /> + +<p>Ninety-seven answers have been received. Of these, 15 are beyond the +reach of discussion, as they give no working. I can but enumerate their +names, 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<span class='pagenum'><a name="Page_97" id="Page_97">[97]</a></span> sort of clue +to the process by which their answers were obtained. In guessing a +conundrum, or in catching a flea, we do not expect the breathless victor +to give us afterwards, in cold blood, a history of the mental or +muscular efforts by which he achieved success; but a mathematical +calculation is another thing. The names of this "mute inglorious" band +are <span class="smcap">Common Sense</span>, D. E. R., <span class="smcap">Douglas</span>, E. L., <span class="smcap">Ellen</span>, I. M. T., J. M. C., +<span class="smcap">Joseph</span>, <span class="smcap">Knot I</span>, <span class="smcap">Lucy</span>, <span class="smcap">Meek</span>, M. F. C., <span class="smcap">Pyramus</span>, <span class="smcap">Shah</span>, <span class="smcap">Veritas</span>.</p> + +<p>Of the eighty-two answers with which the working, or some approach to +it, is supplied, one is wrong: seventeen have given solutions which are +(from one cause or another) practically valueless: the remaining +sixty-four I shall try to arrange in a Class-list, according to the +varying degrees of shortness and neatness to which they seem to have +attained.</p> + +<p>The solitary wrong answer is from <span class="smcap">Nell</span>. To be thus "alone in the crowd" +is a distinction—a painful one, no doubt, but still a distinction. I am +sorry for you, my dear young lady, and I seem to hear your tearful +exclamation, when you read these lines, "Ah! This is the knell of all my +hopes!" Why, oh why, did you assume that the 4th and 5th bags weighed 4 +lbs. each? And why did you not test your answers? However, please try +again: and please don't change your <i>nom-de-plume</i>: let us have <span class="smcap">Nell</span> in +the First Class next time!<span class='pagenum'><a name="Page_98" id="Page_98">[98]</a></span></p> + +<p>The seventeen whose solutions are practically valueless are <span class="smcap">Ardmore</span>, <span class="smcap">A +ready Reckoner</span>, <span class="smcap">Arthur</span>, <span class="smcap">Bog-Lark</span>, <span class="smcap">Bog-Oak</span>, <span class="smcap">Bridget</span>, <span class="smcap">First Attempt</span>, J. L. +C., M. E. T., <span class="smcap">Rose</span>, <span class="smcap">Rowena</span>, <span class="smcap">Sea-Breeze</span>, <span class="smcap">Sylvia</span>, <span class="smcap">Thistledown</span>, +<span class="smcap">Three-Fifths Asleep</span>, <span class="smcap">Vendredi</span>, and <span class="smcap">Winifred</span>. <span class="smcap">Bog-Lark</span> tries it by a sort +of "rule of false," assuming experimentally that Nos. 1, 2, weigh 6 lbs. +each, and having thus produced 17½, instead of 16, as the weight of +1, 3, and 5, she removes "the superfluous pound and a half," but does +not explain how she knows from which to take it. <span class="smcap">Three-fifths Asleep</span> +says that (when in that peculiar state) "it seemed perfectly clear" to +her that, "3 out of the 5 sacks being weighed twice over, <sup>2</sup>⁄<sub>5</sub> of 45 = +27, must be the total weight of the 5 sacks." As to which I can only +say, with the Captain, "it beats me entirely!" <span class="smcap">Winifred</span>, on the plea +that "one must have a starting-point," assumes (what I fear is a mere +guess) that No. 1 weighed 5½ lbs. The rest all do it, wholly or +partly, by guess-work.</p> + +<p>The problem is of course (as any Algebraist sees at once) a case of +"simultaneous simple equations." It is, however, easily soluble by +Arithmetic only; and, when this is the case, I hold that it is bad +workmanship to use the more complex method. I have not, this time, given +more credit to arithmetical solutions; but in future problems I shall +(other things being equal) give the<span class='pagenum'><a name="Page_99" id="Page_99">[99]</a></span> highest marks to those who use the +simplest machinery. I have put into Class I. those whose answers seemed +specially short and neat, and into Class III. those that seemed +specially long or clumsy. Of this last set, A. C. M., <span class="smcap">Furze-Bush</span>, <span class="smcap">James</span>, +<span class="smcap">Partridge</span>, R. W., and <span class="smcap">Waiting for the Train</span>, have sent long wandering +solutions, the substitutions having no definite method, but seeming to +have been made to see what would come of it. <span class="smcap">Chilpome</span> and <span class="smcap">Dublin Boy</span> +omit some of the working. <span class="smcap">Arvon Marlborough Boy</span> only finds the weight of +<i>one</i> sack.<span class='pagenum'><a name="Page_100" id="Page_100">[100]</a></span></p> + + +<p class="center">CLASS LIST</p> + +<p><span style="margin-left: 12.5em;">I.</span></p> + +<p> +<span style="margin-left: 10.5em;">B. E. D.</span><br /> +<span style="margin-left: 10.5em;">C. H.</span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Constance Johnson.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Greystead.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Guy.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Hoopoe.</span></span><br /> +<span style="margin-left: 10.5em;">J. F. A.</span><br /> +<span style="margin-left: 10.5em;">M. A. H.</span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Number Five.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Pedro.</span></span><br /> +<span style="margin-left: 10.5em;">R. E. X.</span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Seven Old Men.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Vis Inertiæ.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Willy B.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Yahoo.</span></span><br /> +</p> + +<p><span style="margin-left: 12.5em;">II.</span></p> + +<p> +<span style="margin-left: 10.5em;"><span class="smcap">American Subscriber.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">An appreciative schoolma'am.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Ayr.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Bradshaw of the Future.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Cheam.</span></span><br /> +<span style="margin-left: 10.5em;">C. M. G.</span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Dinah Mite.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Duckwing.</span></span><br /> +<span style="margin-left: 10.5em;">E. C. M.</span><br /> +<span style="margin-left: 10.5em;">E. N. Lowry.</span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Era.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Euroclydon.</span></span><br /> +<span style="margin-left: 10.5em;">F. H. W.</span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Fifee.</span></span><br /> +<span style="margin-left: 10.5em;">G. E. B.</span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Harlequin.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Hawthorn.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Hough Green.</span></span><br /> +<span style="margin-left: 10.5em;">J. A. B.</span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Jack Tar.</span></span><br /> +<span style="margin-left: 10.5em;">J. B. B.</span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Kgovjni.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Land Lubber.</span></span><br /> +<span class='pagenum'><a name="Page_101" id="Page_101">[101]</a></span><span style="margin-left: 10.5em;">L. D.</span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Magpie.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Mary.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Mhruxi.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Minnie.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Money-Spinner.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Nairam.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Old Cat.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Polichinelle.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Simple Susan.</span></span><br /> +<span style="margin-left: 10.5em;">S. S. G.</span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Thisbe.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Verena.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Wamba.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Wolfe.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Wykehamicus.</span></span><br /> +<span style="margin-left: 10.5em;">Y. M. A. H.</span><br /> +</p> + +<p><span style="margin-left: 12.5em;">III.</span></p> + +<p> +<span style="margin-left: 10.5em;">A. C. M.</span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Arvon Marlborough Boy.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Chilpome.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Dublin Boy.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Furze-Bush.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">James.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Partridge.</span></span><br /> +<span style="margin-left: 10.5em;">R. W.</span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Waiting for the Train.</span></span><br /> +</p> + +<hr style="width: 45%;" /> +<p><span class='pagenum'><a name="Page_102" id="Page_102">[102]</a></span></p> +<h3>ANSWERS TO KNOT V.</h3> + +<p><i>Problem.</i>—To mark pictures, giving 3 x's to 2 or 3, 2 to 4 or 5, and 1 +to 9 or 10; also giving 3 o's to 1 or 2, 2 to 3 or 4 and 1 to 8 or 9; so +as to mark the smallest possible number of pictures, and to give them +the largest possible number of marks.</p> + +<p><i>Answer.</i>—10 pictures; 29 marks; arranged thus:—</p> + +<table summary="diagram1"> +<tr> +<td> x</td> +<td> x</td> +<td> x</td> +<td> x</td> +<td> x</td> +<td> x</td> +<td> x</td> +<td> x</td> +<td> x</td> +<td> o</td> +</tr> +<tr> +<td> x</td> +<td> x</td> +<td> x</td> +<td> x</td> +<td> x</td> +<td> </td> +<td> o</td> +<td> o</td> +<td> o</td> +<td> o</td> +</tr> +<tr> +<td>x</td> +<td>x</td> +<td>o</td> +<td>o</td> +<td>o</td> +<td>o</td> +<td>o</td> +<td>o</td> +<td>o</td> +<td>o</td> +</tr> +</table> + +<p><i>Solution.</i>—By giving all the x's possible, putting into brackets the +optional ones, we get 10 pictures marked thus:—</p> + +<table summary="diagram2"> +<tr> +<td> x</td> +<td> x</td> +<td> x</td> +<td> x</td> +<td> x</td> +<td> x</td> +<td> x</td> +<td> x</td> +<td> x</td> +<td>(x)</td> +</tr> +<tr> +<td> x</td> +<td> x</td> +<td> x</td> +<td> x</td> +<td>(x)</td> +</tr> +<tr> +<td> x</td> +<td> x</td> +<td>(x)</td> +</tr> +</table> + +<p>By then assigning o's in the same way, beginning at the other end, we +get 9 pictures marked thus:—</p> + +<table summary="diagram3"> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td>(o)</td> +<td> o</td> +</tr> +<tr> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td> </td> +<td>(o)</td> +<td> o</td> +<td> o</td> +<td> o</td> +</tr> +<tr> +<td>(o)</td> +<td> o</td> +<td> o</td> +<td> o</td> +<td> o</td> +<td> o</td> +<td> o</td> +<td> o</td> +<td> o</td> +</tr> +</table> + +<p>All we have now to do is to run these two wedges<span class='pagenum'><a name="Page_103" id="Page_103">[103]</a></span> as close together as +they will go, so as to get the minimum number of pictures——erasing +optional marks where by so doing we can run them closer, but otherwise +letting them stand. There are 10 necessary marks in the 1st row, and in +the 3rd; but only 7 in the 2nd. Hence we erase all optional marks in the +1st and 3rd rows, but let them stand in the 2nd.</p> + +<hr style="width: 45%;" /> + +<p>Twenty-two answers have been received. Of these 11 give no working; so, +in accordance with what I announced in my last review of answers, I +leave them unnamed, merely mentioning that 5 are right and 6 wrong.</p> + +<p>Of the eleven answers with which some working is supplied, 3 are wrong. +C. H. begins with the rash assertion that under the given conditions +"the sum is impossible. For," he or she adds (these initialed +correspondents are dismally vague beings to deal with: perhaps "it" +would be a better pronoun), "10 is the least possible number of +pictures" (granted): "therefore we must either give 2 x's to 6, or 2 o's +to 5." Why "must," oh alphabetical phantom? It is nowhere ordained that +every picture "must" have 3 marks! <span class="smcap">Fifee</span> sends a folio page of solution, +which deserved a better fate: she offers 3 answers, in each of which 10 +pictures are<span class='pagenum'><a name="Page_104" id="Page_104">[104]</a></span> marked, with 30 marks; in one she gives 2 x's to 6 +pictures; in another to 7; in the 3rd she gives 2 o's to 5; thus in +every case ignoring the conditions. (I pause to remark that the +condition "2 x's to 4 or 5 pictures" can only mean "<i>either</i> to 4 <i>or +else</i> to 5": if, as one competitor holds, it might mean <i>any</i> number not +less than 4, the words "<i>or</i> 5" would be superfluous.) I. E. A. (I am +happy to say that none of these bloodless phantoms appear this time in +the class-list. Is it IDEA with the "D" left out?) gives 2 x's to 6 +pictures. She then takes me to task for using the word "ought" instead +of "nought." No doubt, to one who thus rebels against the rules laid +down for her guidance, the word must be distasteful. But does not I. E. +A. remember the parallel case of "adder"? That creature was originally +"a nadder": then the two words took to bandying the poor "n" backwards +and forwards like a shuttlecock, the final state of the game being "an +adder." May not "a nought" have similarly become "an ought"? Anyhow, +"oughts and crosses" is a very old game. I don't think I ever heard it +called "noughts and crosses."</p> + +<p>In the following Class-list, I hope the solitary occupant of III. will +sheathe her claws when she hears how narrow an escape she has had of not +being named at all. Her account of the process by which she got the +answer is so meagre that, like the nursery tale of "Jack-a-Minory" (I<span class='pagenum'><a name="Page_105" id="Page_105">[105]</a></span> +trust I. E. A. will be merciful to the spelling), it is scarcely to be +distinguished from "zero."</p> + + +<p class="center">CLASS LIST.</p> + +<p><span style="margin-left: 12.5em;">I.</span></p> + +<p> +<span style="margin-left: 10.5em;"><span class="smcap">Guy.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Old Cat.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Sea-Breeze.</span></span><br /> +</p> + +<p><span style="margin-left: 12.5em;">II.</span></p> + +<p> +<span style="margin-left: 10.5em;"><span class="smcap">Ayr.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Bradshaw of the Future.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">F. Lee.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">H. Vernon.</span></span><br /> +</p> + +<p><span style="margin-left: 12.5em;">III.</span></p> + +<p><span style="margin-left: 10.5em;"><span class="smcap">Cat.</span></span></p> + +<hr style="width: 45%;" /> +<p><span class='pagenum'><a name="Page_106" id="Page_106">[106]</a></span></p> +<h3>ANSWERS TO KNOT VI.</h3> + +<p><i>Problem 1.</i>—<i>A</i> and <i>B</i> began the year with only 1,000<i>l.</i> a-piece. +They borrowed nought; they stole nought. On the next New-Year's Day they +had 60,000<i>l.</i> between them. How did they do it?</p> + +<p><i>Solution.</i>—They went that day to the Bank of England. <i>A</i> stood in +front of it, while <i>B</i> went round and stood behind it.</p> + +<hr style="width: 45%;" /> + +<p>Two answers have been received, both worthy of much honour. <span class="smcap">Addlepate</span> +makes them borrow "0" and steal "0," and uses both cyphers by putting +them at the right-hand end of the 1,000<i>l.</i>, thus producing 100,000<i>l.</i>, +which is well over the mark. But (or to express it in Latin) <span class="smcap">At Spes +infracta</span> has solved it even more ingeniously: with the first cypher she +turns the "1" of the 1,000<i>l.</i> into a "9," and adds the result to the +original sum, thus getting 10,000<i>l.</i>: and in this, by means of the +other "0," she turns the "1" into a "6," thus hitting the exact +60,000<i>l.</i><span class='pagenum'><a name="Page_107" id="Page_107">[107]</a></span></p> + + +<p class="center">CLASS LIST</p> + +<p><span style="margin-left: 12.5em;">I.</span></p> + +<p><span style="margin-left: 10.5em;"><span class="smcap">At Spes Infracta.</span></span></p> + +<p><span style="margin-left: 12.5em;">II.</span></p> + +<p><span style="margin-left: 10.5em;"><span class="smcap">Addlepate.</span></span></p> + +<hr style="width: 45%;" /> + +<p><i>Problem 2.</i>—<i>L</i> makes 5 scarves, while <i>M</i> makes 2: <i>Z</i> makes 4 while +<i>L</i> makes 3. Five scarves of <i>Z</i>'s weigh one of <i>L</i>'s; 5 of <i>M</i>'s weigh +3 of <i>Z</i>'s. One of <i>M</i>'s is as warm as 4 of <i>Z</i>'s: and one of <i>L</i>'s as +warm as 3 of <i>M</i>'s. Which is best, giving equal weight in the result to +rapidity of work, lightness, and warmth?</p> + +<p><i>Answer.</i>—The order is <i>M</i>, <i>L</i>, <i>Z</i>.</p> + +<hr style="width: 45%;" /> + +<p><i>Solution.</i>—As to rapidity (other things being constant) <i>L</i>'s merit is +to <i>M</i>'s in the ratio of 5 to 2: <i>Z</i>'s to <i>L</i>'s in the ratio of 4 to 3. +In order to get one set of 3 numbers fulfilling these conditions, it is +perhaps simplest to take the one that occurs <i>twice</i> as unity, and +reduce the others to fractions: this gives, for <i>L</i>, <i>M</i>, and <i>Z</i>, the +marks 1, <sup>2</sup>⁄<sub>5</sub>, <sup>2</sup>⁄<sub>3</sub>. In estimating for <i>lightness</i>, we observe that the +greater the weight, the less the merit, so that <i>Z</i>'s merit is to <i>L</i>'s +as 5 to 1. Thus the marks for <i>lightness</i> are <sup>1</sup>⁄<sub>5</sub>, <sup>2</sup>⁄<sub>3</sub>, 1. And +similarly, the marks for warmth are 3, 1, ¼. To get the<span class='pagenum'><a name="Page_108" id="Page_108">[108]</a></span> total result, +we must <i>multiply</i> <i>L</i>'s 3 marks together, and do the same for <i>M</i> and +for <i>Z</i>. The final numbers are 1 × <sup>1</sup>⁄<sub>5</sub> × 3, <sup>2</sup>⁄<sub>5</sub> × <sup>2</sup>⁄<sub>3</sub> × 1, +<sup>2</sup>⁄<sub>3</sub> × 1 × +¼; <i>i.e.</i> <sup>3</sup>⁄<sub>5</sub>, +<a name="corr3" id="corr3"></a><a class="correction" href="#cn3" title="changed from '2'"><sup>2</sup>⁄<sub>3</sub></a>, <sup>1</sup>⁄<sub>3</sub>; <i>i.e.</i> multiplying throughout by 15 (which +will not alter the proportion), 9, 10, 5; showing the order of merit to +be <i>M</i>, <i>L</i>, <i>Z</i>.</p> + +<hr style="width: 45%;" /> + +<p>Twenty-nine answers have been received, of which five are right, and +twenty-four wrong. These hapless ones have all (with three exceptions) +fallen into the error of <i>adding</i> the proportional numbers together, for +each candidate, instead of <i>multiplying</i>. <i>Why</i> the latter is right, +rather than the former, is fully proved in text-books, so I will not +occupy space by stating it here: but it can be <i>illustrated</i> very easily +by the case of length, breadth, and depth. Suppose <i>A</i> and <i>B</i> are rival +diggers of rectangular tanks: the amount of work done is evidently +measured by the number of <i>cubical feet</i> dug out. Let <i>A</i> dig a tank 10 +feet long, 10 wide, 2 deep: let <i>B</i> dig one 6 feet long, 5 wide, 10 +deep. The cubical contents are 200, 300; <i>i.e.</i> <i>B</i> is best digger in +the ratio of 3 to 2. Now try marking for length, width, and depth, +separately; giving a maximum mark of 10 to the best in each contest, and +then <i>adding</i> the results!</p> + +<p>Of the twenty-four malefactors, one gives no working, and so has no real +claim to be named; but I break the rule for once, in deference to its +success in Problem 1:<span class='pagenum'><a name="Page_109" id="Page_109">[109]</a></span> he, she, or it, is <span class="smcap">Addlepate</span>. The other +twenty-three may be divided into five groups.</p> + +<p>First and worst are, I take it, those who put the rightful winner +<i>last</i>; arranging them as "Lolo, Zuzu, Mimi." The names of these +desperate wrong-doers are <span class="smcap">Ayr</span>, <span class="smcap">Bradshaw of the Future</span>, <span class="smcap">Furze-bush</span> and +<span class="smcap">Pollux</span> (who send a joint answer), <span class="smcap">Greystead</span>, <span class="smcap">Guy</span>, <span class="smcap">Old Hen</span>, and <span class="smcap">Simple +Susan</span>. The latter was <i>once</i> best of all; the Old Hen has taken +advantage of her simplicity, and beguiled her with the chaff which was +the bane of her own chickenhood.</p> + +<p>Secondly, I point the finger of scorn at those who have put the worst +candidate at the top; arranging them as "Zuzu, Mimi, Lolo." They are +<span class="smcap">Graecia</span>, M. M., <span class="smcap">Old Cat</span>, and R. E. X. "'Tis Greece, but——."</p> + +<p>The third set have avoided both these enormities, and have even +succeeded in putting the worst last, their answer being "Lolo, Mimi, +Zuzu." Their names are <span class="smcap">Ayr</span> (who also appears among the "quite too too"), +<span class="smcap">Clifton C.</span>, F. B., <span class="smcap">Fifee</span>, <span class="smcap">Grig</span>, <span class="smcap">Janet</span>, and <span class="smcap">Mrs. Sairey Gamp</span>. F. B. has +not fallen into the common error; she <i>multiplies</i> together the +proportionate numbers she gets, but in getting them she goes wrong, by +reckoning warmth as a <i>de</i>-merit. Possibly she is "Freshly Burnt," or +comes "From Bombay." <span class="smcap">Janet</span> and <span class="smcap">Mrs. Sairey Gamp</span> have also avoided this +error: the method they have adopted is<span class='pagenum'><a name="Page_110" id="Page_110">[110]</a></span> shrouded in mystery—I scarcely +feel competent to criticize it. <span class="smcap">Mrs. Gamp</span> says "if Zuzu makes 4 while +Lolo makes 3, Zuzu makes 6 while Lolo makes 5 (bad reasoning), while +Mimi makes 2." From this she concludes "therefore Zuzu excels in speed +by 1" (<i>i.e.</i> when compared with Lolo; but what about Mimi?). She then +compares the 3 kinds of excellence, measured on this mystic scale. <span class="smcap">Janet</span> +takes the statement, that "Lolo makes 5 while Mimi makes 2," to prove +that "Lolo makes 3 while Mimi makes 1 and Zuzu 4" (worse reasoning than +<span class="smcap">Mrs. Gamp's</span>), and thence concludes that "Zuzu excels in speed by <sup>1</sup>⁄<sub>8</sub>"! +<span class="smcap">Janet</span> should have been <span class="smcap">Adeline</span>, "mystery of mysteries!"</p> + +<p>The fourth set actually put Mimi at the top, arranging them as "Mimi, +Zuzu, Lolo." They are <span class="smcap">Marquis and Co.</span>, <span class="smcap">Martreb</span>, S. B. B. (first initial +scarcely legible: <i>may</i> be meant for "J"), and <span class="smcap">Stanza</span>.</p> + +<p>The fifth set consist of <span class="smcap">An ancient Fish</span> and <span class="smcap">Camel</span>. These ill-assorted +comrades, by dint of foot and fin, have scrambled into the right answer, +but, as their method is wrong, of course it counts for nothing. Also <span class="smcap">An +ancient Fish</span> has very ancient and fishlike ideas as to <i>how</i> numbers +represent merit: she says "Lolo gains 2½ on Mimi." Two and a half +<i>what</i>? Fish, fish, art thou in thy duty?</p> + +<p>Of the five winners I put <span class="smcap">Balbus</span> and <span class="smcap">The elder Traveller</span> slightly below +the other three—<span class="smcap">Balbus</span> for<span class='pagenum'><a name="Page_111" id="Page_111">[111]</a></span> defective reasoning, the other for scanty +working. <span class="smcap">Balbus</span> gives two reasons for saying that <i>addition</i> of marks is +<i>not</i> the right method, and then adds "it follows that the decision must +be made by <i>multiplying</i> the marks together." This is hardly more +logical than to say "This is not Spring: <i>therefore</i> it must be Autumn."</p> + + +<p class="center">CLASS LIST.</p> + +<p><span style="margin-left: 12.5em;">I.</span></p> + +<p> +<span style="margin-left: 10.5em;"><span class="smcap">Dinah Mite.</span></span><br /> +<span style="margin-left: 10.5em;">E. B. D. L.</span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Joram.</span></span><br /> +</p> + +<p><span style="margin-left: 12.5em;">II.</span></p> + +<p> +<span style="margin-left: 10.5em;"><span class="smcap">Balbus.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">The Elder Traveller.</span></span><br /> +</p> + +<hr style="width: 45%;" /> + +<p>With regard to Knot V., I beg to express to <span class="smcap">Vis Inertiæ</span> and to any +others who, like her, understood the condition to be that <i>every</i> marked +picture must have <i>three</i> marks, my sincere regret that the unfortunate +phrase "<i>fill</i> the columns with oughts and crosses" should have caused +them to waste so much time and trouble. I can only repeat that a +<i>literal</i> interpretation of "fill" would seem to <i>me</i> to require that +<i>every</i> picture in the gallery should be marked. <span class="smcap">Vis Inertiæ</span> would have +been in the First Class if she had sent in the solution she now offers.</p> + +<hr style="width: 45%;" /> +<p><span class='pagenum'><a name="Page_112" id="Page_112">[112]</a></span></p> +<h3>ANSWERS TO KNOT VII.</h3> + +<p><i>Problem.</i>—Given that one glass of lemonade, 3 sandwiches, and 7 +biscuits, cost 1<i>s.</i> 2<i>d.</i>; and that one glass of lemonade, 4 +sandwiches, and 10 biscuits, cost 1<i>s.</i> 5<i>d.</i>: find the cost of (1) a +glass of lemonade, a sandwich, and a biscuit; and (2) 2 glasses of +lemonade, 3 sandwiches, and 5 biscuits.</p> + +<p><i>Answer.</i>—(1) 8<i>d.</i>; (2) 1<i>s.</i> 7<i>d.</i></p> + +<p><i>Solution.</i>—This is best treated algebraically. Let <i>x</i> = the cost (in +pence) of a glass of lemonade, <i>y</i> of a sandwich, and <i>z</i> of a biscuit. +Then we have <i>x</i> + 3<i>y</i> + 7<i>z</i> = 14, and <i>x</i> + 4<i>y</i> + 10<i>z</i> = 17. And we +require the values of <i>x</i> + <i>y</i> + <i>z</i>, and of 2<i>x</i> + 3<i>y</i> + 5<i>z</i>. Now, +from <i>two</i> equations only, we cannot find, <i>separately</i>, the values of +<i>three</i> unknowns: certain <i>combinations</i> of them may, however, be found. +Also we know that we can, by the help of the given equations, eliminate +2 of the 3 unknowns from the quantity whose value is required, which +will then contain one only. If, then, the required value is +ascertainable at all, it can only be by the 3rd unknown vanishing of +itself: otherwise the problem is impossible.<span class='pagenum'><a name="Page_113" id="Page_113">[113]</a></span></p> + +<p>Let us then eliminate lemonade and sandwiches, and reduce everything to +biscuits—a state of things even more depressing than "if all the world +were apple-pie"—by subtracting the 1st equation from the 2nd, which +eliminates lemonade, and gives <i>y</i> + 3<i>z</i> = 3, or <i>y</i> = 3-3<i>z</i>; and then +substituting this value of <i>y</i> in the 1st, which gives <i>x</i>-2<i>z</i> = 5, +<i>i.e.</i> <i>x</i> = 5 + 2<i>z</i>. Now if we substitute these values of <i>x</i>, <i>y</i>, in +the quantities whose values are required, the first becomes (5 + 2<i>z</i>) + +(3-3<i>z</i>) + <i>z</i>, <i>i.e.</i> 8: and the second becomes 2(5 + 2<i>z</i>) + 3(3-3<i>z</i>) ++ 5<i>z</i>, <i>i.e.</i> 19. Hence the answers are (1) 8<i>d.</i>, (2) 1<i>s.</i> 7<i>d.</i></p> + +<hr style="width: 45%;" /> + +<p>The above is a <i>universal</i> method: that is, it is absolutely certain +either to produce the answer, or to prove that no answer is possible. +The question may also be solved by combining the quantities whose values +are given, so as to form those whose values are required. This is merely +a matter of ingenuity and good luck: and as it <i>may</i> fail, even when the +thing is possible, and is of no use in proving it <i>im</i>possible, I cannot +rank this method as equal in value with the other. Even when it +succeeds, it may prove a very tedious process. Suppose the 26 +competitors, who have sent in what I may call <i>accidental</i> solutions, +had had a question to deal with where every number contained 8 or 10 +digits! I suspect it would have been a case of "silvered is the raven +hair" (see<span class='pagenum'><a name="Page_114" id="Page_114">[114]</a></span> "Patience") before any solution would have been hit on by +the most ingenious of them.</p> + +<p>Forty-five answers have come in, of which 44 give, I am happy to say, +some sort of <i>working</i>, and therefore deserve to be mentioned by name, +and to have their virtues, or vices as the case may be, discussed. +Thirteen have made assumptions to which they have no right, and so +cannot figure in the Class-list, even though, in 10 of the +<a name="corr4" id="corr4"></a><a class="correction" href="#cn4" title="changed from '12'">13</a> cases, the +answer is right. Of the remaining 28, no less than 26 have sent in +<i>accidental</i> solutions, and therefore fall short of the highest honours.</p> + +<p>I will now discuss individual cases, taking the worst first, as my +custom is.</p> + +<p><span class="smcap">Froggy</span> gives no working—at least this is all he gives: after stating +the given equations, he says "therefore the difference, 1 sandwich + 3 +biscuits, = 3<i>d.</i>": then follow the amounts of the unknown bills, with +no further hint as to how he got them. <span class="smcap">Froggy</span> has had a <i>very</i> narrow +escape of not being named at all!</p> + +<p>Of those who are wrong, <span class="smcap">Vis Inertiæ</span> has sent in a piece of incorrect +working. Peruse the horrid details, and shudder! She takes <i>x</i> (call it +"<i>y</i>") as the cost of a sandwich, and concludes (rightly enough) that a +biscuit will cost (3-<i>y</i>)/3. She then subtracts the second equation from +the first, and deduces 3<i>y</i> + 7 × (3-<i>y</i>)/3-4<i>y</i> + 10 × (3-<i>y</i>)/3 = 3.<span class='pagenum'><a name="Page_115" id="Page_115">[115]</a></span> +By making two mistakes in this line, she brings out <i>y</i> = <sup>2</sup>⁄<sub>2</sub>. Try it +again, oh <span class="smcap">Vis Inertiæ</span>! Away with <span class="smcap">Inertiæ</span>: infuse a little more <span class="smcap">Vis</span>: and +you will bring out the correct (though uninteresting) result, 0 = 0! +This will show you that it is hopeless to try to coax any one of these 3 +unknowns to reveal its <i>separate</i> value. The other competitor, who is +wrong throughout, is either J. M. C. or T. M. C.: but, whether he be a +Juvenile Mis-Calculator or a True Mathematician Confused, he makes the +answers 7<i>d.</i> and 1<i>s.</i> 5<i>d.</i> He assumes, with Too Much Confidence, that +biscuits were ½<i>d.</i> each, and that Clara paid for 8, though she only +ate 7!</p> + +<p>We will now consider the 13 whose working is wrong, though the answer is +right: and, not to measure their demerits too exactly, I will take them +in alphabetical order. <span class="smcap">Anita</span> finds (rightly) that "1 sandwich and 3 +biscuits cost 3<i>d.</i>," and proceeds "therefore 1 sandwich = 1½<i>d.</i>, 3 +biscuits = 1½<i>d.</i>, 1 lemonade = 6<i>d.</i>" <span class="smcap">Dinah Mite</span> begins like <span class="smcap">Anita</span>: +and thence proves (rightly) that a biscuit costs less than a 1<i>d.</i>: +whence she concludes (wrongly) that it <i>must</i> cost ½<i>d.</i> F. C. W. is +so beautifully resigned to the certainty of a verdict of "guilty," that +I have hardly the heart to utter the word, without adding a "recommended +to mercy owing to extenuating circumstances." But really, you know, +where <i>are</i> the extenuating<span class='pagenum'><a name="Page_116" id="Page_116">[116]</a></span> circumstances? She begins by assuming that +lemonade is 4<i>d.</i> a glass, and sandwiches 3<i>d.</i> each, (making with the 2 +given equations, <i>four</i> conditions to be fulfilled by <i>three</i> miserable +unknowns!). And, having (naturally) developed this into a contradiction, +she then tries 5<i>d.</i> and 2<i>d.</i> with a similar result. (N.B. <i>This</i> +process might have been carried on through the whole of the Tertiary +Period, without gratifying one single Megatherium.) She then, by a +"happy thought," tries half-penny biscuits, and so obtains a consistent +result. This may be a good solution, viewing the problem as a conundrum: +but it is <i>not</i> scientific. <span class="smcap">Janet</span> identifies sandwiches with biscuits! +"One sandwich + 3 biscuits" she makes equal to "4." Four <i>what</i>? <span class="smcap">Mayfair</span> +makes the astounding assertion that the equation, <i>s</i> + 3<i>b</i> = 3, "is +evidently only satisfied by <i>s</i> = <sup>2</sup>⁄<sub>2</sub>, <i>b</i> = ½"! <span class="smcap">Old Cat</span> believes that +the assumption that a sandwich costs 1½<i>d.</i> is "the only way to avoid +unmanageable fractions." But <i>why</i> avoid them? Is there not a certain +glow of triumph in taming such a fraction? "Ladies and gentlemen, the +fraction now before you is one that for years defied all efforts of a +refining nature: it was, in a word, hopelessly vulgar. Treating it as a +circulating decimal (the treadmill of fractions) only made matters +worse. As a last resource, I reduced it to its lowest terms, and +extracted its square root!" Joking<span class='pagenum'><a name="Page_117" id="Page_117">[117]</a></span> apart, let me thank <span class="smcap">Old Cat</span> for some +very kind words of sympathy, in reference to a correspondent (whose name +I am happy to say I have now forgotten) who had found fault with me as a +discourteous critic. O. V. L. is beyond my comprehension. He takes the +given equations as (1) and (2): thence, by the process [(2)-(1)] deduces +(rightly) equation (3) viz. <i>s</i> + 3<i>b</i> = 3: and thence again, by the +process [×33] (a hopeless mystery), deduces 3<i>s</i> + 4<i>b</i> = 4. I have +nothing to say about it: I give it up. <span class="smcap">Sea-Breeze</span> says "it is immaterial +to the answer" (why?) "in what proportion 3<i>d.</i> is divided between the +sandwich and the 3 biscuits": so she assumes <i>s</i> = l½<i>d.</i>, <i>b</i> = +½<i>d.</i> <span class="smcap">Stanza</span> is one of a very irregular metre. At first she (like +<span class="smcap">Janet</span>) identifies sandwiches with biscuits. She then tries two +assumptions (<i>s</i> = 1, <i>b</i> = <sup>2</sup>⁄<sub>3</sub>, and <i>s</i> = ½ <i>b</i> = <sup>2</sup>⁄<sub>6</sub>), and +(naturally) ends in contradictions. Then she returns to the first +assumption, and finds the 3 unknowns separately: <i>quod est absurdum</i>. +<span class="smcap">Stiletto</span> identifies sandwiches and biscuits, as "articles." Is the word +ever used by confectioners? I fancied "What is the next article, Ma'am?" +was limited to linendrapers. <span class="smcap">Two Sisters</span> first assume that biscuits are +4 a penny, and then that they are 2 a penny, adding that "the answer +will of course be the same in both cases." It is a dreamy<span class='pagenum'><a name="Page_118" id="Page_118">[118]</a></span> remark, +making one feel something like Macbeth grasping at the spectral dagger. +"Is this a statement that I see before me?" If you were to say "we both +walked the same way this morning," and <i>I</i> were to say "<i>one</i> of you +walked the same way, but the other didn't," which of the three would be +the most hopelessly confused? <span class="smcap">Turtle Pyate</span> (what <i>is</i> a Turtle Pyate, +please?) and <span class="smcap">Old Crow</span>, who send a joint answer, and Y. Y., adopt the +same method. Y. Y. gets the equation <i>s</i> + 3<i>b</i> = 3: and then says "this +sum must be apportioned in one of the three following ways." It <i>may</i> +be, I grant you: but Y. Y. do you say "must"? I fear it is <i>possible</i> +for Y. Y. to be <i>two</i> Y's. The other two conspirators are less positive: +they say it "can" be so divided: but they add "either of the three +prices being right"! This is bad grammar and bad arithmetic at once, oh +mysterious birds!</p> + +<p>Of those who win honours, <span class="smcap">The Shetland Snark</span> must have the 3rd class all +to himself. He has only answered half the question, viz. the amount of +Clara's luncheon: the two little old ladies he pitilessly leaves in the +midst of their "difficulty." I beg to assure him (with thanks for his +friendly remarks) that entrance-fees and subscriptions are things +unknown in that most economical of clubs, "The Knot-Untiers."</p> + +<p>The authors of the 26 "accidental" solutions differ only in the number +of steps they have taken between the<span class='pagenum'><a name="Page_119" id="Page_119">[119]</a></span> <i>data</i> and the answers. In order +to do them full justice I have arranged the 2nd class in sections, +according to the number of steps. The two Kings are fearfully +deliberate! I suppose walking quick, or taking short cuts, is +inconsistent with kingly dignity: but really, in reading <span class="smcap">Theseus'</span> +solution, one almost fancied he was "marking time," and making no +advance at all! The other King will, I hope, pardon me for having +altered "Coal" into "Cole." King Coilus, or Coil, seems to have reigned +soon after Arthur's time. Henry of Huntingdon identifies him with the +King Coël who first built walls round Colchester, which was named after +him. In the Chronicle of Robert of Gloucester we read:—</p> + +<div class="poem"><div class="stanza"> +<span class="i0">"Aftur Kyng Aruirag, of wam we habbeth y told,<br /></span> +<span class="i0">Marius ys sone was kyng, quoynte mon & bold.<br /></span> +<span class="i0">And ys sone was aftur hym, <i>Coil</i> was ys name,<br /></span> +<span class="i0">Bothe it were quoynte men, & of noble fame."<br /></span> +</div></div> + +<p><span class="smcap">Balbus</span> lays it down as a general principle that "in order to ascertain +the cost of any one luncheon, it must come to the same amount upon two +different assumptions." (<i>Query.</i> Should not "it" be "we"? Otherwise the +<i>luncheon</i> is represented as wishing to ascertain its own cost!) He then +makes two assumptions—one, that sandwiches cost nothing; the other, +that biscuits cost nothing, (either arrangement would lead to the shop +being inconveniently crowded!)—and brings out the unknown<span class='pagenum'><a name="Page_120" id="Page_120">[120]</a></span> luncheons as +8<i>d.</i> and 19<i>d.</i>, on each assumption. He then concludes that this +agreement of results "shows that the answers are correct." Now I propose +to disprove his general law by simply giving <i>one</i> instance of its +failing. One instance is quite enough. In logical language, in order to +disprove a "universal affirmative," it is enough to prove its +contradictory, which is a "particular negative." (I must pause for a +digression on Logic, and especially on Ladies' Logic. The universal +affirmative "everybody says he's a duck" is crushed instantly by proving +the particular negative "Peter says he's a goose," which is equivalent +to "Peter does <i>not</i> say he's a duck." And the universal negative +"nobody calls on her" is well met by the particular affirmative "<i>I</i> +called yesterday." In short, either of two contradictories disproves the +other: and the moral is that, since a particular proposition is much +more easily proved than a universal one, it is the wisest course, in +arguing with a Lady, to limit one's <i>own</i> assertions to "particulars," +and leave <i>her</i> to prove the "universal" contradictory, if she can. You +will thus generally secure a <i>logical</i> victory: a <i>practical</i> victory is +not to be hoped for, since she can always fall back upon the crushing +remark "<i>that</i> has nothing to do with it!"—a move for which Man has not +yet discovered any satisfactory answer. Now let us return to <span class="smcap">Balbus</span>.) +Here is my "particular negative," on which to test his rule. Suppose the +two<span class='pagenum'><a name="Page_121" id="Page_121">[121]</a></span> recorded luncheons to have been "2 buns, one queen-cake, 2 +sausage-rolls, and a bottle of Zoëdone: total, one-and-ninepence," and +"one bun, 2 queen-cakes, a sausage-roll, and a bottle of Zoëdone: total, +one-and-fourpence." And suppose Clara's unknown luncheon to have been "3 +buns, one queen-cake, one sausage-roll, and 2 bottles of Zoëdone:" while +the two little sisters had been indulging in "8 buns, 4 queen-cakes, 2 +sausage-rolls, and 6 bottles of Zoëdone." (Poor souls, how thirsty they +must have been!) If <span class="smcap">Balbus</span> will kindly try this by his principle of "two +assumptions," first assuming that a bun is 1<i>d.</i> and a queen-cake 2<i>d.</i>, +and then that a bun is 3<i>d.</i> and a queen-cake 3<i>d.</i>, he will bring out +the other two luncheons, on each assumption, as "one-and-nine-pence" and +"four-and-ten-pence" respectively, which harmony of results, he will +say, "shows that the answers are correct." And yet, as a matter of fact, +the buns were 2<i>d.</i> each, the queen-cakes 3<i>d.</i>, the sausage-rolls +6<i>d.</i>, and the Zoëdone 2<i>d.</i> a bottle: so that Clara's third luncheon +had cost one-and-sevenpence, and her thirsty friends had spent +<a name="corr5" id="corr5"></a><a class="correction" href="#cn5" title="changed from 'four-and fourpence'">four-and-fourpence</a>!</p> + +<p>Another remark of <span class="smcap">Balbus</span> I will quote and discuss: for I think that it +also may yield a moral for some of my readers. He says "it is the same +thing in substance whether in solving this problem we use words and call +it Arithmetic, or use letters and signs and call it Algebra."<span class='pagenum'><a name="Page_122" id="Page_122">[122]</a></span> Now this +does not appear to me a correct description of the two methods: the +Arithmetical method is that of "synthesis" only; it goes from one known +fact to another, till it reaches its goal: whereas the Algebraical +method is that of "analysis": it begins with the goal, symbolically +represented, and so goes backwards, dragging its veiled victim with it, +till it has reached the full daylight of known facts, in which it can +tear off the veil and say "I know you!"</p> + +<p>Take an illustration. Your house has been broken into and robbed, and +you appeal to the policeman who was on duty that night. "Well, Mum, I +did see a chap getting out over your garden-wall: but I was a good bit +off, so I didn't chase him, like. I just cut down the short way to the +Chequers, and who should I meet but Bill Sykes, coming full split round +the corner. So I just ups and says 'My lad, you're wanted.' That's all I +says. And he says 'I'll go along quiet, Bobby,' he says, 'without the +darbies,' he says." There's your <i>Arithmetical</i> policeman. Now try the +other method. "I seed somebody a running, but he was well gone or ever +<i>I</i> got nigh the place. So I just took a look round in the garden. And I +noticed the foot-marks, where the chap had come right across your +flower-beds. They was good big foot-marks sure-ly. And I noticed as the +left foot went down at the heel, ever so much deeper than the other. And +I says to myself<span class='pagenum'><a name="Page_123" id="Page_123">[123]</a></span> 'The chap's been a big hulking chap: and he goes lame +on his left foot.' And I rubs my hand on the wall where he got over, and +there was soot on it, and no mistake. So I says to myself 'Now where can +I light on a big man, in the chimbley-sweep line, what's lame of one +foot?' And I flashes up permiscuous: and I says 'It's Bill Sykes!' says +I." There is your <i>Algebraical</i> policeman—a higher intellectual type, +to my thinking, than the other.</p> + +<p><span class="smcap">Little Jack's</span> solution calls for a word of praise, as he has written out +what really is an algebraical proof <i>in words</i>, without representing any +of his facts as equations. If it is all his own, he will make a good +algebraist in the time to come. I beg to thank <span class="smcap">Simple Susan</span> for some +kind words of sympathy, to the same effect as those received from <span class="smcap">Old +Cat</span>.</p> + +<p><span class="smcap">Hecla</span> and <span class="smcap">Martreb</span> are the only two who have used a method <i>certain</i> +either to produce the answer, or else to prove it impossible: so they +must share between them the highest honours.<span class='pagenum'><a name="Page_124" id="Page_124">[124]</a></span></p> + + +<p class="center">CLASS LIST.</p> + +<p><span style="margin-left: 12.5em;">I.</span></p> + +<p> +<span style="margin-left: 10.5em;"><span class="smcap">Hecla.</span><br /> +<span style="margin-left: 10.5em;">Martreb.</span></span><br /> +</p> + +<p><span style="margin-left: 12.5em;">II.</span></p> + +<p><span style="margin-left: 10.5em;">§ 1 (2 <i>steps</i>).</span></p> + +<p> +<span style="margin-left: 10.5em;"><span class="smcap">Adelaide.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Clifton C....</span></span><br /> +<span style="margin-left: 10.5em;">E. K. C.</span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Guy.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">L'Inconnu.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Little Jack.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Nil desperandum.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Simple Susan.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Yellow-Hammer.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Woolly One.</span></span><br /> +</p> + +<p><span style="margin-left: 10.5em;">§ 2 (3 <i>steps</i>).</span></p> + +<p> +<span style="margin-left: 10.5em;">A. A.</span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">A Christmas Carol.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Afternoon Tea.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">An appreciative Schoolma'am.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Baby.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Balbus.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Bog-Oak.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">The Red Queen.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Wall-flower.</span></span><br /> +</p> + +<p><span style="margin-left: 10.5em;">§ 3 (4 <i>steps</i>).</span></p> + +<p> +<span style="margin-left: 10.5em;"><span class="smcap">Hawthorn.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Joram.</span></span><br /> +<span style="margin-left: 10.5em;">S. S. G.</span><br /> +</p> + +<p><span style="margin-left: 10.5em;">§ 4 (5 <i>steps</i>).</span></p> + +<p> +<span style="margin-left: 10.5em;"><span class="smcap">A Stepney Coach.</span></span><br /> +</p> + +<p><span style="margin-left: 10.5em;">§ 5 (6 <i>steps</i>).</span></p> + +<p> +<span style="margin-left: 10.5em;"><span class="smcap">Bay Laurel.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Bradshaw of the Future.</span></span><br /> +</p> + +<p><span style="margin-left: 10.5em;">§ 6 (9 <i>steps</i>).</span></p> + +<p> +<span style="margin-left: 10.5em;"><span class="smcap">Old King Cole.</span></span><br /> +</p> + +<p><span style="margin-left: 10.5em;">§ 7 (14 <i>steps</i>).</span></p> + +<p> +<span style="margin-left: 10.5em;"><span class="smcap">Theseus.</span></span><br /> +</p> + +<hr style="width: 45%;" /> +<p><span class='pagenum'><a name="Page_125" id="Page_125">[125]</a></span></p> +<h3>ANSWERS TO CORRESPONDENTS.</h3> + +<p>I have received several letters on the subjects of Knots II. and VI., +which lead me to think some further explanation desirable.</p> + +<p>In Knot II., I had intended the numbering of the houses to begin at one +corner of the Square, and this was assumed by most, if not all, of the +competitors. <span class="smcap">Trojanus</span> however says "assuming, in default of any +information, that the street enters the square in the middle of each +side, it may be supposed that the numbering begins at a street." But +surely the other is the more natural assumption?</p> + +<p>In Knot VI., the first Problem was of course a mere <i>jeu de mots</i>, whose +presence I thought excusable in a series of Problems whose aim is to +entertain rather than to instruct: but it has not escaped the +contemptuous criticisms of two of my correspondents, who seem to think +that Apollo is in duty bound to keep his bow always on the stretch. +Neither of them has guessed it: and this is true human nature. Only the +other day—the 31st of September, to be quite exact—I met my old friend +Brown, and gave him a riddle I had just heard. With one great effort of +his colossal mind, Brown guessed it. "Right!" said I. "Ah," said<span class='pagenum'><a name="Page_126" id="Page_126">[126]</a></span> he, +"it's very neat—very neat. And it isn't an answer that would occur to +everybody. Very neat indeed." A few yards further on, I fell in with +Smith and to him I propounded the same riddle. He frowned over it for a +minute, and then gave it up. Meekly I faltered out the answer. "A poor +thing, sir!" Smith growled, as he turned away. "A very poor thing! I +wonder you care to repeat such rubbish!" Yet Smith's mind is, if +possible, even more colossal than Brown's.</p> + +<p>The second Problem of Knot VI. is an example in ordinary Double Rule of +Three, whose essential feature is that the result depends on the +variation of several elements, which are so related to it that, if all +but one be constant, it varies as that one: hence, if none be constant, +it varies as their product. Thus, for example, the cubical contents of a +rectangular tank vary as its length, if breadth and depth be constant, +and so on; hence, if none be constant, it varies as the product of the +length, breadth, and depth.</p> + +<p>When the result is not thus connected with the varying elements, the +Problem ceases to be Double Rule of Three and often becomes one of great +complexity.</p> + +<p>To illustrate this, let us take two candidates for a prize, <i>A</i> and <i>B</i>, +who are to compete in French, German, and Italian:</p> + +<p>(<i>a</i>) Let it be laid down that the result is to depend<span class='pagenum'><a name="Page_127" id="Page_127">[127]</a></span> on their +<i>relative</i> knowledge of each subject, so that, whether their marks, for +French, be "1, 2" or "100, 200," the result will be the same: and let it +also be laid down that, if they get equal marks on 2 papers, the final +marks are to have the same ratio as those of the 3rd paper. This is a +case of ordinary Double Rule of Three. We multiply <i>A</i>'s 3 marks +together, and do the same for <i>B</i>. Note that, if <i>A</i> gets a single "0," +his final mark is "0," even if he gets full marks for 2 papers while <i>B</i> +gets only one mark for each paper. This of course would be very unfair +on <i>A</i>, though a correct solution under the given conditions.</p> + +<p>(<i>b</i>) The result is to depend, as before, on <i>relative</i> knowledge; but +French is to have twice as much weight as German or Italian. This is an +unusual form of question. I should be inclined to say "the resulting +ratio is to be nearer to the French ratio than if we multiplied as in +(<i>a</i>), and so much nearer that it would be necessary to use the other +multipliers <i>twice</i> to produce the same result as in (<i>a</i>):" <i>e.g.</i> if +the French Ratio were <sup>2</sup>⁄<sub>10</sub>, and the others <sup>2</sup>⁄<sub>9</sub>, <sup>1</sup>⁄<sub>9</sub> so that the ultimate +ratio, by method (<i>a</i>), would be <sup>2</sup>⁄<sub>45</sub>, I should multiply instead by <sup>2</sup>⁄<sub>3</sub>, +<sup>1</sup>⁄<sub>3</sub>, giving the result, <sup>1</sup>⁄<sub>3</sub> which is nearer to <sup>2</sup>⁄<sub>10</sub> than if he had used +method (<i>a</i>).</p> + +<p>(<i>c</i>) The result is to depend on <i>actual</i> amount of knowledge of the 3 +subjects collectively. Here we have<span class='pagenum'><a name="Page_128" id="Page_128">[128]</a></span> to ask two questions. (1) What is +to be the "unit" (<i>i.e.</i> "standard to measure by") in each subject? (2) +Are these units to be of equal, or unequal value? The usual "unit" is +the knowledge shown by answering the whole paper correctly; calling this +"100," all lower amounts are represented by numbers between "0" and +"100." Then, if these units are to be of equal value, we simply add +<i>A</i>'s 3 marks together, and do the same for <i>B</i>.</p> + +<p>(<i>d</i>) The conditions are the same as (<i>c</i>), but French is to have double +weight. Here we simply double the French marks, and add as before.</p> + +<p>(<i>e</i>) French is to have such weight, that, if other marks be equal, the +ultimate ratio is to be that of the French paper, so that a "0" in this +would swamp the candidate: but the other two subjects are only to affect +the result collectively, by the amount of knowledge shown, the two being +reckoned of equal value. Here I should add <i>A</i>'s German and Italian +marks together, and multiply by his French mark.</p> + +<p>But I need not go on: the problem may evidently be set with many varying +conditions, each requiring its own method of solution. The Problem in +Knot VI. was meant to belong to variety (<i>a</i>), and to make this clear, I +inserted the following passage:</p> + +<p>"Usually the competitors differ in one point only. Thus, last year, Fifi +and Gogo made the same number of<span class='pagenum'><a name="Page_129" id="Page_129">[129]</a></span> scarves in the trial week, and they +were equally light; but Fifi's were twice as warm as Gogo's, and she was +pronounced twice as good."</p> + +<p>What I have said will suffice, I hope, as an answer to <span class="smcap">Balbus</span>, who holds +that (<i>a</i>) and (<i>c</i>) are the only possible varieties of the problem, and +that to say "We cannot use addition, therefore we must be intended to +use multiplication," is "no more illogical than, from knowledge that one +was not born in the night, to infer that he was born in the daytime"; +and also to <span class="smcap">Fifee</span>, who says "I think a little more consideration will +show you that our 'error of <i>adding</i> the proportional numbers together +for each candidate instead of <i>multiplying</i>' is no error at all." Why, +even if addition <i>had</i> been the right method to use, not one of the +writers (I speak from memory) showed any consciousness of the necessity +of fixing a "unit" for each subject. "No error at all!" They were +positively steeped in error!</p> + +<p>One correspondent (I do not name him, as the communication is not quite +friendly in tone) writes thus:—"I wish to add, very respectfully, 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 answer. That such a tone must not be" ("be not"?) +"agreeable to<span class='pagenum'><a name="Page_130" id="Page_130">[130]</a></span> the persons concerned who have made mistakes may possibly +have no great weight with you, but I hope you will feel that it would be +as well not to employ it, <i>unless you are quite certain of being correct +yourself</i>." The only instances the writer gives of the "trenchant +expressions" are "hapless" and "malefactors." I beg to assure him (and +any others who may need the assurance: I trust there are none) that all +such words have been used in jest, and with no idea that they could +possibly annoy any one, and that I sincerely regret any annoyance I may +have thus inadvertently given. May I hope that in future they will +recognise the distinction between severe language used in sober earnest, +and the "words of unmeant bitterness," which Coleridge has alluded to in +that lovely passage beginning "A little child, a limber elf"? If the +writer will refer to that passage, or to the preface to "Fire, Famine, +and Slaughter," he will find the distinction, for which I plead, far +better drawn out than I could hope to do in any words of mine.</p> + +<p>The writer's insinuation that I care not how much annoyance I give to my +readers I think it best to pass over in silence; but to his concluding +remark I must entirely demur. 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<span class='pagenum'><a name="Page_131" id="Page_131">[131]</a></span> being correct." I trust that the +knot-untiers and I are not on such terms as those!</p> + +<p>I beg to thank <i>G. B.</i> for the offer of a puzzle—which, however, is too +like the old one "Make four 9's into 100."</p> + +<hr style="width: 45%;" /> +<p><span class='pagenum'><a name="Page_132" id="Page_132">[132]</a></span></p> +<h3>ANSWERS TO KNOT VIII.</h3> + +<p class="fm3">§ 1. <span class="smcap">The Pigs.</span></p> + +<p><i>Problem.</i>—Place twenty-four pigs in four sties so that, as you go +round and round, you may always find the number in each sty nearer to +ten than the number in the last.</p> + +<p><i>Answer.</i>—Place 8 pigs in the first sty, 10 in the second, nothing in +the third, and 6 in the fourth: 10 is nearer ten than 8; nothing is +nearer ten than 10; 6 is nearer ten than nothing; and 8 is nearer ten +than 6.</p> + +<hr style="width: 45%;" /> + +<p>This problem is noticed by only two correspondents. <span class="smcap">Balbus</span> says "it +certainly cannot be solved mathematically, nor do I see how to solve it +by any verbal quibble." <span class="smcap">Nolens Volens</span> makes Her Radiancy change the +direction of going round; and even then is obliged to add "the pigs must +be carried in front of her"!</p> + +<hr style="width: 45%;" /> +<p class="fm3">§ 2. <span class="smcap">The Grurmstipths.</span></p> + +<p><i>Problem.</i>—Omnibuses start from a certain point, both ways, every 15 +minutes. A traveller, starting on<span class='pagenum'><a name="Page_133" id="Page_133">[133]</a></span> foot along with one of them, meets +one in 12½ minutes: when will he be overtaken by one?</p> + +<p><i>Answer.</i>—In 6¼ minutes.</p> + +<hr style="width: 45%;" /> + +<p><i>Solution.</i>—Let "<i>a</i>" be the distance an omnibus goes in 15 minutes, +and "<i>x</i>" the distance from the starting-point to where the traveller is +overtaken. Since the omnibus met is due at the starting-point in 2½ +minutes, it goes in that time as far as the traveller walks in 12½; +<i>i.e.</i> it goes 5 times as fast. Now the overtaking omnibus is "<i>a</i>" +behind the traveller when he starts, and therefore goes "<i>a</i> + <i>x</i>" +while he goes "<i>x</i>." Hence <i>a</i> + <i>x</i> = 5<i>x</i>; <i>i.e.</i> 4<i>x</i> = <i>a</i>, and <i>x</i> += <i>a</i>/4. This distance would be traversed by an omnibus in <sup>15</sup>⁄<sub>4</sub> minutes, +and therefore by the traveller in 5 × <sup>15</sup>⁄<sub>4</sub>. Hence he is overtaken in +18¾ minutes after starting, <i>i.e.</i> in 6¼ minutes after meeting the +omnibus.</p> + +<hr style="width: 45%;" /> + +<p>Four answers have been received, of which two are wrong. <span class="smcap">Dinah Mite</span> +rightly states that the overtaking omnibus reached the point where they +met the other omnibus 5 minutes after they left, but wrongly concludes +that, going 5 times as fast, it would overtake them in another minute. +The travellers are 5-minutes-walk ahead<span class='pagenum'><a name="Page_134" id="Page_134">[134]</a></span> of the omnibus, and must walk +1-4th of this distance farther before the omnibus overtakes them, which +will be 1-5th of the distance traversed by the omnibus in the same time: +this will require 1¼ minutes more. <span class="smcap">Nolens Volens</span> tries it by a +process like "Achilles and the Tortoise." He rightly states that, when +the overtaking omnibus leaves the gate, the travellers are 1-5th of +"<i>a</i>" ahead, and that it will take the omnibus 3 minutes to traverse +this distance; "during which time" the travellers, he tells us, go +1-15th of "<i>a</i>" (this should be 1-25th). The travellers being now 1-15th +of "<i>a</i>" ahead, he concludes that the work remaining to be done is for +the travellers to go 1-60th of "<i>a</i>," while the omnibus goes 1-12th. The +<i>principle</i> is correct, and might have been applied earlier.</p> + + +<p class="center">CLASS LIST.</p> + +<p><span style="margin-left: 12.5em;">I.</span></p> + +<p> +<span style="margin-left: 10.5em;"><span class="smcap">Balbus.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Delta.</span></span><br /> +</p> + +<hr style="width: 45%;" /> +<p><span class='pagenum'><a name="Page_135" id="Page_135">[135]</a></span></p> +<h3>ANSWERS TO KNOT IX.</h3> + +<p class="fm3">§ 1. <span class="smcap">The Buckets.</span></p> + +<p><i>Problem.</i>—Lardner states that a solid, immersed in a fluid, displaces +an amount equal to itself in bulk. How can this be true of a small +bucket floating in a larger one?</p> + +<p><i>Solution.</i>—Lardner means, by "displaces," "occupies a space which +might be filled with water without any change in the surroundings." If +the portion of the floating bucket, which is above the water, could be +annihilated, and the rest of it transformed into water, the surrounding +water would not change its position: which agrees with Lardner's +statement.</p> + +<hr style="width: 45%;" /> + +<p>Five answers have been received, none of which explains the difficulty +arising from the well-known fact that a floating body is the same weight +as the displaced fluid. <span class="smcap">Hecla</span> says that "only that portion of the +smaller bucket which descends below the original level of the water can +be properly said to be immersed, and only an equal bulk of water is +displaced." Hence, according to<span class='pagenum'><a name="Page_136" id="Page_136">[136]</a></span> <span class="smcap">Hecla</span>, a solid, whose weight was equal +to that of an equal bulk of water, would not float till the whole of it +was below "the original level" of the water: but, as a matter of fact, +it would float as soon as it was all under water. <span class="smcap">Magpie</span> says the +fallacy is "the assumption that one body can displace another from a +place where it isn't," and that Lardner's assertion is incorrect, except +when the containing vessel "was originally full to the brim." But the +question of floating depends on the present state of things, not on past +history. <span class="smcap">Old King Cole</span> takes the same view as <span class="smcap">Hecla</span>. <span class="smcap">Tympanum</span> and <span class="smcap">Vindex</span> +assume that "displaced" means "raised above its original level," and +merely explain how it comes to pass that the water, so raised, is less +in bulk than the immersed portion of bucket, and thus land +themselves—or rather set themselves floating—in the same boat as +<span class="smcap">Hecla</span>.</p> + +<p>I regret that there is no Class-list to publish for this Problem.</p> + +<hr style="width: 45%;" /> + + +<p class="fm3">§ 2. <span class="smcap">Balbus' Essay.</span></p> + +<p><i>Problem.</i>—Balbus states that if a certain solid be immersed in a +certain vessel of water, the water will rise through a series of +distances, two inches, one inch, half an inch, &c., which series has no +end. He concludes that the water will rise without limit. Is this true?</p> + +<p><i>Solution.</i>—No. This series can never reach 4 inches,<span class='pagenum'><a name="Page_137" id="Page_137">[137]</a></span> since, however +many terms we take, we are always short of 4 inches by an amount equal +to the last term taken.</p> + +<hr style="width: 45%;" /> + +<p>Three answers have been received—but only two seem to me worthy of +honours.</p> + +<p><span class="smcap">Tympanum</span> says that the statement about the stick "is merely a blind, to +which the old answer may well be applied, <i>solvitur ambulando</i>, or +rather <i>mergendo</i>." I trust <span class="smcap">Tympanum</span> will not test this in his own +person, by taking the place of the man in Balbus' Essay! He would +infallibly be drowned.</p> + +<p><span class="smcap">Old King Cole</span> rightly points out that the series, 2, 1, &c., is a +decreasing Geometrical Progression: while <span class="smcap">Vindex</span> rightly identifies the +fallacy as that of "Achilles and the Tortoise."</p> + + +<p class="center">CLASS LIST.</p> + +<p><span style="margin-left: 12.5em;">I.</span></p> + +<p> +<span style="margin-left: 10.5em;"><span class="smcap">Old King Cole.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Vindex.</span></span><br /> +</p> + +<hr style="width: 45%;" /> + + +<p class="fm3">§ 3. <span class="smcap">The Garden.</span></p> + +<p><i>Problem.</i>—An oblong garden, half a yard longer than wide, consists +entirely of a gravel-walk, spirally arranged, a yard wide and 3,630 +yards long. Find the dimensions of the garden.<span class='pagenum'><a name="Page_138" id="Page_138">[138]</a></span></p> + +<p><i>Answer.</i>—60, 60½.</p> + +<p><i>Solution.</i>—The number of yards and fractions of a yard traversed in +walking along a straight piece of walk, is evidently the same as the +number of square-yards and fractions of a square-yard, contained in that +piece of walk: and the distance, traversed in passing through a +square-yard at a corner, is evidently a yard. Hence the area of the +garden is 3,630 square-yards: <i>i.e.</i>, if <i>x</i> be the width, <i>x</i> (<i>x</i> + +½) = 3,630. Solving this Quadratic, we find <i>x</i> = 60. Hence the +dimensions are 60, 60½.</p> + +<hr style="width: 45%;" /> + +<p>Twelve answers have been received—seven right and five wrong.</p> + +<p>C. G. L., <span class="smcap">Nabob</span>, <span class="smcap">Old Crow</span>, and <span class="smcap">Tympanum</span> assume that the number of yards +in the length of the path is equal to the number of square-yards in the +garden. This is true, but should have been proved. But each is guilty of +darker deeds. C. G. L.'s "working" consists of dividing 3,630 by 60. +Whence came this divisor, oh Segiel? Divination? Or was it a dream? I +fear this solution is worth nothing. <span class="smcap">Old Crow's</span> is shorter, and so (if +possible) worth rather less. He says the answer "is at once seen to be +60 × 60½"! <span class="smcap">Nabob's</span> calculation is short, but "as rich as a Nabob" in +error. He says that the square root of 3,630, multiplied by 2, equals +the<span class='pagenum'><a name="Page_139" id="Page_139">[139]</a></span> length plus the breadth. That is 60.25 × 2 = 120½. His first +assertion is only true of a <i>square</i> garden. His second is irrelevant, +since 60.25 is <i>not</i> the square-root of 3,630! Nay, Bob, this will <i>not</i> +do! <span class="smcap">Tympanum</span> says that, by extracting the square-root of 3,630, we get +60 yards with a remainder of 30/60, or half-a-yard, which we add so as +to make the oblong 60 × 60½. This is very terrible: but worse remains +behind. <span class="smcap">Tympanum</span> proceeds thus:—"But why should there be the half-yard +at all? Because without it there would be no space at all for flowers. +By means of it, we find reserved in the very centre a small plot of +ground, two yards long by half-a-yard wide, the only space not occupied +by walk." But Balbus expressly said that the walk "used up the whole of +the area." Oh, <span class="smcap">Tympanum</span>! My tympa is exhausted: my brain is num! I can +say no more.</p> + +<p><span class="smcap">Hecla</span> indulges, again and again, in that most fatal of all habits in +computation—the making <i>two</i> mistakes which cancel each other. She +takes <i>x</i> as the width of the garden, in yards, and <i>x</i> + ½ as its +length, and makes her first "coil" the sum of <i>x</i>½, <i>x</i>½, <i>x</i>-1, +<i>x</i>-1, <i>i.e.</i> 4<i>x</i>-3: but the fourth term should be <i>x</i>-1½, so that +her first coil is ½ a yard too long. Her second coil is the sum of +<i>x</i>-2½, <i>x</i>-2½, <i>x</i>-3, <i>x</i>-3: here the first term should be <i>x</i>-2 +and the last <i>x</i>-3½: these two<span class='pagenum'><a name="Page_140" id="Page_140">[140]</a></span> mistakes cancel, and this coil is +therefore right. And the same thing is true of every other coil but the +last, which needs an extra half-yard to reach the <i>end</i> of the path: and +this exactly balances the mistake in the first coil. Thus the sum total +of the coils comes right though the working is all wrong.</p> + +<p>Of the seven who are right, <span class="smcap">Dinah Mite</span>, <span class="smcap">Janet</span>, <span class="smcap">Magpie</span>, and <span class="smcap">Taffy</span> make +the same assumption as C. G. L. and Co. They then solve by a Quadratic. +<span class="smcap">Magpie</span> also tries it by Arithmetical Progression, but fails to notice +that the first and last "coils" have special values.</p> + +<p><span class="smcap">Alumnus Etonæ</span> attempts to prove what C. G. L. assumes by a particular +instance, taking a garden 6 by 5½. He ought to have proved it +generally: what is true of one number is not always true of others. <span class="smcap">Old +King Cole</span> solves it by an Arithmetical Progression. It is right, but too +lengthy to be worth as much as a Quadratic.</p> + +<p><span class="smcap">Vindex</span> proves it very neatly, by pointing out that a yard of walk +measured along the middle represents a square yard of garden, "whether +we consider the straight stretches of walk or the square yards at the +angles, 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 direction."<span class='pagenum'><a name="Page_141" id="Page_141">[141]</a></span></p> + + +<p class="center">CLASS LIST.</p> + +<p><span style="margin-left: 12.5em;">I.</span></p> + +<p> +<span style="margin-left: 10.5em;"><span class="smcap">Vindex.</span></span><br /> +</p> + +<p><span style="margin-left: 12.5em;">II.</span></p> + +<p> +<span style="margin-left: 10.5em;"><span class="smcap">Alumnus Etonæ.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Old King Cole.</span></span><br /> +</p> + +<p><span style="margin-left: 12.5em;">III.</span></p> + +<p> +<span style="margin-left: 10.5em;"><span class="smcap">Dinah Mite.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Janet.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Magpie.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Taffy.</span></span><br /> +</p> + +<hr style="width: 45%;" /> +<p><span class='pagenum'><a name="Page_142" id="Page_142">[142]</a></span></p> +<h3>ANSWERS TO KNOT X.</h3> + + +<p class="fm3">§ 1. <span class="smcap">The Chelsea Pensioners.</span></p> + +<p><i>Problem.</i>—If 70 per cent. have lost an eye, 75 per cent. an ear, 80 +per cent. an arm, 85 per cent. a leg: what percentage, <i>at least</i>, must +have lost all four?</p> + +<p><i>Answer.</i>—Ten.</p> + + +<p><i>Solution.</i>—(I adopt that of <span class="smcap">Polar Star</span>, as being better than my own). +Adding the wounds together, we get 70 + 75 + 80 + 85 = 310, among 100 +men; which gives 3 to each, and 4 to 10 men. Therefore the least +percentage is 10.</p> + +<hr style="width: 45%;" /> + +<p>Nineteen answers have been received. One is "5," but, as no working is +given with it, it must, in accordance with the rule, remain "a deed +without a name." <span class="smcap">Janet</span> makes it "35 and <sup>2</sup>⁄<sub>10</sub>ths." I am sorry she has +misunderstood the question, and has supposed that those who had lost an +ear were 75 per cent. <i>of those who had lost an eye</i>; and so on. Of +course, on this supposition, the percentages must all be multiplied +together. This she has<span class='pagenum'><a name="Page_143" id="Page_143">[143]</a></span> done correctly, but I can give her no honours, +as I do not think the question will fairly bear her interpretation, +<span class="smcap">Three Score and Ten</span> makes it "19 and <sup>2</sup>⁄<sub>8</sub>ths." Her solution has given +me—I will not say "many anxious days and sleepless nights," for I wish +to be strictly truthful, but—some trouble in making any sense at all of +it. She makes the number of "pensioners wounded once" to be 310 ("per +cent.," I suppose!): dividing by 4, she gets 77 and a half as "average +percentage:" again dividing by 4, she gets 19 and <sup>2</sup>⁄<sub>8</sub>ths as "percentage +wounded four times." Does she suppose wounds of different kinds to +"absorb" each other, so to speak? Then, no doubt, the <i>data</i> are +equivalent to 77 pensioners with one wound each, and a half-pensioner +with a half-wound. And does she then suppose these concentrated wounds +to be <i>transferable</i>, so that <sup>2</sup>⁄<sub>4</sub>ths of these unfortunates can obtain +perfect health by handing over their wounds to the remaining <sup>1</sup>⁄<sub>4</sub>th? +Granting these suppositions, her answer is right; or rather, <i>if</i> the +question had been "A road is covered with one inch of gravel, along 77 +and a half per cent. of it. How much of it could be covered 4 inches +deep with the same material?" her answer <i>would</i> have been right. But +alas, that <i>wasn't</i> the question! <span class="smcap">Delta</span> makes some most amazing +assumptions: "let every one who has not lost an eye have lost an ear," +"let every one who has not lost both eyes and ears have lost an arm."<span class='pagenum'><a name="Page_144" id="Page_144">[144]</a></span> +Her ideas of a battle-field are grim indeed. Fancy a warrior who would +continue fighting after losing both eyes, both ears, and both arms! This +is a case which she (or "it?") evidently considers <i>possible</i>.</p> + +<p>Next come eight writers who have made the unwarrantable assumption that, +because 70 per cent. have lost an eye, <i>therefore</i> 30 per cent. have +<i>not</i> lost one, so that they have <i>both</i> eyes. This is illogical. If you +give me a bag containing 100 sovereigns, and if in an hour I come to you +(my face <i>not</i> beaming with gratitude nearly so much as when I received +the bag) to say "I am sorry to tell you that 70 of these sovereigns are +bad," do I thereby guarantee the other 30 to be good? Perhaps I have not +tested them yet. The sides of this illogical octagon are as follows, in +alphabetical order:—<span class="smcap">Algernon Bray</span>, <span class="smcap">Dinah Mite</span>, G. S. C., <span class="smcap">Jane E.</span>, J. D. +W., <span class="smcap">Magpie</span> (who makes the delightful remark "therefore 90 per cent. have +two of something," recalling to one's memory that fortunate monarch, +with whom Xerxes was so much pleased that "he gave him ten of +everything!"), S. S. G., and <span class="smcap">Tokio</span>.</p> + +<p><span class="smcap">Bradshaw of the Future</span> and T. R. do the question in a piecemeal +fashion—on the principle that the 70 per cent. and the 75 per cent., +though commenced at opposite ends of the 100, must overlap by <i>at least</i> +45 per cent.; and so on. This is quite correct working, but not, I +think, quite the best way of doing it.<span class='pagenum'><a name="Page_145" id="Page_145">[145]</a></span></p> + +<p>The other five competitors will, I hope, feel themselves sufficiently +glorified by being placed in the first class, without my composing a +Triumphal Ode for each!</p> + + +<p class="center">CLASS LIST.</p> + +<p><span style="margin-left: 12.5em;">I.</span></p> + +<p> +<span style="margin-left: 10.5em;"><span class="smcap">Old Cat.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Old Hen.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Polar Star.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Simple Susan.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">White Sugar.</span></span><br /> +</p> + +<p><span style="margin-left: 12.5em;">II.</span></p> + +<p> +<span style="margin-left: 10.5em;"><span class="smcap">Bradshaw of the Future.</span></span><br /> +<span style="margin-left: 10.5em;">T. R.</span><br /> +</p> + +<p><span style="margin-left: 12.5em;">III.</span></p> + +<p> +<span style="margin-left: 10.5em;"><span class="smcap">Algernon Bray.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Dinah Mite.</span></span><br /> +<span style="margin-left: 10.5em;">G. S. C.</span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Jane E.</span></span><br /> +<span style="margin-left: 10.5em;">J. D. W.</span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Magpie.</span></span><br /> +<span style="margin-left: 10.5em;">S. S. G.</span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Tokio.</span></span><br /> +</p> + + +<p class="fm3">§ 2. <span class="smcap">Change of Day.</span></p> + +<p>I must postpone, <i>sine die</i>, the geographical problem—partly because I +have not yet received the statistics I am hoping for, and partly because +I am myself so entirely puzzled by it; and when an examiner is himself +dimly hovering between a second class and a third how is he to decide +the position of others?</p> + +<p><span class='pagenum'><a name="Page_146" id="Page_146">[146]</a></span></p> +<p class="fm3">§ 3. <span class="smcap">The Sons' Ages.</span></p> + +<p><i>Problem.</i>—"At first, two of the ages are together equal to the third. +A few years afterwards, two of them are together double of the third. +When the number of years since the first occasion is two-thirds of the +sum of the ages on that occasion, one age is 21. What are the other two?</p> + +<p><i>Answer.</i>—"15 and 18."</p> + +<hr style="width: 45%;" /> + +<p><i>Solution.</i>—Let the ages at first be <i>x</i>, <i>y</i>, (<i>x</i> + <i>y</i>). Now, if <i>a</i> ++ <i>b</i> = 2<i>c</i>, then (<i>a</i>-<i>n</i>) + (<i>b</i>-<i>n</i>) = 2(<i>c</i>-<i>n</i>), whatever be the +value of <i>n</i>. Hence the second relationship, if <i>ever</i> true, was +<i>always</i> true. Hence it was true at first. But it cannot be true that +<i>x</i> and <i>y</i> are together double of (<i>x</i> + <i>y</i>). Hence it must be true of +(<i>x</i> + <i>y</i>), together with <i>x</i> or <i>y</i>; and it does not matter which we +take. We assume, then, (<i>x</i> + <i>y</i>) + <i>x</i> = 2<i>y</i>; <i>i.e.</i> <i>y</i> = 2<i>x</i>. +Hence the three ages were, at first, <i>x</i>, 2<i>x</i>, 3<i>x</i>; and the number of +years, since that time is two-thirds of 6<i>x</i>, <i>i.e.</i> is 4<i>x</i>. Hence the +present ages are 5<i>x</i>, 6<i>x</i>, 7<i>x</i>. The ages are clearly <i>integers</i>, +since this is only "the year when one of my sons comes of age." Hence +7<i>x</i> = 21, <i>x</i> = 3, and the other ages are 15, 18.</p> + +<hr style="width: 45%;" /> +<p><span class='pagenum'><a name="Page_147" id="Page_147">[147]</a></span></p> + +<p>Eighteen answers have been received. One of the writers merely asserts +that the first occasion was 12 years ago, that the ages were then 9, 6, +and 3; and that on the second occasion they were 14, 11, and 8! As a +Roman father, I <i>ought</i> to withhold the name of the rash writer; but +respect for age makes me break the rule: it is <span class="smcap">Three Score and Ten</span>. <span class="smcap">Jane +E.</span> also asserts that the ages at first were 9, 6, 3: then she calculates +the present ages, leaving the <i>second</i> occasion unnoticed. <span class="smcap">Old Hen</span> is +nearly as bad; she "tried various numbers till I found one that fitted +<i>all</i> the conditions"; but merely scratching up the earth, and pecking +about, is <i>not</i> the way to solve a problem, oh venerable bird! And close +after <span class="smcap">Old Hen</span> prowls, with hungry eyes, <span class="smcap">Old Cat</span>, who calmly assumes, to +begin with, that the son who comes of age is the <i>eldest</i>. Eat your +bird, Puss, for you will get nothing from me!</p> + +<p>There are yet two zeroes to dispose of. <span class="smcap">Minerva</span> assumes that, on <i>every</i> +occasion, a son comes of age; and that it is only such a son who is +"tipped with gold." Is it wise thus to interpret "now, my boys, +calculate your ages, and you shall have the money"? <span class="smcap">Bradshaw of the +Future</span> says "let" the ages at first be 9, 6, 3, then assumes that the +second occasion was 6 years afterwards, and on these baseless +assumptions brings out the right<span class='pagenum'><a name="Page_148" id="Page_148">[148]</a></span> answers. Guide <i>future</i> travellers, an +thou wilt: thou art no Bradshaw for <i>this</i> Age!</p> + +<p>Of those who win honours, the merely "honourable" are two. <span class="smcap">Dinah Mite</span> +ascertains (rightly) the relationship between the three ages at first, +but then <i>assumes</i> one of them to be "6," thus making the rest of her +solution tentative. M. F. C. does the algebra all right up to the +conclusion that the present ages are 5<i>z</i>, 6<i>z</i>, and 7<i>z</i>; it then +assumes, without giving any reason, that 7<i>z</i> = 21.</p> + +<p>Of the more honourable, <span class="smcap">Delta</span> attempts a novelty—to discover <i>which</i> +son comes of age by elimination: it assumes, successively, that it is +the middle one, and that it is the youngest; and in each case it +<i>apparently</i> brings out an absurdity. Still, as the proof contains the +following bit of algebra, "63 = 7<i>x</i> + 4<i>y</i>; ∴ 21 = <i>x</i> + 4 +sevenths of <i>y</i>," I trust it will admit that its proof is not <i>quite</i> +conclusive. The rest of its work is good. <span class="smcap">Magpie</span> betrays the deplorable +tendency of her tribe—to appropriate any stray conclusion she comes +across, without having any <i>strict</i> logical right to it. Assuming <i>A</i>, +<i>B</i>, <i>C</i>, as the ages at first, and <i>D</i> as the number of the years that +have elapsed since then, she finds (rightly) the 3 equations, 2<i>A</i> = +<i>B</i>, <i>C</i> = <i>B</i> + <i>A</i>, <i>D</i> = 2<i>B</i>. She then says "supposing that <i>A</i> = 1, +then <i>B</i> = 2, <i>C</i> = 3, and <i>D</i> = 4. Therefore for <i>A</i>, <i>B</i>, <i>C</i>, <i>D</i>, +four numbers are wanted which shall be to<span class='pagenum'><a name="Page_149" id="Page_149">[149]</a></span> each other as 1:2:3:4." It is +in the "therefore" that I detect the unconscientiousness of this bird. +The conclusion <i>is</i> true, but this is only because the equations are +"homogeneous" (<i>i.e.</i> having one "unknown" in each term), a fact which I +strongly suspect had not been grasped—I beg pardon, clawed—by her. +Were I to lay this little pitfall, "<i>A</i> + 1 = <i>B</i>, <i>B</i> + 1 = <i>C</i>; +supposing <i>A</i> = 1, then <i>B</i> = 2 and <i>C</i> = 3. <i>Therefore</i> for <i>A</i>, <i>B</i>, +<i>C</i>, three numbers are wanted which shall be to one another as 1:2:3," +would you not flutter down into it, oh <span class="smcap">Magpie</span>, as amiably as a Dove? +<span class="smcap">Simple Susan</span> is anything but simple to <i>me</i>. After ascertaining that the +3 ages at first are as 3:2:1, she says "then, as two-thirds of their +sum, added to one of them, = 21, the sum cannot exceed 30, and +consequently the highest cannot exceed 15." I suppose her (mental) +argument is something like this:—"two-thirds of sum, + one age, = 21; +∴ sum, + 3 halves of one age, = 31 and a half. But 3 halves of +one age cannot be less than 1 and-a-half (here I perceive that <span class="smcap">Simple +Susan</span> would on no account present a guinea to a new-born baby!) hence +the sum cannot exceed 30." This is ingenious, but her proof, after that, +is (as she candidly admits) "clumsy and roundabout." She finds that +there are 5 possible sets of ages, and eliminates four of them. Suppose +that, instead of 5, there had been 5 million possible sets? Would <span class="smcap">Simple +Susan</span> have<span class='pagenum'><a name="Page_150" id="Page_150">[150]</a></span> courageously ordered in the necessary gallon of ink and ream +of paper?</p> + +<p>The solution sent in by C. R. is, like that of <span class="smcap">Simple Susan</span>, partly +tentative, and so does not rise higher than being Clumsily Right.</p> + +<p>Among those who have earned the highest honours, <span class="smcap">Algernon Bray</span> solves +the problem quite correctly, but adds that there is nothing to exclude +the supposition that all the ages were <i>fractional</i>. This would make the +number of answers infinite. Let me meekly protest that I <i>never</i> +intended my readers to devote the rest of their lives to writing out +answers! <span class="smcap">E. M. Rix</span> points out that, if fractional ages be admissible, +any one of the three sons might be the one "come of age"; but she +rightly rejects this supposition on the ground that it would make the +problem indeterminate. <span class="smcap">White Sugar</span> is the only one who has detected an +oversight of mine: I had forgotten the possibility (which of course +ought to be allowed for) that the son, who came of age that <i>year</i>, need +not have done so by that <i>day</i>, so that he <i>might</i> be only 20. This +gives a second solution, viz., 20, 24, 28. Well said, pure Crystal! +Verily, thy "fair discourse hath been as sugar"!</p> + +<p><span class='pagenum'><a name="Page_151" id="Page_151">[151]</a></span></p> +<p class="center">CLASS LIST.</p> + +<p><span style="margin-left: 12.5em;">I.</span></p> + +<p> +<span style="margin-left: 10.5em;"><span class="smcap">Algernon Bray.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">An Old Fogey.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">E. M. Rix.</span></span><br /> +<span style="margin-left: 10.5em;">G. S. C.</span><br /> +<span style="margin-left: 10.5em;">S. S. G.</span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Tokio.</span></span><br /> +<span style="margin-left: 10.5em;">T. R.</span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">White Sugar.</span></span><br /> +</p> + +<p><span style="margin-left: 12.5em;">II.</span></p> + +<p> +<span style="margin-left: 10.5em;">C. R.</span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Delta.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Magpie.</span></span><br /> +<span style="margin-left: 10.5em;"><span class="smcap">Simple Susan.</span></span><br /> +</p> + +<p><span style="margin-left: 12.5em;">III.</span></p> + +<p> +<span style="margin-left: 10.5em;"><span class="smcap">Dinah Mite.</span></span><br /> +<span style="margin-left: 10.5em;">M. F. C.</span><br /> +</p> + +<hr style="width: 45%;" /> + +<p>I have received more than one remonstrance on my assertion, in the +Chelsea Pensioners' problem, that it was illogical to assume, from the +<i>datum</i> "70 p. c. have lost an eye," that 30 p. c. have <i>not</i>. <span class="smcap">Algernon +Bray</span> states, as a parallel case, "suppose Tommy's father gives him 4 +apples, and he eats one of them, how many has he left?" and says "I +think we are justified in answering, 3." I think so too. There is no +"must" here, and the <i>data</i> are evidently meant to fix the answer<span class='pagenum'><a name="Page_152" id="Page_152">[152]</a></span> +<i>exactly</i>: but, if the question were set me "how many <i>must</i> he have +left?", I should understand the <i>data</i> to be that his father gave him 4 +<i>at least</i>, but <i>may</i> have given him more.</p> + +<p>I take this opportunity of thanking those who have sent, along with +their answers to the Tenth Knot, regrets that there are no more Knots to +come, or petitions that I should recall my resolution to bring them to +an end. I am most grateful for their kind words; but I think it wisest +to end what, at best, was but a lame attempt. "The stretched metre of an +antique song" is beyond my compass; and my puppets were neither +distinctly <i>in</i> my life (like those I now address), nor yet (like Alice +and the Mock Turtle) distinctly <i>out</i> of it. Yet let me at least fancy, +as I lay down the pen, that I carry with me into my silent life, dear +reader, a farewell smile from your unseen face, and a kindly farewell +pressure from your unfelt hand! And so, good night! Parting is such +sweet sorrow, that I shall say "good night!" till it be morrow.</p> + +<p class="fm3">THE END</p> + +<p class="fm4">LONDON: RICHARD CLAY AND SONS, PRINTERS.</p> + +<hr style="width: 65%;" /> +<p class="author">[TURN OVER.</p> + + + +<hr style="width: 65%;" /> +<h2><a name="WORKS_BY_LEWIS_CARROLL" id="WORKS_BY_LEWIS_CARROLL"></a>WORKS BY LEWIS CARROLL.</h2> + + +<p>ALICE'S ADVENTURES IN WONDERLAND. +With Forty-two Illustrations by <span class="smcap">Tenniel</span>. Crown 8vo, cloth, gilt edges, +price 6<i>s.</i> Seventy-fifth Thousand.</p> + +<p>TRANSLATIONS OF THE SAME—into French, by <span class="smcap">Henri Bué</span>—into German, by +<span class="smcap">Antonie Zimmermann</span>—and into Italian, by <span class="smcap">T. Pietrocòla Rossetti</span>—with +<span class="smcap">Tenniel's</span> Illustrations. Crown 8vo, cloth, gilt edges, price 6<i>s.</i> each.</p> + +<p>THROUGH THE LOOKING-GLASS, AND WHAT ALICE FOUND THERE. With Fifty +Illustrations by <span class="smcap">Tenniel</span>. Crown 8vo, cloth, gilt edges, price 6<i>s.</i> +Fifty-sixth Thousand.</p> + +<p>RHYME? AND REASON? With Sixty-five Illustrations by <span class="smcap">Arthur B. Frost</span>, and +Nine by <span class="smcap">Henry Holiday</span>. (This book is a reprint, with a few additions, of +the comic portion of "Phantasmagoria and other Poems," and of "The +Hunting of the Snark." Mr. Frost's pictures are new.) Crown 8vo, cloth, +coloured edges, price 7<i>s.</i> +<a name="corr6" id="corr6"></a><a class="correction" href="#cn6" title="changed from 'Fifth'">Fifty</a> Thousand.</p> + +<p>A TANGLED TALE. Reprinted from <i>The Monthly Packet</i>. With Six +Illustrations by Arthur B. Frost. Crown 8vo, 4<i>s.</i> 6<i>d.</i></p> + +<hr style="width: 45%;" /> + +<p>N.B. In selling the above-mentioned books to the Trade, Messrs. +Macmillan and Co. will abate 2<i>d.</i> in the shilling (no odd copies), and +allow 5 per cent. discount for payment within six months, and 10 per +cent. for cash. In selling them to the Public (for cash only) they will +allow 10 per cent. discount.</p> + +<hr style="width: 45%;" /> + +<p><span class="smcap">Mr. Lewis Carroll</span>, having been requested to allow "<span class="smcap">An Easter Greeting</span>" +(a leaflet, addressed to children, and frequently given with his books) +to be sold separately, has arranged with Messrs. Harrison, of 59, Pall +Mall, who will supply a single copy for 1<i>d.</i>, or 12 for 9<i>d.</i>, or 100 +for 5<i>s.</i></p> + +<hr style="width: 45%;" /> +<p class="fm2">MACMILLAN AND CO., LONDON.</p> + +<p class="fm3">LONDON: RICHARD CLAY AND SONS, PRINTERS.</p> + +<hr style="width: 65%;" /> +<div class="transnote"> +<h3>Transcriber's note<a name="tnotes" id="tnotes"></a></h3> + +<p> +The following changes have been made to the text:</p> + + +<p>Page 88: "he corners of the" changed to +"<a name="cn1" id="cn1"></a><a href="#corr1">the</a> corners of the".</p> + +<p>Page 95: "Aix-le-Bains" changed to +"<a name="cn2" id="cn2"></a><a href="#corr2">Aix-les-Bains</a>".</p> + +<p>Page 108: "<sup>3</sup>⁄<sub>5</sub>, 2, <sup>1</sup>⁄<sub>3</sub>;" changed to "<sup>3</sup>⁄<sub>5</sub>, +<a name="cn3" id="cn3"></a><a href="#corr3"><sup>2</sup>⁄<sub>3</sub></a>, <sup>1</sup>⁄<sub>3</sub>;.</p> + +<p>Page 114: "10 of the 12 cases" changed to "10 of the +<a name="cn4" id="cn4"></a><a href="#corr4">13</a> cases".</p> + +<p>Page 121: "four-and fourpence" changed to +"<a name="cn5" id="cn5"></a><a href="#corr5">four-and-fourpence</a>".</p> + +<p>Last page: "Fifth Thousand" changed to +"<a name="cn6" id="cn6"></a><a href="#corr6">Fifty</a> Thousand".</p> +</div> + +<hr style="width: 65%;" /> +<div class="transnote"> +<h3>Music Transcriber's note<a name="mtnotes" id="mtnotes"></a></h3> + +<p>The following corrections have been made to the <a name="cn7" id="cn7"></a><a href="#corr7">music</a>:</p> + +<p>Bar 2 - The dotted notes should not be dotted.</p> + +<p>Bar 3 - The first note should be a dotted eighth, not a dotted quarter.</p> + +<p>Bar 5 - The dotted notes should not be dotted.</p> + +<p>Bar 6 - The first note should be a dotted eighth, not a dotted quarter.</p> +</div> + + + + + + + +<pre> + + + + + +End of the Project Gutenberg EBook of A Tangled Tale, by Lewis Carroll + +*** END OF THIS PROJECT GUTENBERG EBOOK A TANGLED TALE *** + +***** This file should be named 29042-h.htm or 29042-h.zip ***** +This and all associated files of various formats will be found in: + https://www.gutenberg.org/2/9/0/4/29042/ + +Produced by Chris Curnow, Carla Foust, Lindy Walsh and the +Online Distributed Proofreading Team at https://www.pgdp.net. +Music transcribed by Linda Cantoni. + + +Updated editions will replace the previous one--the old editions +will be renamed. + +Creating the works from public domain print editions means that no +one owns a United States copyright in these works, so the Foundation +(and you!) can copy and distribute it in the United States without +permission and without paying copyright royalties. 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