[illustration: alice in the room of the duchess.] _the "storyland" series_ alice's adventures in wonderland sam'l gabriel sons & company new york copyright, , by sam'l gabriel sons & company new york alice's adventures in wonderland [illustration] i--down the rabbit-hole alice was beginning to get very tired of sitting by her sister on the bank, and of having nothing to do. once or twice she had peeped into the book her sister was reading, but it had no pictures or conversations in it, "and what is the use of a book," thought alice, "without pictures or conversations?" so she was considering in her own mind (as well as she could, for the day made her feel very sleepy and stupid), whether the pleasure of making a daisy-chain would be worth the trouble of getting up and picking the daisies, when suddenly a white rabbit with pink eyes ran close by her. there was nothing so very remarkable in that, nor did alice think it so very much out of the way to hear the rabbit say to itself, "oh dear! oh dear! i shall be too late!" but when the rabbit actually took a watch out of its waistcoat-pocket and looked at it and then hurried on, alice started to her feet, for it flashed across her mind that she had never before seen a rabbit with either a waistcoat-pocket, or a watch to take out of it, and, burning with curiosity, she ran across the field after it and was just in time to see it pop down a large rabbit-hole, under the hedge. in another moment, down went alice after it! [illustration] the rabbit-hole went straight on like a tunnel for some way and then dipped suddenly down, so suddenly that alice had not a moment to think about stopping herself before she found herself falling down what seemed to be a very deep well. either the well was very deep, or she fell very slowly, for she had plenty of time, as she went down, to look about her. first, she tried to make out what she was coming to, but it was too dark to see anything; then she looked at the sides of the well and noticed that they were filled with cupboards and book-shelves; here and there she saw maps and pictures hung upon pegs. she took down a jar from one of the shelves as she passed. it was labeled "orange marmalade," but, to her great disappointment, it was empty; she did not like to drop the jar, so managed to put it into one of the cupboards as she fell past it. down, down, down! would the fall never come to an end? there was nothing else to do, so alice soon began talking to herself. "dinah'll miss me very much to-night, i should think!" (dinah was the cat.) "i hope they'll remember her saucer of milk at tea-time. dinah, my dear, i wish you were down here with me!" alice felt that she was dozing off, when suddenly, thump! thump! down she came upon a heap of sticks and dry leaves, and the fall was over. alice was not a bit hurt, and she jumped up in a moment. she looked up, but it was all dark overhead; before her was another long passage and the white rabbit was still in sight, hurrying down it. there was not a moment to be lost. away went alice like the wind and was just in time to hear it say, as it turned a corner, "oh, my ears and whiskers, how late it's getting!" she was close behind it when she turned the corner, but the rabbit was no longer to be seen. she found herself in a long, low hall, which was lit up by a row of lamps hanging from the roof. there were doors all 'round the hall, but they were all locked; and when alice had been all the way down one side and up the other, trying every door, she walked sadly down the middle, wondering how she was ever to get out again. suddenly she came upon a little table, all made of solid glass. there was nothing on it but a tiny golden key, and alice's first idea was that this might belong to one of the doors of the hall; but, alas! either the locks were too large, or the key was too small, but, at any rate, it would not open any of them. however, on the second time 'round, she came upon a low curtain she had not noticed before, and behind it was a little door about fifteen inches high. she tried the little golden key in the lock, and to her great delight, it fitted! [illustration] alice opened the door and found that it led into a small passage, not much larger than a rat-hole; she knelt down and looked along the passage into the loveliest garden you ever saw. how she longed to get out of that dark hall and wander about among those beds of bright flowers and those cool fountains, but she could not even get her head through the doorway. "oh," said alice, "how i wish i could shut up like a telescope! i think i could, if i only knew how to begin." alice went back to the table, half hoping she might find another key on it, or at any rate, a book of rules for shutting people up like telescopes. this time she found a little bottle on it ("which certainly was not here before," said alice), and tied 'round the neck of the bottle was a paper label, with the words "drink me" beautifully printed on it in large letters. "no, i'll look first," she said, "and see whether it's marked '_poison_' or not," for she had never forgotten that, if you drink from a bottle marked "poison," it is almost certain to disagree with you, sooner or later. however, this bottle was _not_ marked "poison," so alice ventured to taste it, and, finding it very nice (it had a sort of mixed flavor of cherry-tart, custard, pineapple, roast turkey, toffy and hot buttered toast), she very soon finished it off. * * * * * "what a curious feeling!" said alice. "i must be shutting up like a telescope!" and so it was indeed! she was now only ten inches high, and her face brightened up at the thought that she was now the right size for going through the little door into that lovely garden. after awhile, finding that nothing more happened, she decided on going into the garden at once; but, alas for poor alice! when she got to the door, she found she had forgotten the little golden key, and when she went back to the table for it, she found she could not possibly reach it: she could see it quite plainly through the glass and she tried her best to climb up one of the legs of the table, but it was too slippery, and when she had tired herself out with trying, the poor little thing sat down and cried. "come, there's no use in crying like that!" said alice to herself rather sharply. "i advise you to leave off this minute!" she generally gave herself very good advice (though she very seldom followed it), and sometimes she scolded herself so severely as to bring tears into her eyes. soon her eye fell on a little glass box that was lying under the table: she opened it and found in it a very small cake, on which the words "eat me" were beautifully marked in currants. "well, i'll eat it," said alice, "and if it makes me grow larger, i can reach the key; and if it makes me grow smaller, i can creep under the door: so either way i'll get into the garden, and i don't care which happens!" she ate a little bit and said anxiously to herself, "which way? which way?" holding her hand on the top of her head to feel which way she was growing; and she was quite surprised to find that she remained the same size. so she set to work and very soon finished off the cake. [illustration] ii--the pool of tears "curiouser and curiouser!" cried alice (she was so much surprised that for the moment she quite forgot how to speak good english). "now i'm opening out like the largest telescope that ever was! good-by, feet! oh, my poor little feet, i wonder who will put on your shoes and stockings for you now, dears? i shall be a great deal too far off to trouble myself about you." just at this moment her head struck against the roof of the hall; in fact, she was now rather more than nine feet high, and she at once took up the little golden key and hurried off to the garden door. poor alice! it was as much as she could do, lying down on one side, to look through into the garden with one eye; but to get through was more hopeless than ever. she sat down and began to cry again. she went on shedding gallons of tears, until there was a large pool all 'round her and reaching half down the hall. after a time, she heard a little pattering of feet in the distance and she hastily dried her eyes to see what was coming. it was the white rabbit returning, splendidly dressed, with a pair of white kid-gloves in one hand and a large fan in the other. he came trotting along in a great hurry, muttering to himself, "oh! the duchess, the duchess! oh! _won't_ she be savage if i've kept her waiting!" when the rabbit came near her, alice began, in a low, timid voice, "if you please, sir--" the rabbit started violently, dropped the white kid-gloves and the fan and skurried away into the darkness as hard as he could go. [illustration] alice took up the fan and gloves and she kept fanning herself all the time she went on talking. "dear, dear! how queer everything is to-day! and yesterday things went on just as usual. _was_ i the same when i got up this morning? but if i'm not the same, the next question is, 'who in the world am i?' ah, _that's_ the great puzzle!" as she said this, she looked down at her hands and was surprised to see that she had put on one of the rabbit's little white kid-gloves while she was talking. "how _can_ i have done that?" she thought. "i must be growing small again." she got up and went to the table to measure herself by it and found that she was now about two feet high and was going on shrinking rapidly. she soon found out that the cause of this was the fan she was holding and she dropped it hastily, just in time to save herself from shrinking away altogether. "that _was_ a narrow escape!" said alice, a good deal frightened at the sudden change, but very glad to find herself still in existence. "and now for the garden!" and she ran with all speed back to the little door; but, alas! the little door was shut again and the little golden key was lying on the glass table as before. "things are worse than ever," thought the poor child, "for i never was so small as this before, never!" as she said these words, her foot slipped, and in another moment, splash! she was up to her chin in salt-water. her first idea was that she had somehow fallen into the sea. however, she soon made out that she was in the pool of tears which she had wept when she was nine feet high. [illustration] just then she heard something splashing about in the pool a little way off, and she swam nearer to see what it was: she soon made out that it was only a mouse that had slipped in like herself. "would it be of any use, now," thought alice, "to speak to this mouse? everything is so out-of-the-way down here that i should think very likely it can talk; at any rate, there's no harm in trying." so she began, "o mouse, do you know the way out of this pool? i am very tired of swimming about here, o mouse!" the mouse looked at her rather inquisitively and seemed to her to wink with one of its little eyes, but it said nothing. "perhaps it doesn't understand english," thought alice. "i dare say it's a french mouse, come over with william the conqueror." so she began again: "où est ma chatte?" which was the first sentence in her french lesson-book. the mouse gave a sudden leap out of the water and seemed to quiver all over with fright. "oh, i beg your pardon!" cried alice hastily, afraid that she had hurt the poor animal's feelings. "i quite forgot you didn't like cats." "not like cats!" cried the mouse in a shrill, passionate voice. "would _you_ like cats, if you were me?" "well, perhaps not," said alice in a soothing tone; "don't be angry about it. and yet i wish i could show you our cat dinah. i think you'd take a fancy to cats, if you could only see her. she is such a dear, quiet thing." the mouse was bristling all over and she felt certain it must be really offended. "we won't talk about her any more, if you'd rather not." "we, indeed!" cried the mouse, who was trembling down to the end of its tail. "as if _i_ would talk on such a subject! our family always _hated_ cats--nasty, low, vulgar things! don't let me hear the name again!" [illustration: alice at the mad tea party.] "i won't indeed!" said alice, in a great hurry to change the subject of conversation. "are you--are you fond--of--of dogs? there is such a nice little dog near our house, i should like to show you! it kills all the rats and--oh, dear!" cried alice in a sorrowful tone. "i'm afraid i've offended it again!" for the mouse was swimming away from her as hard as it could go, and making quite a commotion in the pool as it went. so she called softly after it, "mouse dear! do come back again, and we won't talk about cats, or dogs either, if you don't like them!" when the mouse heard this, it turned 'round and swam slowly back to her; its face was quite pale, and it said, in a low, trembling voice, "let us get to the shore and then i'll tell you my history and you'll understand why it is i hate cats and dogs." it was high time to go, for the pool was getting quite crowded with the birds and animals that had fallen into it; there were a duck and a dodo, a lory and an eaglet, and several other curious creatures. alice led the way and the whole party swam to the shore. [illustration] iii--a caucus-race and a long tale they were indeed a queer-looking party that assembled on the bank--the birds with draggled feathers, the animals with their fur clinging close to them, and all dripping wet, cross and uncomfortable. [illustration] the first question, of course, was how to get dry again. they had a consultation about this and after a few minutes, it seemed quite natural to alice to find herself talking familiarly with them, as if she had known them all her life. at last the mouse, who seemed to be a person of some authority among them, called out, "sit down, all of you, and listen to me! _i'll_ soon make you dry enough!" they all sat down at once, in a large ring, with the mouse in the middle. "ahem!" said the mouse with an important air. "are you all ready? this is the driest thing i know. silence all 'round, if you please! 'william the conqueror, whose cause was favored by the pope, was soon submitted to by the english, who wanted leaders, and had been of late much accustomed to usurpation and conquest. edwin and morcar, the earls of mercia and northumbria'--" "ugh!" said the lory, with a shiver. "--'and even stigand, the patriotic archbishop of canterbury, found it advisable'--" "found _what_?" said the duck. "found _it_," the mouse replied rather crossly; "of course, you know what 'it' means." "i know what 'it' means well enough, when _i_ find a thing," said the duck; "it's generally a frog or a worm. the question is, what did the archbishop find?" the mouse did not notice this question, but hurriedly went on, "'--found it advisable to go with edgar atheling to meet william and offer him the crown.'--how are you getting on now, my dear?" it continued, turning to alice as it spoke. "as wet as ever," said alice in a melancholy tone; "it doesn't seem to dry me at all." "in that case," said the dodo solemnly, rising to its feet, "i move that the meeting adjourn, for the immediate adoption of more energetic remedies--" "speak english!" said the eaglet. "i don't know the meaning of half those long words, and, what's more, i don't believe you do either!" "what i was going to say," said the dodo in an offended tone, "is that the best thing to get us dry would be a caucus-race." "what _is_ a caucus-race?" said alice. [illustration] "why," said the dodo, "the best way to explain it is to do it." first it marked out a race-course, in a sort of circle, and then all the party were placed along the course, here and there. there was no "one, two, three and away!" but they began running when they liked and left off when they liked, so that it was not easy to know when the race was over. however, when they had been running half an hour or so and were quite dry again, the dodo suddenly called out, "the race is over!" and they all crowded 'round it, panting and asking, "but who has won?" this question the dodo could not answer without a great deal of thought. at last it said, "_everybody_ has won, and _all_ must have prizes." "but who is to give the prizes?" quite a chorus of voices asked. "why, _she_, of course," said the dodo, pointing to alice with one finger; and the whole party at once crowded 'round her, calling out, in a confused way, "prizes! prizes!" alice had no idea what to do, and in despair she put her hand into her pocket and pulled out a box of comfits (luckily the salt-water had not got into it) and handed them 'round as prizes. there was exactly one a-piece, all 'round. the next thing was to eat the comfits; this caused some noise and confusion, as the large birds complained that they could not taste theirs, and the small ones choked and had to be patted on the back. however, it was over at last and they sat down again in a ring and begged the mouse to tell them something more. "you promised to tell me your history, you know," said alice, "and why it is you hate--c and d," she added in a whisper, half afraid that it would be offended again. "mine is a long and a sad tale!" said the mouse, turning to alice and sighing. "it _is_ a long tail, certainly," said alice, looking down with wonder at the mouse's tail, "but why do you call it sad?" and she kept on puzzling about it while the mouse was speaking, so that her idea of the tale was something like this:-- "fury said to a mouse, that he met in the house, 'let us both go to law: _i_ will prosecute _you_.-- come, i'll take no denial: we must have the trial; for really this morning i've nothing to do.' said the mouse to the cur, 'such a trial, dear sir, with no jury or judge, would be wasting our breath.' 'i'll be judge, i'll be jury,' said cunning old fury; 'i'll try the whole cause, and condemn you to death.'" "you are not attending!" said the mouse to alice, severely. "what are you thinking of?" "i beg your pardon," said alice very humbly, "you had got to the fifth bend, i think?" "you insult me by talking such nonsense!" said the mouse, getting up and walking away. "please come back and finish your story!" alice called after it. and the others all joined in chorus, "yes, please do!" but the mouse only shook its head impatiently and walked a little quicker. "i wish i had dinah, our cat, here!" said alice. this caused a remarkable sensation among the party. some of the birds hurried off at once, and a canary called out in a trembling voice, to its children, "come away, my dears! it's high time you were all in bed!" on various pretexts they all moved off and alice was soon left alone. "i wish i hadn't mentioned dinah! nobody seems to like her down here and i'm sure she's the best cat in the world!" poor alice began to cry again, for she felt very lonely and low-spirited. in a little while, however, she again heard a little pattering of footsteps in the distance and she looked up eagerly. [illustration] [illustration] iv--the rabbit sends in a little bill it was the white rabbit, trotting slowly back again and looking anxiously about as it went, as if it had lost something; alice heard it muttering to itself, "the duchess! the duchess! oh, my dear paws! oh, my fur and whiskers! she'll get me executed, as sure as ferrets are ferrets! where _can_ i have dropped them, i wonder?" alice guessed in a moment that it was looking for the fan and the pair of white kid-gloves and she very good-naturedly began hunting about for them, but they were nowhere to be seen--everything seemed to have changed since her swim in the pool, and the great hall, with the glass table and the little door, had vanished completely. very soon the rabbit noticed alice, and called to her, in an angry tone, "why, mary ann, what _are_ you doing out here? run home this moment and fetch me a pair of gloves and a fan! quick, now!" "he took me for his housemaid!" said alice, as she ran off. "how surprised he'll be when he finds out who i am!" as she said this, she came upon a neat little house, on the door of which was a bright brass plate with the name "w. rabbit" engraved upon it. she went in without knocking and hurried upstairs, in great fear lest she should meet the real mary ann and be turned out of the house before she had found the fan and gloves. by this time, alice had found her way into a tidy little room with a table in the window, and on it a fan and two or three pairs of tiny white kid-gloves; she took up the fan and a pair of the gloves and was just going to leave the room, when her eyes fell upon a little bottle that stood near the looking-glass. she uncorked it and put it to her lips, saying to herself, "i do hope it'll make me grow large again, for, really, i'm quite tired of being such a tiny little thing!" before she had drunk half the bottle, she found her head pressing against the ceiling, and had to stoop to save her neck from being broken. she hastily put down the bottle, remarking, "that's quite enough--i hope i sha'n't grow any more." alas! it was too late to wish that! she went on growing and growing and very soon she had to kneel down on the floor. still she went on growing, and, as a last resource, she put one arm out of the window and one foot up the chimney, and said to herself, "now i can do no more, whatever happens. what _will_ become of me?" [illustration] luckily for alice, the little magic bottle had now had its full effect and she grew no larger. after a few minutes she heard a voice outside and stopped to listen. "mary ann! mary ann!" said the voice. "fetch me my gloves this moment!" then came a little pattering of feet on the stairs. alice knew it was the rabbit coming to look for her and she trembled till she shook the house, quite forgetting that she was now about a thousand times as large as the rabbit and had no reason to be afraid of it. presently the rabbit came up to the door and tried to open it; but as the door opened inwards and alice's elbow was pressed hard against it, that attempt proved a failure. alice heard it say to itself, "then i'll go 'round and get in at the window." "_that_ you won't!" thought alice; and after waiting till she fancied she heard the rabbit just under the window, she suddenly spread out her hand and made a snatch in the air. she did not get hold of anything, but she heard a little shriek and a fall and a crash of broken glass, from which she concluded that it was just possible it had fallen into a cucumber-frame or something of that sort. next came an angry voice--the rabbit's--"pat! pat! where are you?" and then a voice she had never heard before, "sure then, i'm here! digging for apples, yer honor!" "here! come and help me out of this! now tell me, pat, what's that in the window?" "sure, it's an arm, yer honor!" "well, it's got no business there, at any rate; go and take it away!" there was a long silence after this and alice could only hear whispers now and then, and at last she spread out her hand again and made another snatch in the air. this time there were _two_ little shrieks and more sounds of broken glass. "i wonder what they'll do next!" thought alice. "as for pulling me out of the window, i only wish they _could_!" she waited for some time without hearing anything more. at last came a rumbling of little cart-wheels and the sound of a good many voices all talking together. she made out the words: "where's the other ladder? bill's got the other--bill! here, bill! will the roof bear?--who's to go down the chimney?--nay, _i_ sha'n't! _you_ do it! here, bill! the master says you've got to go down the chimney!" alice drew her foot as far down the chimney as she could and waited till she heard a little animal scratching and scrambling about in the chimney close above her; then she gave one sharp kick and waited to see what would happen next. the first thing she heard was a general chorus of "there goes bill!" then the rabbit's voice alone--"catch him, you by the hedge!" then silence and then another confusion of voices--"hold up his head--brandy now--don't choke him--what happened to you?" last came a little feeble, squeaking voice, "well, i hardly know--no more, thank ye. i'm better now--all i know is, something comes at me like a jack-in-the-box and up i goes like a sky-rocket!" after a minute or two of silence, they began moving about again, and alice heard the rabbit say, "a barrowful will do, to begin with." "a barrowful of _what_?" thought alice. but she had not long to doubt, for the next moment a shower of little pebbles came rattling in at the window and some of them hit her in the face. alice noticed, with some surprise, that the pebbles were all turning into little cakes as they lay on the floor and a bright idea came into her head. "if i eat one of these cakes," she thought, "it's sure to make _some_ change in my size." so she swallowed one of the cakes and was delighted to find that she began shrinking directly. as soon as she was small enough to get through the door, she ran out of the house and found quite a crowd of little animals and birds waiting outside. they all made a rush at alice the moment she appeared, but she ran off as hard as she could and soon found herself safe in a thick wood. [illustration: "the duchess tucked her arm affectionately into alice's."] "the first thing i've got to do," said alice to herself, as she wandered about in the wood, "is to grow to my right size again; and the second thing is to find my way into that lovely garden. i suppose i ought to eat or drink something or other, but the great question is 'what?'" alice looked all around her at the flowers and the blades of grass, but she could not see anything that looked like the right thing to eat or drink under the circumstances. there was a large mushroom growing near her, about the same height as herself. she stretched herself up on tiptoe and peeped over the edge and her eyes immediately met those of a large blue caterpillar, that was sitting on the top, with its arms folded, quietly smoking a long hookah and taking not the smallest notice of her or of anything else. [illustration] v--advice from a caterpillar at last the caterpillar took the hookah out of its mouth and addressed alice in a languid, sleepy voice. "who are _you_?" said the caterpillar. [illustration] alice replied, rather shyly, "i--i hardly know, sir, just at present--at least i know who i _was_ when i got up this morning, but i think i must have changed several times since then." "what do you mean by that?" said the caterpillar, sternly. "explain yourself!" "i can't explain _myself_, i'm afraid, sir," said alice, "because i'm not myself, you see--being so many different sizes in a day is very confusing." she drew herself up and said very gravely, "i think you ought to tell me who _you_ are, first." "why?" said the caterpillar. as alice could not think of any good reason and the caterpillar seemed to be in a _very_ unpleasant state of mind, she turned away. "come back!" the caterpillar called after her. "i've something important to say!" alice turned and came back again. "keep your temper," said the caterpillar. "is that all?" said alice, swallowing down her anger as well as she could. "no," said the caterpillar. it unfolded its arms, took the hookah out of its mouth again, and said, "so you think you're changed, do you?" "i'm afraid, i am, sir," said alice. "i can't remember things as i used--and i don't keep the same size for ten minutes together!" "what size do you want to be?" asked the caterpillar. "oh, i'm not particular as to size," alice hastily replied, "only one doesn't like changing so often, you know. i should like to be a _little_ larger, sir, if you wouldn't mind," said alice. "three inches is such a wretched height to be." "it is a very good height indeed!" said the caterpillar angrily, rearing itself upright as it spoke (it was exactly three inches high). in a minute or two, the caterpillar got down off the mushroom and crawled away into the grass, merely remarking, as it went, "one side will make you grow taller, and the other side will make you grow shorter." "one side of _what_? the other side of _what_?" thought alice to herself. "of the mushroom," said the caterpillar, just as if she had asked it aloud; and in another moment, it was out of sight. alice remained looking thoughtfully at the mushroom for a minute, trying to make out which were the two sides of it. at last she stretched her arms 'round it as far as they would go, and broke off a bit of the edge with each hand. "and now which is which?" she said to herself, and nibbled a little of the right-hand bit to try the effect. the next moment she felt a violent blow underneath her chin--it had struck her foot! she was a good deal frightened by this very sudden change, as she was shrinking rapidly; so she set to work at once to eat some of the other bit. her chin was pressed so closely against her foot that there was hardly room to open her mouth; but she did it at last and managed to swallow a morsel of the left-hand bit.... "come, my head's free at last!" said alice; but all she could see, when she looked down, was an immense length of neck, which seemed to rise like a stalk out of a sea of green leaves that lay far below her. "where _have_ my shoulders got to? and oh, my poor hands, how is it i can't see you?" she was delighted to find that her neck would bend about easily in any direction, like a serpent. she had just succeeded in curving it down into a graceful zigzag and was going to dive in among the leaves, when a sharp hiss made her draw back in a hurry--a large pigeon had flown into her face and was beating her violently with its wings. [illustration] "serpent!" cried the pigeon. "i'm _not_ a serpent!" said alice indignantly. "let me alone!" "i've tried the roots of trees, and i've tried banks, and i've tried hedges," the pigeon went on, "but those serpents! there's no pleasing them!" alice was more and more puzzled. "as if it wasn't trouble enough hatching the eggs," said the pigeon, "but i must be on the look-out for serpents, night and day! and just as i'd taken the highest tree in the wood," continued the pigeon, raising its voice to a shriek, "and just as i was thinking i should be free of them at last, they must needs come wriggling down from the sky! ugh, serpent!" "but i'm _not_ a serpent, i tell you!" said alice. "i'm a--i'm a--i'm a little girl," she added rather doubtfully, as she remembered the number of changes she had gone through that day. "you're looking for eggs, i know _that_ well enough," said the pigeon; "and what does it matter to me whether you're a little girl or a serpent?" "it matters a good deal to _me_," said alice hastily; "but i'm not looking for eggs, as it happens, and if i was, i shouldn't want _yours_--i don't like them raw." "well, be off, then!" said the pigeon in a sulky tone, as it settled down again into its nest. alice crouched down among the trees as well as she could, for her neck kept getting entangled among the branches, and every now and then she had to stop and untwist it. after awhile she remembered that she still held the pieces of mushroom in her hands, and she set to work very carefully, nibbling first at one and then at the other, and growing sometimes taller and sometimes shorter, until she had succeeded in bringing herself down to her usual height. it was so long since she had been anything near the right size that it felt quite strange at first. "the next thing is to get into that beautiful garden--how _is_ that to be done, i wonder?" as she said this, she came suddenly upon an open place, with a little house in it about four feet high. "whoever lives there," thought alice, "it'll never do to come upon them _this_ size; why, i should frighten them out of their wits!" she did not venture to go near the house till she had brought herself down to nine inches high. vi--pig and pepper for a minute or two she stood looking at the house, when suddenly a footman in livery came running out of the wood (judging by his face only, she would have called him a fish)--and rapped loudly at the door with his knuckles. it was opened by another footman in livery, with a round face and large eyes like a frog. [illustration] the fish-footman began by producing from under his arm a great letter, and this he handed over to the other, saying, in a solemn tone, "for the duchess. an invitation from the queen to play croquet." the frog-footman repeated, in the same solemn tone, "from the queen. an invitation for the duchess to play croquet." then they both bowed low and their curls got entangled together. when alice next peeped out, the fish-footman was gone, and the other was sitting on the ground near the door, staring stupidly up into the sky. alice went timidly up to the door and knocked. "there's no sort of use in knocking," said the footman, "and that for two reasons. first, because i'm on the same side of the door as you are; secondly, because they're making such a noise inside, no one could possibly hear you." and certainly there _was_ a most extraordinary noise going on within--a constant howling and sneezing, and every now and then a great crash, as if a dish or kettle had been broken to pieces. "how am i to get in?" asked alice. "_are_ you to get in at all?" said the footman. "that's the first question, you know." alice opened the door and went in. the door led right into a large kitchen, which was full of smoke from one end to the other; the duchess was sitting on a three-legged stool in the middle, nursing a baby; the cook was leaning over the fire, stirring a large caldron which seemed to be full of soup. "there's certainly too much pepper in that soup!" alice said to herself, as well as she could for sneezing. even the duchess sneezed occasionally; and as for the baby, it was sneezing and howling alternately without a moment's pause. the only two creatures in the kitchen that did _not_ sneeze were the cook and a large cat, which was grinning from ear to ear. "please would you tell me," said alice, a little timidly, "why your cat grins like that?" "it's a cheshire-cat," said the duchess, "and that's why." "i didn't know that cheshire-cats always grinned; in fact, i didn't know that cats _could_ grin," said alice. "you don't know much," said the duchess, "and that's a fact." just then the cook took the caldron of soup off the fire, and at once set to work throwing everything within her reach at the duchess and the baby--the fire-irons came first; then followed a shower of saucepans, plates and dishes. the duchess took no notice of them, even when they hit her, and the baby was howling so much already that it was quite impossible to say whether the blows hurt it or not. "oh, _please_ mind what you're doing!" cried alice, jumping up and down in an agony of terror. "here! you may nurse it a bit, if you like!" the duchess said to alice, flinging the baby at her as she spoke. "i must go and get ready to play croquet with the queen," and she hurried out of the room. alice caught the baby with some difficulty, as it was a queer-shaped little creature and held out its arms and legs in all directions. "if i don't take this child away with me," thought alice, "they're sure to kill it in a day or two. wouldn't it be murder to leave it behind?" she said the last words out loud and the little thing grunted in reply. "if you're going to turn into a pig, my dear," said alice, "i'll have nothing more to do with you. mind now!" alice was just beginning to think to herself, "now, what am i to do with this creature, when i get it home?" when it grunted again so violently that alice looked down into its face in some alarm. this time there could be _no_ mistake about it--it was neither more nor less than a pig; so she set the little creature down and felt quite relieved to see it trot away quietly into the wood. alice was a little startled by seeing the cheshire-cat sitting on a bough of a tree a few yards off. the cat only grinned when it saw her. "cheshire-puss," began alice, rather timidly, "would you please tell me which way i ought to go from here?" "in _that_ direction," the cat said, waving the right paw 'round, "lives a hatter; and in _that_ direction," waving the other paw, "lives a march hare. visit either you like; they're both mad." "but i don't want to go among mad people," alice remarked. "oh, you can't help that," said the cat; "we're all mad here. do you play croquet with the queen to-day?" "i should like it very much," said alice, "but i haven't been invited yet." "you'll see me there," said the cat, and vanished. alice had not gone much farther before she came in sight of the house of the march hare; it was so large a house that she did not like to go near till she had nibbled some more of the left-hand bit of mushroom. vii--a mad tea-party there was a table set out under a tree in front of the house, and the march hare and the hatter were having tea at it; a dormouse was sitting between them, fast asleep. the table was a large one, but the three were all crowded together at one corner of it. "no room! no room!" they cried out when they saw alice coming. "there's _plenty_ of room!" said alice indignantly, and she sat down in a large arm-chair at one end of the table. the hatter opened his eyes very wide on hearing this, but all he said was "why is a raven like a writing-desk?" "i'm glad they've begun asking riddles--i believe i can guess that," she added aloud. "do you mean that you think you can find out the answer to it?" said the march hare. "exactly so," said alice. "then you should say what you mean," the march hare went on. "i do," alice hastily replied; "at least--at least i mean what i say--that's the same thing, you know." "you might just as well say," added the dormouse, which seemed to be talking in its sleep, "that 'i breathe when i sleep' is the same thing as 'i sleep when i breathe!'" "it _is_ the same thing with you," said the hatter, and he poured a little hot tea upon its nose. the dormouse shook its head impatiently and said, without opening its eyes, "of course, of course; just what i was going to remark myself." [illustration] "have you guessed the riddle yet?" the hatter said, turning to alice again. "no, i give it up," alice replied. "what's the answer?" "i haven't the slightest idea," said the hatter. "nor i," said the march hare. alice gave a weary sigh. "i think you might do something better with the time," she said, "than wasting it in asking riddles that have no answers." "take some more tea," the march hare said to alice, very earnestly. "i've had nothing yet," alice replied in an offended tone, "so i can't take more." "you mean you can't take _less_," said the hatter; "it's very easy to take _more_ than nothing." at this, alice got up and walked off. the dormouse fell asleep instantly and neither of the others took the least notice of her going, though she looked back once or twice; the last time she saw them, they were trying to put the dormouse into the tea-pot. [illustration: the trial of the knave of hearts.] "at any rate, i'll never go _there_ again!" said alice, as she picked her way through the wood. "it's the stupidest tea-party i ever was at in all my life!" just as she said this, she noticed that one of the trees had a door leading right into it. "that's very curious!" she thought. "i think i may as well go in at once." and in she went. once more she found herself in the long hall and close to the little glass table. taking the little golden key, she unlocked the door that led into the garden. then she set to work nibbling at the mushroom (she had kept a piece of it in her pocket) till she was about a foot high; then she walked down the little passage; and _then_--she found herself at last in the beautiful garden, among the bright flower-beds and the cool fountains. viii--the queen's croquet ground a large rose-tree stood near the entrance of the garden; the roses growing on it were white, but there were three gardeners at it, busily painting them red. suddenly their eyes chanced to fall upon alice, as she stood watching them. "would you tell me, please," said alice, a little timidly, "why you are painting those roses?" five and seven said nothing, but looked at two. two began, in a low voice, "why, the fact is, you see, miss, this here ought to have been a _red_ rose-tree, and we put a white one in by mistake; and, if the queen was to find it out, we should all have our heads cut off, you know. so you see, miss, we're doing our best, afore she comes, to--" at this moment, five, who had been anxiously looking across the garden, called out, "the queen! the queen!" and the three gardeners instantly threw themselves flat upon their faces. there was a sound of many footsteps and alice looked 'round, eager to see the queen. first came ten soldiers carrying clubs, with their hands and feet at the corners: next the ten courtiers; these were ornamented all over with diamonds. after these came the royal children; there were ten of them, all ornamented with hearts. next came the guests, mostly kings and queens, and among them alice recognized the white rabbit. then followed the knave of hearts, carrying the king's crown on a crimson velvet cushion; and last of all this grand procession came the king and the queen of hearts. when the procession came opposite to alice, they all stopped and looked at her, and the queen said severely, "who is this?" she said it to the knave of hearts, who only bowed and smiled in reply. "my name is alice, so please your majesty," said alice very politely; but she added to herself, "why, they're only a pack of cards, after all!" "can you play croquet?" shouted the queen. the question was evidently meant for alice. "yes!" said alice loudly. "come on, then!" roared the queen. "it's--it's a very fine day!" said a timid voice to alice. she was walking by the white rabbit, who was peeping anxiously into her face. "very," said alice. "where's the duchess?" "hush! hush!" said the rabbit. "she's under sentence of execution." "what for?" said alice. "she boxed the queen's ears--" the rabbit began. "get to your places!" shouted the queen in a voice of thunder, and people began running about in all directions, tumbling up against each other. however, they got settled down in a minute or two, and the game began. alice thought she had never seen such a curious croquet-ground in her life; it was all ridges and furrows. the croquet balls were live hedgehogs, and the mallets live flamingos and the soldiers had to double themselves up and stand on their hands and feet, to make the arches. the players all played at once, without waiting for turns, quarrelling all the while and fighting for the hedgehogs; and in a very short time, the queen was in a furious passion and went stamping about and shouting, "off with his head!" or "off with her head!" about once in a minute. "they're dreadfully fond of beheading people here," thought alice; "the great wonder is that there's anyone left alive!" she was looking about for some way of escape, when she noticed a curious appearance in the air. "it's the cheshire-cat," she said to herself; "now i shall have somebody to talk to." "how are you getting on?" said the cat. "i don't think they play at all fairly," alice said, in a rather complaining tone; "and they all quarrel so dreadfully one can't hear oneself speak--and they don't seem to have any rules in particular." "how do you like the queen?" said the cat in a low voice. "not at all," said alice. [illustration] alice thought she might as well go back and see how the game was going on. so she went off in search of her hedgehog. the hedgehog was engaged in a fight with another hedgehog, which seemed to alice an excellent opportunity for croqueting one of them with the other; the only difficulty was that her flamingo was gone across to the other side of the garden, where alice could see it trying, in a helpless sort of way, to fly up into a tree. she caught the flamingo and tucked it away under her arm, that it might not escape again. just then alice ran across the duchess (who was now out of prison). she tucked her arm affectionately into alice's and they walked off together. alice was very glad to find her in such a pleasant temper. she was a little startled, however, when she heard the voice of the duchess close to her ear. "you're thinking about something, my dear, and that makes you forget to talk." "the game's going on rather better now," alice said, by way of keeping up the conversation a little. "'tis so," said the duchess; "and the moral of that is--'oh, 'tis love, 'tis love that makes the world go 'round!'" "somebody said," alice whispered, "that it's done by everybody minding his own business!" "ah, well! it means much the same thing," said the duchess, digging her sharp little chin into alice's shoulder, as she added "and the moral of _that_ is--'take care of the sense and the sounds will take care of themselves.'" to alice's great surprise, the duchess's arm that was linked into hers began to tremble. alice looked up and there stood the queen in front of them, with her arms folded, frowning like a thunderstorm! "now, i give you fair warning," shouted the queen, stamping on the ground as she spoke, "either you or your head must be off, and that in about half no time. take your choice!" the duchess took her choice, and was gone in a moment. "let's go on with the game," the queen said to alice; and alice was too much frightened to say a word, but slowly followed her back to the croquet-ground. all the time they were playing, the queen never left off quarreling with the other players and shouting, "off with his head!" or "off with her head!" by the end of half an hour or so, all the players, except the king, the queen and alice, were in custody of the soldiers and under sentence of execution. then the queen left off, quite out of breath, and walked away with alice. alice heard the king say in a low voice to the company generally, "you are all pardoned." suddenly the cry "the trial's beginning!" was heard in the distance, and alice ran along with the others. ix--who stole the tarts? the king and queen of hearts were seated on their throne when they arrived, with a great crowd assembled about them--all sorts of little birds and beasts, as well as the whole pack of cards: the knave was standing before them, in chains, with a soldier on each side to guard him; and near the king was the white rabbit, with a trumpet in one hand and a scroll of parchment in the other. in the very middle of the court was a table, with a large dish of tarts upon it. "i wish they'd get the trial done," alice thought, "and hand 'round the refreshments!" the judge, by the way, was the king and he wore his crown over his great wig. "that's the jury-box," thought alice; "and those twelve creatures (some were animals and some were birds) i suppose they are the jurors." just then the white rabbit cried out "silence in the court!" "herald, read the accusation!" said the king. [illustration] on this, the white rabbit blew three blasts on the trumpet, then unrolled the parchment-scroll and read as follows: "the queen of hearts, she made some tarts, all on a summer day; the knave of hearts, he stole those tarts and took them quite away!" "call the first witness," said the king; and the white rabbit blew three blasts on the trumpet and called out, "first witness!" the first witness was the hatter. he came in with a teacup in one hand and a piece of bread and butter in the other. "you ought to have finished," said the king. "when did you begin?" the hatter looked at the march hare, who had followed him into the court, arm in arm with the dormouse. "fourteenth of march, i _think_ it was," he said. "give your evidence," said the king, "and don't be nervous, or i'll have you executed on the spot." this did not seem to encourage the witness at all; he kept shifting from one foot to the other, looking uneasily at the queen, and, in his confusion, he bit a large piece out of his teacup instead of the bread and butter. just at this moment alice felt a very curious sensation--she was beginning to grow larger again. the miserable hatter dropped his teacup and bread and butter and went down on one knee. "i'm a poor man, your majesty," he began. "you're a _very_ poor _speaker_," said the king. "you may go," said the king, and the hatter hurriedly left the court. "call the next witness!" said the king. the next witness was the duchess's cook. she carried the pepper-box in her hand and the people near the door began sneezing all at once. "give your evidence," said the king. "sha'n't," said the cook. the king looked anxiously at the white rabbit, who said, in a low voice, "your majesty must cross-examine _this_ witness." "well, if i must, i must," the king said. "what are tarts made of?" "pepper, mostly," said the cook. for some minutes the whole court was in confusion and by the time they had settled down again, the cook had disappeared. "never mind!" said the king, "call the next witness." alice watched the white rabbit as he fumbled over the list. imagine her surprise when he read out, at the top of his shrill little voice, the name "alice!" x--alice's evidence "here!" cried alice. she jumped up in such a hurry that she tipped over the jury-box, upsetting all the jurymen on to the heads of the crowd below. "oh, i _beg_ your pardon!" she exclaimed in a tone of great dismay. "the trial cannot proceed," said the king, "until all the jurymen are back in their proper places--_all_," he repeated with great emphasis, looking hard at alice. "what do you know about this business?" the king said to alice. "nothing whatever," said alice. the king then read from his book: "rule forty-two. _all persons more than a mile high to leave the court_." "_i'm_ not a mile high," said alice. "nearly two miles high," said the queen. [illustration] "well, i sha'n't go, at any rate," said alice. the king turned pale and shut his note-book hastily. "consider your verdict," he said to the jury, in a low, trembling voice. "there's more evidence to come yet, please your majesty," said the white rabbit, jumping up in a great hurry. "this paper has just been picked up. it seems to be a letter written by the prisoner to--to somebody." he unfolded the paper as he spoke and added, "it isn't a letter, after all; it's a set of verses." "please, your majesty," said the knave, "i didn't write it and they can't prove that i did; there's no name signed at the end." "you _must_ have meant some mischief, or else you'd have signed your name like an honest man," said the king. there was a general clapping of hands at this. "read them," he added, turning to the white rabbit. there was dead silence in the court whilst the white rabbit read out the verses. "that's the most important piece of evidence we've heard yet," said the king. "_i_ don't believe there's an atom of meaning in it," ventured alice. "if there's no meaning in it," said the king, "that saves a world of trouble, you know, as we needn't try to find any. let the jury consider their verdict." "no, no!" said the queen. "sentence first--verdict afterwards." "stuff and nonsense!" said alice loudly. "the idea of having the sentence first!" "hold your tongue!" said the queen, turning purple. "i won't!" said alice. "off with her head!" the queen shouted at the top of her voice. nobody moved. "who cares for _you_?" said alice (she had grown to her full size by this time). "you're nothing but a pack of cards!" [illustration] at this, the whole pack rose up in the air and came flying down upon her; she gave a little scream, half of fright and half of anger, and tried to beat them off, and found herself lying on the bank, with her head in the lap of her sister, who was gently brushing away some dead leaves that had fluttered down from the trees upon her face. "wake up, alice dear!" said her sister. "why, what a long sleep you've had!" "oh, i've had such a curious dream!" said alice. and she told her sister, as well as she could remember them, all these strange adventures of hers that you have just been reading about. alice got up and ran off, thinking while she ran, as well she might, what a wonderful dream it had been. [illustration] pg-i symbolic logic by lewis carroll pg-ii pg-iii pg-iv a syllogism worked out. that story of yours, about your once meeting the sea-serpent, always sets me off yawning; i never yawn, unless when i'm listening to something totally devoid of interest. the premisses, separately. ·---------------· ·---------------· |( ) | ( )| | | | | ·---|---· | | ·---|---· | | | (#) | | | | |( )| | |---|---|---|---| |---|---|---|---| | | | | | | | |( )| | | ·---|---· | | ·---|---· | | | | | | | ·---------------· ·---------------· the premisses, combined. ·---------------· |( ) | ( )| | ·---|---· | | |(#)|( )| | |---|---|---|---| | | |( )| | | ·---|---· | | | | ·---------------· the conclusion. ·-------· |(#)|( )| |---|---| | | | ·-------· that story of yours, about your once meeting the sea-serpent, is totally devoid of interest. pg-v symbolic logic _part i_ elementary by lewis carroll second thousand fourth edition _price two shillings_ london macmillan and co., limited new york: the macmillan company _all rights reserved_ pg-vi richard clay and sons, limited, london and bungay pg-vii advertisement. an envelope, containing two blank diagrams (biliteral and triliteral) and counters ( red and grey), may be had, from messrs. macmillan, for _d._, by post _d._ * * * * * i shall be grateful to any reader of this book who will point out any mistakes or misprints he may happen to notice in it, or any passage which he thinks is not clearly expressed. * * * * * i have a quantity of ms. in hand for parts ii and iii, and hope to be able----should life, and health, and opportunity, be granted to me, to publish them in the course of the next few years. their contents will be as follows:-- _part ii. advanced._ further investigations in the subjects of part i. propositions of other forms (such as "not-all x are y"). triliteral and multiliteral propositions (such as "all abc are de"). hypotheticals. dilemmas. &c. &c. _part iii. transcendental._ analysis of a proposition into its elements. numerical and geometrical problems. the theory of inference. the construction of problems. and many other _curiosa logica_. pg-viii preface to the fourth edition. the chief alterations, since the first edition, have been made in the chapter on 'classification' (pp. , ) and the book on 'propositions' (pp. to ). the chief additions have been the questions on words and phrases, added to the examination-papers at p. , and the notes inserted at pp. , . in book i, chapter ii, i have adopted a new definition of 'classification', which enables me to regard the whole universe as a 'class,' and thus to dispense with the very awkward phrase 'a set of things.' in the chapter on 'propositions of existence' i have adopted a new 'normal form,' in which the class, whose existence is affirmed or denied, is regarded as the _predicate_, instead of the _subject_, of the proposition, thus evading a very subtle difficulty which besets the other form. these subtle difficulties seem to lie at the root of every tree of knowledge, and they are _far_ more hopeless to grapple with than any that occur in its higher branches. for example, the difficulties of the forty-seventh proposition of euclid are mere child's play compared with the mental torture endured in the effort to think out the essential nature of a straight line. and, in the present work, the difficulties of the " liars" problem, at p. , are "trifles, light as air," compared with the bewildering question "what is a thing?" in the chapter on 'propositions of relation' i have inserted a new section, containing the proof that a proposition, beginning with "all," is a _double_ proposition (a fact that is quite independent of the arbitrary rule, laid down in the next section, that such a proposition is to be understood as implying the actual _existence_ of its subject). this proof was given, in the earlier editions, incidentally, in the course of the discussion of the biliteral diagram: but its _proper_ place, in this treatise, is where i have now introduced it. pg-ix in the sorites-examples, i have made a good many verbal alterations, in order to evade a difficulty, which i fear will have perplexed some of the readers of the first three editions. some of the premisses were so worded that their terms were not specieses of the univ. named in the dictionary, but of a larger class, of which the univ. was only a portion. in all such cases, it was intended that the reader should perceive that what was asserted of the larger class was thereby asserted of the univ., and should ignore, as superfluous, all that it asserted of its _other_ portion. thus, in ex. , the univ. was stated to be "ducks in this village," and the third premiss was "mrs. bond has no gray ducks," i.e. "no gray ducks are ducks belonging to mrs. bond." here the terms are _not_ specieses of the univ., but of the larger class "ducks," of which the univ. is only a portion: and it was intended that the reader should perceive that what is here asserted of "ducks" is thereby asserted of "ducks in this village." and should treat this premiss as if it were "mrs. bond has no gray ducks in this village," and should ignore, as superfluous, what it asserts as to the _other_ portion of the class "ducks," viz. "mrs. bond has no gray ducks _out of_ this village". in the appendix i have given a new version of the problem of the "five liars." my object, in doing so, is to escape the subtle and mysterious difficulties which beset all attempts at regarding a proposition as being its own subject, or a set of propositions as being subjects for one another. it is certainly, a most bewildering and unsatisfactory theory: one cannot help feeling that there is a great lack of _substance_ in all this shadowy host----that, as the procession of phantoms glides before us, there is not _one_ that we can pounce upon, and say "_here_ is a proposition that _must_ be either true or false!"----that it is but a barmecide feast, to which we have been bidden----and that its prototype is to be found in that mythical island, whose inhabitants "earned a precarious living by taking in each others' washing"! by simply translating "telling truths" into "taking _both_ of condiments (salt and mustard)," "telling lies" into "taking _neither_ of them" and "telling a truth and a lie (order not specified)" into "taking only _one_ condiment (it is not specified _which_)," i have escaped all those metaphysical puzzles, and have produced a problem which, when translated into a set of symbolized premisses, furnishes the very same _data_ as were furnished by the problem of the "five liars." pg-x the coined words, introduced in previous editions, such as "eliminands" and "retinends", perhaps hardly need any apology: they were indispensable to my system: but the new plural, here used for the first time, viz. "soriteses", will, i fear, be condemned as "bad english", unless i say a word in its defence. we have _three_ singular nouns, in english, of plural _form_, "series", "species", and "sorites": in all three, the awkwardness, of using the same word for both singular and plural, must often have been felt: this has been remedied, in the case of "series" by coining the plural "serieses", which has already found its way into the dictionaries: so i am no rash innovator, but am merely "following suit", in using the new plural "soriteses". in conclusion, let me point out that even those, who are obliged to study _formal_ logic, with a view to being able to answer examination-papers in that subject, will find the study of _symbolic_ logic most helpful for this purpose, in throwing light upon many of the obscurities with which formal logic abounds, and in furnishing a delightfully easy method of _testing_ the results arrived at by the cumbrous processes which formal logic enforces upon its votaries. this is, i believe, the very first attempt (with the exception of my own little book, _the game of logic_, published in , a very incomplete performance) that has been made to _popularise_ this fascinating subject. it has cost me _years_ of hard work: but if it should prove, as i hope it may, to be of _real_ service to the young, and to be taken up, in high schools and in private families, as a valuable addition to their stock of healthful mental recreations, such a result would more than repay ten times the labour that i have expended on it. l. c. , bedford street, strand. _christmas, ._ pg-xi introduction. _to learners._ [n.b. some remarks, addressed to _teachers_, will be found in the appendix, at p. .] the learner, who wishes to try the question _fairly_, whether this little book does, or does not, supply the materials for a most interesting mental recreation, is _earnestly_ advised to adopt the following rules:-- ( ) begin at the _beginning_, and do not allow yourself to gratify a mere idle curiosity by dipping into the book, here and there. this would very likely lead to your throwing it aside, with the remark "this is _much_ too hard for me!", and thus losing the chance of adding a very _large_ item to your stock of mental delights. this rule (of not _dipping_) is very _desirable_ with _other_ kinds of books----such as novels, for instance, where you may easily spoil much of the enjoyment you would otherwise get from the story, by dipping into it further on, so that what the author meant to be a pleasant surprise comes to you as a matter of course. some people, i know, make a practice of looking into vol. iii first, just to see how the story ends: and perhaps it _is_ as well just to know that all ends _happily_----that the much-persecuted lovers _do_ marry after all, that he is proved to be quite innocent of the murder, that the wicked cousin is completely foiled in his plot and gets the punishment he deserves, and that the rich uncle in india (_qu._ why in _india_? _ans._ because, somehow, uncles never _can_ get rich anywhere else) dies at exactly the right moment----before taking the trouble to read vol. i. this, i say, is _just_ permissible with a _novel_, where vol. iii has a _meaning_, even for those who have not read the earlier part of the story; but, with a _scientific_ book, it is sheer insanity: you will find the latter part _hopelessly_ unintelligible, if you read it before reaching it in regular course. pg-xii ( ) don't begin any fresh chapter, or section, until you are certain that you _thoroughly_ understand the whole book _up to that point_, and that you have worked, correctly, most if not all of the examples which have been set. so long as you are conscious that all the land you have passed through is absolutely _conquered_, and that you are leaving no unsolved difficulties _behind_ you, which will be sure to turn up again later on, your triumphal progress will be easy and delightful. otherwise, you will find your state of puzzlement get worse and worse as you proceed, till you give up the whole thing in utter disgust. ( ) when you come to any passage you don't understand, _read it again_: if you _still_ don't understand it, _read it again_: if you fail, even after _three_ readings, very likely your brain is getting a little tired. in that case, put the book away, and take to other occupations, and next day, when you come to it fresh, you will very likely find that it is _quite_ easy. ( ) if possible, find some genial friend, who will read the book along with you, and will talk over the difficulties with you. _talking_ is a wonderful smoother-over of difficulties. when _i_ come upon anything----in logic or in any other hard subject----that entirely puzzles me, i find it a capital plan to talk it over, _aloud_, even when i am all alone. one can explain things so _clearly_ to one's self! and then, you know, one is so _patient_ with one's self: one _never_ gets irritated at one's own stupidity! if, dear reader, you will faithfully observe these rules, and so give my little book a really _fair_ trial, i promise you, most confidently, that you will find symbolic logic to be one of the most, if not _the_ most, fascinating of mental recreations! in this first part, i have carefully avoided all difficulties which seemed to me to be beyond the grasp of an intelligent child of (say) twelve or fourteen years of age. i have myself taught most of its contents, _vivâ voce_, to _many_ children, and have found them take a real intelligent interest in the subject. for those, who succeed in mastering part i, and who begin, like oliver, "asking for more," i hope to provide, in part ii, some _tolerably_ hard nuts to crack----nuts that will require all the nut-crackers they happen to possess! pg-xiii mental recreation is a thing that we all of us need for our mental health; and you may get much healthy enjoyment, no doubt, from games, such as back-gammon, chess, and the new game "halma". but, after all, when you have made yourself a first-rate player at any one of these games, you have nothing real to _show_ for it, as a _result!_ you enjoyed the game, and the victory, no doubt, _at the time_: but you have no _result_ that you can treasure up and get real _good_ out of. and, all the while, you have been leaving unexplored a perfect _mine_ of wealth. once master the machinery of symbolic logic, and you have a mental occupation always at hand, of absorbing interest, and one that will be of real _use_ to you in _any_ subject you may take up. it will give you clearness of thought----the ability to _see your way_ through a puzzle----the habit of arranging your ideas in an orderly and get-at-able form----and, more valuable than all, the power to detect _fallacies_, and to tear to pieces the flimsy illogical arguments, which you will so continually encounter in books, in newspapers, in speeches, and even in sermons, and which so easily delude those who have never taken the trouble to master this fascinating art. _try it._ that is all i ask of you! l. c. , bedford street, strand. _february , ._ pg-xiv pg-xv contents. =book i.= =things and their attributes.= chapter i. _introductory._ page '=things=' '=attributes=' " '=adjuncts=' " chapter ii. _classification._ '=classification=' ½ '=class=' " '=peculiar=' attributes " '=genus=' " '=species=' " '=differentia=' " '=real=' and '=unreal=', or '=imaginary=', classes '=individual=' " a class regarded as a single thing ½ pg-xvi chapter iii. _division._ § . _introductory._ '=division=' '=codivisional=' classes " § . _dichotomy._ '=dichotomy=' ½ arbitrary limits of classes " subdivision of classes chapter iv. _names._ '=name=' ½ '=real=' and '=unreal=' names " three ways of expressing a name " two senses in which a plural name may be used chapter v. _definitions._ '=definition=' examples worked as models " pg-xvii =book ii.= =propositions.= chapter i. _propositions generally._ § . _introductory._ technical meaning of "some" '=proposition=' " '=normal form=' of a proposition " '=subject=', '=predicate=', and '=terms=' § . _normal form of a proposition._ its four parts:-- ( ) '=sign of quantity=' ( ) name of subject " ( ) '=copula=' " ( ) name of predicate " § . _various kinds of propositions._ three kinds of propositions:-- ( ) begins with "some". called a '=particular=' proposition: also a proposition '=in i=' ( ) begins with "no". called a '=universal negative=' proposition: also a proposition '=in e=' " ( ) begins with "all". called a '=universal affirmative=' proposition: also a proposition '=in a=' " pg-xviii a proposition, whose subject is an individual, is to be regarded as universal " two kinds of propositions, 'propositions of existence', and 'propositions of relation' " chapter ii. _propositions of existence._ '=proposition of existence =' chapter iii. _propositions of relation._ § . _introductory._ '=proposition of relation=' '=universe of discourse=,' or '=univ.=' " § . _reduction of a proposition of relation to normal form._ rules examples worked " § . _a proposition of relation, beginning with "all", is a double proposition._ its equivalence to _two_ propositions pg-xix § . _what is implied, in a proposition of relation, as to the reality of its terms?_ propositions beginning with "some" " " "no" " " " "all" " § . _translation of a proposition of relation into one or more propositions of existence._ rules examples worked " =book iii.= =the biliteral diagram.= chapter i. _symbols and cells._ the diagram assigned to a certain set of things, viz. our univ. univ. divided into 'the x-class' and 'the x'-class' the north and south halves assigned to these two classes " the x-class subdivided into 'the xy-class' and 'the xy'-class' " the north-west and north-east cells assigned to these two classes " the x'-class similarly divided " the south-west and south-east cells similarly assigned " the west and east halves have thus been assigned to 'the y-class' and 'the y'-class' =table i.= attributes of classes, and compartments, or cells, assigned to them pg-xx chapter ii. _counters._ meaning of a red counter placed in a cell " " " " on a partition " american phrase "=sitting on the fence=" " meaning of a grey counter placed in a cell " chapter iii. _representation of propositions._ § . _introductory._ the word "things" to be henceforwards omitted '=uniliteral=' proposition " '=biliteral=' do. " proposition '=in terms of=' certain letters " § . _representation of propositions of existence._ the proposition "some x exist" three other similar propositions " the proposition "no x exist" " three other similar propositions the proposition "some xy exist" " three other similar propositions " the proposition "no xy exist" " three other similar propositions " the proposition "no x exist" is _double_, and is equivalent to the two propositions "no xy exist" and "no xy' exist" pg-xxi § . _representation of propositions of relations._ the proposition "some x are y" " three other similar propositions " the proposition "some y are x" three other similar propositions " trio of equivalent propositions, viz. "some xy exist" = "some x are y" = "some y are x" " '=converse=' propositions, and '=conversion=' " three other similar trios the proposition "no x are y" " three other similar propositions " the proposition "no y are x" " three other similar propositions " trio of equivalent propositions, viz. "no xy exist" = "no x are y" = "no y are x" three other similar trios " the proposition "all x are y" is _double_, and is equivalent to the two propositions "some x are y" and "no x are _y'_" " seven other similar propositions =tables ii, iii.= representation of propositions of existence and relation , chapter iv. _interpretation of biliteral diagram, when marked with counters._ ·-------· |(.)| | interpretation of |---|---| | | | ·-------· and of three other similar arrangements " pg-xxii ·-------· |( )| | interpretation of |---|---| " | | | ·-------· and of three other similar arrangements " ·-------· | (.) | interpretation of |---|---| | | | ·-------· and of three other similar arrangements " ·-------· |(.)|(.)| interpretation of |---|---| " | | | ·-------· and of three other similar arrangements " ·-------· |( )|( )| interpretation of |---|---| " | | | ·-------· and of three other similar arrangements " ·-------· |(.)|( )| interpretation of |---|---| " | | | ·-------· and of seven other similar arrangements =book iv.= =the triliteral diagram.= chapter i. _symbols and cells._ change of biliteral into triliteral diagram the xy-class subdivided into 'the xym-class' and 'the xym'-class' pg-xxiii the inner and outer cells of the north-west quarter assigned to these classes " the xy'-class, the x'y-class, and the x'y'-class similarly subdivided " the inner and outer cells of the north-east, the south-west, and the south-east quarter similarly assigned " the inner square and the outer border have thus been assigned to 'the m-class' and 'the _m'_-class' " rules for finding readily the compartment, or cell, assigned to any given attribute or attributes " =table iv.= attributes of classes, and compartments, or cells, assigned to them chapter ii. _representation of propositions in terms of x and m, or of y and m._ § . _representation of propositions of existence in terms of x and m, or of y and m._ the proposition "some xm exist" seven other similar propositions " the proposition "no xm exist" seven other similar propositions " § . _representation of propositions of relation in terms of x and m, or of y and m._ the pair of converse propositions "some x are m" = "some m are x" " seven other similar pairs " the pair of converse propositions "no x are m" = "no m are x" " seven other similar pairs " the proposition "all x are m" fifteen other similar propositions " =tables v, vi, vii, viii.= representations of propositions in terms of x and m, or of y and m to pg-xxiv chapter iii. _representation of two propositions of relation, one in terms of x and m, and the other in terms of y and m, on the same diagram._ the digits "i" and "o" to be used instead of red and grey counters rules " examples worked " chapter iv. _interpretation, in terms of x and y, of triliteral diagram, when marked with counters or digits._ rules examples worked =book v.= =syllogisms.= chapter i. _introductory._ '=syllogism=' '=premisses=' " '=conclusion=' " '=eliminands=' '=retinends=' " '=consequent=' " the symbol ".'." " specimen-syllogisms " pg-xxv chapter ii. _problems in syllogisms._ § . _introductory._ '=concrete=' and '=abstract=' propositions method of translating a proposition from concrete into abstract form " two forms of problems " § . _given a pair of propositions of relation, which contain between them a pair of codivisional classes, and which are proposed as premisses: to ascertain what conclusion, if any, is consequent from them._ rules examples worked fully " the same worked briefly, as models § . _given a trio of propositions of relation, of which every two contain a pair of codivisional classes, and which are proposed as a syllogism: to ascertain whether the proposed conclusion is consequent from the proposed premisses, and, if so, whether it is complete._ rules examples worked briefly, as models " pg-xxvi =book vi.= =the method of subscripts.= chapter i. _introductory._ meaning of x_{ }, xy_{ }, &c. '=entity=' " meaning of x_{ }, xy_{ }, &c. " '=nullity=' " the symbols "+" and "¶" " '=like=' and '=unlike=' signs " chapter ii. _representation of propositions of relation._ the pair of converse propositions "some x are y" = "some y are x" three other similar pairs " the pair of converse propositions "no x are y" = "no y are x" " three other similar pairs " the proposition "all x are y" the proposition "all x are y" is _double_, and is equivalent to the two propositions "some x exist" and "no x and y'" " seven other similar propositions " rule for translating "all x are y" from abstract into subscript form, and _vice versâ_ " pg-xxvii chapter iii. _syllogisms._ § . _representation of syllogisms._ rules § . _formulæ for syllogisms._ three formulæ worked out:-- fig. i. xm_{ } + ym'_{ } ¶ xy_{ } its two variants (a) and (b) " fig. ii. xm_{ } + ym_{ } ¶ x'y_{ } fig. iii. xm_{ } + ym_{ } + m_{ } ¶ x'y'_{ } =table ix.= formulæ and rules examples worked briefly, as models " § . _fallacies._ '=fallacy=' method of finding forms of fallacies forms best stated in _words_ " three forms of fallacies:-- ( ) fallacy of like eliminands not asserted to exist " ( ) fallacy of unlike eliminands with an entity-premiss ( ) fallacy of two entity-premisses " § . _method of proceeding with a given pair of propositions._ rules pg-xxviii =book vii.= =soriteses.= chapter i. _introductory._ '=sorites=' '=premisses=' " '=partial conclusion=' " '=complete conclusion=' (or '=conclusion=') " '=eliminands=' " '=retinends=' " '=consequent=' " the symbol ".'." " specimen-soriteses chapter ii. _problems in soriteses._ § . _introductory._ form of problem two methods of solution " § . _solution by method of separate syllogisms._ rules example worked " pg-xxix § . _solution by method of underscoring._ '=underscoring=' subscripts to be omitted " example worked fully example worked briefly, as model seventeen examination-papers =book viii.= =examples, with answers and solutions.= chapter i. _examples._ § . _propositions of relation, to be reduced to normal form_ § . _pairs of abstract propositions, one in terms of x and m, and the other in terms of y and m, to be represented on the same triliteral diagram_ § . _marked triliteral diagrams, to be interpreted in terms of x and y_ § . _pairs of abstract propositions, proposed as premisses: conclusions to be found_ pg-xxx § . _pairs of concrete propositions, proposed as premisses: conclusions to be found_ § . _trios of abstract propositions, proposed as syllogisms: to be examined_ § . _trios of concrete propositions, proposed as syllogisms: to be examined_ § . _sets of abstract propositions, proposed as premisses for soriteses: conclusions to be found_ § . _sets of concrete propositions, proposed as premisses for soriteses: conclusions to be found_ chapter ii. _answers._ answers to § § §§ , § § § §§ , pg-xxxi chapter iii. _solutions._ § . _propositions of relation reduced to normal form._ solutions for § § . _method of diagrams._ solutions for § nos. to § " to § " to § " to § . _method of subscripts._ solutions for § § nos. to §§ , , , to =notes= =appendix, addressed to teachers= =notes to appendix= =index.= § . tables § . words &c. explained " pg-xxxii pg book i. things and their attributes. chapter i. _introductory._ the universe contains '=things=.' [for example, "i," "london," "roses," "redness," "old english books," "the letter which i received yesterday."] things have '=attributes=.' [for example, "large," "red," "old," "which i received yesterday."] one thing may have many attributes; and one attribute may belong to many things. [thus, the thing "a rose" may have the attributes "red," "scented," "full-blown," &c.; and the attribute "red" may belong to the things "a rose," "a brick," "a ribbon," &c.] any attribute, or any set of attributes, may be called an '=adjunct=.' [this word is introduced in order to avoid the constant repetition of the phrase "attribute or set of attributes." thus, we may say that a rose has the attribute "red" (or the adjunct "red," whichever we prefer); or we may say that it has the adjunct "red, scented and full-blown."] pg ½ chapter ii. _classification._ 'classification,' or the formation of classes, is a mental process, in which we imagine that we have put together, in a group, certain things. such a group is called a '=class=.' this process may be performed in three different ways, as follows:-- ( ) we may imagine that we have put together all things. the class so formed (i.e. the class "things") contains the whole universe. ( ) we may think of the class "things," and may imagine that we have picked out from it all the things which possess a certain adjunct _not_ possessed by the whole class. this adjunct is said to be '=peculiar=' to the class so formed. in this case, the class "things" is called a '=genus=' with regard to the class so formed: the class, so formed, is called a '=species=' of the class "things": and its peculiar adjunct is called its '=differentia='. pg as this process is entirely _mental_, we can perform it whether there _is_, or _is not_, an _existing_ thing which possesses that adjunct. if there _is_, the class is said to be '=real='; if not, it is said to be '=unreal=', or '=imaginary=.' [for example, we may imagine that we have picked out, from the class "things," all the things which possess the adjunct "material, artificial, consisting of houses and streets"; and we may thus form the real class "towns." here we may regard "things" as a _genus_, "towns" as a _species_ of things, and "material, artificial, consisting of houses and streets" as its _differentia_. again, we may imagine that we have picked out all the things which possess the adjunct "weighing a ton, easily lifted by a baby"; and we may thus form the _imaginary_ class "things that weigh a ton and are easily lifted by a baby."] ( ) we may think of a certain class, _not_ the class "things," and may imagine that we have picked out from it all the members of it which possess a certain adjunct _not_ possessed by the whole class. this adjunct is said to be '=peculiar=' to the smaller class so formed. in this case, the class thought of is called a '=genus=' with regard to the smaller class picked out from it: the smaller class is called a '=species=' of the larger: and its peculiar adjunct is called its '=differentia='. [for example, we may think of the class "towns," and imagine that we have picked out from it all the towns which possess the attribute "lit with gas"; and we may thus form the real class "towns lit with gas." here we may regard "towns" as a _genus_, "towns lit with gas" as a _species_ of towns, and "lit with gas" as its _differentia_. if, in the above example, we were to alter "lit with gas" into "paved with gold," we should get the _imaginary_ class "towns paved with gold."] a class, containing only _one_ member is called an '=individual=.' [for example, the class "towns having four million inhabitants," which class contains only _one_ member, viz. "london."] pg ½ hence, any single thing, which we can name so as to distinguish it from all other things, may be regarded as a one-member class. [thus "london" may be regarded as the one-member class, picked out from the class "towns," which has, as its differentia, "having four million inhabitants."] a class, containing two or more members, is sometimes regarded as _one single thing_. when so regarded, it may possess an adjunct which is _not_ possessed by any member of it taken separately. [thus, the class "the soldiers of the tenth regiment," when regarded as _one single thing_, may possess the attribute "formed in square," which is _not_ possessed by any member of it taken separately.] pg chapter iii. _division._ § . _introductory._ 'division' is a mental process, in which we think of a certain class of things, and imagine that we have divided it into two or more smaller classes. [thus, we might think of the class "books," and imagine that we had divided it into the two smaller classes "bound books" and "unbound books," or into the three classes, "books priced at less than a shilling," "shilling-books," "books priced at more than a shilling," or into the twenty-six classes, "books whose names begin with _a_," "books whose names begin with _b_," &c.] a class, that has been obtained by a certain division, is said to be 'codivisional' with every class obtained by that division. [thus, the class "bound books" is codivisional with each of the two classes, "bound books" and "unbound books." similarly, the battle of waterloo may be said to have been "contemporary" with every event that happened in .] hence a class, obtained by division, is codivisional with itself. [thus, the class "bound books" is codivisional with itself. similarly, the battle of waterloo may be said to have been "contemporary" with itself.] pg ½ § . _dichotomy._ if we think of a certain class, and imagine that we have picked out from it a certain smaller class, it is evident that the _remainder_ of the large class does _not_ possess the differentia of that smaller class. hence it may be regarded as _another_ smaller class, whose differentia may be formed, from that of the class first picked out, by prefixing the word "not"; and we may imagine that we have _divided_ the class first thought of into _two_ smaller classes, whose differentiæ are _contradictory_. this kind of division is called '=dichotomy='. [for example, we may divide "books" into the two classes whose differentiæ are "old" and "not-old."] in performing this process, we may sometimes find that the attributes we have chosen are used so loosely, in ordinary conversation, that it is not easy to decide _which_ of the things belong to the one class and _which_ to the other. in such a case, it would be necessary to lay down some arbitrary _rule_, as to _where_ the one class should end and the other begin. [thus, in dividing "books" into "old" and "not-old," we may say "let all books printed before a.d. , be regarded as 'old,' and all others as 'not-old'."] henceforwards let it be understood that, if a class of things be divided into two classes, whose differentiæ have contrary meanings, each differentia is to be regarded as equivalent to the other with the word "not" prefixed. [thus, if "books" be divided into "old" and "new" the attribute "old" is to be regarded as equivalent to "not-new," and the attribute "new" as equivalent to "not-old."] pg after dividing a class, by the process of _dichotomy_, into two smaller classes, we may sub-divide each of these into two still smaller classes; and this process may be repeated over and over again, the number of classes being doubled at each repetition. [for example, we may divide "books" into "old" and "new" (i.e. "_not_-old"): we may then sub-divide each of these into "english" and "foreign" (i.e. "_not_-english"), thus getting _four_ classes, viz. ( ) old english; ( ) old foreign; ( ) new english; ( ) new foreign. if we had begun by dividing into "english" and "foreign," and had then sub-divided into "old" and "new," the four classes would have been ( ) english old; ( ) english new; ( ) foreign old; ( ) foreign new. the reader will easily see that these are the very same four classes which we had before.] pg ½ chapter iv. _names._ the word "thing", which conveys the idea of a thing, _without_ any idea of an adjunct, represents _any_ single thing. any other word (or phrase), which conveys the idea of a thing, _with_ the idea of an adjunct represents _any_ thing which possesses that adjunct; i.e., it represents any member of the class to which that adjunct is _peculiar_. such a word (or phrase) is called a '=name='; and, if there be an existing thing which it represents, it is said to be a name of that thing. [for example, the words "thing," "treasure," "town," and the phrases "valuable thing," "material artificial thing consisting of houses and streets," "town lit with gas," "town paved with gold," "old english book."] just as a class is said to be _real_, or _unreal_, according as there _is_, or _is not_, an existing thing in it, so also a name is said to be _real_, or _unreal_, according as there _is_, or _is not_, an existing thing represented by it. [thus, "town lit with gas" is a _real_ name: "town paved with gold" is an _unreal_ name.] every name is either a substantive only, or else a phrase consisting of a substantive and one or more adjectives (or phrases used as adjectives). every name, except "thing", may usually be expressed in three different forms:-- (_a_) the substantive "thing", and one or more adjectives (or phrases used as adjectives) conveying the ideas of the attributes; pg (_b_) a substantive, conveying the idea of a thing with the ideas of _some_ of the attributes, and one or more adjectives (or phrases used as adjectives) conveying the ideas of the _other_ attributes; (_c_) a substantive conveying the idea of a thing with the ideas of _all_ the attributes. [thus, the phrase "material living thing, belonging to the animal kingdom, having two hands and two feet" is a name expressed in form (_a_). if we choose to roll up together the substantive "thing" and the adjectives "material, living, belonging to the animal kingdom," so as to make the new substantive "animal," we get the phrase "animal having two hands and two feet," which is a name (representing the same thing as before) expressed in form (_b_). and, if we choose to roll up the whole phrase into one word, so as to make the new substantive "man," we get a name (still representing the very same thing) expressed in form (_c_).] a name, whose substantive is in the _plural_ number, may be used to represent either ( ) members of a class, _regarded as separate things_; or ( ) a whole class, _regarded as one single thing_. [thus, when i say "some soldiers of the tenth regiment are tall," or "the soldiers of the tenth regiment are brave," i am using the name "soldiers of the tenth regiment" in the _first_ sense; and it is just the same as if i were to point to each of them _separately_, and to say "_this_ soldier of the tenth regiment is tall," "_that_ soldier of the tenth regiment is tall," and so on. but, when i say "the soldiers of the tenth regiment are formed in square," i am using the phrase in the _second_ sense; and it is just the same as if i were to say "the _tenth regiment_ is formed in square."] pg chapter v. _definitions._ it is evident that every member of a _species_ is _also_ a member of the _genus_ out of which that species has been picked, and that it possesses the _differentia_ of that species. hence it may be represented by a name consisting of two parts, one being a name representing any member of the _genus_, and the other being the _differentia_ of that species. such a name is called a '=definition=' of any member of that species, and to give it such a name is to '=define=' it. [thus, we may define a "treasure" as a "valuable thing." in this case we regard "things" as the _genus_, and "valuable" as the _differentia_.] the following examples, of this process, may be taken as models for working others. [note that, in each definition, the substantive, representing a member (or members) of the _genus_, is printed in capitals.] . define "a treasure." _ans._ "a valuable thing." . define "treasures." _ans._ "valuable things." . define "a town." _ans._ "a material artificial thing, consisting of houses and streets." pg . define "men." _ans._ "material, living things, belonging to the animal kingdom, having two hands and two feet"; or else "animals having two hands and two feet." . define "london." _ans._ "the material artificial thing, which consists of houses and streets, and has four million inhabitants"; or else "the town which has four million inhabitants." [note that we here use the article "the" instead of "a", because we happen to know that there is only _one_ such thing. the reader can set himself any number of examples of this process, by simply choosing the name of any common thing (such as "house," "tree," "knife"), making a definition for it, and then testing his answer by referring to any english dictionary.] pg book ii. propositions. chapter i. _propositions generally._ § . _introductory._ note that the word "some" is to be regarded, henceforward, as meaning "one or more." the word 'proposition,' as used in ordinary conversation, may be applied to _any_ word, or phrase, which conveys any information whatever. [thus the words "yes" and "no" are propositions in the ordinary sense of the word; and so are the phrases "you owe me five farthings" and "i don't!" such words as "oh!" or "never!", and such phrases as "fetch me that book!" "which book do you mean?" do not seem, at first sight, to convey any _information_; but they can easily be turned into equivalent forms which do so, viz. "i am surprised," "i will never consent to it," "i order you to fetch me that book," "i want to know which book you mean."] but a '=proposition=,' as used in this first part of "symbolic logic," has a peculiar form, which may be called its '=normal form='; and if any proposition, which we wish to use in an argument, is not in normal form, we must reduce it to such a form, before we can use it. pg a '=proposition=,' when in normal form, asserts, as to certain two classes, which are called its '=subject=' and '=predicate=,' either ( ) that _some_ members of its subject are members of its predicate; or ( ) that _no_ members of its subject are members of its predicate; or ( ) that _all_ members of its subject are members of its predicate. the subject and the predicate of a proposition are called its '=terms=.' two propositions, which convey the _same_ information, are said to be '=equivalent='. [thus, the two propositions, "i see john" and "john is seen by me," are equivalent.] § . _normal form of a proposition._ a proposition, in normal form, consists of four parts, viz.-- ( ) the word "some," or "no," or "all." (this word, which tells us _how many_ members of the subject are also members of the predicate, is called the '=sign of quantity=.') ( ) name of subject. ( ) the verb "are" (or "is"). (this is called the '=copula=.') ( ) name of predicate. pg § . _various kinds of propositions._ a proposition, that begins with "some", is said to be '=particular=.' it is also called 'a proposition =in i=.' [note, that it is called 'particular,' because it refers to a _part_ only of the subject.] a proposition, that begins with "no", is said to be '=universal negative=.' it is also called 'a proposition =in e=.' a proposition, that begins with "all", is said to be '=universal affirmative=.' it is also called 'a proposition =in a=.' [note, that they are called 'universal', because they refer to the _whole_ of the subject.] a proposition, whose subject is an _individual_, is to be regarded as _universal_. [let us take, as an example, the proposition "john is not well". this of course implies that there is an _individual_, to whom the speaker refers when he mentions "john", and whom the listener _knows_ to be referred to. hence the class "men referred to by the speaker when he mentions 'john'" is a one-member class, and the proposition is equivalent to "_all_ the men, who are referred to by the speaker when he mentions 'john', are not well."] propositions are of two kinds, 'propositions of existence' and 'propositions of relation.' these shall be discussed separately. pg chapter ii. _propositions of existence._ a '=proposition of existence=', when in normal form, has, for its _subject_, the class "existing things". its sign of quantity is "some" or "no". [note that, though its sign of quantity tells us _how many_ existing things are members of its predicate, it does _not_ tell us the _exact_ number: in fact, it only deals with _two_ numbers, which are, in ascending order, " " and " or more."] it is called "a proposition of existence" because its effect is to assert the _reality_ (i.e. the real _existence_), or else the _imaginariness_, of its predicate. [thus, the proposition "some existing things are honest men" asserts that the class "honest men" is _real_. this is the _normal_ form; but it may also be expressed in any one of the following forms:-- ( ) "honest men exist"; ( ) "some honest men exist"; ( ) "the class 'honest men' exists"; ( ) "there are honest men"; ( ) "there are some honest men". similarly, the proposition "no existing things are men fifty feet high" asserts that the class "men feet high" is _imaginary_. this is the _normal_ form; but it may also be expressed in any one of the following forms:-- ( ) "men feet high do not exist"; ( ) "no men feet high exist"; ( ) "the class 'men feet high' does not exist"; ( ) "there are not any men feet high"; ( ) "there are no men feet high."] pg chapter iii. _propositions of relation._ § . _introductory._ a =proposition of relation=, of the kind to be here discussed, has, for its terms, two specieses of the same genus, such that each of the two names conveys the idea of some attribute _not_ conveyed by the other. [thus, the proposition "some merchants are misers" is of the right kind, since "merchants" and "misers" are specieses of the same genus "men"; and since the name "merchants" conveys the idea of the attribute "mercantile", and the name "misers" the idea of the attribute "miserly", each of which ideas is _not_ conveyed by the other name. but the proposition "some dogs are setters" is _not_ of the right kind, since, although it is true that "dogs" and "setters" are specieses of the same genus "animals", it is _not_ true that the name "dogs" conveys the idea of any attribute not conveyed by the name "setters". such propositions will be discussed in part ii.] the genus, of which the two terms are specieses, is called the '=universe of discourse=,' or (more briefly) the '=univ.=' the sign of quantity is "some" or "no" or "all". [note that, though its sign of quantity tells us _how many_ members of its subject are _also_ members of its predicate, it does not tell us the _exact_ number: in fact, it only deals with _three_ numbers, which are, in ascending order, " ", " or more", "the total number of members of the subject".] it is called "a proposition of relation" because its effect is to assert that a certain _relationship_ exists between its terms. pg § . _reduction of a proposition of relation to normal form._ the rules, for doing this, are as follows:-- ( ) ascertain what is the _subject_ (i.e., ascertain what class we are _talking about_); ( ) if the verb, governed by the subject, is _not_ the verb "are" (or "is"), substitute for it a phrase beginning with "are" (or "is"); ( ) ascertain what is the _predicate_ (i.e., ascertain what class it is, which is asserted to contain _some_, or _none_, or _all_, of the members of the subject); ( ) if the name of each term is _completely expressed_ (i.e. if it contains a substantive), there is no need to determine the 'univ.'; but, if either name is _incompletely expressed_, and contains _attributes_ only, it is then necessary to determine a 'univ.', in order to insert its name as the substantive. ( ) ascertain the _sign of quantity_; ( ) arrange in the following order:-- sign of quantity, subject, copula, predicate. [let us work a few examples, to illustrate these rules. ( ) "some apples are not ripe." ( ) the subject is "apples." ( ) the verb is "are." ( ) the predicate is "not-ripe * * *." (as no substantive is expressed, and we have not yet settled what the univ. is to be, we are forced to leave a blank.) ( ) let univ. be "fruit." ( ) the sign of quantity is "some." ( ) the proposition now becomes "some | apples | are | not-ripe fruit." pg ( ) "none of my speculations have brought me as much as per cent." ( ) the subject is "my speculations." ( ) the verb is "have brought," for which we substitute the phrase "are * * * that have brought". ( ) the predicate is "* * * that have brought &c." ( ) let univ. be "transactions." ( ) the sign of quantity is "none of." ( ) the proposition now becomes "none of | my speculations | are | transactions that have brought me as much as per cent." ( ) "none but the brave deserve the fair." to begin with, we note that the phrase "none but the brave" is equivalent to "no _not_-brave." ( ) the subject has for its _attribute_ "not-brave." but no _substantive_ is supplied. so we express the subject as "not-brave * * *." ( ) the verb is "deserve," for which we substitute the phrase "are deserving of". ( ) the predicate is "* * * deserving of the fair." ( ) let univ. be "persons." ( ) the sign of quantity is "no." ( ) the proposition now becomes "no | not-brave persons | are | persons deserving of the fair." ( ) "a lame puppy would not say "thank you" if you offered to lend it a skipping-rope." ( ) the subject is evidently "lame puppies," and all the rest of the sentence must somehow be packed into the predicate. ( ) the verb is "would not say," &c., for which we may substitute the phrase "are not grateful for." ( ) the predicate may be expressed as "* * * not grateful for the loan of a skipping-rope." ( ) let univ. be "puppies." ( ) the sign of quantity is "all." ( ) the proposition now becomes "all | lame puppies | are | puppies not grateful for the loan of a skipping-rope." pg ( ) "no one takes in the _times_, unless he is well-educated." ( ) the subject is evidently persons who are not well-educated ("no _one_" evidently means "no _person_"). ( ) the verb is "takes in," for which we may substitute the phrase "are persons taking in." ( ) the predicate is "persons taking in the _times_." ( ) let univ. be "persons." ( ) the sign of quantity is "no." ( ) the proposition now becomes "no | persons who are not well-educated | are | persons taking in the _times_." ( ) "my carriage will meet you at the station." ( ) the subject is "my carriage." this, being an 'individual,' is equivalent to the class "my carriages." (note that this class contains only _one_ member.) ( ) the verb is "will meet", for which we may substitute the phrase "are * * * that will meet." ( ) the predicate is "* * * that will meet you at the station." ( ) let univ. be "things." ( ) the sign of quantity is "all." ( ) the proposition now becomes "all | my carriages | are | things that will meet you at the station." ( ) "happy is the man who does not know what 'toothache' means!" ( ) the subject is evidently "the man &c." (note that in this sentence, the _predicate_ comes first.) at first sight, the subject seems to be an '_individual_'; but on further consideration, we see that the article "the" does _not_ imply that there is only _one_ such man. hence the phrase "the man who" is equivalent to "all men who". ( ) the verb is "are." ( ) the predicate is "happy * * *." ( ) let univ. be "men." ( ) the sign of quantity is "all." ( ) the proposition now becomes "all | men who do not know what 'toothache' means | are | happy men." pg ( ) "some farmers always grumble at the weather, whatever it may be." ( ) the subject is "farmers." ( ) the verb is "grumble," for which we substitute the phrase "are * * * who grumble." ( ) the predicate is "* * * who always grumble &c." ( ) let univ. be "persons." ( ) the sign of quantity is "some." ( ) the proposition now becomes "some | farmers | are | persons who always grumble at the weather, whatever it may be." ( ) "no lambs are accustomed to smoke cigars." ( ) the subject is "lambs." ( ) the verb is "are." ( ) the predicate is "* * * accustomed &c." ( ) let univ. be "animals." ( ) the sign of quantity is "no." ( ) the proposition now becomes "no | lambs | are | animals accustomed to smoke cigars." ( ) "i ca'n't understand examples that are not arranged in regular order, like those i am used to." ( ) the subject is "examples that," &c. ( ) the verb is "i ca'n't understand," which we must alter, so as to have "examples," instead of "i," as the nominative case. it may be expressed as "are not understood by me." ( ) the predicate is "* * * not understood by me." ( ) let univ. be "examples." ( ) the sign of quantity is "all." ( ) the proposition now becomes "all | examples that are not arranged in regular order like those i am used to | are | examples not understood by me."] pg § . _a proposition of relation, beginning with "all", is a double proposition._ a proposition of relation, beginning with "all", asserts (as we already know) that "_all_ members of the subject are members of the predicate". this evidently contains, as a _part_ of what it tells us, the smaller proposition "_some_ members of the subject are members of the predicate". [thus, the proposition "_all_ bankers are rich men" evidently contains the smaller proposition "_some_ bankers are rich men".] the question now arises "what is the _rest_ of the information which this proposition gives us?" in order to answer this question, let us begin with the smaller proposition, "_some_ members of the subject are members of the predicate," and suppose that this is _all_ we have been told; and let us proceed to inquire what _else_ we need to be told, in order to know that "_all_ members of the subject are members of the predicate". [thus, we may suppose that the proposition "_some_ bankers are rich men" is all the information we possess; and we may proceed to inquire what _other_ proposition needs to be added to it, in order to make up the entire proposition "_all_ bankers are rich men".] let us also suppose that the 'univ.' (i.e. the genus, of which both the subject and the predicate are specieses) has been divided (by the process of _dichotomy_) into two smaller classes, viz. ( ) the predicate; ( ) the class whose differentia is _contradictory_ to that of the predicate. [thus, we may suppose that the genus "men," (of which both "bankers" and "rich men" are specieses) has been divided into the two smaller classes, "rich men", "poor men".] pg now we know that _every_ member of the subject is (as shown at p. ) a member of the univ. hence _every_ member of the subject is either in class ( ) or else in class ( ). [thus, we know that _every_ banker is a member of the genus "men". hence, _every_ banker is either in the class "rich men", or else in the class "poor men".] also we have been told that, in the case we are discussing, _some_ members of the subject are in class ( ). what _else_ do we need to be told, in order to know that _all_ of them are there? evidently we need to be told that _none_ of them are in class ( ); i.e. that _none_ of them are members of the class whose differentia is _contradictory_ to that of the predicate. [thus, we may suppose we have been told that _some_ bankers are in the class "rich men". what _else_ do we need to be told, in order to know that _all_ of them are there? evidently we need to be told that _none_ of them are in the class "_poor_ men".] hence a proposition of relation, beginning with "all", is a _double_ proposition, and is '=equivalent=' to (i.e. gives the same information as) the _two_ propositions ( ) "_some_ members of the subject are members of the predicate"; ( ) "_no_ members of the subject are members of the class whose differentia is _contradictory_ to that of the predicate". [thus, the proposition "_all_ bankers are rich men" is a _double_ proposition, and is equivalent to the _two_ propositions ( ) "_some_ bankers are rich men"; ( ) "_no_ bankers are _poor_ men".] pg § . _what is implied, in a proposition of relation, as to the reality of its terms?_ note that the rules, here laid down, are _arbitrary_, and only apply to part i of my "symbolic logic." a proposition of relation, beginning with "some", is henceforward to be understood as asserting that there are _some existing things_, which, being members of the subject, are also members of the predicate; i.e. that _some existing things_ are members of _both_ terms at once. hence it is to be understood as implying that _each_ term, taken by itself, is _real_. [thus, the proposition "some rich men are invalids" is to be understood as asserting that _some existing things_ are "rich invalids". hence it implies that _each_ of the two classes, "rich men" and "invalids", taken by itself, is _real_.] a proposition of relation, beginning with "no", is henceforward to be understood as asserting that there are _no existing things_ which, being members of the subject, are also members of the predicate; i.e. that _no existing things_ are members of _both_ terms at once. but this implies nothing as to the _reality_ of either term taken by itself. [thus, the proposition "no mermaids are milliners" is to be understood as asserting that _no existing things_ are "mermaid-milliners". but this implies nothing as to the _reality_, or the _unreality_, of either of the two classes, "mermaids" and "milliners", taken by itself. in this case as it happens, the subject is _imaginary_, and the predicate _real_.] a proposition of relation, beginning with "all", contains (see § ) a similar proposition beginning with "some". hence it is to be understood as implying that _each_ term, taken by itself, is _real_. [thus, the proposition "all hyænas are savage animals" contains the proposition "some hyænas are savage animals". hence it implies that _each_ of the two classes, "hyænas" and "savage animals", taken by itself, is _real_.] pg § . _translation of a proposition of relation into one or more propositions of existence._ we have seen that a proposition of relation, beginning with "some," asserts that _some existing things_, being members of its subject, are _also_ members of its predicate. hence, it asserts that some existing things are members of _both_; i.e. it asserts that some existing things are members of the class of things which have _all_ the attributes of the subject and the predicate. hence, to translate it into a proposition of existence, we take "existing things" as the new _subject_, and things, which have _all_ the attributes of the subject and the predicate, as the new predicate. similarly for a proposition of relation beginning with "no". a proposition of relation, beginning with "all", is (as shown in § ) equivalent to _two_ propositions, one beginning with "some" and the other with "no", each of which we now know how to translate. [let us work a few examples, to illustrate these rules. ( ) "some apples are not ripe." here we arrange thus:-- "some" _sign of quantity_. "existing things" _subject_. "are" _copula_. "not-ripe apples" _predicate_. or thus:-- "some | existing things | are | not-ripe apples." pg ( ) "some farmers always grumble at the weather, whatever it may be." here we arrange thus:-- "some | existing things | are | farmers who always grumble at the weather, whatever it may be." ( ) "no lambs are accustomed to smoke cigars." here we arrange thus:-- "no | existing things |are | lambs accustomed to smoke cigars." ( ) "none of my speculations have brought me as much as per cent." here we arrange thus:-- "no | existing things | are | speculations of mine, which have brought me as much as per cent." ( ) "none but the brave deserve the fair." here we note, to begin with, that the phrase "none but the brave" is equivalent to "no not-brave men." we then arrange thus:-- "no | existing things | are | not-brave men deserving of the fair." ( ) "all bankers are rich men." this is equivalent to the two propositions "some bankers are rich men" and "no bankers are poor men." here we arrange thus:-- "some | existing things | are | rich bankers"; and "no | existing things | are | poor bankers."] [work examples § = =, - (p. ).] pg book iii. the biliteral diagram. ·-------------· | | | | xy | xy' | | | | |------|------| | | | | x'y | x'y' | | | | ·-------------· chapter i. _symbols and cells._ first, let us suppose that the above diagram is an enclosure assigned to a certain class of things, which we have selected as our 'universe of discourse.' or, more briefly, as our 'univ'. [for example, we might say "let univ. be 'books'"; and we might imagine the diagram to be a large table, assigned to all "books."] [the reader is strongly advised, in reading this chapter, _not_ to refer to the above diagram, but to draw a large one for himself, _without any letters_, and to have it by him while he reads, and keep his finger on that particular _part_ of it, about which he is reading.] pg secondly, let us suppose that we have selected a certain adjunct, which we may call "x," and have divided the large class, to which we have assigned the whole diagram, into the two smaller classes whose differentiæ are "x" and "not-x" (which we may call "x'"), and that we have assigned the _north_ half of the diagram to the one (which we may call "the class of x-things," or "the x-class"), and the _south_ half to the other (which we may call "the class of x'-things," or "the x'-class"). [for example, we might say "let x mean 'old,' so that x' will mean 'new'," and we might suppose that we had divided books into the two classes whose differentiæ are "old" and "new," and had assigned the _north_ half of the table to "_old_ books" and the _south_ half to "_new_ books."] thirdly, let us suppose that we have selected another adjunct, which we may call "y", and have subdivided the x-class into the two classes whose differentiæ are "y" and "y'", and that we have assigned the north-_west_ cell to the one (which we may call "the xy-class"), and the north-_east_ cell to the other (which we may call "the xy'-class"). [for example, we might say "let y mean 'english,' so that y' will mean 'foreign'", and we might suppose that we had subdivided "old books" into the two classes whose differentiæ are "english" and "foreign", and had assigned the north-_west_ cell to "old _english_ books", and the north-_east_ cell to "old _foreign_ books."] fourthly, let us suppose that we have subdivided the x'-class in the same manner, and have assigned the south-_west_ cell to the x'y-class, and the south-_east_ cell to the x'y'-class. [for example, we might suppose that we had subdivided "new books" into the two classes "new _english_ books" and "new _foreign_ books", and had assigned the south-_west_ cell to the one, and the south-_east_ cell to the other.] it is evident that, if we had begun by dividing for y and y', and had then subdivided for x and x', we should have got the _same_ four classes. hence we see that we have assigned the _west_ half to the y-class, and the _east_ half to the y'-class. pg [thus, in the above example, we should find that we had assigned the _west_ half of the table to "_english_ books" and the _east_ half to "_foreign_ books." ·-------------------· | old | old | | english | foreign | | books | books | |---------|---------| | new | new | | english | foreign | | books | books | ·-------------------· we have, in fact, assigned the four quarters of the table to four different classes of books, as here shown.] the reader should carefully remember that, in such a phrase as "the x-things," the word "things" means that particular _kind_ of things, to which the whole diagram has been assigned. [thus, if we say "let univ. be 'books'," we mean that we have assigned the whole diagram to "books." in that case, if we took "x" to mean "old", the phrase "the x-things" would mean "the old books."] the reader should not go on to the next chapter until he is _quite familiar_ with the _blank_ diagram i have advised him to draw. he ought to be able to name, _instantly_, the _adjunct_ assigned to any compartment named in the right-hand column of the following table. also he ought to be able to name, _instantly_, the _compartment_ assigned to any adjunct named in the left-hand column. to make sure of this, he had better put the book into the hands of some genial friend, while he himself has nothing but the blank diagram, and get that genial friend to question him on this table, _dodging_ about as much as possible. the questions and answers should be something like this:-- pg table i. ·----------------------------------------· | _adjuncts_ | _compartments, or cells,_ | | _of_ | _assigned to them._ | | _classes._ | | |------------|---------------------------| | x | north half. | | x' | south " | | y | west " | | y' | east " | |------------|---------------------------| | xy | north-west cell. | | xy' | " east " | | x'y | south-west " | | x'y' | " east " | ·----------------------------------------· q. "adjunct for west half?" a. "y." q. "compartment for xy'?" a. "north-east cell." q. "adjunct for south-west cell?" a. "x'y." &c., &c. after a little practice, he will find himself able to do without the blank diagram, and will be able to see it _mentally_ ("in my mind's eye, horatio!") while answering the questions of his genial friend. when _this_ result has been reached, he may safely go on to the next chapter. pg chapter ii. _counters._ let us agree that a _red_ counter, placed within a cell, shall mean "this cell is _occupied_" (i.e. "there is at least _one_ thing in it"). let us also agree that a _red_ counter, placed on the partition between two cells, shall mean "the compartment, made up of these two cells, is _occupied_; but it is not known _whereabouts_, in it, its occupants are." hence it may be understood to mean "at least _one_ of these two cells is occupied: possibly _both_ are." our ingenious american cousins have invented a phrase to describe the condition of a man who has not yet made up his mind _which_ of two political parties he will join: such a man is said to be "=sitting on the fence=." this phrase exactly describes the condition of the red counter. let us also agree that a _grey_ counter, placed within a cell, shall mean "this cell is _empty_" (i.e. "there is _nothing_ in it"). [the reader had better provide himself with red counters and grey ones.] pg chapter iii. _representation of propositions._ § . _introductory._ henceforwards, in stating such propositions as "some x-things exist" or "no x-things are y-things", i shall omit the word "things", which the reader can supply for himself, and shall write them as "some x exist" or "no x are y". [note that the word "things" is here used with a special meaning, as explained at p. .] a proposition, containing only _one_ of the letters used as symbols for attributes, is said to be '=uniliteral='. [for example, "some x exist", "no y' exist", &c.] a proposition, containing _two_ letters, is said to be ='biliteral'=. [for example, "some xy' exist", "no x' are y", &c.] a proposition is said to be '=in terms of=' the letters it contains, whether with or without accents. [thus, "some xy' exist", "no x' are y", &c., are said to be _in terms of_ x and y.] pg § . _representation of propositions of existence._ let us take, first, the proposition "some x exist". [note that this proposition is (as explained at p. ) equivalent to "some existing things are x-things."] this tells us that there is at least _one_ thing in the north half; that is, that the north half is _occupied_. and this we can evidently represent by placing a _red_ counter (here represented by a _dotted_ circle) on the partition which divides the north half. ·-------· | (.) | |---|---| | | | ·-------· [in the "books" example, this proposition would be "some old books exist".] similarly we may represent the three similar propositions "some x' exist", "some y exist", and "some y' exist". [the reader should make out all these for himself. in the "books" example, these propositions would be "some new books exist", &c.] let us take, next, the proposition "no x exist". this tells us that there is _nothing_ in the north half; that is, that the north half is _empty_; that is, that the north-west cell and the north-east cell are both of them _empty_. and this we can represent by placing _two grey_ counters in the north half, one in each cell. ·-------· |( )|( )| |---|---| | | | ·-------· [the reader may perhaps think that it would be enough to place a _grey_ counter on the partition in the north half, and that, just as a _red_ counter, so placed, would mean "this half is _occupied_", so a _grey_ one would mean "this half is _empty_". this, however, would be a mistake. we have seen that a _red_ counter, so placed, would mean "at least _one_ of these two cells is occupied: possibly _both_ are." hence a _grey_ one would merely mean "at least _one_ of these two cells is empty: possibly _both_ are". but what we have to represent is, that both cells are _certainly_ empty: and this can only be done by placing a _grey_ counter in _each_ of them. in the "books" example, this proposition would be "no old books exist".] pg similarly we may represent the three similar propositions "no x' exist", "no y exist", and "no y' exist". [the reader should make out all these for himself. in the "books" example, these three propositions would be "no new books exist", &c.] let us take, next, the proposition "some xy exist". this tells us that there is at least _one_ thing in the north-west cell; that is, that the north-west cell is _occupied_. and this we can represent by placing a _red_ counter in it. ·-------· |(.)| | |---|---| | | | ·-------· [in the "books" example, this proposition would be "some old english books exist".] similarly we may represent the three similar propositions "some xy' exist", "some x'y exist", and "some x'y' exist". [the reader should make out all these for himself. in the "books" example, these three propositions would be "some old foreign books exist", &c.] let us take, next, the proposition "no xy exist". this tells us that there is _nothing_ in the north-west cell; that is, that the north-west cell is _empty_. and this we can represent by placing a _grey_ counter in it. ·-------· |( )| | |---|---| | | | ·-------· [in the "books" example, this proposition would be "no old english books exist".] similarly we may represent the three similar propositions "no xy' exist", "no x'y exist", and "no x'y' exist". [the reader should make out all these for himself. in the "books" example, these three propositions would be "no old foreign books exist", &c.] pg we have seen that the proposition "no x exist" may be represented by placing _two grey_ counters in the north half, one in each cell. ·-------· |( )|( )| |---|---| | | | ·-------· we have also seen that these two _grey_ counters, taken _separately_, represent the two propositions "no xy exist" and "no xy' exist". hence we see that the proposition "no x exist" is a _double_ proposition, and is equivalent to the _two_ propositions "no xy exist" and "no xy' exist". [in the "books" example, this proposition would be "no old books exist". hence this is a _double_ proposition, and is equivalent to the _two_ propositions "no old _english_ books exist" and "no old _foreign_ books exist".] § . _representation of propositions of relation._ let us take, first, the proposition "some x are y". this tells us that at least _one_ thing, in the _north_ half, is also in the _west_ half. hence it must be in the space _common_ to them, that is, in the _north-west cell_. hence the north-west cell is _occupied_. and this we can represent by placing a _red_ counter in it. ·-------· |(.)| | |---|---| | | | ·-------· [note that the _subject_ of the proposition settles which _half_ we are to use; and that the _predicate_ settles in which _portion_ of it we are to place the red counter. in the "books" example, this proposition would be "some old books are english".] similarly we may represent the three similar propositions "some x are y'", "some x' are y", and "some x' are y'". [the reader should make out all these for himself. in the "books" example, these three propositions would be "some old books are foreign", &c.] pg let us take, next, the proposition "some y are x". this tells us that at least _one_ thing, in the _west_ half, is also in the _north_ half. hence it must be in the space _common_ to them, that is, in the _north-west cell_. hence the north-west cell is _occupied_. and this we can represent by placing a _red_ counter in it. ·-------· |(.)| | |---|---| | | | ·-------· [in the "books" example, this proposition would be "some english books are old".] similarly we may represent the three similar propositions "some y are x'", "some y' are x", and "some y' are x'". [the reader should make out all these for himself. in the "books" example, these three propositions would be "some english books are new", &c.] we see that this _one_ diagram has now served to represent no less than _three_ propositions, viz. ( ) "some xy exist; ( ) some x are y; ( ) some y are x". ·-------· |(.)| | |---|---| | | | ·-------· hence these three propositions are equivalent. [in the "books" example, these propositions would be ( ) "some old english books exist; ( ) some old books are english; ( ) some english books are old".] the two equivalent propositions, "some x are y" and "some y are x", are said to be '=converse=' to each other; and the process, of changing one into the other, is called '=converting=', or '=conversion='. [for example, if we were told to convert the proposition "some apples are not ripe," we should first choose our univ. (say "fruit"), and then complete the proposition, by supplying the substantive "fruit" in the predicate, so that it would be "some apples are not-ripe fruit"; and we should then convert it by interchanging its terms, so that it would be "some not-ripe fruit are apples".] pg similarly we may represent the three similar trios of equivalent propositions; the whole set of _four_ trios being as follows:-- ( ) "some xy exist" = "some x are y" = "some y are x". ( ) "some xy' exist" = "some x are y'" = "some y' are x". ( ) "some x'y exist" = "some x' are y" = "some y are x'". ( ) "some x'y' exist" = "some x' are y'" = "some y' are x'". let us take, next, the proposition "no x are y". this tell us that no thing, in the _north_ half, is also in the _west_ half. hence there is _nothing_ in the space _common_ to them, that is, in the _north-west cell_. hence the north-west cell is _empty_. and this we can represent by placing a _grey_ counter in it. ·-------· |( )| | |---|---| | | | ·-------· [in the "books" example, this proposition would be "no old books are english".] similarly we may represent the three similar propositions "no x are y'", and "no x' are y", and "no x' are y'". [the reader should make out all these for himself. in the "books" example, these three propositions would be "no old books are foreign", &c.] let us take, next, the proposition "no y are x". this tells us that no thing, in the _west_ half, is also in the _north_ half. hence there is _nothing_ in the space _common_ to them, that is, in the _north-west cell_. that is, the north-west cell is _empty_. and this we can represent by placing a _grey_ counter in it. ·-------· |( )| | |---|---| | | | ·-------· [in the "books" example, this proposition would be "no english books are old".] similarly we may represent the three similar propositions "no y are x'", "no y' are x", and "no y' are x'". [the reader should make out all these for himself. in the "books" example, these three propositions would be "no english books are new", &c.] pg ·-------· |( )| | |---|---| | | | ·-------· we see that this _one_ diagram has now served to present no less than _three_ propositions, viz. ( ) "no xy exist; ( ) no x are y; ( ) no y are x." hence these three propositions are equivalent. [in the "books" example, these propositions would be ( ) "no old english books exist; ( ) no old books are english; ( ) no english books are old".] the two equivalent propositions, "no x are y" and "no y are x", are said to be 'converse' to each other. [for example, if we were told to convert the proposition "no porcupines are talkative", we should first choose our univ. (say "animals"), and then complete the proposition, by supplying the substantive "animals" in the predicate, so that it would be "no porcupines are talkative animals", and we should then convert it, by interchanging its terms, so that it would be "no talkative animals are porcupines".] similarly we may represent the three similar trios of equivalent propositions; the whole set of _four_ trios being as follows:-- ( ) "no xy exist" = "no x are y" = "no y are x". ( ) "no xy' exist" = "no x are y'" = "no y' are x". ( ) "no x'y exist" = "no x' are y" = "no y are x'". ( ) "no x'y' exist" = "no x' are y'" = "no y' are x'". let us take, next, the proposition "all x are y". we know (see p. ) that this is a _double_ proposition, and equivalent to the _two_ propositions "some x are y" and "no x are y'", each of which we already know how to represent. ·-------· |(.)|( )| |---|---| | | | ·-------· [note that the _subject_ of the given proposition settles which _half_ we are to use; and that its _predicate_ settles in which _portion_ of that half we are to place the red counter.] pg table ii. ·-----------------------------------------------------· | | ·-------· | | ·-------· | | | | (.) | | | |( )|( )| | | some x exist | |---|---| | no x exist | |---|---| | | | | | | | | | | | | | | ·-------· | | ·-------· | |---------------|-----------|-------------|-----------| | | ·-------· | | ·-------· | | | | | | | | | | | | | some x' exist | |---|---| | no x' exist | |---|---| | | | | (.) | | | |( )|( )| | | | ·-------· | | ·-------· | |---------------|-----------|-------------|-----------| | | ·-------· | | ·-------· | | | | | | | | |( )| | | | some y exist | |(.)|---| | no y exist | |---|---| | | | | | | | | |( )| | | | | ·-------· | | ·-------· | |---------------|-----------|-------------|-----------| | | ·-------· | | ·-------· | | | | | | | | | |( )| | | some y' exist | |---|(.)| | no y' exist | |---|---| | | | | | | | | | |( )| | | | ·-------· | | ·-------· | ·-----------------------------------------------------· similarly we may represent the seven similar propositions "all x are y'", "all x' are y", "all x' are y'", "all y are x", "all y are x'", "all y' are x", and "all y' are x'". let us take, lastly, the double proposition "some x are y and some are y'", each part of which we already know how to represent. ·-------· |(.)|(.)| |---|---| | | | ·-------· similarly we may represent the three similar propositions, "some x' are y and some are y'", "some y are x and some are x'", "some y' are x and some are x'". the reader should now get his genial friend to question him, severely, on these two tables. the _inquisitor_ should have the tables before him: but the _victim_ should have nothing but a blank diagram, and the counters with which he is to represent the various propositions named by his friend, e.g. "some y exist", "no y' are x", "all x are y", &c. &c. pg table iii. ·-------------------------------------------------------------· | | ·-------· | | ·-------· | | some xy exist | |(.)| | | | |(.)|( )| | |  = some x are y | |---|---| | all x are y | |---|---| | |  = some y are x | | | | | | | | | | | | ·-------· | | ·-------· | |------------------|-----------|------------------|-----------| | | ·-------· | | ·-------· | | some xy' exist | | |(.)| | | |( )|(.)| | |  = some x are y' | |---|---| | all x are y' | |---|---| | |  = some y' are x | | | | | | | | | | | | ·-------· | | ·-------· | |------------------|-----------|------------------|-----------| | | ·-------· | | ·-------· | | some x'y exist | | | | | | | | | | |  = some x' are y | |---|---| | all x' are y | |---|---| | |  = some y are x' | |(.)| | | | |(.)|( )| | | | ·-------· | | ·-------· | |------------------|-----------|------------------|-----------| | | ·-------· | | ·-------· | | some x'y' exist | | | | | | | | | | |  = some x' are y'| |---|---| | all x' are y' | |---|---| | |  = some y' are x'| | |(.)| | | |( )|(.)| | | | ·-------· | | ·-------· | ·-------------------------------------------------------------· ·-------------------------------------------------------------· | | ·-------· | | ·-------· | | no xy exist | |( )| | | | |(.)| | | |  = no x are y | |---|---| | all y are x | |---|---| | |  = no y are x | | | | | | |( )| | | | | ·-------· | | ·-------· | |------------------|-----------|------------------|-----------| | | ·-------· | | ·-------· | | no xy' exist | | |( )| | | |( )| | | |  = no x are y' | |---|---| | all y are x' | |---|---| | |  = no y' are x | | | | | | |(.)| | | | | ·-------· | | ·-------· | |------------------|-----------|------------------|-----------| | | ·-------· | | ·-------· | | no x'y exist | | | | | | | |(.)| | |  = no x' are y | |---|---| | all y' are x | |---|---| | |  = no y are x' | |( )| | | | | |( )| | | | ·-------· | | ·-------· | |------------------|-----------|------------------|-----------| | | ·-------· | | ·-------· | | no x'y' exist | | | | | | | |( )| | |  = no x' are y' | |---|---| | all y' are x' | |---|---| | |  = no y' are x' | | |( )| | | | |(.)| | | | ·-------· | | ·-------· | ·-------------------------------------------------------------· ·-------------------------------------------------------------· | | ·-------· | | ·-------· | | | |(.)|(.)| | | |(.)| | | | some x are y, | |---|---| | some y are x | |---|---| | | and some are y' | | | | | and some are x' | |(.)| | | | | ·-------· | | ·-------· | |------------------|-----------|------------------|-----------| | | ·-------· | | ·-------· | | | | | | | | | |(.)| | | some x' are y, | |---|---| | some y' are x | |---|---| | | and some are y' | |(.)|(.)| | and some are x' | | |(.)| | | | ·-------· | | ·-------· | ·-------------------------------------------------------------· pg chapter iv. _interpretation of biliteral diagram when marked with counters._ the diagram is supposed to be set before us, with certain counters placed upon it; and the problem is to find out what proposition, or propositions, the counters represent. as the process is simply the reverse of that discussed in the previous chapter, we can avail ourselves of the results there obtained, as far as they go. first, let us suppose that we find a _red_ counter placed in the north-west cell. ·-------· |(.)| | |---|---| | | | ·-------· we know that this represents each of the trio of equivalent propositions "some xy exist" = "some x are y" = "some y are x". similarly we may interpret a _red_ counter, when placed in the north-east, or south-west, or south-east cell. next, let us suppose that we find a _grey_ counter placed in the north-west cell. ·-------· |( )| | |---|---| | | | ·-------· we know that this represents each of the trio of equivalent propositions "no xy exist" = "no x are y" = "no y are x". similarly we may interpret a _grey_ counter, when placed in the north-east, or south-west, or south-east cell. pg next, let us suppose that we find a _red_ counter placed on the partition which divides the north half. ·-------· | (.) | |---|---| | | | ·-------· we know that this represents the proposition "some x exist." similarly we may interpret a _red_ counter, when placed on the partition which divides the south, or west, or east half. * * * * * next, let us suppose that we find _two red_ counters placed in the north half, one in each cell. ·-------· |(.)|(.)| |---|---| | | | ·-------· we know that this represents the _double_ proposition "some x are y and some are y'". similarly we may interpret _two red_ counters, when placed in the south, or west, or east half. * * * * * next, let us suppose that we find _two grey_ counters placed in the north half, one in each cell. ·-------· |( )|( )| |---|---| | | | ·-------· we know that this represents the proposition "no x exist". similarly we may interpret _two grey_ counters, when placed in the south, or west, or east half. * * * * * lastly, let us suppose that we find a _red_ and a _grey_ counter placed in the north half, the _red_ in the north-_west_ cell, and the _grey_ in the north-_east_ cell. ·-------· |(.)|( )| |---|---| | | | ·-------· we know that this represents the proposition, "all x are y". [note that the _half_, occupied by the two counters, settles what is to be the _subject_ of the proposition, and that the _cell_, occupied by the _red_ counter, settles what is to be its _predicate_.] pg similarly we may interpret a _red_ and a _grey_ counter, when placed in any one of the seven similar positions red in north-east, grey in north-west; red in south-west, grey in south-east; red in south-east, grey in south-west; red in north-west, grey in south-west; red in south-west, grey in north-west; red in north-east, grey in south-east; red in south-east, grey in north-east. once more the genial friend must be appealed to, and requested to examine the reader on tables ii and iii, and to make him not only _represent_ propositions, but also _interpret_ diagrams when marked with counters. the questions and answers should be like this:-- q. represent "no x' are y'." a. grey counter in s.e. cell. q. interpret red counter on e. partition. a. "some y' exist." q. represent "all y' are x." a. red in n.e. cell; grey in s.e. q. interpret grey counter in s.w. cell. a. "no x'y exist" = "no x' are y" = "no y are x'". &c., &c. at first the examinee will need to have the board and counters before him; but he will soon learn to dispense with these, and to answer with his eyes shut or gazing into vacancy. [work examples § = =, - (p. ).] pg book iv. the triliteral diagram. ·-----------------· ·-----------------· | | | | xy | xy' | | | | | m' | m' | | xy | xy' | | ·----|----· | | | | | | xy | xy'| | | | | | | m | m | | |--------|--------| |---|----|----|---| | | | | |x'y |x'y'| | | | | | | m | m | | | x'y | x'y' | | ·----|----· | | | | | x'y | x'y' | | | | | m' | m' | ·-----------------· ·-----------------· chapter i. _symbols and cells._ first, let us suppose that the above _left_-hand diagram is the biliteral diagram that we have been using in book iii., and that we change it into a _triliteral_ diagram by drawing an _inner square_, so as to divide each of its cells into portions, thus making cells altogether. the _right_-hand diagram shows the result. [the reader is strongly advised, in reading this chapter, _not_ to refer to the above diagrams, but to make a large copy of the right-hand one for himself, _without any letters_, and to have it by him while he reads, and keep his finger on that particular _part_ of it, about which he is reading.] pg secondly, let us suppose that we have selected a certain adjunct, which we may call "m", and have subdivided the xy-class into the two classes whose differentiæ are m and m', and that we have assigned the n.w. _inner_ cell to the one (which we may call "the class of xym-things", or "the xym-class"), and the n.w. _outer_ cell to the other (which we may call "the class of xym'-things", or "the xym'-class"). [thus, in the "books" example, we might say "let m mean 'bound', so that m' will mean 'unbound'", and we might suppose that we had subdivided the class "old english books" into the two classes, "old english bound books" and "old english unbound books", and had assigned the n.w. _inner_ cell to the one, and the n.w. _outer_ cell to the other.] thirdly, let us suppose that we have subdivided the xy'-class, the x'y-class, and the x'y'-class in the same manner, and have, in each case, assigned the _inner_ cell to the class possessing the attribute m, and the _outer_ cell to the class possessing the attribute m'. [thus, in the "books" example, we might suppose that we had subdivided the "new english books" into the two classes, "new english bound books" and "new english unbound books", and had assigned the s.w. _inner_ cell to the one, and the s.w. _outer_ cell to the other.] it is evident that we have now assigned the _inner square_ to the m-class, and the _outer border_ to the m'-class. [thus, in the "books" example, we have assigned the _inner square_ to "bound books" and the _outer border_ to "unbound books".] when the reader has made himself familiar with this diagram, he ought to be able to find, in a moment, the compartment assigned to a particular _pair_ of attributes, or the cell assigned to a particular _trio_ of attributes. the following rules will help him in doing this:-- ( ) arrange the attributes in the order x, y, m. pg ( ) take the _first_ of them and find the compartment assigned to it. ( ) then take the _second_, and find what _portion_ of that compartment is assigned to it. ( ) treat the _third_, if there is one, in the same way. [for example, suppose we have to find the compartment assigned to ym. we say to ourselves "y has the _west_ half; and m has the _inner_ portion of that west half." again, suppose we have to find the cell assigned to x'ym'. we say to ourselves "x' has the _south_ half; y has the _west_ portion of that south half, i.e. has the _south-west quarter_; and m' has the _outer_ portion of that south-west quarter."] the reader should now get his genial friend to question him on the table given on the next page, in the style of the following specimen-dialogue. q. adjunct for south half, inner portion? a. x'm. q. compartment for m'? a. the outer border. q. adjunct for north-east quarter, outer portion? a. xy'm'. q. compartment for ym? a. west half, inner portion. q. adjunct for south half? a. x'. q. compartment for x'y'm? a. south-east quarter, inner portion. &c. &c. pg table iv. ·-----------------------------------------------· | adjunct | | | of | compartments, or cells, assigned | | classes. | to them. | |----------|------------------------------------| | x | north half. | | x' | south " | | y | west " | | y' | east " | | m | inner square. | | m' | outer border. | |----------|------------------------------------| | xy | north-west quarter. | | xy' | " east " | | x'y | south-west " | | x'y' | " east " | | xm | north half, inner portion. | | xm' | " " outer " | | x'm | south " inner " | | x'm' | " " outer " | | ym | west " inner " | | ym' | " " outer " | | y'm | east " inner " | | y'm' | " " outer " | |----------|------------------------------------| | xym | north-west quarter, inner portion. | | xym' | " " " outer " | | xy'm | " east " inner " | | xy'm' | " " " outer " | | x'ym | south-west " inner " | | x'ym' | " " " outer " | | x'y'm | " east " inner " | | x'y'm' | " " " outer " | ·-----------------------------------------------· pg chapter ii. _representation of propositions in terms of x and m, or of y and m._ § . _representation of propositions of existence in terms of x and m, or of y and m._ let us take, first, the proposition "some xm exist". [note that the _full_ meaning of this proposition is (as explained at p. ) "some existing things are xm-things".] this tells us that there is at least _one_ thing in the inner portion of the north half; that is, that this compartment is _occupied_. and this we can evidently represent by placing a _red_ counter on the partition which divides it. ·---------------· | | | | ·---|---· | | | (.) | | |---|---|---|---| | | | | | | ·---|---· | | | | ·---------------· [in the "books" example, this proposition would mean "some old bound books exist" (or "there are some old bound books").] similarly we may represent the seven similar propositions, "some xm' exist", "some x'm exist", "some x'm' exist", "some ym exist", "some ym' exist", "some y'm exist", and "some y'm' exist". pg let us take, next, the proposition "no xm exist". this tells us that there is _nothing_ in the inner portion of the north half; that is, that this compartment is _empty_. and this we can represent by placing _two grey_ counters in it, one in each cell. ·---------------· | | | | ·---|---· | | |( )|( )| | |---|---|---|---| | | | | | | ·---|---· | | | | ·---------------· similarly we may represent the seven similar propositions, in terms of x and m, or of y and m, viz. "no xm' exist", "no x'm exist", &c. * * * * * these sixteen propositions of existence are the only ones that we shall have to represent on this diagram. § . _representation of propositions of relation in terms of x and m, or of y and m._ let us take, first, the pair of converse propositions "some x are m" = "some m are x." we know that each of these is equivalent to the proposition of existence "some xm exist", which we already know how to represent. ·---------------· | | | | ·---|---· | | | (.) | | |---|---|---|---| | | | | | | ·---|---· | | | | ·---------------· similarly for the seven similar pairs, in terms of x and m, or of y and m. let us take, next, the pair of converse propositions "no x are m" = "no m are x." we know that each of these is equivalent to the proposition of existence "no xm exist", which we already know how to represent. ·---------------· | | | | ·---|---· | | |( )|( )| | |---|---|---|---| | | | | | | ·---|---· | | | | ·---------------· similarly for the seven similar pairs, in terms of x and m, or of y and m. pg let us take, next, the proposition "all x are m." ·---------------· |( ) | ( )| | ·---|---· | | | (.) | | |---|---|---|---| | | | | | | ·---|---· | | | | ·---------------· we know (see p. ) that this is a _double_ proposition, and equivalent to the _two_ propositions "some x are m" and "no x are m' ", each of which we already know how to represent. similarly for the fifteen similar propositions, in terms of x and m, or of y and m. these thirty-two propositions of relation are the only ones that we shall have to represent on this diagram. the reader should now get his genial friend to question him on the following four tables. the victim should have nothing before him but a blank triliteral diagram, a red counter, and grey ones, with which he is to represent the various propositions named by the inquisitor, _e.g._ "no y' are m", "some xm' exist", &c., &c. pg table v. ·---------------------------------------------------------· | ·---------------· some xm exist | ·---------------· | | | | |  = some x are m | | | | | | | ·---|---· |  = some m are x | | ·---|---· | | | | | (.) | | | | |( )|( )| | | | |---|---|---|---| ·-----------------· |---|---|---|---| | | | | | | | | no xm exist | | | | | | | | ·---|---· | |  = no x are m | ·---|---· | | | | | | |  = no m are x | | | | | ·---------------· | ·---------------· | |---------------------------------------------------------| | ·---------------· some xm' exist | ·---------------· | | | (.) |  = some x are m' | |( ) | ( )| | | | ·---|---· |  = some m' are x | | ·---|---· | | | | | | | | | | | | | | | | |---|---|---|---| ·-----------------· |---|---|---|---| | | | | | | | | no xm' exist | | | | | | | | ·---|---· | |  = no x are m' | ·---|---· | | | | | | |  = no m' are x | | | | | ·---------------· | ·---------------· | |---------------------------------------------------------| | ·---------------· some x'm exist | ·---------------· | | | | |  = some x' are m | | | | | | | ·---|---· |  = some m are x' | | ·---|---· | | | | | | | | | | | | | | | | |---|---|---|---| ·-----------------· |---|---|---|---| | | | | (.) | | | no x'm exist | |( )|( )| | | | | ·---|---· | |  = no x' are m | ·---|---· | | | | | | |  = no m are x' | | | | | ·---------------· | ·---------------· | |---------------------------------------------------------| | ·---------------· some x'm' exist | ·---------------· | | | | |  = some x' are m'| | | | | | | ·---|---· |  = some m' are x'| | ·---|---· | | | | | | | | | | | | | | | | |---|---|---|---| ·-----------------· |---|---|---|---| | | | | | | | | no x'm' exist | | | | | | | | ·---|---· | |  = no x' are m' | ·---|---· | | | | (.) | |  = no m' are x' |( ) | ( )| | | ·---------------· | ·---------------· | ·---------------------------------------------------------· pg table vi. ·---------------------------------------------------------· | ·---------------· some ym exist | ·---------------· | | | | |  = some y are m | | | | | | | ·---|---· |  = some m are y | | ·---|---· | | | | | | | | | | |( )| | | | | |---|(.)|---|---| ·-----------------· |---|---|---|---| | | | | | | | | no ym exist | |( )| | | | | | ·---|---· | |  = no y are m | ·---|---· | | | | | | |  = no m are y | | | | | ·---------------· | ·---------------· | |---------------------------------------------------------| | ·---------------· some ym' exist | ·---------------· | | | | |  = some y are m' | |( ) | | | | | ·---|---· |  = some m' are y | | ·---|---· | | | | | | | | | | | | | | | | |(.)|---|---|---| ·-----------------· |---|---|---|---| | | | | | | | | no ym' exist | | | | | | | | ·---|---· | |  = no y are m' | ·---|---· | | | | | | |  = no m' are y |( ) | | | | ·---------------· | ·---------------· | |---------------------------------------------------------| | ·---------------· some y'm exist | ·---------------· | | | | |  = some y' are m | | | | | | | ·---|---· |  = some m are y' | | ·---|---· | | | | | | | | | | | |( )| | | | |---|---|(.)|---| ·-----------------· |---|---|---|---| | | | | | | | | no y'm exist | | |( )| | | | | ·---|---· | |  = no y' are m | ·---|---· | | | | | | |  = no m are y' | | | | | ·---------------· | ·---------------· | |---------------------------------------------------------| | ·---------------· some y'm' exist | ·---------------· | | | | |  = some y' are m'| | | ( )| | | | ·---|---· |  = some m' are y'| | ·---|---· | | | | | | | | | | | | | | | | |---|---|---|(.)| ·-----------------· |---|---|---|---| | | | | | | | | no y'm' exist | | | | | | | | ·---|---· | |  = no y' are m' | ·---|---· | | | | | | |  = no m' are y' | | ( )| | | ·---------------· | ·---------------· | ·---------------------------------------------------------· pg table vii. ·---------------------------------------------------------· | ·---------------· | ·---------------· | | |( ) | ( )| all x are m | | (.) | | | | ·---|---· | | | ·---|---· | | | | | (.) | | | | |( )|( )| | | | |---|---|---|---| ·-----------------· |---|---|---|---| | | | | | | | | | | | | | | | | ·---|---· | | all x are m' | ·---|---· | | | | | | | | | | | | ·---------------· | ·---------------· | |---------------------------------------------------------| | ·---------------· | ·---------------· | | | | | all x' are m | | | | | | | ·---|---· | | | ·---|---· | | | | | | | | | | | | | | | | |---|---|---|---| ·-----------------· |---|---|---|---| | | | | (.) | | | | |( )|( )| | | | | ·---|---· | | all x' are m' | ·---|---· | | | |( ) | ( )| | | (.) | | | ·---------------· | ·---------------· | |---------------------------------------------------------| | ·---------------· | ·---------------· | | | | | all m are x | | | | | | | ·---|---· | | | ·---|---· | | | | | (.) | | | | |( )|( )| | | | |---|---|---|---| ·-----------------· |---|---|---|---| | | | |( )|( )| | | | | (.) | | | | | ·---|---· | | all m are x' | ·---|---· | | | | | | | | | | | | ·---------------· | ·---------------· | |---------------------------------------------------------| | ·---------------· | ·---------------· | | | (.) | all m' are x | |( ) | ( )| | | | ·---|---· | | | ·---|---· | | | | | | | | | | | | | | | | |---|---|---|---| ·-----------------· |---|---|---|---| | | | | | | | | | | | | | | | | ·---|---· | | all m' are x' | ·---|---· | | | |( ) | ( )| | | (.) | | | ·---------------· | ·---------------· | ·---------------------------------------------------------· pg table viii. ·---------------------------------------------------------· | ·---------------· | ·---------------· | | |( ) | | all y are m | | | | | | | ·---|---· | | | ·---|---· | | | | | | | | | | |( )| | | | | |---|(.)|---|---| ·-----------------· |(.)|---|---|---| | | | | | | | | | |( )| | | | | | ·---|---· | | all y are m' | ·---|---· | | | |( ) | | | | | | | | ·---------------· | ·---------------· | |---------------------------------------------------------| | ·---------------· | ·---------------· | | | | ( )| all y' are m | | | | | | | ·---|---· | | | ·---|---· | | | | | | | | | | | |( )| | | | |---|---|(.)|---| ·-----------------· |---|---|---|(.)| | | | | | | | | | | |( )| | | | | ·---|---· | | all y' are m' | ·---|---· | | | | | ( )| | | | | | | ·---------------· | ·---------------· | |---------------------------------------------------------| | ·---------------· | ·---------------· | | | | | all m are y | | | | | | | ·---|---· | | | ·---|---· | | | | | |( )| | | | |( )| | | | | |---|(.)|---|---| ·-----------------· |---|---|(.)|---| | | | | |( )| | | | |( )| | | | | | ·---|---· | | all m are y' | ·---|---· | | | | | | | | | | | | ·---------------· | ·---------------· | |---------------------------------------------------------| | ·---------------· | ·---------------· | | | | ( )| all m' are y | |( ) | | | | | ·---|---· | | | ·---|---· | | | | | | | | | | | | | | | | |(.)|---|---|---| ·-----------------· |---|---|---|(.)| | | | | | | | | | | | | | | | | ·---|---· | | all m' are y' | ·---|---· | | | | | ( )| | |( ) | | | | ·---------------· | ·---------------· | ·---------------------------------------------------------· pg chapter iii. _representation of two propositions of relation, one in terms of x and m, and the other in terms of y and m, on the same diagram._ the reader had better now begin to draw little diagrams for himself, and to mark them with the digits "i" and "o", instead of using the board and counters: he may put a "i" to represent a _red_ counter (this may be interpreted to mean "there is at least _one_ thing here"), and a "o" to represent a _grey_ counter (this may be interpreted to mean "there is _nothing_ here"). the pair of propositions, that we shall have to represent, will always be, one in terms of x and m, and the other in terms of y and m. when we have to represent a proposition beginning with "all", we break it up into the _two_ propositions to which it is equivalent. when we have to represent, on the same diagram, propositions, of which some begin with "some" and others with "no", we represent the _negative_ ones _first_. this will sometimes save us from having to put a "i" "on a fence" and afterwards having to shift it into a cell. [let us work a few examples. ( ) "no x are m'; no y' are m". let us first represent "no x are m'". this gives us diagram a. then, representing "no y' are m" on the same diagram, we get diagram b. pg a b ·---------------· ·---------------· |(o) | (o)| |(o) | (o)| | ·---|---· | | ·---|---· | | | | | | | | |(o)| | |---|---|---|---| |---|---|---|---| | | | | | | | |(o)| | | ·---|---· | | ·---|---· | | | | | | | ·---------------· ·---------------· ( ) "some m are x; no m are y". if, neglecting the rule, we were begin with "some m are x", we should get diagram a. and if we were then to take "no m are y", which tells us that the inner n.w. cell is _empty_, we should be obliged to take the "i" off the fence (as it no longer has the choice of _two_ cells), and to put it into the inner n.e. cell, as in diagram c. this trouble may be saved by beginning with "no m are y", as in diagram b. and _now_, when we take "some m are x", there is no fence to sit on! the "i" has to go, at once, into the n.e. cell, as in diagram c. a b c ·---------------· ·---------------· ·---------------· | | | | | | | | | | ·---|---· | | ·---|---· | | ·---|---· | | | (i) | | | |(o)| | | | |(o)|(i)| | |---|---|---|---| |---|---|---|---| |---|---|---|---| | | | | | | |(o)| | | | |(o)| | | | ·---|---· | | ·---|---· | | ·---|---· | | | | | | | | | | ·---------------· ·---------------· ·---------------· ( ) "no x' are m'; all m are y". here we begin by breaking up the second into the two propositions to which it is equivalent. thus we have _three_ propositions to represent, viz.-- ( ) "no x' are m'; ( ) some m are y; ( ) no m are y'". these we will take in the order , , . first we take no. ( ), viz. "no x' are m'". this gives us diagram a. pg adding to this, no. ( ), viz. "no m are y'", we get diagram b. this time the "i", representing no. ( ), viz. "some m are y," has to sit on the fence, as there is no "o" to order it off! this gives us diagram c. a b c ·---------------· ·---------------· ·---------------· | | | | | | | | | | ·---|---· | | ·---|---· | | ·---|---· | | | | | | | | |(o)| | | | |(o)| | |---|---|---|---| |---|---|---|---| |---|(i)|---|---| | | | | | | | |(o)| | | | |(o)| | | ·---|---· | | ·---|---· | | ·---|---· | |(o) | (o)| |(o) | (o)| |(o) | (o)| ·---------------· ·---------------· ·---------------· ( ) "all m are x; all y are m". here we break up _both_ propositions, and thus get _four_ to represent, viz.-- ( ) "some m are x; ( ) no m are x'; ( ) some y are m; ( ) no y are m'". these we will take in the order , , , . first we take no. ( ), viz. "no m are x'". this gives us diagram a. to this we add no. ( ), viz. "no y are m'", and thus get diagram b. if we were to add to this no. ( ), viz. "some m are x", we should have to put the "i" on a fence: so let us try no. ( ) instead, viz. "some y are m". this gives us diagram c. and now there is no need to trouble about no. ( ), as it would not add anything to our information to put a "i" on the fence. the diagram _already_ tells us that "some m are x".] a b c ·---------------· ·---------------· ·---------------· | | | |(o) | | |(o) | | | ·---|---· | | ·---|---· | | ·---|---· | | | | | | | | | | | | |(i)| | | |---|---|---|---| |---|---|---|---| |---|---|---|---| | |(o)|(o)| | | |(o)|(o)| | | |(o)|(o)| | | ·---|---· | | ·---|---· | | ·---|---· | | | | |(o) | | |(o) | | ·---------------· ·---------------· ·---------------· [work examples § = =, - (p. ); § = =, - (p. ).] pg chapter iv. _interpretation, in terms of x and y, of triliteral diagram, when marked with counters or digits._ the problem before us is, given a marked triliteral diagram, to ascertain _what_ propositions of relation, in terms of x and y, are represented on it. the best plan, for a _beginner_, is to draw a _biliteral_ diagram alongside of it, and to transfer, from the one to the other, all the information he can. he can then read off, from the biliteral diagram, the required propositions. after a little practice, he will be able to dispense with the biliteral diagram, and to read off the result from the triliteral diagram itself. to _transfer_ the information, observe the following rules:-- ( ) examine the n.w. quarter of the triliteral diagram. ( ) if it contains a "i", in _either_ cell, it is certainly _occupied_, and you may mark the n.w. quarter of the biliteral diagram with a "i". ( ) if it contains _two_ "o"s, one in _each_ cell, it is certainly _empty_, and you may mark the n.w. quarter of the biliteral diagram with a "o". pg ( ) deal in the same way with the n.e., the s.w., and the s.e. quarter. [let us take, as examples, the results of the four examples worked in the previous chapters. ( ) ·---------------· |(o) | (o)| | ·---|---· | | | |(o)| | |---|---|---|---| | | |(o)| | | ·---|---· | | | | ·---------------· in the n.w. quarter, only _one_ of the two cells is marked as _empty_: so we do not know whether the n.w. quarter of the biliteral diagram is _occupied_ or _empty_: so we cannot mark it. ·-------· | |(o)| |---|---| | | | ·-------· in the n.e. quarter, we find _two_ "o"s: so _this_ quarter is certainly _empty_; and we mark it so on the biliteral diagram. in the s.w. quarter, we have no information _at all_. in the s.e. quarter, we have not enough to use. we may read off the result as "no x are y'", or "no y' are x," whichever we prefer. ( ) ·---------------· | | | | ·---|---· | | |(o)|(i)| | |---|---|---|---| | |(o)| | | | ·---|---· | | | | ·---------------· in the n.w. quarter, we have not enough information to use. in the n.e. quarter, we find a "i". this shows us that it is _occupied_: so we may mark the n.e. quarter on the biliteral diagram with a "i". ·-------· | |(i)| |---|---| | | | ·-------· in the s.w. quarter, we have not enough information to use. in the s.e. quarter, we have none at all. we may read off the result as "some x are y'", or "some y' are x", whichever we prefer. pg ( ) ·---------------· | | | | ·---|---· | | | |(o)| | |---|(i)|---|---| | | |(o)| | | ·---|---· | |(o) | (o)| ·---------------· in the n.w. quarter, we have _no_ information. (the "i", sitting on the fence, is of no use to us until we know on _which_ side he means to jump down!) in the n.e. quarter, we have not enough information to use. neither have we in the s.w. quarter. ·-------· | | | |---|---| | |(o)| ·-------· the s.e. quarter is the only one that yields enough information to use. it is certainly _empty_: so we mark it as such on the biliteral diagram. we may read off the results as "no x' are y'", or "no y' are x'", whichever we prefer. ( ) ·---------------· |(o) | | | ·---|---· | | |(i)| | | |---|---|---|---| | |(o)|(o)| | | ·---|---· | |(o) | | ·---------------· the n.w. quarter is _occupied_, in spite of the "o" in the outer cell. so we mark it with a "i" on the biliteral diagram. the n.e. quarter yields no information. ·-------· |(i)| | |---|---| | | | ·-------· the s.w. quarter is certainly _empty_. so we mark it as such on the biliteral diagram. ·-------· |(i)| | |---|---| |(o)| | ·-------· the s.e. quarter does not yield enough information to use. we read off the result as "all y are x."] [review tables v, vi (pp. , ). work examples § = =, - (p. ); § = =, - (p. ); § = =, - (p. ).] pg book v. syllogisms. chapter i. _introductory_ when a trio of biliteral propositions of relation is such that ( ) all their six terms are species of the same genus, ( ) every two of them contain between them a pair of codivisional classes, ( ) the three propositions are so related that, if the first two were true, the third would be true, the trio is called a '=syllogism='; the genus, of which each of the six terms is a species, is called its ='universe of discourse=', or, more briefly, its '=univ.='; the first two propositions are called its '=premisses=', and the third its '=conclusion='; also the pair of codivisional terms in the premisses are called its '=eliminands=', and the other two its '=retinends='. the conclusion of a syllogism is said to be '=consequent=' from its premisses: hence it is usual to prefix to it the word "therefore" (or the symbol ".'."). pg [note that the 'eliminands' are so called because they are _eliminated_, and do not appear in the conclusion; and that the 'retinends' are so called because they are _retained_, and _do_ appear in the conclusion. note also that the question, whether the conclusion is or is not _consequent_ from the premisses, is not affected by the _actual_ truth or falsity of any of the trio, but depends entirely on their _relationship to each other_. as a specimen-syllogism, let us take the trio "no x-things are m-things; no y-things are m'-things. no x-things are y-things." which we may write, as explained at p. , thus:-- "no x are m; no y are m'. no x are y". here the first and second contain the pair of codivisional classes m and m'; the first and third contain the pair x and x; and the second and third contain the pair y and y. also the three propositions are (as we shall see hereafter) so related that, if the first two were true, the third would also be true. hence the trio is a _syllogism_; the two propositions, "no x are m" and "no y are m'", are its _premisses_; the proposition "no x are y" is its _conclusion_; the terms m and m' are its _eliminands_; and the terms x and y are its _retinends_. hence we may write it thus:-- "no x are m; no y are m'. .'. no x are y". as a second specimen, let us take the trio "all cats understand french; some chickens are cats. some chickens understand french". these, put into normal form, are "all cats are creatures understanding french; some chickens are cats. some chickens are creatures understanding french". here all the six terms are species of the genus "creatures." also the first and second propositions contain the pair of codivisional classes "cats" and "cats"; the first and third contain the pair "creatures understanding french" and "creatures understanding french"; and the second and third contain the pair "chickens" and "chickens". pg also the three propositions are (as we shall see at p. ) so related that, if the first two were true, the third would be true. (the first two are, as it happens, _not_ strictly true in _our_ planet. but there is nothing to hinder them from being true in some _other_ planet, say _mars_ or _jupiter_--in which case the third would _also_ be true in that planet, and its inhabitants would probably engage chickens as nursery-governesses. they would thus secure a singular _contingent_ privilege, unknown in england, namely, that they would be able, at any time when provisions ran short, to utilise the nursery-governess for the nursery-dinner!) hence the trio is a _syllogism_; the genus "creatures" is its 'univ.'; the two propositions, "all cats understand french" and "some chickens are cats", are its _premisses_, the proposition "some chickens understand french" is its _conclusion_; the terms "cats" and "cats" are its _eliminands_; and the terms, "creatures understanding french" and "chickens", are its _retinends_. hence we may write it thus:-- "all cats understand french; some chickens are cats; .'. some chickens understand french".] pg chapter ii. _problems in syllogisms._ § . _introductory._ when the terms of a proposition are represented by _words_, it is said to be '=concrete='; when by _letters_, '=abstract=.' to translate a proposition from concrete into abstract form, we fix on a univ., and regard each term as a _species_ of it, and we choose a letter to represent its _differentia_. [for example, suppose we wish to translate "some soldiers are brave" into abstract form. we may take "men" as univ., and regard "soldiers" and "brave men" as _species_ of the _genus_ "men"; and we may choose x to represent the peculiar attribute (say "military") of "soldiers," and y to represent "brave." then the proposition may be written "some military men are brave men"; _i.e._ "some x-men are y-men"; _i.e._ (omitting "men," as explained at p. ) "some x are y." in practice, we should merely say "let univ. be "men", x = soldiers, y = brave", and at once translate "some soldiers are brave" into "some x are y."] the problems we shall have to solve are of two kinds, viz. ( ) "given a pair of propositions of relation, which contain between them a pair of codivisional classes, and which are proposed as premisses: to ascertain what conclusion, if any, is consequent from them." ( ) "given a trio of propositions of relation, of which every two contain a pair of codivisional classes, and which are proposed as a syllogism: to ascertain whether the proposed conclusion is consequent from the proposed premisses, and, if so, whether it is _complete_." these problems we will discuss separately. pg § . _given a pair of propositions of relation, which contain between them a pair of codivisional classes, and which are proposed as premisses: to ascertain what conclusion, if any, is consequent from them._ the rules, for doing this, are as follows:-- ( ) determine the 'universe of discourse'. ( ) construct a dictionary, making m and m (or m and m') represent the pair of codivisional classes, and x (or x') and y (or y') the other two. ( ) translate the proposed premisses into abstract form. ( ) represent them, together, on a triliteral diagram. ( ) ascertain what proposition, if any, in terms of x and y, is _also_ represented on it. ( ) translate this into concrete form. it is evident that, if the proposed premisses were true, this other proposition would _also_ be true. hence it is a _conclusion_ consequent from the proposed premisses. [let us work some examples. ( ) "no son of mine is dishonest; people always treat an honest man with respect". taking "men" as univ., we may write these as follows:-- "no sons of mine are dishonest men; all honest men are men treated with respect". we can now construct our dictionary, viz. m = honest; x = sons of mine; y = treated with respect. (note that the expression "x = sons of mine" is an abbreviated form of "x = the differentia of 'sons of mine', when regarded as a species of 'men'".) the next thing is to translate the proposed premisses into abstract form, as follows:-- "no x are m'; all m are y". pg next, by the process described at p. , we represent these on a triliteral diagram, thus:-- ·---------------· |(o) | (o)| | ·---|---· | | | |(o)| | |---|(i)|---|---| | | |(o)| | | ·---|---· | | | | ·---------------· next, by the process described at p. , we transfer to a biliteral diagram all the information we can. ·-------· | |(o)| |---|---| | | | ·-------· the result we read as "no x are y'" or as "no y' are x," whichever we prefer. so we refer to our dictionary, to see which will look best; and we choose "no x are y'", which, translated into concrete form, is "no son of mine fails to be treated with respect". ( ) "all cats understand french; some chickens are cats". taking "creatures" as univ., we write these as follows:-- "all cats are creatures understanding french; some chickens are cats". we can now construct our dictionary, viz. m = cats; x = understanding french; y = chickens. the proposed premisses, translated into abstract form, are "all m are x; some y are m". in order to represent these on a triliteral diagram, we break up the first into the two propositions to which it is equivalent, and thus get the _three_ propositions ( ) "some m are x; ( ) no m are x'; ( ) some y are m". the rule, given at p. , would make us take these in the order , , . this, however, would produce the result ·-----------------· | | | | ·----|----· | | |(i)(i) | | | |----|----| | | |(o) | (o)| | | ·----|----· | | | | ·-----------------· pg so it would be better to take them in the order , , . nos. ( ) and ( ) give us the result here shown; and now we need not trouble about no. ( ), as the proposition "some m are x" is _already_ represented on the diagram. ·---------------· | | | | ·---|---· | | |(i)| | | |---|---|---|---| | |(o)|(o)| | | ·---|---· | | | | ·---------------· transferring our information to a biliteral diagram, we get ·-------· |(i)| | |---|---| | | | ·-------· this result we can read either as "some x are y" or "some y are x". after consulting our dictionary, we choose "some y are x", which, translated into concrete form, is "some chickens understand french." ( ) "all diligent students are successful; all ignorant students are unsuccessful". let univ. be "students"; m = successful; x = diligent; y = ignorant. these premisses, in abstract form, are "all x are m; all y are m'". these, broken up, give us the four propositions ( ) "some x are m; ( ) no x are m'; ( ) some y are m'; ( ) no y are m". which we will take in the order , , , . representing these on a triliteral diagram, we get ·---------------· |(o) | (o)| | ·---|---· | | |(o)|(i)| | |---|---|---|---| | |(o)| | | | ·---|---· | |(i) | | ·---------------· and this information, transferred to a biliteral diagram, is ·-------· |(o)|(i)| |---|---| |(i)| | ·-------· here we get _two_ conclusions, viz. "all x are y'; all y are x'." pg and these, translated into concrete form, are "all diligent students are (not-ignorant, i.e.) learned; all ignorant students are (not-diligent, i.e.) idle". (see p. .) ( ) "of the prisoners who were put on their trial at the last assizes, all, against whom the verdict 'guilty' was returned, were sentenced to imprisonment; some, who were sentenced to imprisonment, were also sentenced to hard labour". let univ. be "the prisoners who were put on their trial at the last assizes"; m = who were sentenced to imprisonment; x = against whom the verdict 'guilty' was returned; y = who were sentenced to hard labour. the premisses, translated into abstract form, are "all x are m; some m are y". breaking up the first, we get the three ( ) "some x are m; ( ) no x are m'; ( ) some m are y". representing these, in the order , , , on a triliteral diagram, we get ·---------------· |(o) | (o)| | ·---|---· | | | (i) | | |---|(i)|---|---| | | | | | | ·---|---· | | | | ·---------------· here we get no conclusion at all. you would very likely have guessed, if you had seen _only_ the premisses, that the conclusion would be "some, against whom the verdict 'guilty' was returned, were sentenced to hard labour". but this conclusion is not even _true_, with regard to the assizes i have here invented. "not _true!_" you exclaim. "then who _were_ they, who were sentenced to imprisonment and were also sentenced to hard labour? they _must_ have had the verdict 'guilty' returned against them, or how could they be sentenced?" well, it happened like _this_, you see. they were three ruffians, who had committed highway-robbery. when they were put on their trial, they _pleaded_ 'guilty'. so no _verdict_ was returned at all; and they were sentenced at once.] i will now work out, in their briefest form, as models for the reader to imitate in working examples, the above four concrete problems. pg ( ) [see p. ] "no son of mine is dishonest; people always treat an honest man with respect." univ. "men"; m = honest; x = my sons; y = treated with respect. ·---------------· ·-------· "no x are m'; |(o) | (o)| | |(o)| all m are y." | ·---|---· | |---|---| | | |(o)| | | | | |---|(i)|---|---| ·-------· | | |(o)| | | ·---|---· | .'. "no x are y'." | | | ·---------------· i.e. "no son of mine ever fails to be treated with respect." ( ) [see p. ] "all cats understand french; some chickens are cats". univ. "creatures"; m = cats; x = understanding french; y = chickens. ·---------------· ·-------· "all m are x; | | | |(i)| | some y are m." | ·---|---· | |---|---| | |(i)| | | | | | |---|---|---|---| ·-------· | |(o)|(o)| | | ·---|---· | .'. "some y are x." | | | ·---------------· i.e. "some chickens understand french." ( ) [see p. ] "all diligent students are successful; all ignorant students are unsuccessful". univ. "students"; m = successful; x = diligent; y = ignorant. ·---------------· ·-------· "all x are m; |(o) | (o)| |(o)|(i)| all y are m'." | ·---|---· | |---|---| | |(o)|(i)| | |(i)| | |---|---|---|---| ·-------· | |(o)| | | | ·---|---· | .'. "all x are y'; |(i) | | all y are x'." ·---------------· i.e. "all diligent students are learned; and all ignorant students are idle". pg ( ) [see p. ] "of the prisoners who were put on their trial at the last assizes, all, against whom the verdict 'guilty' was returned, were sentenced to imprisonment; some, who were sentenced to imprisonment, were also sentenced to hard labour". univ. "prisoners who were put on their trial at the last assizes", m = sentenced to imprisonment; x = against whom the verdict 'guilty' was returned; y = sentenced to hard labour. ·---------------· "all x are m; |(o) | (o)| some m are y." | ·---|---· | | | (i) | | |---|(i)|---|---| | | | | | | ·---|---· | there is no | | | conclusion. ·---------------· [review tables vii, viii (pp. , ). work examples § = =, - (p. ); § = =, - (p. ); § = =, - (p. ).] pg § . _given a trio of propositions of relation, of which every two contain a pair of codivisional classes, and which are proposed as a syllogism; to ascertain whether the proposed conclusion is consequent from the proposed premisses, and, if so, whether it is complete._ the rules, for doing this, are as follows:-- ( ) take the proposed premisses, and ascertain, by the process described at p. , what conclusion, if any, is consequent from them. ( ) if there be _no_ conclusion, say so. ( ) if there _be_ a conclusion, compare it with the proposed conclusion, and pronounce accordingly. i will now work out, in their briefest form, as models for the reader to imitate in working examples, six problems. ( ) "all soldiers are strong; all soldiers are brave. some strong men are brave." univ. "men"; m = soldiers; x = strong; y = brave. pg ·---------------· ·-------· "all m are x; | | | |(i)| | all m are y. | ·---|---· | |---|---| some x are y." | |(i)|(o)| | | | | |---|---|---|---| ·-------· | |(o)|(o)| | | ·---|---· | .'. "some x are y." | | | ·---------------· hence proposed conclusion is right. ( ) "i admire these pictures; when i admire anything i wish to examine it thoroughly. i wish to examine some of these pictures thoroughly." univ. "things"; m = admired by me; x = these pictures; y = things which i wish to examine thoroughly. ·---------------· ·-------· "all x are m; |(o) | (o)| |(i)|(o)| all m are y. | ·---|---· | |---|---| some x are y." | |(i)|(o)| | | | | |---|---|---|---| ·-------· | | |(o)| | | ·---|---· | .'. "all x are y." | | | ·---------------· hence proposed conclusion is _incomplete_, the _complete_ one being "i wish to examine _all_ these pictures thoroughly". ( ) "none but the brave deserve the fair; some braggarts are cowards. some braggarts do not deserve the fair." univ. "persons"; m = brave; x = deserving of the fair; y = braggarts. ·---------------· ·-------· "no m' are x; |(o) | (o)| | | | some y are m'. | ·---|---· | |---|---| some y are x'." | | | | | |(i)| | |---|---|---|---| ·-------· | | | | | | ·---|---· | .'. "some y are x'." |(i) | | ·---------------· hence proposed conclusion is right. pg ( ) "all soldiers can march; some babies are not soldiers. some babies cannot march". univ. "persons"; m = soldiers; x = able to march; y = babies. ·---------------· "all m are x; | | | some y are m'. | ·---|---· | some y are x'." | | (i) | | |(i)|---|---|---| | |(o)|(o)| | | ·---|---· | there is no | | | conclusion. ·---------------· ( ) "all selfish men are unpopular; all obliging men are popular. all obliging men are unselfish". univ. "men"; m = popular; x = selfish; y = obliging. ·---------------· ·-------· "all x are m'; |(o) | (i)| |(o)|(i)| all y are m. | ·---|---· | |---|---| all y are x'." | |(o)|(o)| | |(i)| | |---|---|---|---| ·-------· | |(i)| | | | ·---|---· | .'. "all x are y'; |(o) | | all y are x'." ·---------------· hence proposed conclusion is _incomplete_, the _complete_ one containing, in addition, "all selfish men are disobliging". ( ) "no one, who means to go by the train and cannot get a conveyance, and has not enough time to walk to the station, can do without running; this party of tourists mean to go by the train and cannot get a conveyance, but they have plenty of time to walk to the station. this party of tourists need not run." univ. "persons meaning to go by the train, and unable to get a conveyance"; m = having enough time to walk to the station; x = needing to run; y = these tourists. pg ·---------------· "no m' are x'; |(o) | | all y are m. | ·---|---· | all y are x'." | | | | | |---|(i)|---|---| | | | | | | ·---|---· | there is no |(o) | (o)| conclusion. ·---------------· [here is _another_ opportunity, gentle reader, for playing a trick on your innocent friend. put the proposed syllogism before him, and ask him what he thinks of the conclusion. he will reply "why, it's perfectly correct, of course! and if your precious logic-book tells you it _isn't_, don't believe it! you don't mean to tell me those tourists _need_ to run? if _i_ were one of them, and knew the _premisses_ to be true, i should be _quite_ clear that i _needn't_ run--and i _should walk!_" and _you_ will reply "but suppose there was a mad bull behind you?" and then your innocent friend will say "hum! ha! i must think that over a bit!" you may then explain to him, as a convenient _test_ of the soundness of a syllogism, that, if circumstances can be invented which, without interfering with the truth of the _premisses_, would make the _conclusion_ false, the syllogism _must_ be unsound.] [review tables v-viii (pp. - ). work examples § = =, - (p. ); § = =, - (p. ); § = =, - (p. ); § = =, - (pp. , ).] pg book vi. the method of subscripts. chapter i. _introductory._ let us agree that "x_{ }" shall mean "some existing things have the attribute x", i.e. (more briefly) "some x exist"; also that "xy_{ }" shall mean "some xy exist", and so on. such a proposition may be called an '=entity=.' [note that, when there are _two_ letters in the expression, it does not in the least matter which stands _first_: "xy_{ }" and "yx_{ }" mean exactly the same.] also that "x_{ }" shall mean "no existing things have the attribute x", i.e. (more briefly) "no x exist"; also that "xy_{ }" shall mean "no xy exist", and so on. such a proposition may be called a '=nullity='. also that "+" shall mean "and". [thus "ab_{ } + cd_{ }" means "some ab exist and no cd exist".] also that "¶" shall mean "would, if true, prove". [thus, "x_{ } ¶ xy_{ }" means "the proposition 'no x exist' would, if true, prove the proposition 'no xy exist'".] when two letters are both of them accented, or both _not_ accented, they are said to have '=like signs=', or to be '=like=': when one is accented, and the other not, they are said to have '=unlike signs=', or to be '=unlike='. pg chapter ii. _representation of propositions of relation._ let us take, first, the proposition "some x are y". this, we know, is equivalent to the proposition of existence "some xy exist". (see p. .) hence it may be represented by the expression "xy_{ }". the converse proposition "some y are x" may of course be represented by the _same_ expression, viz. "xy_{ }". similarly we may represent the three similar pairs of converse propositions, viz.-- "some x are y'"  = "some y' are x", "some x' are y"  = "some y are x'", "some x' are y'" = "some y' are x'". let us take, next, the proposition "no x are y". this, we know, is equivalent to the proposition of existence "no xy exist". (see p. .) hence it may be represented by the expression "xy_{ }". the converse proposition "no y are x" may of course be represented by the _same_ expression, viz. "xy_{ }". similarly we may represent the three similar pairs of converse propositions, viz.-- "no x are y'"  = "no y' are x", "no x' are y"  = "no y are x'", "no x' are y'" = "no y' are x'". pg let us take, next, the proposition "all x are y". now it is evident that the double proposition of existence "some x exist and no xy' exist" tells us that _some_ x-things exist, but that _none_ of them have the attribute y': that is, it tells us that _all_ of them have the attribute y: that is, it tells us that "all x are y". also it is evident that the expression "x_{ } + xy'_{ }" represents this double proposition. hence it also represents the proposition "all x are y". [the reader will perhaps be puzzled by the statement that the proposition "all x are y" is equivalent to the double proposition "some x exist and no xy' exist," remembering that it was stated, at p. , to be equivalent to the double proposition "some x are y and no x are y'" (i.e. "some xy exist and no xy' exist"). the explanation is that the proposition "some xy exist" contains _superfluous information_. "some x exist" is enough for our purpose.] this expression may be written in a shorter form, viz. "x_{ }y'_{ }", since _each_ subscript takes effect back to the _beginning_ of the expression. similarly we may represent the seven similar propositions "all x are y'", "all x' are y", "all x' are y'", "all y are x", "all y are x'", "all y' are x", and "all y' are x'". [the reader should make out all these for himself.] it will be convenient to remember that, in translating a proposition, beginning with "all", from abstract form into subscript form, or _vice versâ_, the predicate _changes sign_ (that is, changes from positive to negative, or else from negative to positive). [thus, the proposition "all y are x'" becomes "y_{ }x_{ }", where the predicate changes from x' to x. again, the expression "x'_{ }y'_{ }" becomes "all x' are y", where the predicate changes for y' to y.] pg chapter iii. _syllogisms._ § . _representation of syllogisms._ we already know how to represent each of the three propositions of a syllogism in subscript form. when that is done, all we need, besides, is to write the three expressions in a row, with "+" between the premisses, and "¶" before the conclusion. [thus the syllogism "no x are m'; all m are y. .'. no x are y'." may be represented thus:-- xm'_{ } + m_{ }y'_{ } ¶ xy'_{ } when a proposition has to be translated from concrete form into subscript form, the reader will find it convenient, just at first, to translate it into _abstract_ form, and _thence_ into subscript form. but, after a little practice, he will find it quite easy to go straight from concrete form to subscript form.] pg § . _formulæ for solving problems in syllogisms._ when once we have found, by diagrams, the conclusion to a given pair of premisses, and have represented the syllogism in subscript form, we have a _formula_, by which we can at once find, without having to use diagrams again, the conclusion to any _other_ pair of premisses having the _same_ subscript forms. [thus, the expression xm_{ } + ym'_{ } ¶ xy_{ } is a formula, by which we can find the conclusion to any pair of premisses whose subscript forms are xm_{ } + ym'_{ } for example, suppose we had the pair of propositions "no gluttons are healthy; no unhealthy men are strong". proposed as premisses. taking "men" as our 'universe', and making m = healthy; x = gluttons; y = strong; we might translate the pair into abstract form, thus:-- "no x are m; no m' are y". these, in subscript form, would be xm_{ } + m'y_{ } which are identical with those in our _formula_. hence we at once know the conclusion to be xy_{ } that is, in abstract form, "no x are y"; that is, in concrete form, "no gluttons are strong".] i shall now take three different forms of pairs of premisses, and work out their conclusions, once for all, by diagrams; and thus obtain some useful formulæ. i shall call them "fig. i", "fig. ii", and "fig. iii". pg fig. i. this includes any pair of premisses which are both of them nullities, and which contain unlike eliminands. the simplest case is ·---------------· ·-------· xm_{ } + ym'_{ } |(o) | | |(o)| | | ·---|---· | |---|---| | |(o)|(o)| | | | | |---|---|---|---| ·-------· | | | | | | ·---|---· | .'. xy_{ } |(o) | | ·---------------· in this case we see that the conclusion is a nullity, and that the retinends have kept their signs. and we should find this rule to hold good with _any_ pair of premisses which fulfil the given conditions. [the reader had better satisfy himself of this, by working out, on diagrams, several varieties, such as m_{ }x_{ } + ym'_{ } (which ¶ xy_{ }) xm'_{ } + m_{ }y_{ } (which ¶ xy_{ }) x'm_{ } + ym'_{ } (which ¶ x'y_{ }) m'_{ }x'_{ } + m_{ }y'_{ } (which ¶ x'y'_{ }).] if either retinend is asserted in the _premisses_ to exist, of course it may be so asserted in the _conclusion_. hence we get two _variants_ of fig. i, viz. (a) where _one_ retinend is so asserted; (b) where _both_ are so asserted. [the reader had better work out, on diagrams, examples of these two variants, such as m_{ }x_{ } + y_{ }m'_{ } (which proves y_{ }x_{ }) x_{ }m'_{ } + m_{ }y_{ } (which proves x_{ }y_{ }) x'_{ }m_{ } + y_{ }m'_{ } (which proves x'_{ }y_{ } + y_{ }x'_{ }).] the formula, to be remembered, is xm_{ } + ym'_{ } ¶ xy_{ } with the following two rules:-- ( ) _two nullities, with unlike eliminands, yield a nullity, in which both retinends keep their signs._ pg ( ) _a retinend, asserted in the premisses to exist, may be so asserted in the conclusion._ [note that rule ( ) is merely the formula expressed in words.] fig. ii. this includes any pair of premisses, of which one is a nullity and the other an entity, and which contain like eliminands. the simplest case is xm_{ } + ym_{ } ·---------------· ·-------· | | | | | | | ·---|---· | |---|---| | |(o)|(o)| | |(i)| | |---|---|---|---| ·-------· | |(i)| | | | ·---|---· | .'. x'y_{ } | | | ·---------------· in this case we see that the conclusion is an entity, and that the nullity-retinend has changed its sign. and we should find this rule to hold good with _any_ pair of premisses which fulfil the given conditions. [the reader had better satisfy himself of this, by working out, on diagrams, several varieties, such as x'm_{ } + ym_{ } (which ¶ xy_{ }) x_{ }m'_{ } + y'm'_{ } (which ¶ x'y'_{ }) m_{ }x_{ } + y'm_{ } (which ¶ x'y'_{ }).] the formula, to be remembered, is, xm_{ } + ym_{ } ¶ x'y_{ } with the following rule:-- _a nullity and an entity, with like eliminands, yield an entity, in which the nullity-retinend changes its sign._ [note that this rule is merely the formula expressed in words.] pg fig. iii. this includes any pair of premisses which are both of them nullities, and which contain like eliminands asserted to exist. the simplest case is xm_{ } + ym_{ } + m_{ } [note that "m_{ }" is here stated _separately_, because it does not matter in which of the two premisses it occurs: so that this includes the _three_ forms "m_{ }x_{ } + ym_{ }", "xm_{ } + m_{ }y_{ }", and "m_{ }x_{ } + m_{ }y_{ }".] ·---------------· ·-------· | | | | | | | ·---|---· | |---|---| | |(o)|(o)| | | |(i)| |---|---|---|---| ·-------· | |(o)|(i)| | | ·---|---· | .'. x'y'_{ } | | | ·---------------· in this case we see that the conclusion is an entity, and that _both_ retinends have changed their signs. and we should find this rule to hold good with _any_ pair of premisses which fulfil the given conditions. [the reader had better satisfy himself of this, by working out, on diagrams, several varieties, such as x'm_{ } + m_{ }y_{ } (which ¶ xy'_{ }) m'_{ }x_{ } + m'y'_{ } (which ¶ x'y_{ }) m_{ }x'_{ } + m_{ }y'_{ } (which ¶ xy_{ }).] the formula, to be remembered, is xm_{ } + ym_{ } + m_{ } ¶ x'y'_{ } with the following rule (which is merely the formula expressed in words):-- _two nullities, with like eliminands asserted to exist, yield an entity, in which both retinends change their signs._ * * * * * in order to help the reader to remember the peculiarities and formulæ of these three figures, i will put them all together in one table. pg table ix. _______________________________________________________ | | | fig. i. | | | | xm_{ } + ym'_{ } ¶ xy_{ } | | | | two nullities, with unlike eliminands, yield a | | nullity, in which both retinends keep their signs. | | | | a retinend, asserted in the premisses to exist, may | | be so asserted in the conclusion. | |_______________________________________________________| | | | fig. ii. | | | | xm_{ } + ym_{ } ¶ x'y_{ } | | | | a nullity and an entity, with like eliminands, | | yield an entity, in which the nullity-retinend | | changes its sign. | |_______________________________________________________| | | | fig. iii. | | | | xm_{ } + ym_{ } + m_{ } ¶ x'y'_{ } | | | | two nullities, with like eliminands asserted | | to exist, yield an entity, in which both retinends | | change their signs. | |_______________________________________________________| i will now work out, by these formulæ, as models for the reader to imitate, some problems in syllogisms which have been already worked, by diagrams, in book v., chap. ii. ( ) [see p. ] "no son of mine is dishonest; people always treat an honest man with respect." univ. "men"; m = honest; x = my sons; y = treated with respect. xm'_{ } + m_{ }y'_{ } ¶ xy'_{ } [fig. i. _i.e._ "no son of mine ever fails to be treated with respect." pg ( ) [see p. ] "all cats understand french; some chickens are cats." univ. "creatures"; m = cats; x = understanding french; y = chickens. m_{ }x'_{ } + ym_{ } ¶ xy_{ } [fig. ii. _i.e._ "some chickens understand french." ( ) [see p. ] "all diligent students are successful; all ignorant students are unsuccessful." univ. "students"; m = successful; x = diligent; y = ignorant. x_{ }m'_{ } + y_{ }m_{ } ¶ x_{ }y_{ } + y_{ }x_{ } [fig. i (b). _i.e._ "all diligent students are learned; and all ignorant students are idle." ( ) [see p. ] "all soldiers are strong; all soldiers are brave. some strong men are brave." univ. "men"; m = soldiers; x = strong; y = brave. m_{ }x'_{ } + m_{ }y'_{ } ¶ xy_{ } [fig. iii. hence proposed conclusion is right. ( ) [see p. ] "i admire these pictures; when i admire anything, i wish to examine it thoroughly. i wish to examine some of these pictures thoroughly." univ. "things"; m = admired by me; x = these; y = things which i wish to examine thoroughly. x_{ }m'_{ } + m_{ }y'_{ } ¶ x_{ }y'_{ } [fig. i (a). hence proposed conclusion, xy_{ }, is _incomplete_, the _complete_ one being "i wish to examine _all_ these pictures thoroughly." pg ( ) [see p. ] "none but the brave deserve the fair; some braggarts are cowards. some braggarts do not deserve the fair." univ. "persons"; m = brave; x = deserving of the fair; y = braggarts. m'x_{ } + ym'_{ } ¶ x'y_{ } [fig. ii. hence proposed conclusion is right. ( ) [see p. ] "no one, who means to go by the train and cannot get a conveyance, and has not enough time to walk to the station, can do without running; this party of tourists mean to go by the train and cannot get a conveyance, but they have plenty of time to walk to the station. this party of tourists need not run." univ. "persons meaning to go by the train, and unable to get a conveyance"; m = having enough time to walk to the station; x = needing to run; y = these tourists. m'x'_{ } + y_{ }m'_{ } do not come under any of the three figures. hence it is necessary to return to the method of diagrams, as shown at p. . hence there is no conclusion. [work examples § = =, - (p. ); § = =, - (pp. , ); § = =, - (p. ); § = =, - (pp. , ). also read note (a), at p. .] pg § . _fallacies._ any argument which _deceives_ us, by seeming to prove what it does not really prove, may be called a '=fallacy=' (derived from the latin verb _fallo_ "i deceive"): but the particular kind, to be now discussed, consists of a pair of propositions, which are proposed as the premisses of a syllogism, but yield no conclusion. when each of the proposed premisses is a proposition in _i_, or _e_, or _a_, (the only kinds with which we are now concerned,) the fallacy may be detected by the 'method of diagrams,' by simply setting them out on a triliteral diagram, and observing that they yield no information which can be transferred to the biliteral diagram. but suppose we were working by the 'method of _subscripts_,' and had to deal with a pair of proposed premisses, which happened to be a 'fallacy,' how could we be certain that they would not yield any conclusion? our best plan is, i think, to deal with _fallacies_ in the same was as we have already dealt with _syllogisms_: that is, to take certain forms of pairs of propositions, and to work them out, once for all, on the triliteral diagram, and ascertain that they yield _no_ conclusion; and then to record them, for future use, as _formulæ for fallacies_, just as we have already recorded our three _formulæ for syllogisms_. pg now, if we were to record the two sets of formulæ in the _same_ shape, viz. by the method of subscripts, there would be considerable risk of confusing the two kinds. hence, in order to keep them distinct, i propose to record the formulæ for _fallacies_ in _words_, and to call them "forms" instead of "formulæ." let us now proceed to find, by the method of diagrams, three "forms of fallacies," which we will then put on record for future use. they are as follows:-- ( ) fallacy of like eliminands not asserted to exist. ( ) fallacy of unlike eliminands with an entity-premiss. ( ) fallacy of two entity-premisses. these shall be discussed separately, and it will be seen that each fails to yield a conclusion. ( ) _fallacy of like eliminands not asserted to exist._ it is evident that neither of the given propositions can be an _entity_, since that kind asserts the _existence_ of both of its terms (see p. ). hence they must both be _nullities_. hence the given pair may be represented by (xm_{ } + ym_{ }), with or without x_{ }, y_{ }. these, set out on triliteral diagrams, are xm_{ } + ym_{ } x_{ }m_{ } + ym_{ } ·---------------· ·---------------· | | | | (i) | | ·---|---· | | ·---|---· | | |(o)|(o)| | | |(o)|(o)| | |---|---|---|---| |---|---|---|---| | |(o)| | | | |(o)| | | | ·---|---· | | ·---|---· | | | | | | | ·---------------· ·---------------· xm_{ } + y_{ }m_{ } x_{ }m_{ } + y_{ }m_{ } ·---------------· ·---------------· | | | | (i) | | ·---|---· | | ·---|---· | | |(o)|(o)| | | |(o)|(o)| | |(i)|---|---|---| |(i)|---|---|---| | |(o)| | | | |(o)| | | | ·---|---· | | ·---|---· | | | | | | | ·---------------· ·---------------· pg ( ) _fallacy of unlike eliminands with an entity-premiss._ here the given pair may be represented by (xm_{ } + ym'_{ }) with or without x_{ } or m_{ }. these, set out on triliteral diagrams, are xm_{ } + ym'_{ } x_{ }m_{ } + ym'_{ } m_{ }x_{ } + ym'_{ } ·---------------· ·---------------· ·---------------· | | | | (i) | | | | | ·---|---· | | ·---|---· | | ·---|---· | | |(o)|(o)| | | |(o)|(o)| | | |(o)|(o)| | |(i)|---|---|---| |(i)|---|---|---| |(i)|---|---|---| | | | | | | | | | | | | (i) | | | ·---|---· | | ·---|---· | | ·---|---· | | | | | | | | | | ·---------------· ·---------------· ·---------------· ( ) _fallacy of two entity-premisses._ here the given pair may be represented by either (xm_{ } + ym_{ }) or (xm_{ } + ym'_{ }). these, set out on triliteral diagrams, are xm_{ } + ym_{ } xm_{ } + ym'_{ } ·---------------· ·---------------· | | | | | | | ·---|---· | | ·---|---· | | | (i) | | | | (i) | | |---|(i)|---|---| |(i)|---|---|---| | | | | | | | | | | | ·---|---· | | ·---|---· | | | | | | | ·---------------· ·---------------· pg § . _method of proceeding with a given pair of propositions._ let us suppose that we have before us a pair of propositions of relation, which contain between them a pair of codivisional classes, and that we wish to ascertain what conclusion, if any, is consequent from them. we translate them, if necessary, into subscript-form, and then proceed as follows:-- ( ) we examine their subscripts, in order to see whether they are (a) a pair of nullities; or (b) a nullity and an entity; or (c) a pair of entities. ( ) if they are a pair of nullities, we examine their eliminands, in order to see whether they are unlike or like. if their eliminands are _unlike_, it is a case of fig. i. we then examine their retinends, to see whether one or both of them are asserted to _exist_. if one retinend is so asserted, it is a case of fig. i (a); if both, it is a case of fig. i (b). if their eliminands are like, we examine them, in order to see whether either of them is asserted to exist. if so, it is a case of fig. iii.; if not, it is a case of "fallacy of like eliminands not asserted to exist." ( ) if they are a nullity and an entity, we examine their eliminands, in order to see whether they are like or unlike. if their eliminands are like, it is a case of fig. ii.; if _unlike_, it is a case of "fallacy of unlike eliminands with an entity-premiss." ( ) if they are a pair of entities, it is a case of "fallacy of two entity-premisses." [work examples § = =, - (p. ); § = =, - (p. ); § = =, - (p. ); § = =, - (p. ).] pg book vii. soriteses. chapter i. _introductory._ when a set of three or more biliteral propositions are such that all their terms are species of the same genus, and are also so related that two of them, taken together, yield a conclusion, which, taken with another of them, yields another conclusion, and so on, until all have been taken, it is evident that, if the original set were true, the last conclusion would _also_ be true. such a set, with the last conclusion tacked on, is called a '=sorites='; the original set of propositions is called its '=premisses='; each of the intermediate conclusions is called a '=partial conclusion=' of the sorites; the last conclusion is called its '=complete conclusion=,' or, more briefly, its '=conclusion='; the genus, of which all the terms are species, is called its '=universe of discourse=', or, more briefly, its '=univ.='; the terms, used as eliminands in the syllogisms, are called its '=eliminands='; and the two terms, which are retained, and therefore appear in the conclusion, are called its '=retinends='. [note that each _partial_ conclusion contains one or two _eliminands_; but that the _complete_ conclusion contains _retinends_ only.] the conclusion is said to be '=consequent=' from the premisses; for which reason it is usual to prefix to it the word "therefore" (or the symbol ".'."). [note that the question, whether the conclusion is or is not _consequent_ from the premisses, is not affected by the _actual_ truth or falsity of any one of the propositions which make up the sorites, by depends entirely on their _relationship to one another_. pg as a specimen-sorites, let us take the following set of propositions:-- ( ) "no a are b'; ( ) all b are c; ( ) all c are d; ( ) no e' are a'; ( ) all h are e'". here the first and second, taken together, yield "no a are c'". this, taken along with the third, yields "no a are d'". this, taken along with the fourth, yields "no d' are e'". and this, taken along with the fifth, yields "all h are d". hence, if the original set were true, this would _also_ be true. hence the original set, with this tacked on, is a _sorites_; the original set is its _premisses_; the proposition "all h are d" is its _conclusion_; the terms a, b, c, e are its _eliminands_; and the terms d and h are its _retinends_. hence we may write the whole sorites thus:-- "no a are b'; all b are c; all c are d; no e' are a'; all h are e'. .'. all h are d". in the above sorites, the partial conclusions are the positions "no a are e'", "no a are d'", "no d' are e'"; but, if the premisses were arranged in other ways, other partial conclusions might be obtained. thus, the order yields the partial conclusions "no c' are b'", "all h are b", "all h are c". there are altogether _nine_ partial conclusions to this sorites, which the reader will find it an interesting task to make out for himself.] pg chapter ii. _problems in soriteses._ § . _introductory._ the problems we shall have to solve are of the following form:-- "given three or more propositions of relation, which are proposed as premisses: to ascertain what conclusion, if any, is consequent from them." we will limit ourselves, at present, to problems which can be worked by the formulæ of fig. i. (see p. .) those, that require _other_ formulæ, are rather too hard for beginners. such problems may be solved by either of two methods, viz. ( ) the method of separate syllogisms; ( ) the method of underscoring. these shall be discussed separately. pg § . _solution by method of separate syllogisms._ the rules, for doing this, are as follows:-- ( ) name the 'universe of discourse'. ( ) construct a dictionary, making a, b, c, &c. represent the terms. ( ) put the proposed premisses into subscript form. ( ) select two which, containing between them a pair of codivisional classes, can be used as the premisses of a syllogism. ( ) find their conclusion by formula. ( ) find a third premiss which, along with this conclusion, can be used as the premisses of a second syllogism. ( ) find a second conclusion by formula. ( ) proceed thus, until all the proposed premisses have been used. ( ) put the last conclusion, which is the complete conclusion of the sorites, into concrete form. [as an example of this process, let us take, as the proposed set of premisses, ( ) "all the policemen on this beat sup with our cook; ( ) no man with long hair can fail to be a poet; ( ) amos judd has never been in prison; ( ) our cook's 'cousins' all love cold mutton; ( ) none but policemen on this beat are poets; ( ) none but her 'cousins' ever sup with our cook; ( ) men with short hair have all been in prison." univ. "men"; a = amos judd; b = cousins of our cook; c = having been in prison; d = long-haired; e = loving cold mutton; h = poets; k = policemen on this beat; l = supping with our cook pg we now have to put the proposed premisses into _subscript_ form. let us begin by putting them into _abstract_ form. the result is ( ) "all k are l; ( ) no d are h'; ( ) all a are c'; ( ) all b are e; ( ) no k' are h; ( ) no b' are l; ( ) all d' are c." and it is now easy to put them into _subscript_ form, as follows:-- ( ) k_{ }l'_{ } ( ) dh'_{ } ( ) a_{ }c_{ } ( ) b_{ }e'_{ } ( ) k'h_{ } ( ) b'l_{ } ( ) d'_{ }c'_{ } we now have to find a pair of premisses which will yield a conclusion. let us begin with no. ( ), and look down the list, till we come to one which we can take along with it, so as to form premisses belonging to fig. i. we find that no. ( ) will do, since we can take k as our eliminand. so our first syllogism is ( ) k_{ }l'_{ } ( ) k'h_{ } .'. l'h_{ } ... ( ) we must now begin again with l'h_{ } and find a premiss to go along with it. we find that no. ( ) will do, h being our eliminand. so our next syllogism is ( ) l'h_{ } ( ) dh'_{ } .'. l'd_{ } ... ( ) we have now used up nos. ( ), ( ), and ( ), and must search among the others for a partner for l'd_{ }. we find that no. ( ) will do. so we write ( ) l'd_{ } ( ) b'l_{ } .'. db'_{ } ... ( ) now what can we take along with db'_{ }? no. ( ) will do. ( ) db'_{ } ( ) b_{ }e'_{ } .'. de'_{ } ... ( ) pg along with this we may take no. ( ). ( ) de'_{ } ( ) d'_{ }c'_{ } .'. c'e'_{ } ... ( ) and along with this we may take no. ( ). ( ) c'e'_{ } ( ) a_{ }c_{ } .'. a_{ }e'_{ } this complete conclusion, translated into _abstract_ form, is "all a are e"; and this, translated into _concrete_ form, is "amos judd loves cold mutton." in actually _working_ this problem, the above explanations would, of course, be omitted, and all, that would appear on paper, would be as follows:-- ( ) k_{ }l'_{ } ( ) dh'_{ } ( ) a_{ }c_{ } ( ) b_{ }e'_{ } ( ) k'h_{ } ( ) b'l_{ } ( ) d'_{ }c'_{ } ( ) k_{ }l'_{ } ( ) k'h_{ } .'. l'h_{ } ... ( ) ( ) l'h_{ } ( ) dh'_{ } .'. l'd_{ } ... ( ) ( ) l'd_{ } ( ) b'l_{ } .'. db'_{ } ... ( ) ( ) db'_{ } ( ) b_{ }e'_{ } .'. de'_{ } ... ( ) ( ) de'_{ } ( ) d'_{ }c'_{ } .'. c'e'_{ } ... ( ) ( ) c'e'_{ } ( ) a_{ }c_{ } .'. a_{ }e'_{ } note that, in working a sorites by this process, we may begin with _any_ premiss we choose.] pg § . _solution by method of underscoring._ consider the pair of premisses xm_{ } + ym'_{ } which yield the conclusion xy_{ } we see that, in order to get this conclusion, we must eliminate m and m', and write x and y together in one expression. now, if we agree to _mark_ m and m' as eliminated, and to read the two expressions together, as if they were written in one, the two premisses will then exactly represent the _conclusion_, and we need not write it out separately. let us agree to mark the eliminated letters by _underscoring_ them, putting a _single_ score under the _first_, and a _double_ one under the _second_. the two premisses now become xm_{ } + ym'_{ } - = which we read as "xy_{ }". in copying out the premisses for underscoring, it will be convenient to _omit all subscripts_. as to the " 's" we may always _suppose_ them written, and, as to the " 's", we are not concerned to know _which_ terms are asserted to _exist_, except those which appear in the _complete_ conclusion; and for _them_ it will be easy enough to refer to the original list. pg [i will now go through the process of solving, by this method, the example worked in § . the data are k_{ }l'_{ } + dh'_{ } + a_{ }c_{ } + b_{ }e'_{ } + k'h_{ } + b'l_{ } + d'_{ }c'_{ } the reader should take a piece of paper, and write out this solution for himself. the first line will consist of the above data; the second must be composed, bit by bit, according to the following directions. we begin by writing down the first premiss, with its numeral over it, but omitting the subscripts. we have now to find a premiss which can be combined with this, _i.e._, a premiss containing either k' or l. the first we find is no. ; and this we tack on, with a +. to get the _conclusion_ from these, k and k' must be eliminated, and what remains must be taken as one expression. so we _underscore_ them, putting a _single_ score under k, and a _double_ one under k'. the result we read as l'h. we must now find a premiss containing either l or h'. looking along the row, we fix on no. , and tack it on. now these nullities are really equivalent to (l'h + dh'), in which h and h' must be eliminated, and what remains taken as one expression. so we _underscore_ them. the result reads as l'd. we now want a premiss containing l or d'. no. will do. these nullities are really equivalent to (l'd + b'l). so we underscore l' and l. the result reads as db'. we now want a premiss containing d' or b. no. will do. here we underscore b' and b. the result reads as de'. we now want a premiss containing d' or e. no. will do. here we underscore d and d'. the result reads as c'e'. we now want a premiss containing c or e. no. will do--in fact _must_ do, as it is the only one left. here we underscore c' and c; and, as the whole thing now reads as e'a, we tack on e'a_{ } as the _conclusion_, with a ¶. we now look along the row of data, to see whether e' or a has been given as _existent_. we find that a has been so given in no. . so we add this fact to the conclusion, which now stands as ¶ e'a_{ } + a_{ }, _i.e._ ¶ a_{ }e'_{ }; i.e. "all a are e." if the reader has faithfully obeyed the above directions, his written solution will now stand as follows:-- k_{ }l'_{ } + dh'_{ } + a_{ }c_{ } + b_{ }e'_{ } + k'h_{ } + b'l_{ } + d'_{ }c'_{ } kl' + k'h + dh' + b'l + be' + d'c' + ac -- = - -= - = = = - = ¶ e'a_{ } + a_{ } _i.e._ ¶ a_{ }e'_{ }; _i.e._ "all a are e." pg the reader should now take a second piece of paper, and copy the data only, and try to work out the solution for himself, beginning with some other premiss. if he fails to bring out the conclusion a_{ }e'_{ }, i would advise him to take a third piece of paper, and _begin again_!] i will now work out, in its briefest form, a sorites of premisses, to serve as a model for the reader to imitate in working examples. ( ) "i greatly value everything that john gives me; ( ) nothing but this bone will satisfy my dog; ( ) i take particular care of everything that i greatly value; ( ) this bone was a present from john; ( ) the things, of which i take particular care, are things i do _not_ give to my dog". univ. "things"; a = given by john to me; b = given by me to my dog; c = greatly valued by me; d = satisfactory to my dog; e = taken particular care of by me; h = this bone. a_{ }c'_{ } + h'd_{ } + c_{ }e'_{ } + h_{ }a'_{ } + e_{ }b_{ } ac' + ce' + ha' + h'd + eb ¶ db_{ } -- =- -= = = i.e. "nothing, that i give my dog, satisfies him," or, "my dog is not satisfied with _anything_ that i give him!" [note that, in working a sorites by this process, we may begin with _any_ premiss we choose. for instance, we might begin with no. , and the result would then be eb + ce' + ac' + ha' + h'd ¶ bd_{ }] - -= -= -= = [work examples § = =, - (p. ); § = =, - (p. ); § = =, - (p. ); § = =, - (p. ); § = =, - , , , , (pp. , ); § = =, - , , , , (pp. , , , ).] pg the reader, who has successfully grappled with all the examples hitherto set, and who thirsts, like alexander the great, for "more worlds to conquer," may employ his spare energies on the following examination-papers. he is recommended not to attempt more than _one_ paper on any one day. the answers to the questions about words and phrases may be found by referring to the index at p. . i. § = =, (p. ); § = =, - (p. ); § = =, , (p. ); § = =, (p. ); § = =, , (p. ); § = =, , , (pp. , , ). what is 'classification'? and what is a 'class'? ii. § = =, (p. ); § = =, - (pp. , ); § = =, (p. ); § = =, , (p. ); § = =, , (p. ); § = =, , , (pp. , , ). what are 'genus', 'species', and 'differentia'? iii. § = =, (p. ); § = =, - (p. ); § = =, , (p. ); § = =, (p. ); § = =, , (p. ); § = =, , , (pp. , , ). what are 'real' and 'imaginary' classes? iv. § = =, (p. ); § = =, - (p. ); § = =, (p. ); § = =, , (p. ); § = =, , (p. ); § = =, , , (pp. , , ). what is 'division'? when are classes said to be 'codivisional'? v. § = =, (p. ); § = =, - (p. ); § = =, , (p. ); § = =, (p. ); § = =, , (p. ); § = =, , , (pp. , , ). what is 'dichotomy'? what arbitrary rule does it sometimes require? pg vi. § = =, (p. ); § = =, - (p. ); § = =, (p. ); § = =, , (p. ); § = =, (p. ); § = =, , , (pp. , , ). what is a 'definition'? vii. § = =, (p. ); § = =, - (pp. , ); § = =, , (p. ); § = =, (p. ); § = =, (p. ); § = =, , , (pp. , , ). what are the 'subject' and the 'predicate' of a proposition? what is its 'normal' form? viii. § = =, (p. ); § = =, - (p. ); § = =, (p. ); § = =, , (p. ); § = =, (p. ); § = =, , , (pp. , , ). what is a proposition 'in _i_'? 'in _e_'? and 'in _a_'? ix. § = =, (p. ); § = =, - (p. ); § = =, , (p. ); § = =, (p. ); § = =, (p. ); § = =, , , (pp. , , ). what is the 'normal' form of a proposition of existence? x. § = =, (p. ); § = =, - (p. ); § = =, (p. ); § = =, , (p. ); § = =, (p. ); § = =, , , (pp. , , ). what is the 'universe of discourse'? xi. § = =, (p. ); § = =, - (p. ); § = =, , (p. ); § = =, (p. ); § = =, (p. ); § = =, , , (pp. , , ). what is implied, in a proposition of relation, as to the reality of its terms? xii. § = =, (p. ); § = =, - (p. ); § = =, (p. ); § = =, , (pp. , ); § = =, (p. ); § = =, , , (pp. , , ). explain the phrase "sitting on the fence". xiii. § = =, - (p. ); § = =, , (p. ); § = =, (p. ); § = =, (p. ); § = =, , , (pp. , , ). what are 'converse' propositions? xiv. § = =, - (p. ); § = =, (p. ); § = =, , (p. ); § = =, (p. ); § = =, , , (pp. , , ). what are 'concrete' and 'abstract' propositions? pg xv. § = =, - (p. ); § = =, , (p. ); § = =, (p. ); § = =, (p. ); § = =, , , (pp. , , ). what is a 'syllogism'? and what are its 'premisses' and its 'conclusion'? xvi. § = =, - (p. ); § = =, (p. ); § = =, , (p. ); § = =, (p. ); § = =, , , (pp. , , ). what is a 'sorites'? and what are its 'premisses', its 'partial conclusions', and its 'complete conclusion'? xvii. § = =, - (p. ); § = =, (p. ); § = =, (p. ); § = =, (p. ); § = =, , , , (pp. , , ). what are the 'universe of discourse', the 'eliminands', and the 'retinends', of a syllogism? and of a sorites? pg book viii. examples, answers, and solutions. [n.b. reference tags for examples, answers & solutions will be found in the right margin.] chapter i. _examples._ § . ex _propositions of relation, to be reduced to normal form._ . i have been out for a walk. . i am feeling better. . no one has read the letter but john. . neither you nor i are old. . no fat creatures run well. . none but the brave deserve the fair. . no one looks poetical unless he is pale. . some judges lose their tempers. . i never neglect important business. . what is difficult needs attention. . what is unwholesome should be avoided. . all the laws passed last week relate to excise. . logic puzzles me. . there are no jews in the house. . some dishes are unwholesome if not well-cooked. . unexciting books make one drowsy. . when a man knows what he's about, he can detect a sharper. . you and i know what we're about. . some bald people wear wigs. . those who are fully occupied never talk about their grievances. . no riddles interest me if they can be solved. pg § . ex _pairs of abstract propositions, one in terms of x and m, and the other in terms of y and m, to be represented on the same triliteral diagram._ . no x are m; no m' are y. . no x' are m'; all m' are y. . some x' are m; no m are y. . all m are x; all m' are y'. . all m' are x; all m' are y'. . all x' are m'; no y' are m. . all x are m; all y' are m'. . some m' are x'; no m are y. . all m are x'; no m are y. . no m are x'; no y are m'. . no x' are m'; no m are y. . some x are m; all y' are m. . all x' are m; no m are y. . some x are m'; all m are y. . no m' are x'; all y are m. . all x are m'; no y are m. . some m' are x; no m' are y'. . all x are m'; some m' are y'. . all m are x; some m are y'. . no x' are m; some y are m. . some x' are m'; all y' are m. . no m are x; some m are y. . no m' are x; all y are m'. . all m are x; no y' are m'. . some m are x; no y' are m. . all m' are x'; some y are m'. . some m are x'; no y' are m'. . no x are m'; all m are y'. . no x' are m; no m are y'. . no x are m; some y' are m'. . some m' are x; all y' are m; . all x are m'; all y are m. pg § . ex _marked triliteral diagrams, to be interpreted in terms of x and y._ ·---------------· |(o) | | | ·---|---· | | |(i)|(o)| | |---|---|---|---| | | |(o)| | | ·---|---· | |(o) | | ·---------------· ·---------------· |(o) | | | ·---|---· | | | |(o)| | |---|---|---|(i)| | | |(o)| | | ·---|---· | |(o) | | ·---------------· ·---------------· | | (o)| | ·---|---· | | |(o)|(o)| | |---|---|---|---| | | |(i)| | | ·---|---· | | | (o)| ·---------------· ·---------------· |(o) | (o)| | ·---|---· | | |(o)| | | |---|---|(i)|---| | |(o)| | | | ·---|---· | | | | ·---------------· ·---------------· | | (i)| | ·---|---· | | | |(o)| | |---|---|---|---| | | |(o)| | | ·---|---· | |(o) | (o)| ·---------------· ·---------------· |(o) | (o)| | ·---|---· | | | | | | |---|---|---|---| | | |(o)| | | ·---|---· | |(i) | (o)| ·---------------· ·---------------· |(o) | (o)| | ·---|---· | | |(i)|(o)| | |---|---|---|---| | | |(o)| | | ·---|---· | | | | ·---------------· ·---------------· |(i) | | | ·---|---· | | |(o)| | | |---|---|---|---| | |(o)|(i)| | | ·---|---· | |(o) | | ·---------------· ·---------------· | | | | ·---|---· | | |(o)| | | |---|---|---|---| | |(o)|(i)| | | ·---|---· | |(o) | (o)| ·---------------· ·---------------· |(o) | (o)| | ·---|---· | | |(o)|(i)| | |---|---|---|---| | |(o)| | | | ·---|---· | | | | ·---------------· ·---------------· |(o) | (o)| | ·---|---· | | | (i) | | |---|---|---|---| | |(o)| | | | ·---|---· | | | (o)| ·---------------· ·---------------· |(o) | | | ·---|---· | | | | | | |---|(i)|---|---| | | | | | | ·---|---· | |(o) | (i)| ·---------------· ·---------------· |(o) | (i)| | ·---|---· | | | |(o)| | |---|---|---|---| | | |(o)| | | ·---|---· | |(o) | | ·---------------· ·---------------· |(o) | (o)| | ·---|---· | | | |(o)| | |---|---|---|---| | | |(o)| | | ·---|---· | | (i) | ·---------------· ·---------------· | | | | ·---|---· | | |(i)| | | |---|---|---|---| | |(o)|(o)| | | ·---|---· | | | | ·---------------· ·---------------· |(i) | | | ·---|---· | | | | | | |---|---|---|---| | |(o)| | | | ·---|---· | |(o) | (o)| ·---------------· ·---------------· | | (o)| | ·---|---· | | | |(i)| | |---|---|---|---| | |(o)|(o)| | | ·---|---· | |(i) | (o)| ·---------------· ·---------------· |(o) | (o)| | ·---|---· | | |(o)|(i)| | |---|---|---|---| | |(o)| | | | ·---|---· | |(i) | | ·---------------· ·---------------· |(o) | (o)| | ·---|---· | | |(i)|(o)| | |---|---|---|---| | | |(o)| | | ·---|---· | | | (i)| ·---------------· ·---------------· |(o) | | | ·---|---· | | |(o)|(o)| | |---|---|---|---| | |(i)| | | | ·---|---· | |(o) | | ·---------------· pg § . ex _pairs of abstract propositions, proposed as premisses: conclusions to be found._ . no m are x'; all m' are y. . no m' are x; some m' are y'. . all m' are x; all m' are y'. . no x' are m'; all y' are m. . some m are x'; no y are m. . no x' are m; no m are y. . no m are x'; some y' are m. . all m' are x'; no m' are y. . some x' are m'; no m are y'. . all x are m; all y' are m'. . no m are x; all y' are m'. . no x are m; all y are m. . all m' are x; no y are m. . all m are x; all m' are y. . no x are m; no m' are y. . all x are m'; all y are m. . no x are m; all m' are y. . no x are m'; no m are y. . all m are x; all m are y'. . no m are x; all m' are y. . all x are m; some m' are y. . some x are m; all y are m. . all m are x; some y are m. . no x are m; all y are m. . some m are x'; all y' are m'. . no m are x'; all y are m. . all x are m'; all y' are m. . all m are x'; some m are y. . no m are x; all y are m'. . all x are m'; some y are m. . all x are m; all y are m. . no x are m'; all m are y. . no m are x; no m are y. . no m are x'; some y are m. . no m are x; all y are m. . all m are x'; some y are m. . all m are x; no y are m. . no m are x; no m' are y. . some m are x'; no m are y. . no x' are m; all y' are m. . all x are m'; no y are m'. . no m' are x; no y are m. pg § . ex _pairs of concrete propositions, proposed as premisses: conclusions to be found._ . i have been out for a walk; i am feeling better. . no one has read the letter but john; no one, who has _not_ read it, knows what it is about. . those who are not old like walking; you and i are young. . your course is always honest; your course is always the best policy. . no fat creatures run well; some greyhounds run well. . some, who deserve the fair, get their deserts; none but the brave deserve the fair. . some jews are rich; all esquimaux are gentiles. . sugar-plums are sweet; some sweet things are liked by children. . john is in the house; everybody in the house is ill. . umbrellas are useful on a journey; what is useless on a journey should be left behind. . audible music causes vibration in the air; inaudible music is not worth paying for. . some holidays are rainy; rainy days are tiresome. . no frenchmen like plumpudding; all englishmen like plumpudding. . no portrait of a lady, that makes her simper or scowl, is satisfactory; no photograph of a lady ever fails to make her simper or scowl. . all pale people are phlegmatic; no one looks poetical unless he is pale. . no old misers are cheerful; some old misers are thin. . no one, who exercises self-control, fails to keep his temper; some judges lose their tempers. pg . all pigs are fat; nothing that is fed on barley-water is fat. . all rabbits, that are not greedy, are black; no old rabbits are free from greediness. . some pictures are not first attempts; no first attempts are really good. . i never neglect important business; your business is unimportant. . some lessons are difficult; what is difficult needs attention. . all clever people are popular; all obliging people are popular. . thoughtless people do mischief; no thoughtful person forgets a promise. . pigs cannot fly; pigs are greedy. . all soldiers march well; some babies are not soldiers. . no bride-cakes are wholesome; what is unwholesome should be avoided. . john is industrious; no industrious people are unhappy. . no philosophers are conceited; some conceited persons are not gamblers. . some excise laws are unjust; all the laws passed last week relate to excise. . no military men write poetry; none of my lodgers are civilians. . no medicine is nice; senna is a medicine. . some circulars are not read with pleasure; no begging-letters are read with pleasure. . all britons are brave; no sailors are cowards. . nothing intelligible ever puzzles _me_; logic puzzles me. . some pigs are wild; all pigs are fat. pg . all wasps are unfriendly; all unfriendly creatures are unwelcome. . no old rabbits are greedy; all black rabbits are greedy. . some eggs are hard-boiled; no eggs are uncrackable. . no antelope is ungraceful; graceful creatures delight the eye. . all well-fed canaries sing loud; no canary is melancholy if it sings loud. . some poetry is original; no original work is producible at will. . no country, that has been explored, is infested by dragons; unexplored countries are fascinating. . no coals are white; no niggers are white. . no bridges are made of sugar; some bridges are picturesque. . no children are patient; no impatient person can sit still. . no quadrupeds can whistle; some cats are quadrupeds. . bores are terrible; you are a bore. . some oysters are silent; no silent creatures are amusing. . there are no jews in the house; no gentiles have beards a yard long. . canaries, that do not sing loud, are unhappy; no well-fed canaries fail to sing loud. . all my sisters have colds; no one can sing who has a cold. . all that is made of gold is precious; some caskets are precious. . some buns are rich; all buns are nice. . all my cousins are unjust; all judges are just. pg . pain is wearisome; no pain is eagerly wished for. . all medicine is nasty; senna is a medicine. . some unkind remarks are annoying; no critical remarks are kind. . no tall men have woolly hair; niggers have woolly hair. . all philosophers are logical; an illogical man is always obstinate. . john is industrious; all industrious people are happy. . these dishes are all well-cooked; some dishes are unwholesome if not well-cooked. . no exciting books suit feverish patients; unexciting books make one drowsy. . no pigs can fly; all pigs are greedy. . when a man knows what he's about, he can detect a sharper; you and i know what we're about. . some dreams are terrible; no lambs are terrible. . no bald creature needs a hairbrush; no lizards have hair. . all battles are noisy; what makes no noise may escape notice. . all my cousins are unjust; no judges are unjust. . all eggs can be cracked; some eggs are hard-boiled. . prejudiced persons are untrustworthy; some unprejudiced persons are disliked. . no dictatorial person is popular; she is dictatorial. . some bald people wear wigs; all your children have hair. . no lobsters are unreasonable; no reasonable creatures expect impossibilities. pg . no nightmare is pleasant; unpleasant experiences are not eagerly desired. . no plumcakes are wholesome; some wholesome things are nice. . nothing that is nice need be shunned; some kinds of jam are nice. . all ducks waddle; nothing that waddles is graceful. . sandwiches are satisfying; nothing in this dish is unsatisfying. . no rich man begs in the street; those who are not rich should keep accounts. . spiders spin webs; some creatures, that do not spin webs, are savage. . some of these shops are not crowded; no crowded shops are comfortable. . prudent travelers carry plenty of small change; imprudent travelers lose their luggage. . some geraniums are red; all these flowers are red. . none of my cousins are just; all judges are just. . no jews are mad; all my lodgers are jews. . busy folk are not always talking about their grievances; discontented folk are always talking about their grievances. . none of my cousins are just; no judges are unjust. . all teetotalers like sugar; no nightingale drinks wine. . no riddles interest me if they can be solved; all these riddles are insoluble. . all clear explanations are satisfactory; some excuses are unsatisfactory. . all elderly ladies are talkative; all good-tempered ladies are talkative. . no kind deed is unlawful; what is lawful may be done without scruple. pg . no babies are studious; no babies are good violinists. . all shillings are round; all these coins are round. . no honest men cheat; no dishonest men are trustworthy. . none of my boys are clever; none of my girls are greedy. . all jokes are meant to amuse; no act of parliament is a joke. . no eventful tour is ever forgotten; uneventful tours are not worth writing a book about. . all my boys are disobedient; all my girls are discontented. . no unexpected pleasure annoys me; your visit is an unexpected pleasure. § . ex _trios of abstract propositions, proposed as syllogisms: to be examined._ . some x are m; no m are y'. some x are y. . all x are m; no y are m'. no y are x'. . some x are m'; all y' are m. some x are y. . all x are m; no y are m. all x are y'. . some m' are x'; no m' are y. some x' are y'. . no x' are m; all y are m'. all y are x'. . some m' are x'; all y' are m'. some x' are y'. . no m' are x'; all y' are m'. all y' are x. . some m are x'; no m are y. some x' are y'. . all m' are x'; all m' are y. some y are x'. . all x are m'; some y are m. some y are x'. . no x are m; no m' are y'. no x are y'. . no x are m; all y' are m. all y' are x'. . all m' are x'; all m' are y. some y are x'. . some m are x'; all y are m'. some x' are y'. . no x' are m; all y' are m'. some y' are x. . no m' are x; all m' are y'. some x' are y'. pg . no x' are m; some m are y. some x are y. . some m are x; all m are y. some y are x'. . no x' are m'; some m' are y'. some x are y'. . no m are x; all m are y'. some x' are y'. . all x' are m; some y are m'. all x' are y'. . all m are x; no m' are y'. no x' are y'. . all x are m'; all m' are y. all x are y. . no x are m'; all m are y. no x are y'. . all m are x'; all y are m. all y are x'. . all x are m; no m are y'. all x are y. . all x are m; no y' are m'. all x are y. . no x' are m; no m' are y'. no x' are y'. . all x are m; all m are y'. all x are y'. . all x' are m'; no y' are m'. all x' are y. . no x are m; no y' are m'. no x are y'. . all m are x'; all y' are m. all y' are x'. . all x are m'; some y are m'. some y are x. . some x are m; all m are y. some x are y. . all m are x'; all y are m. all y are x'. . no m are x'; all m are y'. some x are y'. . no x are m; no m are y'. no x are y'. . no m are x; some m are y'. some x' are y'. . no m are x'; some y are m. some x are y. § . ex _trios of concrete propositions, proposed as syllogisms: to be examined._ . no doctors are enthusiastic; you are enthusiastic. you are not a doctor. . dictionaries are useful; useful books are valuable. dictionaries are valuable. . no misers are unselfish; none but misers save egg-shells. no unselfish people save egg-shells. . some epicures are ungenerous; all my uncles are generous. my uncles are not epicures. pg . gold is heavy; nothing but gold will silence him. nothing light will silence him. . some healthy people are fat; no unhealthy people are strong. some fat people are not strong. . "i saw it in a newspaper." "all newspapers tell lies." it was a lie. . some cravats are not artistic; i admire anything artistic. there are some cravats that i do not admire. . his songs never last an hour; a song, that lasts an hour, is tedious. his songs are never tedious. . some candles give very little light; candles are _meant_ to give light. some things, that are meant to give light, give very little. . all, who are anxious to learn, work hard; some of these boys work hard. some of these boys are anxious to learn. . all lions are fierce; some lions do not drink coffee. some creatures that drink coffee are not fierce. . no misers are generous; some old men are ungenerous. some old men are misers. . no fossil can be crossed in love; an oyster may be crossed in love. oysters are not fossils. . all uneducated people are shallow; students are all educated. no students are shallow. . all young lambs jump; no young animals are healthy, unless they jump. all young lambs are healthy. . ill-managed business is unprofitable; railways are never ill-managed. all railways are profitable. . no professors are ignorant; all ignorant people are vain. no professors are vain. pg . a prudent man shuns hyænas; no banker is imprudent. no banker fails to shun hyænas. . all wasps are unfriendly; no puppies are unfriendly. puppies are not wasps. . no jews are honest; some gentiles are rich. some rich people are dishonest. . no idlers win fame; some painters are not idle. some painters win fame. . no monkeys are soldiers; all monkeys are mischievous. some mischievous creatures are not soldiers. . all these bonbons are chocolate-creams; all these bonbons are delicious. chocolate-creams are delicious. . no muffins are wholesome; all buns are unwholesome. buns are not muffins. . some unauthorised reports are false; all authorised reports are trustworthy. some false reports are not trustworthy. . some pillows are soft; no pokers are soft. some pokers are not pillows. . improbable stories are not easily believed; none of his stories are probable. none of his stories are easily believed. . no thieves are honest; some dishonest people are found out. some thieves are found out. . no muffins are wholesome; all puffy food is unwholesome. all muffins are puffy. . no birds, except peacocks, are proud of their tails; some birds, that are proud of their tails, cannot sing. some peacocks cannot sing. . warmth relieves pain; nothing, that does not relieve pain, is useful in toothache. warmth is useful in toothache. pg . no bankrupts are rich; some merchants are not bankrupts. some merchants are rich. . bores are dreaded; no bore is ever begged to prolong his visit. no one, who is dreaded, is ever begged to prolong his visit. . all wise men walk on their feet; all unwise men walk on their hands. no man walks on both. . no wheelbarrows are comfortable; no uncomfortable vehicles are popular. no wheelbarrows are popular. . no frogs are poetical; some ducks are unpoetical. some ducks are not frogs. . no emperors are dentists; all dentists are dreaded by children. no emperors are dreaded by children. . sugar is sweet; salt is not sweet. salt is not sugar. . every eagle can fly; some pigs cannot fly. some pigs are not eagles. § . ex _sets of abstract propositions, proposed as premisses for soriteses: conclusions to be found._ [n.b. at the end of this section instructions are given for varying these examples.] . . no c are d; . all a are d; . all b are c. . . all d are b; . no a are c'; . no b are c. . . no b are a; . no c are d'; . all d are b. . . no b are c; . all a are b; . no c' are d. . . all b' are a'; . no b are c; . no a' are d. . . all a are b'; . no b' are c; . all d are a. . . no d are b'; . all b are a; . no c are d'. . . no b' are d; . no a' are b; . all c are d. pg . . all b' are a; . no a are d; . all b are c. . . no c are d; . all b are c; . no a are d'. . . no b are c; . all d are a; . all c' are a'. . . no c are b'; . all c' are d'; . all b are a. . . all d are e; . all c are a; . no b are d'; . all e are a'. . . all e are b; . all a are e; . all d are b'; . all a' are c; . . no b' are d; . all e are c; . all b are a; . all d' are c'. . . no a' are e; . all d are c'; . all a are b; . all e' are d. . . all d are c; . all a are e; . no b are d'; . all c are e'. . . all a are b; . all d are e; . all a' are c'; . no b are e. . . no b are c; . all e are h; . all a are b; . no d are h; . all e' are c. . . no d are h'; . no c are e; . all h are b; . no a are d'; . no b are e'. . . all b are a; . no d are h; . no c are e; . no a are h'; . all c' are b. . . all e are d'; . no b' are h'; . all c' are d; . all a are e; . no c are h. . . all b' are a'; . no d are e'; . all h are b'; . no c are e; . all d' are a. . . all h' are k'; . no b' are a; . all c are d; . all e are h'; . no d are k'; . no b are c'. . . all a are d; . all k are b; . all e are h; . no a' are b; . all d are c; . all h are k. . . all a' are h; . no d' are k'; . all e are b'; . no h are k; . all a are e; . no b' are d. . . all c are d'; . no h are b; . all a' are k; . no c are e'; . all b' are d; . no a are c'. . . no a' are k; . all e are b; . no h are k'; . no d' are c; . no a are b; . all c' are h. . . no e are k; . no b' are m; . no a are c'; . all h' are e; . all d are k; . no c are b; . all d' are l; . no h are m'. . . all n are m; . all a' are e; . no c' are l; . all k are r'; . no a are h'; . no d are l'; . no c are n'; . all e are b; . all m are r; . all h are d. [n.b. in each example, in sections and , it is possible to begin with _any_ premiss, at pleasure, and thus to get as many different solutions (all of course yielding the _same_ complete conclusion) as there are premisses in the example. hence § really contains different examples, and § contains .] pg § . ex _sets of concrete propositions, proposed as premisses for soriteses: conclusions to be found._ = .= ( ) babies are illogical; ( ) nobody is despised who can manage a crocodile; ( ) illogical persons are despised. univ. "persons"; a = able to manage a crocodile; b = babies; c = despised; d = logical. = .= ( ) my saucepans are the only things i have that are made of tin; ( ) i find all _your_ presents very useful; ( ) none of my saucepans are of the slightest use. univ. "things of mine"; a = made of tin; b = my saucepans; c = useful; d = your presents. = .= ( ) no potatoes of mine, that are new, have been boiled; ( ) all my potatoes in this dish are fit to eat; ( ) no unboiled potatoes of mine are fit to eat. univ. "my potatoes"; a = boiled; b = eatable; c = in this dish; d = new. = .= ( ) there are no jews in the kitchen; ( ) no gentiles say "shpoonj"; ( ) my servants are all in the kitchen. univ. "persons"; a = in the kitchen; b = jews; c = my servants; d = saying "shpoonj." = .= ( ) no ducks waltz; ( ) no officers ever decline to waltz; ( ) all my poultry are ducks. univ. "creatures"; a = ducks; b = my poultry; c = officers; d = willing to waltz. = .= ( ) every one who is sane can do logic; ( ) no lunatics are fit to serve on a jury; ( ) none of _your_ sons can do logic. univ. "persons"; a = able to do logic; b = fit to serve on a jury; c = sane; d = your sons. pg = .= ( ) there are no pencils of mine in this box; ( ) no sugar-plums of mine are cigars; ( ) the whole of my property, that is not in this box, consists of cigars. univ. "things of mine"; a = cigars; b = in this box; c = pencils; d = sugar-plums. = .= ( ) no experienced person is incompetent; ( ) jenkins is always blundering; ( ) no competent person is always blundering. univ. "persons"; a = always blundering; b = competent; c = experienced; d = jenkins. = .= ( ) no terriers wander among the signs of the zodiac; ( ) nothing, that does not wander among the signs of the zodiac, is a comet; ( ) nothing but a terrier has a curly tail. univ. "things"; a = comets; b = curly-tailed; c = terriers; d = wandering among the signs of the zodiac. = .= ( ) no one takes in the _times_, unless he is well-educated; ( ) no hedge-hogs can read; ( ) those who cannot read are not well-educated. univ. "creatures"; a = able to read; b = hedge-hogs; c = taking in the times; d = well-educated. = .= ( ) all puddings are nice; ( ) this dish is a pudding; ( ) no nice things are wholesome. univ. "things"; a = nice; b = puddings; c = this dish; d = wholesome. = .= ( ) my gardener is well worth listening to on military subjects; ( ) no one can remember the battle of waterloo, unless he is very old; ( ) nobody is really worth listening to on military subjects, unless he can remember the battle of waterloo. univ. "persons"; a = able to remember the battle of waterloo; b = my gardener; c = well worth listening to on military subjects; d = very old. pg = .= ( ) all humming-birds are richly coloured; ( ) no large birds live on honey; ( ) birds that do not live on honey are dull in colour. univ. "birds"; a = humming-birds; b = large; c = living on honey; d = richly coloured. = .= ( ) no gentiles have hooked noses; ( ) a man who is a good hand at a bargain always makes money; ( ) no jew is ever a bad hand at a bargain. univ. "persons"; a = good hands at a bargain; b = hook-nosed; c = jews; d = making money. = .= ( ) all ducks in this village, that are branded 'b,' belong to mrs. bond; ( ) ducks in this village never wear lace collars, unless they are branded 'b'; ( ) mrs. bond has no gray ducks in this village. univ. "ducks in this village"; a = belonging to mrs. bond; b = branded 'b'; c = gray; d = wearing lace-collars. = .= ( ) all the old articles in this cupboard are cracked; ( ) no jug in this cupboard is new; ( ) nothing in this cupboard, that is cracked, will hold water. univ. "things in this cupboard"; a = able to hold water; b = cracked; c = jugs; d = old. = .= ( ) all unripe fruit is unwholesome; ( ) all these apples are wholesome; ( ) no fruit, grown in the shade, is ripe. univ. "fruit"; a = grown in the shade; b = ripe; c = these apples; d = wholesome. = .= ( ) puppies, that will not lie still, are always grateful for the loan of a skipping-rope; ( ) a lame puppy would not say "thank you" if you offered to lend it a skipping-rope. ( ) none but lame puppies ever care to do worsted-work. univ. "puppies"; a = caring to do worsted-work; b = grateful for the loan of a skipping-rope; c = lame; d = willing to lie still. pg = .= ( ) no name in this list is unsuitable for the hero of a romance; ( ) names beginning with a vowel are always melodious; ( ) no name is suitable for the hero of a romance, if it begins with a consonant. univ. "names"; a = beginning with a vowel; b = in this list; c = melodious; d = suitable for the hero of a romance. = .= ( ) all members of the house of commons have perfect self-command; ( ) no m.p., who wears a coronet, should ride in a donkey-race; ( ) all members of the house of lords wear coronets. univ. "m.p.'s"; a = belonging to the house of commons; b = having perfect self-command; c = one who may ride in a donkey-race; d = wearing a coronet. = .= ( ) no goods in this shop, that have been bought and paid for, are still on sale; ( ) none of the goods may be carried away, unless labeled "sold"; ( ) none of the goods are labeled "sold," unless they have been bought and paid for. univ. "goods in this shop"; a = allowed to be carried away; b = bought and paid for; c = labeled "sold"; d = on sale. = .= ( ) no acrobatic feats, that are not announced in the bills of a circus, are ever attempted there; ( ) no acrobatic feat is possible, if it involves turning a quadruple somersault; ( ) no impossible acrobatic feat is ever announced in a circus bill. univ. "acrobatic feats"; a = announced in the bills of a circus; b = attempted in a circus; c = involving the turning of a quadruple somersault; d = possible. = .= ( ) nobody, who really appreciates beethoven, fails to keep silence while the moonlight-sonata is being played; ( ) guinea-pigs are hopelessly ignorant of music; ( ) no one, who is hopelessly ignorant of music, ever keeps silence while the moonlight-sonata is being played. univ. "creatures"; a = guinea-pigs; b = hopelessly ignorant of music; c = keeping silence while the moonlight-sonata is being played; d = really appreciating beethoven. pg = .= ( ) coloured flowers are always scented; ( ) i dislike flowers that are not grown in the open air; ( ) no flowers grown in the open air are colourless. univ. "flowers"; a = coloured; b = grown in the open air; c = liked by me; d = scented. = .= ( ) showy talkers think too much of themselves; ( ) no really well-informed people are bad company; ( ) people who think too much of themselves are not good company. univ. "persons"; a = good company; b = really well-informed; c = showy talkers; d = thinking too much of one's self. = .= ( ) no boys under are admitted to this school as boarders; ( ) all the industrious boys have red hair; ( ) none of the day-boys learn greek; ( ) none but those under are idle. univ. "boys in this school"; a = boarders; b = industrious; c = learning greek; d = red-haired; e = under . = .= ( ) the only articles of food, that my doctor allows me, are such as are not very rich; ( ) nothing that agrees with me is unsuitable for supper; ( ) wedding-cake is always very rich; ( ) my doctor allows me all articles of food that are suitable for supper. univ. "articles of food"; a = agreeing with me; b = allowed by my doctor; c = suitable for supper; d = very rich; e = wedding-cake. = .= ( ) no discussions in our debating-club are likely to rouse the british lion, so long as they are checked when they become too noisy; ( ) discussions, unwisely conducted, endanger the peacefulness of our debating-club; ( ) discussions, that go on while tomkins is in the chair, are likely to rouse the british lion; ( ) discussions in our debating-club, when wisely conducted, are always checked when they become too noisy. univ. "discussions in our debating-club"; a = checked when too noisy; b = dangerous to the peacefulness of our debating-club; c = going on while tomkins is in the chair; d = likely to rouse the british lion; e = wisely conducted. pg = .= ( ) all my sons are slim; ( ) no child of mine is healthy who takes no exercise; ( ) all gluttons, who are children of mine, are fat; ( ) no daughter of mine takes any exercise. univ. "my children"; a = fat; b = gluttons; c = healthy; d = sons; e = taking exercise. = .= ( ) things sold in the street are of no great value; ( ) nothing but rubbish can be had for a song; ( ) eggs of the great auk are very valuable; ( ) it is only what is sold in the street that is really _rubbish_. univ. "things"; a = able to be had for a song; b = eggs of the great auk; c = rubbish; d = sold in the street; e = very valuable. = .= ( ) no books sold here have gilt edges, except what are in the front shop; ( ) all the _authorised_ editions have red labels; ( ) all the books with red labels are priced at s. and upwards; ( ) none but _authorised_ editions are ever placed in the front shop. univ. "books sold here"; a = authorised editions; b = gilt-edged; c = having red labels; d = in the front shop; e = priced at s. and upwards. = .= ( ) remedies for bleeding, which fail to check it, are a mockery; ( ) tincture of calendula is not to be despised; ( ) remedies, which will check the bleeding when you cut your finger, are useful; ( ) all mock remedies for bleeding are despicable. univ. "remedies for bleeding"; a = able to check bleeding; b = despicable; c = mockeries; d = tincture of calendula; e = useful when you cut your finger. = .= ( ) none of the unnoticed things, met with at sea, are mermaids; ( ) things entered in the log, as met with at sea, are sure to be worth remembering; ( ) i have never met with anything worth remembering, when on a voyage; ( ) things met with at sea, that are noticed, are sure to be recorded in the log; univ. "things met with at sea"; a = entered in log; b = mermaids; c = met with by me; d = noticed; e = worth remembering. pg = .= ( ) the only books in this library, that i do _not_ recommend for reading, are unhealthy in tone; ( ) the bound books are all well-written; ( ) all the romances are healthy in tone; ( ) i do not recommend you to read any of the unbound books. univ. "books in this library"; a = bound; b = healthy in tone; c = recommended by me; d = romances; e = well-written. = .= ( ) no birds, except ostriches, are feet high; ( ) there are no birds in this aviary that belong to any one but _me_; ( ) no ostrich lives on mince-pies; ( ) i have no birds less than feet high. univ. "birds"; a = in this aviary; b = living on mince-pies; c = my; d =  feet high; e = ostriches. = .= ( ) a plum-pudding, that is not really solid, is mere porridge; ( ) every plum-pudding, served at my table, has been boiled in a cloth; ( ) a plum-pudding that is mere porridge is indistinguishable from soup; ( ) no plum-puddings are really solid, except what are served at _my_ table. univ. "plum-puddings"; a = boiled in a cloth; b = distinguishable from soup; c = mere porridge; d = really solid; e = served at my table. = .= ( ) no interesting poems are unpopular among people of real taste; ( ) no modern poetry is free from affectation; ( ) all _your_ poems are on the subject of soap-bubbles; ( ) no affected poetry is popular among people of real taste; ( ) no ancient poem is on the subject of soap-bubbles. univ. "poems"; a = affected; b = ancient; c = interesting; d = on the subject of soap-bubbles; e = popular among people of real taste; h = written by you. = .= ( ) all the fruit at this show, that fails to get a prize, is the property of the committee; ( ) none of my peaches have got prizes; ( ) none of the fruit, sold off in the evening, is unripe; ( ) none of the ripe fruit has been grown in a hot-house; ( ) all fruit, that belongs to the committee, is sold off in the evening. univ. "fruit at this show"; a = belonging to the committee; b = getting prizes; c = grown in a hot-house; d = my peaches; e = ripe; h = sold off in the evening. pg = .= ( ) promise-breakers are untrustworthy; ( ) wine-drinkers are very communicative; ( ) a man who keeps his promises is honest; ( ) no teetotalers are pawnbrokers; ( ) one can always trust a very communicative person. univ. "persons"; a = honest; b = pawnbrokers; c = promise-breakers; d = trustworthy; e = very communicative; h = wine-drinkers. = .= ( ) no kitten, that loves fish, is unteachable; ( ) no kitten without a tail will play with a gorilla; ( ) kittens with whiskers always love fish; ( ) no teachable kitten has green eyes; ( ) no kittens have tails unless they have whiskers. univ. "kittens"; a = green-eyed; b = loving fish; c = tailed; d = teachable; e = whiskered; h = willing to play with a gorilla. = .= ( ) all the eton men in this college play cricket; ( ) none but the scholars dine at the higher table; ( ) none of the cricketers row; ( ) _my_ friends in this college all come from eton; ( ) all the scholars are rowing-men. univ. "men in this college"; a = cricketers; b = dining at the higher table; c = etonians; d = my friends; e = rowing-men; h = scholars. = .= ( ) there is no box of mine here that i dare open; ( ) my writing-desk is made of rose-wood; ( ) all my boxes are painted, except what are here; ( ) there is no box of mine that i dare not open, unless it is full of live scorpions; ( ) all my rose-wood boxes are unpainted. univ. "my boxes"; a = boxes that i dare open; b = full of live scorpions; c = here; d = made of rose-wood; e = painted; h = writing-desks. = .= ( ) gentiles have no objection to pork; ( ) nobody who admires pigsties ever reads hogg's poems; ( ) no mandarin knows hebrew; ( ) every one, who does not object to pork, admires pigsties; ( ) no jew is ignorant of hebrew. univ. "persons"; a = admiring pigsties; b = jews; c = knowing hebrew; d = mandarins; e = objecting to pork; h = reading hogg's poems. pg = .= ( ) all writers, who understand human nature, are clever; ( ) no one is a true poet unless he can stir the hearts of men; ( ) shakespeare wrote "hamlet"; ( ) no writer, who does not understand human nature, can stir the hearts of men; ( ) none but a true poet could have written "hamlet."; univ. "writers"; a = able to stir the hearts of men; b = clever; c = shakespeare; d = true poets; e = understanding human nature; h = writer of 'hamlet.' = .= ( ) i despise anything that cannot be used as a bridge; ( ) everything, that is worth writing an ode to, would be a welcome gift to me; ( ) a rainbow will not bear the weight of a wheel-barrow; ( ) whatever can be used as a bridge will bear the weight of a wheel-barrow; ( ) i would not take, as a gift, a thing that i despise. univ. "things"; a = able to bear the weight of a wheel-barrow; b = acceptable to me; c = despised by me; d = rainbows; e = useful as a bridge; h = worth writing an ode to. = .= ( ) when i work a logic-example without grumbling, you may be sure it is one that i can understand; ( ) these soriteses are not arranged in regular order, like the examples i am used to; ( ) no easy example ever make my head ache; ( ) i ca'n't understand examples that are not arranged in regular order, like those i am used to; ( ) i never grumble at an example, unless it gives me a headache. univ. "logic-examples worked by me"; a = arranged in regular order, like the examples i am used to; b = easy; c = grumbled at by me; d = making my head ache; e = these soriteses; h = understood by me. = .= ( ) every idea of mine, that cannot be expressed as a syllogism, is really ridiculous; ( ) none of my ideas about bath-buns are worth writing down; ( ) no idea of mine, that fails to come true, can be expressed as a syllogism; ( ) i never have any really ridiculous idea, that i do not at once refer to my solicitor; ( ) my dreams are all about bath-buns; ( ) i never refer any idea of mine to my solicitor, unless it is worth writing down. univ. "my ideas"; a = able to be expressed as a syllogism; b = about bath-buns; c = coming true; d = dreams; e = really ridiculous h = referred to my solicitor; k = worth writing down. pg = .= ( ) none of the pictures here, except the battle-pieces, are valuable; ( ) none of the unframed ones are varnished; ( ) all the battle-pieces are painted in oils; ( ) all those that have been sold are valuable; ( ) all the english ones are varnished; ( ) all those in frames have been sold. univ. "the pictures here"; a = battle-pieces; b = english; c = framed; d = oil-paintings; e = sold; h = valuable; k = varnished. = .= ( ) animals, that do not kick, are always unexcitable; ( ) donkeys have no horns; ( ) a buffalo can always toss one over a gate; ( ) no animals that kick are easy to swallow; ( ) no hornless animal can toss one over a gate; ( ) all animals are excitable, except buffaloes. univ. "animals"; a = able to toss one over a gate; b = buffaloes; c = donkeys; d = easy to swallow; e = excitable; h = horned; k = kicking. = .= ( ) no one, who is going to a party, ever fails to brush his hair; ( ) no one looks fascinating, if he is untidy; ( ) opium-eaters have no self-command; ( ) every one, who has brushed his hair, looks fascinating; ( ) no one wears white kid gloves, unless he is going to a party; ( ) a man is always untidy, if he has no self-command. univ. "persons"; a = going to a party; b = having brushed one's hair; c = having self-command; d = looking fascinating; e = opium-eaters; h = tidy; k = wearing white kid gloves. = .= ( ) no husband, who is always giving his wife new dresses, can be a cross-grained man; ( ) a methodical husband always comes home for his tea; ( ) no one, who hangs up his hat on the gas-jet, can be a man that is kept in proper order by his wife; ( ) a good husband is always giving his wife new dresses; ( ) no husband can fail to be cross-grained, if his wife does not keep him in proper order; ( ) an unmethodical husband always hangs up his hat on the gas-jet. univ. "husbands"; a = always coming home for his tea; b = always giving his wife new dresses; c = cross-grained; d = good; e = hanging up his hat on the gas-jet; h = kept in proper order; k = methodical. pg = .= ( ) everything, not absolutely ugly, may be kept in a drawing-room; ( ) nothing, that is encrusted with salt, is ever quite dry; ( ) nothing should be kept in a drawing-room, unless it is free from damp; ( ) bathing-machines are always kept near the sea; ( ) nothing, that is made of mother-of-pearl, can be absolutely ugly; ( ) whatever is kept near the sea gets encrusted with salt. univ. "things"; a = absolutely ugly; b = bathing-machines; c = encrusted with salt; d = kept near the sea; e = made of mother-of-pearl; h = quite dry; k = things that may be kept in a drawing-room. = .= ( ) i call no day "unlucky," when robinson is civil to me; ( ) wednesdays are always cloudy; ( ) when people take umbrellas, the day never turns out fine; ( ) the only days when robinson is uncivil to me are wednesdays; ( ) everybody takes his umbrella with him when it is raining; ( ) my "lucky" days always turn out fine. univ. "days"; a = called by me 'lucky'; b = cloudy; c = days when people take umbrellas; d = days when robinson is civil to me; e = rainy; h = turning out fine; k = wednesdays. = .= ( ) no shark ever doubts that it is well fitted out; ( ) a fish, that cannot dance a minuet, is contemptible; ( ) no fish is quite certain that it is well fitted out, unless it has three rows of teeth; ( ) all fishes, except sharks, are kind to children; ( ) no heavy fish can dance a minuet; ( ) a fish with three rows of teeth is not to be despised. univ. "fishes"; a = able to dance a minuet; b = certain that he is well fitted out; c = contemptible; d = having rows of teeth; e = heavy; h = kind to children; k = sharks. = .= ( ) all the human race, except my footmen, have a certain amount of common-sense; ( ) no one, who lives on barley-sugar, can be anything but a mere baby; ( ) none but a hop-scotch player knows what real happiness is; ( ) no mere baby has a grain of common sense; ( ) no engine-driver ever plays hop-scotch; ( ) no footman of mine is ignorant of what true happiness is. univ. "human beings"; a = engine-drivers; b = having common sense; c = hop-scotch players; d = knowing what real happiness is; e = living on barley-sugar; h = mere babies; k = my footmen. pg = .= ( ) i trust every animal that belongs to me; ( ) dogs gnaw bones; ( ) i admit no animals into my study, unless they will beg when told to do so; ( ) all the animals in the yard are mine; ( ) i admit every animal, that i trust, into my study; ( ) the only animals, that are really willing to beg when told to do so, are dogs. univ. "animals"; a = admitted to my study; b = animals that i trust; c = dogs; d = gnawing bones; e = in the yard; h = my; k = willing to beg when told. = .= ( ) animals are always mortally offended if i fail to notice them; ( ) the only animals that belong to _me_ are in that field; ( ) no animal can guess a conundrum, unless it has been properly trained in a board-school; ( ) none of the animals in that field are badgers; ( ) when an animal is mortally offended, it always rushes about wildly and howls; ( ) i never notice any animal, unless it belongs to me; ( ) no animal, that has been properly trained in a board-school, ever rushes about wildly and howls. univ. "animals"; a = able to guess a conundrum; b = badgers; c = in that field; d = mortally offended; e = my; h = noticed by me; k = properly trained in a board-school; l = rushing about wildly and howling. = .= ( ) i never put a cheque, received by me, on that file, unless i am anxious about it; ( ) all the cheques received by me, that are not marked with a cross, are payable to bearer; ( ) none of them are ever brought back to me, unless they have been dishonoured at the bank; ( ) all of them, that are marked with a cross, are for amounts of over £ ; ( ) all of them, that are not on that file, are marked "not negotiable"; ( ) no cheque of yours, received by me, has ever been dishonoured; ( ) i am never anxious about a cheque, received by me, unless it should happen to be brought back to me; ( ) none of the cheques received by me, that are marked "not negotiable," are for amounts of over £ . univ. "cheques received by me"; a = brought back to me; b = cheques that i am anxious about; c = honoured; d = marked with a cross; e = marked 'not negotiable'; h = on that file; k = over £ ; l = payable to bearer; m = your. pg = .= ( ) all the dated letters in this room are written on blue paper; ( ) none of them are in black ink, except those that are written in the third person; ( ) i have not filed any of them that i can read; ( ) none of them, that are written on one sheet, are undated; ( ) all of them, that are not crossed, are in black ink; ( ) all of them, written by brown, begin with "dear sir"; ( ) all of them, written on blue paper, are filed; ( ) none of them, written on more than one sheet, are crossed; ( ) none of them, that begin with "dear sir," are written in the third person. univ. "letters in this room"; a = beginning with "dear sir"; b = crossed; c = dated; d = filed; e = in black ink; h = in third person; k = letters that i can read; l = on blue paper; m = on one sheet; n = written by brown. = .= ( ) the only animals in this house are cats; ( ) every animal is suitable for a pet, that loves to gaze at the moon; ( ) when i detest an animal, i avoid it; ( ) no animals are carnivorous, unless they prowl at night; ( ) no cats fails to kill mice; ( ) no animals ever take to me, except what are in this house; ( ) kangaroos are not suitable for pets; ( ) none but carnivora kill mice; ( ) i detest animals that do not take to me; ( ) animals, that prowl at night, always love to gaze at the moon. univ. "animals"; a = avoided by me; b = carnivora; c = cats; d = detested by me; e = in this house; h = kangaroos; k = killing mice; l = loving to gaze at the moon; m = prowling at night; n = suitable for pets; r = taking to me. pg chapter ii. _answers._ _answers to § ._ an . "all" _sign of quantity._ "persons represented by the name 'i'" (or "i's") _subject._ "are" _copula._ "persons who have been out for a walk" _predicate._ or, more briefly, "all | 'i's | are | persons who have been out for a walk". . "all | 'i's | are | persons who feel better". . "no | persons who are not 'john' | are | persons who have read the letter". . "no | members of the class 'you and i' | are | old persons". . "no | fat creatures | are | creatures that run well". . "no | not-brave persons | are | persons deserving of the fair". . "no | not-pale persons | are | persons who look poetical". . "some | judges | are | persons who lose their tempers". . "all | 'i's | are | persons who do not neglect important business". . "all | difficult things | are | things that need attention". . "all | unwholesome things | are | things that should be avoided". . "all | laws passed last week | are | laws relating to excise". . "all | logical studies | are | things that puzzle me". . "no | persons in the house | are | jews". . "some | not well-cooked dishes | are | unwholesome dishes". . "all | unexciting books | are | books that make one drowsy". . "all | men who know what they're about | are | men who can detect a sharper". . "all | members of the class 'you and i' | are | persons who know what they're about". . "some | bald persons | are | persons accustomed to wear wigs". . "all | fully occupied persons | are | persons who do not talk about their grievances". . "no | riddles that can be solved | are | riddles that interest me". pg _answers to § ._ an ·---------------· |(o) | | | ·---|---· | | |(o)|(o)| | |---|---|---|---| | | | | | | ·---|---· | |(o) | | ·---------------· ·---------------· |(i) | (o)| | ·---|---· | | | | | | |---|---|---|---| | | | | | | ·---|---· | |(o) | (o)| ·---------------· ·---------------· | | | | ·---|---· | | |(o)| | | |---|---|---|---| | |(o)|(i)| | | ·---|---· | | | | ·---------------· ·---------------· |(o) | | | ·---|---· | | | (i) | | |---|---|---|(i)| | |(o)|(o)| | | ·---|---· | |(o) | | ·---------------· ·---------------· |(o) | (i)| | ·---|---· | | | | | | |---|---|---|---| | | | | | | ·---|---· | |(o) | (o)| ·---------------· ·---------------· | | | | ·---|---· | | | |(o)| | |---|---|---|---| | |(o)|(o)| | | ·---|---· | | (i) | ·---------------· ·---------------· |(o) | (o)| | ·---|---· | | |(i)|(o)| | |---|---|---|---| | | |(o)| | | ·---|---· | | | (i)| ·---------------· ·---------------· | | | | ·---|---· | | |(o)| | | |---|---|---|---| | |(o)| | | | ·---|---· | | (i) | ·---------------· ·---------------· | | | | ·---|---· | | |(o)|(o)| | |---|---|---|---| | |(o)|(i)| | | ·---|---· | | | | ·---------------· ·---------------· |(o) | | | ·---|---· | | | | | | |---|---|---|---| | |(o)|(o)| | | ·---|---· | |(o) | | ·---------------· ·---------------· | | | | ·---|---· | | |(o)| | | |---|---|---|---| | |(o)| | | | ·---|---· | |(o) | (o)| ·---------------· ·---------------· | | (o)| | ·---|---· | | | (i) | | |---|---|(i)|---| | | | | | | ·---|---· | | | (o)| ·---------------· ·---------------· | | | | ·---|---· | | | |(o)| | |---|---|---|---| | |(i)|(o)| | | ·---|---· | |(o) | (o)| ·---------------· ·---------------· | (i) | | ·---|---· | | | |(o)| | |---|(i)|---|---| | | |(o)| | | ·---|---· | | | | ·---------------· ·---------------· |(o) | | | ·---|---· | | | | | | |---|(i)|---|---| | | | | | | ·---|---· | |(o) | (o)| ·---------------· ·---------------· | (i) | | ·---|---· | | |(o)|(o)| | |---|---|---|---| | |(o)| | | | ·---|---· | | | | ·---------------· ·---------------· |(i) | (o)| | ·---|---· | | | | | | |---|---|---|---| | | | | | | ·---|---· | | | (o)| ·---------------· ·---------------· | (i) | | ·---|---· | | |(o)|(o)| | |---|---|---|(i)| | | | | | | ·---|---· | | | | ·---------------· ·---------------· | | | | ·---|---· | | | |(i)| | |---|---|---|---| | |(o)|(o)| | | ·---|---· | | | | ·---------------· ·---------------· | | | | ·---|---· | | |(i)| | | |---|---|---|---| | |(o)|(o)| | | ·---|---· | | | | ·---------------· ·---------------· | | (o)| | ·---|---· | | | | | | |---|---|(i)|---| | | | | | | ·---|---· | |(i) | (o)| ·---------------· ·---------------· | | | | ·---|---· | | |(o)|(o)| | |---|---|---|---| | |(i)| | | | ·---|---· | | | | ·---------------· ·---------------· |(o) | (o)| | ·---|---· | | |(o)| | | |---|---|---|---| | |(o)| | | | ·---|---· | |(i) | | ·---------------· ·---------------· | | (o)| | ·---|---· | | | (i) | | |---|---|---|---| | |(o)|(o)| | | ·---|---· | | | (o)| ·---------------· pg ·---------------· | | | | ·---|---· | | |(i)|(o)| | |---|---|---|---| | | |(o)| | | ·---|---· | | | | ·---------------· ·---------------· |(o) | (o)| | ·---|---· | | | | | | |---|---|---|---| | | | | | | ·---|---· | |(i) | | ·---------------· ·---------------· | | (o)| | ·---|---· | | | | | | |---|---|---|---| | | (i) | | | ·---|---· | | | (o)| ·---------------· ·---------------· |(o) | (o)| | ·---|---· | | |(o)| | | |---|---|(i)|---| | |(o)| | | | ·---|---· | | | | ·---------------· ·---------------· | | | | ·---|---· | | | |(o)| | |---|---|---|---| | |(o)|(o)| | | ·---|---· | | | | ·---------------· ·---------------· | | | | ·---|---· | | |(o)|(o)| | |---|---|---|(i)| | | | | | | ·---|---· | | | | ·---------------· ·---------------· |(i) | (o)| | ·---|---· | | | | | | |---|---|(i)|---| | | | | | | ·---|---· | | | (o)| ·---------------· ·---------------· |(o) | (i)| | ·---|---· | | |(o)|(o)| | |---|---|---|---| | |(i)| | | | ·---|---· | |(o) | | ·---------------· _answers to § ._ an . some xy exist, or some x are y, or some y are x. . no information. . all y' are x'. . no xy exist, &c. . all y' are x. . all x' are y. . all x are y. . all x' are y', and all y are x. . all x' are y'. . all x are y'. . no information. . some x'y' exist, &c. . some xy' exist, &c. . no xy' exist, &c. . some xy exist, &c. . all y are x. . all x' are y, and all y' are x. . all x are y', and all y are x'. . all x are y, and all y' are x'. . all y are x'. _answers to § ._ an . no x' are y'. . some x' are y'. . some x are y'. . [no concl. fallacy of like eliminands not asserted to exist.] . some x' are y'. . [no concl. fallacy of like eliminands not asserted to exist.] . some x are y'. . some x' are y'. . [no concl. fallacy of unlike eliminands with an entity-premiss.] . all x are y, and all y' are x'. . [no concl. fallacy of like eliminands not asserted to exist.] . all y are x'. . no x' are y. . no x' are y'. . no x are y. . all x are y', and all y are x'. pg . no x are y'. . no x are y. . some x are y'. . no x are y'. . some y are x'. . [no concl. fallacy of unlike eliminands with an entity-premiss.] . some x are y. . all y are x'. . some y are x'. . all y are x. . all x are y, and all y' are x'. . some y are x'. . [no concl. fallacy of like eliminands not asserted to exist.] . some y are x'. . [no concl. fallacy of like eliminands not asserted to exist.] . no x are y'. . [no concl. fallacy of like eliminands not asserted to exist.] . some x are y. . all y are x'. . some y are x'. . some x are y'. . no x are y. . some x' are y'. . all y' are x. . all x are y'. . no x are y. _answers to § ._ an . somebody who has been out for a walk is feeling better. . no one but john knows what the letter is about. . you and i like walking. . honesty is sometimes the best policy. . some greyhounds are not fat. . some brave persons get their deserts. . some rich persons are not esquimaux. . [no concl. fallacy of unlike eliminands with an entity-premiss.] . john is ill. . some things, that are not umbrellas, should be left behind on a journey. . no music is worth paying for, unless it causes vibration in the air. . some holidays are tiresome. . englishmen are not frenchmen. . no photograph of a lady is satisfactory. . no one looks poetical unless he is phlegmatic. . some thin persons are not cheerful. . some judges do not exercise self-control. . pigs are not fed on barley-water. . some black rabbits are not old. pg . [no concl. fallacy of unlike eliminands with an entity-premiss.] . [no concl. fallacy of like eliminands not asserted to exist.] . some lessons need attention. . [no concl. fallacy of like eliminands not asserted to exist.] . no one, who forgets a promise, fails to do mischief. . some greedy creatures cannot fly. . [no concl. fallacy of unlike eliminands with an entity-premiss.] . no bride-cakes are things that need not be avoided. . john is happy. . some people, who are not gamblers, are not philosophers. . [no concl. fallacy of unlike eliminands with an entity-premiss.] . none of my lodgers write poetry. . senna is not nice. . [no concl. fallacy of unlike eliminands with an entity-premiss.] . [no concl. fallacy of like eliminands not asserted to exist.] . logic is unintelligible. . some wild creatures are fat. . all wasps are unwelcome. . all black rabbits are young. . some hard-boiled things can be cracked. . no antelopes fail to delight the eye. . all well-fed canaries are cheerful. . some poetry is not producible at will. . no country infested by dragons fails to be fascinating. . [no concl. fallacy of like eliminands not asserted to exist.] . some picturesque things are not made of sugar. . no children can sit still. . some cats cannot whistle. . you are terrible. . some oysters are not amusing. . nobody in the house has a beard a yard long. . some ill-fed canaries are unhappy. . my sisters cannot sing. . [no concl. fallacy of unlike eliminands with an entity-premiss.] . some rich things are nice. . my cousins are none of them judges, and judges are none of them cousins of mine. . something wearisome is not eagerly wished for. . senna is nasty. . [no concl. fallacy of unlike eliminands with an entity-premiss.] . niggers are not any of them tall. . some obstinate persons are not philosophers. . john is happy. . some unwholesome dishes are not present here (i.e. cannot be spoken of as "these"). . no books suit feverish patients unless they make one drowsy. . some greedy creatures cannot fly. . you and i can detect a sharper. . some dreams are not lambs. pg . no lizard needs a hairbrush. . some things, that may escape notice, are not battles. . my cousins are not any of them judges. . some hard-boiled things can be cracked. . [no concl. fallacy of unlike eliminands with an entity-premiss.] . she is unpopular. . some people, who wear wigs, are not children of yours. . no lobsters expect impossibilities. . no nightmare is eagerly desired. . some nice things are not plumcakes. . some kinds of jam need not be shunned. . all ducks are ungraceful. . [no concl. fallacy of like eliminands not asserted to exist.] . no man, who begs in the street, should fail to keep accounts. . some savage creatures are not spiders. . [no concl. fallacy of unlike eliminands with an entity-premiss.] . no travelers, who do not carry plenty of small change, fail to lose their luggage. . [no concl. fallacy of unlike eliminands with an entity-premiss.] . judges are none of them cousins of mine. . all my lodgers are sane. . those who are busy are contented, and discontented people are not busy. . none of my cousins are judges. . no nightingale dislikes sugar. . [no concl. fallacy of like eliminands not asserted to exist.] . some excuses are not clear explanations. . [no concl. fallacy of like eliminands not asserted to exist.] . no kind deed need cause scruple. . [no concl. fallacy of like eliminands not asserted to exist.] . [no concl. fallacy of like eliminands not asserted to exist.] . no cheats are trustworthy. . no clever child of mine is greedy. . some things, that are meant to amuse, are not acts of parliament. . no tour, that is ever forgotten, is worth writing a book about. . no obedient child of mine is contented. . your visit does not annoy me. _answers to § ._ an . conclusion right. . no concl. fallacy of like eliminands not asserted to exist. . concl. right. . concl. right. . concl. right. . no concl. fallacy of like eliminands not asserted to exist. . no concl. fallacy of unlike eliminands with an entity-premiss. . concl. right. . concl. right. . concl. right. . concl. right. . concl. right. . concl. right. . concl. right. . concl. right. pg . no concl. fallacy of like eliminands not asserted to exist. . concl. right. . concl. right. . concl. right. . concl. right. . concl. right. . concl. wrong: the right one is "some x are y." . concl. right. . concl. right. . concl. right. . concl. right. . concl. right. . no concl. fallacy of like eliminands not asserted to exist. . concl. right. . concl. right. . concl. right. . concl. right. . concl. right. . no concl. fallacy of unlike eliminands with an entity-premiss. . concl. right. . concl. right. . concl. right. . no concl. fallacy of like eliminands not asserted to exist. . concl. right. . concl. right. _answers to § ._ an . concl. right. . concl. right. . concl. right. . concl. wrong: right one is "some epicures are not uncles of mine." . concl. right. . no concl. fallacy of unlike eliminands with an entity-premiss. . concl. wrong: right one is "the publication, in which i saw it, tells lies." . no concl. fallacy of unlike eliminands with an entity-premiss. . concl. wrong: right one is "some tedious songs are not his." . concl. right. . no concl. fallacy of unlike eliminands with an entity-premiss. . concl. wrong: right one is "some fierce creatures do not drink coffee." . no concl. fallacy of unlike eliminands with an entity-premiss. . concl. right. . concl. wrong: right one is "some shallow persons are not students." . no concl. fallacy of like eliminands not asserted to exist. . concl. wrong: right one is "some business, other than railways, is unprofitable." . concl. wrong: right one is "some vain persons are not professors." . concl. right. . concl. wrong: right one is "wasps are not puppies." . no concl. fallacy of unlike eliminands with an entity-premiss. . no concl. same fallacy. . concl. right. . concl. wrong: right one is "some chocolate-creams are delicious." . no concl. fallacy of like eliminands not asserted to exist. . no concl. fallacy of unlike eliminands with an entity-premiss. . concl. wrong: right one is "some pillows are not pokers." . concl. right. . no concl. fallacy of unlike eliminands with an entity-premiss. . no concl. fallacy of like eliminands not asserted to exist. . concl. right. . no concl. fallacy of like eliminands not asserted to exist. . no concl. fallacy of unlike eliminands with an entity-premiss. pg . concl. wrong: right one is "some dreaded persons are not begged to prolong their visits." . concl. wrong: right one is "no man walks on neither." . concl. right. . no concl. fallacy of unlike eliminands with an entity-premiss. . concl. wrong: right one is "some persons, dreaded by children, are not emperors." . concl. incomplete: the omitted portion is "sugar is not salt." . concl. right. _answers to § ._ an . a_{ }b_{ } + b_{ }a_{ }. . d_{ }a_{ }. . ac_{ }. . a_{ }d_{ }. . cd_{ }. . d_{ }c_{ }. . a'c_{ }. . c_{ }a'_{ }. . c'd_{ }. . b_{ }a_{ }. . d_{ }b_{ }. . a'd_{ }. . e_{ }b_{ }. . d_{ }e'_{ }. . e_{ }a'_{ }. . b'c_{ }. . a_{ }b_{ }. . d_{ }c_{ }. . a_{ }d_{ }. . ac_{ }. . de_{ }. . a_{ }b'_{ }. . h_{ }c_{ }. . e_{ }a_{ }. . e_{ }c'_{ }. . e_{ }c'_{ }. . hk'_{ }. . e_{ }d'_{ }. . l'a_{ }. . k_{ }b'_{ }. _answers to § ._ an . babies cannot manage crocodiles. . _your_ presents to me are not made of tin. . all my potatoes in this dish are old ones. . my servants never say "shpoonj." . my poultry are not officers. . none of _your_ sons are fit to serve on a jury. . no pencils of mine are sugar-plums. . jenkins is inexperienced. . no comet has a curly tail. . no hedge-hog takes in the _times_. . this dish is unwholesome. . my gardener is very old. . all humming-birds are small. . no one with a hooked nose ever fails to make money. . no gray ducks in this village wear lace collars. . no jug in this cupboard will hold water. . these apples were grown in the sun. . puppies, that will not lie still, never care to do worsted work. . no name in this list is unmelodious. . no m.p. should ride in a donkey-race, unless he has perfect self-command. . no goods in this shop, that are still on sale, may be carried away. pg . no acrobatic feat, which involves turning a quadruple somersault, is ever attempted in a circus. . guinea-pigs never really appreciate beethoven. . no scentless flowers please me. . showy talkers are not really well-informed. . none but red-haired boys learn greek in this school. . wedding-cake always disagrees with me. . discussions, that go on while tomkins is in the chair, endanger the peacefulness of our debating-club. . all gluttons, who are children of mine, are unhealthy. . an egg of the great auk is not to be had for a song. . no books sold here have gilt edges, unless they are priced at s. and upwards. . when you cut your finger, you will find tincture of calendula useful. . _i_ have never come across a mermaid at sea. . all the romances in this library are well-written. . no bird in this aviary lives on mince-pies. . no plum-pudding, that has not been boiled in a cloth, can be distinguished from soup. . all _your_ poems are uninteresting. . none of my peaches have been grown in a hot-house. . no pawnbroker is dishonest. . no kitten with green eyes will play with a gorilla. . all _my_ friends dine at the lower table. . my writing-desk is full of live scorpions. . no mandarin ever reads hogg's poems. . shakespeare was clever. . rainbows are not worth writing odes to. . these sorites-examples are difficult. . all my dreams come true. . all the english pictures here are painted in oils. . donkeys are not easy to swallow. . opium-eaters never wear white kid gloves. . a good husband always comes home for his tea. . bathing-machines are never made of mother-of-pearl. . rainy days are always cloudy. . no heavy fish is unkind to children. . no engine-driver lives on barley-sugar. . all the animals in the yard gnaw bones. . no badger can guess a conundrum. . no cheque of yours, received by me, is payable to order. . i cannot read any of brown's letters. . i always avoid a kangaroo. pg chapter iii. _solutions._ § . _propositions of relation reduced to normal form._ _solutions for § ._ sl . the univ. is "persons." the individual "i" may be regarded as a class, of persons, whose peculiar attribute is "represented by the name 'i'", and may be called the class of "i's". it is evident that this class cannot possibly contain more than one member: hence the sign of quantity is "all". the verb "have been" may be replaced by the phrase "are persons who have been". the proposition may be written thus:-- "all" _sign of quantity_. "i's" _subject_. "are" _copula_. "persons who have been out for a walk" _predicate_. or, more briefly, "all | i's | are | persons who have been out for a walk". . the univ. and the subject are the same as in ex. . the proposition may be written "all | i's | are | persons who feel better". . univ. is "persons". the subject is evidently the class of persons from which john is _excluded_; _i.e._ it is the class containing all persons who are _not_ "john". the sign of quantity is "no". the verb "has read" may be replaced by the phrase "are persons who have read". the proposition may be written "no | persons who are not 'john' | are | persons who have read the letter". . univ. is "persons". the subject is evidently the class of persons whose only two members are "you and i". hence the sign of quantity is "no". the proposition may be written "no | members of the class 'you and i' | are | old persons". pg . univ. is "creatures". the verb "run well" may be replaced by the phrase "are creatures that run well". the proposition may be written "no | fat creatures | are | creatures that run well". . univ. is "persons". the subject is evidently the class of persons who are _not_ brave. the verb "deserve" may be replaced by the phrase "are deserving of". the proposition may be written "no | not-brave persons | are | persons deserving of the fair". . univ. is "persons". the phrase "looks poetical" evidently belongs to the _predicate_; and the _subject_ is the class, of persons, whose peculiar attribute is "_not_-pale". the proposition may be written "no | not-pale persons | are | persons who look poetical". . univ. is "persons". the proposition may be written "some | judges | are | persons who lose their tempers". . univ. is "persons". the phrase "never neglect" is merely a stronger form of the phrase "am a person who does not neglect". the proposition may be written "all | 'i's' | are | persons who do not neglect important business". . univ. is "things". the phrase "what is difficult" (_i.e._ "that which is difficult") is equivalent to the phrase "all difficult things". the proposition may be written "all | difficult things | are | things that need attention". . univ. is "things". the phrase "what is unwholesome" may be interpreted as in ex. . the proposition may be written "all | unwholesome things | are | things that should be avoided". . univ. is "laws". the predicate is evidently a class whose peculiar attribute is "relating to excise". the proposition may be written "all | laws passed last week | are | laws relating to excise". . univ. is "things". the subject is evidently the class, of studies, whose peculiar attribute is "logical"; hence the sign of quantity is "all". the proposition may be written "all | logical studies | are | things that puzzle me". . univ. is "persons". the subject is evidently "persons in the house". the proposition may be written "no | persons in the house | are | jews". . univ. is "dishes". the phrase "if not well-cooked" is equivalent to the attribute "not well-cooked". the proposition may be written "some | not well-cooked dishes | are | unwholesome dishes". pg . univ. is "books". the phrase "make one drowsy" may be replaced by the phrase "are books that make one drowsy". the sign of quantity is evidently "all". the proposition may be written "all | unexciting books | are | books that make one drowsy". . univ. is "men". the subject is evidently "a man who knows what he's about"; and the word "when" shows that the proposition is asserted of _every_ such man, _i.e._ of _all_ such men. the verb "can" may be replaced by "are men who can". the proposition may be written "all | men who know what they're about | are | men who can detect a sharper". . the univ. and the subject are the same as in ex. . the proposition may be written "all | members of the class 'you and i' | are | persons who know what they're about". . univ. is "persons". the verb "wear" may be replaced by the phrase "are accustomed to wear". the proposition may be written "some | bald persons | are | persons accustomed to wear wigs". . univ. is "persons". the phrase "never talk" is merely a stronger form of "are persons who do not talk". the proposition may be written "all | fully occupied persons | are | persons who do not talk about their grievances". . univ. is "riddles". the phrase "if they can be solved" is equivalent to the attribute "that can be solved". the proposition may be written "no | riddles that can be solved | are | riddles that interest me". § . _method of diagrams._ _solutions for § , nos. - ._ sl -a ·---------------· ·-------· . no m are x'; | | (o)| | | | all m' are y. | ·---|---· | |---|---| | | | | | | |(o)| |(i)|---|---|---| ·-------· | |(o)|(o)| | | ·---|---· | .'. no x' are y'. | | (o)| ·---------------· pg ·---------------· ·-------· . no m' are x; |(o) | (o)| | | | some m' are y'. | ·---|---· | |---|---| | | | | | | |(i)| |---|---|---|---| ·-------· | | | | | | ·---|---· | .'. some x are y'. | | (i)| ·---------------· ·---------------· ·-------· . all m' are x; |(o) | (i)| | |(i)| all m' are y'. | ·---|---· | |---|---| | | | | | | | | |---|---|---|---| ·-------· | | | | | | ·---|---· | .'. some x are y'. |(o) | (o)| ·---------------· ·---------------· . no x' are m'; | | (o)| all y' are m. | ·---|---· | | | | | | |---|---|(i)|---| there is no conclusion. | | | | | | ·---|---· | |(o) | (o)| ·---------------· ·---------------· ·-------· . some m are x'; | | | | | | no y are m. | ·---|---· | |---|---| | |(o)| | | | |(i)| |---|---|---|---| ·-------· | |(o)|(i)| | | ·---|---· | .'. some x' are y'. | | | ·---------------· ·---------------· . no x' are m; | | | no m are y. | ·---|---· | | |(o)| | | |---|---|---|---| there is no conclusion. | |(o)|(o)| | | ·---|---· | | | | ·---------------· ·---------------· ·-------· . no m are x'; | | | | |(i)| some y' are m. | ·---|---· | |---|---| | | |(i)| | | | | |---|---|---|---| ·-------· | |(o)|(o)| | | ·---|---· | .'. some x are y'. | | | ·---------------· ·---------------· ·-------· . all m' are x'; |(o) | (o)| | | | no m' are y. | ·---|---· | |---|---| | | | | | | |(i)| |---|---|---|---| ·-------· | | | | | | ·---|---· | .'. some x' are y'. |(o) | (i)| ·---------------· pg ·---------------· . some x' are m'; | | | no m are y'. | ·---|---· | | | |(o)| | |---|---|---|---| there is no conclusion. | | |(o)| | | ·---|---· | | (i) | ·---------------· ·---------------· ·-------· . all x are m; |(o) | (o)| |(i)|(o)| all y' are m'. | ·---|---· | |---|---| | |(i)|(o)| | | |(i)| |---|---|---|---| ·-------· | | |(o)| | | ·---|---· | .'. all x are y; | | (i)| all y' are x'. ·---------------· ·---------------· . no m are x; | | | all y' are m'. | ·---|---· | | |(o)|(o)| | |---|---|---|(i)| there is no conclusion. | | |(o)| | | ·---|---· | | | | ·---------------· ·---------------· ·-------· . no x are m; |(o) | | |(o)| | all y are m. | ·---|---· | |---|---| | |(o)|(o)| | |(i)| | |---|---|---|---| ·-------· | |(i)| | | | ·---|---· | .'. all y are x'. |(o) | | ·---------------· _solutions for § , nos. - ._ sl -a . i have been out for a walk; i am feeling better. univ. is "persons"; m = the class of i's; x = persons who have been out for a walk; y = persons who are feeling better. ·---------------· ·-------· all m are x; | | | |(i)| | all m are y. | ·---|---· | |---|---| | |(i)|(o)| | | | | |---|---|---|---| ·-------· | |(o)|(o)| | | ·---|---· | .'. some x are y. | | | ·---------------· i.e. somebody, who has been out for a walk, is feeling better. pg . no one has read the letter but john; no one, who has _not_ read it, knows what it is about. univ. is "persons"; m = persons who have read the letter; x = the class of johns; y = persons who know what the letter is about. ·---------------· ·-------· no x' are m; |(o) | | | | | no m' are y. | ·---|---· | |---|---| | | | | | |(o)| | |---|---|---|---| ·-------· | |(o)|(o)| | | ·---|---· | .'. no x' are y. |(o) | | ·---------------· i.e. no one, but john, knows what the letter is about. . those who are not old like walking; you and i are young. univ. is "persons"; m = old; x = persons who like walking; y = you and i. ·---------------· ·-------· all m' are x; |(i) | | |(i)| | all y are m'. | ·---|---· | |---|---| | |(o)| | | |(o)| | |---|---|---|---| ·-------· | |(o)| | | | ·---|---· | .'. all y are x. |(o) | (o)| ·---------------· i.e. you and i like walking. . your course is always honest; your course is always the best policy. univ. is "courses"; m = your; x = honest; y = courses which are the best policy. ·---------------· ·-------· all m are x; | | | |(i)| | all m are y. | ·---|---· | |---|---| | |(i)|(o)| | | | | |---|---|---|---| ·-------· | |(o)|(o)| | | ·---|---· | .'. some x are y. | | | ·---------------· i.e. honesty is sometimes the best policy. . no fat creatures run well; some greyhounds run well. univ. is "creatures"; m = creatures that run well; x = fat; y = greyhounds. ·---------------· ·-------· no x are m; | | | | | | some y are m. | ·---|---· | |---|---| | |(o)|(o)| | |(i)| | |---|---|---|---| ·-------· | |(i)| | | | ·---|---· | .'. some y are x'. | | | ·---------------· i.e. some greyhounds are not fat. pg . some, who deserve the fair, get their deserts; none but the brave deserve the fair. univ. is "persons"; m = persons who deserve the fair; x = persons who get their deserts; y = brave. ·---------------· ·-------· some m are x; | | | |(i)| | no y' are m. | ·---|---· | |---|---| | |(i)|(o)| | | | | |---|---|---|---| ·-------· | | |(o)| | | ·---|---· | .'. some y are x. | | | ·---------------· i.e. some brave persons get their deserts. . some jews are rich; all esquimaux are gentiles. univ. is "persons"; m = jews; x = rich; y = esquimaux. ·---------------· ·-------· some m are x; | | | | |(i)| all y are m'. | ·---|---· | |---|---| | |(o)|(i)| | | | | |(i)|---|---|---| ·-------· | |(o)| | | | ·---|---· | .'. some x are y'. | | | ·---------------· i.e. some rich persons are not esquimaux. . sugar-plums are sweet; some sweet things are liked by children. univ. is "things"; m = sweet; x = sugar-plums; y = things that are liked by children. ·---------------· all x are m; |(o) | (o)| some m are y. | ·---|---· | | | (i) | | |---|(i)|---|---| | | | | | | ·---|---· | | | | ·---------------· there is no conclusion. . john is in the house; everybody in the house is ill. univ. is "persons"; m = persons in the house; x = the class of johns; y = ill. ·---------------· ·-------· all x are m; |(o) | (o)| |(i)|(o)| all m are y. | ·---|---· | |---|---| | |(i)|(o)| | | | | |---|---|---|---| ·-------· | | |(o)| | | ·---|---· | .'. all x are y. | | | ·---------------· i.e. john is ill. pg . umbrellas are useful on a journey; what is useless on a journey should be left behind. univ. is "things"; m = useful on a journey; x = umbrellas; y = things that should be left behind. ·---------------· ·-------· all x are m; |(o) | (o)| | | | all m' are y. | ·---|---· | |---|---| | | (i) | | |(i)| | |---|---|---|---| ·-------· | | | | | | ·---|---· | .'. some x' are y. |(i) | (o)| ·---------------· i.e. some things, that are not umbrellas, should be left behind on a journey. . audible music causes vibration in the air; inaudible music is not worth paying for. univ. is "music"; m = audible; x = music that causes vibration in the air; y = worth paying for. ·---------------· ·-------· all m are x; |(o) | | | | | all m' are y'. | ·---|---· | |---|---| | | (i) | | |(o)| | |---|---|---|(i)| ·-------· | |(o)|(o)| | | ·---|---· | .'. no x' are y. |(o) | | ·---------------· i.e. no music is worth paying for, unless it causes vibration in the air. . some holidays are rainy; rainy days are tiresome. univ. is "days"; m = rainy; x = holidays; y = tiresome. ·---------------· ·-------· some x are m; | | | |(i)| | all m are y. | ·---|---· | |---|---| | |(i)|(o)| | | | | |---|---|---|---| ·-------· | | |(o)| | | ·---|---· | .'. some x are y. | | | ·---------------· i.e. some holidays are tiresome. _solutions for § , nos. - ._ sl -a . some x are m; no m are y'. some x are y. ·---------------· ·-------· | | | |(i)| | | ·---|---· | |---|---| | |(i)|(o)| | | | | |---|---|---|---| ·-------· | | |(o)| | | ·---|---· | hence proposed conclusion is right. | | | ·---------------· pg . all x are m; no y are m'. no y are x'. ·---------------· |(o) | (o)| | ·---|---· | | | (i) | | |---|---|---|---| there is no conclusion. | | | | | | ·---|---· | |(o) | | ·---------------· . some x are m'; all y' are m. some x are y. ·---------------· ·-------· |(i) | (o)| |(i)| | | ·---|---· | |---|---| | | | | | | | | |---|---|(i)|---| ·-------· | | | | | | ·---|---· | hence proposed conclusion is right. | | (o)| ·---------------· . all x are m; no y are m. all x are y'. ·---------------· ·-------· |(o) | (o)| |(o)|(i)| | ·---|---· | |---|---| | |(o)|(i)| | | | | |---|---|---|---| ·-------· | |(o)| | | | ·---|---· | hence proposed conclusion is right. | | | ·---------------· . some m' are x'; no m' are y. some x' are y'. ·---------------· ·-------· |(o) | | | | | | ·---|---· | |---|---| | | | | | | |(i)| |---|---|---|---| ·-------· | | | | | | ·---|---· | hence proposed conclusion is right. |(o) | (i)| ·---------------· . no x' are m; all y are m'. all y are x. ·---------------· | | | | ·---|---· | | |(o)| | | |(i)|---|---|---| there is no conclusion. | |(o)|(o)| | | ·---|---· | | | | ·---------------· pg . some m' are x'; all y' are m'. some x' are y'. ·---------------· | | | | ·---|---· | | | |(o)| | |---|---|---|(i)| there is no conclusion. | | |(o)| | | ·---|---· | | (i) | ·---------------· . no m' are x'; all y' are m'. all y' are x. ·---------------· ·-------· | | (i)| | |(i)| | ·---|---· | |---|---| | | |(o)| | | |(o)| |---|---|---|---| ·-------· | | |(o)| | | ·---|---· | hence proposed conclusion is right. |(o) | (o)| ·---------------· . some m are x'; no m are y. some x' are y'. ·---------------· ·-------· | | | | | | | ·---|---· | |---|---| | |(o)| | | | |(i)| |---|---|---|---| ·-------· | |(o)|(i)| | | ·---|---· | hence proposed conclusion is right. | | | ·---------------· . all m' are x'; all m are y. some y are x'. ·---------------· ·-------· |(o) | (o)| | | | | ·---|---· | |---|---| | | | | | |(i)| | |---|---|---|---| ·-------· | | | | | | ·---|---· | hence proposed conclusion is right. |(i) | (o)| ·---------------· pg _solutions for § , nos. - ._ sl -a . no doctors are enthusiastic; you are enthusiastic. you are not a doctor. univ. "persons"; m = enthusiastic; x = doctors; y = you. ·---------------· ·-------· |(o) | | |(o)| | | ·---|---· | |---|---| no x are m; | |(o)|(o)| | |(i)| | all y are m. |---|---|---|---| ·-------· all y are x'. | |(i)| | | | ·---|---· | .'. all y are x'. |(o) | | ·---------------· hence proposed conclusion is right. . all dictionaries are useful; useful books are valuable. dictionaries are valuable. univ. "books"; m = useful; x = dictionaries; y = valuable. ·---------------· ·-------· |(o) | (o)| |(i)|(o)| | ·---|---· | |---|---| all x are m; | |(i)|(o)| | | | | all m are y. |---|---|---|---| ·-------· all x are y. | | |(o)| | | ·---|---· | .'. all x are y. | | | ·---------------· hence proposed conclusion is right. . no misers are unselfish; none but misers save egg-shells. no unselfish people save egg-shells. univ. "people"; m = misers; x = selfish; y = people who save egg-shells. ·---------------· ·-------· |(o) | | | | | | ·---|---· | |---|---| no m are x'; | | | | | |(o)| | no m' are y. |---|---|---|---| ·-------· no x' are y. | |(o)|(o)| | | ·---|---· | .'. no x' are y. |(o) | | ·---------------· hence proposed conclusion is right. pg . some epicures are ungenerous; all my uncles are generous. my uncles are not epicures. univ. "persons"; m = generous; x = epicures; y = my uncles. ·---------------· ·-------· |(o) | (i)| | |(i)| | ·---|---· | |---|---| some x are m'. | | | | | | | | all y are m. |---|(i)|---|---| ·-------· all y are x'. | | | | | | ·---|---· | .'. some x are y'. |(o) | | ·---------------· hence proposed conclusion is wrong, the right one being "some epicures are not uncles of mine." . gold is heavy; nothing but gold will silence him. nothing light will silence him. univ. "things"; m = gold; x = heavy; y = able to silence him. ·---------------· ·-------· |(o) | | | | | | ·---|---· | |---|---| all m are x; | | (i) | | |(o)| | no m' are y. |---|---|---|---| ·-------· no x' are y. | |(o)|(o)| | | ·---|---· | .'. no x' are y. |(o) | | ·---------------· hence proposed conclusion is right. . some healthy people are fat; no unhealthy people are strong. some fat people are not strong. univ. "persons"; m = healthy; x = fat; y = strong. ·---------------· |(o) | | | ·---|---· | some m are x; | | (i) | | no m' are y. |---|---|---|---| there is no conclusion. some x are y'.| | | | | | ·---|---· | |(o) | | ·---------------· pg § . _method of subscripts._ _solutions for § ._ sl -b . mx'_{ } + m'_{ }y'_{ } ¶ x'y'_{ } [fig. i. i.e. "no x' are y'." . m'x_{ } + m'y'_{ } ¶ x'y'_{ } [fig. ii. i.e. "some x' are y'." . m'_{ }x'_{ } + m'_{ }y_{ } ¶ xy'_{ } [fig. iii. i.e. "some x are y'." . x'm'_{ } + y'_{ }m'_{ } ¶ nothing. [fallacy of like eliminands not asserted to exist.] . mx'_{ } + ym_{ } ¶ x'y'_{ } [fig. ii. i.e. "some x' are y'." . x'm_{ } + my_{ } ¶ nothing. [fallacy of like eliminands not asserted to exist.] . mx'_{ } + y'm_{ } ¶ xy'_{ } [fig. ii. i.e. "some x are y'." . m'_{ }x_{ } + m'y_{ } ¶ x'y'_{ } [fig. iii. i.e. "some x' are y'." . x'm'_{ } + my_{ } ¶ nothing. [fallacy of unlike eliminands with an entity-premiss.] . x_{ }m'_{ } + y'_{ }m_{ } ¶ x_{ }y'_{ } + y'_{ }x_{ } [fig. i (b). i.e. "all x are y, and all y' are x'." . mx_{ } + y'_{ }m_{ } ¶ nothing. [fallacy of like eliminands not asserted to exist.] . xm_{ } + y_{ }m'_{ } ¶ y_{ }x_{ } [fig. i (a). i.e. "all y are x'." . m'_{ }x'_{ } + ym_{ } ¶ x'y_{ } [fig. i. i.e. "no x' are y." . m_{ }x'_{ } + m'_{ }y'_{ } ¶ x'y'_{ } [fig. i. i.e. "no x' are y'." . xm_{ } + m'y_{ } ¶ xy_{ } [fig. i. i.e. "no x are y." . x_{ }m_{ } + y_{ }m'_{ } ¶ (x_{ }y_{ } + y_{ }x_{ }) [fig. i (b). i.e. "all x are y' and all y are x'." . xm_{ } + m'_{ }y'_{ } ¶ xy'_{ } [fig. i. i.e. "no x are y'." . xm'_{ } + my_{ } ¶ xy_{ } [fig. i. i.e. "no x are y." . m_{ }x'_{ } + m_{ }y_{ } ¶ xy'_{ } [fig. iii. i.e. "some x are y'." . mx_{ } + m'_{ }y'_{ } ¶ xy'_{ } [fig. i. i.e. "no x are y'." . x_{ }m'_{ } + m'y_{ } ¶ x'y_{ } [fig. ii. i.e. "some x' are y." . xm_{ } + y_{ }m'_{ } ¶ nothing. [fallacy of unlike eliminands with an entity-premiss.] . m_{ }x'_{ } + ym_{ } ¶ xy_{ } [fig. ii. i.e. "some x are y." . xm_{ } + y_{ }m'_{ } ¶ y_{ }x_{ } [fig. i (a). i.e. "all y are x'." . mx'_{ } + my'_{ } ¶ x'y_{ } [fig. ii. i.e. "some x' are y." . mx'_{ } + y_{ }m'_{ } ¶ y_{ }x'_{ } [fig. i (a). i.e. "all y are x." . x_{ }m_{ } + y'_{ }m'_{ } ¶ (x_{ }y'_{ } + y'_{ }x_{ }) [fig. i (b). i.e. "all x are y, and all y' are x'." . m_{ }x_{ } + my_{ } ¶ x'y_{ } [fig. ii. i.e. "some x' are y." . mx_{ } + y_{ }m_{ } ¶ nothing. [fallacy of like eliminands not asserted to exist.] . x_{ }m_{ } + ym_{ } ¶ x'y_{ } [fig. ii. i.e. "some y are x'." . x_{ }m'_{ } + y_{ }m'_{ } ¶ nothing. [fallacy of like eliminands not asserted to exist.] pg . xm'_{ } + m_{ }y'_{ } ¶ xy'_{ } [fig. i. i.e. "no x are y'." . mx_{ } + my_{ } ¶ nothing. [fallacy of like eliminands not asserted to exist.] . mx'_{ } + ym_{ } ¶ xy_{ } [fig. ii. i.e. "some x are y." . mx_{ } + y_{ }m'_{ } ¶ y_{ }x_{ } [fig. i (a). i.e. "all y are x'." . m_{ }x_{ } + ym_{ } ¶ x'y_{ } [fig. ii. i.e. "some x' are y." . m_{ }x'_{ } + ym_{ } ¶ xy'_{ } [fig. iii. i.e. "some x are y'." . mx_{ } + m'y_{ } ¶ xy_{ } [fig. i. i.e. "no x are y." . mx'_{ } + my_{ } ¶ x'y'_{ } [fig. ii. i.e. "some x' are y'." . x'm_{ } + y'_{ }m'_{ } ¶ y'_{ }x'_{ } [fig. i (a). i.e. "all y' are x." . x_{ }m_{ } + ym'_{ } ¶ x_{ }y_{ } [fig. i (a). i.e. "all x are y'." . m'x_{ } + ym_{ } ¶ xy_{ } [fig. i. i.e. "no x are y." _solutions for § , nos. - ._ sl -b . no frenchmen like plumpudding; all englishmen like plumpudding. univ. "men"; m = liking plumpudding; x = french; y = english. xm_{ } + y_{ }m'_{ } ¶ y_{ }x_{ } [fig. i (a). i.e. englishmen are not frenchmen. . no portrait of a lady, that makes her simper or scowl, is satisfactory; no photograph of a lady ever fails to make her simper or scowl. univ. "portraits of ladies"; m = making the subject simper or scowl; x = satisfactory; y = photographic. mx_{ } + ym'_{ } ¶ xy_{ } [fig. i. i.e. no photograph of a lady is satisfactory. . all pale people are phlegmatic; no one looks poetical unless he is pale. univ. "people"; m = pale; x = phlegmatic; y = looking poetical. m_{ }x'_{ } + m'y_{ } ¶ x'y_{ } [fig. i. i.e. no one looks poetical unless he is phlegmatic. . no old misers are cheerful; some old misers are thin. univ. "persons"; m = old misers; x = cheerful; y = thin. mx_{ } + my_{ } ¶ x'y_{ } [fig. ii. i.e. some thin persons are not cheerful. . no one, who exercises self-control, fails to keep his temper; some judges lose their tempers. univ. "persons"; m = keeping their tempers; x = exercising self-control; y = judges. xm'_{ } + ym'_{ } ¶ x'y_{ } [fig. ii. i.e. some judges do not exercise self-control. pg . all pigs are fat; nothing that is fed on barley-water is fat. univ. is "things"; m = fat; x = pigs; y = fed on barley-water. x_{ }m'_{ } + ym_{ } ¶ x_{ }y_{ } [fig. i (a). i.e. pigs are not fed on barley-water. . all rabbits, that are not greedy, are black; no old rabbits are free from greediness. univ. is "rabbits"; m = greedy; x = black; y = old. m'_{ }x'_{ } + ym'_{ } ¶ xy'_{ } [fig. iii. i.e. some black rabbits are not old. . some pictures are not first attempts; no first attempts are really good. univ. is "things"; m = first attempts; x = pictures; y = really good. xm'_{ } + my_{ } ¶ nothing. [fallacy of unlike eliminands with an entity-premiss.] . i never neglect important business; your business is unimportant. univ. is "business"; m = important; x = neglected by me; y = your. mx_{ } + y_{ }m_{ } ¶ nothing. [fallacy of like eliminands not asserted to exist.] . some lessons are difficult; what is difficult needs attention. univ. is "things"; m = difficult; x = lessons; y = needing attention. xm_{ } + m_{ }y'_{ } ¶ xy_{ } [fig. ii. i.e. some lessons need attention. . all clever people are popular; all obliging people are popular. univ. is "people"; m = popular; x = clever; y = obliging. x_{ }m'_{ } + y_{ }m'_{ } ¶ nothing. [fallacy of like eliminands not asserted to exist.] . thoughtless people do mischief; no thoughtful person forgets a promise. univ. is "persons"; m = thoughtful; x = mischievous; y = forgetful of promises. m'_{ }x'_{ } + my_{ } ¶ x'y_{ } i.e. no one, who forgets a promise, fails to do mischief. _solutions for § ._ sl -b . xm_{ } + my'_{ } ¶ xy_{ } [fig. ii.] concl. right. . x_{ }m'_{ } + ym'_{ } fallacy of like eliminands not asserted to exist. . xm'_{ } + y'_{ }m'_{ } ¶ xy_{ } [fig. ii.] concl. right. pg . x_{ }m'_{ } + ym_{ } ¶ x_{ }y_{ } [fig. i (a).] concl. right. . m'x'_{ } + m'y_{ } ¶ x'y'_{ } [fig. ii.] " . x'm_{ } + y_{ }m_{ } fallacy of like eliminands not asserted to exist. . m'x'_{ } + y'_{ }m_{ } fallacy of unlike eliminands with an entity-premiss. . m'x'_{ } + y'_{ }m_{ } ¶ y'_{ }x'_{ } [fig. i (a).] concl. right. . mx'_{ } + my_{ } ¶ x'y'_{ } [fig. ii.] " . m'_{ }x_{ } + m'_{ }y'_{ } ¶ x'y_{ } [fig. iii.] " . x_{ }m_{ } + ym_{ } ¶ x'y_{ } [fig. ii.] " . xm_{ } + m'y'_{ } ¶ xy'_{ } [fig. i.] " . xm_{ } + y'_{ }m'_{ } ¶ y'_{ }x_{ } [fig. i (a).] " . m'_{ }x_{ } + m'_{ }y'_{ } ¶ x'y_{ } [fig. iii.] " . mx'_{ } + y_{ }m_{ } ¶ x'y'_{ } [fig. ii.] " . x'm_{ } + y'_{ }m_{ } fallacy of like eliminands not asserted to exist. . m'x_{ } + m'_{ }y_{ } ¶ x'y'_{ } [fig. iii.] concl. right. . x'm_{ } + my_{ } ¶ xy_{ } [fig. ii.] " . mx'_{ } + m_{ }y'_{ } ¶ x'y_{ } [ " ] " . x'm'_{ } + m'y'_{ } ¶ xy'_{ } [ " ] " . mx_{ } + m_{ }y_{ } ¶ x'y'_{ } [fig. iii.] " . x'_{ }m'_{ } + ym'_{ } ¶ xy_{ } [fig. ii.] concl. wrong: the right one is "some x are y." . m_{ }x'_{ } + m'y'_{ } ¶ x'y'_{ } [fig. i.] concl. right. . x_{ }m_{ } + m'_{ }y'_{ } ¶ x_{ }y'_{ } [fig. i (a).] " . xm'_{ } + m_{ }y'_{ } ¶ xy'_{ } [fig. i.] " . m_{ }x_{ } + y_{ }m'_{ } ¶ y_{ }x_{ } [fig. i (a).] " . x_{ }m'_{ } + my'_{ } ¶ x_{ }y'_{ } [ " ] " . x_{ }m'_{ } + y'm'_{ } fallacy of like eliminands not asserted to exist. . x'm_{ } + m'y'_{ } ¶ x'y'_{ } [fig. i.] concl. right. . x_{ }m'_{ } + m_{ }y_{ } ¶ x_{ }y_{ } [fig. i (a).] " . x'_{ }m_{ } + y'm'_{ } ¶ x'_{ }y'_{ } [ " ] " . xm_{ } + y'm'_{ } ¶ xy'_{ } [fig. i.] " . m_{ }x_{ } + y'_{ }m'_{ } ¶ y'_{ }x_{ } [fig. i (a).] " . x_{ }m_{ } + ym'_{ } fallacy of unlike eliminands with an entity-premiss. . xm_{ } + m_{ }y'_{ } ¶ xy_{ } [fig. ii.] concl. right. . m_{ }x_{ } + y_{ }m'_{ } ¶ y_{ }x_{ } [fig. i (a).] " . mx'_{ } + m_{ }y_{ } ¶ xy'_{ } [fig. iii.] " . xm_{ } + my'_{ } fallacy of like eliminands not asserted to exist. . mx_{ } + my'_{ } ¶ x'y'_{ } [fig. ii.] concl. right. . mx'_{ } + ym_{ } ¶ xy_{ } [fig. ii.] " pg _solutions for § ._ sl -b . no doctors are enthusiastic; you are enthusiastic. you are not a doctor. univ. "persons"; m = enthusiastic; x = doctors; y = you. xm_{ } + y_{ }m'_{ } ¶ y_{ }x_{ } [fig. i (a). conclusion right. . dictionaries are useful; useful books are valuable. dictionaries are valuable. univ. "books"; m = useful; x = dictionaries; y = valuable. x_{ }m'_{ } + m_{ }y'_{ } ¶ x_{ }y'_{ } [fig. i (a). conclusion right. . no misers are unselfish; none but misers save egg-shells. no unselfish people save egg-shells. univ. "people"; m = misers; x = selfish; y = people who save egg-shells. mx'_{ } + m'y_{ } ¶ x'y_{ } [fig. i. conclusion right. . some epicures are ungenerous; all my uncles are generous. my uncles are not epicures. univ. "persons"; m = generous; x = epicures; y = my uncles. xm'_{ } + y_{ }m'_{ } ¶ xy'_{ } [fig. ii. conclusion wrong: right one is "some epicures are not uncles of mine." . gold is heavy; nothing but gold will silence him. nothing light will silence him. univ. "things"; m = gold; x = heavy; y = able to silence him. m_{ }x'_{ } + m'y_{ } ¶ x'y_{ } [fig. i. conclusion right. . some healthy people are fat; no unhealthy people are strong. some fat people are not strong. univ. "people"; m = healthy; x = fat; y = strong. mx_{ } + m'y_{ } no conclusion. [fallacy of unlike eliminands with an entity-premiss.] . i saw it in a newspaper; all newspapers tell lies. it was a lie. univ. "publications"; m = newspapers; x = publications in which i saw it; y = telling lies. x_{ }m'_{ } + m_{ }y'_{ } ¶ x_{ }y'_{ } [fig. i (a). conclusion wrong: right one is "the publication, in which i saw it, tells lies." pg . some cravats are not artistic; i admire anything artistic. there are some cravats that i do not admire. univ. "things"; m = artistic; x = cravats; y = things that i admire. xm_{ } + m_{ }y_{ } no conclusion. [fallacy of unlike eliminands with an entity-premiss.] . his songs never last an hour. a song, that lasts an hour, is tedious. his songs are never tedious. univ. "songs"; m = lasting an hour; x = his; y = tedious. x_{ }m_{ } + m_{ }y'_{ } ¶ x'y_{ } [fig. iii. conclusion wrong: right one is "some tedious songs are not his." . some candles give very little light; candles are meant to give light. some things, that are meant to give light, give very little. univ. "things"; m = candles; x = giving &c.; y = meant &c. mx_{ } + m_{ }y'_{ } ¶ xy_{ } [fig. ii. conclusion right. . all, who are anxious to learn, work hard. some of these boys work hard. some of these boys are anxious to learn. univ. "persons"; m = hard-working; x = anxious to learn; y = these boys. x_{ }m'_{ } + ym_{ } no conclusion. [fallacy of unlike eliminands with an entity-premiss.] . all lions are fierce; some lions do not drink coffee. some creatures that drink coffee are not fierce. univ. "creatures"; m = lions; x = fierce; y = creatures that drink coffee. m_{ }x'_{ } + my'_{ } ¶ xy'_{ } [fig. ii. conclusion wrong: right one is "some fierce creatures do not drink coffee." . no misers are generous; some old men are ungenerous. some old men are misers. univ. "persons"; m = generous; x = misers; y = old men. xm_{ } + ym'_{ } no conclusion. [fallacy of unlike eliminands with an entity-premiss.] . no fossil can be crossed in love; an oyster may be crossed in love. oysters are not fossils. univ. "things"; m = things that can be crossed in love; x = fossils; y = oysters. xm_{ } + y_{ }m'_{ } ¶ y_{ }x_{ } [fig. i (a). conclusion right. pg . all uneducated people are shallow; students are all educated. no students are shallow. univ. "people"; m = educated; x = shallow; y = students. m'_{ }x'_{ } + y_{ }m'_{ } ¶ xy'_{ } [fig. iii. conclusion wrong: right one is "some shallow people are not students." . all young lambs jump; no young animals are healthy, unless they jump. all young lambs are healthy. univ. "young animals"; m = young animals that jump; x = lambs; y = healthy. x_{ }m'_{ } + m'y_{ } no conclusion. [fallacy of like eliminands not asserted to exist.] . ill-managed business is unprofitable; railways are never ill-managed. all railways are profitable. univ. "business"; m = ill-managed; x = profitable; y = railways. m_{ }x_{ } + y_{ }m_{ } ¶ x'y'_{ } [fig. iii. conclusion wrong: right one is "some business, other than railways, is profitable." . no professors are ignorant; all ignorant people are vain. no professors are vain. univ. "people"; m = ignorant; x = professors; y = vain. xm_{ } + m_{ }y'_{ } ¶ x'y_{ } [fig. iii. conclusion wrong: right one is "some vain persons are not professors." . a prudent man shuns hyænas. no banker is imprudent. no banker fails to shun hyænas. univ. "men"; m = prudent; x = shunning hyænas; y = bankers. m_{ }x'_{ } + ym'_{ } ¶ x'y_{ } [fig. i. conclusion right. . all wasps are unfriendly; no puppies are unfriendly. no puppies are wasps. univ. "creatures"; m = friendly; x = wasps; y = puppies. x_{ }m_{ } + ym'_{ } ¶ x_{ }y_{ } [fig. i (a). conclusion incomplete: complete one is "wasps are not puppies". . no jews are honest; some gentiles are rich. some rich people are dishonest. univ. "persons"; m = jews; x = honest; y = rich. mx_{ } + m'y_{ } no conclusion. [fallacy of unlike eliminands with an entity-premiss.] pg . no idlers win fame; some painters are not idle. some painters win fame. univ. "persons"; m = idlers; x = persons who win fame; y = painters. mx_{ } + ym'_{ } no conclusion. [fallacy of unlike eliminands with an entity-premiss.] . no monkeys are soldiers; all monkeys are mischievous. some mischievous creatures are not soldiers. univ. "creatures"; m = monkeys; x = soldiers; y = mischievous. mx_{ } + m_{ }y'_{ } ¶ x'y_{ } [fig. iii. conclusion right. . all these bonbons are chocolate-creams; all these bonbons are delicious. chocolate-creams are delicious. univ. "food"; m = these bonbons; x = chocolate-creams; y = delicious. m_{ }x'_{ } + m_{ }y'_{ } ¶ xy_{ } [fig. iii. conclusion wrong, being in excess of the right one, which is "some chocolate-creams are delicious." . no muffins are wholesome; all buns are unwholesome. buns are not muffins. univ. "food"; m = wholesome; x = muffins; y = buns. xm_{ } + y_{ }m_{ } no conclusion. [fallacy of like eliminands not asserted to exist.] . some unauthorised reports are false; all authorised reports are trustworthy. some false reports are not trustworthy. univ. "reports"; m = authorised; x = true; y = trustworthy. m'x'_{ } + m_{ }y'_{ } no conclusion. [fallacy of unlike eliminands with an entity-premiss.] . some pillows are soft; no pokers are soft. some pokers are not pillows. univ. "things"; m = soft; x = pillows; y = pokers. xm_{ } + ym_{ } ¶ xy'_{ } [fig. ii. conclusion wrong: right one is "some pillows are not pokers." . improbable stories are not easily believed; none of his stories are probable. none of his stories are easily believed. univ. "stories"; m = probable; x = easily believed; y = his. m'_{ }x_{ } + ym_{ } ¶ xy_{ } [fig. i. conclusion right. pg . no thieves are honest; some dishonest people are found out. some thieves are found out. univ. "people"; m = honest; x = thieves; y = found out. xm_{ } + m'y_{ } no conclusion. [fallacy of unlike eliminands with an entity-premiss.] . no muffins are wholesome; all puffy food is unwholesome. all muffins are puffy. univ. is "food"; m = wholesome; x = muffins; y = puffy. xm_{ } + y_{ }m_{ } no conclusion. [fallacy of like eliminands not asserted to exist.] . no birds, except peacocks, are proud of their tails; some birds, that are proud of their tails, cannot sing. some peacocks cannot sing. univ. "birds"; m = proud of their tails; x = peacocks; y = birds that cannot sing. x'm_{ } + my'_{ } ¶ xy'_{ } [fig. ii. conclusion right. . warmth relieves pain; nothing, that does not relieve pain, is useful in toothache. warmth is useful in toothache. univ. "applications"; m = relieving pain; x = warmth; y = useful in toothache. x_{ }m'_{ } + m'y_{ } no conclusion. [fallacy of like eliminands not asserted to exist.] . no bankrupts are rich; some merchants are not bankrupts. some merchants are rich. univ. "persons"; m = bankrupts; x = rich; y = merchants. mx_{ } + ym'_{ } no conclusion. [fallacy of unlike eliminands with an entity-premiss.] . bores are dreaded; no bore is ever begged to prolong his visit. no one, who is dreaded, is ever begged to prolong his visit. univ. "persons"; m = bores; x = dreaded; y = begged to prolong their visits. m_{ }x'_{ } + my_{ } ¶ xy'_{ } [fig. iii. conclusion wrong: the right one is "some dreaded persons are not begged to prolong their visits." . all wise men walk on their feet; all unwise men walk on their hands. no man walks on both. univ. "men"; m = wise; x = walking on their feet; y = walking on their hands. m_{ }x'_{ } + m'_{ }y'_{ } ¶ x'y'_{ } [fig. i. conclusion wrong: right one is "no man walks on neither." pg . no wheelbarrows are comfortable; no uncomfortable vehicles are popular. no wheelbarrows are popular. univ. "vehicles"; m = comfortable; x = wheelbarrows; y = popular. xm_{ } + m'x_{ } ¶ xy_{ } [fig. i. conclusion right. . no frogs are poetical; some ducks are unpoetical. some ducks are not frogs. univ. "creatures"; m = poetical; x = frogs; y = ducks. xm_{ } + ym'_{ } no conclusion. [fallacy of unlike eliminands with an entity-premiss.] . no emperors are dentists; all dentists are dreaded by children. no emperors are dreaded by children. univ. "persons"; m = dentists; x = emperors; y = dreaded by children. xm_{ } + m_{ }y'_{ } ¶ x'y_{ } [fig. iii. conclusion wrong: right one is "some persons, dreaded by children, are not emperors." . sugar is sweet; salt is not sweet. salt is not sugar. univ. "things"; m = sweet; x = sugar; y = salt. x_{ }m'_{ } + y_{ }m_{ } ¶ (x_{ }y_{ } + y_{ }x_{ }) [fig. i (b). conclusion incomplete: omitted portion is "sugar is not salt." . every eagle can fly; some pigs cannot fly. some pigs are not eagles. univ. "creatures"; m = creatures that can fly; x = eagles; y = pigs. x_{ }m'_{ } + ym'_{ } ¶ x'y_{ } [fig. ii. conclusion right. _solutions for § ._ sl . cd_{ } + a_{ }d'_{ } + b_{ }c'_{ }; cd + ad' + bc' ¶ ab_{ } + a_{ } + b_{ } -- = = i.e. ¶ a_{ }b_{ } + b_{ }a_{ } . d_{ }b'_{ } + ac'_{ } + bc_{ }; db' + bc + ac' ¶ da_{ } + d_{ } i.e. ¶ d_{ }a_{ } - =- = . ba_{ } + cd'_{ } + d_{ }b'_{ }; ba + db' + cd' ¶ ac_{ } - -= = . bc_{ } + a_{ }b'_{ } + c'd_{ }; bc + ab' + c'd ¶ ad_{ } + a_{ } i.e. ¶ a_{ }d_{ } -- = = pg . b'_{ }a_{ } + bc_{ } + a'd_{ }; b'a + bc + a'd ¶ cd_{ } - - = = . a_{ }b_{ } + b'c_{ } + d_{ }a'_{ }; ab + b'c + da' ¶ cd_{ } + d_{ } i.e. ¶ d_{ }c_{ } -- = = . db'_{ } + b_{ }a'_{ } + cd'_{ }; db' + ba' + cd' ¶ a'c_{ } -- = = . b'd_{ } + a'b_{ } + c_{ }d'_{ }; b'd + a'b + cd' ¶ a'c_{ } + c_{ } i.e. ¶ c_{ }a'_{ } - - = = . b'_{ }a'_{ } + ad_{ } + b_{ }c'_{ }; b'a' + ad + bc' ¶ dc'_{ } - - = = . cd_{ } + b_{ }c'_{ } + ad'_{ }; cd + bc' + ad' ¶ ba_{ } + b_{ } i.e. ¶ b_{ }a_{ } -- = = . bc_{ } + d_{ }a'_{ } + c'_{ }a_{ }; bc + c'a + da' ¶ bd_{ } + d_{ } i.e. ¶ d_{ }b_{ } - = - = . cb'_{ } + c'_{ }d_{ } + b_{ }a'_{ }; cb' + c'd + ba' ¶ da'_{ } -- = = . d_{ }e'_{ } + c_{ }a'_{ } + bd'_{ } + e_{ }a_{ }; de' + bd' + ea + ca' ¶ bc_{ } + c_{ } i.e. ¶ c_{ }b_{ } -- = =- = . c_{ }b'_{ } + a_{ }e'_{ } + d_{ }b_{ } + a'_{ }c'_{ }; cb' + db + a'c' + ae' ¶ de'_{ } + d_{ } i.e. ¶ d_{ }e'_{ } -- = - = = . b'd_{ } + e_{ }c'_{ } + b_{ }a'_{ } + d'_{ }c_{ }; b'd + ba' + d'c + ec' ¶ a'e_{ } + e_{ } i.e. ¶ e_{ }a'_{ } - - = = - = . a'e_{ } + d_{ }c_{ } + a_{ }b'_{ } + e'_{ }d'_{ }; a'e + ab' + e'd' + dc ¶ b'c_{ } - - = = - = . d_{ }c'_{ } + a_{ }e'_{ } + bd'_{ } + c_{ }e_{ }; dc' + bd' + ce + ae' ¶ ba_{ } + a_{ } i.e. ¶ a_{ }b_{ } -- = =- = . a_{ }b'_{ } + d_{ }e'_{ } + a'_{ }c_{ } + be_{ }; ab' + a'c + be + de' ¶ cd_{ } + d_{ } i.e. ¶ d_{ }c_{ } -- = =- = . bc_{ } + e_{ }h'_{ } + a_{ }b'_{ } + dh_{ } + e'_{ }c'_{ }; bc + ab' + e'c' + eh' + dh ¶ ad_{ } + a_{ } -- = - = =- = i.e. ¶ a_{ }d_{ } . dh'_{ } + ce_{ } + h_{ }b'_{ } + ad'_{ } + be'_{ }; dh' + hb' + ad' + be' + ce ¶ ac_{ } -- =- = =- = . b_{ }a'_{ } + dh_{ } + ce_{ } + ah'_{ } + c'_{ }b'_{ }; ba' + ah' + dh + c'b' + ce ¶ de_{ } -- =- = - = = . e_{ }d_{ } + b'h'_{ } + c'_{ }d'_{ } + a_{ }e'_{ } + ch_{ }; ed + c'd' + ae' + ch + b'h' ¶ ab'_{ } + a_{ } -- - = = =- = i.e. ¶ a_{ }b_{ } pg . b'_{ }a_{ } + de'_{ } + h_{ }b_{ } + ce_{ } + d'_{ }a'_{ }; b'a + hb + d'a' + de' + ce - - = - = =- = ¶ hc_{ } + h_{ } i.e. ¶ h_{ }c_{ } . h'_{ }k_{ } + b'a_{ } + c_{ }d'_{ } + e_{ }h_{ } + dk'_{ } + bc'_{ }; h'k + eh + dk' + cd' + bc' + b'a ¶ ea_{ } + e_{ } - - = -= -= -= = i.e. ¶ e_{ }a_{ } . a_{ }d'_{ } + k_{ }b'_{ } + e_{ }h'_{ } + a'b_{ } + d_{ }c'_{ } + h_{ }k'_{ }; ad' + a'b + kb' + dc' + hk' + eh' ¶ c'e_{ } + e_{ } -- = - -= = -= = i.e. ¶ e_{ }c'_{ } . a'_{ }h'_{ } + d'k'_{ } + e_{ }b_{ } + hk_{ } + a_{ }c'_{ } + b'd_{ }; a'h' + hk + d'k' + ac' + b'd + eb ¶ c'e_{ } + e_{ } - - =- - = = - = = i.e. ¶ e_{ }c'_{ } . e_{ }d_{ } + hb_{ } + a'_{ }k'_{ } + ce'_{ } + b'_{ }d'_{ } + ac'_{ }; ed + ce' + b'd' + hb + ac' + a'k' ¶ hk'_{ } -- -= - = = -= = . a'k_{ } + e_{ }b'_{ } + hk'_{ } + d'c_{ } + ab_{ } + c'_{ }h'_{ }; a'k + hk' + ab + eb' + c'h' + d'c ¶ ed'_{ } + e_{ } - - -= =- = - = = i.e. ¶ e_{ }d'_{ } . ek_{ } + b'm_{ } + ac'_{ } + h'_{ }e'_{ } + d_{ }k'_{ } + cb_{ } + d'_{ }l'_{ } + hm'_{ }; ek + h'e' + dk' + d'l' + hm' + b'm + cb + ac' -- - = -= = =- - = -= = ¶ l'a_{ } . n_{ }m'_{ } + a'_{ }e'_{ } + c'l_{ } + k_{ }r_{ } + ah'_{ } + dl'_{ } + cn'_{ } + e_{ }b'_{ } + m_{ }r'_{ } + h_{ }d'_{ }; nm' + cn' + c'l + dl' + mr' + kr + hd' + ah' + a'e' + eb' -- -= = - -= =- = -= -= = - = ¶ kb'_{ } + k_{ } i.e. ¶ k_{ }b'_{ } _solutions for § ._ sl . b_{ }d_{ } + ac_{ } + d'_{ }c'_{ }; bd + d'c' + ac ¶ ba_{ } + b_{ }, i.e. ¶ b_{ }a_{ } - = - = i.e. babies cannot manage crocodiles. . a_{ }b'_{ } + d_{ }c'_{ } + bc_{ }; ab' + bc + dc' ¶ ad_{ } + d_{ }, i.e. ¶ d_{ }a_{ } - =- = i.e. _your_ presents to me are not made of tin. pg . da_{ } + c_{ }b'_{ } + a'b_{ }; da + a'b + cb' ¶ dc_{ } + c_{ }, i.e. ¶ c_{ }d_{ } - = - = i.e. all my potatoes in this dish are old ones. . ba_{ } + b'd_{ } + c_{ }a'_{ }; ba + b'd + ca' ¶ dc_{ } + c_{ }, i.e. ¶ c_{ }d_{ } -- = = i.e. my servants never say "shpoonj." . ad_{ } + cd'_{ } + b_{ }a'_{ }; ad + cd' + ba' ¶ cb_{ } + b_{ }, i.e. ¶ b_{ }c_{ } -- = = i.e. my poultry are not officers. . c_{ }a'_{ } + c'b_{ } + da_{ }; ca' + c'b + da ¶ bd_{ } -- = = i.e. none of your sons are fit to serve on a jury. . cb_{ } + da_{ } + b'_{ }a'_{ }; cb + b'a' + da ¶ cd_{ } - = - = i.e. no pencils of mine are sugarplums. . cb'_{ } + d_{ }a'_{ } + ba_{ }; cb' + ba + da' ¶ cd_{ } + d_{ }, i.e. ¶ d_{ }c_{ } - =- = i.e. jenkins is inexperienced. . cd_{ } + d'a_{ } + c'b_{ }; cd + d'a + c'b ¶ ab_{ } -- = = i.e. no comet has a curly tail. . d'c_{ } + ba_{ } + a'_{ }d_{ }; d'c + a'd + ba ¶ cb_{ } - - = = i.e. no hedgehog takes in the _times_. . b_{ }a'_{ } + c_{ }b'_{ } + ad_{ }; ba' + cb' + ad ¶ cd_{ } + c_{ }, i.e. ¶ c_{ }d_{ } -- = = i.e. this dish is unwholesome. . b_{ }c'_{ } + d'a_{ } + a'c_{ }; bc' + a'c + d'a ¶ bd'_{ } + b_{ }, i.e. ¶ b_{ }d'_{ } - - = = i.e. my gardener is very old. . a_{ }d'_{ } + bc_{ } + c'_{ }d_{ }; ad' + c'd + bc ¶ ab_{ } + a_{ }, i.e. ¶ a_{ }b_{ } - - = = i.e. all humming-birds are small. pg . c'b_{ } + a_{ }d'_{ } + ca'_{ }; c'b + ca' + ad' ¶ bd'_{ } - =- = i.e. no one with a hooked nose ever fails to make money. . b_{ }a'_{ } + b'_{ }d_{ } + ca_{ }; ba' + b'd + ca ¶ dc_{ } -- = = i.e. no gray ducks in this village wear lace collars. . d_{ }b'_{ } + cd'_{ } + ba_{ }; db' + cd' + ba ¶ ca_{ } -- = = i.e. no jug in this cupboard will hold water. . b'_{ }d_{ } + c_{ }d'_{ } + ab_{ }; b'd + cd' + ab ¶ ca_{ } + c_{ }, i.e. ¶ c_{ }a_{ } - - = = i.e. these apples were grown in the sun. . d'_{ }b'_{ } + c_{ }b_{ } + c'a_{ }; d'b' + cb + c'a ¶ d'a_{ } + d'_{ }, i.e. ¶ d'_{ }a_{ } - -= = i.e. puppies, that will not lie still, never care to do worsted-work. . bd'_{ } + a_{ }c'_{ } + a'd_{ }; bd' + a'd + ac' ¶ bc'_{ } - - = = i.e. no name in this list is unmelodious. . a_{ }b'_{ } + dc_{ } + a'_{ }d'_{ }; ab' + a'd' + dc ¶ b'c_{ } - = - = i.e. no m.p. should ride in a donkey-race, unless he has perfect self-command. . bd_{ } + c'a_{ } + b'c_{ }; bd + b'c + c'a ¶ da_{ } - = - = i.e. no goods in this shop, that are still on sale, may be carried away. . a'b_{ } + cd_{ } + d'a_{ }; a'b + d'a + cd ¶ bc_{ } - - = = i.e. no acrobatic feat, which involves turning a quadruple somersault, is ever attempted in a circus. . dc'_{ } + a_{ }b'_{ } + bc_{ }; dc' + bc + ab' ¶ da_{ } + a_{ }, i.e. ¶ a_{ }d_{ } - -= = i.e. guinea-pigs never really appreciate beethoven. pg . a_{ }d'_{ } + b'_{ }c_{ } + ba'_{ }; ad' + ba' + b'c ¶ d'c_{ } - -= = i.e. no scentless flowers please me. . c_{ }d'_{ } + ba'_{ } + d_{ }a_{ }; cd' + da + ba' ¶ cb_{ } + c_{ }, i.e. ¶ c_{ }b_{ } - =- = i.e. showy talkers are not really well-informed. . ea_{ } + b_{ }d'_{ } + a'_{ }c_{ } + e'b'_{ }; ea + a'c + e'b' + bd' ¶ cd'_{ } -- = = - = i.e. none but red-haired boys learn greek in this school. . b_{ }d_{ } + ac'_{ } + e_{ }d'_{ } + c_{ }b'_{ }; bd + ed' + cb' + ac' ¶ ea_{ } + e_{ }, i.e. ¶ e_{ }a_{ } -- = -= = i.e. wedding-cake always disagrees with me. . ad_{ } + e'_{ }b'_{ } + c_{ }d'_{ } + e_{ }a'_{ }; ad + cd' + ea' + e'b' ¶ cb'_{ } + c_{ }, i.e. ¶ c_{ }b'_{ } -- = -= = i.e. discussions, that go on while tomkins is in the chair, endanger the peacefulness of our debating-club. . d_{ }a_{ } + e'c_{ } + b_{ }a'_{ } + d'e_{ }; da + ba' + d'e + e'c ¶ bc_{ } + b_{ }, i.e. ¶ b_{ }c_{ } -- = = - = i.e. all gluttons in my family are unhealthy. . d_{ }e_{ } + c'a_{ } + b_{ }e'_{ } + c_{ }d'_{ }; de + be' + cd' + c'a ¶ ba_{ } + b_{ }, i.e. ¶ b_{ }a_{ } -- = -= = i.e. an egg of the great auk is not to be had for a song. . d'b_{ } + a_{ }c'_{ } + c_{ }e'_{ } + a'd_{ }; d'b + a'd + ac' + ce' ¶ be'_{ } - - = =- = i.e. no books sold here have gilt edges unless they are priced at s. and upwards. . a'_{ }c'_{ } + d_{ }b_{ } + a_{ }e'_{ } + c_{ }b'_{ }; a'c' + ae' + cb' + db ¶ e'd_{ } + d_{ }, i.e. ¶ d_{ }e'_{ } - - = =- = i.e. when you cut your finger, you will find tincture of calendula useful. . d'b_{ } + a_{ }e'_{ } + ec_{ } + d_{ }a'_{ }; d'b + da' + ae' + ec ¶ bc_{ } - =- =- = i.e. _i_ have never come across a mermaid at sea. pg . c'_{ }b_{ } + a_{ }e'_{ } + d_{ }b'_{ } + a'_{ }c_{ }; c'b + db' + a'c + ae' ¶ de'_{ } + d_{ }, i.e. ¶ d_{ }e'_{ } - - = - = = i.e. all the romances in this library are well-written. . e'd_{ } + c'a_{ } + eb_{ } + d'c_{ }; e'd + eb + d'c + c'a ¶ ba_{ } - - = = - = i.e. no bird in this aviary lives on mince-pies. . d'_{ }c'_{ } + e_{ }a'_{ } + c_{ }b_{ } + e'd_{ }; d'c' + cb + e'd + ea' ¶ ba'_{ } - - = - = = i.e. no plum-pudding, that has not been boiled in a cloth, can be distinguished from soup. . ce'_{ } + b'a'_{ } + h_{ }d'_{ } + ae_{ } + bd_{ }; ce' + ae + b'a' + bd + hd' ¶ ch_{ } + h_{ }, i.e. ¶ h_{ }c_{ } - -= - = =- = i.e. all _your_ poems are uninteresting. . b'_{ }a'_{ } + db_{ } + he'_{ } + ec_{ } + a_{ }h'_{ }; b'a' + db + ah' + he' + ec ¶ dc_{ } - - = =- =- = i.e. none of my peaches have been grown in a hothouse. . c_{ }d_{ } + h_{ }e'_{ } + c'_{ }a'_{ } + h'b_{ } + e_{ }d'_{ }; cd + c'a' + ed' + he' + h'b ¶ a'b_{ } -- = -= -= = i.e. no pawnbroker is dishonest. . bd'_{ } + c'h_{ } + e_{ }b'_{ } + da_{ } + e'c_{ }; bd' + eb' + da + e'c + c'h ¶ ah_{ } -- -= = = - = i.e. no kitten with green eyes will play with a gorilla. . c_{ }a'_{ } + h'b_{ } + ae_{ } + d_{ }c'_{ } + h_{ }e'_{ }; ca' + ae + dc' + he' + h'b ¶ db_{ } + d_{ }, i.e. ¶ d_{ }b_{ } -- =- = -= = i.e. all _my_ friends in this college dine at the lower table. . ca_{ } + h_{ }d'_{ } + c'_{ }e'_{ } + b'a'_{ } + d_{ }e_{ }; ca + c'e' + b'a' + de + hd' ¶ b'h_{ } + h_{ }, -- = - = -= = i.e. ¶ h_{ }b'_{ } i.e. my writing-desk is full of live scorpions. . b'_{ }e_{ } + ah_{ } + dc_{ } + e'_{ }a'_{ } + bc'_{ } b'e + e'a' + ah + bc' + dc ¶ hd_{ } - - = - = =- = i.e. no mandarin ever reads hogg's poems. pg . e_{ }b'_{ } + a'd_{ } + c_{ }h'_{ } + e'a_{ } + d'h_{ }; eb' + e'a + a'd + d'h + ch' ¶ b'c_{ } + c_{ }, - = - = - = - = i.e. ¶ c_{ }b'_{ } i.e. shakespeare was clever. . e'_{ }c'_{ } + hb'_{ } + d_{ }a_{ } + e_{ }a'_{ } + c_{ }b_{ }; e'c' + ea' + da + cb + hb' ¶ dh_{ } + d_{ }, i.e. ¶ d_{ }h_{ } - - =- = =- = i.e. rainbows are not worth writing odes to. . c'_{ }h'_{ } + e_{ }a_{ } + bd_{ } + a'_{ }h_{ } + d'c_{ }; c'h' + a'h + ea + d'c + bd ¶ eb_{ } + e_{ }, i.e. ¶ e_{ }b_{ } - - - = = - = = i.e. these sorites-examples are difficult. . a'_{ }e'_{ } + bk_{ } + c'a_{ } + eh'_{ } + d_{ }b'_{ } + k'h_{ }; a'e' + c'a + eh' + k'h + bk + db' ¶ c'd_{ } + d_{ }, - - = =- - = -= = i.e. ¶ d_{ }c'_{ } i.e. all my dreams come true. . a'h_{ } + c'k_{ } + a_{ }d'_{ } + e_{ }h'_{ } + b_{ }k'_{ } + c_{ }e'_{ }; a'h + ad' + eh' + ce' + c'k + bk' ¶ d'b_{ } + b_{ }, - - = -= -= = - = i.e. ¶ b_{ }d'_{ } i.e. all the english pictures here are painted in oils. . k'_{ }e_{ } + c_{ }h_{ } + b_{ }a'_{ } + kd_{ } + h'a_{ } + b'_{ }e'_{ }; k'e + kd + b'e' + ba' + h'a + ch ¶ dc_{ } + c_{ }, - - = - = =- - = = i.e. ¶ c_{ }d_{ } i.e. donkeys are not easy to swallow. . ab'_{ } + h'd_{ } + e_{ }c_{ } + b_{ }d'_{ } + a'k_{ } + c'_{ }h_{ }; ab' + bd' + h'd + a'k + c'h + ec ¶ ke_{ } + e_{ }, -- =- - = = - = = i.e. ¶ e_{ }k_{ } i.e. opium-eaters never wear white kid gloves. . bc_{ } + k_{ }a'_{ } + eh_{ } + d_{ }b'_{ } + h'c'_{ } + k'_{ }e'_{ }; bc + db' + h'c' + eh + k'e' + ka' ¶ da'_{ } + d_{ }, -- = - = -= - = = i.e. ¶ d_{ }a'_{ } i.e. a good husband always comes home for his tea. . a'_{ }k'_{ } + ch_{ } + h'k_{ } + b_{ }d'_{ } + ea_{ } + d_{ }c'_{ } a'k' + h'k + ch + dc' + bd' + ea ¶ be_{ } + b_{ }, - - - = -= -= = = i.e. ¶ b_{ }e_{ } i.e. bathing-machines are never made of mother-of-pearl. pg . da'_{ } + k_{ }b'_{ } + c_{ }h_{ } + d'_{ }k'_{ } + e_{ }c'_{ } + a_{ }h'_{ }; da' + d'k' + kb' + ah' + ch + ec' -- = - = =- -= = ¶ b'e_{ } + e_{ }, i.e. ¶ e_{ }b'_{ } i.e. rainy days are always cloudy. . kb'_{ } + a'_{ }c'_{ } + d'b_{ } + k'_{ }h'_{ } + ea_{ } + d_{ }c_{ }; kb' + d'b + k'h' + dc + a'c' + ea -- - = = =- - = = ¶ h'e_{ } i.e. no heavy fish is unkind to children. . k'_{ }b'_{ } + eh'_{ } + c'd_{ } + hb_{ } + ac_{ } + kd'_{ }; k'b' + hb + eh' + kd' + c'd + ac ¶ ea_{ } - - -= = =- - = = i.e. no engine-driver lives on barley-sugar. . h_{ }b'_{ } + c_{ }d'_{ } + k'a_{ } + e_{ }h'_{ } + b_{ }a'_{ } + k_{ }c'_{ }; hb' + eh' + ba' + k'a + kc' + cd' -- = =- - = =- = ¶ ed'_{ } + e_{ }, i.e. ¶ e_{ }d'_{ } i.e. all the animals in the yard gnaw bones. . h'_{ }d'_{ } + e_{ }c'_{ } + k'a_{ } + cb_{ } + d_{ }l'_{ } + e'h_{ } + kl_{ }; h'd' + dl' + kl + k'a + e'h + ec' + cb ¶ ab_{ } - - =- -= = - = =- = i.e. no badger can guess a conundrum. . b'h_{ } + d'_{ }l'_{ } + ca_{ } + d_{ }k'_{ } + h'_{ }e'_{ } + mc'_{ } + a'b_{ } + ek_{ }; b'h + h'e' + a'b + ca + mc' + ek + dk' + d'l' ¶ ml'_{ } - - = - - = -= = =- -= = i.e. no cheque of yours, received by me, is payable to order. . c_{ }l'_{ } + h'e_{ } + kd_{ } + mc'_{ } + b'_{ }e'_{ } + n_{ }a'_{ } + l_{ }d'_{ } + m'b_{ } + ah_{ }; cl' + mc' + ld' + kd + m'b + b'e' + h'e + ah + na' -- -= =- = = - = - - = -= = ¶ kn_{ } i.e. i cannot read any of brown's letters. . e_{ }c'_{ } + l_{ }n'_{ } + d_{ }a'_{ } + m'b_{ } + ck'_{ } + e'r_{ } + h_{ }n_{ } + b'k_{ } + r'_{ }d'_{ } + m_{ }l'_{ }; ec' + ck' + e'r + b'k + m'b + r'd' + da' + ml' + ln' + hn -- =- = - - = - = = - = =- =- = ¶ a'h_{ } + h_{ }, i.e. ¶ h_{ }a'_{ } i.e. i always avoid a kangaroo. pg notes. (a) [see p. ]. one of the favourite objections, brought against the science of logic by its detractors, is that a syllogism has no real validity as an argument, since it involves the fallacy of _petitio principii_ (i.e. "begging the question", the essence of which is that the whole conclusion is involved in _one_ of the premisses). this formidable objection is refuted, with beautiful clearness and simplicity, by these three diagrams, which show us that, in each of the three figures, the conclusion is really involved in the _two_ premisses taken together, each contributing its share. thus, in fig. i., the premiss xm_{ } empties the _inner_ cell of the n.w. quarter, while the premiss ym_{ } empties its _outer_ cell. hence it needs the _two_ premisses to empty the _whole_ of the n.w. quarter, and thus to prove the conclusion xy_{ }. again, in fig. ii., the premiss xm_{ } empties the inner cell of the n.w. quarter. the premiss ym_{ } merely tells us that the inner portion of the w. half is _occupied_, so that we may place a 'i' in it, _somewhere_; but, if this were the _whole_ of our information, we should not know in _which_ cell to place it, so that it would have to 'sit on the fence': it is only when we learn, from the other premiss, that the _upper_ of these two cells is _empty_, that we feel authorised to place the 'i' in the _lower_ cell, and thus to prove the conclusion x'y_{ }. lastly, in fig. iii., the information, that m _exists_, merely authorises us to place a 'i' _somewhere_ in the inner square----but it has large choice of fences to sit upon! it needs the premiss xm_{ } to drive it out of the n. half of that square; and it needs the premiss ym_{ } to drive it out of the w. half. hence it needs the _two_ premisses to drive it into the inner portion of the s.e. quarter, and thus to prove the conclusion x'y'_{ }. pg appendix, addressed to teachers. § . _introductory._ there are several matters, too hard to discuss with _learners_, which nevertheless need to be explained to any _teachers_, into whose hands this book may fall, in order that they may thoroughly understand what my symbolic method _is_, and in what respects it differs from the many other methods already published. these matters are as follows:-- the "existential import" of propositions. the use of "is-not" (or "are-not") as a copula. the theory "two negative premisses prove nothing." euler's method of diagrams. venn's method of diagrams. my method of diagrams. the solution of a syllogism by various methods. my method of treating syllogisms and sorites. some account of parts ii, iii. § . _the "existential import" of propositions._ the writers, and editors, of the logical text-books which run in the ordinary grooves----to whom i shall hereafter refer by the (i hope inoffensive) title "the logicians"----take, on this subject, what seems to me to be a more humble position than is at all necessary. they speak of the copula of a proposition "with bated breath", almost as if it were a living, conscious entity, capable of declaring for itself what it chose to mean, and that we, poor human creatures, had nothing to do but to ascertain _what_ was its sovereign will and pleasure, and submit to it. pg in opposition to this view, i maintain that any writer of a book is fully authorised in attaching any meaning he likes to any word or phrase he intends to use. if i find an author saying, at the beginning of his book, "let it be understood that by the word '_black_' i shall always mean '_white_', and that by the word '_white_' i shall always mean '_black_'," i meekly accept his ruling, however injudicious i may think it. and so, with regard to the question whether a proposition is or is not to be understood as asserting the existence of its subject, i maintain that every writer may adopt his own rule, provided of course that it is consistent with itself and with the accepted facts of logic. let us consider certain views that may _logically_ be held, and thus settle which of them may _conveniently_ be held; after which i shall hold myself free to declare which of them _i_ intend to hold. the _kinds_ of propositions, to be considered, are those that begin with "some", with "no", and with "all". these are usually called propositions "in _i_", "in _e_", and "in _a_". first, then, a proposition in _i_ may be understood as asserting, or else as _not_ asserting, the existence of its subject. (by "existence" i mean of course whatever kind of existence suits its nature. the two propositions, "_dreams_ exist" and "_drums_ exist", denote two totally different kinds of "existence". a _dream_ is an aggregate of ideas, and exists only in the _mind of a dreamer_: whereas a _drum_ is an aggregate of wood and parchment, and exists in _the hands of a drummer_.) first, let us suppose that _i_ "asserts" (i.e. "asserts the existence of its subject"). here, of course, we must regard a proposition in _a_ as making the _same_ assertion, since it necessarily _contains_ a proposition in _i_. we now have _i_ and _a_ "asserting". does this leave us free to make what supposition we choose as to _e_? my answer is "no. we are tied down to the supposition that _e_ does _not_ assert." this can be proved as follows:-- if possible, let _e_ "assert". then (taking x, y, and z to represent attributes) we see that, if the proposition "no xy are z" be true, some things exist with the attributes x and y: i.e. "some x are y." pg also we know that, if the proposition "some xy are z" be true, the same result follows. but these two propositions are contradictories, so that one or other of them _must_ be true. hence this result is _always_ true: i.e. the proposition "some x are y" is _always_ true! _quod est absurdum._ (see note (a), p. ). we see, then, that the supposition "_i_ asserts" necessarily leads to "_a_ asserts, but _e_ does not". and this is the _first_ of the various views that may conceivably be held. next, let us suppose that _i_ does _not_ "assert." and, along with this, let us take the supposition that _e_ _does_ "assert." hence the proposition "no x are y" means "some x exist, and none of them are y": i.e. "_all_ of them are _not_-y," which is a proposition in _a_. we also know, of course, that the proposition "all x are not-y" proves "no x are y." now two propositions, each of which proves the other, are _equivalent_. hence every proposition in _a_ is equivalent to one in _e_, and therefore "_asserts_". hence our _second_ conceivable view is "_e_ and _a_ assert, but _i_ does not." this view does not seen to involve any necessary contradiction with itself or with the accepted facts of logic. but, when we come to _test_ it, as applied to the actual _facts_ of life, we shall find i think, that it fits in with them so badly that its adoption would be, to say the least of it, singularly inconvenient for ordinary folk. let me record a little dialogue i have just held with my friend jones, who is trying to form a new club, to be regulated on strictly _logical_ principles. _author._ "well, jones! have you got your new club started yet?" _jones_ (_rubbing his hands_). "you'll be glad to hear that some of the members (mind, i only say '_some_') are millionaires! rolling in gold, my boy!" _author._ "that sounds well. and how many members have entered?" _jones_ (_staring_). "none at all. we haven't got it started yet. what makes you think we have?" _author._ "why, i thought you said that some of the members----" pg _jones_ (_contemptuously_). "you don't seem to be aware that we're working on strictly _logical_ principles. a _particular_ proposition does _not_ assert the existence of its subject. i merely meant to say that we've made a rule not to admit _any_ members till we have at least _three_ candidates whose incomes are over ten thousand a year!" _author._ "oh, _that's_ what you meant, is it? let's hear some more of your rules." _jones._ "another is, that no one, who has been convicted seven times of forgery, is admissible." _author._ "and here, again, i suppose you don't mean to assert there _are_ any such convicts in existence?" _jones._ "why, that's exactly what i _do_ mean to assert! don't you know that a universal negative _asserts_ the existence of its subject? _of course_ we didn't make that rule till we had satisfied ourselves that there are several such convicts now living." the reader can now decide for himself how far this _second_ conceivable view would fit in with the facts of life. he will, i think, agree with me that jones' view, of the 'existential import' of propositions, would lead to some inconvenience. thirdly, let us suppose that neither _i_ nor _e_ "asserts". now the supposition that the two propositions, "some x are y" and "no x are not-y", do _not_ "assert", necessarily involves the supposition that "all x are y" does _not_ "assert", since it would be absurd to suppose that they assert, when combined, more than they do when taken separately. hence the _third_ (and last) of the conceivable views is that neither _i_, nor _e_, nor _a_, "asserts". the advocates of this third view would interpret the proposition "some x are y" to mean "if there _were_ any x in existence, some of them _would_ be y"; and so with _e_ and _a_. it admits of proof that this view, as regards _a_, conflicts with the accepted facts of logic. let us take the syllogism _darapti_, which is universally accepted as valid. its form is "all m are x; all m are y. .'. some y are x". pg this they would interpret as follows:-- "if there were any m in existence, all of them would be x; if there were any m in existence, all of them would be y. .'. if there were any y in existence, some of them would be x". that this conclusion does _not_ follow has been so briefly and clearly explained by mr. keynes (in his "formal logic", dated , pp. , ), that i prefer to quote his words:-- "_let no proposition imply the existence either of its subject or of its predicate._ "take, as an example, a syllogism in _darapti_:-- '_all m is p_, _all m is s_, _.'. some s is p_.' "taking s, m, p, as the minor, middle, and major terms respectively, the conclusion will imply that, if there is an s, there is some p. will the premisses also imply this? if so, then the syllogism is valid; but not otherwise. "the conclusion implies that if s exists p exists; but, consistently with the premisses, s may be existent while m and p are both non-existent. an implication is, therefore, contained in the conclusion, which is not justified by the premisses." this seems to _me_ entirely clear and convincing. still, "to make sicker", i may as well throw the above (_soi-disant_) syllogism into a concrete form, which will be within the grasp of even a _non_-logical reader. let us suppose that a boys' school has been set up, with the following system of rules:-- "all boys in the first (the highest) class are to do french, greek, and latin. all in the second class are to do greek only. all in the third class are to do latin only." suppose also that there _are_ boys in the third class, and in the second; but that no boy has yet risen into the first. it is evident that there are no boys in the school doing french: still we know, by the rules, what would happen if there _were_ any. pg we are authorised, then, by the _data_, to assert the following two propositions:-- "if there were any boys doing french, all of them would be doing greek; if there were any boys doing french, all of them would be doing latin." and the conclusion, according to "the logicians" would be "if there were any boys doing latin, some of them would be doing greek." here, then, we have two _true_ premisses and a _false_ conclusion (since we know that there _are_ boys doing latin, and that _none_ of them are doing greek). hence the argument is _invalid_. similarly it may be shown that this "non-existential" interpretation destroys the validity of _disamis_, _datisi_, _felapton_, and _fresison_. some of "the logicians" will, no doubt, be ready to reply "but we are not _aldrichians_! why should _we_ be responsible for the validity of the syllogisms of so antiquated an author as aldrich?" very good. then, for the _special_ benefit of these "friends" of mine (with what ominous emphasis that name is sometimes used! "i must have a private interview with _you_, my young _friend_," says the bland dr. birch, "in my library, at a.m. tomorrow. and you will please to be _punctual_!"), for their _special_ benefit, i say, i will produce _another_ charge against this "non-existential" interpretation. it actually invalidates the ordinary process of "conversion", as applied to proposition in '_i_'. _every_ logician, aldrichian or otherwise, accepts it as an established fact that "some x are y" may be legitimately converted into "some y are x." but is it equally clear that the proposition "if there _were_ any x, some of them _would_ be y" may be legitimately converted into "if there _were_ any y, some of them would be x"? i trow not. the example i have already used----of a boys' school with a non-existent first class----will serve admirably to illustrate this new flaw in the theory of "the logicians." pg let us suppose that there is yet _another_ rule in this school, viz. "in each class, at the end of the term, the head boy and the second boy shall receive prizes." this rule entirely authorises us to assert (in the sense in which "the logicians" would use the words) "some boys in the first class will receive prizes", for this simply means (according to them) "if there _were_ any boys in the first class, some of them _would_ receive prizes." now the converse of this proposition is, of course, "some boys, who will receive prizes, are in the first class", which means (according to "the logicians") "if there _were_ any boys about to receive prizes, some of them _would_ be in the first class" (which class we know to be _empty_). of this pair of converse propositions, the first is undoubtedly _true_: the second, _as_ undoubtedly, _false_. it is always sad to see a batsman knock down his own wicket: one pities him, as a man and a brother, but, as a _cricketer_, one can but pronounce him "out!" we see, then, that, among all the conceivable views we have here considered, there are only _two_ which can _logically_ be held, viz. _i_ and _a_ "assert", but _e_ does not. _e_ and _a_ "assert", but _i_ does not. the _second_ of these i have shown to involve great practical inconvenience. the _first_ is the one adopted in this book. (see p. .) some further remarks on this subject will be found in note (b), at p. . § . _the use of "is-not" (or "are-not") as a copula._ is it better to say "john _is-not_ in-the-house" or "john _is_ not-in-the-house"? "some of my acquaintances _are-not_ men-i-should-like-to-be-seen-with" or "some of my acquaintances _are_ men-i-should-_not_-like-to-be-seen-with"? that is the sort of question we have now to discuss. pg this is no question of logical right and wrong: it is merely a matter of _taste_, since the two forms mean exactly the same thing. and here, again, "the logicians" seem to me to take much too humble a position. when they are putting the final touches to the grouping of their proposition, just before the curtain goes up, and when the copula----always a rather fussy 'heavy father', asks them "am _i_ to have the 'not', or will you tack it on to the predicate?" they are much too ready to answer, like the subtle cab-driver, "leave it to _you_, sir!" the result seems to be, that the grasping copula constantly gets a "not" that had better have been merged in the predicate, and that propositions are differentiated which had better have been recognised as precisely similar. surely it is simpler to treat "some men are jews" and "some men are gentiles" as being both of them, _affirmative_ propositions, instead of translating the latter into "some men are-not jews", and regarding it as a _negative_ propositions? the fact is, "the logicians" have somehow acquired a perfectly _morbid_ dread of negative attributes, which makes them shut their eyes, like frightened children, when they come across such terrible propositions as "all not-x are y"; and thus they exclude from their system many very useful forms of syllogisms. under the influence of this unreasoning terror, they plead that, in dichotomy by contradiction, the _negative_ part is too large to deal with, so that it is better to regard each thing as either included in, or excluded from, the _positive_ part. i see no force in this plea: and the facts often go the other way. as a personal question, dear reader, if _you_ were to group your acquaintances into the two classes, men that you _would_ like to be seen with, and men that you would _not_ like to be seen with, do you think the latter group would be so _very_ much the larger of the two? for the purposes of symbolic logic, it is so _much_ the most convenient plan to regard the two sub-divisions, produced by dichotomy, on the _same_ footing, and to say, of any thing, either that it "is" in the one, or that it "is" in the other, that i do not think any reader of this book is likely to demur to my adopting that course. pg § . _the theory that "two negative premisses prove nothing"._ this i consider to be _another_ craze of "the logicians", fully as morbid as their dread of a negative attribute. it is, perhaps, best refuted by the method of _instantia contraria_. take the following pairs of premisses:-- "none of my boys are conceited; none of my girls are greedy". "none of my boys are clever; none but a clever boy could solve this problem". "none of my boys are learned; some of my boys are not choristers". (this last proposition is, in _my_ system, an _affirmative_ one, since i should read it "are not-choristers"; but, in dealing with "the logicians," i may fairly treat it as a _negative_ one, since _they_ would read it "are-not choristers".) if you, dear reader, declare, after full consideration of these pairs of premisses, that you cannot deduce a conclusion from _any_ of them----why, all i can say is that, like the duke in patience, you "will have to be contented with our heart-felt sympathy"! [see note (c), p. .] § . _euler's method of diagrams._ diagrams seem to have been used, at first, to represent _propositions_ only. in euler's well-known circles, each was supposed to contain a class, and the diagram consisted of two circles, which exhibited the relations, as to inclusion and exclusion, existing between the two classes. _____ _/ ___ \_ / / y \ \ | \___/ | \_ x _/ \_____/ thus, the diagram, here given, exhibits the two classes, whose respective attributes are x and y, as so related to each other that the following propositions are all simultaneously true:--"all x are y", "no x are not-y", "some x are y", "some y are not-x", "some not-y are not-x", and, of course, the converses of the last four. pg _____ _/ ___ \_ / / y \ \ | \___/ | \_ x _/ \_____/ similarly, with this diagram, the following propositions are true:--"all y are x", "no y are not-x", "some y are x", "some x are not-y", "some not-x are not-y", and, of course, the converses of the last four. _____ _____ _/ \_ _/ \_ / \ / \ | x | | y | \_ _/ \_ _/ \_____/ \_____/ similarly, with this diagram, the following are true:--"all x are not-y", "all y are not-x", "no x are y", "some x are not-y", "some y are not-x", "some not-x are not-y", and the converses of the last four. _____ _____ _/ \_/ \_ / / \ \ | x | | y | \_ \_/ _/ \_____/ \_____/ similarly, with this diagram, the following are true:--"some x are y", "some x are not-y", "some not-x are y", "some not-x are not-y", and of course, their four converses. note that _all_ euler's diagrams assert "some not-x are not-y." apparently it never occured to him that it might _sometimes_ fail to be true! now, to represent "all x are y", the _first_ of these diagrams would suffice. similarly, to represent "no x are y", the _third_ would suffice. but to represent any _particular_ proposition, at least _three_ diagrams would be needed (in order to include all the possible cases), and, for "some not-x are not-y", all the _four_. § . _venn's method of diagrams._ let us represent "not-x" by "x'". mr. venn's method of diagrams is a great advance on the above method. he uses the last of the above diagrams to represent _any_ desired relation between x and y, by simply shading a compartment known to be _empty_, and placing a + in one known to be _occupied_. thus, he would represent the three propositions "some x are y", "no x are y", and "all x are y", as follows:-- _____ _____ _/ \_/ \_ / / \ \ | | + | | \_ \_/ _/ \_____/ \_____/ _____ _____ _/ \_/ \_ / /#\ \ | |###| | \_ \#/ _/ \_____/ \_____/ _____ _____ _/#####\_/ \_ /#######/ \ \ |#######| + | | \#######\_/ _/ \#####/ \_____/ pg it will be seen that, of the _four_ classes, whose peculiar sets of attributes are xy, xy', x'y, and x'y', only _three_ are here provided with closed compartments, while the _fourth_ is allowed the rest of the infinite plane to range about in! this arrangement would involve us in very serious trouble, if we ever attempted to represent "no x' are y'." mr. venn _once_ (at p. ) encounters this awful task; but evades it, in a quite masterly fashion, by the simple foot-note "we have not troubled to shade the outside of this diagram"! to represent _two_ propositions (containing a common term) _together_, a _three_-letter diagram is needed. this is the one used by mr. venn. _____ _/ \_ _/___ x ___\_ _/| \_/ |\_ / \_ / \ _/ \ | \|___|/ | \_ m \_/ y _/ \_____/ \_____/ here, again, we have only _seven_ closed compartments, to accommodate the _eight_ classes whose peculiar sets of attributes are xym, xym', &c. "with four terms in request," mr. venn says, "the most simple and symmetrical diagram seems to me that produced by making four ellipses intersect one another in the desired manner". this, however, provides only _fifteen_ closed compartments. b ____ ____ c / \ / \ a ___/___ \/ ___\___ d / \ \ /\ / / \ / \ \/ \/ / \ \ \ /\ /\ / / \ \/ \/ \/ / \ /\ /\ /\ / \ \ \/ \/ / / \ \/\ /\/ / \ /\_\/_/\ / \__\______/__/ for _five_ letters, "the simplest diagram i can suggest," mr. venn says, "is one like this (the small ellipse in the centre is to be regarded as a portion of the _outside_ of c; i.e. its four component portions are inside b and d but are no part of c). it must be admitted that such a diagram is not quite so simple to draw as one might wish it to be; but then consider what the alternative is of one undertakes to deal with five terms and all their combinations--nothing short of the disagreeable task of writing out, or in some way putting before us, all the combinations involved." b c d ______ ____ ______ ______/_ \/ \/ _\______ a / / \ /\ /\ / \ \ e / / \ / \ / \ / \ \ / \ \/ \/ \/ / \ / \ /\ /\ /\ / \ / \ / \ / \ / \ / \ | \/ \/ __ \/ \/ | | /\ /\/ \/\ /\ | | / \ / /\ /\ \ / \ | | | \/ | \/ | \/ | | | | /\ | /\ | /\ | | | \ / \ \/ \/ / \ / | \ \/ \/\__/\/ \/ / \ /\ /\ /\ /\ / \ / \ / \ / \ / \ / \ / \/ \/ \/ \ / \ \ /\ /\ /\ / / \ \ / \ / \ / \ / / \ \/ \/ \/ \/ / \ /\____/\____/\____/\ / \ / \ / \| |/ \____________________/ pg this diagram gives us closed compartments. for _six_ letters, mr. venn suggests that we might use _two_ diagrams, like the above, one for the f-part, and the other for the not-f-part, of all the other combinations. "this", he says, "would give the desired subdivisions." this, however, would only give closed compartments, and _one_ infinite area, which the two classes, a'b'c'd'e'f and a'b'c'd'e'f', would have to share between them. beyond _six_ letters mr. venn does not go. § . _my method of diagrams._ my method of diagrams _resembles_ mr. venn's, in having separate compartments assigned to the various classes, and in marking these compartments as _occupied_ or as _empty_; but it _differs_ from his method, in assigning a _closed_ area to the _universe of discourse_, so that the class which, under mr. venn's liberal sway, has been ranging at will through infinite space, is suddenly dismayed to find itself "cabin'd, cribb'd, confined", in a limited cell like any other class! also i use _rectilinear_, instead of _curvilinear_, figures; and i mark an _occupied_ cell with a 'i' (meaning that there is at least _one_ thing in it), and an _empty_ cell with a 'o' (meaning that there is _no_ thing in it). for _two_ letters, i use this diagram, in which the north half is assigned to 'x', the south to 'not-x' (or 'x''), the west to y, and the east to y'. thus the n.w. cell contains the xy-class, the n.e. cell the xy'-class, and so on. ·-------· | | | |---|---| | | | ·-------· for _three_ letters, i subdivide these four cells, by drawing an _inner_ square, which i assign to m, the _outer_ border being assigned to m'. i thus get _eight_ cells that are needed to accommodate the eight classes, whose peculiar sets of attributes are xym, xym', &c. ·---------------· | | | | ·---|---· | | | | | | |---|---|---|---| | | | | | | ·---|---· | | | | ·---------------· this last diagram is the most complex that i use in the _elementary_ part of my 'symbolic logic.' but i may as well take this opportunity of describing the more complex ones which will appear in part ii. pg for _four_ letters (which i call a, b, c, d) i use this diagram; assigning the north half to a (and of course the _rest_ of the diagram to a'), the west half to b, the horizontal oblong to c, and the upright oblong to d. we have now got cells. ·---------------------· | | | | ·---|---· | | | | | | | ·---|---|---|---· | | | | | | | | |--|---|---|---|---|--| | | | | | | | | ·---|---|---|---· | | | | | | | ·---|---· | | | | ·---------------------· for _five_ letters (adding e) i subdivide the cells of the previous diagram by _oblique_ partitions, assigning all the _upper_ portions to e, and all the _lower_ portions to e'. here, i admit, we lose the advantage of having the e-class all _together_, "in a ring-fence", like the other classes. still, it is very easy to find; and the operation, of erasing it, is nearly as easy as that of erasing any other class. we have now got cells. ·---------------------· | / | / | | / ·---|---· / | | / | / | / | / | | ·---|---|---|---· | | | / | / | / | / | | |--|---|---|---|---|--| | | / | / | / | / | | | ·---|---|---|---· | | / | / | / | / | | / ·---|---· / | | / | / | ·---------------------· for _six_ letters (adding h, as i avoid _tailed_ letters) i substitute upright crosses for the oblique partitions, assigning the portions, into which each of the cells is thus divided, to the four classes eh, eh', e'h, e'h'. we have now got cells. #=============================# h | h | h h | #=====h=====# | h h | h | h | h | h h-----|--h--|--h--|--h--|-----h h | h | h | h | h h #=====h=====h=====h=====# h h h | h | h | h | h h h h--|--h--|--h--|--h--|--h h h h | h | h | h | h h h==h=====h=====h=====h=====h--h h h | h | h | h | h h h h--|--h--|--h--|--h--|--h h h h | h | h | h | h h h #=====h=====h=====h=====# h h | h | h | h | h h-----|--h--|--h--|--h--|-----h h | h | h | h | h h | #=====h=====# | h h | h | h #=============================# pg for _seven_ letters (adding k) i add, to each upright cross, a little inner square. all these little squares are assigned to the k-class, and all outside them to the k'-class; so that little cells (into which each of the cells is divided) are respectively assigned to the classes ehk, ehk', &c. we have now got cells. #=====================================================# h | h | h h | #===========h===========# | h h | h | h | h | h h ·--|--· h ·--|--· h ·--|--· h ·--|--· h h | | | h | | | h | | | h | | | h h-----|--|--|--h--|--|--|--h--|--|--|--h--|--|--|-----h h | | | h | | | h | | | h | | | h h ·--|--· h ·--|--· h ·--|--· h ·--|--· h h | h | h | h | h h #===========h===========h===========h===========# h h h | h | h | h | h h h h ·--|--· h ·--|--· h ·--|--· h ·--|--· h h h h | | | h | | | h | | | h | | | h h h h--|--|--|--h--|--|--|--h--|--|--|--h--|--|--|--h h h h | | | h | | | h | | | h | | | h h h h ·--|--· h ·--|--· h ·--|--· h ·--|--· h h h h | h | h | h | h h h==h===========h===========h===========h===========h==h h h | h | h | h | h h h h ·--|--· h ·--|--· h ·--|--· h ·--|--· h h h h | | | h | | | h | | | h | | | h h h h--|--|--|--h--|--|--|--h--|--|--|--h--|--|--|--h h h h | | | h | | | h | | | h | | | h h h h ·--|--· h ·--|--· h ·--|--· h ·--|--· h h h h | h | h | h | h h h #===========h===========h===========h===========# h h | h | h | h | h h ·--|--· h ·--|--· h ·--|--· h ·--|--· h h | | | h | | | h | | | h | | | h h-----|--|--|--h--|--|--|--h--|--|--|--h--|--|--|-----h h | | | h | | | h | | | h | | | h h ·--|--· h ·--|--· h ·--|--· h ·--|--· h h | h | h | h | h h | #===========h===========# | h h | h | h ·=====================================================# for _eight_ letters (adding l) i place, in each of the cells, a _lattice_, which is a reduced copy of the whole diagram; and, just as the large cells of the whole diagram are assigned to the classes abcd, abcd', &c., so the little cells of each lattice are assigned to the classes ehkl, ehkl', &c. thus, the lattice in the n.w. corner serves to accommodate the classes abc'd'ehkl, abc'd'eh'kl', &c. this octoliteral diagram (see next page) contains cells. for _nine_ letters, i place octoliteral diagrams side by side, assigning one of them to m, and the other to m'. we have now got cells. pg #=====================================================================# h | h | h h | #===============h===============# | h h | h | h | h | h h ·--|--· h ·--|--· h ·--|--· h ·--|--· h h | | | h | | | h | | | h | | | h h ·--|--|--|--· h ·--|--|--|--· h ·--|--|--|--· h ·--|--|--|--· h h | | | | | h | | | | | h | | | | | h | | | | | h h----|--|--|--|--|-h-|--|--|--|--|-h-|--|--|--|--|-h-|--|--|--|--|----h h | | | | | h | | | | | h | | | | | h | | | | | h h ·--|--|--|--· h ·--|--|--|--· h ·--|--|--|--· h ·--|--|--|--· h h | | | h | | | h | | | h | | | h h ·--|--· h ·--|--· h ·--|--· h ·--|--· h h | h | h | h | h h #===============h===============h===============h===============# h h h | h | h | h | h h h h ·--|--· h ·--|--· h ·--|--· h ·--|--· h h h h | | | h | | | h | | | h | | | h h h h ·--|--|--|--· h ·--|--|--|--· h ·--|--|--|--· h ·--|--|--|--· h h h h | | | | | h | | | | | h | | | | | h | | | | | h h h h-|--|--|--|--|-h-|--|--|--|--|-h-|--|--|--|--|-h-|--|--|--|--|-h h h h | | | | | h | | | | | h | | | | | h | | | | | h h h h ·--|--|--|--· h ·--|--|--|--· h ·--|--|--|--· h ·--|--|--|--· h h h h | | | h | | | h | | | h | | | h h h h ·--|--· h ·--|--· h ·--|--· h ·--|--· h h h h | h | h | h | h h h==h===============h===============h===============h===============h==h h h | h | h | h | h h h h ·--|--· h ·--|--· h ·--|--· h ·--|--· h h h h | | | h | | | h | | | h | | | h h h h ·--|--|--|--· h ·--|--|--|--· h ·--|--|--|--· h ·--|--|--|--· h h h h | | | | | h | | | | | h | | | | | h | | | | | h h h h-|--|--|--|--|-h-|--|--|--|--|-h-|--|--|--|--|-h-|--|--|--|--|-h h h h | | | | | h | | | | | h | | | | | h | | | | | h h h h ·--|--|--|--· h ·--|--|--|--· h ·--|--|--|--· h ·--|--|--|--· h h h h | | | h | | | h | | | h | | | h h h h ·--|--· h ·--|--· h ·--|--· h ·--|--· h h h h | h | h | h | h h h #===============h===============h===============h===============# h h | h | h | h | h h ·--|--· h ·--|--· h ·--|--· h ·--|--· h h | | | h | | | h | | | h | | | h h ·--|--|--|--· h ·--|--|--|--· h ·--|--|--|--· h ·--|--|--|--· h h | | | | | h | | | | | h | | | | | h | | | | | h h----|--|--|--|--|-h-|--|--|--|--|-h-|--|--|--|--|-h-|--|--|--|--|----h h | | | | | h | | | | | h | | | | | h | | | | | h h ·--|--|--|--· h ·--|--|--|--· h ·--|--|--|--· h ·--|--|--|--· h h | | | h | | | h | | | h | | | h h ·--|--· h ·--|--· h ·--|--· h ·--|--· h h | h | h | h | h h | #===============h===============# | h h | h | h #=====================================================================# finally, for _ten_ letters, i arrange octoliteral diagrams, like the above, in a square, assigning them to the classes mn, mn', m'n, m'n'. we have now got cells. § . _solution of a syllogism by various methods._ the best way, i think, to exhibit the differences between these various methods of solving syllogisms, will be to take a concrete example, and solve it by each method in turn. let us take, as our example, no. (see p. ). "no philosophers are conceited; some conceited persons are not gamblers. .'. some persons, who are not gamblers, are not philosophers." pg ( ) _solution by ordinary method._ these premisses, as they stand, will give no conclusion, as they are both negative. if by 'permutation' or 'obversion', we write the minor premiss thus, 'some conceited persons are not-gamblers,' we can get a conclusion in _fresison_, viz. "no philosophers are conceited; some conceited persons are not-gamblers. .'. some not-gamblers are not philosophers" this can be proved by reduction to _ferio_, thus:-- "no conceited persons are philosophers; some not-gamblers are conceited. .'. some not-gamblers are not philosophers". the validity of _ferio_ follows directly from the axiom '_de omni et nullo_'. ( ) _symbolic representation._ before proceeding to discuss other methods of solution, it is necessary to translate our syllogism into an _abstract_ form. let us take "persons" as our 'universe of discourse'; and let x = "philosophers", m = "conceited", and y = "gamblers." then the syllogism may be written thus:-- "no x are m; some m are y'. .'. some y' are x'." ( ) _solution by euler's method of diagrams._ the major premiss requires only _one_ diagram, viz. _____ _____ _/ \_ _/ \_ / \ / \ | x | | m | \_ _/ \_ _/ \_____/ \_____/ pg the minor requires _three_, viz. _____ _____ _/ \_ _/ \_ / \ / \ | y | | m | \_ _/ \_ _/ \_____/ \_____/ _____ _____ _/ \_/ \_ / / \ \ | y | | m | \_ \_/ _/ \_____/ \_____/ _____ _/ ___ \_ / / y \ \ | \___/ | \_ m _/ \_____/ the combination of major and minor, in every possible way requires _nine_, viz. figs. and give _____ _____ _____ _/ \_ _/ \_ _/ \_ / \ / \ / \ | x | | y | | m | \_ _/ \_ _/ \_ _/ \_____/ \_____/ \_____/ _____ _____ _____ _/ \_/ \_ _/ \_ / / \ \ / \ | x | | y | | m | \_ \_/ _/ \_ _/ \_____/ \_____/ \_____/ _____ _____ _/ \_ _/ \_ / \ / \ | xy | | m | \_ _/ \_ _/ \_____/ \_____/ _____ _____ _/ ___ \_ _/ \_ / / x \ \ / \ | \___/ | | m | \_ y _/ \_ _/ \_____/ \_____/ _____ _____ _/ ___ \_ _/ \_ / / y \ \ / \ | \___/ | | m | \_ x _/ \_ _/ \_____/ \_____/ figs. and give _____ _____ _____ _/ \_ _/ \_/ \_ / \ / / \ \ | x | | y | | m | \_ _/ \_ \_/ _/ \_____/ \_____/ \_____/ _____ _____ _____ _/ \_/ \_/ \_ / / \ / \ \ | x | | y | | m | \_ \_/ \_/ _/ \_____/ \_____/ \_____/ _____ _____ _/___ \_/ \_ / / x \ / \ \ | \___/| | m | \_ y \_/ _/ \_____/ \_____/ figs. and give _____ _____ _/ \_ _/ ___ \_ / \ / / y \ \ | x | | \___/ | \_ _/ \_ m _/ \_____/ \_____/ from this group (figs. to ) we have, by disregarding m, to find the relation of x and y. on examination we find that figs. , , express the relation of entire mutual exclusion; that figs. , express partial inclusion and partial exclusion; that fig. expresses coincidence; that figs. , express entire inclusion of x in y; and that fig. expresses entire inclusion of y in x. pg we thus get five biliteral diagrams for x and y, viz. _____ _____ _/ \_ _/ \_ / \ / \ | x | | y | \_ _/ \_ _/ \_____/ \_____/ _____ _____ _/ \_/ \_ / / \ \ | x | | y | \_ \_/ _/ \_____/ \_____/ _____ _/ \_ / \ | xy | \_ _/ \_____/ _____ _/ ___ \_ / / x \ \ | \___/ | \_ y _/ \_____/ _____ _/ ___ \_ / / y \ \ | \___/ | \_ x _/ \_____/ where the only proposition, represented by them all, is "some not-y are not-x," i.e. "some persons, who are not gamblers, are not philosophers"----a result which euler would hardly have regarded as a _valuable_ one, since he seems to have assumed that a proposition of this form is _always_ true! ( ) _solution by venn's method of diagrams._ the following solution has been kindly supplied to me mr. venn himself. "the minor premiss declares that some of the constituents in my' must be saved: mark these constituents with a cross. _____ _/ + \_ _/___ ___\_ _/|##+#\_/ |\_ / \###/#\ _/ \ | + \|_#_|/ | \_ m \_/ y _/ \_____/ \_____/ the major declares that all xm must be destroyed; erase it. then, as some my' is to be saved, it must clearly be my'x'. that is, there must exist my'x'; or eliminating m, y'x'. in common phraseology, 'some y' are x',' or, 'some not-gamblers are not-philosophers.'" pg ( ) _solution by my method of diagrams._ the first premiss asserts that no xm exist: so we mark the xm-compartment as empty, by placing a 'o' in each of its cells. the second asserts that some my' exist: so we mark the my'-compartment as occupied, by placing a 'i' in its only available cell. ·---------------· | | | | ·---|---· | | |(o)|(o)| | |---|---|---|---| | | |(i)| | | ·---|---· | | | | ·---------------· the only information, that this gives us as to x and y, is that the x'y'-compartment is _occupied_, i.e. that some x'y' exist. hence "some x' are y'": i.e. "some persons, who are not philosophers, are not gamblers". ( ) _solution by my method of subscripts._ xm_{ } + my'_{ } ¶ x'y'_{ } i.e. "some persons, who are not philosophers, are not gamblers." § . _my method of treating syllogisms and sorites._ of all the strange things, that are to be met with in the ordinary text-books of formal logic, perhaps the strangest is the violent contrast one finds to exist between their ways of dealing with these two subjects. while they have elaborately discussed no less than _nineteen_ different forms of _syllogisms_----each with its own special and exasperating rules, while the whole constitute an almost useless machine, for practical purposes, many of the conclusions being incomplete, and many quite legitimate forms being ignored----they have limited _sorites_ to _two_ forms only, of childish simplicity; and these they have dignified with special _names_, apparently under the impression that no other possible forms existed! as to _syllogisms_, i find that their nineteen forms, with about a score of others which they have ignored, can all be arranged under _three_ forms, each with a very simple rule of its own; and the only question the reader has to settle, in working any one of the examples given at p. of this book, is "does it belong to fig. i., ii., or iii.?" pg as to _sorites_, the only two forms, recognised by the text-books, are the _aristotelian_, whose premisses are a series of propositions in a, so arranged that the predicate of each is the subject of the next, and the _goclenian_, whose premisses are the very same series, written backwards. goclenius, it seems, was the first who noticed the startling fact that it does not affect the force of a syllogism to invert the order of its premisses, and who applied this discovery to a sorites. if we assume (as surely we may?) that he is the _same_ man as that transcendent genius who first noticed that times is the same thing as times , we may apply to him what somebody (edmund yates, i think it was) has said of tupper, viz., "here is a man who, beyond all others of his generation, has been favoured with glimpses of the obvious!" these puerile----not to say infantine----forms of a sorites i have, in this book, ignored from the very first, and have not only admitted freely propositions in _e_, but have purposely stated the premisses in random order, leaving to the reader the useful task of arranging them, for himself, in an order which can be worked as a series of regular syllogisms. in doing this, he can begin with _any one_ of them he likes. i have tabulated, for curiosity, the various orders in which the premisses of the aristotelian sorites . all a are b; . all b are c; . all c are d; . all d are e; . all e are h. .'. all a are h. may be syllogistically arranged, and i find there are no less than _sixteen_ such orders, viz., , , , , , , , , , , , , , , , . of these the _first_ and the _last_ have been dignified with names; but the other _fourteen_----first enumerated by an obscure writer on logic, towards the end of the nineteenth century----remain without a name! pg § . _some account of parts ii, iii._ in part ii. will be found some of the matters mentioned in this appendix, viz., the "existential import" of propositions, the use of a _negative_ copula, and the theory that "two negative premisses prove nothing." i shall also extend the range of syllogisms, by introducing propositions containing alternatives (such as "not-all x are y"), propositions containing or more terms (such as "all ab are c", which, taken along with "some bc' are d", would prove "some d are a'"), &c. i shall also discuss sorites containing entities, and the _very_ puzzling subjects of hypotheticals and dilemmas. i hope, in the course of part ii., to go over all the ground usually traversed in the text-books used in our schools and universities, and to enable my readers to solve problems of the same kind as, and far harder than, those that are at present set in their examinations. in part iii. i hope to deal with many curious and out-of-the-way subjects, some of which are not even alluded to in any of the treatises i have met with. in this part will be found such matters as the analysis of propositions into their elements (let the reader, who has never gone into this branch of the subject, try to make out for himself what _additional_ proposition would be needed to convert "some a are b" into "some a are bc"), the treatment of numerical and geometrical problems, the construction of problems, and the solution of syllogisms and sorites containing propositions more complex than any that i have used in part ii. i will conclude with eight problems, as a taste of what is coming in part ii. i shall be very glad to receive, from any reader, who thinks he has solved any one of them (more especially if he has done so _without_ using any method of symbols), what he conceives to be its complete conclusion. it may be well to explain what i mean by the _complete_ conclusion of a syllogism or a sorites. i distinguish their terms as being of two kinds----those which _can_ be eliminated (e.g. the middle term of a syllogism), which i call the "eliminands," and those which _cannot_, which i call the "retinends"; and i do not call the conclusion _complete_, unless it states _all_ the relations among the retinends only, which can be deduced from the premisses. pg . all the boys, in a certain school, sit together in one large room every evening. they are of no less than _five_ nationalities----english, scotch, welsh, irish, and german. one of the monitors (who is a great reader of wilkie collins' novels) is very observant, and takes ms. notes of almost everything that happens, with the view of being a good sensational witness, in case any conspiracy to commit a murder should be on foot. the following are some of his notes:-- ( ) whenever some of the english boys are singing "rule britannia", and some not, some of the monitors are wide-awake; ( ) whenever some of the scotch are dancing reels, and some of the irish fighting, some of the welsh are eating toasted cheese; ( ) whenever all the germans are playing chess, some of the eleven are _not_ oiling their bats; ( ) whenever some of the monitors are asleep, and some not, some of the irish are fighting; ( ) whenever some of the germans are playing chess, and none of the scotch are dancing reels, some of the welsh are _not_ eating toasted cheese; ( ) whenever some of the scotch are _not_ dancing reels, and some of the irish _not_ fighting, some of the germans are playing chess; ( ) whenever some of the monitors are awake, and some of the welsh are eating toasted cheese, none of the scotch are dancing reels; ( ) whenever some of the germans are _not_ playing chess, and some of the welsh are _not_ eating toasted cheese, none of the irish are fighting; pg ( ) whenever all the english are singing "rule britannia," and some of the scotch are _not_ dancing reels, none of the germans are playing chess; ( ) whenever some of the english are singing "rule britannia", and some of the monitors are asleep, some of the irish are _not_ fighting; ( ) whenever some of the monitors are awake, and some of the eleven are _not_ oiling their bats, some of the scotch are dancing reels; ( ) whenever some of the english are singing "rule britannia", and some of the scotch are _not_ dancing reels, * * * * here the ms. breaks off suddenly. the problem is to complete the sentence, if possible. [n.b. in solving this problem, it is necessary to remember that the proposition "all x are y" is a _double_ proposition, and is equivalent to "some x are y, and none are y'." see p. .] . ( ) a logician, who eats pork-chops for supper, will probably lose money; ( ) a gambler, whose appetite is not ravenous, will probably lose money; ( ) a man who is depressed, having lost money and being likely to lose more, always rises at a.m.; ( ) a man, who neither gambles nor eats pork-chops for supper, is sure to have a ravenous appetite; ( ) a lively man, who goes to bed before a.m., had better take to cab-driving; ( ) a man with a ravenous appetite, who has not lost money and does not rise at a.m., always eats pork-chops for supper; ( ) a logician, who is in danger of losing money, had better take to cab-driving; ( ) an earnest gambler, who is depressed though he has not lost money, is in no danger of losing any; ( ) a man, who does not gamble, and whose appetite is not ravenous, is always lively; pg ( ) a lively logician, who is really in earnest, is in no danger of losing money; ( ) a man with a ravenous appetite has no need to take to cab-driving, if he is really in earnest; ( ) a gambler, who is depressed though in no danger of losing money, sits up till a.m. ( ) a man, who has lost money and does not eat pork-chops for supper, had better take to cab-driving, unless he gets up at a.m. ( ) a gambler, who goes to bed before a.m., need not take to cab-driving, unless he has a ravenous appetite; ( ) a man with a ravenous appetite, who is depressed though in no danger of losing, is a gambler. univ. "men"; a = earnest; b = eating pork-chops for supper; c = gamblers; d = getting up at ; e = having lost money; h = having a ravenous appetite; k = likely to lose money; l = lively; m = logicians; n = men who had better take to cab-driving; r = sitting up till . [n.b. in this problem, clauses, beginning with "though", are intended to be treated as _essential_ parts of the propositions in which they occur, just as if they had begun with "and".] . ( ) when the day is fine, i tell froggy "you're quite the dandy, old chap!"; ( ) whenever i let froggy forget that £ he owes me, and he begins to strut about like a peacock, his mother declares "he shall _not_ go out a-wooing!"; ( ) now that froggy's hair is out of curl, he has put away his gorgeous waistcoat; ( ) whenever i go out on the roof to enjoy a quiet cigar, i'm sure to discover that my purse is empty; ( ) when my tailor calls with his little bill, and i remind froggy of that £ he owes me, he does _not_ grin like a hyæna; pg ( ) when it is very hot, the thermometer is high; ( ) when the day is fine, and i'm not in the humour for a cigar, and froggy is grinning like a hyæna, i never venture to hint that he's quite the dandy; ( ) when my tailor calls with his little bill and finds me with an empty purse, i remind froggy of that £ he owes me; ( ) my railway-shares are going up like anything! ( ) when my purse is empty, and when, noticing that froggy has got his gorgeous waistcoat on, i venture to remind him of that £ he owes me, things are apt to get rather warm; ( ) now that it looks like rain, and froggy is grinning like a hyæna, i can do without my cigar; ( ) when the thermometer is high, you need not trouble yourself to take an umbrella; ( ) when froggy has his gorgeous waistcoat on, but is _not_ strutting about like a peacock, i betake myself to a quiet cigar; ( ) when i tell froggy that he's quite the dandy, he grins like a hyæna; ( ) when my purse is tolerably full, and froggy's hair is one mass of curls, and when he is _not_ strutting about like a peacock, i go out on the roof; ( ) when my railway-shares are going up, and when it is chilly and looks like rain, i have a quiet cigar; ( ) when froggy's mother lets him go a-wooing, he seems nearly mad with joy, and puts on a waistcoat that is gorgeous beyond words; ( ) when it is going to rain, and i am having a quiet cigar, and froggy is _not_ intending to go a-wooing, you had better take an umbrella; ( ) when my railway-shares are going up, and froggy seems nearly mad with joy, _that_ is the time my tailor always chooses for calling with his little bill; ( ) when the day is cool and the thermometer low, and i say nothing to froggy about his being quite the dandy, and there's not the ghost of a grin on his face, i haven't the heart for my cigar! pg . ( ) any one, fit to be an m.p., who is not always speaking, is a public benefactor; ( ) clear-headed people, who express themselves well, have had a good education; ( ) a woman, who deserves praise, is one who can keep a secret; ( ) people, who benefit the public, but do not use their influence for good purpose, are not fit to go into parliament; ( ) people, who are worth their weight in gold and who deserve praise, are always unassuming; ( ) public benefactors, who use their influence for good objects, deserve praise; ( ) people, who are unpopular and not worth their weight in gold, never can keep a secret; ( ) people, who can talk for ever and are fit to be members of parliament, deserve praise; ( ) any one, who can keep a secret and who is unassuming, is a never-to-be-forgotten public benefactor; ( ) a woman, who benefits the public, is always popular; ( ) people, who are worth their weight in gold, who never leave off talking, and whom it is impossible to forget, are just the people whose photographs are in all the shop-windows; ( ) an ill-educated woman, who is not clear-headed, is not fit to go into parliament; ( ) any one, who can keep a secret and is not for ever talking, is sure to be unpopular; ( ) a clear-headed person, who has influence and uses it for good objects, is a public benefactor; ( ) a public benefactor, who is unassuming, is not the sort of person whose photograph is in every shop-window; ( ) people, who can keep a secret and who use their influence for good purposes, are worth their weight in gold; ( ) a person, who has no power of expression and who cannot influence others, is certainly not a _woman_; pg ( ) people, who are popular and worthy of praise, either are public benefactors or else are unassuming. univ. "persons"; a = able to keep a secret; b = clear-headed; c = constantly talking; d = deserving praise; e = exhibited in shop-windows; h = expressing oneself well; k = fit to be an m.p.; l = influential; m = never-to-be-forgotten; n = popular; r = public benefactors; s = unassuming; t = using one's influence for good objects; v = well-educated; w = women; z = worth one's weight in gold. . six friends, and their six wives, are staying in the same hotel; and they all walk out daily, in parties of various size and composition. to ensure variety in these daily walks, they have agree to observe the following rules:-- ( ) if acres is with (i.e. is in the same party with) his wife, and barry with his, and eden with mrs. hall, cole must be with mrs. dix; ( ) if acres is with his wife, and hall with his, and barry with mrs. cole, dix must _not_ be with mrs. eden; ( ) if cole and dix and their wives are all in the same party, and acres _not_ with mrs. barry, eden must _not_ be with mrs. hall; ( ) if acres is with his wife, and dix with his, and barry _not_ with mrs. cole, eden must be with mrs. hall; ( ) if eden is with his wife, and hall with his, and cole with mrs. dix, acres must _not_ be with mrs. barry; ( ) if barry and cole and their wives are all in the same party, and eden _not_ with mrs. hall, dix must be with mrs. eden. the problem is to prove that there must be, every day, at least _one_ married couple who are not in the same party. pg . after the six friends, named in problem , had returned from their tour, three of them, barry, cole, and dix, agreed, with two other friends of theirs, lang and mill, that the five should meet, every day, at a certain _table d'hôte_. remembering how much amusement they had derived from their code of rules for walking-parties, they devised the following rules to be observed whenever beef appeared on the table:-- ( ) if barry takes salt, then either cole or lang takes _one_ only of the two condiments, salt and mustard: if he takes mustard, then either dix takes neither condiment, or mill takes both. ( ) if cole takes salt, then either barry takes only _one_ condiment, or mill takes neither: if he takes mustard, then either dix or lang takes both. ( ) if dix takes salt, then either barry takes neither condiment or cole take both: if he takes mustard, then either lang or mill takes neither. ( ) if lang takes salt, then barry or dix takes only _one_ condiment: if he takes mustard, then either cole or mill takes neither. ( ) if mill takes salt, then either barry or lang takes both condiments: if he takes mustard, then either cole or dix takes only _one_. the problem is to discover whether these rules are _compatible_; and, if so, what arrangements are possible. [n.b. in this problem, it is assumed that the phrase "if barry takes salt" allows of _two_ possible cases, viz. ( ) "he takes salt _only_"; ( ) "he takes _both_ condiments". and so with all similar phrases. it is also assumed that the phrase "either cole or lang takes _one_ only of the two condiments" allows _three_ possible cases, viz. ( ) "cole takes _one_ only, lang takes both or neither"; ( ) "cole takes both or neither, lang takes _one_ only"; ( ) "cole takes _one_ only, lang takes _one_ only". and so with all similar phrases. it is also assumed that every rule is to be understood as implying the words "and _vice versâ_." thus the first rule would imply the addition "and, if either cole or lang takes only _one_ condiment, then barry takes salt."] pg . ( ) brothers, who are much admired, are apt to be self-conscious; ( ) when two men of the same height are on opposite sides in politics, if one of them has his admirers, so also has the other; ( ) brothers, who avoid general society, look well when walking together; ( ) whenever you find two men, who differ in politics and in their views of society, and who are not both of them ugly, you may be sure that they look well when walking together; ( ) ugly men, who look well when walking together, are not both of them free from self-consciousness; ( ) brothers, who differs in politics, and are not both of them handsome, never give themselves airs; ( ) john declines to go into society, but never gives himself airs; ( ) brothers, who are apt to be self-conscious, though not _both_ of them handsome, usually dislike society; ( ) men of the same height, who do not give themselves airs, are free from self-consciousness; ( ) men, who agree on questions of art, though they differ in politics, and who are not both of them ugly, are always admired; ( ) men, who hold opposite views about art and are not admired, always give themselves airs; ( ) brothers of the same height always differ in politics; ( ) two handsome men, who are neither both of them admired nor both of them self-conscious, are no doubt of different heights; ( ) brothers, who are self-conscious, and do not both of them like society, never look well when walking together. [n.b. see note at end of problem .] pg . ( ) a man can always master his father; ( ) an inferior of a man's uncle owes that man money; ( ) the father of an enemy of a friend of a man owes that man nothing; ( ) a man is always persecuted by his son's creditors; ( ) an inferior of the master of a man's son is senior to that man; ( ) a grandson of a man's junior is not his nephew; ( ) a servant of an inferior of a friend of a man's enemy is never persecuted by that man; ( ) a friend of a superior of the master of a man's victim is that man's enemy; ( ) an enemy of a persecutor of a servant of a man's father is that man's friend. the problem is to deduce some fact about great-grandsons. [n.b. in this problem, it is assumed that all the men, here referred to, live in the same town, and that every pair of them are either "friends" or "enemies," that every pair are related as "senior and junior", "superior and inferior", and that certain pairs are related as "creditor and debtor", "father and son", "master and servant", "persecutor and victim", "uncle and nephew".] . "jack sprat could eat no fat: his wife could eat no lean: and so, between them both, they licked the platter clean." solve this as a sorites-problem, taking lines and as the conclusion to be proved. it is permitted to use, as premisses, not only all that is here _asserted_, but also all that we may reasonably understand to be _implied_. pg notes to appendix. (a) [see p. , line .] it may, perhaps, occur to the reader, who has studied formal logic that the argument, here applied to the propositions i and e, will apply equally well to the propositions i and a (since, in the ordinary text-books, the propositions "all xy are z" and "some xy are not z" are regarded as contradictories). hence it may appear to him that the argument might have been put as follows:-- "we now have i and a 'asserting.' hence, if the proposition 'all xy are z' be true, some things exist with the attributes x and y: i.e. 'some x are y.' "also we know that, if the proposition 'some xy are not-z' be true the same result follows. "but these two propositions are contradictories, so that one or other of them _must_ be true. hence this result is always true: i.e. the proposition 'some x are y' is _always_ true! "_quod est absurdum._ hence i _cannot_ assert." this matter will be discussed in part ii; but i may as well give here what seems to me to be an irresistable proof that this view (that _a_ and _i_ are contradictories), though adopted in the ordinary text-books, is untenable. the proof is as follows:-- with regard to the relationship existing between the class 'xy' and the two classes 'z' and 'not-z', there are _four_ conceivable states of things, viz. ( ) some xy are z, and some are not-z; ( ) " " none " ( ) no xy " some " ( ) " " none " of these four, no. ( ) is equivalent to "all xy are z", no. ( ) is equivalent to "all xy are not-z", and no. ( ) is equivalent to "no xy exist." now it is quite undeniable that, of these _four_ states of things, each is, _a priori_, _possible_, some _one must_ be true, and the other three _must_ be false. hence the contradictory to ( ) is "either ( ) or ( ) or ( ) is true." now the assertion "either ( ) or ( ) is true" is equivalent to "some xy are not-z"; and the assertion "( ) is true" is equivalent to "no xy exist." hence the contradictory to "all xy are z" may be expressed as the alternative proposition "either some xy are not-z, or no xy exist," but _not_ as the categorical proposition "some y are not-z." pg (b) [see p. , at end of section .] there are yet _other_ views current among "the logicians", as to the "existential import" of propositions, which have not been mentioned in this section. one is, that the proposition "some x are y" is to be interpreted, neither as "some x _exist_ and are y", nor yet as "if there _were_ any x in existence, some of them _would_ be y", but merely as "some x _can be_ y; i.e. the attributes x and y are _compatible_". on _this_ theory, there would be nothing offensive in my telling my friend jones "some of your brothers are swindlers"; since, if he indignantly retorted "what do you _mean_ by such insulting language, you scoundrel?", i should calmly reply "i merely mean that the thing is _conceivable_----that some of your brothers _might possibly_ be swindlers". but it may well be doubted whether such an explanation would _entirely_ appease the wrath of jones! another view is, that the proposition "all x are y" _sometimes_ implies the actual _existence_ of x, and _sometimes_ does _not_ imply it; and that we cannot tell, without having it in _concrete_ form, _which_ interpretation we are to give to it. _this_ view is, i think, strongly supported by common usage; and it will be fully discussed in part ii: but the difficulties, which it introduces, seem to me too formidable to be even alluded to in part i, which i am trying to make, as far as possible, easily intelligible to mere _beginners_. (c) [see p. , § .] the three conclusions are "no conceited child of mine is greedy"; "none of my boys could solve this problem"; "some unlearned boys are not choristers." pg index. § . _tables._ i. biliteral diagram. attributes of classes, and compartments, or cells, assigned to them ii. do. representation of uniliteral propositions of existence iii. do. representation of biliteral propositions of existence and of relation iv. triliteral diagram. attributes of classes, and compartments, or cells, assigned to them v. do. representation of particular and universal negative propositions, of existence and of relation, in terms of x and m vi. do. do., in terms of y and m vii. do. representation of universal affirmative propositions of relation, in terms of x and m viii. do. do. in terms of y and m ix. method of subscripts. formulæ and rules for syllogisms § . _words &c. explained._ 'abstract' proposition 'adjuncts' 'affirmative' proposition 'attributes' 'biliteral' diagram " proposition 'class' ½ classes, arbitrary limits of ½ " , subdivision of pg 'classification' ½ 'codivisional' classes 'complete' conclusion of a sorites 'conclusion' of a sorites " " " syllogism 'concrete' proposition 'consequent' in a sorites " " syllogism 'converse' propositions 'conversion' of a proposition " 'copula' of a proposition 'definition' 'dichotomy' ½ 'differentia' ½ 'division' 'eliminands' of a sorites " " syllogism 'entity' 'equivalent' propositions 'fallacy' 'genus' ½ 'imaginary' class " " name ½ 'individual' 'like', and 'unlike', signs of terms 'name' 'negative' proposition 'normal' form of a proposition " " of existence " " of relation 'nullity' 'partial' conclusion of a sorites 'particular' proposition 'peculiar' attributes ½ 'predicate' of a proposition " of a proposition of existence " " relation 'premisses' of a sorites " " syllogism pg 'proposition' " 'in _i_', 'in _e_', and 'in _a_' " 'in terms of' certain letters " of existence " of relation 'real' class ½ 'retinends' of a sorites " " syllogism 'sign of quantity' in a proposition 'sitting on the fence' 'some', technical meaning of 'sorites' 'species' ½ 'subject' of a proposition " " of existence " " of relation 'subscripts' of terms 'syllogism' symbol ".'." 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(first published in .) crown vo, cloth, gilt edges, price _s._ _d._ net. twelfth thousand. n.b.--this book contains pages--nearly as much as the two alice' books put together. sylvie and bruno concluded. with forty-six illustrations by harry furniss. (first published in .) crown vo, cloth, gilt edges, price _s._ _d._ net. third thousand. n.b.--this book contains pages. original games and puzzles. crown vo, cloth, gilt edges. [_in preparation._ three sunsets, and other poems. with twelve illustrations by e. gertrude thomson. fcap. to, cloth, gilt edges. n.b.--this will be a reprint, possibly with a few additions, of the serious portion of "phantasmagoria, and other poems," published in . * * * * * advice to writers. buy "the wonderland case for postage-stamps," invented by lewis carrol, oct. , , size inches by , containing separate pockets for stamps of different values, coloured pictorial surprises taken from _alice in wonderland_, and or wise words about letter-writing. it is published by messrs. emberlin & son, magdalene street, oxford. price _s._ n.b.--if ordered by post, an additional payment will be required, to cover cost of postage, as follows:-- one copy, ½_d._ two or three do., _d._ four do., ½_d._ five to fourteen do., _d._ each subsequent fourteen or fraction thereof, ½_d._ * * * * * * * * * * transcriber's note this book makes extensive use of page references. to assist the reader, page markers in the forms "pg-ix", "pg " & "px- " have been included in the right margin at points corresponding closely to the tops of the original pages. these may be searched for to locate the material referred to. in the main section, these page markers are always given with digits including, if necessary, leading zeroes. this book contains a large number of line drawn illustrations which are un-credited. as these cannot be rendered here in the original manner they have been reproduced as well as possible in the manner known as "ascii art". a number of transcription errors were found in the original book. as these were clearly not part of the author's intention they have, as far as possible, been identified and corrected in accordance with the methods given by the author. these corrections are listed here below with their locations and original text. in these notes the word 'natural' identifies a letter symbol occurring without a prime mark. page viii: " " corrected to " ". page viii: " " corrected to " ". page xv, '=real=' and '=unreal=', or '=imaginary=', classes: " ½" corrected to " ". page xix, propositions beginning with "some": " " corrected to " ". page xix, rules: " " corrected to " ". page xix, the west and east halves ...: " " corrected to " ". page xxi, the proposition "all x are y" ...: was originally shown as occurring in page . page xxii, and of three other similar arrangements: " " corrected to " ". page xxiii, the proposition 'no xm exist': " " corrected to " ". page : "contruct" corrected to "construct". page , paper v: missing word "it" supplied. page , # : "no h are k-natural" corrected to "no h are k-prime". page , # : "no a are h-natural" corrected to "no a are h-prime". page , # : "no c are n-natural" corrected to "no c are n-prime". page , # (d): "mortally offended if i fail to notice them" corrected to "mortally offended". page , # : "some x-prime are y" corrected to "some x-natural are y". page , # : "x_{ }m-prime_{ }" corrected to "x_{ }m-natural_{ }". page , # : "xm-natural_{ }" corrected to "xm-prime_{ }". page , # ( / ): "be" corrected to "be-sub:zero". page , # ( / ): "ch" corrected to "ch-sub:zero". page , # ( / ): second underline corrected from single to double. page , # ( / ): first underline corrected from single to double. page , # (¶ ): "h_{ }c" corrected to "h_{ }c-sub:zero". page , # ( / ): "a_{ }c-natural_{ }" corrected to "a_{ }c-prime_{ }". page , # (¶ ): "e_{ }c-natural_{ }" corrected to "e_{ }c-prime_{ }". page , # ( / ): first underline corrected from single to double. page , # ( / ): "e'd-prime" corrected to "e'd-natural". page , # ( / ): "c-prime+d" corrected to "c-natural+d". page , # : "a" and "b" interchanged. page , # ( / ): "b-natural_{ }e_{ }" corrected to "b-prime_{ }e_{ }". page , # ( / ): "h'k-prime_{ }" corrected to "h'k-natural_{ }". page , # ( / ): "c'd-prime" corrected to "c'd-natural". page , # ( / ): "h-natural+d'" corrected to "h-prime+d'". in the original book at the top of page , the following text occurred: [n.b. the numbers at the foot of each page indicate the pages where the corresponding matter may be found.] in accordance with the un-paged medium here this has been changed to: [n.b. reference tags for examples, answers & solutions will be found in the right margin.] the part of the book to which this relates contains, by sections, "examples" (exercises for the student), "answers" (to the examples) & "solutions" (worked answers). all example sections have corresponding answer sections. for sections & , worked solutions are not supplied; for sections - , solutions are given by different methods. in association with this, the original text contained editorial notes at the foot of each page giving the page numbers for the related sections. in this version, these notes are replaced with marginal tags such as ex , an , sl , & sl -a and sl -b which are placed at the top of each section to identify the current location. as with page tags, these may be searched for to locate the material refered to. (a search for "sl " should find successively both "sl -a" & "sl -b".) these tags are unique regardless of case. music transcribed by linda cantoni. transcriber's note minor punctuation errors have been changed without notice. printer errors have been changed and are listed at the end. all other inconsistencies are as in the original. in this version the square root symbol is indicated by [** sqrt], the superscript by ^, and the therefore symbol by [** therefore]. a tangled tale [decoration] [illustration: "at a pace of six miles in the hour." _frontispiece._] a tangled tale by lewis carroll _with six illustrations_ by arthur b. frost hoc meum tale quale est accipe. _second thousand._ london macmillan and co. [_all rights reserved_] richard clay & sons, bread street hill, london, e.c. _and bungay, suffolk_. to my pupil. beloved pupil! tamed by thee, addish-, subtrac-, multiplica-tion, division, fractions, rule of three, attest thy deft manipulation! then onward! let the voice of fame from age to age repeat thy story, till thou hast won thyself a name exceeding even euclid's glory! preface. this tale originally appeared as a serial in _the monthly packet_, beginning in april, . 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. l. c. _october, ._ contents. knot page i. excelsior ii. eligible apartments iii. mad mathesis iv. the dead reckoning v. oughts and crosses vi. her radiancy vii. petty cash viii. de omnibus rebus ix. a serpent with corners x. chelsea buns answers to knot i. " " ii. " " iii. " " iv. " " v. " " vi. " " vii. " " viii. " " ix. " " x. a tangled tale. knot i. excelsior. "goblin, lead them up and down." 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. as is always the case under such circumstances, the younger knight was the first to break the silence. "a goodly pace, i trow!" he exclaimed. "we sped not thus in the ascent!" "goodly, indeed!" the other echoed with a groan. "we clomb it but at three miles in the hour." "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. "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!" "'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!" "he will chide our tardy return," was the grave reply, "and such a rebuke will be meet." "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!" "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 added in an undertone, "by the time we regain our hostelry. full many a mile shall we have plodded this day!" "how many? how many?" cried the eager youth, ever athirst for knowledge. the old man was silent. "tell me," he answered, after a moment's thought, "what time it was 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." 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. knot ii. eligible apartments. "straight down the crooked lane, and all round the square." "let's ask balbus about it," said hugh. "all right," said lambert. "_he_ can guess it," said hugh. "rather," said lambert. no more words were needed: the two brothers understood each other perfectly. [illustration: "balbus was assisting his mother-in-law to convince the dragon."] 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 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. 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 different houses, all announcing in flaming capitals "eligible apartments." "so there's plenty of choice, after all, you see," said spokesman hugh in conclusion. "that doesn't follow from the data," said balbus, as he rose from the easy chair, where he had been dozing over _the little mendip gazette_. "they may be all single rooms. however, we may as well see them. i shall be glad to stretch my legs a bit." 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----_what_ was that name again, lambert?" ("kgovjni," said lambert.) "well, yes. the governor of----what-you-may-call-it----wants to give a _very_ 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, and his brother-in-law's father: and we're to guess how many guests there will be." there was an anxious pause. "_how_ 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----" "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." "it _is_ a square!" was balbus' first cry of delight, as he gazed around him. "beautiful! beau-ti-ful! equilateral! _and_ rectangular!" 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. "see, boys!" he cried. "twenty doors on a side! what symmetry! each side divided into twenty-one equal parts! it's delicious!" "shall i knock, or ring?" said hugh, looking in some perplexity at a square brass plate which bore the simple inscription "ring also." "both," said balbus. "that's an ellipsis, my boy. did you never see an ellipsis before?" "i couldn't hardly read it," said hugh, evasively. "it's no good having an ellipsis, if they don't keep it clean." "which there is _one_ room, gentlemen," said the smiling landlady. "and a sweet room too! as snug a little back-room----" "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?" "which indeed there _is_, gentlemen!" the landlady indignantly protested, as she drew up the blind, and indicated the back garden. "cabbages, i perceive," said balbus. "well, they're green, at any rate." "which the greens at the shops," their hostess explained, "are by no means dependable upon. here you has them on the premises, _and_ of the best." "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. this landlady was grave and stern. "i've nobbut one room left," she told them: "and it gives on the back-gyardin." "but there are cabbages?" balbus suggested. 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." "a singular advantage," said balbus: and, after the usual questions, they went on to fifty-two. "and i'd gladly accommodate you all, if i could," was the greeting that met them. "we are but mortal," ("irrelevant!" muttered balbus) "and i've let all my rooms but one." "which one is a back-room, i perceive," said balbus: "and looking out on--on cabbages, i presume?" "yes, indeed, sir!" said their hostess. "whatever _other_ folks may do, _we_ grows our own. for the shops----" "an excellent arrangement!" balbus interrupted. "then one can really depend on their being good. does the window open?" the usual questions were answered satisfactorily: but this time hugh added one of his own invention--"does the cat scratch?" the landlady looked round suspiciously, as if to make sure the cat was not listening, "i will not deceive you, gentlemen," she said. "it _do_ 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!" "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!" at number seventy-three they found only a small shy girl to show the house, who said "yes'm" in answer to all questions. "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?" "yes'm," said the girl. "well, you may tell your mistress we will take the room, and that her plan of growing her own cabbages is simply _admirable_!" "yes'm," said the girl, as she showed them out. "one day-room and three bed-rooms," said balbus, as they returned to the hotel. "we will take as our day-room the one that gives us the least walking to do to get to it." "must we walk from door to door, and count the steps?" said lambert. "no, no! figure it out, my boys, figure it out!" balbus gaily exclaimed, as he put pens, ink, and paper before his hapless pupils, and left the room. "i say! it'll be a job!" said hugh. "rather!" said lambert. knot iii. mad mathesis. "i waited for the train." "well, they call me so because i _am_ a little mad, i suppose," 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 _that_), and i never play lawn-tennis. i can't cook an omelette. i can't even set a broken limb! _there's_ an ignoramus for you!" 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 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. "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 _most_ 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." "they part to meet again," said clara, her eyes filling with tears at the romantic thought. "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 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!" "i never make bets," clara said very gravely. "our excellent preceptress has often warned us----" "you'd be none the worse if you did!" mad mathesis interrupted. "in fact, you'd be the better, i'm certain!" "neither does our excellent preceptress approve of puns," said clara. "but we'll have a match, if you like. let me choose my train," she added after a brief mental calculation, "and i'll engage to meet exactly half as many again as you do." "not if you count fair," mad mathesis bluntly interrupted. "remember, we only count the trains we meet _on the way_. you mustn't count the one that starts as you start, nor the one that arrives as you arrive." "that will only make the difference of _one_ 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." 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. "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. 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. "mind where you go, dear!" cried her aunt, checking her just in time. "one step more, and you'd have been in that pail of cold water!" "i know, i know," clara said, dreamily. "the pale, the cold, and the moony----" "take your places on the spring-boards!" shouted a porter. "what are _they_ for!" clara asked in a terrified whisper. "merely to help us into the trains." the elder lady spoke with the nonchalance of one quite used 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. 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. "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." clara brightened up. "i shall win _that_," she exclaimed eagerly, "if i may choose my train!" another shriek of engine whistles, another upheaving of spring-boards, another living avalanche plunging into two trains as they flashed by: and the travellers were off again. each gazed eagerly from her carriage window, holding up her 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 _two_ sighs--of relief. "one!" clara murmured to herself. "won! it's a word of good omen. _this_ time, at any rate, the victory will be mine!" but _was_ it? knot iv. the dead reckoning. "i did dream of money-bags to-night." noonday on the open sea within a few degrees of the equator is apt to 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. along with their armour, the tourists had laid aside the antiquated speech it had pleased them to affect while in knightly disguise, and had returned to the ordinary style of two country gentlemen of the twentieth century. stretched on a pile of cushions, under the shade of a huge umbrella, they were lazily watching some native fishermen, who had come on 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. "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. "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. the captain paused in his march, and towered over the travellers--tall, grave, and serenely self-satisfied. "fishermen," he explained, "are often passengers in my ship. these five are from mhruxi--the 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." 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. 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. when next the captain's heavy footfall passed, the younger man roused himself to speak. "_what_ did you call the place those fellows came from, captain?" he asked. "mhruxi, sir." "and the one we are bound for?" the captain took a long breath, plunged into the word, and came out of it nobly. "they call it kgovjni, sir." "k--i give it up!" the young man faintly said. he stretched out his hand for a glass of iced water which 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. his father courteously tried to make amends for his _nonchalance_. "whereabouts are we now, captain?" said he, "have you any idea?" the captain cast a pitying look on the ignorant landsman. "i could tell you _that_, sir," he said, in a tone of lofty condescension, "to an inch!" "you don't say so!" the old man remarked, in a tone of languid surprise. "and mean so," persisted the captain. "why, what do you suppose would become of my ship, if i were to lose my longitude and my latitude? could _you_ make anything of my dead reckoning?" "nobody could, i'm sure!" the other heartily rejoined. but he had overdone it. "it's _perfectly_ intelligible," the captain said, in an offended tone, "to any one that understands such things." with these words he moved away, and began giving orders to the men, who were preparing to hoist the jib. 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. 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. the old man explained it to the captain. "let us make it up among us," he added in conclusion. "ten pounds will do it, i think you said?" [illustration] but the captain put aside the suggestion with a wave of the hand. "no, sir!" he said, in his grandest manner. "you will excuse me, i am sure; but these are my passengers. the accident has happened on board my ship, and under my orders. it is for me to 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." 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. 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. "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!" "if they didn't have five separate weighings, of course you can't value them separately," the youth hastily decided. "let's hear all about it," was the old man's more cautious remark. "they _did_ have five separate weighings," the captain said, "but--well, it beats _me_ entirely!" he added, in a sudden burst of candour. "here'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 _three_ sacks to weigh it down--that's the first, third and fifth--and _they_ weighed sixteen pounds. there, gentlemen! did you ever hear anything like _that_?" 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_ contemplate _myself_!" knot v. oughts and crosses. "look here, upon this picture, and on this." "and what made you choose the first train, goosey?" said mad mathesis, as they got into the cab. "couldn't you count better than _that_?" "i took an extreme case," was the tearful reply. "our excellent preceptress always says 'when in doubt, my dears, take an extreme case.' and i _was_ in doubt." "does it always succeed?" her aunt enquired. clara sighed. "not _always_," she reluctantly admitted. "and i can'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 _vice versâ_.' and i thought they wouldn't know what '_vice versâ_' meant: so i explained it to them. i 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. _why_ wasn't it?" she added plaintively. 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." "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." "an extreme case, indeed," her aunt remarked with admirable gravity: "and a most dangerous state of things!" "and i thought, if i went with a tortoise, there would be only _one_ hare to meet: but if i went with the hare--you know there were _crowds_ of tortoises!" "it wasn't a bad idea," said the elder lady, as they left the cab, at the entrance of burlington house. "you shall have another chance to-day. we'll have a match in marking pictures." clara brightened up. "i should like to try again, very much," she said. "i'll take more care this time. how are we to play?" to this question mad mathesis made no reply: she was busy drawing 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----" "do you mean _only_ two crosses?" said clara. "or may i count the three-cross pictures among the two-cross pictures?" "of course you may," said her aunt. "any one, that has _three_ eyes, may be said to have _two_ eyes, i suppose?" clara followed her aunt's dreamy gaze across the crowded gallery, half-dreading to find that there was a three-eyed person in sight. "and you must give one cross to nine or ten." "and which wins the match?" clara asked, as she carefully entered these conditions on a blank leaf in her catalogue. "whichever marks fewest pictures." "but suppose we marked the same number?" "then whichever uses most marks." clara considered. "i don't think it's much of a match," she said. "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." "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 _too_ hard on the r.a.'s." 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!" her aunt smiled grimly. "we can begin _here_," 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." "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!" "mind what you say, my dear!" her aunt interposed. "it's by an r.a.!" but clara was quite reckless. "i don't care who it's by!" she cried. "and i shall give it three bad marks!" aunt and niece soon drifted away from each other in the crowd, and 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 _can't_ find the one i want!" she exclaimed at last, almost crying with vexation. "what is it you want to find, my dear?" the voice was strange to clara, but so sweet and gentle that she felt attracted to the owner of it, 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. "i was looking for a picture," she said, "that has a good subject--and that's well arranged--but badly coloured." 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 _was_ the subject?" "was it an elephant, for instance?" the other sister suggested. they were still in sight of lieutenant brown. "i don't know, indeed!" clara impetuously replied. "you know it doesn't matter a bit what the subject _is_, so long as it's a good one!" once more the sisters exchanged looks of alarm, and one of them whispered something to the other, of which clara caught only the one word "mad." "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 _there_--just three pictures beyond lieutenant brown." "ah, well! then you'd better go to her, my dear!" her new friend said, soothingly. "_she'll_ find you the picture you want. good-bye, dear!" "good-bye, dear!" echoed the other sister, "mind you don't lose sight of your aunt!" and the pair trotted off into another room, leaving clara rather perplexed at their manner. "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----" knot vi. her radiancy. "one piecee thing that my have got, maskee[a] that thing my no can do. you talkee you no sabey what? bamboo." they landed, and were at once conducted to the palace. about half way they were met by the governor, who welcomed them in english--a great relief to our travellers, whose guide could speak nothing but kgovjnian. "i don't half like the way they grin at us as we go by!" the old man whispered to his son. "and why do they say 'bamboo!' so often?" "it alludes to a local custom," replied the governor, who had overheard the question. "such persons as happen in any way to displease her radiancy are usually beaten with rods." [illustration: "why do they say 'bamboo!' so often?"] the old man shuddered. "a most objectional local custom!" he remarked with strong emphasis. "i wish we had never landed! did you notice that black fellow, norman, opening his great mouth at us? i verily believe he would like to eat us!" norman appealed to the governor, who was walking at his other side. "do they often eat distinguished strangers here?" he said, in as indifferent a tone as he could assume. "not often--not ever!" was the welcome reply. "they are not good for it. pigs we eat, for they are fat. this old man is thin." "and thankful to be so!" muttered the elder traveller. "beaten we shall be without a doubt. it's a comfort to know it won't be beaten without the b! my dear boy, just look at the peacocks!" they were now walking between two unbroken lines of those gorgeous birds, each held in check, by means of a golden collar and chain, by a black slave, who stood well behind, so as not to interrupt the view of the glittering tail, with its network of rustling feathers and its hundred eyes. the governor smiled proudly. "in your honour," he said, "her radiancy has ordered up ten thousand additional peacocks. she will, no doubt, decorate you, before you go, with the usual star and feathers." "it'll be star without the s!" faltered one of his hearers. "come, come! don't lose heart!" said the other. "all this is full of charm for me." "you are young, norman," sighed his father; "young and light-hearted. for me, it is charm without the c." "the old one is sad," the governor remarked with some anxiety. "he has, without doubt, effected some fearful crime?" "but i haven't!" the poor old gentleman hastily exclaimed. "tell him i haven't, norman!" "he has not, as yet," norman gently explained. and the governor repeated, in a satisfied tone, "not as yet." "yours is a wondrous country!" the governor resumed, after a pause. "now here is a letter from a friend of mine, a merchant, in london. he and his brother went there a year ago, with a thousand pounds apiece; and on new-year's-day they had sixty thousand pounds between them!" "how did they do it?" norman eagerly exclaimed. even the elder traveller looked excited. the governor handed him the open letter. "anybody can do it, when once they know how," so ran this oracular document. "we borrowed nought: we stole nought. we began the year with only a thousand pounds apiece: and last new-year's-day we had sixty thousand pounds between us--sixty thousand golden sovereigns!" norman looked grave and thoughtful as he handed back the letter. his father hazarded one guess. "was it by gambling?" "a kgovjnian never gambles," said the governor gravely, as he ushered them through the palace gates. they followed him in silence down a long passage, and soon found themselves in a lofty hall, lined entirely with peacocks' feathers. in the centre was a pile of crimson cushions, which almost concealed the figure of her radiancy--a plump little damsel, in a robe of green satin dotted with silver stars, whose pale round face lit up for a moment with a half-smile as the travellers bowed before her, and then relapsed into the exact expression of a wax doll, while she languidly murmured a word or two in the kgovjnian dialect. the governor interpreted. "her radiancy welcomes you. she notes the impenetrable placidity of the old one, and the imperceptible acuteness of the youth." here the little potentate clapped her hands, and a troop of slaves instantly appeared, carrying trays of coffee and sweetmeats, which they offered to the guests, who had, at a signal from the governor, seated themselves on the carpet. "sugar-plums!" muttered the old man. "one might as well be at a confectioner's! ask for a penny bun, norman!" "not so loud!" his son whispered. "say something complimentary!" for the governor was evidently expecting a speech. "we thank her exalted potency," the old man timidly began. "we bask in the light of her smile, which----" "the words of old men are weak!" the governor interrupted angrily. "let the youth speak!" "tell her," cried norman, in a wild burst of eloquence, "that, like two grasshoppers in a volcano, we are shrivelled up in the presence of her spangled vehemence!" "it is well," said the governor, and translated this into kgovjnian. "i am now to tell you," he proceeded, "what her radiancy requires of you 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." the old man groaned. "all is lost!" he wildly exclaimed. but norman heeded him not: he had taken out his note-book, and was calmly jotting down the particulars. "three they be," the governor proceeded, "lolo, mimi, and zuzu. lolo makes scarves while mimi makes ; but zuzu makes while lolo makes ! again, so fairylike is zuzu's handiwork, of her scarves weigh no more than one of lolo's; yet mimi's is lighter still-- of hers will but balance of zuzu's! and for warmth one of mimi's is equal to of zuzu's; yet one of lolo's is as warm as of mimi's!" here the little lady once more clapped her hands. "it is our signal of dismissal!" the governor hastily said. "pay her radiancy your farewell compliments--and walk out backwards." the walking part was all the elder tourist could manage. norman simply said "tell her radiancy we are transfixed by the spectacle of her serene brilliance, and bid an agonized farewell to her condensed milkiness!" "her radiancy is pleased," the governor reported, after duly translating this. "she casts on you a glance from her imperial eyes, and is confident that you will catch it!" "that i warrant we shall!" the elder traveller moaned to himself distractedly. once more they bowed low, and then followed the governor down a winding staircase to the imperial dungeon, which they found to be lined with coloured marble, lighted from the roof, and splendidly though not luxuriously furnished with a bench of polished malachite. "i trust you will 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. "i told you how it would end!" moaned the elder traveller, wringing his hands, and quite forgetting in his anguish that he had himself proposed the expedition, and had never predicted anything of the sort. "oh that we were well out of this miserable business!" "courage!" cried the younger cheerily. "_hæc olim meminisse juvabit!_ the end of all this will be glory!" "glory without the l!" was all the poor old man could say, as he rocked himself to and fro on the malachite bench. "glory without the l!" footnote: [footnote a: "_maskee_," in pigeon-english, means "_without_."] knot vii. petty cash. "base is the slave that pays." "aunt mattie!" "my child?" "_would_ you mind writing it down at once? i shall be quite _certain_ to forget it if you don't!" "my dear, we really must wait till the cab stops. how can i possibly write anything in the midst of all this jolting?" "but _really_ i shall be forgetting it!" clara's voice took the plaintive tone that her aunt never knew how to resist, and with a sigh the old lady drew forth her ivory tablets and prepared to record the amount that clara had just spent at the confectioner's shop. her expenditure was always made out of her aunt's purse, but the poor girl knew, by bitter experience, that sooner or later "mad mathesis" would expect an exact account of every penny that had gone, and she waited, with ill-concealed impatience, while the old lady turned the tablets over and over, till she had found the one headed "petty cash." "here's the place," she said at last, "and here we have yesterday's luncheon duly entered. _one glass lemonade_ (why can't you drink water, like me?) _three sandwiches_ (they never put in half mustard enough. i told the young woman so, to her face; and she tossed her head--like her impudence!) _and seven biscuits_. _total one-and-two-pence._ well, now for to-day's?" "one glass of lemonade----" clara was beginning to say, when suddenly the cab drew up, and a courteous railway-porter was handing out the bewildered girl before she had had time to finish her sentence. her aunt pocketed the tablets instantly. "business first," she said: "petty cash--which is a form of pleasure, whatever _you_ 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. "my dear, you really must cultivate a more capacious mind!" was all the consolation she vouchsafed to the poor girl. "are not the tablets of your memory wide enough to contain the record of one single luncheon?" "not wide enough! not half wide enough!" was the passionate reply. the words came in aptly enough, but the voice was not that of clara, and both ladies turned in some surprise to see who it was that had so suddenly struck into their conversation. a fat little old lady was standing at the door of a cab, helping the driver to extricate what seemed an exact duplicate of herself: it would have been no easy task to decide which was the fatter, or which looked the more good-humoured of the two sisters. "i tell you the cab-door isn't half wide enough!" she repeated, as her sister finally emerged, somewhat after the fashion of a pellet from a pop-gun, and she turned to appeal to clara. "is it, dear?" she said, trying hard to bring a frown into a face that dimpled all over with smiles. "some folks is too wide for 'em," growled the cab-driver. [illustration: "i tell you the cab-door isn't half wide enough!"] "don't provoke me, man!" cried the little old lady, in what she meant for a tempest of fury. "say another word and i'll put you into the county court, and sue you for a _habeas corpus_!" the cabman touched his hat, and marched off, grinning. "nothing like a little law to cow the ruffians, my dear!" she remarked confidentially to clara. "you saw how he quailed when i mentioned the _habeas corpus_? not that i've any idea what it means, but it sounds very grand, doesn't it?" "it's very provoking," clara replied, a little vaguely. "very!" the little old lady eagerly repeated. "and we're very much provoked indeed. aren't we, sister?" "i never was so provoked in all my life!" the fatter sister assented, radiantly. by this time clara had recognised her picture-gallery acquaintances, and, drawing her aunt aside, she hastily whispered her reminiscences. "i met them first in the royal academy--and they were very kind to me--and they were lunching at the next table to us, just now, you know--and they tried to help me to find the picture i wanted--and i'm sure they're dear old things!" "friends of yours, are they?" said mad mathesis. "well, i like their looks. you can be civil to them, while i get the tickets. but do try and arrange your ideas a little more chronologically!" and so it came to pass that the four ladies found themselves seated side by side on the same bench waiting for the train, and chatting as if they had known one another for years. "now this i call quite a remarkable coincidence!" exclaimed the smaller and more talkative of the two sisters--the one whose legal knowledge had annihilated the cab-driver. "not only that we should be waiting for the same train, and at the same station--_that_ would be curious enough--but actually on the same day, and the same hour of the day! that's what strikes _me_ 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-- "and me too, sister!" "those are not _independent_ coincidences----" mad mathesis was just beginning, when clara ventured to interpose. "there's no jolting here," she pleaded meekly. "_would_ you mind writing it down now?" out came the ivory tablets once more. "what was it, then?" said her aunt. "one glass of lemonade, one sandwich, one biscuit--oh dear me!" cried poor clara, the historical tone suddenly changing to a wail of agony. "toothache?" said her aunt calmly, as she wrote down the items. the two sisters instantly opened their reticules and produced two different remedies for neuralgia, each marked "unequalled." "it isn't that!" said poor clara. "thank you very much. it's only that i _can't_ remember how much i paid!" "well, try and make it out, then," said her aunt. "you've got yesterday's luncheon to help you, you know. and here's the luncheon we had the day before--the first day we went to that shop--_one glass lemonade_, _four sandwiches_, _ten biscuits_. _total, one-and-fivepence._" she handed the tablets to clara, who gazed at them with eyes so dim with tears that she did not at first notice that she was holding them upside down. the two sisters had been listening to all this with the deepest interest, and at this juncture the smaller one softly laid her hand on clara's arm. "do you know, my dear," she said coaxingly, "my sister and i are in the very same predicament! quite identically the very same predicament! aren't we, sister?" "quite identically and absolutely the very----" began the fatter sister, but she was constructing her sentence on too large a scale, and the little one would not wait for her to finish it. "yes, my dear," she resumed; "we were lunching at the very same shop as you were--and we had two glasses of lemonade and three sandwiches and five biscuits--and neither of us has the least idea what we paid. have we, sister?" "quite identically and absolutely----" murmured the other, who evidently considered that she was now a whole sentence in arrears, and that she ought to discharge one obligation before contracting any fresh liabilities; but the little lady broke in again, and she retired from the conversation a bankrupt. "_would_ you make it out for us, my dear?" pleaded the little old lady. "you can do arithmetic, i trust?" her aunt said, a little anxiously, as clara turned from one tablet to another, vainly trying to collect her thoughts. her mind was a blank, and all human expression was rapidly fading out of her face. a gloomy silence ensued. knot viii. de omnibus rebus. "this little pig went to market: this little pig staid at home." "by her radiancy's express command," said the governor, as he conducted the travellers, for the last time, from the imperial presence, "i shall now have the ecstasy of escorting you as far as the outer gate of the military quarter, where the agony of parting--if indeed nature can survive the shock--must be endured! from that gate grurmstipths start every quarter of an hour, both ways----" "would you mind repeating that word?" said norman. "grurm----?" "grurmstipths," the governor repeated. "you call them omnibuses in england. they run both ways, and you can travel by one of them all the way down to the harbour." the old man breathed a sigh of relief; four hours of courtly ceremony had wearied him, and he had been in constant terror lest something should call into use the ten thousand additional bamboos. in another minute they were crossing a large quadrangle, paved with marble, and tastefully decorated with a pigsty in each corner. soldiers, carrying pigs, were marching in all directions: and in the middle stood a gigantic officer giving orders in a voice of thunder, which made itself heard above all the uproar of the pigs. "it is the commander-in-chief!" the governor hurriedly whispered to his companions, who at once followed his example in prostrating themselves before the great man. the commander gravely bowed in return. he was covered with gold lace from head to foot: his face wore an expression of deep misery: and he had a little black pig under each arm. still the gallant fellow did his best, in the midst of the orders he was every moment issuing to his men, to bid a courteous farewell to the departing guests. "farewell, oh old one--carry these three to the 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. "he is in distress," the governor explained as they left the court. "her radiancy has commanded him to place twenty-four pigs in those four sties, so that, as she goes round the court, she may always find the number in each sty nearer to ten than the number in the last." "does she call ten nearer to ten than nine is?" said norman. "surely," said the governor. "her radiancy would admit that ten is nearer to ten than nine is--and also nearer than eleven is." "then i think it can be done," said norman. the governor shook his head. "the commander has been transferring them in vain for four months," he said. "what hope remains? and her radiancy has ordered up ten thousand additional----" "the pigs don't seem to enjoy being transferred," the old man hastily interrupted. he did not like the subject of bamboos. "they are only _provisionally_ 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." "of course she would only go _once_ round?" said norman. "alas, no!" sighed their conductor. "round and round. round and round. these are her radiancy's own words. but oh, agony! here is the outer gate, and we must part!" he sobbed as he shook hands with them, and the next moment was briskly walking away. "he _might_ have waited to see us off!" said the old man, piteously. "and he needn't have begun whistling the very _moment_ he left us!" said the young one, severely. "but look sharp--here are two what's-his-names in the act of starting!" unluckily, the sea-bound omnibus was full. "never mind!" said norman, cheerily. "we'll walk on till the next one overtakes us." they trudged on in silence, both thinking over the military problem, till they met an omnibus coming from the sea. the elder traveller took out his watch. "just twelve minutes and a half since we started," he remarked in an absent manner. suddenly the vacant face brightened; the old man had an idea. "my boy!" he shouted, bringing his hand down upon norman's shoulder so suddenly as for a moment to transfer his centre of gravity beyond the base of support. thus taken off his guard, the young man wildly staggered forwards, and seemed about to plunge into space: but in another moment he had gracefully recovered himself. "problem in precession and nutation," he remarked--in tones where filial respect only just managed to conceal a shade of annoyance. "what is it?" he hastily added, fearing his father might have been taken ill. "will you have some brandy?" "when will the next omnibus overtake us? when? when?" the old man cried, growing more excited every moment. norman looked gloomy. "give me time," he 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. knot ix. a serpent with corners. "water, water, every where, nor any drop to drink." "it'll just take one more pebble." "what ever _are_ you doing with those buckets?" the speakers were hugh and lambert. place, the beach of little mendip. time, . , p.m. hugh was floating a bucket in another a size larger, and trying how many pebbles it would carry without sinking. lambert was lying on his back, doing nothing. for the next minute or two hugh was silent, evidently deep in thought. suddenly he started. "i say, look here, lambert!" he cried. "if it's alive, and slimy, and with legs, i don't care to," said lambert. "didn't balbus say this morning that, if a body is immersed in liquid, it displaces as much liquid as is equal to its own bulk?" said hugh. "he said things of that sort," lambert vaguely replied. "well, just look here a minute. here's the little bucket almost 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 _that_ water is the same bulk as the little bucket?" "course it is," said lambert. "well, look here again!" cried hugh, triumphantly, as he poured the water from the big bucket into the little one. "why, it doesn't half fill it!" "that's _its_ business," said lambert. "if balbus says it's the same bulk, why, it _is_ the same bulk, you know." "well, i don't believe it," said hugh. "you needn't," said lambert. "besides, it's dinner-time. come along." they found balbus waiting dinner for them, and to him hugh at once propounded his difficulty. "let's get you helped first," said balbus, briskly cutting away at the joint. "you know the old proverb 'mutton first, mechanics afterwards'?" the boys did _not_ 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. "a friend of mine has a flower-garden--a very pretty one, though no great size--" "how big is it?" said hugh. "that's what _you_ have to find out!" balbus gaily replied. "all _i_ tell you is that it is oblong in shape--just half a yard longer than its width--and that a gravel-walk, one yard wide, begins at one corner and runs all round it." "joining into itself?" said hugh. "_not_ joining into itself, young man. just before doing _that_, it turns a corner, and runs round the garden again, alongside of the first portion, and then inside that again, winding in and in, and each lap touching the last one, till it has used up the whole of the area." "like a serpent with corners?" said lambert. "exactly so. and if you walk the whole length of it, to the last inch, keeping in the centre of the path, it's exactly two miles and half a furlong. now, while you find out the length and breadth of the garden, i'll see if i can think out that sea-water puzzle." "you said it was a flower-garden?" hugh inquired, as balbus was leaving the room. "i did," said balbus. "where do the flowers grow?" said hugh. but balbus thought it best not to hear the question. he left the boys to their problem, and, in the silence of his own room, set himself to unravel hugh's mechanical paradox. "to fix our thoughts," he murmured to himself, as, with hands deep-buried in his pockets, he paced up and down the room, "we will take a cylindrical glass jar, with a scale of inches marked up the side, and fill it with water up to the -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 _half_ a pint of water, and plunge inches of it into the water, so that the end of the cylinder comes down to the -inch mark. well, that displaces 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 -inch mark. but unfortunately there _is_ more cylinder, occupying half the space between the -inch and the -inch marks, so that only _one_ 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 -inch mark. but unfortunately----shade of newton!" he exclaimed, in sudden accents of terror. "when _does_ the water stop rising?" a bright idea struck him. "i'll write a little essay on it," he said. * * * * * _balbus's essay._ "when a solid is immersed in a liquid, it is well known that it displaces a portion of the liquid equal to itself in bulk, and that the level of the liquid rises just so much as it would rise if a quantity of liquid had been added to it, equal in bulk to the solid. lardner says, precisely the same process occurs when a solid is _partially_ immersed: the quantity of liquid displaced, in this case, equalling the portion of the solid which is immersed, and the rise of the level being in proportion. "suppose a solid held above the surface of a liquid and partially immersed: a portion of the liquid is displaced, and the level of the liquid rises. but, by this rise of level, a little bit more of the solid is of course immersed, and so there is a new displacement of a second portion of the liquid, and a consequent rise of level. again, this second rise of level causes a yet further immersion, and by consequence another displacement of liquid and another rise. it is self-evident that this process must continue till the entire solid is immersed, and that the liquid will then begin to immerse whatever holds the solid, which, being connected with it, must for the time be considered a part of it. if you hold a stick, six feet long, with its end in a tumbler of water, and wait long enough, you must eventually be immersed. the question as to the source from which the water is supplied--which belongs to a high branch of mathematics, and is therefore beyond our present scope--does not apply 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."[b] * * * * * "there's a fallacy _somewhere_," he murmured drowsily, as he stretched his long legs upon the sofa. "i must think it over again." he closed his eyes, in order to concentrate his attention more perfectly, and for the next hour or so his slow and regular breathing bore witness to the careful deliberation with which he was investigating this new and perplexing view of the subject. [illustration: "he remains steadfast and unmoved."] footnote: [footnote b: _note by the writer._--for the above essay i am indebted to a dear friend, now deceased.] knot x. chelsea buns. "yea, buns, and buns, and buns!" old song. "how very, very sad!" exclaimed clara; and the eyes of the gentle girl filled with tears as she spoke. "sad--but very curious when you come to look at it arithmetically," was her aunt's less romantic reply. "some of them have lost an arm in their country's service, some a leg, some an ear, some an eye----" "and some, perhaps, _all_!" clara murmured dreamily, as they passed the long rows of weather-beaten heroes basking in the sun. "did you notice that very old one, with a red face, who was drawing a map in the dust with his wooden leg, and all the others watching? i _think_ it was a plan of a battle----" "the battle of trafalgar, no doubt," her aunt interrupted, briskly. "hardly that, i think," clara ventured to say. "you see, in that case, he couldn't well be alive----" "couldn't well be alive!" the old lady contemptuously repeated. "he's as lively as you and me put together! why, if drawing a map in the dust--with one's wooden leg--doesn't prove one to be alive, perhaps you'll kindly mention what _does_ prove it!" clara did not see her way out of it. logic had never been her _forte_. "to return to the arithmetic," mad mathesis resumed--the eccentric old lady never let slip an opportunity of driving her niece into a calculation--"what percentage do you suppose must have lost all four--a leg, an arm, an eye, and an ear?" "how _can_ i tell?" gasped the terrified girl. she knew well what was coming. "you can't, of course, without _data_," her aunt replied: "but i'm just going to give you----" "give her a chelsea bun, miss! that's what most young ladies likes best!" the voice was rich and musical, and the speaker dexterously whipped back the snowy cloth that covered his basket, and disclosed a tempting array of the familiar square buns, joined together in rows, richly egged and browned, and glistening in the sun. "no, sir! i shall give her nothing so indigestible! be off!" the old lady waved her parasol threateningly: but nothing seemed to disturb the good-humour of the jolly old man, who marched on, chanting his melodious refrain:-- [music: chel-sea buns! chel-sea buns hot! chel-sea buns! pi-ping hot! chel-sea buns hot! chel-sea buns!] "far too indigestible, my love!" said the old lady. "percentages will agree with you ever so much better!" clara sighed, and there was a hungry look in her eyes as she watched the basket lessening in the distance: but she meekly listened to the relentless old lady, who at once proceeded to count off the _data_ on her fingers. "say that per cent. have lost an eye-- per cent. an ear-- per cent. an arm-- per cent. a leg--that'll do it beautifully. now, my dear, what percentage, _at least_, must have lost all four?" no more conversation occurred--unless a smothered exclamation of "piping hot!" which escaped from clara's lips as the basket vanished round a corner could be counted as such--until they reached the old chelsea mansion, where clara's father was then staying, with his three sons and their old tutor. balbus, lambert, and hugh had entered the house only a few minutes before them. they had been out walking, and hugh had been propounding a difficulty which had reduced lambert to the depths of gloom, and had even puzzled balbus. "it changes from wednesday to thursday at midnight, doesn't it?" hugh had begun. "sometimes," said balbus, cautiously. "always," said lambert, decisively. "_sometimes_," balbus gently insisted. "six midnights out of seven, it changes to some other name." "i meant, of course," hugh corrected himself, "when it _does_ change from wednesday to thursday, it does it at midnight--and _only_ at midnight." "surely," said balbus. lambert was silent. "well, now, suppose it's midnight here in chelsea. then it's wednesday _west_ of chelsea (say in ireland or america) where midnight hasn't arrived yet: and it's thursday _east_ of chelsea (say in germany or russia) where midnight has just passed by?" "surely," balbus said again. even lambert nodded this time. "but it isn't midnight, anywhere else; so it can't be changing from one day to another anywhere else. and yet, if ireland and america and so on call it wednesday, and germany and russia and so on call it thursday, there _must_ be some place--not chelsea--that has different days on the two sides of it. and the worst of it is, the people _there_ get their days in the wrong order: they've got wednesday _east_ of them, and thursday _west_--just as if their day had changed from thursday to wednesday!" "i've heard that puzzle before!" cried lambert. "and i'll tell you the explanation. when a ship goes round the world from east to west, we know that it loses a day in its reckoning: so that when it gets home, and calls its day wednesday, it finds people here calling it thursday, because we've had one more midnight than the ship has had. and when you go the other way round you gain a day." "i know all that," said hugh, in reply to this not very lucid explanation: "but it doesn't help me, because the ship hasn't proper days. one way round, you get more than twenty-four hours to the day, and the other way you get less: so of course the names get wrong: but people that live on in one place always get twenty-four hours to the day." "i suppose there _is_ such a place," balbus said, meditatively, "though i never heard of it. and the people must find it very queer, as hugh says, to have the old day _east_ of them, and the new one _west_: because, when midnight comes round to them, with the new day in front of it and the old one behind it, one doesn't see exactly what happens. i must think it over." so they had entered the house in the state i have described--balbus puzzled, and lambert buried in gloomy thought. "yes, m'm, master _is_ at home, m'm," said the stately old butler. (n.b.--it is only a butler of experience who can manage a series of three m's together, without any interjacent vowels.) "and the _ole_ party is a-waiting for you in the libery." "i don't like his calling your father an _old_ party," mad mathesis whispered to her niece, as they crossed the hall. and clara had only just time to whisper in reply "he meant the _whole_ party," before they were ushered into the library, and the sight of the five solemn faces there assembled chilled her into silence. her father sat at the head of the table, and mutely signed to the ladies to take the two vacant chairs, one on each side of him. his three sons and balbus completed the party. writing materials had been arranged round the table, after the fashion of a ghostly banquet: the butler had evidently bestowed much thought on the grim device. sheets of quarto paper, each flanked by a pen on one side and a pencil on the other, represented the plates--penwipers did duty for rolls of bread--while ink-bottles stood in the places usually occupied by wine-glasses. the _pièce de resistance_ was a large green baize bag, which gave forth, as the old man restlessly lifted it from side to side, a charming jingle, as of innumerable golden guineas. "sister, daughter, sons--and balbus--," the old man began, so nervously, that balbus put in a gentle "hear, hear!" while hugh drummed on the table with his fists. this disconcerted the unpractised orator. "sister--" he began again, then paused a moment, moved the bag to the other side, and went on with a rush, "i mean--this being--a critical occasion--more or less--being the year when one of my sons comes of age--" he paused again in some confusion, having evidently got into the middle of his speech sooner than he intended: but it was too late to go back. "hear, hear!" cried balbus. "quite so," said the old gentleman, recovering his self-possession a little: "when first i began this annual custom--my friend balbus will correct me if i am wrong--" (hugh whispered "with a strap!" but nobody heard him except lambert, who only frowned and shook his head at him) "--this annual custom of giving each of my sons as many guineas as would represent his age--it was a critical time--so balbus informed me--as the ages of two of you were together equal to that of the third--so on that occasion i made a speech----" he paused so long that balbus thought it well to come to the rescue with the words "it was a most----" but the old man checked him with a warning look: "yes, made a speech," he repeated. "a few years after that, balbus pointed out--i say pointed out--" ("hear, hear"! cried balbus. "quite so," said the grateful old man.) "--that it was _another_ critical occasion. the ages of two of you were together _double_ that of the third. so i made another speech--another speech. and now again it's a critical occasion--so balbus says--and i am making----" (here mad mathesis pointedly referred to her watch) "all the haste i can!" the old man cried, with wonderful presence of mind. "indeed, sister, i'm coming to the point now! the number of years that have passed since that first occasion is just two-thirds of the number of guineas i then gave you. now, my boys, calculate your ages from the _data_, and you shall have the money!" "but we _know_ our ages!" cried hugh. "silence, sir!" thundered the old man, rising to his full height (he was exactly five-foot five) in his indignation. "i say you must use the _data_ only! you mustn't even assume _which_ it is that comes of age!" he clutched the bag as he spoke, and with tottering steps (it was about as much as he could do to carry it) he left the room. "and _you_ shall have a similar _cadeau_," the old lady whispered to her niece, "when you've calculated that percentage!" and she followed her brother. nothing could exceed the solemnity with which the old couple had risen from the table, and yet was it--was it a _grin_ with which the father turned away from his unhappy sons? could it be--could it be a _wink_ with which the aunt abandoned her despairing niece? and were those--were those sounds of suppressed _chuckling_ which floated into the room, just before balbus (who had followed them out) closed the door? surely not: and yet the butler told the cook--but no, that was merely idle gossip, and i will not repeat it. the shades of evening granted their unuttered petition, and "closed not o'er" them (for the butler brought in the lamp): the same obliging shades left them a "lonely bark" (the wail of a dog, in the back-yard, baying the moon) for "awhile": but neither "morn, alas," (nor any other epoch) seemed likely to "restore" them--to that peace of mind which had once been theirs ere ever these problems had swooped upon them, and crushed them with a load of unfathomable mystery! "it's hardly fair," muttered hugh, "to give us such a jumble as this to work out!" "fair?" clara echoed, bitterly. "well!" and to all my readers i can but repeat the last words of gentle clara-- fare-well! appendix. "a knot!" said alice. "oh, do let me help to undo it!" answers to knot i. _problem._--"two travellers spend from o'clock till in walking along a level road, up a hill, and home again: their pace on the level being miles an hour, up hill , and down hill . find distance walked: also (within half an hour) time of reaching top of hill." _answer._--" miles: half-past ." * * * * * _solution._--a level mile takes / of an hour, up hill / , down hill / . hence to go and return over the same mile, whether on the level or on the hill-side, takes / an hour. hence in hours they went miles out and back. if the miles out had been nearly all level, they would have taken a little over hours; if nearly all up hill, a little under . hence - / hours must be within / an hour of the time taken in reaching the peak; thus, as they started at , they got there within / an hour of / past . * * * * * twenty-seven answers have come in. of these, are right, partially right, and wrong. the give the _distance_ correctly, but they have failed to grasp the fact that the top of the hill might have been reached at _any_ moment between o'clock and . the two wrong answers are from gerty vernon and a nihilist. the former makes the distance " miles," while her revolutionary companion puts it at " ." gerty vernon says "they had to go miles along the plain, and got to the foot of the hill at o'clock." they _might_ have done so, i grant; but you have no ground for saying they _did_ so. "it was - / miles to the top of the hill, and they reached that at / before o'clock." here you go wrong in your arithmetic, and i must, however reluctantly, bid you farewell. - / miles, at miles an hour, would _not_ require - / hours. a nihilist says "let _x_ denote the whole number of miles; _y_ the number of hours to hill-top; [** therefore] _y_ = number of miles to hill-top, and _x_- _y_ = number of miles on the other side." you bewilder me. the other side of _what_? "of the hill," you say. but then, how did they get home again? however, to accommodate your views we will build a new hostelry at the foot of the hill on the opposite side, and also assume (what i grant you is _possible_, though it is not _necessarily_ true) that there was no level road at all. even then you go wrong. you say "_y_ = - (_x_ - _y_)/ , ..... (i); _x_/ - / = ..... (ii)." i grant you (i), but i deny (ii): it rests on the assumption that to go _part_ of the time at miles an hour, and the rest at miles an hour, comes to the same result as going the _whole_ time at - / miles an hour. but this would only be true if the "_part_" were an exact _half_, i.e., if they went up hill for hours, and down hill for the other : which they certainly did _not_ do. the sixteen, who are partially right, are agnes bailey, f. k., fifee, g. e. b., h. p., kit, m. e. t., mysie, a mother's son, nairam, a redruthian, a socialist, spear maiden, t. b. c., vis inertiÃ�, and yak. of these, f. k., fifee, t. b. c., and vis inertiÃ� 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 miles of level road--and so on, all leading to _particular_ times being fixed for reaching the hill-top. the most curious assumption is that of agnes bailey, who says "let _x_ = number of hours occupied in ascent; then _x_/ = hours occupied in descent; and _x_/ = hours occupied on the level." i suppose you were thinking of the relative _rates_, up hill and on the level; which we might express by saying that, if they went _x_ miles up hill in a certain time, they would go _x_/ miles on the level _in the same time_. you have, in fact, assumed that they took _the same time_ on the level that they took in ascending the hill. fifee assumes that, when the aged knight said they had gone "four miles in the hour" on the level, he meant that four miles was the _distance_ gone, not merely the rate. this would have been--if fifee will excuse the slang expression--a "sell," ill-suited to the dignity of the hero. and now "descend, ye classic nine!" who have solved the whole problem, and let me sing your praises. your names are blithe, e. w., l. b., a marlborough boy, o. v. l., putney walker, rose, sea breeze, simple susan, and money spinner. (these last two i count as one, as they send a joint answer.) rose and simple susan and co. do not actually state that the hill-top was reached some time between and , but, as they have clearly grasped the fact that a mile, ascended and descended, took the same time as two level miles, i mark them as "right." a marlborough boy and putney walker deserve honourable mention for their algebraical solutions being the only two who have perceived that the question leads to _an indeterminate equation_. e. w. brings a charge of untruthfulness against the aged knight--a serious charge, for he was the very pink of chivalry! she says "according to the data given, the time at the summit affords no clue to the total distance. it does not enable us to state precisely to an inch how much level and how much hill there was on the road." "fair damsel," the aged knight replies, "--if, as i surmise, thy initials denote early womanhood--bethink thee that the word 'enable' is thine, not mine. i did but ask the time of reaching the hill-top as my _condition_ for further parley. if _now_ thou wilt not grant that i am a truth-loving man, then will i affirm that those same initials denote envenomed wickedness!" class list. i. a marlborough boy. putney walker. ii. blithe. e. w. l. b. o. v. l. rose. sea breeze. {simple susan. {money-spinner. blithe has made so ingenious an addition to the problem, and simple susan and co. have solved it in such tuneful verse, that i record both their answers in full. i have altered a word or two in blithe's--which i trust she will excuse; it did not seem quite clear as it stood. * * * * * "yet stay," said the youth, as a gleam of inspiration lighted up the relaxing muscles of his quiescent features. "stay. methinks it matters little _when_ we reached that summit, the crown of our toil. for in the space of time wherein we clambered up one mile and bounded down the same on our return, we could have trudged the _twain_ on the level. we have plodded, then, four-and-twenty miles in these six mortal hours; for never a moment did we stop for catching of fleeting breath or for gazing on the scene around!" "very good," said the old man. "twelve miles out and twelve miles in. and we reached the top some time between six and seven of the clock. now mark me! for every five minutes that had fled since six of the clock when we stood on yonder peak, so many miles had we toiled upwards on the dreary mountainside!" the youth moaned and rushed into the hostel. blithe. the elder and the younger knight, they sallied forth at three; how far they went on level ground it matters not to me; what time they reached the foot of hill, when they began to mount, are problems which i hold to be of very small account. the moment that each waved his hat upon the topmost peak-- to trivial query such as this no answer will i seek. yet can i tell the distance well they must have travelled o'er: on hill and plain, 'twixt three and nine, the miles were twenty-four. four miles an hour their steady pace along the level track, three when they climbed--but six when they came swiftly striding back adown the hill; and little skill it needs, methinks, to show, up hill and down together told, four miles an hour they go. for whether long or short the time upon the hill they spent, two thirds were passed in going up, one third in the descent. two thirds at three, one third at six, if rightly reckoned o'er, will make one whole at four--the tale is tangled now no more. simple susan. money spinner. answers to knot ii. § . the dinner party. _problem._--"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." _answer._--"one." * * * * * in this genealogy, males are denoted by capitals, and females by small letters. the governor is e and his guest is c. a = a | +------+-+----+ | | | b = b d = d c = c | | | | +---++--+ +-+-+ | | | | | | e = e | g = g | f ========= f ten answers have been received. of these, one is wrong, galanthus nivalis major, who insists on inviting _two_ guests, one being the governor's _wife's brother's father_. if she had taken his _sister's husband's father_ instead, she would have found it possible to reduce the guests to _one_. of the nine who send right answers, sea-breeze 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! bog-oak and bradshaw of the future use genealogies which require people instead of , by inviting the governor's _father's sister's husband_ instead of his _father's wife's brother_. i cannot think this so good a solution as one that requires only . caius and valentine deserve special mention as the only two who have supplied genealogies. class list. i. bee. caius. m. m. matthew matticks. old cat. valentine. ii. bog-oak. bradshaw of the future. iii. sea-breeze. § . the lodgings. _problem._--"a square has doors on each side, which contains equal parts. they are numbered all round, beginning at one corner. from which of the four, nos. , , , , is the sum of the distances, to the other three, least?" _answer._--"from no. ." * * * * * [illustration] let a be no. , b no. , c no. , and d no. . then ab = [** sqrt]( ^{ } + ^{ }) = [** sqrt] = ; ac = ; ad = [** sqrt]( ^{ } + ^{ }) = [** sqrt] = + (n.b. _i.e._ "between and .") bc = [** sqrt]( ^{ } + ^{ }) = [** sqrt] = ; bd = [** sqrt]( ^{ } + ^{ }) = [** sqrt] = +; cd = [** sqrt]( ^{ } + ^{ }) = [** sqrt] = +; hence sum of distances from a is between and ; from b, between and ; from c, between and ; from d, between and . (why not "between and "? make this out for yourselves.) hence the sum is least for a. * * * * * twenty-five solutions have been received. of these, must be marked " ," are partly right, and right. of the , i may dismiss alphabetical phantom, bog-oak, dinah mite, fifee, galanthus nivalis major (i fear the cold spring has blighted our snowdrop), guy, h.m.s. pinafore, janet, and valentine with the simple remark that they insist on the unfortunate lodgers _keeping to the pavement_. (i used the words "crossed to number seventy-three" for the special purpose of showing that _short cuts_ were possible.) sea-breeze 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. is the house, "as the diagram shows." i cannot see _how_ it does so. old cat assumes that the house _must_ be no. or no. . she does not explain how she estimates the distances. bee's arithmetic is faulty: she makes [** sqrt] + [** sqrt] + [** sqrt] = . (i suppose you mean [** sqrt] , which would be a little nearer the truth. but roots cannot be added in this manner. do you think [** sqrt] + [** sqrt] is , or even [** sqrt] ?) but ayr's 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 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. , , , : this will make your premiss true, and your conclusion false. similarly, for the second, try nos. , , , .) of the five partly-right solutions, rags and tatters and mad hatter (who send one answer between them) make no. units from the corner instead of . cheam, e. r. d. l., and meggy potts leave openings at the corners of the square, which are not in the _data_: moreover cheam gives values for the distances without any hint that they are only _approximations_. crophi and mophi make the bold and unfounded assumption that there were really houses on each side, instead of as stated by balbus. "we may assume," they add, "that the doors of nos. , , , , are invisible from the centre of the square"! what is there, i wonder, that crophi and mophi would _not_ assume? of the five who are wholly right, i think bradshaw of the future, caius, clifton c., and martreb deserve special praise for their full _analytical_ solutions. matthew matticks picks out no. , and proves it to be the right house in two ways, very neatly and ingeniously, but _why_ he picks it out does not appear. it is an excellent _synthetical_ proof, but lacks the analysis which the other four supply. class list. i. bradshaw of the future caius. clifton c. martreb. ii. matthew matticks. iii. cheam. crophi and mophi. e. r. d. l. meggy potts. {rags and tatters. {mad hatter. a remonstrance has reached me from scrutator on the subject of knot i., which he declares was "no problem at all." "two questions," he says, "are put. to solve one there is no data: the other answers itself." as to the first point, scrutator is mistaken; there _are_ (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. answers to knot iii. _problem._--( ) "two travellers, starting at the same time, went opposite ways round a circular railway. trains start each way every minutes, the easterly ones going round in hours, the westerly in . how many trains did each meet on the way, not counting trains met at the terminus itself?" ( ) "they went round, as before, each traveller counting as 'one' the train containing the other traveller. how many did each meet?" _answers._--( ) . ( ) the easterly traveller met ; the other . * * * * * the trains one way took minutes, the other way . let us take the l. c. m., , and divide the railway into units. then one set of trains went at the rate of units a minute and at intervals of units; the other at the rate of units a minute and at intervals of units. an easterly train starting has units between it and the first train it will meet: it does - ths of this while the other does - ths, and thus meets it at the end of units, and so all the way round. a westerly train starting has units between it and the first train it will meet: it does - ths of this while the other does - ths, and thus meets it at the end of units, and so all the way round. hence if the railway be divided, by posts, into parts, each containing units, trains meet at every post, and, in ( ), each traveller passes posts in going round, and so meets trains. but, in ( ), the easterly traveller only begins to count after traversing - ths of the journey, _i.e._, on reaching the th post, and so counts posts: similarly the other counts . they meet at the end of - ths of hours, or - ths of hours, _i.e._, minutes. * * * * * forty-five answers have been received. of these are beyond the reach of discussion, as they give no working. i can but enumerate their names. ardmore, e. a., f. a. d., l. d., matthew matticks, m. e. t., poo-poo, and the red queen are all wrong. beta and rowena have got ( ) right and ( ) wrong. cheeky bob and nairam 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 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.] of the answers for which the working is given, are wrong; half-wrong and half-right; right, except that they cherish the delusion that it was _clara_ who travelled in the easterly train--a point which the data do not enable us to settle; and wholly right. the wrong answers are from bo-peep, financier, i. w. t., kate b., m. a. h., q. y. z., sea-gull, thistledown, tom-quad, and an unsigned one. bo-peep rightly says that the easterly traveller met all trains which started during the hours of her trip, as well as all which started during the previous hours, _i.e._, all which started at the commencements of periods of minutes each; and she is right in striking out the one she met at the moment of starting; but wrong in striking out the _last_ train, for she did not meet this at the terminus, but minutes before she got there. she makes the same mistake in ( ). financier 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 minutes and - / seconds. kate b. thinks the trains which are met on starting and on arriving are _never_ 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. sea-gull seems to think that, in ( ), the easterly train _stood still_ for hours; and says that, in ( ), the travellers met at the end of minutes seconds. thistledown nobly confesses to having tried no calculation, but merely having drawn a picture of the railway and counted the trains; in ( ), she counts wrong; in ( ) she makes them meet in minutes. tom-quad omits ( ): in ( ) he makes clara count the train she met on her arrival. the unsigned one is also unintelligible; it states that the travellers go " - th more than the total distance to be traversed"! the "clara" theory, already referred to, is adopted by of these, viz., bo-peep, financier, kate b., tom-quad, and the nameless writer. the half-right answers are from bog-oak, bridget, castor, cheshire cat, g. e. b., guy, mary, m. a. h., old maid, r. w., and vendredi. all these adopt the "clara" theory. castor omits ( ). vendredi gets ( ) right, but in ( ) makes the same mistake as bo-peep. i notice in your solution a marvellous proportion-sum:--" miles: hours :: one mile: seconds." may i venture to advise your acquiring, as soon as possible, an utter disbelief in the possibility of a ratio existing between _miles_ and _hours_? do not be disheartened by your two friends' sarcastic remarks on your "roundabout ways." their short method, of adding and , has the slight disadvantage of bringing the answer wrong: even a "roundabout" method is better than _that_! m. a. h., in ( ), makes the travellers count "one" _after_ they met, not _when_ they met. cheshire cat and old maid get " " as answer for ( ), by forgetting to strike out the train met on arrival. the others all get " " in various ways. bog-oak, guy, and r. w. divide the trains which the westerly traveller has to meet into sets, viz., those already on the line, which they (rightly) make " ," and those which started during her hours' journey (exclusive of train met on arrival), which they (wrongly) make " "; and they make a similar mistake with the easterly train. bridget (rightly) says that the westerly traveller met a train every minutes for hours, but (wrongly) makes the number " "; it should be " ." g. e. b. adopts bo-peep's method, but (wrongly) strikes out (for the easterly traveller) the train which started at the _commencement_ of the previous hours. mary thinks a train, met on arrival, must not be counted, even when met on a _previous_ occasion. the , who are wholly right but for the unfortunate "clara" theory, are f. lee, g. s. c., and x. a. b. and now "descend, ye classic ten!" who have solved the whole problem. your names are aix-les-bains, algernon bray (thanks for a friendly remark, which comes with a heart-warmth that not even the atlantic could chill), arvon, bradshaw of the future, fifee, h. l. r., j. l. o., omega, s. s. g., and waiting for the train. several of these have put clara, provisionally, into the easterly train: but they seem to have understood that the data do not decide that point. class list. i. aix-les-bains. algernon bray. bradshaw of the future. fifee. h. l. r. omega. s. s. g. waiting for the train. ii. arvon. j. l. o. iii. f. lee. g. s. c. x. a. b. answers to knot iv. _problem._--"there are sacks, of which nos. , , weigh lbs.; nos. , , - / lbs.; nos. , , - / lbs.; nos. , , lbs.; nos. , , , lbs. required the weight of each sack." _answer._--" - / , - / , , - / , - / ." * * * * * the sum of all the weighings, lbs., includes sack no. _thrice_ and each other _twice_. deducting twice the sum of the st and th weighings, we get lbs. for _thrice_ no. , _i.e._, lbs. for no. . hence, the nd and rd weighings give - / lbs., - / lbs. for nos. , ; and hence again, the st and th weighings give - / lbs., - / lbs., for nos. , . * * * * * ninety-seven answers have been received. of these, 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 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 common sense, d. e. r., douglas, e. l., ellen, i. m. t., j. m. c., joseph, knot i, lucy, meek, m. f. c., pyramus, shah, veritas. of the eighty-two answers with which the working, or some approach 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. the solitary wrong answer is from nell. to be thus "alone in the crowd" is a distinction--a painful one, no doubt, but still a distinction. i 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 th and th bags weighed lbs. each? and why did you not test your answers? however, please try again: and please don't change your _nom-de-plume_: let us have nell in the first class next time! the seventeen whose solutions are practically valueless are ardmore, a ready reckoner, arthur, bog-lark, bog-oak, bridget, first attempt, j. l. c., m. e. t., rose, rowena, sea-breeze, sylvia, thistledown, three-fifths asleep, vendredi, and winifred. bog-lark tries it by a sort of "rule of false," assuming experimentally that nos. , , weigh lbs. each, and having thus produced - / , instead of , as the weight of , , and , she removes "the superfluous pound and a half," but does not explain how she knows from which to take it. three-fifths asleep says that (when in that peculiar state) "it seemed perfectly clear" to her that, " out of the sacks being weighed twice over, / of = , must be the total weight of the sacks." as to which i can only say, with the captain, "it beats me entirely!" winifred, on the plea that "one must have a starting-point," assumes (what i fear is a mere guess) that no. weighed - / lbs. the rest all do it, wholly or partly, by guess-work. the problem is of course (as any algebraist sees at once) a case of "simultaneous simple equations." it is, however, easily soluble 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 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., furze-bush, james, partridge, r. w., and waiting for the train, have sent long wandering solutions, the substitutions having no definite method, but seeming to have been made to see what would come of it. chilpome and dublin boy omit some of the working. arvon marlborough boy only finds the weight of _one_ sack. class list i. b. e. d. c. h. constance johnson. greystead. guy. hoopoe. j. f. a. m. a. h. number five. pedro. r. e. x. seven old men. vis inertiÃ�. willy b. yahoo. ii. american subscriber. an appreciative schoolma'am. ayr. bradshaw of the future. cheam. c. m. g. dinah mite. duckwing. e. c. m. e. n. lowry. era. euroclydon. f. h. w. fifee. g. e. b. harlequin. hawthorn. hough green. j. a. b. jack tar. j. b. b. kgovjni. land lubber. l. d. magpie. mary. mhruxi. minnie. money-spinner. nairam. old cat. polichinelle. simple susan. s. s. g. thisbe. verena. wamba. wolfe. wykehamicus. y. m. a. h. iii. a. c. m. arvon marlborough boy. chilpome. dublin boy. furze-bush. james. partridge. r. w. waiting for the train. answers to knot v. _problem._--to mark pictures, giving x's to or , to or , and to or ; also giving o's to or , to or and to or ; so as to mark the smallest possible number of pictures, and to give them the largest possible number of marks. _answer._-- pictures; marks; arranged thus:-- x x x x x x x x x o x x x x x o o o o x x o o o o o o o o _solution._--by giving all the x's possible, putting into brackets the optional ones, we get pictures marked thus:-- x x x x x x x x x (x) x x x x (x) x x (x) by then assigning o's in the same way, beginning at the other end, we get pictures marked thus:-- (o) o (o) o o o (o) o o o o o o o o all we have now to do is to run these two wedges as close together 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 necessary marks in the st row, and in the rd; but only in the nd. hence we erase all optional marks in the st and rd rows, but let them stand in the nd. * * * * * twenty-two answers have been received. of these give no working; so, in accordance with what i announced in my last review of answers, i leave them unnamed, merely mentioning that are right and wrong. of the eleven answers with which some working is supplied, 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), " is the least possible number of pictures" (granted): "therefore we must either give x's to , or o's to ." why "must," oh alphabetical phantom? it is nowhere ordained that every picture "must" have marks! fifee sends a folio page of solution, which deserved a better fate: she offers answers, in each of which pictures are marked, with marks; in one she gives x's to pictures; in another to ; in the rd she gives o's to ; thus in every case ignoring the conditions. (i pause to remark that the condition " x's to or pictures" can only mean "_either_ to _or else_ to ": if, as one competitor holds, it might mean _any_ number not less than , the words "_or_ " 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 x's to 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." 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 trust i. e. a. will be merciful to the spelling), it is scarcely to be distinguished from "zero." class list. i. guy. old cat. sea-breeze. ii. ayr. bradshaw of the future. f. lee. h. vernon. iii. cat. answers to knot vi. _problem ._--_a_ and _b_ began the year with only , _l._ a-piece. they borrowed nought; they stole nought. on the next new-year's day they had , _l._ between them. how did they do it? _solution._--they went that day to the bank of england. _a_ stood in front of it, while _b_ went round and stood behind it. * * * * * two answers have been received, both worthy of much honour. addlepate makes them borrow " " and steal " ," and uses both cyphers by putting them at the right-hand end of the , _l._, thus producing , _l._, which is well over the mark. but (or to express it in latin) at spes infracta has solved it even more ingeniously: with the first cypher she turns the " " of the , _l._ into a " ," and adds the result to the original sum, thus getting , _l._: and in this, by means of the other " ," she turns the " " into a " ," thus hitting the exact , _l._ class list i. at spes infracta. ii. addlepate. * * * * * _problem ._--_l_ makes scarves, while _m_ makes : _z_ makes while _l_ makes . five scarves of _z_'s weigh one of _l_'s; of _m_'s weigh of _z_'s. one of _m_'s is as warm as of _z_'s: and one of _l_'s as warm as of _m_'s. which is best, giving equal weight in the result to rapidity of work, lightness, and warmth? _answer._--the order is _m_, _l_, _z_. * * * * * _solution._--as to rapidity (other things being constant) _l_'s merit is to _m_'s in the ratio of to : _z_'s to _l_'s in the ratio of to . in order to get one set of numbers fulfilling these conditions, it is perhaps simplest to take the one that occurs _twice_ as unity, and reduce the others to fractions: this gives, for _l_, _m_, and _z_, the marks , / , / . in estimating for _lightness_, we observe that the greater the weight, the less the merit, so that _z_'s merit is to _l_'s as to . thus the marks for _lightness_ are / , / , . and similarly, the marks for warmth are , , / . to get the total result, we must _multiply_ _l_'s marks together, and do the same for _m_ and for _z_. the final numbers are Ã� / Ã� , / Ã� / Ã� , / Ã� Ã� / ; _i.e._ / , / , / ; _i.e._ multiplying throughout by (which will not alter the proportion), , , ; showing the order of merit to be _m_, _l_, _z_. * * * * * 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 _adding_ the proportional numbers together, for each candidate, instead of _multiplying_. _why_ 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 _illustrated_ very easily by the case of length, breadth, and depth. suppose _a_ and _b_ are rival diggers of rectangular tanks: the amount of work done is evidently measured by the number of _cubical feet_ dug out. let _a_ dig a tank feet long, wide, deep: let _b_ dig one feet long, wide, deep. the cubical contents are , ; _i.e._ _b_ is best digger in the ratio of to . now try marking for length, width, and depth, separately; giving a maximum mark of to the best in each contest, and then _adding_ the results! 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 : he, she, or it, is addlepate. the other twenty-three may be divided into five groups. first and worst are, i take it, those who put the rightful winner _last_; arranging them as "lolo, zuzu, mimi." the names of these desperate wrong-doers are ayr, bradshaw of the future, furze-bush and pollux (who send a joint answer), greystead, guy, old hen, and simple susan. the latter was _once_ 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. 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 graecia, m. m., old cat, and r. e. x. "'tis greece, but----." 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 ayr (who also appears among the "quite too too"), clifton c., f. b., fifee, grig, janet, and mrs. sairey gamp. f. b. has not fallen into the common error; she _multiplies_ together the proportionate numbers she gets, but in getting them she goes wrong, by reckoning warmth as a _de_-merit. possibly she is "freshly burnt," or comes "from bombay." janet and mrs. sairey gamp have also avoided this error: the method they have adopted is shrouded in mystery--i scarcely feel competent to criticize it. mrs. gamp says "if zuzu makes while lolo makes , zuzu makes while lolo makes (bad reasoning), while mimi makes ." from this she concludes "therefore zuzu excels in speed by " (_i.e._ when compared with lolo; but what about mimi?). she then compares the kinds of excellence, measured on this mystic scale. janet takes the statement, that "lolo makes while mimi makes ," to prove that "lolo makes while mimi makes and zuzu " (worse reasoning than mrs. gamp's), and thence concludes that "zuzu excels in speed by / "! janet should have been adeline, "mystery of mysteries!" the fourth set actually put mimi at the top, arranging them as "mimi, zuzu, lolo." they are marquis and co., martreb, s. b. b. (first initial scarcely legible: _may_ be meant for "j"), and stanza. the fifth set consist of an ancient fish and camel. 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 an ancient fish has very ancient and fishlike ideas as to _how_ numbers represent merit: she says "lolo gains - / on mimi." two and a half _what_? fish, fish, art thou in thy duty? of the five winners i put balbus and the elder traveller slightly below the other three--balbus for defective reasoning, the other for scanty working. balbus gives two reasons for saying that _addition_ of marks is _not_ the right method, and then adds "it follows that the decision must be made by _multiplying_ the marks together." this is hardly more logical than to say "this is not spring: _therefore_ it must be autumn." class list. i. dinah mite. e. b. d. l. joram. ii. balbus. the elder traveller. * * * * * with regard to knot v., i beg to express to vis inertiÃ� and to any others who, like her, understood the condition to be that _every_ marked picture must have _three_ marks, my sincere regret that the unfortunate phrase "_fill_ the columns with oughts and crosses" should have caused them to waste so much time and trouble. i can only repeat that a _literal_ interpretation of "fill" would seem to _me_ to require that _every_ picture in the gallery should be marked. vis inertiÃ� would have been in the first class if she had sent in the solution she now offers. answers to knot vii. _problem._--given that one glass of lemonade, sandwiches, and biscuits, cost _s._ _d._; and that one glass of lemonade, sandwiches, and biscuits, cost _s._ _d._: find the cost of ( ) a glass of lemonade, a sandwich, and a biscuit; and ( ) glasses of lemonade, sandwiches, and biscuits. _answer._--( ) _d._; ( ) _s._ _d._ _solution._--this is best treated algebraically. let _x_ = the cost (in pence) of a glass of lemonade, _y_ of a sandwich, and _z_ of a biscuit. then we have _x_ + _y_ + _z_ = , and _x_ + _y_ + _z_ = . and we require the values of _x_ + _y_ + _z_, and of _x_ + _y_ + _z_. now, from _two_ equations only, we cannot find, _separately_, the values of _three_ unknowns: certain _combinations_ of them may, however, be found. also we know that we can, by the help of the given equations, eliminate of the 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 rd unknown vanishing of itself: otherwise the problem is impossible. 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 st equation from the nd, which eliminates lemonade, and gives _y_ + _z_ = , or _y_ = - _z_; and then substituting this value of _y_ in the st, which gives _x_- _z_ = , _i.e._ _x_ = + _z_. now if we substitute these values of _x_, _y_, in the quantities whose values are required, the first becomes ( + _z_) + ( - _z_) + _z_, _i.e._ : and the second becomes ( + _z_) + ( - _z_) + _z_, _i.e._ . hence the answers are ( ) _d._, ( ) _s._ _d._ * * * * * the above is a _universal_ 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 _may_ fail, even when the thing is possible, and is of no use in proving it _im_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 competitors, who have sent in what i may call _accidental_ solutions, had had a question to deal with where every number contained or digits! i suspect it would have been a case of "silvered is the raven hair" (see "patience") before any solution would have been hit on by the most ingenious of them. forty-five answers have come in, of which give, i am happy to say, some sort of _working_, and therefore deserve to be mentioned by name, and to have their virtues, or vices as the case may be, discussed. thirteen have made assumptions to which they have no right, and so cannot figure in the class-list, even though, in of the cases, the answer is right. of the remaining , no less than have sent in _accidental_ solutions, and therefore fall short of the highest honours. i will now discuss individual cases, taking the worst first, as my custom is. froggy gives no working--at least this is all he gives: after stating the given equations, he says "therefore the difference, sandwich + biscuits, = _d._": then follow the amounts of the unknown bills, with no further hint as to how he got them. froggy has had a _very_ narrow escape of not being named at all! of those who are wrong, vis inertiÃ� has sent in a piece of incorrect working. peruse the horrid details, and shudder! she takes _x_ (call it "_y_") as the cost of a sandwich, and concludes (rightly enough) that a biscuit will cost ( -_y_)/ . she then subtracts the second equation from the first, and deduces _y_ + Ã� ( -_y_)/ - _y_ + Ã� ( -_y_)/ = . by making two mistakes in this line, she brings out _y_ = / . try it again, oh vis inertiÃ�! away with inertiÃ�: infuse a little more vis: and you will bring out the correct (though uninteresting) result, = ! this will show you that it is hopeless to try to coax any one of these unknowns to reveal its _separate_ 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 _d._ and _s._ _d._ he assumes, with too much confidence, that biscuits were / _d._ each, and that clara paid for , though she only ate ! we will now consider the whose working is wrong, though the answer is right: and, not to measure their demerits too exactly, i will take them in alphabetical order. anita finds (rightly) that " sandwich and biscuits cost _d._," and proceeds "therefore sandwich = - / _d._, biscuits = - / _d._, lemonade = _d._" dinah mite begins like anita: and thence proves (rightly) that a biscuit costs less than a _d._: whence she concludes (wrongly) that it _must_ cost / _d._ 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 _are_ the extenuating circumstances? she begins by assuming that lemonade is _d._ a glass, and sandwiches _d._ each, (making with the given equations, _four_ conditions to be fulfilled by _three_ miserable unknowns!). and, having (naturally) developed this into a contradiction, she then tries _d._ and _d._ with a similar result. (n.b. _this_ 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 _not_ scientific. janet identifies sandwiches with biscuits! "one sandwich + biscuits" she makes equal to " ." four _what_? mayfair makes the astounding assertion that the equation, _s_ + _b_ = , "is evidently only satisfied by _s_ = / , _b_ = / "! old cat believes that the assumption that a sandwich costs - / _d._ is "the only way to avoid unmanageable fractions." but _why_ 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 apart, let me thank old cat 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 ( ) and ( ): thence, by the process [( )-( )] deduces (rightly) equation ( ) viz. _s_ + _b_ = : and thence again, by the process [Ã� ] (a hopeless mystery), deduces _s_ + _b_ = . i have nothing to say about it: i give it up. sea-breeze says "it is immaterial to the answer" (why?) "in what proportion _d._ is divided between the sandwich and the biscuits": so she assumes _s_ = l- / _d._, _b_ = / _d._ stanza is one of a very irregular metre. at first she (like janet) identifies sandwiches with biscuits. she then tries two assumptions (_s_ = , _b_ = / , and _s_ = / _b_ = / ), and (naturally) ends in contradictions. then she returns to the first assumption, and finds the unknowns separately: _quod est absurdum_. stiletto 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. two sisters first assume that biscuits are a penny, and then that they are a penny, adding that "the answer will of course be the same in both cases." it is a dreamy remark, making one feel something like macbeth grasping at the spectral dagger. "is this a statement that i see before me?" if you were to say "we both walked the same way this morning," and _i_ were to say "_one_ of you walked the same way, but the other didn't," which of the three would be the most hopelessly confused? turtle pyate (what _is_ a turtle pyate, please?) and old crow, who send a joint answer, and y. y., adopt the same method. y. y. gets the equation _s_ + _b_ = : and then says "this sum must be apportioned in one of the three following ways." it _may_ be, i grant you: but y. y. do you say "must"? i fear it is _possible_ for y. y. to be _two_ y's. the other two conspirators are less positive: they say it "can" be so divided: but they add "either of the three prices being right"! this is bad grammar and bad arithmetic at once, oh mysterious birds! of those who win honours, the shetland snark must have the rd 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." the authors of the "accidental" solutions differ only in the number of steps they have taken between the _data_ and the answers. in order to do them full justice i have arranged the nd 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 theseus' 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:-- "aftur kyng aruirag, of wam we habbeth y told, marius ys sone was kyng, quoynte mon & bold. and ys sone was aftur hym, _coil_ was ys name, bothe it were quoynte men, & of noble fame." balbus lays it down as a general principle that "in order to ascertain the cost of any one luncheon, it must come to the same amount upon two different assumptions." (_query._ should not "it" be "we"? otherwise the _luncheon_ 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 luncheons as _d._ and _d._, 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 _one_ 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 _not_ say he's a duck." and the universal negative "nobody calls on her" is well met by the particular affirmative "_i_ called yesterday." in short, either of two contradictories disproves 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 _own_ assertions to "particulars," and leave _her_ to prove the "universal" contradictory, if she can. you will thus generally secure a _logical_ victory: a _practical_ victory is not to be hoped for, since she can always fall back upon the crushing remark "_that_ has nothing to do with it!"--a move for which man has not yet discovered any satisfactory answer. now let us return to balbus.) here is my "particular negative," on which to test his rule. suppose the two recorded luncheons to have been " buns, one queen-cake, sausage-rolls, and a bottle of zoëdone: total, one-and-ninepence," and "one bun, queen-cakes, a sausage-roll, and a bottle of zoëdone: total, one-and-fourpence." and suppose clara's unknown luncheon to have been " buns, one queen-cake, one sausage-roll, and bottles of zoëdone:" while the two little sisters had been indulging in " buns, queen-cakes, sausage-rolls, and bottles of zoëdone." (poor souls, how thirsty they must have been!) if balbus will kindly try this by his principle of "two assumptions," first assuming that a bun is _d._ and a queen-cake _d._, and then that a bun is _d._ and a queen-cake _d._, 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 _d._ each, the queen-cakes _d._, the sausage-rolls _d._, and the zoëdone _d._ a bottle: so that clara's third luncheon had cost one-and-sevenpence, and her thirsty friends had spent four-and-fourpence! another remark of balbus 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." 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!" 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 _arithmetical_ policeman. now try the other method. "i seed somebody a running, but he was well gone or ever _i_ got nigh the place. so i just took a look round in the garden. and 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 '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 _algebraical_ policeman--a higher intellectual type, to my thinking, than the other. little jack's solution calls for a word of praise, as he has written out what really is an algebraical proof _in words_, 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 simple susan for some kind words of sympathy, to the same effect as those received from old cat. hecla and martreb are the only two who have used a method _certain_ either to produce the answer, or else to prove it impossible: so they must share between them the highest honours. class list. i. hecla. martreb. ii. § ( _steps_). adelaide. clifton c.... e. k. c. guy. l'inconnu. little jack. nil desperandum. simple susan. yellow-hammer. woolly one. § ( _steps_). a. a. a christmas carol. afternoon tea. an appreciative schoolma'am. baby. balbus. bog-oak. the red queen. wall-flower. § ( _steps_). hawthorn. joram. s. s. g. § ( _steps_). a stepney coach. § ( _steps_). bay laurel. bradshaw of the future. § ( _steps_). old king cole. § ( _steps_). theseus. answers to correspondents. i have received several letters on the subjects of knots ii. and vi., which lead me to think some further explanation desirable. in knot ii., i had intended the numbering of the houses to begin at one corner of the square, and this was assumed by most, if not all, of the competitors. trojanus 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? in knot vi., the first problem was of course a mere _jeu de mots_, 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 st 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 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. 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. 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. to illustrate this, let us take two candidates for a prize, _a_ and _b_, who are to compete in french, german, and italian: (_a_) let it be laid down that the result is to depend on their _relative_ knowledge of each subject, so that, whether their marks, for french, be " , " or " , ," the result will be the same: and let it also be laid down that, if they get equal marks on papers, the final marks are to have the same ratio as those of the rd paper. this is a case of ordinary double rule of three. we multiply _a_'s marks together, and do the same for _b_. note that, if _a_ gets a single " ," his final mark is " ," even if he gets full marks for papers while _b_ gets only one mark for each paper. this of course would be very unfair on _a_, though a correct solution under the given conditions. (_b_) the result is to depend, as before, on _relative_ 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 (_a_), and so much nearer that it would be necessary to use the other multipliers _twice_ to produce the same result as in (_a_):" _e.g._ if the french ratio were / , and the others / , / so that the ultimate ratio, by method (_a_), would be / , i should multiply instead by / , / , giving the result, / which is nearer to / than if he had used method (_a_). (_c_) the result is to depend on _actual_ amount of knowledge of the subjects collectively. here we have to ask two questions. ( ) what is to be the "unit" (_i.e._ "standard to measure by") in each subject? ( ) 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 " ," all lower amounts are represented by numbers between " " and " ." then, if these units are to be of equal value, we simply add _a_'s marks together, and do the same for _b_. (_d_) the conditions are the same as (_c_), but french is to have double weight. here we simply double the french marks, and add as before. (_e_) 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 " " 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 _a_'s german and italian marks together, and multiply by his french mark. 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 (_a_), and to make this clear, i inserted the following passage: "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." what i have said will suffice, i hope, as an answer to balbus, who holds that (_a_) and (_c_) 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 fifee, who says "i think a little more consideration will show you that our 'error of _adding_ the proportional numbers together for each candidate instead of _multiplying_' is no error at all." why, even if addition _had_ 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! 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 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, _unless you are quite certain of being correct yourself_." 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. 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 being correct." i trust that the knot-untiers and i are not on such terms as those! i beg to thank _g. b._ for the offer of a puzzle--which, however, is too like the old one "make four 's into ." answers to knot viii. § . the pigs. _problem._--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. _answer._--place pigs in the first sty, in the second, nothing in the third, and in the fourth: is nearer ten than ; nothing is nearer ten than ; is nearer ten than nothing; and is nearer ten than . * * * * * this problem is noticed by only two correspondents. balbus says "it certainly cannot be solved mathematically, nor do i see how to solve it by any verbal quibble." nolens volens 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"! § . the grurmstipths. _problem._--omnibuses start from a certain point, both ways, every minutes. a traveller, starting on foot along with one of them, meets one in - / minutes: when will he be overtaken by one? _answer._--in - / minutes. * * * * * _solution._--let "_a_" be the distance an omnibus goes in minutes, and "_x_" the distance from the starting-point to where the traveller is overtaken. since the omnibus met is due at the starting-point in - / minutes, it goes in that time as far as the traveller walks in - / ; _i.e._ it goes times as fast. now the overtaking omnibus is "_a_" behind the traveller when he starts, and therefore goes "_a_ + _x_" while he goes "_x_." hence _a_ + _x_ = _x_; _i.e._ _x_ = _a_, and _x_ = _a_/ . this distance would be traversed by an omnibus in / minutes, and therefore by the traveller in Ã� / . hence he is overtaken in - / minutes after starting, _i.e._ in - / minutes after meeting the omnibus. four answers have been received, of which two are wrong. dinah mite rightly states that the overtaking omnibus reached the point where they met the other omnibus minutes after they left, but wrongly concludes that, going times as fast, it would overtake them in another minute. the travellers are -minutes-walk ahead of the omnibus, and must walk - th of this distance farther before the omnibus overtakes them, which will be - th of the distance traversed by the omnibus in the same time: this will require - / minutes more. nolens volens tries it by a process like "achilles and the tortoise." he rightly states that, when the overtaking omnibus leaves the gate, the travellers are - th of "_a_" ahead, and that it will take the omnibus minutes to traverse this distance; "during which time" the travellers, he tells us, go - th of "_a_" (this should be - th). the travellers being now - th of "_a_" ahead, he concludes that the work remaining to be done is for the travellers to go - th of "_a_," while the omnibus goes - th. the _principle_ is correct, and might have been applied earlier. class list. i. balbus. delta. answers to knot ix. § . the buckets. _problem._--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? _solution._--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. * * * * * 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. hecla says that "only that portion of the smaller bucket which descends below the original level of the water can be properly said to be immersed, and only an equal bulk of water is displaced." hence, according to hecla, 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. magpie 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. old king cole takes the same view as hecla. tympanum and vindex 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 hecla. i regret that there is no class-list to publish for this problem. * * * * * § . balbus' essay. _problem._--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? _solution._--no. this series can never reach inches, since, however many terms we take, we are always short of inches by an amount equal to the last term taken. * * * * * three answers have been received--but only two seem to me worthy of honours. tympanum says that the statement about the stick "is merely a blind, to which the old answer may well be applied, _solvitur ambulando_, or rather _mergendo_." i trust tympanum will not test this in his own person, by taking the place of the man in balbus' essay! he would infallibly be drowned. old king cole rightly points out that the series, , , &c., is a decreasing geometrical progression: while vindex rightly identifies the fallacy as that of "achilles and the tortoise." class list. i. old king cole. vindex. * * * * * § . the garden. _problem._--an oblong garden, half a yard longer than wide, consists entirely of a gravel-walk, spirally arranged, a yard wide and , yards long. find the dimensions of the garden. _answer._-- , - / . _solution._--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 , square-yards: _i.e._, if _x_ be the width, _x_ (_x_ + / ) = , . solving this quadratic, we find _x_ = . hence the dimensions are , - / . * * * * * twelve answers have been received--seven right and five wrong. c. g. l., nabob, old crow, and tympanum 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 , by . whence came this divisor, oh segiel? divination? or was it a dream? i fear this solution is worth nothing. old crow's is shorter, and so (if possible) worth rather less. he says the answer "is at once seen to be Ã� - / "! nabob's calculation is short, but "as rich as a nabob" in error. he says that the square root of , , multiplied by , equals the length plus the breadth. that is . Ã� = - / . his first assertion is only true of a _square_ garden. his second is irrelevant, since . is _not_ the square-root of , ! nay, bob, this will _not_ do! tympanum says that, by extracting the square-root of , , we get yards with a remainder of / , or half-a-yard, which we add so as to make the oblong Ã� - / . this is very terrible: but worse remains behind. tympanum 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, tympanum! my tympa is exhausted: my brain is num! i can say no more. hecla indulges, again and again, in that most fatal of all habits in computation--the making _two_ mistakes which cancel each other. she takes _x_ as the width of the garden, in yards, and _x_ + / as its length, and makes her first "coil" the sum of _x_- / , _x_- / , _x_- , _x_- , _i.e._ _x_- : but the fourth term should be _x_- - / , so that her first coil is / a yard too long. her second coil is the sum of _x_- - / , _x_- - / , _x_- , _x_- : here the first term should be _x_- and the last _x_- - / : these two 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 _end_ 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. of the seven who are right, dinah mite, janet, magpie, and taffy make the same assumption as c. g. l. and co. they then solve by a quadratic. magpie also tries it by arithmetical progression, but fails to notice that the first and last "coils" have special values. alumnus etonÃ� attempts to prove what c. g. l. assumes by a particular instance, taking a garden by - / . he ought to have proved it generally: what is true of one number is not always true of others. old king cole solves it by an arithmetical progression. it is right, but too lengthy to be worth as much as a quadratic. vindex 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." class list. i. vindex. ii. alumnus etonÃ�. old king cole. iii. dinah mite. janet. magpie. taffy. answers to knot x. § . the chelsea pensioners. _problem._--if per cent. have lost an eye, per cent. an ear, per cent. an arm, per cent. a leg: what percentage, _at least_, must have lost all four? _answer._--ten. * * * * * _solution._--(i adopt that of polar star, as being better than my own). adding the wounds together, we get + + + = , among men; which gives to each, and to men. therefore the least percentage is . * * * * * nineteen answers have been received. one is " ," but, as no working is given with it, it must, in accordance with the rule, remain "a deed without a name." janet makes it " and / ths." i am sorry she has misunderstood the question, and has supposed that those who had lost an ear were per cent. _of those who had lost an eye_; and so on. of course, on this supposition, the percentages must all be multiplied together. this she has done correctly, but i can give her no honours, as i do not think the question will fairly bear her interpretation, three score and ten makes it " and / 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 ("per cent.," i suppose!): dividing by , she gets and a half as "average percentage:" again dividing by , she gets and / ths as "percentage wounded four times." does she suppose wounds of different kinds to "absorb" each other, so to speak? then, no doubt, the _data_ are equivalent to pensioners with one wound each, and a half-pensioner with a half-wound. and does she then suppose these concentrated wounds to be _transferable_, so that / ths of these unfortunates can obtain perfect health by handing over their wounds to the remaining / th? granting these suppositions, her answer is right; or rather, _if_ the question had been "a road is covered with one inch of gravel, along and a half per cent. of it. how much of it could be covered inches deep with the same material?" her answer _would_ have been right. but alas, that _wasn't_ the question! delta 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." 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 _possible_. next come eight writers who have made the unwarrantable assumption that, because per cent. have lost an eye, _therefore_ per cent. have _not_ lost one, so that they have _both_ eyes. this is illogical. if you give me a bag containing sovereigns, and if in an hour i come to you (my face _not_ beaming with gratitude nearly so much as when i received the bag) to say "i am sorry to tell you that of these sovereigns are bad," do i thereby guarantee the other to be good? perhaps i have not tested them yet. the sides of this illogical octagon are as follows, in alphabetical order:--algernon bray, dinah mite, g. s. c., jane e., j. d. w., magpie (who makes the delightful remark "therefore 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 tokio. bradshaw of the future and t. r. do the question in a piecemeal fashion--on the principle that the per cent. and the per cent., though commenced at opposite ends of the , must overlap by _at least_ per cent.; and so on. this is quite correct working, but not, i think, quite the best way of doing it. 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! class list. i. old cat. old hen. polar star. simple susan. white sugar. ii. bradshaw of the future. t. r. iii. algernon bray. dinah mite. g. s. c. jane e. j. d. w. magpie. s. s. g. tokio. § . change of day. i must postpone, _sine die_, 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? § . the sons' ages. _problem._--"at first, two of the ages are together equal to the third. a few years afterwards, two of them are together double of the third. when the number of years since the first occasion is two-thirds of the sum of the ages on that occasion, one age is . what are the other two? _answer._--" and ." * * * * * _solution._--let the ages at first be _x_, _y_, (_x_ + _y_). now, if _a_ + _b_ = _c_, then (_a_-_n_) + (_b_-_n_) = (_c_-_n_), whatever be the value of _n_. hence the second relationship, if _ever_ true, was _always_ true. hence it was true at first. but it cannot be true that _x_ and _y_ are together double of (_x_ + _y_). hence it must be true of (_x_ + _y_), together with _x_ or _y_; and it does not matter which we take. we assume, then, (_x_ + _y_) + _x_ = _y_; _i.e._ _y_ = _x_. hence the three ages were, at first, _x_, _x_, _x_; and the number of years, since that time is two-thirds of _x_, _i.e._ is _x_. hence the present ages are _x_, _x_, _x_. the ages are clearly _integers_, since this is only "the year when one of my sons comes of age." hence _x_ = , _x_ = , and the other ages are , . * * * * * eighteen answers have been received. one of the writers merely asserts that the first occasion was years ago, that the ages were then , , and ; and that on the second occasion they were , , and ! as a roman father, i _ought_ to withhold the name of the rash writer; but respect for age makes me break the rule: it is three score and ten. jane e. also asserts that the ages at first were , , : then she calculates the present ages, leaving the _second_ occasion unnoticed. old hen is nearly as bad; she "tried various numbers till i found one that fitted _all_ the conditions"; but merely scratching up the earth, and pecking about, is _not_ the way to solve a problem, oh venerable bird! and close after old hen prowls, with hungry eyes, old cat, who calmly assumes, to begin with, that the son who comes of age is the _eldest_. eat your bird, puss, for you will get nothing from me! there are yet two zeroes to dispose of. minerva assumes that, on _every_ occasion, a son comes of age; and that it is only such a son who is "tipped with gold." is it wise thus to interpret "now, my boys, calculate your ages, and you shall have the money"? bradshaw of the future says "let" the ages at first be , , , then assumes that the second occasion was years afterwards, and on these baseless assumptions brings out the right answers. guide _future_ travellers, an thou wilt: thou art no bradshaw for _this_ age! of those who win honours, the merely "honourable" are two. dinah mite ascertains (rightly) the relationship between the three ages at first, but then _assumes_ one of them to be " ," 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 _z_, _z_, and _z_; it then assumes, without giving any reason, that _z_ = . of the more honourable, delta attempts a novelty--to discover _which_ 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 _apparently_ brings out an absurdity. still, as the proof contains the following bit of algebra, " = _x_ + _y_; [** therefore] = _x_ + sevenths of _y_," i trust it will admit that its proof is not _quite_ conclusive. the rest of its work is good. magpie betrays the deplorable tendency of her tribe--to appropriate any stray conclusion she comes across, without having any _strict_ logical right to it. assuming _a_, _b_, _c_, as the ages at first, and _d_ as the number of the years that have elapsed since then, she finds (rightly) the equations, _a_ = _b_, _c_ = _b_ + _a_, _d_ = _b_. she then says "supposing that _a_ = , then _b_ = , _c_ = , and _d_ = . therefore for _a_, _b_, _c_, _d_, four numbers are wanted which shall be to each other as : : : ." it is in the "therefore" that i detect the unconscientiousness of this bird. the conclusion _is_ true, but this is only because the equations are "homogeneous" (_i.e._ having one "unknown" in each term), a fact which i strongly suspect had not been grasped--i beg pardon, clawed--by her. were i to lay this little pitfall, "_a_ + = _b_, _b_ + = _c_; supposing _a_ = , then _b_ = and _c_ = . _therefore_ for _a_, _b_, _c_, three numbers are wanted which shall be to one another as : : ," would you not flutter down into it, oh magpie, as amiably as a dove? simple susan is anything but simple to _me_. after ascertaining that the ages at first are as : : , she says "then, as two-thirds of their sum, added to one of them, = , the sum cannot exceed , and consequently the highest cannot exceed ." i suppose her (mental) argument is something like this:--"two-thirds of sum, + one age, = ; [** therefore] sum, + halves of one age, = and a half. but halves of one age cannot be less than and-a-half (here i perceive that simple susan would on no account present a guinea to a new-born baby!) hence the sum cannot exceed ." this is ingenious, but her proof, after that, is (as she candidly admits) "clumsy and roundabout." she finds that there are possible sets of ages, and eliminates four of them. suppose that, instead of , there had been million possible sets? would simple susan have courageously ordered in the necessary gallon of ink and ream of paper? the solution sent in by c. r. is, like that of simple susan, partly tentative, and so does not rise higher than being clumsily right. among those who have earned the highest honours, algernon bray solves the problem quite correctly, but adds that there is nothing to exclude the supposition that all the ages were _fractional_. this would make the number of answers infinite. let me meekly protest that i _never_ intended my readers to devote the rest of their lives to writing out answers! e. m. rix points out that, if fractional ages be admissible, any one of the three sons might be the one "come of age"; but she rightly rejects this supposition on the ground that it would make the problem indeterminate. white sugar 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 _year_, need not have done so by that _day_, so that he _might_ be only . this gives a second solution, viz., , , . well said, pure crystal! verily, thy "fair discourse hath been as sugar"! class list. i. algernon bray. an old fogey. e. m. rix. g. s. c. s. s. g. tokio. t. r. white sugar. ii. c. r. delta. magpie. simple susan. iii. dinah mite. m. f. c. * * * * * i have received more than one remonstrance on my assertion, in the chelsea pensioners' problem, that it was illogical to assume, from the _datum_ " p. c. have lost an eye," that p. c. have _not_. algernon bray states, as a parallel case, "suppose tommy's father gives him apples, and he eats one of them, how many has he left?" and says "i think we are justified in answering, ." i think so too. there is no "must" here, and the _data_ are evidently meant to fix the answer _exactly_: but, if the question were set me "how many _must_ he have left?", i should understand the _data_ to be that his father gave him _at least_, but _may_ have given him more. 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 _in_ my life (like those i now address), nor yet (like alice and the mock turtle) distinctly _out_ of it. yet let me at least fancy, as i lay down the pen, that i carry with me into my silent life, 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. the end london: richard clay and sons, printers. [turn over. works by lewis carroll. alice's adventures in wonderland. with forty-two illustrations by tenniel. crown vo, cloth, gilt edges, price _s._ seventy-fifth thousand. translations of the same--into french, by henri buÃ�--into german, by antonie zimmermann--and into italian, by t. pietrocÃ�la rossetti--with tenniel's illustrations. crown vo, cloth, gilt edges, price _s._ each. through the looking-glass, and what alice found there. with fifty illustrations by tenniel. crown vo, cloth, gilt edges, price _s._ fifty-sixth thousand. rhyme? and reason? with sixty-five illustrations by arthur b. frost, and nine by henry holiday. (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 vo, cloth, coloured edges, price _s._ fifty thousand. a tangled tale. reprinted from _the monthly packet_. with six illustrations by arthur b. frost. crown vo, _s._ _d._ * * * * * n.b. in selling the above-mentioned books to the trade, messrs. macmillan and co. will abate _d._ in the shilling (no odd copies), and allow per cent. discount for payment within six months, and per cent. for cash. in selling them to the public (for cash only) they will allow per cent. discount. * * * * * mr. lewis carroll, having been requested to allow "an easter greeting" (a leaflet, addressed to children, and frequently given with his books) to be sold separately, has arranged with messrs. harrison, of , pall mall, who will supply a single copy for _d._, or for _d._, or for _s._ macmillan and co., london. london: richard clay and sons, printers. * * * * * transcriber's note the following changes have been made to the text: page : "he corners of the" changed to "the corners of the". page : "aix-le-bains" changed to "aix-les-bains". page : " / , , / " changed to " / , / , / ". page : " of the cases" changed to " of the cases". page : "four-and fourpence" changed to "four-and-fourpence". last page: "fifth thousand" changed to "fifty thousand". brett fishburne the game of logic by lewis carroll --------------------- | | | | | | | -----x------ | | | | | | | | | | | |---y-----m------y'---| | | | | | | | | | | | -----x'----- | | | | | | | --------------------- colours for ------------- counters | | | ___ | x | | | | see the sun is overhead, |--y-------y'-| shining on us, full and | | | red! | x' | | | | now the sun is gone away, ------------- and the empty sky is grey! ___ the game of logic by lewis carrol to my child-friend. i charm in vain; for never again, all keenly as my glance i bend, will memory, goddess coy, embody for my joy departed days, nor let me gaze on thee, my fairy friend! yet could thy face, in mystic grace, a moment smile on me, 'twould send far-darting rays of light from heaven athwart the night, by which to read in very deed thy spirit, sweetest friend! so may the stream of life's long dream flow gently onward to its end, with many a floweret gay, adown its willowy way: may no sigh vex, no care perplex, my loving little friend! nota bene. with each copy of this book is given an envelope, containing a diagram (similar to the frontispiece) on card, and nine counters, four red and five grey. the envelope, &c. can be had separately, at d. each. the author will be very grateful for suggestions, especially from beginners in logic, of any alterations, or further explanations, that may seem desirable. letters should be addressed to him at " , bedford street, covent garden, london." preface "there foam'd rebellious logic, gagg'd and bound." this game requires nine counters--four of one colour and five of another: say four red and five grey. besides the nine counters, it also requires one player, at least. i am not aware of any game that can be played with less than this number: while there are several that require more: take cricket, for instance, which requires twenty-two. how much easier it is, when you want to play a game, to find one player than twenty-two. at the same time, though one player is enough, a good deal more amusement may be got by two working at it together, and correcting each other's mistakes. a second advantage, possessed by this game, is that, besides being an endless source of amusement (the number of arguments, that may be worked by it, being infinite), it will give the players a little instruction as well. but is there any great harm in that, so long as you get plenty of amusement? contents. chapter page i. new lamps for old. . propositions . . . . . . . . syllogisms . . . . . . . . . fallacies . . . . . . . . ii. cross questions. . elementary . . . . . . . . . half of smaller diagram. propositions to be represented . . . . . . do. symbols to be interpreted. . . smaller diagram. propositions to be represented . . . . . . . . do. symbols to be interpreted. . . larger diagram. propositions to be represented . . . . . . . . both diagrams to be employed . . iii. crooked answers. . elementary . . . . . . . . . half of smaller diagram. propositions represented . . . . . . . . do. symbols interpreted . . . . smaller diagram. propositions represented. . do. symbols interpreted . . . . larger diagram. propositions represented. . both diagrams employed . . . . iv. hit or miss . . . . . . . . . chapter i. new lamps for old. "light come, light go." _________ . propositions. "some new cakes are nice." "no new cakes are nice." "all new cakes are nice." there are three 'propositions' for you--the only three kinds we are going to use in this game: and the first thing to be done is to learn how to express them on the board. let us begin with "some new cakes are nice." but before doing so, a remark has to be made--one that is rather important, and by no means easy to understand all in a moment: so please to read this very carefully. the world contains many things (such as "buns", "babies", "beetles". "battledores". &c.); and these things possess many attributes (such as "baked", "beautiful", "black", "broken", &c.: in fact, whatever can be "attributed to", that is "said to belong to", any thing, is an attribute). whenever we wish to mention a thing, we use a substantive: when we wish to mention an attribute, we use an adjective. people have asked the question "can a thing exist without any attributes belonging to it?" it is a very puzzling question, and i'm not going to try to answer it: let us turn up our noses, and treat it with contemptuous silence, as if it really wasn't worth noticing. but, if they put it the other way, and ask "can an attribute exist without any thing for it to belong to?", we may say at once "no: no more than a baby could go a railway-journey with no one to take care of it!" you never saw "beautiful" floating about in the air, or littered about on the floor, without any thing to be beautiful, now did you? and now what am i driving at, in all this long rigmarole? it is this. you may put "is" or "are" between names of two things (for example, "some pigs are fat animals"), or between the names of two attributes (for example, "pink is light-red"), and in each case it will make good sense. but, if you put "is" or "are" between the name of a thing and the name of an attribute (for example, "some pigs are pink"), you do not make good sense (for how can a thing be an attribute?) unless you have an understanding with the person to whom you are speaking. and the simplest understanding would, i think, be this--that the substantive shall be supposed to be repeated at the end of the sentence, so that the sentence, if written out in full, would be "some pigs are pink (pigs)". and now the word "are" makes quite good sense. thus, in order to make good sense of the proposition "some new cakes are nice", we must suppose it to be written out in full, in the form "some new cakes are nice (cakes)". now this contains two 'terms'--"new cakes" being one of them, and "nice (cakes)" the other. "new cakes," being the one we are talking about, is called the 'subject' of the proposition, and "nice (cakes)" the 'predicate'. also this proposition is said to be a 'particular' one, since it does not speak of the whole of its subject, but only of a part of it. the other two kinds are said to be 'universal', because they speak of the whole of their subjects--the one denying niceness, and the other asserting it, of the whole class of "new cakes". lastly, if you would like to have a definition of the word 'proposition' itself, you may take this:--"a sentence stating that some, or none, or all, of the things belonging to a certain class, called its 'subject', are also things belonging to a certain other class, called its 'predicate'". you will find these seven words--proposition, attribute, term, subject, predicate, particular, universal--charmingly useful, if any friend should happen to ask if you have ever studied logic. mind you bring all seven words into your answer, and you friend will go away deeply impressed--'a sadder and a wiser man'. now please to look at the smaller diagram on the board, and suppose it to be a cupboard, intended for all the cakes in the world (it would have to be a good large one, of course). and let us suppose all the new ones to be put into the upper half (marked 'x'), and all the rest (that is, the not-new ones) into the lower half (marked 'x''). thus the lower half would contain elderly cakes, aged cakes, ante-diluvian cakes--if there are any: i haven't seen many, myself--and so on. let us also suppose all the nice cakes to be put into the left-hand half (marked 'y'), and all the rest (that is, the not-nice ones) into the right-hand half (marked 'y''). at present, then, we must understand x to mean "new", x' "not-new", y "nice", and y' "not-nice." and now what kind of cakes would you expect to find in compartment no. ? it is part of the upper half, you see; so that, if it has any cakes in it, they must be new: and it is part of the left-hand half; so that they must be nice. hence if there are any cakes in this compartment, they must have the double 'attribute' "new and nice": or, if we use letters, the must be "x y." observe that the letters x, y are written on two of the edges of this compartment. this you will find a very convenient rule for knowing what attributes belong to the things in any compartment. take no. , for instance. if there are any cakes there, they must be "x' y", that is, they must be "not-new and nice." now let us make another agreement--that a red counter in a compartment shall mean that it is 'occupied', that is, that there are some cakes in it. (the word 'some,' in logic, means 'one or more' so that a single cake in a compartment would be quite enough reason for saying "there are some cakes here"). also let us agree that a grey counter in a compartment shall mean that it is 'empty', that is that there are no cakes in it. in the following diagrams, i shall put ' ' (meaning 'one or more') where you are to put a red counter, and ' ' (meaning 'none') where you are to put a grey one. as the subject of our proposition is to be "new cakes", we are only concerned, at present, with the upper half of the cupboard, where all the cakes have the attribute x, that is, "new." now, fixing our attention on this upper half, suppose we found it marked like this, ----------- | | | | | | | | | ----------- that is, with a red counter in no. . what would this tell us, with regard to the class of "new cakes"? would it not tell us that there are some of them in the x y-compartment? that is, that some of them (besides having the attribute x, which belongs to both compartments) have the attribute y (that is, "nice"). this we might express by saying "some x-cakes are y-(cakes)", or, putting words instead of letters, "some new cakes are nice (cakes)", or, in a shorter form, "some new cakes are nice". at last we have found out how to represent the first proposition of this section. if you have not clearly understood all i have said, go no further, but read it over and over again, till you do understand it. after that is once mastered, you will find all the rest quite easy. it will save a little trouble, in doing the other propositions, if we agree to leave out the word "cakes" altogether. i find it convenient to call the whole class of things, for which the cupboard is intended, the 'universe.' thus we might have begun this business by saying "let us take a universe of cakes." (sounds nice, doesn't it?) of course any other things would have done just as well as cakes. we might make propositions about "a universe of lizards", or even "a universe of hornets". (wouldn't that be a charming universe to live in?) so far, then, we have learned that ----------- | | | | | | | | | ----------- means "some x and y," i.e. "some new are nice." i think you will see without further explanation, that ----------- | | | | | | | | | ----------- means "some x are y'," i.e. "some new are not-nice." now let us put a grey counter into no. , and ask ourselves the meaning of ----------- | | | | | | | | | ----------- this tells us that the x y-compartment is empty, which we may express by "no x are y", or, "no new cakes are nice". this is the second of the three propositions at the head of this section. in the same way, ----------- | | | | | | | | | ----------- would mean "no x are y'," or, "no new cakes are not-nice." what would you make of this, i wonder? ----------- | | | | | | | | | ----------- i hope you will not have much trouble in making out that this represents a double proposition: namely, "some x are y, and some are y'," i.e. "some new are nice, and some are not-nice." the following is a little harder, perhaps: ----------- | | | | | | | | | ----------- this means "no x are y, and none are y'," i.e. "no new are nice, and none are not-nice": which leads to the rather curious result that "no new exist," i.e. "no cakes are new." this is because "nice" and "not-nice" make what we call an 'exhaustive' division of the class "new cakes": i.e. between them, they exhaust the whole class, so that all the new cakes, that exist, must be found in one or the other of them. and now suppose you had to represent, with counters the contradictory to "no cakes are new", which would be "some cakes are new", or, putting letters for words, "some cakes are x", how would you do it? this will puzzle you a little, i expect. evidently you must put a red counter somewhere in the x-half of the cupboard, since you know there are some new cakes. but you must not put it into the left-hand compartment, since you do not know them to be nice: nor may you put it into the right-hand one, since you do not know them to be not-nice. what, then, are you to do? i think the best way out of the difficulty is to place the red counter on the division-line between the xy-compartment and the xy'-compartment. this i shall represent (as i always put ' ' where you are to put a red counter) by the diagram ----------- | | | | - - | | | | ----------- our ingenious american cousins have invented a phrase to express the position of a man who wants to join one or the other of two parties--such as their two parties 'democrats' and 'republicans'--but can't make up his mind which. such a man is said to be "sitting on the fence." now that is exactly the position of the red counter you have just placed on the division-line. he likes the look of no. , and he likes the look of no. , and he doesn't know which to jump down into. so there he sits astride, silly fellow, dangling his legs, one on each side of the fence! now i am going to give you a much harder one to make out. what does this mean? ----------- | | | | | | | | | ----------- this is clearly a double proposition. it tells us not only that "some x are y," but also the "no x are not y." hence the result is "all x are y," i.e. "all new cakes are nice", which is the last of the three propositions at the head of this section. we see, then, that the universal proposition "all new cakes are nice" consists of two propositions taken together, namely, "some new cakes are nice," and "no new cakes are not-nice." in the same way ----------- | | | | | | | | | ----------- would mean "all x are y' ", that is, "all new cakes are not-nice." now what would you make of such a proposition as "the cake you have given me is nice"? is it particular or universal? "particular, of course," you readily reply. "one single cake is hardly worth calling 'some,' even." no, my dear impulsive reader, it is 'universal'. remember that, few as they are (and i grant you they couldn't well be fewer), they are (or rather 'it is') all that you have given me! thus, if (leaving 'red' out of the question) i divide my universe of cakes into two classes--the cakes you have given me (to which i assign the upper half of the cupboard), and those you haven't given me (which are to go below)--i find the lower half fairly full, and the upper one as nearly as possible empty. and then, when i am told to put an upright division into each half, keeping the nice cakes to the left, and the not-nice ones to the right, i begin by carefully collecting all the cakes you have given me (saying to myself, from time to time, "generous creature! how shall i ever repay such kindness?"), and piling them up in the left-hand compartment. and it doesn't take long to do it! here is another universal proposition for you. "barzillai beckalegg is an honest man." that means "all the barzillai beckaleggs, that i am now considering, are honest men." (you think i invented that name, now don't you? but i didn't. it's on a carrier's cart, somewhere down in cornwall.) this kind of universal proposition (where the subject is a single thing) is called an 'individual' proposition. now let us take "nice cakes" as the subject of proposition: that is, let us fix our thoughts on the left-hand half of the cupboard, where all the cakes have attribute y, that is, "nice." ----- suppose we find it marked like this:-- | | | | what would that tell us? | | ----- | | | | | | ----- i hope that it is not necessary, after explaining the horizontal oblong so fully, to spend much time over the upright one. i hope you will see, for yourself, that this means "some y are x", that is, "some nice cakes are new." "but," you will say, "we have had this case before. you put a red counter into no. , and you told us it meant 'some new cakes are nice'; and now you tell us that it means 'some nice cakes are new'! can it mean both?" the question is a very thoughtful one, and does you great credit, dear reader! it does mean both. if you choose to take x (that is, "new cakes") as your subject, and to regard no. as part of a horizontal oblong, you may read it "some x are y", that is, "some new cakes are nice": but, if you choose to take y (that is, "nice cake") as your subject, and to regard no. as part of an upright oblong, then you may read it "some y are x", that is, "some nice cakes are new". they are merely two different ways of expressing the very same truth. without more words, i will simply set down the other ways in which this upright oblong might be marked, adding the meaning in each case. by comparing them with the various cases of the horizontal oblong, you will, i hope, be able to understand them clearly. you will find it a good plan to examine yourself on this table, by covering up first one column and then the other, and 'dodging about', as the children say. also you will do well to write out for yourself two other tables--one for the lower half of the cupboard, and the other for its right-hand half. and now i think we have said all we need to say about the smaller diagram, and may go on to the larger one. _________________________________________________ | symbols. | meanings. _______________|_________________________________ ----- | | | | | | | some y are x'; | | | i.e. some nice are not-new. ----- | | | | | | | | | | ----- | | ----- | | | | no y are x; | | | i.e. no nice are new. | | | ----- | [observe that this is merely another way of | | | expressing "no new are nice."] | | | | | | ----- | | ----- | | | | | | | no y are x'; | | | i.e. no nice are not-new. ----- | | | | | | | | | | ----- | | ----- | | | | | | | some y are x, and some are x'; | | | i.e. some nice are new, and some are ----- | not-new. | | | | | | | | | ----- | | ----- | | | | | | | no y are x, and none are x'; i.e. no y | | | exist; ----- | i.e. no cakes are nice. | | | | | | | | | ----- | | ----- | | | | | | | all y are x; | | | i.e. all nice are new. ----- | | | | | | | | | | ----- | | ----- | | | | | | | all y are x'; | | | i.e. all nice are not-new. ----- | | | | | | | | | | ----- | _______________|_________________________________ this may be taken to be a cupboard divided in the same way as the last, but also divided into two portions, for the attribute m. let us give to m the meaning "wholesome": and let us suppose that all wholesome cakes are placed inside the central square, and all the unwholesome ones outside it, that is, in one or other of the four queer-shaped outer compartments. we see that, just as, in the smaller diagram, the cakes in each compartment had two attributes, so, here, the cakes in each compartment have three attributes: and, just as the letters, representing the two attributes, were written on the edges of the compartment, so, here, they are written at the corners. (observe that m' is supposed to be written at each of the four outer corners.) so that we can tell in a moment, by looking at a compartment, what three attributes belong to the things in it. for instance, take no. . here we find x, y', m, at the corners: so we know that the cakes in it, if there are any, have the triple attribute, 'xy'm', that is, "new, not-nice, and wholesome." again, take no. . here we find, at the corners, x', y', m': so the cakes in it are "not-new, not-nice, and unwholesome." (remarkably untempting cakes!) it would take far too long to go through all the propositions, containing x and y, x and m, and y and m which can be represented on this diagram (there are ninety-six altogether, so i am sure you will excuse me!) and i must content myself with doing two or three, as specimens. you will do well to work out a lot more for yourself. taking the upper half by itself, so that our subject is "new cakes", how are we to represent "no new cakes are wholesome"? this is, writing letters for words, "no x are m." now this tells us that none of the cakes, belonging to the upper half of the cupboard, are to be found inside the central square: that is, the two compartments, no. and no. , are empty. and this, of course, is represented by ------------------- | | | | _____|_____ | | | | | | | | | | | | | | | | ------------------- and now how are we to represent the contradictory proposition "some x are m"? this is a difficulty i have already considered. i think the best way is to place a red counter on the division-line between no. and no. , and to understand this to mean that one of the two compartments is 'occupied,' but that we do not at present know which. this i shall represent thus:-- ------------------- | | | | _____|_____ | | | | | | | | - - | | | | | | | ------------------- now let us express "all x are m." this consists, we know, of two propositions, "some x are m," and "no x are m'." let us express the negative part first. this tells us that none of the cakes, belonging to the upper half of the cupboard, are to be found outside the central square: that is, the two compartments, no. and no. , are empty. this, of course, is represented by ------------------- | | | | _____|_____ | | | | | | | | | | | | | | | | ------------------- but we have yet to represent "some x are m." this tells us that there are some cakes in the oblong consisting of no. and no. : so we place our red counter, as in the previous example, on the division-line between no. and no. , and the result is ------------------- | | | | _____|_____ | | | | | | | | - - | | | | | | | ------------------- now let us try one or two interpretations. what are we to make of this, with regard to x and y? ------------------- | | | | _____|_____ | | | | | | | | | | | | | | | | ------------------- this tells us, with regard to the xy'-square, that it is wholly 'empty', since both compartments are so marked. with regard to the xy-square, it tells us that it is 'occupied'. true, it is only one compartment of it that is so marked; but that is quite enough, whether the other be 'occupied' or 'empty', to settle the fact that there is something in the square. if, then, we transfer our marks to the smaller diagram, so as to get rid of the m-subdivisions, we have a right to mark it ----------- | | | | | | | | | ----------- which means, you know, "all x are y." the result would have been exactly the same, if the given oblong had been marked thus:-- ------------------- | | | | _____|_____ | | | | | | | | | | | | | | | | ------------------- once more: how shall we interpret this, with regard to x and y? ------------------- | | | | _____|_____ | | | | | | | | | | | | | | | | ------------------- this tells us, as to the xy-square, that one of its compartments is 'empty'. but this information is quite useless, as there is no mark in the other compartment. if the other compartment happened to be 'empty' too, the square would be 'empty': and, if it happened to be 'occupied', the square would be 'occupied'. so, as we do not know which is the case, we can say nothing about this square. the other square, the xy'-square, we know (as in the previous example) to be 'occupied'. if, then, we transfer our marks to the smaller diagram, we get merely this:-- ----------- | | | | | | | | | ----------- which means, you know, "some x are y'." these principles may be applied to all the other oblongs. for instance, to represent "all y' are m'" we should mark the ------- right-hand upright oblong (the one | | that has the attribute y') thus:-- |--- | | | | |---|- -| | | | |--- | | | ------- and, if we were told to interpret the lower half of the cupboard, marked as follows, with regard to x and y, ------------------- | | | | | | | | | | | | | | | | -----|----- | | | | ------------------- we should transfer it to the smaller diagram thus, ----------- | | | | | | | | | ----------- and read it "all x' are y." two more remarks about propositions need to be made. one is that, in every proposition beginning with "some" or "all", the actual existence of the 'subject' is asserted. if, for instance, i say "all misers are selfish," i mean that misers actually exist. if i wished to avoid making this assertion, and merely to state the law that miserliness necessarily involves selfishness, i should say "no misers are unselfish" which does not assert that any misers exist at all, but merely that, if any did exist, they would be selfish. the other is that, when a proposition begins with "some" or "no", and contains more that two attributes, these attributes may be re-arranged, and shifted from one term to the other, "ad libitum." for example, "some abc are def" may be re-arranged as "some bf are acde," each being equivalent to "some things are abcdef". again "no wise old men are rash and reckless gamblers" may be re-arranged as "no rash old gamblers are wise and reckless," each being equivalent to "no men are wise old rash reckless gamblers." . syllogisms now suppose we divide our universe of things in three ways, with regard to three different attributes. out of these three attributes, we may make up three different couples (for instance, if they were a, b, c, we might make up the three couples ab, ac, bc). also suppose we have two propositions given us, containing two of these three couples, and that from them we can prove a third proposition containing the third couple. (for example, if we divide our universe for m, x, and y; and if we have the two propositions given us, "no m are x'" and "all m' are y", containing the two couples mx and my, it might be possible to prove from them a third proposition, containing x and y.) in such a case we call the given propositions 'the premisses', the third one 'the conclusion' and the whole set 'a syllogism'. evidently, one of the attributes must occur in both premisses; or else one must occur in one premiss, and its contradictory in the other. in the first case (when, for example, the premisses are "some m are x" and "no m are y'") the term, which occurs twice, is called 'the middle term', because it serves as a sort of link between the other two terms. in the second case (when, for example, the premisses are "no m are x'" and "all m' are y") the two terms, which contain these contradictory attributes, may be called 'the middle terms'. thus, in the first case, the class of "m-things" is the middle term; and, in the second case, the two classes of "m-things" and "m'-things" are the middle terms. the attribute, which occurs in the middle term or terms, disappears in the conclusion, and is said to be "eliminated", which literally means "turned out of doors". now let us try to draw a conclusion from the two premisses-- "some new cakes are unwholesome; no nice cakes are unwholesome." in order to express them with counters, we need to divide cakes in three different ways, with regard to newness, to niceness, and to wholesomeness. for this we must use the larger diagram, making x mean "new", y "nice", and m "wholesome". (everything inside the central square is supposed to have the attribute m, and everything outside it the attribute m', i.e. "not-m".) you had better adopt the rule to make m mean the attribute which occurs in the middle term or terms. (i have chosen m as the symbol, because 'middle' begins with 'm'.) now, in representing the two premisses, i prefer to begin with the negative one (the one beginning with "no"), because grey counters can always be placed with certainty, and will then help to fix the position of the red counters, which are sometimes a little uncertain where they will be most welcome. let us express, the "no nice cakes are unwholesome (cakes)", i.e. "no y-cakes are m'-(cakes)". this tells us that none of the cakes belonging to the y-half of the cupboard are in its m'-compartments (i.e. the ones outside the central square). hence the two compartments, no. and no. , are both 'empty'; and we must place a grey counter in each of them, thus:-- ----------- | | | | --|-- | | | | | | |--|-----|--| | | | | | | --|-- | | | | ----------- we have now to express the other premiss, namely, "some new cakes are unwholesome (cakes)", i.e. "some x-cakes are m'-(cakes)". this tells us that some of the cakes in the x-half of the cupboard are in its m'-compartments. hence one of the two compartments, no. and no. , is 'occupied': and, as we are not told in which of these two compartments to place the red counter, the usual rule would be to lay it on the division-line between them: but, in this case, the other premiss has settled the matter for us, by declaring no. to be empty. hence the red counter has no choice, and must go into no. , thus:-- ----------- | | | | --|-- | | | | | | |--|-----|--| | | | | | | --|-- | | | | ----------- and now what counters will this information enable us to place in the smaller diagram, so as to get some proposition involving x and y only, leaving out m? let us take its four compartments, one by one. first, no. . all we know about this is that its outer portion is empty: but we know nothing about its inner portion. thus the square may be empty, or it may have something in it. who can tell? so we dare not place any counter in this square. secondly, what of no. ? here we are a little better off. we know that there is something in it, for there is a red counter in its outer portion. it is true we do not know whether its inner portion is empty or occupied: but what does that matter? one solitary cake, in one corner of the square, is quite sufficient excuse for saying "this square is occupied", and for marking it with a red counter. as to no. , we are in the same condition as with no. --we find it partly 'empty', but we do not know whether the other part is empty or occupied: so we dare not mark this square. and as to no. , we have simply no information at all. the result is ------- | | | |---|---| | | | ------- our 'conclusion', then, must be got out of the rather meager piece of information that there is a red counter in the xy'-square. hence our conclusion is "some x are y' ", i.e. "some new cakes are not-nice (cakes)": or, if you prefer to take y' as your subject, "some not-nice cakes are new (cakes)"; but the other looks neatest. we will now write out the whole syllogism, putting the symbol &there [*] for "therefore", and omitting "cakes", for the sake of brevity, at the end of each proposition. [*][note from brett: the use of "&there " is a rather arbitrary selection. there is no font available in general practice which renders the "therefore" symbol correction (three dots in a triangular formation). this can be done, however, in html, so if this document is read in a browser, then the symbol will be properly recognized. this is a poor man's excuse.] "some new cakes are unwholesome; no nice cakes are unwholesome &there some new cakes are not-nice." and you have now worked out, successfully, your first 'syllogism'. permit me to congratulate you, and to express the hope that it is but the beginning of a long and glorious series of similar victories! we will work out one other syllogism--a rather harder one than the last--and then, i think, you may be safely left to play the game by yourself, or (better) with any friend whom you can find, that is able and willing to take a share in the sport. let us see what we can make of the two premisses-- "all dragons are uncanny; all scotchmen are canny." remember, i don't guarantee the premisses to be facts. in the first place, i never even saw a dragon: and, in the second place, it isn't of the slightest consequence to us, as logicians, whether our premisses are true or false: all we have to do is to make out whether they lead logically to the conclusion, so that, if they were true, it would be true also. you see, we must give up the "cakes" now, or our cupboard will be of no use to us. we must take, as our 'universe', some class of things which will include dragons and scotchmen: shall we say 'animals'? and, as "canny" is evidently the attribute belonging to the 'middle terms', we will let m stand for "canny", x for "dragons", and y for "scotchmen". so that our two premisses are, in full, "all dragon-animals are uncanny (animals); all scotchman-animals are canny (animals)." and these may be expressed, using letters for words, thus:-- "all x are m'; all y are m." the first premiss consists, as you already know, of two parts:-- "some x are m'," and "no x are m." and the second also consists of two parts:-- "some y are m," and "no y are m'." let us take the negative portions first. we have, then, to mark, on the larger diagram, first, "no x are m", and secondly, "no y are m'". i think you will see, without further explanation, that the two results, separately, are ----------- ----------- | | | | | | | --|-- | | --|-- | | | | | | | | | | | |--|--|--|--| |--|--|--|--| | | | | | | | | | | | --|-- | | --|-- | | | | | | | ----------- ----------- and that these two, when combined, give us ----------- | | | | --|-- | | | | | | |--|--|--|--| | | | | | | --|-- | | | | ----------- we have now to mark the two positive portions, "some x are m'" and "some y are m". the only two compartments, available for things which are xm', are no. and no. . of these, no. is already marked as 'empty'; so our red counter must go into no. . similarly, the only two, available for ym, are no. and no. . of these, no. is already marked as 'empty'; so our red counter must go into no. . the final result is ----------- | | | | --|-- | | | | | | |--|--|--|--| | | | | | | --|-- | | | | ----------- and now how much of this information can usefully be transferred to the smaller diagram? let us take its four compartments, one by one. as to no. ? this, we see, is wholly 'empty'. (so mark it with a grey counter.) as to no. ? this, we see, is 'occupied'. (so mark it with a red counter.) as to no. ? ditto, ditto. as to no. ? no information. the smaller diagram is now pretty liberally marked:-- ------- | | | |---|---| | | | ------- and now what conclusion can we read off from this? well, it is impossible to pack such abundant information into one proposition: we shall have to indulge in two, this time. first, by taking x as subject, we get "all x are y'", that is, "all dragons are not-scotchmen": secondly, by taking y as subject, we get "all y are x'", that is, "all scotchmen are not-dragons". let us now write out, all together, our two premisses and our brace of conclusions. "all dragons are uncanny; all scotchmen are canny. &there all dragons are not-scotchmen; all scotchmen are not-dragons." let me mention, in conclusion, that you may perhaps meet with logical treatises in which it is not assumed that any thing exists at all, by "some x are y" is understood to mean "the attributes x, y are compatible, so that a thing can have both at once", and "no x are y" to mean "the attributes x, y are incompatible, so that nothing can have both at once". in such treatises, propositions have quite different meanings from what they have in our 'game of logic', and it will be well to understand exactly what the difference is. first take "some x are y". here we understand "are" to mean "are, as an actual fact"--which of course implies that some x-things exist. but they (the writers of these other treatises) only understand "are" to mean "can be", which does not at all imply that any exist. so they mean less than we do: our meaning includes theirs (for of course "some x are y" includes "some x can be y"), but theirs does not include ours. for example, "some welsh hippopotami are heavy" would be true, according to these writers (since the attributes "welsh" and "heavy" are quite compatible in a hippopotamus), but it would be false in our game (since there are no welsh hippopotami to be heavy). secondly, take "no x are y". here we only understand "are" to mean "are, as an actual fact"--which does not at all imply that no x can be y. but they understand the proposition to mean, not only that none are y, but that none can possibly be y. so they mean more than we do: their meaning includes ours (for of course "no x can be y" includes "no x are y"), but ours does not include theirs. for example, "no policemen are eight feet high" would be true in our game (since, as an actual fact, no such splendid specimens are ever found), but it would be false, according to these writers (since the attributes "belonging to the police force" and "eight feet high" are quite compatible: there is nothing to prevent a policeman from growing to that height, if sufficiently rubbed with rowland's macassar oil--which said to make hair grow, when rubbed on hair, and so of course will make a policeman grow, when rubbed on a policeman). thirdly, take "all x are y", which consists of the two partial propositions "some x are y" and "no x are y'". here, of course, the treatises mean less than we do in the first part, and more than we do in the second. but the two operations don't balance each other--any more than you can console a man, for having knocked down one of his chimneys, by giving him an extra door-step. if you meet with syllogisms of this kind, you may work them, quite easily, by the system i have given you: you have only to make 'are' mean 'are capable of being', and all will go smoothly. for "some x are y" will become "some x are capable of being y", that is, "the attributes x, y are compatible". and "no x are y" will become "no x are capable of being y", that is, "the attributes x, y are incompatible". and, of course, "all x are y" will become "some x are capable of being y, and none are capable of being y'", that is, "the attributes x, y are compatible, and the attributes x, y' are incompatible." in using the diagrams for this system, you must understand a red counter to mean "there may possibly be something in this compartment," and a grey one to mean "there cannot possibly be anything in this compartment." . fallacies. and so you think, do you, that the chief use of logic, in real life, is to deduce conclusions from workable premisses, and to satisfy yourself that the conclusions, deduced by other people, are correct? i only wish it were! society would be much less liable to panics and other delusions, and political life, especially, would be a totally different thing, if even a majority of the arguments, that scattered broadcast over the world, were correct! but it is all the other way, i fear. for one workable pair of premisses (i mean a pair that lead to a logical conclusion) that you meet with in reading your newspaper or magazine, you will probably find five that lead to no conclusion at all: and, even when the premisses are workable, for one instance, where the writer draws a correct conclusion, there are probably ten where he draws an incorrect one. in the first case, you may say "the premisses are fallacious": in the second, "the conclusion is fallacious." the chief use you will find, in such logical skill as this game may teach you, will be in detecting 'fallacies' of these two kinds. the first kind of fallacy--'fallacious premisses'--you will detect when, after marking them on the larger diagram, you try to transfer the marks to the smaller. you will take its four compartments, one by one, and ask, for each in turn, "what mark can i place here?"; and in every one the answer will be "no information!", showing that there is no conclusion at all. for instance, "all soldiers are brave; some englishmen are brave. &there some englishmen are soldiers." looks uncommonly like a syllogism, and might easily take in a less experienced logician. but you are not to be caught by such a trick! you would simply set out the premisses, and would then calmly remark "fallacious premisses!": you wouldn't condescend to ask what conclusion the writer professed to draw--knowing that, whatever it is, it must be wrong. you would be just as safe as that wise mother was, who said "mary, just go up to the nursery, and see what baby's doing, and tell him not to do it!" the other kind of fallacy--'fallacious conclusion'--you will not detect till you have marked both diagrams, and have read off the correct conclusion, and have compared it with the conclusion which the writer has drawn. but mind, you mustn't say "fallacious conclusion," simply because it is not identical with the correct one: it may be a part of the correct conclusion, and so be quite correct, as far as it goes. in this case you would merely remark, with a pitying smile, "defective conclusion!" suppose, of example, you were to meet with this syllogism:-- "all unselfish people are generous; no misers are generous. &there no misers are unselfish." the premisses of which might be thus expressed in letters:-- "all x' are m; no y are m." here the correct conclusion would be "all x' are y'" (that is, "all unselfish people are not misers"), while the conclusion, drawn by the writer, is "no y are x'," (which is the same as "no x' are y," and so is part of "all x' are y'.") here you would simply say "defective conclusion!" the same thing would happen, if you were in a confectioner's shop, and if a little boy were to come in, put down twopence, and march off triumphantly with a single penny-bun. you would shake your head mournfully, and would remark "defective conclusion! poor little chap!" and perhaps you would ask the young lady behind the counter whether she would let you eat the bun, which the little boy had paid for and left behind him: and perhaps she would reply "sha'n't!" but if, in the above example, the writer had drawn the conclusion "all misers are selfish" (that is, "all y are x"), this would be going beyond his legitimate rights (since it would assert the existence of y, which is not contained in the premisses), and you would very properly say "fallacious conclusion!" now, when you read other treatises on logic, you will meet with various kinds of (so-called) 'fallacies' which are by no means always so. for example, if you were to put before one of these logicians the pair of premisses "no honest men cheat; no dishonest men are trustworthy." and were to ask him what conclusion followed, he would probably say "none at all! your premisses offend against two distinct rules, and are as fallacious as they can well be!" then suppose you were bold enough to say "the conclusion is 'no men who cheat are trustworthy'," i fear your logical friend would turn away hastily--perhaps angry, perhaps only scornful: in any case, the result would be unpleasant. i advise you not to try the experiment! "but why is this?" you will say. "do you mean to tell us that all these logicians are wrong?" far from it, dear reader! from their point of view, they are perfectly right. but they do not include, in their system, anything like all the possible forms of syllogisms. they have a sort of nervous dread of attributes beginning with a negative particle. for example, such propositions as "all not-x are y," "no x are not-y," are quite outside their system. and thus, having (from sheer nervousness) excluded a quantity of very useful forms, they have made rules which, though quite applicable to the few forms which they allow of, are no use at all when you consider all possible forms. let us not quarrel with them, dear reader! there is room enough in the world for both of us. let us quietly take our broader system: and, if they choose to shut their eyes to all these useful forms, and to say "they are not syllogisms at all!" we can but stand aside, and let them rush upon their fate! there is scarcely anything of yours, upon which it is so dangerous to rush, as your fate. you may rush upon your potato-beds, or your strawberry-beds, without doing much harm: you may even rush upon your balcony (unless it is a new house, built by contract, and with no clerk of the works) and may survive the foolhardy enterprise: but if you once rush upon your fate--why, you must take the consequences! chapter ii. cross questions. "the man in the wilderness asked of me 'how many strawberries grow in the sea?'" __________ . elementary. . what is an 'attribute'? give examples. . when is it good sense to put "is" or "are" between two names? give examples. . when is it not good sense? give examples. . when it is not good sense, what is the simplest agreement to make, in order to make good sense? . explain 'proposition', 'term', 'subject', and 'predicate'. give examples. . what are 'particular' and 'universal' propositions? give examples. . give a rule for knowing, when we look at the smaller diagram, what attributes belong to the things in each compartment. . what does "some" mean in logic? [see pp. , ] . in what sense do we use the word 'universe' in this game? . what is a 'double' proposition? give examples. . when is a class of things said to be 'exhaustively' divided? give examples. . explain the phrase "sitting on the fence." . what two partial propositions make up, when taken together, "all x are y"? . what are 'individual' propositions? give examples. . what kinds of propositions imply, in this game, the existence of their subjects? . when a proposition contains more than two attributes, these attributes may in some cases be re-arranged, and shifted from one term to the other. in what cases may this be done? give examples. __________ break up each of the following into two partial propositions: . all tigers are fierce. . all hard-boiled eggs are unwholesome. . i am happy. . john is not at home. __________ [see pp. , ] . give a rule for knowing, when we look at the larger diagram, what attributes belong to the things contained in each compartment. . explain 'premisses', 'conclusion', and 'syllogism'. give examples. . explain the phrases 'middle term' and 'middle terms'. . in marking a pair of premisses on the larger diagram, why is it best to mark negative propositions before affirmative ones? . why is it of no consequence to us, as logicians, whether the premisses are true or false? . how can we work syllogisms in which we are told that "some x are y" is to be understood to mean "the attribute x, y are compatible", and "no x are y" to mean "the attributes x, y are incompatible"? . what are the two kinds of 'fallacies'? . how may we detect 'fallacious premisses'? . how may we detect a 'fallacious conclusion'? . sometimes the conclusion, offered to us, is not identical with the correct conclusion, and yet cannot be fairly called 'fallacious'. when does this happen? and what name may we give to such a conclusion? [see pp. - ] . half of smaller diagram. propositions to be represented. ----------- | | | | x | | | | --y-----y'- __________ . some x are not-y. . all x are not-y. . some x are y, and some are not-y. . no x exist. . some x exist. . no x are not-y. . some x are not-y, and some x exist. __________ taking x="judges"; y="just"; . no judges are just. . some judges are unjust. . all judges are just. __________ taking x="plums"; y="wholesome"; . some plums are wholesome. . there are no wholesome plums. . plums are some of them wholesome, and some not. . all plums are unwholesome. [see pp. , ] ----- | | | x | | |--y--| | | | x' | | ----- __________ taking y="diligent students"; x="successful"; . no diligent students are unsuccessful. . all diligent students are successful. . no students are diligent. . there are some diligent, but unsuccessful, students. . some students are diligent. [see pp. , ] . half of smaller diagram. symbols to be interpreted. __________ ----------- | | | | x | | | | --y-----y'- __________ ------- ------- | | | | | | . | | | . | | | | | | | | | ------- ------- ------- ------- | | | | | | . | - | . | | | | | | | | | ------- ------- __________ taking x="good riddles"; y="hard"; ------- ------- | | | | | | . | | | . | | | | | | | | | ------- ------- ------- ------- | | | | | | . | | | . | | | | | | | | | ------- ------- __________ [see pp. , ] taking x="lobster"; y="selfish"; ------- ------- | | | | | | . | | | . | | | | | | | | | ------- ------- ------- ------- | | | | | | . | | | . | | | | | | | | | ------- ------- __________ ----- | | x | | | |--y'-| | | x' | | | ----- taking y="healthy people"; x="happy"; --- --- --- --- | | | | | | | | . |---| . |- -| . |---| . |---| | | | | | | | | --- --- --- --- [see p. ] . smaller diagram. propositions to be represented. ----------- | | | | x | |--y--|--y'-| | x' | | | | ----------- __________ . all y are x. . some y are not-x. . no not-x are not-y. . some x are not-y. . some not-y are x. . no not-x are y. . some not-x are not-y. . all not-x are not-y. . some not-y exist. . no not-x exist. . some y are x, and some are not-x. . all x are y, and all not-y are not-x. [see pp. , ] taking "nations" as universe; x="civilised"; y="warlike"; . no uncivilised nation is warlike. . all unwarlike nations are uncivilised. . some nations are unwarlike. . all warlike nations are civilised, and all civilised nations are warlike. . no nation is uncivilised. __________ taking "crocodiles" as universe; x="hungry"; and y="amiable"; . all hungry crocodiles are unamiable. . no crocodiles are amiable when hungry. . some crocodiles, when not hungry, are amiable; but some are not. . no crocodiles are amiable, and some are hungry. . all crocodiles, when not hungry, are amiable; and all unamiable crocodiles are hungry. . some hungry crocodiles are amiable, and some that are not hungry are unamiable. [see pp. , ] . smaller diagram. symbols to be interpreted. __________ ----------- | | | | x | |--y--|--y'-| | x' | | | | ----------- __________ ------- ------- | | | | | | . |---|---| . |---|---| | | | | | | ------- ------- ------- ------- | | | | | | . |---|---| . |---|---| | | | | | | ------- ------- __________ taking "houses" as universe; x="built of brick"; and y="two-storied"; interpret ------- ------- | | | | | | . |---|---| . |---|---| | | | | - | ------- ---|--- ------- ------- | | | | | | . |---|---| . |---|---| | | | | | | ------- ------- [see p. ] taking "boys" as universe; x="fat"; and y="active"; interpret ------- ------- | | | | | | . |---|---| . |---|---| | | | | | | ------- ------- ------- ------- | | | | | | . |---|---| . |---|---| | | | | | | ------- ------- __________ taking "cats" as universe; x="green-eyed"; and y="good-tempered"; interpret ------- ------- | | | | | | . |---|---| . |---|---| | | | | | | ------- ------- ------- ------- | | | | | | . |---|---| . |---|---| | | | | | | ------- ------- [see pp. , ] . larger diagram. propositions to be represented. __________ ----------- | | | | --x-- | | | | | | |--y--m--y'-| | | | | | | --x'- | | | | ----------- __________ . no x are m. . some y are m'. . all m are x'. . no m' are y'. . no m are x; all y are m. . some x are m; no y are m. . all m are x'; no m are y. . no x' are m; no y' are m'. [see pp. , ] taking "rabbits" as universe; m="greedy"; x="old"; and y="black"; represent . no old rabbits are greedy. . some not-greedy rabbits are black. . all white rabbits are free from greediness. . all greedy rabbits are young. . no old rabbits are greedy; all black rabbits are greedy. . all rabbits, that are not greedy, are black; no old rabbits are free from greediness. __________ taking "birds" as universe; m="that sing loud"; x="well-fed"; and y="happy"; represent . all well-fed birds sing loud; no birds, that sing loud, are unhappy. . all birds, that do not sing loud, are unhappy; no well-fed birds fail to sing loud. __________ taking "persons" as universe; m="in the house"; x="john"; and y="having a tooth-ache"; represent . john is in the house; everybody in the house is suffering from tooth-ache. . there is no one in the house but john; nobody, out of the house, has a tooth-ache. __________ [see pp. - ] taking "persons" as universe; m="i"; x="that has taken a walk"; y="that feels better"; represent . i have been out for a walk; i feel much better. __________ choosing your own 'universe' &c., represent . i sent him to bring me a kitten; he brought me a kettle by mistake. [see pp. , ] . both diagrams to be employed. __________ ----------- | | | ----------- | --x-- | | | | | | | | | | x | |--y--m--y'-| |--y--|--y'-| | | | | | | x' | | --x'- | | | | | | | ----------- ----------- __________ n.b. in each question, a small diagram should be drawn, for x and y only, and marked in accordance with the given large diagram: and then as many propositions as possible, for x and y, should be read off from this small diagram. ----------- ----------- | | | | | | | --|-- | | --|-- | | | | | | | | | | | . |--|--|--|--| . |--|--|--|--| | | | | | | | | | | | --|-- | | --|-- | | | | | | | ----------- ----------- [see p. ] ----------- ----------- | | | | | | | --|-- | | --|-- | | | | | | | | | | | . |--|--|--|--| . |--|--|--|--| | | | | | | | | | | | --|-- | | --|-- | | | | | | | ----------- ----------- __________ mark, in a large diagram, the following pairs of propositions from the preceding section: then mark a small diagram in accordance with it, &c. . no. . [see p. ] . no. . . no. . . no. . . no. . . no. . [see p. ] . no. . . no. . __________ mark, on a large diagram, the following pairs of propositions: then mark a small diagram, &c. these are, in fact, pairs of premisses for syllogisms: and the results, read off from the small diagram, are the conclusions. . no exciting books suit feverish patients; unexciting books make one drowsy. . some, who deserve the fair, get their deserts; none but the brave deserve the fair. . no children are patient; no impatient person can sit still. [see pp. - ] . all pigs are fat; no skeletons are fat. . no monkeys are soldiers; all monkeys are mischievous. . none of my cousins are just; no judges are unjust. . some days are rainy; rainy days are tiresome. . all medicine is nasty; senna is a medicine. . some jews are rich; all patagonians are gentiles. . all teetotalers like sugar; no nightingale drinks wine. . no muffins are wholesome; all buns are unwholesome. . no fat creatures run well; some greyhounds run well. . all soldiers march; some youths are not soldiers. . sugar is sweet; salt is not sweet. . some eggs are hard-boiled; no eggs are uncrackable. . there are no jews in the house; there are no gentiles in the garden. [see pp. - ] . all battles are noisy; what makes no noise may escape notice. . no jews are mad; all rabbis are jews. . there are no fish that cannot swim; some skates are fish. . all passionate people are unreasonable; some orators are passionate. [see pp. - ] chapter iii. crooked answers. "i answered him, as i thought good, 'as many as red-herrings grow in the wood'." __________ . elementary. . whatever can be "attributed to", that is "said to belong to", a thing, is called an 'attribute'. for example, "baked", which can (frequently) be attributed to "buns", and "beautiful", which can (seldom) be attributed to "babies". . when they are the names of two things (for example, "these pigs are fat animals"), or of two attributes (for example, "pink is light red"). . when one is the name of a thing, and the other the name of an attribute (for example, "these pigs are pink"), since a thing cannot actually be an attribute. . that the substantive shall be supposed to be repeated at the end of the sentence (for example, "these pigs are pink (pigs)"). . a 'proposition' is a sentence stating that some, or none, or all, of the things belonging to a certain class, called the 'subject', are also things belonging to a certain other class, called the 'predicate'. for example, "some new cakes are not nice", that is (written in full) "some new cakes are not nice cakes"; where the class "new cakes" is the subject, and the class "not-nice cakes" is the predicate. . a proposition, stating that some of the things belonging to its subject are so-and-so, is called 'particular'. for example, "some new cakes are nice", "some new cakes are not nice." a proposition, stating that none of the things belonging to its subject, or that all of them, are so-and-so, is called 'universal'. for example, "no new cakes are nice", "all new cakes are not nice". . the things in each compartment possess two attributes, whose symbols will be found written on two of the edges of that compartment. . "one or more." . as a name of the class of things to which the whole diagram is assigned. . a proposition containing two statements. for example, "some new cakes are nice and some are not-nice." . when the whole class, thus divided, is "exhausted" among the sets into which it is divided, there being no member of it which does not belong to some one of them. for example, the class "new cakes" is "exhaustively" divided into "nice" and "not-nice" since every new cake must be one or the other. . when a man cannot make up his mind which of two parties he will join, he is said to be "sitting on the fence"--not being able to decide on which side he will jump down. . "some x are y" and "no x are y'". . a proposition, whose subject is a single thing, is called 'individual'. for example, "i am happy", "john is not at home". these are universal propositions, being the same as "all the i's that exist are happy", "all the johns, that i am now considering, are not at home". . propositions beginning with "some" or "all". . when they begin with "some" or "no". for example, "some abc are def" may be re-arranged as "some bf are acde", each being equivalent to "some abcdef exist". . some tigers are fierce, no tigers are not-fierce. . some hard-boiled eggs are unwholesome, no hard-boiled eggs are wholesome. . some i's are happy, no i's are unhappy. . some johns are not at home, no johns are at home. . the things, in each compartment of the larger diagram, possess three attributes, whose symbols will be found written at three of the corners of the compartment (except in the case of m', which is not actually inserted in the diagram, but is supposed to stand at each of its four outer corners). . if the universe of things be divided with regard to three different attributes; and if two propositions be given, containing two different couples of these attributes; and if from these we can prove a third proposition, containing the two attributes that have not yet occurred together; the given propositions are called 'the premisses', the third one 'the conclusion', and the whole set 'a syllogism'. for example, the premisses might be "no m are x'" and "all m' are y"; and it might be possible to prove from them a conclusion containing x and y. . if an attribute occurs in both premisses, the term containing it is called 'the middle term'. for example, if the premisses are "some m are x" and "no m are y'", the class of "m-things" is 'the middle term.' if an attribute occurs in one premiss, and its contradictory in the other, the terms containing them may be called 'the middle terms'. for example, if the premisses are "no m are x'" and "all m' are y", the two classes of "m-things" and "m'-things" may be called 'the middle terms'. . because they can be marked with certainty: whereas affirmative propositions (that is, those that begin with "some" or "all") sometimes require us to place a red counter 'sitting on a fence'. . because the only question we are concerned with is whether the conclusion follows logically from the premisses, so that, if they were true, it also would be true. . by understanding a red counter to mean "this compartment can be occupied", and a grey one to mean "this compartment cannot be occupied" or "this compartment must be empty". . 'fallacious premisses' and 'fallacious conclusion'. . by finding, when we try to transfer marks from the larger diagram to the smaller, that there is 'no information' for any of its four compartments. . by finding the correct conclusion, and then observing that the conclusion, offered to us, is neither identical with it nor a part of it. . when the offered conclusion is part of the correct conclusion. in this case, we may call it a 'defective conclusion'. . half of smaller diagram. propositions represented. __________ ------- ------- | | | | | | . | | | . | | | | | | | | | ------- ------- ------- ------- | | | | | | . | | | . | | | | | | | | | ------- ------- ------- ------- | | | | | | . | | . | | | | | | | | | ------- ------- ------- | | | . | | | it might be thought that the proper | | | ------- ------- | | | diagram would be | |, in order to express "some | | | ------- x exist": but this is really contained in "some x are y'." to put a red counter on the division-line would only tell us "one of the compartments is occupied", which we know already, in knowing that one is occupied. ------- | | | . no x are y. i.e. | | | | | | ------- ------- | | | . some x are y'. i.e. | | | | | | ------- ------- | | | . all x are y. i.e. | | | | | | ------- ------- | | | . some x are y. i.e. | | | | | | ------- ------- | | | . no x are y. i.e. | | | | | | ------- ------- | | | . some x are y, and some are y'. i.e. | | | | | | ------- ------- | | | . all x are y'. i.e. | | | | | | ------- --- | | . no y are x'. i.e. |---| | | --- --- | | . all y are x. i.e. |---| | | --- --- | | . no y exist. i.e. |---| | | --- --- | | . some y are x'. i.e. |---| | | --- --- | | . some y exist. i.e. |- -| | | --- . half of smaller diagram. symbols interpreted. __________ . no x are y'. . no x exist. . some x exist. . all x are y'. . some x are y. i.e. some good riddles are hard. . all x are y. i.e. all good riddles are hard. . no x exist. i.e. no riddles are good. . no x are y. i.e. no good riddles are hard. . some x are y'. i.e. some lobsters are unselfish. . no x are y. i.e. no lobsters are selfish. . all x are y'. i.e. all lobsters are unselfish. . some x are y, and some are y'. i.e. some lobsters are selfish, and some are unselfish. . all y' are x'. i.e. all invalids are unhappy. . some y' exist. i.e. some people are unhealthy. . some y' are x, and some are x'. i.e. some invalids are happy, and some are unhappy. . no y' exist. i.e. nobody is unhealthy. . smaller diagram. propositions represented. __________ ------- ------- | | | | | | . |---|---| . |---|---| | | | | | | ------- ------- ------- ------- | | | | | | . |---|---| . |---|---| | | | | | | ------- ------- ------- ------- | | | | | | . |---|---| . |---|---| | | | | | | ------- ------- ------- ------- | | | | | | . |---|---| . |---|---| | | | | | | ------- ------- ------- ------- | | | | | | . |---|- -| . |---|---| | | | | | | ------- ------- ------- ------- | | | | | | . |---|---| . |---|---| | | | | | | ------- ------- ------- | | | . no x' are y. i.e. |---|---| | | | ------- ------- | | | . all y' are x'. i.e. |---|---| | | | ------- ------- | | | . some y' exist. i.e. |---|- -| | | | ------- ------- | | | . all y are x, and all x are y. i.e. |---|---| | | | ------- ------- | | | . no x' exist. i.e. |---|---| | | | ------- ------- | | | . all x are y'. i.e. |---|---| | | | ------- ------- | | | . no x are y. i.e. |---|---| | | | ------- ------- | | | . some x' are y, and some are y'. i.e. |---|---| | | | ------- ------- | | | . no y exist, and some x exist. i.e. |---|---| | | | ------- ------- | | | . all x' are y, and all y' are x. i.e. |---|---| | | | ------- ------- | | | . some x are y, and some x' are y'. i.e. |---|---| | | | ------- . smaller diagram. symbols interpreted. __________ . some y are not-x, or, some not-x are y. . no not-x are not-y, or, no not-y are not-x. . no not-y are x. . no not-x exist. i.e. no things are not-x. . no y exist. i.e. no houses are two-storied. . some x' exist. i.e. some houses are not built of brick. . no x are y'. or, no y' are x. i.e. no houses, built of brick, are other than two-storied. or, no houses, that are not two-storied, are built of brick. . all x' are y'. i.e. all houses, that are not built of brick, are not two-storied. . some x are y, and some are y'. i.e. some fat boys are active, and some are not. . all y' are x'. i.e. all lazy boys are thin. . all x are y', and all y' are x. i.e. all fat boys are lazy, and all lazy ones are fat. . all y are x, and all x' are y. i.e. all active boys are fat, and all thin ones are lazy. . no x exist, and no y' exist. i.e. no cats have green eyes, and none have bad tempers. . some x are y', and some x' are y. or some y are x', and some y' are x. i.e. some green-eyed cats are bad-tempered, and some, that have not green eyes, are good-tempered. or, some good-tempered cats have not green eyes, and some bad-tempered ones have green eyes. . some x are y, and no x' are y'. or, some y are x, and no y' are x'. i.e. some green-eyed cats are good-tempered, and none, that are not green-eyed, are bad-tempered. or, some good-tempered cats have green eyes, and none, that are bad-tempered, have not green eyes. . all x are y', and all x' are y. or, all y are x', and all y' are x. i.e. all green-eyed cats are bad-tempered and all, that have not green eyes, are good-tempered. or, all good-tempered ones have eyes that are not green, and all bad-tempered ones have green eyes. . larger diagram. propositions represented. __________ --------------- --------------- | | | | | | | ---|--- | | ---|--- | | | | | | | | | | | . |---|---|---|---| . |- -|---|---|---| | | | | | | | | | | | ---|--- | | ---|--- | | | | | | | --------------- --------------- --------------- --------------- | | | | | | | ---|--- | | ---|--- | | | | | | | | | | | . |---|---|---|---| . |---|---|---|---| | | - | | | | | | | | ---|--- | | ---|--- | | | | | | | --------------- --------------- --------------- --------------- | | | | | | | ---|--- | | ---|--- | | | | | | | | | | | . |---|---|---|---| . |---|---|---|---| | | | | | | | | | | | ---|--- | | ---|--- | | | | | | | --------------- --------------- --------------- --------------- | | | | | | | ---|--- | | ---|--- | | | | | | | | | | | . |---|---|---|---| . |---|---|---|---| | | | | | | | | | | | ---|--- | | ---|--- | | | | | | | --------------- --------------- --------------- | | | | ---|--- | | | | | | . no x are m. i.e. |---|---|---|---| | | | | | | ---|--- | | | | --------------- --------------- | | | | ---|--- | | | | | | . some m' are y. i.e. |- -|---|---|---| | | | | | | ---|--- | | | | --------------- --------------- | | | | ---|--- | | | | | | . all y' are m'. i.e. |---|---|---|- -| | | | | | | ---|--- | | | | --------------- --------------- | | | | ---|--- | | | | | | . all m are x'. i.e. |---|---|---|---| | | | | | ---|--- | | | | --------------- --------------- | | | | ---|--- | | | | | | . no x are m; i.e. |---|---|---|---| all y are m. | | | | | | ---|--- | | | | --------------- --------------- | | | | ---|--- | | | | | | . all m' are y; i.e. |---|---|---|---| no x are m'. | | | | | | ---|--- | | | | --------------- --------------- | | | | ---|--- | | | | | | . all x are m; i.e. |---|---|---|---| no m are y'. | | | | | | ---|--- | | | | --------------- --------------- | | | | ---|--- | | | | | | . all m' are y'; i.e. |---|---|---|---| no x are m'. | | | | | | ---|--- | | | | --------------- --------------- | | | | ---|--- | | | | | | . all x are m; i.e. |---|---|---|---| all m are y. | | | | | | ---|--- | [see remarks on no. , p. .] | | | --------------- --------------- | | | | ---|--- | | | | | | . no x' are m; i.e. |---|---|---|---| no m' are y. | | | | | | ---|--- | | | | --------------- --------------- | | | | ---|--- | | | | | | . all m are x; i.e. |---|---|---|---| all m are y. | | | | | | ---|--- | | | | --------------- . we had better take "persons" as universe. we may choose "myself" as 'middle term', in which case the premisses will take the form i am a-person-who-sent-him-to-bring-a-kitten; i am a-person-to-whom-he-brought-a-kettle-by-mistake. or we may choose "he" as 'middle term', in which case the premisses will take the form he is a-person-whom-i-sent-to-bring-me-a-kitten; he is a-person-who-brought-me-a-kettle-by-mistake. the latter form seems best, as the interest of the anecdote clearly depends on his stupidity--not on what happened to me. let us then make m = "he"; x = "persons whom i sent, &c."; and y = "persons who brought, &c." hence, all m are x; all m are y. and the required diagram is --------------- | | | | ---|--- | | | | | | |---|---|---|---| | | | | | | ---|--- | | | | --------------- . both diagrams employed. ------- | | | . |---|---| i.e. all y are x'. | | | ------- ------- | | | . |---|---| i.e. some x are y'; or, some y' are x. | | | ------- ------- | | | . |---|---| i.e. some y are x'; or, some x' are y. | | | ------- ------- | | | . |---|---| i.e. no x' are y'; or, no y' are x'. | | | ------- ------- | | | . |---|---| i.e. all y are x'. i.e. all black rabbits | | | are young. ------- ------- | | | . |---|---| i.e. some y are x'. i.e. some black | | | rabbits are young. ------- ------- | | | . |---|---| i.e. all x are y. i.e. all well-fed birds | | | are happy. ------- ------- | | | i.e. some x' are y'. i.e. some birds, . |---|---| that are not well-fed, are unhappy; | | | or, some unhappy birds are not ------- well-fed. ------- | | | . |---|---| i.e. all x are y. i.e. john has got a | | | tooth-ache. ------- ------- | | | . |---|---| i.e. no x' are y. i.e. no one, but john, | | | has got a tooth-ache. ------- ------- | | | . |---|---| i.e. some x are y. i.e. some one, who | | | has taken a walk, feels better. ------- ------- | | | i.e. some x are y. i.e. some one, . |---|---| whom i sent to bring me a kitten, | | | brought me a kettle by mistake. ------- --------------- | | | | ---|--- | | | | | | . |- -|---|---|---| ------- | | | | | | | | | ---|--- | |---|---| | | | | | | --------------- ------- let "books" be universe; m="exciting", x="that suit feverish patients"; y="that make one drowsy". no m are x; &there no y' are x. all m' are y. i.e. no books suit feverish patients, except such as make one drowsy. --------------- | | | | ---|--- | | | | | | . |---|---|---|---| ------- | | | | | | | | | ---|--- | |---|---| | | | | | | --------------- ------- let "persons" be universe; m="that deserve the fair"; x="that get their deserts"; y="brave". some m are x; &there some y are x. no y' are m. i.e. some brave persons get their deserts. --------------- | | | | ---|--- | | | | | | . |---|---|---|---| ------- | | | | | | | | | ---|--- | |---|---| | | | | | | --------------- ------- let "persons" be universe; m="patient"; x="children"; y="that can sit still". no x are m; &there no x are y. no m' are y. i.e. no children can sit still. --------------- | | | | ---|--- | | | | | | . |---|---|---|---| ------- | | | | | | | | | ---|--- | |---|---| | | | | | | --------------- ------- let "things" be universe; m="fat"; x="pigs"; y="skeletons". all x are m; &there all x are y'. no y are m. i.e. all pigs are not-skeletons. --------------- | | | | ---|--- | | | | | | . |---|---|---|---| ------- | | | | | | | | | ---|--- | |---|---| | | | | | | --------------- ------- let "creatures" be universe; m="monkeys"; x="soldiers"; y="mischievous". no m are x; &there some y are x'. all m are y. i.e. some mischievous creatures are not soldiers. --------------- | | | | ---|--- | | | | | | . |---|---|---|---| ------- | | | | | | | | | ---|--- | |---|---| | | | | | | --------------- ------- let "persons" be universe; m="just"; x="my cousins"; y="judges". no x are m; &there no x are y. no y are m'. i.e. none of my cousins are judges. --------------- | | | | ---|--- | | | | | | . |---|---|---|---| ------- | | | | | | | | | ---|--- | |---|---| | | | | | | --------------- ------- let "periods" be universe; m="days"; x="rainy"; y="tiresome". some m are x; &there some x are y. all xm are y. i.e. some rainy periods are tiresome. n.b. these are not legitimate premisses, since the conclusion is really part of the second premiss, so that the first premiss is superfluous. this may be shown, in letters, thus:-- "all xm are y" contains "some xm are y", which contains "some x are y". or, in words, "all rainy days are tiresome" contains "some rainy days are tiresome", which contains "some rainy periods are tiresome". moreover, the first premiss, besides being superfluous, is actually contained in the second; since it is equivalent to "some rainy days exist", which, as we know, is implied in the proposition "all rainy days are tiresome". altogether, a most unsatisfactory pair of premisses! --------------- | | | | ---|--- | | | | | | . |---|---|---|---| ------- | | | | | | | | | ---|--- | |---|---| | | | | | | --------------- ------- let "things" be universe; m="medicine"; x="nasty"; y="senna". all m are x; &there all y are x. all y are m. i.e. senna is nasty. [see remarks on no. , p .] --------------- | | | | ---|--- | | | | | | . |- -|---|---|---| ------- | | | | | | | | | ---|--- | |---|---| | | | | | | --------------- ------- let "persons" be universe; m="jews"; x="rich"; y="patagonians". some m are x; &there some x are y'. all y are m'. i.e. some rich persons are not patagonians. --------------- | | | | ---|--- | | | - | | . |---|---|---|---| ------- | | | | | | | | | ---|--- | |---|---| | | | | | | --------------- ------- let "creatures" be universe; m="teetotalers"; x="that like sugar"; y="nightingales". all m are x; &there no y are x'. no y are m'. i.e. no nightingales dislike sugar. --------------- | | | | ---|--- | | | | | | . |- -|---|---|---| ------- | | | | | | | | | ---|--- | |---|---| | | | | | | --------------- ------- let "food" be universe; m="wholesome"; x="muffins"; y="buns". no x are m; all y are m. there is 'no information' for the smaller diagram; so no conclusion can be drawn. --------------- | | | | ---|--- | | | | | | . |---|---|---|---| ------- | | | | | | | | | ---|--- | |---|---| | | | | | | --------------- ------- let "creatures" be universe; m="that run well"; x="fat"; y="greyhounds". no x are m; &there some y are x'. some y are m. i.e. some greyhounds are not fat. --------------- | | | | ---|--- | | | - | | . |- -|---|---|---| ------- | | | | | | | | | ---|--- | |---|---| | | | | | | --------------- ------- let "persons" be universe; m="soldiers"; x="that march"; y="youths". all m are x; some y are m'. there is 'no information' for the smaller diagram; so no conclusion can be drawn. --------------- | | | | ---|--- | | | | | | . |---|---|---|---| ------- | | | | | | | | | ---|--- | |---|---| | | | | | | --------------- ------- let "food" be universe; m="sweet"; x="sugar"; y="salt". all x are m; &there all x are y'. all y are m'. all y are x'. i.e. sugar is not salt. salt is not sugar. --------------- | | | | ---|--- | | | | | | . |---|---|---|---| ------- | | | | | | | | | ---|--- | |---|---| | | | | | | --------------- ------- let "things" be universe; m="eggs"; x="hard-boiled"; y="crackable". some m are x; &there some x are y. no m are y'. i.e. some hard-boiled things can be cracked. --------------- | | | | ---|--- | | | | | | . |---|---|---|---| ------- | | | | | | | | | ---|--- | |---|---| | | | | | | --------------- ------- let "persons" be universe; m="jews"; x="that are in the house"; y="that are in the garden". no m are x; &there no x are y. no m' are y. i.e. no persons, that are in the house, are also in the garden. --------------- | | | | ---|--- | | | - | | . |---|---|---|---| ------- | | | | | | | | | ---|--- | |---|---| | | | | | | --------------- ------- let "things" be universe; m="noisy"; x="battles"; y="that may escape notice". all x are m; &there some x' are y. all m' are y. i.e. some things, that are not battles, may escape notice. --------------- | | | | ---|--- | | | | | | . |---|---|---|---| ------- | | | | | | | | | ---|--- | |---|---| | | | | | | --------------- ------- let "persons" be universe; m="jews"; x="mad"; y="rabbis". no m are x; &there all y are x'. all y are m. i.e. all rabbis are sane. --------------- | | | | ---|--- | | | | | | . |---|---|---|---| ------- | | | | | | | | | ---|--- | |---|---| | | | | | | --------------- ------- let "things" be universe; m="fish"; x="that can swim"; y="skates". no m are x'; &there some y are x. some y are m. i.e. some skates can swim. --------------- | | | | ---|--- | | | | | | . |---|---|---|---| ------- | | | | | | | | | ---|--- | |---|---| | | | | | | --------------- ------- let "people" be universe; m="passionate"; x="reasonable"; y="orators". all m are x'; &there some y are x'. some y are m. i.e. some orators are unreasonable. [see remarks on no. , p. .] chapter iv. hit or miss. "thou canst not hit it, hit it, hit it, thou canst not hit it, my good man." __________ . pain is wearisome; no pain is eagerly wished for. . no bald person needs a hair-brush; no lizards have hair. . all thoughtless people do mischief; no thoughtful person forgets a promise. . i do not like john; some of my friends like john. . no potatoes are pine-apples; all pine-apples are nice. . no pins are ambitious; no needles are pins. . all my friends have colds; no one can sing who has a cold. . all these dishes are well-cooked; some dishes are unwholesome if not well-cooked. . no medicine is nice; senna is a medicine. . some oysters are silent; no silent creatures are amusing. . all wise men walk on their feet; all unwise men walk on their hands. . "mind your own business; this quarrel is no business of yours." . no bridges are made of sugar; some bridges are picturesque. . no riddles interest me that can be solved; all these riddles are insoluble. . john is industrious; all industrious people are happy. . no frogs write books; some people use ink in writing books. . no pokers are soft; all pillows are soft. . no antelope is ungraceful; graceful animals delight the eye. . some uncles are ungenerous; all merchants are generous. . no unhappy people chuckle; no happy people groan. . audible music causes vibration in the air; inaudible music is not worth paying for. . he gave me five pounds; i was delighted. . no old jews are fat millers; all my friends are old millers. . flour is good for food; oatmeal is a kind of flour. . some dreams are terrible; no lambs are terrible. . no rich man begs in the street; all who are not rich should keep accounts. . no thieves are honest; some dishonest people are found out. . all wasps are unfriendly; all puppies are friendly. . all improbable stories are doubted; none of these stories are probable. . "he told me you had gone away." "he never says one word of truth." . his songs never last an hour; a song, that lasts an hour, is tedious. . no bride-cakes are wholesome; unwholesome food should be avoided. . no old misers are cheerful; some old misers are thin. . all ducks waddle; nothing that waddles is graceful. . no professors are ignorant; some ignorant people are conceited. . toothache is never pleasant; warmth is never unpleasant. . bores are terrible; you are a bore. . some mountains are insurmountable; all stiles can be surmounted. . no frenchmen like plumpudding; all englishmen like plumpudding. . no idlers win fame; some painters are not idle. . no lobsters are unreasonable; no reasonable creatures expect impossibilities. . no kind deed is unlawful; what is lawful may be done without fear. . no fossils can be crossed in love; any oyster may be crossed in love. . "this is beyond endurance!" "well, nothing beyond endurance has ever happened to me." . all uneducated men are shallow; all these students are educated. . all my cousins are unjust; no judges are unjust. . no country, that has been explored, is infested by dragons; unexplored countries are fascinating. . no misers are generous; some old men are not generous. . a prudent man shuns hyaenas; no banker is imprudent. . some poetry is original; no original work is producible at will. . no misers are unselfish; none but misers save egg-shells. . all pale people are phlegmatic; no one, who is not pale, looks poetical. . all spiders spin webs; some creatures, that do not spin webs, are savage. . none of my cousins are just; all judges are just. . john is industrious; no industrious people are unhappy. . umbrellas are useful on a journey; what is useless on a journey should be left behind. . some pillows are soft; no pokers are soft. . i am old and lame; no old merchant is a lame gambler. . no eventful journey is ever forgotten; uneventful journeys are not worth writing a book about. . sugar is sweet; some sweet things are liked by children. . richard is out of temper; no one but richard can ride that horse. . all jokes are meant to amuse; no act of parliament is a joke. . "i saw it in a newspaper." "all newspapers tell lies." . no nightmare is pleasant; unpleasant experiences are not anxiously desired. . prudent travellers carry plenty of small change; imprudent travellers lose their luggage. . all wasps are unfriendly; no puppies are unfriendly. . he called here yesterday; he is no friend of mine. . no quadrupeds can whistle; some cats are quadrupeds. . no cooked meat is sold by butchers; no uncooked meat is served at dinner. . gold is heavy; nothing but gold will silence him. . some pigs are wild; there are no pigs that are not fat. . no emperors are dentists; all dentists are dreaded by children. . all, who are not old, like walking; neither you nor i are old. . all blades are sharp; some grasses are blades. . no dictatorial person is popular; she is dictatorial. . some sweet things are unwholesome; no muffins are sweet. . no military men write poetry; no generals are civilians. . bores are dreaded; a bore is never begged to prolong his visit. . all owls are satisfactory; some excuses are unsatisfactory. . all my cousins are unjust; all judges are just. . some buns are rich; all buns are nice. . no medicine is nice; no pills are unmedicinal. . some lessons are difficult; what is difficult needs attention. . no unexpected pleasure annoys me; your visit is an unexpected pleasure. . caterpillars are not eloquent; jones is eloquent. . some bald people wear wigs; all your children have hair. . all wasps are unfriendly; unfriendly creatures are always unwelcome. . no bankrupts are rich; some merchants are not bankrupts. . weasels sometimes sleep; all animals sometimes sleep. . ill-managed concerns are unprofitable; railways are never ill-managed. . everybody has seen a pig; nobody admires a pig. ______________ extract a pair of premisses out of each of the following: and deduce the conclusion, if there is one:-- . "the lion, as any one can tell you who has been chased by them as often as i have, is a very savage animal: and there are certain individuals among them, though i will not guarantee it as a general law, who do not drink coffee." . "it was most absurd of you to offer it! you might have known, if you had had any sense, that no old sailors ever like gruel!" "but i thought, as he was an uncle of yours--" "an uncle of mine, indeed! stuff!" "you may call it stuff, if you like. all i know is, my uncles are all old men: and they like gruel like anything!" "well, then your uncles are--" . "do come away! i can't stand this squeezing any more. no crowded shops are comfortable, you know very well." "well, who expects to be comfortable, out shopping?" "why, i do, of course! and i'm sure there are some shops, further down the street, that are not crowded. so--" . "they say no doctors are metaphysical organists: and that lets me into a little fact about you, you know." "why, how do you make that out? you never heard me play the organ." "no, doctor, but i've heard you talk about browning's poetry: and that showed me that you're metaphysical, at any rate. so--" ___________________ extract a syllogism out of each of the following: and test its correctness:-- . "don't talk to me! i've known more rich merchants than you have: and i can tell you not one of them was ever an old miser since the world began!" "and what has that got to do with old mr. brown?" "why, isn't he very rich?" "yes, of course he is. and what then?" "why, don't you see that it's absurd to call him a miserly merchant? either he's not a merchant, or he's not a miser!" . "it is so kind of you to enquire! i'm really feeling a great deal better to-day." "and is it nature, or art, that is to have the credit of this happy change?" "art, i think. the doctor has given me some of that patent medicine of his." "well, i'll never call him a humbug again. there's somebody, at any rate, that feels better after taking his medicine!" . "no, i don't like you one bit. and i'll go and play with my doll. dolls are never unkind." "so you like a doll better than a cousin? oh you little silly!" "of course i do! cousins are never kind--at least no cousins i've ever seen." "well, and what does that prove, i'd like to know! if you mean that cousins aren't dolls, who ever said they were?" . "what are you talking about geraniums for? you can't tell one flower from another, at this distance! i grant you they're all red flowers: it doesn't need a telescope to know that." "well, some geraniums are red, aren't they?" "i don't deny it. and what then? i suppose you'll be telling me some of those flowers are geraniums!" "of course that's what i should tell you, if you'd the sense to follow an argument! but what's the good of proving anything to you, i should like to know?" . "boys, you've passed a fairly good examination, all things considered. now let me give you a word of advice before i go. remember that all, who are really anxious to learn, work hard." "i thank you, sir, in the name of my scholars! and proud am i to think there are some of them, at least, that are really anxious to learn." "very glad to hear it: and how do you make it out to be so?" "why, sir, i know how hard they work--some of them, that is. who should know better?" ___________________ extract from the following speech a series of syllogisms, or arguments having the form of syllogisms: and test their correctness. it is supposed to be spoken by a fond mother, in answer to a friend's cautious suggestion that she is perhaps a little overdoing it, in the way of lessons, with her children. . "well, they've got their own way to make in the world. we can't leave them a fortune apiece. and money's not to be had, as you know, without money's worth: they must work if they want to live. and how are they to work, if they don't know anything? take my word for it, there's no place for ignorance in these times! and all authorities agree that the time to learn is when you're young. one's got no memory afterwards, worth speaking of. a child will learn more in an hour than a grown man in five. so those, that have to learn, must learn when they're young, if ever they're to learn at all. of course that doesn't do unless children are healthy: i quite allow that. well, the doctor tells me no children are healthy unless they've got a good colour in their cheeks. and only just look at my darlings! why, their cheeks bloom like peonies! well, now, they tell me that, to keep children in health, you should never give them more than six hours altogether at lessons in the day, and at least two half-holidays in the week. and that's exactly our plan i can assure you! we never go beyond six hours, and every wednesday and saturday, as ever is, not one syllable of lessons do they do after their one o'clock dinner! so how you can imagine i'm running any risk in the education of my precious pets is more than i can understand, i promise you!" the end. generously made available by the internet archive.) three sunsets and other poems [illustration] three sunsets and other poems by lewis carroll _with twelve fairy-fancies_ by _e. gertrude thomson_ _price four shillings net_ london macmillan and co., limited new york: the macmillan company _all rights reserved_ richard clay and sons, limited, london and bungay. preface. nearly the whole of this volume is a reprint of the serious portion of _phantasmagoria and other poems_, which was first published in and has long been out of print. "the path of roses" was written soon after the crimean war, when the name of florence nightingale had already become a household-word. "only a woman's hair" was suggested by a circumstance mentioned in _the life of dean swift_, viz., that, after his death, a small packet was found among his papers, containing a single lock of hair and inscribed with those words. "after three days" was written after seeing holman hunt's picture, _the finding of christ in the temple_. the two poems, "far away" and "a song of love", are reprinted from _sylvie and bruno_ and _sylvie and bruno concluded_, books whose high price (made necessary by the great cost of production) has, i fear, put them out of the reach of most of my readers. "a lesson in latin" is reprinted from _the jabberwock_, a magazine got up among the members of "the girls' latin school, boston, u.s.a." the only poems, here printed for the first time, are put together under the title of "puck lost and found," having been inscribed in two books--_fairies_, a poem by allingham, illustrated by miss e. gertrude thomson, and _merry elves_, a story-book, by whom written i do not know, illustrated by c. o. murray--which were presented to a little girl and boy, as a sort of memento of a visit paid by them to the author one day, on which occasion he taught them the pastime--dear to the hearts of children--of folding paper-"pistols," which can be made to imitate, fairly well, the noise of a real one. _jan., ._ contents. page three sunsets the path of roses the valley of the shadow of death solitude far away beatrice stolen waters the willow-tree only a woman's hair the sailor's wife after three days faces in the fire a lesson in latin puck lost and found a song of love list of illustrations. fairies and nautilus _front._ fairies in boat fairies and bower sleeping fairies fairy riding on cray-fish fairies and squirrel fairies and jonquils fairies and frog fairy on mushroom fairies riding on fish fairy and wasp fairies under mushroom three sunsets. he saw her once, and in the glance, a moment's glance of meeting eyes, his heart stood still in sudden trance: he trembled with a sweet surprise-- all in the waning light she stood, the star of perfect womanhood. that summer-eve his heart was light: with lighter step he trod the ground: and life was fairer in his sight, and music was in every sound: he blessed the world where there could be so beautiful a thing as she. there once again, as evening fell and stars were peering overhead, two lovers met to bid farewell: the western sun gleamed faint and red, lost in a drift of purple cloud that wrapped him like a funeral-shroud. long time the memory of that night-- the hand that clasped, the lips that kissed, the form that faded from his sight slow sinking through the tearful mist-- in dreamy music seemed to roll through the dark chambers of his soul. so after many years he came a wanderer from a distant shore: the street, the house, were still the same, but those he sought were there no more: his burning words, his hopes and fears, unheeded fell on alien ears. only the children from their play would pause the mournful tale to hear, shrinking in half-alarm away, or, step by step, would venture near to touch with timid curious hands that strange wild man from other lands. he sat beside the busy street, there, where he last had seen her face: and thronging memories, bitter-sweet, seemed yet to haunt the ancient place: her footfall ever floated near: her voice was ever in his ear. he sometimes, as the daylight waned and evening mists began to roll, in half-soliloquy complained of that black shadow on his soul, and blindly fanned, with cruel care, the ashes of a vain despair. the summer fled: the lonely man still lingered out the lessening days; still, as the night drew on, would scan each passing face with closer gaze-- till, sick at heart, he turned away, and sighed "she will not come to-day." so by degrees his spirit bent to mock its own despairing cry, in stern self-torture to invent new luxuries of agony, and people all the vacant space with visions of her perfect face. then for a moment she was nigh, he heard no step, but she was there; as if an angel suddenly were bodied from the viewless air, and all her fine ethereal frame should fade as swiftly as it came. so, half in fancy's sunny trance, and half in misery's aching void with set and stony countenance his bitter being he enjoyed, and thrust for ever from his mind the happiness he could not find. as when the wretch, in lonely room, to selfish death is madly hurled, the glamour of that fatal fume shuts out the wholesome living world-- so all his manhood's strength and pride one sickly dream had swept aside. yea, brother, and we passed him there, but yesterday, in merry mood, and marveled at the lordly air that shamed his beggar's attitude, nor heeded that ourselves might be wretches as desperate as he; who let the thought of bliss denied make havoc of our life and powers, and pine, in solitary pride, for peace that never shall be ours, because we will not work and wait in trustful patience for our fate. and so it chanced once more that she came by the old familiar spot: the face he would have died to see bent o'er him, and he knew it not; too rapt in selfish grief to hear, even when happiness was near. and pity filled her gentle breast for him that would not stir nor speak the dying crimson of the west, that faintly tinged his haggard cheek, fell on her as she stood, and shed a glory round the patient head. ah, let him wake! the moments fly: this awful tryst may be the last. and see, the tear, that dimmed her eye, had fallen on him ere she passed-- she passed: the crimson paled to gray: and hope departed with the day. the heavy hours of night went by, and silence quickened into sound, and light slid up the eastern sky, and life began its daily round-- but light and life for him were fled: his name was numbered with the dead. _nov., ._ [illustration] the path of roses. in the dark silence of an ancient room, whose one tall window fronted to the west, where, through laced tendrils of a hanging vine, the sunset-glow was fading into night, sat a pale lady, resting weary hands upon a great clasped volume, and her face within her hands. not as in rest she bowed, but large hot tears were coursing down her cheek, and her low-panted sobs broke awefully upon the sleeping echoes of the night. soon she unclasp'd the volume once again, and read the words in tone of agony, as in self-torture, weeping as she read:-- _"he crowns the glory of his race: he prayeth but in some fit place to meet his foeman face to face: "and, battling for the true, the right, from ruddy dawn to purple night, to perish in the midmost fight: "where hearts are fierce and hands are strong, where peals the bugle loud and long, where blood is dropping in the throng: "still, with a dim and glazing eye, to watch the tide of victory, to hear in death the battle-cry: "then, gathered grandly to his grave, to rest among the true and brave, in holy ground, where yew-trees wave: "where, from church-windows sculptured fair, float out upon the evening air the note of praise, the voice of prayer: "where no vain marble mockery insults with loud and boastful lie the simple soldier's memory: "where sometimes little children go, and read, in whisper'd accent slow, the name of him who sleeps below."_ her voice died out: like one in dreams she sat. "alas!" she sighed. "for what can woman do? her life is aimless, and her death unknown: hemmed in by social forms she pines in vain. man has his work, but what can woman do?" and answer came there from the creeping gloom, the creeping gloom that settled into night: "peace! for thy lot is other than a man's: his is a path of thorns: he beats them down: he faces death: he wrestles with despair. thine is of roses, to adorn and cheer his lonely life, and hide the thorns in flowers." she spake again: in bitter tone she spake: "aye, as a toy, the puppet of an hour, or a fair posy, newly plucked at morn, but flung aside and withered ere the night." and answer came there from the creeping gloom, the creeping gloom that blackened into night: "so shalt thou be the lamp to light his path, what time the shades of sorrow close around." and, so it seemed to her, an awful light pierced slowly through the darkness, orbed, and grew, until all passed away--the ancient room-- the sunlight dying through the trellised vine-- the one tall window--all had passed away, and she was standing on the mighty hills. beneath, around, and far as eye could see, squadron on squadron, stretched opposing hosts, ranked as for battle, mute and motionless. anon a distant thunder shook the ground, the tramp of horses, and a troop shot by-- plunged headlong in that living sea of men-- plunged to their death: back from that fatal field a scattered handful, fighting hard for life, broke through the serried lines; but, as she gazed, they shrank and melted, and their forms grew thin-- grew pale as ghosts when the first morning ray dawns from the east--the trumpet's brazen blare died into silence--and the vision passed-- passed to a room where sick and dying lay in long, sad line--there brooded fear and pain-- darkness was there, the shade of azrael's wing. but there was one that ever, to and fro, moved with light footfall: purely calm her face, and those deep steadfast eyes that starred the gloom: still, as she went, she ministered to each comfort and counsel; cooled the fevered brow with softest touch, and in the listening ear of the pale sufferer whispered words of peace. the dying warrior, gazing as she passed, clasped his thin hands and blessed her. bless her too, thou, who didst bless the merciful of old! so prayed the lady, watching tearfully her gentle moving onward, till the night had veiled her wholly, and the vision passed. then once again the solemn whisper came: "so in the darkest path of man's despair, where war and terror shake the troubled earth, lies woman's mission; with unblenching brow to pass through scenes of horror and affright where men grow sick and tremble: unto her all things are sanctified, for all are good. nothing so mean, but shall deserve her care: nothing so great, but she may bear her part. no life is vain: each hath his place assigned: do thou thy task, and leave the rest to god." and there was silence, but the lady made no answer, save one deeply-breathed "amen." and she arose, and in that darkening room stood lonely as a spirit of the night-- stood calm and fearless in the gathered night-- and raised her eyes to heaven. there were tears upon her face, but in her heart was peace, peace that the world nor gives nor takes away! _april , ._ [illustration] the valley of the shadow of death. hark, _said the dying man_, _and sighed_, to that complaining tone-- like sprite condemned, each eventide, to walk the world alone. at sunset, when the air is still, i hear it creep from yonder hill: it breathes upon me, dead and chill, a moment, and is gone. my son, it minds me of a day left half a life behind, that i have prayed to put away for ever from my mind. but bitter memory will not die: it haunts my soul when none is nigh: i hear its whisper in the sigh of that complaining wind. and now in death my soul is fain to tell the tale of fear that hidden in my breast hath lain through many a weary year: yet time would fail to utter all-- the evil spells that held me thrall, and thrust my life from fall to fall, thou needest not to hear. the spells that bound me with a chain, sin's stern behests to do, till pleasure's self, invoked in vain, a heavy burden grew-- till from my spirit's fevered eye, a hunted thing, i seemed to fly through the dark woods that underlie yon mountain-range of blue. deep in those woods i found a vale no sunlight visiteth, nor star, nor wandering moonbeam pale; where never comes the breath of summer-breeze--there in mine ear, even as i lingered half in fear, i heard a whisper, cold and clear, "this is the gate of death. "o bitter is it to abide in weariness alway: at dawn to sigh for eventide, at eventide for day. thy noon hath fled: thy sun hath shone. the brightness of thy day is gone: what need to lag and linger on till life be cold and gray? "o well," it said, "beneath yon pool, in some still cavern deep, the fevered brain might slumber cool, the eyes forget to weep: within that goblet's mystic rim are draughts of healing, stored for him whose heart is sick, whose sight is dim, who prayeth but to sleep!" the evening-breeze went moaning by, like mourner for the dead, and stirred, with shrill complaining sigh, the tree-tops overhead: my guardian-angel seemed to stand and mutely wave a warning hand-- with sudden terror all unmanned, i turned myself and fled! a cottage-gate stood open wide: soft fell the dying ray on two fair children, side by side, that rested from their play-- together bent the earnest head, as ever and anon they read from one dear book: the words they said come back to me to-day. like twin cascades on mountain-stair together wandered down the ripples of the golden hair, the ripples of the brown: while, through the tangled silken haze, blue eyes looked forth in eager gaze, more starlike than the gems that blaze about a monarch's crown. my son, there comes to each an hour when sinks the spirit's pride-- when weary hands forget their power the strokes of death to guide: in such a moment, warriors say, a word the panic-rout may stay, a sudden charge redeem the day and turn the living tide. i could not see, for blinding tears, the glories of the west: a heavenly music filled mine ears, a heavenly peace my breast. "come unto me, come unto me-- all ye that labour, unto me-- ye heavy-laden, come to me-- and i will give you rest." the night drew onward: thin and blue the evening mists arise to bathe the thirsty land in dew, as erst in paradise-- while, over silent field and town, the deep blue vault of heaven looked down; not, as of old, in angry frown, but bright with angels' eyes. blest day! then first i heard the voice that since hath oft beguiled these eyes from tears, and bid rejoice this heart with anguish wild-- thy mother, boy, thou hast not known; so soon she left me here to moan-- left me to weep and watch, alone, our one beloved child. though, parted from my aching sight, like homeward-speeding dove, she passed into the perfect light that floods the world above; yet our twin spirits, well i know-- though one abide in pain below-- love, as in summers long ago, and evermore shall love. so with a glad and patient heart i move toward mine end: the streams, that flow awhile apart, shall both in ocean blend. i dare not weep: i can but bless the love that pitied my distress, and lent me, in life's wilderness, so sweet and true a friend. but if there be--o if there be a truth in what they say, that angel-forms we cannot see go with us on our way; then surely she is with me here, i dimly feel her spirit near-- the morning-mists grow thin and clear, and death brings in the day. _april, ._ [illustration] solitude. i love the stillness of the wood: i love the music of the rill: i love to couch in pensive mood upon some silent hill. scarce heard, beneath yon arching trees, the silver-crested ripples pass; and, like a mimic brook, the breeze whispers among the grass. here from the world i win release, nor scorn of men, nor footstep rude, break in to mar the holy peace of this great solitude. here may the silent tears i weep lull the vexed spirit into rest, as infants sob themselves to sleep upon a mother's breast. but when the bitter hour is gone, and the keen throbbing pangs are still, oh sweetest then to couch alone upon some silent hill! to live in joys that once have been, to put the cold world out of sight, and deck life's drear and barren scene with hues of rainbow-light. for what to man the gift of breath, if sorrow be his lot below; if all the day that ends in death be dark with clouds of woe? shall the poor transport of an hour repay long years of sore distress-- the fragrance of a lonely flower make glad the wilderness? ye golden hours of life's young spring, of innocence, of love and truth! bright, beyond all imagining, thou fairy-dream of youth! i'd give all wealth that years have piled, the slow result of life's decay, to be once more a little child for one bright summer-day. _march , ._ far away. he stept so lightly to the land, all in his manly pride: he kissed her cheek, he clasped her hand; yet still she glanced aside. "too gay he seems," she darkly dreams, "too gallant and too gay, to think of me--poor simple me-- when he is far away!" "i bring my love this goodly pearl across the seas," he said: "a gem to deck the dearest girl that ever sailor wed!" she holds it tight: her eyes are bright: her throbbing heart would say "he thought of me--he thought of me-- when he was far away!" the ship has sailed into the west: her ocean-bird is flown: a dull dead pain is in her breast, and she is weak and lone: but there's a smile upon her face, a smile that seems to say "he'll think of me--he'll think of me-- when he is far away! "though waters wide between us glide, our lives are warm and near: no distance parts two faithful hearts-- two hearts that love so dear: and i will trust my sailor-lad, for ever and a day, to think of me--to think of me-- when he is far away!" [illustration] beatrice. in her eyes is the living light of a wanderer to earth from a far celestial height: summers five are all the span-- summers five since time began to veil in mists of human night a shining angel-birth. does an angel look from her eyes? will she suddenly spring away, and soar to her home in the skies? beatrice! blessing and blessed to be! beatrice! still, as i gaze on thee, visions of two sweet maids arise, whose life was of yesterday: of a beatrice pale and stern, with the lips of a dumb despair, with the innocent eyes that yearn-- yearn for the young sweet hours of life, far from sorrow and far from strife, for the happy summers, that never return, when the world seemed good and fair: of a beatrice glorious, bright-- of a sainted, ethereal maid, whose blue eyes are deep fountains of light, cheering the poet that broodeth apart, filling with gladness his desolate heart, like the moon when she shines thro' a cloudless night on a world of silence and shade. and the visions waver and faint, and the visions vanish away that my fancy delighted to paint-- she is here at my side, a living child, with the glowing cheek and the tresses wild, nor death-pale martyr, nor radiant saint, yet stainless and bright as they. for i think, if a grim wild beast were to come from his charnel-cave, from his jungle-home in the east-- stealthily creeping with bated breath, stealthily creeping with eyes of death-- he would all forget his dream of the feast, and crouch at her feet a slave. she would twine her hand in his mane: she would prattle in silvery tone, like the tinkle of summer-rain-- questioning him with her laughing eyes, questioning him with a glad surprise, till she caught from those fierce eyes again the love that lit her own. and be sure, if a savage heart, in a mask of human guise, were to come on her here apart-- bound for a dark and a deadly deed, hurrying past with pitiless speed-- he would suddenly falter and guiltily start at the glance of her pure blue eyes. nay, be sure, if an angel fair, a bright seraph undefiled, were to stoop from the trackless air, fain would she linger in glad amaze-- lovingly linger to ponder and gaze, with a sister's love and a sister's care, on the happy, innocent child. _dec. , ._ [illustration] stolen waters. the light was faint, and soft the air that breathed around the place; and she was lithe, and tall, and fair, and with a wayward grace her queenly head she bare. with glowing cheek, with gleaming eye, she met me on the way: my spirit owned the witchery within her smile that lay: i followed her, i knew not why. the trees were thick with many a fruit, the grass with many a flower: my soul was dead, my tongue was mute, in that accursëd hour. and, in my dream, with silvery voice, she said, or seemed to say, "youth is the season to rejoice--" i could not choose but stay: i could not say her nay. she plucked a branch above her head, with rarest fruitage laden: "drink of the juice, sir knight," she said: "'tis good for knight and maiden." oh, blind mine eye that would not trace-- oh, deaf mine ear that would not heed-- the mocking smile upon her face, the mocking voice of greed! i drank the juice; and straightway felt a fire within my brain: my soul within me seemed to melt in sweet delirious pain. "sweet is the stolen draught," she said: "hath sweetness stint or measure? pleasant the secret hoard of bread: what bars us from our pleasure?" "yea, take we pleasure while we may," i heard myself replying. in the red sunset, far away, my happier life was dying: my heart was sad, my voice was gay. and unawares, i knew not how, i kissed her dainty finger-tips, i kissed her on the lily brow, i kissed her on the false, false lips-- that burning kiss, i feel it now! "true love gives true love of the best: then take," i cried, "my heart to thee!" the very heart from out my breast i plucked, i gave it willingly: her very heart she gave to me-- then died the glory from the west. in the gray light i saw her face, and it was withered, old, and gray; the flowers were fading in their place, were fading with the fading day. forth from her, like a hunted deer, through all that ghastly night i fled, and still behind me seemed to hear her fierce unflagging tread; and scarce drew breath for fear. yet marked i well how strangely seemed the heart within my breast to sleep: silent it lay, or so i dreamed, with never a throb or leap. for hers was now my heart, she said, the heart that once had been mine own: and in my breast i bore instead a cold, cold heart of stone. so grew the morning overhead. the sun shot downward through the trees his old familiar flame: all ancient sounds upon the breeze from copse and meadow came-- but i was not the same. they call me mad: i smile, i weep, uncaring how or why: yea, when one's heart is laid asleep, what better than to die? so that the grave be dark and deep. to die! to die? and yet, methinks, i drink of life, to-day, deep as the thirsty traveler drinks of fountain by the way: my voice is sad, my heart is gay. when yestereve was on the wane, i heard a clear voice singing so sweetly that, like summer-rain, my happy tears came springing: my human heart returned again. _"a rosy child, sitting and singing, in a garden fair, the joy of hearing, seeing, the simple joy of being-- or twining rosebuds in the golden hair that ripples free and wild. "a sweet pale child-- wearily looking to the purple west-- waiting the great for-ever that suddenly shall sever the cruel chains that hold her from her rest-- by earth-joys unbeguiled. "an angel-child-- gazing with living eyes on a dead face: the mortal form forsaken, that none may now awaken, that lieth painless, moveless in her place, as though in death she smiled! "be as a child-- so shalt thou sing for very joy of breath-- so shalt thou wait thy dying, in holy transport lying-- so pass rejoicing through the gate of death, in garment undefiled."_ then call me what they will, i know that now my soul is glad: if this be madness, better so, far better to be mad, weeping or smiling as i go. for if i weep, it is that now i see how deep a loss is mine, and feel how brightly round my brow the coronal might shine, had i but kept mine early vow: and if i smile, it is that now i see the promise of the years-- the garland waiting for my brow, that must be won with tears, with pain--with death--i care not how. _may , ._ [illustration] the willow-tree. the morn was bright, the steeds were light, the wedding guests were gay: young ellen stood within the wood and watched them pass away. she scarcely saw the gallant train: the tear-drop dimmed her ee: unheard the maiden did complain beneath the willow-tree. "oh robin, thou didst love me well, till, on a bitter day, she came, the lady isabel, and stole thy heart away. my tears are vain: i live again in days that used to be, when i could meet thy welcome feet beneath the willow-tree. "oh willow gray, i may not stay till spring renew thy leaf; but i will hide myself away, and nurse a lonely grief. it shall not dim life's joy for him: my tears he shall not see: while he is by, i'll come not nigh my weeping willow-tree. "but when i die, oh let me lie beneath thy loving shade, that he may loiter careless by, where i am lowly laid. and let the white white marble tell, if he should stoop to see, 'here lies a maid that loved thee well, beneath the willow-tree.'" . only a woman's hair. 'only a woman's hair'! fling it aside! a bubble on life's mighty stream: heed it not, man, but watch the broadening tide bright with the western beam. nay! in those words there rings from other years the echo of a long low cry, where a proud spirit wrestles with its tears in loneliest agony. and, as i touch that lock, strange visions throng upon my soul with dreamy grace-- of woman's hair, the theme of poet's song in every time and place. a child's bright tresses, by the breezes kissed to sweet disorder as she flies, veiling, beneath a cloud of golden mist, flushed cheek and laughing eyes-- or fringing, like a shadow, raven-black, the glory of a queen-like face-- or from a gipsy's sunny brow tossed back in wild and wanton grace-- or crown-like on the hoary head of age, whose tale of life is well-nigh told-- or, last, in dreams i make my pilgrimage to bethany of old. i see the feast--the purple and the gold-- the gathering crowd of pharisees, whose scornful eyes are centred to behold yon woman on her knees. the stifled sob rings strangely on mine ears, wrung from the depth of sin's despair: and still she bathes the sacred feet with tears, and wipes them with her hair. he scorned not then the simple loving deed of her, the lowest and the last; then scorn not thou, but use with earnest heed this relic of the past. the eyes that loved it once no longer wake: so lay it by with reverent care-- touching it tenderly for sorrow's sake-- it is a woman's hair. _feb. , ._ [illustration] the sailor's wife. see! there are tears upon her face-- tears newly shed, and scarcely dried: close, in an agonised embrace, she clasps the infant at her side. peace dwells in those soft-lidded eyes, those parted lips that faintly smile-- peace, the foretaste of paradise, in heart too young for care or guile. no peace that mother's features wear; but quivering lip, and knotted brow, and broken mutterings, all declare the fearful dream that haunts her now. the storm-wind, rushing through the sky, wails from the depths of cloudy space; shrill, piercing as the seaman's cry when death and he are face to face. familiar tones are in the gale: they ring upon her startled ear: and quick and low she pants the tale that tells of agony and fear: "still that phantom-ship is nigh-- with a vexed and life-like motion, all beneath an angry sky, rocking on an angry ocean. "round the straining mast and shrouds throng the spirits of the storm: darkly seen through driving clouds, bends each gaunt and ghastly form. "see! the good ship yields at last! dumbly yields, and fights no more; driving, in the frantic blast, headlong on the fatal shore. "hark! i hear her battered side, with a low and sullen shock, dashed, amid the foaming tide, full upon a sunken rock. "his face shines out against the sky, like a ghost, so cold and white; with a dead despairing eye gazing through the gathered night. "is he watching, through the dark where a mocking ghostly hand points a faint and feeble spark glimmering from the distant land? "sees he, in this hour of dread, hearth and home and wife and child? loved ones who, in summers fled, clung to him and wept and smiled? "reeling sinks the fated bark to her tomb beneath the wave: must he perish in the dark-- not a hand stretched out to save? "see the spirits, how they crowd! watching death with eyes that burn! waves rush in----" she shrieks aloud, ere her waking sense return. the storm is gone: the skies are clear: hush'd is that bitter cry of pain: the only sound, that meets her ear, the heaving of the sullen main. though heaviness endure the night, yet joy shall come with break of day: she shudders with a strange delight-- the fearful dream is pass'd away. she wakes: the grey dawn streaks the dark: with early song the copses ring: far off she hears the watch-dog bark a joyful bark of welcoming! _feb. , ._ [illustration] after three days. i stood within the gate of a great temple, 'mid the living stream of worshipers that thronged its regal state fair-pictured in my dream. jewels and gold were there; and floors of marble lent a crystal sheen to body forth, as in a lower air, the wonders of the scene. such wild and lavish grace had whispers in it of a coming doom; as richest flowers lie strown about the face of her that waits the tomb. the wisest of the land had gathered there, three solemn trysting-days, for high debate: men stood on either hand to listen and to gaze. the aged brows were bent, bent to a frown, half thought, and half annoy, that all their stores of subtlest argument were baffled by a boy. in each averted face i marked but scorn and loathing, till mine eyes fell upon one that stirred not in his place, tranced in a dumb surprise. surely within his mind strange thoughts are born, until he doubts the lore of those old men, blind leaders of the blind, whose kingdom is no more. surely he sees afar a day of death the stormy future brings; the crimson setting of the herald-star that led the eastern kings. thus, as a sunless deep mirrors the shining heights that crown the bay, so did my soul create anew in sleep the picture seen by day. gazers came and went-- a restless hum of voices marked the spot-- in varying shades of critic discontent prating they knew not what. "where is the comely limb, the form attuned in every perfect part, the beauty that we should desire in him?" ah! fools and slow of heart! look into those deep eyes, deep as the grave, and strong with love divine; those tender, pure, and fathomless mysteries, that seem to pierce through thine. look into those deep eyes, stirred to unrest by breath of coming strife, until a longing in thy soul arise that this indeed were life: that thou couldst find him there, bend at his sacred feet thy willing knee, and from thy heart pour out the passionate prayer "lord, let me follow thee!" but see the crowd divide: mother and sire have found their lost one now: the gentle voice, that fain would seem to chide whispers "son, why hast thou"-- in tone of sad amaze-- "thus dealt with us, that art our dearest thing? behold, thy sire and i, three weary days, have sought thee sorrowing." and i had stayed to hear the loving words "how is it that ye sought?"-- but that the sudden lark, with matins clear, severed the links of thought. then over all there fell shadow and silence; and my dream was fled, as fade the phantoms of a wizard's cell when the dark charm is said. yet, in the gathering light, i lay with half-shut eyes that would not wake, lovingly clinging to the skirts of night for that sweet vision's sake. _feb. , ._ [illustration] faces in the fire. the night creeps onward, sad and slow: in these red embers' dying glow the forms of fancy come and go. an island-farm--broad seas of corn stirred by the wandering breath of morn-- the happy spot where i was born. the picture fadeth in its place: amid the glow i seem to trace the shifting semblance of a face. 'tis now a little childish form-- red lips for kisses pouted warm-- and elf-locks tangled in the storm. 'tis now a grave and gentle maid, at her own beauty half afraid, shrinking, and willing to be stayed. oh, time was young, and life was warm, when first i saw that fairy-form, her dark hair tossing in the storm. and fast and free these pulses played, when last i met that gentle maid-- when last her hand in mine was laid. those locks of jet are turned to gray, and she is strange and far away that might have been mine own to-day-- that might have been mine own, my dear, through many and many a happy year-- that might have sat beside me here. ay, changeless through the changing scene, the ghostly whisper rings between, the dark refrain of 'might have been.' the race is o'er i might have run: the deeds are past i might have done; and sere the wreath i might have won. sunk is the last faint flickering blaze: the vision of departed days is vanished even as i gaze. the pictures, with their ruddy light, are changed to dust and ashes white, and i am left alone with night. _jan., ._ [illustration] a lesson in latin. our latin books, in motley row, invite us to our task-- gay horace, stately cicero: yet there's one verb, when once we know, no higher skill we ask: this ranks all other lore above-- we've learned "'_amare_' means '_to love_'!" so, hour by hour, from flower to flower, we sip the sweets of life: till, all too soon, the clouds arise, and flaming cheeks and flashing eyes proclaim the dawn of strife: with half a smile and half a sigh, "_amare! bitter one!_" we cry. last night we owned, with looks forlorn, "too well the scholar knows there is no rose without a thorn"-- but peace is made! we sing, this morn, "no thorn without a rose!" our latin lesson is complete: we've learned that love is bitter-sweet! _may, ._ puck lost and found. puck has fled the haunts of men: ridicule has made him wary: in the woods, and down the glen, no one meets a fairy! "cream!" the greedy goblin cries-- empties the deserted dairy-- steals the spoons, and off he flies. still we seek our fairy! ah! what form is entering? lovelit eyes and laughter airy! is not this a better thing, child, whose visit thus i sing, even than a fairy? _nov. , ._ puck has ventured back agen: ridicule no more affrights him: in the very haunts of men newer sport delights him. capering lightly to and fro, ever frolicking and funning-- "crack!" the mimic pistols go! hark! the noise is stunning! all too soon will childhood gay realise life's sober sadness. let's be merry while we may, innocent and happy fay! elves were made for gladness! _nov. , ._ [illustration] a song of love. say, what is the spell, when her fledgelings are cheeping, that lures the bird home to her nest? or wakes the tired mother, whose infant is weeping, to cuddle and croon it to rest? what the magic that charms the glad babe in her arms, till it cooes with the voice of the dove? 'tis a secret, and so let us whisper it low-- and the name of the secret is love! for i think it is love, for i feel it is love, for i'm sure it is nothing but love! say, whence is the voice that, when anger is burning, bids the whirl of the tempest to cease? that stirs the vexed soul with an aching--a yearning for the brotherly hand-grip of peace? whence the music that fills all our being--that thrills around us, beneath, and above? 'tis a secret: none knows how it comes, how it goes-- but the name of the secret is love! for i think it is love, for i feel it is love, for i'm sure it is nothing but love! say, whose is the skill that paints valley and hill, like a picture so fair to the sight? that flecks the green meadow with sunshine and shadow, till the little lambs leap with delight? 'tis a secret untold to hearts cruel and cold, though 'tis sung, by the angels above, in notes that ring clear for the ears that can hear-- and the name of the secret is love! for i think it is love, for i feel it is love, for i'm sure it is nothing but love! _oct., ._ the end. [turn over. works by lewis carroll. alice's adventures in wonderland. with forty-two illustrations by tenniel. (first published in .) crown vo, cloth, gilt edges, price _s._ net. eighty-sixth thousand. the same; people's edition. (first published in .) crown vo, cloth, price _s._ _d._ net. seventieth thousand. aventures d'alice au pays des merveilles. traduit de l'anglais par henri buÃ�. ouvrage illustré de vignettes par john tenniel. (first published in .) crown vo, cloth, gilt edges, price _s._ net. second thousand. alice's abenteuer im wunderland. aus dem englischen von antonie zimmermann. mit illustrationen von john tenniel. (first published in .) crown vo, cloth, gilt edges, price _s._ net. le avventure d'alice nel paese delle meraviglie. tradotte dall' inglese da t. pietrocola-rossetti. con vignette di giovanni tenniel. (first published in .) crown vo, cloth, gilt edges, price _s._ net. alice's adventures under ground. being a facsimile of the original ms. book, which was afterwards developed into "alice's adventures in wonderland." with thirty-seven illustrations by the author. (begun, july, ; finished, feb. ; first published, in facsimile, in .) crown vo, cloth, gilt edges, price _s._ net. third thousand. the nursery "alice." containing twenty coloured enlargements from tenniel's illustrations to "alice's adventures in wonderland." with text adapted to nursery readers. cover designed by e. gertrude thomson. (first published in .) to, boards, price _s._ net. eleventh thousand. through the looking glass, and what alice found there. with fifty illustrations by tenniel. 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(first published in , being a reprint, with a few additions, of the comic portion of "phantasmagoria and other poems," published in , and of "the hunting of the snark," published in .) crown vo, cloth, gilt edges, price _s._ net. sixth thousand. symbolic logic. in three parts, which will be issued separately:-- part i. elementary. (first published in .) crown vo, limp cloth, price _s._, net. second thousand, fourth edition. part ii. advanced. } } [_in preparation._ part iii. transcendental. } n.b.--an envelope, containing two blank diagrams (biliteral and triliteral) and counters ( red and grey) can be had for _d._, by post _d._ a tangled tale. reprinted from _the monthly packet_. with six illustrations by arthur b. frost. (first published in .) crown vo, cloth, gilt edges, price _s._ _d._ net. fourth thousand. sylvie and bruno. with forty-six illustrations by harry furniss. (first published in .) crown vo, cloth, gilt edges, price _s._ _d._ net. thirteenth thousand. n.b.--this book contains pages--nearly as much as the two "alice" books put together. sylvie and bruno concluded. with forty-six illustrations by harry furniss. (first published in .) crown vo, cloth, gilt edges, price _s._ _d._ net. third thousand. n.b.--this book contains pages. original games and puzzles. crown vo, cloth, gilt edges. [_in preparation._ three sunsets, and other poems. with twelve fairy-fancies by e. gertrude thomson. (first published in .) fcap. to, cloth, gilt edges, price _s._, net. n.b.--this is a reprint, with a few additions, of the serious portion of "phantasmagoria, and other poems," published in . macmillan and co., limited, london. advice to writers. buy "the wonderland case for postage stamps," invented by lewis carroll, oct. , , size inches by , containing separate pockets for stamps of different values, coloured pictorial surprises taken from _alice in wonderland_, and or wise words about letter-writing. it is published by messrs. emberlin & son, magdalen street, oxford. price _s._ n.b.--if ordered by post, an additional payment will be required, to cover cost of postage, as follows:-- one copy, - / _d._ two or three do., _d._ four do., - / _d._ five to fourteen do., _d._ each subsequent fourteen or fraction thereof, - / _d._ rhyme? and reason? [illustration: "upon a battlement." _see_ p. .] rhyme? and reason? by lewis carroll _with sixty-five illustrations_ by arthur b. frost _and nine_ by henry holiday i have had nor rhyme nor reason _price seven shillings_ london macmillan and co. [_all rights reserved_] london: r. clay, sons, and taylor bread street hill, e.c. inscribed to a dear child: in memory of golden summer hours and whispers of a summer sea. * * * * * girt with a boyish garb for boyish task, eager she wields her spade: yet loves as well rest on a friendly knee, intent to ask the tale one loves to tell. rude scoffer of the seething outer strife, unmeet to read her pure and simple spright, deem, if thou wilt, such hours a waste of life, empty of all delight! chat on, sweet maid, and rescue from annoy hearts that by wiser talk are unbeguiled; ah, happy he who owns that tenderest joy, the heart-love of a child! away, fond thoughts, and vex my soul no more! work claims my wakeful nights, my busy days albeit bright memories of that sunlit shore yet haunt my dreaming gaze! [of the following poems, echoes, a game of fives, the last three of the four riddles, and fame's penny-trumpet, are here published for the first time. the others have all appeared before, as have also the illustrations to the hunting of the snark.] contents. page phantasmagoria, in seven cantos:-- i. the trystyng ii. hys fyve rules iii. scarmoges iv. hys nouryture v. byckerment vi. dyscomfyture vii. sad souvenaunce echoes a sea dirge y{e} carpette knyghte hiawatha's photographing melancholetta a valentine the three voices:-- the first voice the second voice the third voice tÈma con variaziÓni a game of fives poeta fit, non nascitur the hunting of the snark, an agony in eight fits:-- i. the landing ii. the bellman's speech iii. the baker's tale iv. the hunting v. the beaver's lesson vi. the barrister's dream vii. the banker's fate viii. the vanishing size and tears atalanta in camden town the lang coortin' four riddles fame's penny-trumpet phantasmagoria. canto i. the trystyng. one winter night, at half-past nine, cold, tired, and cross, and muddy, i had come home, too late to dine, and supper, with cigars and wine, was waiting in the study. there was a strangeness in the room, and something white and wavy was standing near me in the gloom-- _i_ took it for the carpet-broom left by that careless slavey. [illustration] but presently the thing began to shiver and to sneeze: on which i said "come, come, my man! that's a most inconsiderate plan. less noise there, if you please!" "i've caught a cold," the thing replies, "out there upon the landing." i turned to look in some surprise, and there, before my very eyes, a little ghost was standing! he trembled when he caught my eye, and got behind a chair. "how came you here," i said, "and why? i never saw a thing so shy. come out! don't shiver there!" he said "i'd gladly tell you how, and also tell you why; but" (here he gave a little bow) "you're in so bad a temper now, you'd think it all a lie. "and as to being in a fright, allow me to remark that ghosts have just as good a right, in every way, to fear the light, as men to fear the dark." "no plea," said i, "can well excuse such cowardice in you: for ghosts can visit when they choose, whereas we humans ca'n't refuse to grant the interview." he said "a flutter of alarm is not unnatural, is it? i really feared you meant some harm: but, now i see that you are calm, let me explain my visit. "houses are classed, i beg to state, according to the number of ghosts that they accommodate: (the tenant merely counts as _weight_, with coals and other lumber). "this is a 'one-ghost' house, and you when you arrived last summer, may have remarked a spectre who was doing all that ghosts can do to welcome the new-comer. "in villas this is always done-- however cheaply rented: for, though of course there's less of fun when there is only room for one, ghosts have to be contented. "that spectre left you on the third-- since then you've not been haunted: for, as he never sent us word, 'twas quite by accident we heard that any one was wanted. "a spectre has first choice, by right, in filling up a vacancy; then phantom, goblin, elf, and sprite-- if all these fail them, they invite the nicest ghoul that they can see. "the spectres said the place was low, and that you kept bad wine: so, as a phantom had to go, and i was first, of course, you know, i couldn't well decline." "no doubt," said i, "they settled who was fittest to be sent: yet still to choose a brat like you, to haunt a man of forty-two, was no great compliment!" "i'm not so young, sir," he replied, "as you might think. the fact is, in caverns by the water-side, and other places that i've tried, i've had a lot of practice: "but i have never taken yet a strict domestic part, and in my flurry i forget the five good rules of etiquette we have to know by heart." my sympathies were warming fast towards the little fellow: he was so utterly aghast at having found a man at last, and looked so scared and yellow. [illustration: "in caverns by the water-side"] "at least," i said, "i'm glad to find a ghost is not a _dumb_ thing! but pray sit down: you'll feel inclined (if, like myself, you have not dined) to take a snack of something: "though, certainly, you don't appear a thing to offer _food_ to! and then i shall be glad to hear-- if you will say them loud and clear-- the rules that you allude to." "thanks! you shall hear them by and by this _is_ a piece of luck!" "what may i offer you?" said i. "well, since you _are_ so kind, i'll try a little bit of duck. "_one_ slice! and may i ask you for another drop of gravy?" i sat and looked at him in awe, for certainly i never saw a thing so white and wavy. [illustration] and still he seemed to grow more white, more vapoury, and wavier-- seen in the dim and flickering light, as he proceeded to recite his "maxims of behaviour." canto ii. hys fyve rules. "my first--but don't suppose," he said, "i'm setting you a riddle-- is--if your victim be in bed, don't touch the curtains at his head, but take them in the middle, "and wave them slowly in and out, while drawing them asunder; and in a minute's time, no doubt, he'll raise his head and look about with eyes of wrath and wonder. "and here you must on no pretence make the first observation. wait for the victim to commence: no ghost of any common sense begins a conversation. [illustration] "if he should say '_how came you here?_' (the way that _you_ began, sir,) in such a case your course is clear-- '_on the bat's back, my little dear!_' is the appropriate answer. "if after this he says no more, you'd best perhaps curtail your exertions--go and shake the door, and then, if he begins to snore, you'll know the thing's a failure. "by day, if he should be alone-- at home or on a walk-- you merely give a hollow groan, to indicate the kind of tone in which you mean to talk. "but if you find him with his friends, the thing is rather harder. in such a case success depends on picking up some candle-ends, or butter, in the larder. "with this you make a kind of slide (it answers best with suet), on which you must contrive to glide, and swing yourself from side to side-- one soon learns how to do it. "the second tells us what is right in ceremonious calls:-- '_first burn a blue or crimson light_' (a thing i quite forgot to-night), '_then scratch the door or walls._'" [illustration: "and swing yourself from side to side"] i said "you'll visit _here_ no more, if you attempt the guy. i'll have no bonfires on _my_ floor-- and, as for scratching at the door, i'd like to see you try!" "the third was written to protect the interests of the victim, and tells us, as i recollect, _to treat him with a grave respect, and not to contradict him_." "that's plain," said i, "as tare and tret, to any comprehension: i only wish _some_ ghosts i've met would not so _constantly_ forget the maxim that you mention!" "perhaps," he said, "_you_ first transgressed the laws of hospitality: all ghosts instinctively detest the man that fails to treat his guest with proper cordiality. [illustration] "if you address a ghost as 'thing!' or strike him with a hatchet, he is permitted by the king to drop all _formal_ parleying-- and then you're _sure_ to catch it! "the fourth prohibits trespassing where other ghosts are quartered: and those convicted of the thing (unless when pardoned by the king) must instantly be slaughtered. "that simply means 'be cut up small': ghosts soon unite anew: the process scarcely hurts at all-- not more than when _you're_ what you call 'cut up' by a review. "the fifth is one you may prefer that i should quote entire:-- _the king must be addressed as 'sir.' this, from a simple courtier, is all the laws require_: "_but, should you wish to do the thing with out-and-out politeness, accost him as 'my goblin king!' and always use, in answering, the phrase 'your royal whiteness!'_ "i'm getting rather hoarse, i fear, after so much reciting: so, if you don't object, my dear, we'll try a glass of bitter beer-- i think it looks inviting." [illustration] canto iii. scarmoges. "and did you really walk," said i, "on such a wretched night? i always fancied ghosts could fly-- if not exactly in the sky, yet at a fairish height." "it's very well," said he, "for kings to soar above the earth: but phantoms often find that wings-- like many other pleasant things-- cost more than they are worth. "spectres of course are rich, and so can buy them from the elves: but _we_ prefer to keep below-- they're stupid company, you know. for any but themselves: [illustration] "for, though they claim to be exempt from pride, they treat a phantom as something quite beneath contempt-- just as no turkey ever dreamt of noticing a bantam." "they seem too proud," said i, "to go to houses such as mine. pray, how did they contrive to know so quickly that 'the place was low,' and that i 'kept bad wine'?" "inspector kobold came to you--" the little ghost began. here i broke in--"inspector who? inspecting ghosts is something new! explain yourself my man!" "his name is kobold," said my guest: "one of the spectre order: you'll very often see him dressed in a yellow gown, a crimson vest, and a night-cap with a border. "he tried the brocken business first, but caught a sort of chill; so came to england to be nursed, and here it took the form of _thirst_, which he complains of still. [illustration: "and here it took the form of _thirst_"] "port-wine, he says, when rich and sound, warms his old bones like nectar: and as the inns, where it is found, are his especial hunting-ground, we call him the _inn-spectre_." i bore it--bore it like a man-- this agonizing witticism! and nothing could be sweeter than my temper, till the ghost began some most provoking criticism. "cooks need not be indulged in waste; yet still you'd better teach them dishes should have _some sort_ of taste. pray, why are all the cruets placed where nobody can reach them? "that man of yours will never earn his living as a waiter! is that queer _thing_ supposed to burn? (it's far too dismal a concern to call a moderator). "the duck was tender, but the peas were very much too old: and just remember, if you please, the _next_ time you have toasted cheese, don't let them send it cold. "you'd find the bread improved, i think, by getting better flour: and have you anything to drink that looks a _little_ less like ink, and isn't _quite_ so sour?" then, peering round with curious eyes, he muttered "goodness gracious!" and so went on to criticise-- "your room's an inconvenient size: it's neither snug nor spacious. "that narrow window, i expect, serves but to let the dusk in--" "but please," said i, "to recollect 'twas fashioned by an architect who pinned his faith on ruskin!" "i don't care who he was, sir, or on whom he pinned his faith! constructed by whatever law, so poor a job i never saw, as i'm a living wraith! "what a re-markable cigar! how much are they a dozen?" i growled "no matter what they are! you're getting as familiar as if you were my cousin! "now that's a thing _i will not stand_, and so i tell you flat." "aha," said he, "we're getting grand!" (taking a bottle in his hand) "i'll soon arrange for _that_!" and here he took a careful aim, and gaily cried "here goes!" i tried to dodge it as it came, but somehow caught it, all the same, exactly on my nose. and i remember nothing more that i can clearly fix, till i was sitting on the floor, repeating "two and five are four, but _five and two_ are six." what really passed i never learned, nor guessed: i only know that, when at last my sense returned, the lamp, neglected, dimly burned-- the fire was getting low-- through driving mists i seemed to see a thing that smirked and smiled: and found that he was giving me a lesson in biography, as if i were a child. [illustration] canto iv. hys nouryture. "oh, when i was a little ghost, a merry time had we! each seated on his favourite post, we chumped and chawed the buttered toast they gave us for our tea." "that story is in print!" i cried. "don't say it's not, because it's known as well as bradshaw's guide!" (the ghost uneasily replied he hardly thought it was). "it's not in nursery rhymes? and yet i almost think it is-- 'three little ghosteses' were set 'on posteses,' you know, and ate their 'buttered toasteses.' "i have the book; so, if you doubt it--" i turned to search the shelf. "don't stir!" he cried. "we'll do without it; i now remember all about it; i wrote the thing myself. "it came out in a 'monthly,' or at least my agent said it did: some literary swell, who saw it, thought it seemed adapted for the magazine he edited. "my father was a brownie, sir; my mother was a fairy. the notion had occurred to her, the children would be happier, if they were taught to vary. "the notion soon became a craze; and, when it once began, she brought us all out in different ways-- one was a pixy, two were fays, another was a banshee; "the fetch and kelpie went to school, and gave a lot of trouble; next came a poltergeist and ghoul, and then two trolls (which broke the rule), a goblin, and a double-- "(if that's a snuff-box on the shelf," he added with a yawn, "i'll take a pinch)--next came an elf, and then a phantom (that's myself), and last, a leprechaun. "one day, some spectres chanced to call, dressed in the usual white: i stood and watched them in the hall, and couldn't make them out at all, they seemed so strange a sight. [illustration] "i wondered what on earth they were, that looked all head and sack; but mother told me not to stare, and then she twitched me by the hair, and punched me in the back. "since then i've often wished that i had been a spectre born. but what's the use?" (he heaved a sigh). "_they_ are the ghost-nobility, and look on _us_ with scorn. "my phantom-life was soon begun: when i was barely six, i went out with an older one-- and just at first i thought it fun, and learned a lot of tricks. "i've haunted dungeons, castles, towers-- wherever i was sent: i've often sat and howled for hours, drenched to the skin with driving showers, upon a battlement. "it's quite old-fashioned now to groan when you begin to speak: this is the newest thing in tone--" and here (it chilled me to the bone) he gave an _awful_ squeak. "perhaps," he added, "to _your_ ear that sounds an easy thing? try it yourself, my little dear! it took _me_ something like a year, with constant practising. "and when you've learned to squeak, my man and caught the double sob, you're pretty much where you began: just try and gibber if you can! that's something _like_ a job! "_i've_ tried it, and can only say i'm sure you couldn't do it, e- ven if you practised night and day, unless you have a turn that way, and natural ingenuity. "shakspeare i think it is who treats of ghosts, in days of old, who 'gibbered in the roman streets,' dressed, if you recollect, in sheets-- they must have found it cold. "i've often spent ten pounds on stuff, in dressing as a double; but, though it answers as a puff, it never has effect enough to make it worth the trouble. "long bills soon quenched the little thirst i had for being funny. the setting-up is always worst: such heaps of things you want at first, one must be made of money! [illustration] "for instance, take a haunted tower, with skull, cross-bones, and sheet; blue lights to burn (say) two an hour, condensing lens of extra power, and set of chains complete: "what with the things you have to hire-- the fitting on the robe-- and testing all the coloured fire-- the outfit of itself would tire the patience of a job! "and then they're so fastidious, the haunted-house committee: i've often known them make a fuss because a ghost was french, or russ, or even from the city! "some dialects are objected to-- for one, the _irish_ brogue is: and then, for all you have to do, one pound a week they offer you, and find yourself in bogies!" canto v. byckerment. "don't they consult the 'victims,' though?" i said. "they should, by rights, give them a chance--because, you know, the tastes of people differ so, especially in sprites." the phantom shook his head and smiled. "consult them? not a bit! 'twould be a job to drive one wild, to satisfy one single child-- there'd be no end to it!" "of course you can't leave _children_ free," said i, "to pick and choose: but, in the case of men like me, i think 'mine host' might fairly be allowed to state his views." he said "it really wouldn't pay-- folk are so full of fancies. we visit for a single day, and whether then we go, or stay, depends on circumstances. "and, though we don't consult 'mine host' before the thing's arranged, still, if he often quits his post, or is not a well-mannered ghost, then you can have him changed. "but if the host's a man like you-- i mean a man of sense; and if the house is not too new--" "why, what has _that_," said i, "to do with ghost's convenience?" "a new house does not suit, you know-- it's such a job to trim it: but, after twenty years or so, the wainscotings begin to go, so twenty is the limit." "to trim" was not a phrase i could remember having heard: "perhaps," i said, "you'll be so good as tell me what is understood exactly by that word?" [illustration] "it means the loosening all the doors," the ghost replied, and laughed: "it means the drilling holes by scores in all the skirting-boards and floors, to make a thorough draught. "you'll sometimes find that one or two are all you really need to let the wind come whistling through-- but _here_ there'll be a lot to do!" i faintly gasped "indeed! "if i'd been rather later, i'll be bound," i added, trying (most unsuccessfully) to smile, "you'd have been busy all this while, trimming and beautifying?" "why, no," said he; "perhaps i should have stayed another minute-- but still no ghost, that's any good, without an introduction would have ventured to begin it. "the proper thing, as you were late, was certainly to go: but, with the roads in such a state, i got the knight-mayor's leave to wait for half an hour or so." "who's the knight-mayor?" i cried. instead of answering my question, "well! if you don't know _that_," he said, "either you never go to bed, or you've a grand digestion! "he goes about and sits on folk that eat too much at night: his duties are to pinch, and poke, and squeeze them till they nearly choke." (i said "it serves them right!") "and folk that sup on things like these--" he muttered, "eggs and bacon-- lobster--and duck--and toasted cheese-- if they don't get an awful squeeze, i'm very much mistaken! "he is immensely fat, and so well suits the occupation: in point of fact, if you must know, we used to call him, years ago, _the mayor and corporation_! [illustration: "he goes about and sits on folk"] "the day he was elected mayor i _know_ that every sprite meant to vote for _me_, but did not dare-- he was so frantic with despair and furious with excitement. [illustration] "when it was over, for a whim, he ran to tell the king; and being the reverse of slim, a two-mile trot was not for him a very easy thing. "so, to reward him for his run (as it was baking hot, and he was over twenty stone), the king proceeded, half in fun, to knight him on the spot." "'twas a great liberty to take!" (i fired up like a rocket). "he did it just for punning's sake: 'the man,' says johnson, 'that would make a pun, would pick a pocket!'" "a man," said he, "is not a king." i argued for a while, and did my best to prove the thing-- the phantom merely listening with a contemptuous smile. at last, when, breath and patience spent, i had recourse to smoking-- "your _aim_," he said, "is excellent: but--when you call it _argument_-- of course you're only joking?" [illustration] stung by his cold and snaky eye, i roused myself at length to say "at least i do defy the veriest sceptic to deny that union is strength!" "that's true enough," said he, "yet stay--" i listened in all meekness-- "_union_ is strength, i'm bound to say; in fact, the thing's as clear as day; but _onions_--are a weakness." canto vi. dyscomfyture. as one who strives a hill to climb, who never climbed before: who finds it, in a little time, grow every moment less sublime, and votes the thing a bore: yet, having once begun to try, dares not desert his quest, but, climbing, ever keeps his eye on one small hut against the sky, wherein he hopes to rest: who climbs till nerve and force are spent, with many a puff and pant: who still, as rises the ascent, in language grows more violent, although in breath more scant: [illustration] who, climbing, gains at length the place that crowns the upward track; and, entering with unsteady pace, receives a buffet in the face that lands him on his back: and feels himself, like one in sleep, glide swiftly down again, a helpless weight, from steep to steep, till, with a headlong giddy sweep, he drops upon the plain-- so i, that had resolved to bring conviction to a ghost, and found it quite a different thing from any human arguing, yet dared not quit my post but, keeping still the end in view to which i hoped to come, i strove to prove the matter true by putting everything i knew into an axiom: commencing every single phrase with 'therefore' or 'because,' i blindly reeled, a hundred ways, about the syllogistic maze, unconscious where i was. quoth he "that's regular clap-trap: don't bluster any more. now _do_ be cool and take a nap! such a ridiculous old chap was never seen before! "you're like a man i used to meet, who got one day so furious in arguing, the simple heat scorched both his slippers off his feet!" i said "_that's very curious!_" [illustration: "scorched both his slippers off his feet"] "well, it _is_ curious, i agree, and sounds perhaps like fibs: but still it's true as true can be-- as sure as your name's tibbs," said he. i said "my name's _not_ tibbs." "_not_ tibbs!" he cried--his tone became a shade or two less hearty-- "why, no," said i. "my proper name is tibbets--" "tibbets?" "aye, the same." "why, then you're not the party!" with that he struck the board a blow that shivered half the glasses. "why couldn't you have told me so three quarters of an hour ago, you prince of all the asses? "to walk four miles through mud and rain, to spend the night in smoking, and then to find that it's in vain-- and i've to do it all again-- it's really _too_ provoking! [illustration] "don't talk!" he cried, as i began to mutter some excuse. "who can have patience with a man that's got no more discretion than an idiotic goose? "to keep me waiting here, instead of telling me at once that this was not the house!" he said. "there, that'll do--be off to bed! don't gape like that, you dunce!" "it's very fine to throw the blame on _me_ in such a fashion! why didn't you enquire my name the very minute that you came?" i answered in a passion. "of course it worries you a bit to come so far on foot-- but how was _i_ to blame for it?" "well, well!" said he. "i must admit that isn't badly put. "and certainly you've given me the best of wine and victual-- excuse my violence," said he, "but accidents like this, you see, they put one out a little. "'twas _my_ fault after all, i find-- shake hands, old turnip-top!" the name was hardly to my mind, but, as no doubt he meant it kind, i let the matter drop. "good-night, old turnip-top, good-night! when i am gone, perhaps they'll send you some inferior sprite, who'll keep you in a constant fright and spoil your soundest naps. "tell him you'll stand no sort of trick; then, if he leers and chuckles, you just be handy with a stick (mind that it's pretty hard and thick) and rap him on the knuckles! "then carelessly remark 'old coon! perhaps you're not aware that, if you don't behave, you'll soon be chuckling to another tune-- and so you'd best take care!' "that's the right way to cure a sprite of such-like goings-on-- but gracious me! it's getting light! good-night, old turnip-top, good-night!" a nod, and he was gone. [illustration] canto vii. sad souvenaunce. [illustration] "what's this?" i pondered. "have i slept? or can i have been drinking?" but soon a gentler feeling crept upon me, and i sat and wept an hour or so, like winking. "no need for bones to hurry so!" i sobbed. "in fact, i doubt if it was worth his while to go-- and who is tibbs, i'd like to know, to make such work about? "if tibbs is anything like me, it's _possible_," i said, "he won't be over-pleased to be dropped in upon at half-past three, after he's snug in bed. "and if bones plagues him anyhow-- squeaking and all the rest of it, as he was doing here just now-- _i_ prophesy there'll be a row, and tibbs will have the best of it!" then, as my tears could never bring the friendly phantom back, it seemed to me the proper thing to mix another glass, and sing the following coronach. [illustration: "and tibbs will have the best of it"] '_and art thou gone, beloved ghost? best of familiars! nay then, farewell, my duckling roast, farewell, farewell, my tea and toast, my meerschaum and cigars!_ '_the hues of life are dull and gray, the sweets of life insipid, when thou, my charmer, art away-- old brick, or rather, let me say, old parallelepiped!_' instead of singing verse the third, i ceased--abruptly, rather: but, after such a splendid word, i felt that it would be absurd to try it any farther. so with a yawn i went my way to seek the welcome downy, and slept, and dreamed till break of day of poltergeist and fetch and fay and leprechaun and brownie! for years i've not been visited by any kind of sprite; yet still they echo in my head, those parting words, so kindly said, "old turnip-top, good-night!" [illustration] echoes. lady clara vere de vere was eight years old, she said: every ringlet, lightly shaken, ran itself in golden thread. she took her little porringer: of me she shall not win renown: for the baseness of its nature shall have strength to drag her down. "sisters and brothers, little maid? there stands the inspector at thy door: like a dog, he hunts for boys who know not two and two are four." "kind words are more than coronets," she said, and wondering looked at me: "it is the dead unhappy night, and i must hurry home to tea." a sea dirge. [illustration] there are certain things--as, a spider, a ghost, the income-tax, gout, an umbrella for three-- that i hate, but the thing that i hate the most is a thing they call the sea. pour some salt water over the floor-- ugly i'm sure you'll allow it to be: suppose it extended a mile or more, _that's_ very like the sea. beat a dog till it howls outright-- cruel, but all very well for a spree: suppose that he did so day and night, _that_ would be like the sea. i had a vision of nursery-maids; tens of thousands passed by me-- all leading children with wooden spades, and this was by the sea. who invented those spades of wood? who was it cut them out of the tree? none, i think, but an idiot could-- or one that loved the sea. it is pleasant and dreamy, no doubt, to float with 'thoughts as boundless, and souls as free': but, suppose you are very unwell in the boat, how do you like the sea? [illustration: "and this was by the sea"] there is an insect that people avoid (whence is derived the verb 'to flee'). where have you been by it most annoyed? in lodgings by the sea. if you like your coffee with sand for dregs, a decided hint of salt in your tea, and a fishy taste in the very eggs-- by all means choose the sea. and if, with these dainties to drink and eat, you prefer not a vestige of grass or tree, and a chronic state of wet in your feet, then--i recommend the sea. for _i_ have friends who dwell by the coast-- pleasant friends they are to me! it is when i am with them i wonder most that any one likes the sea. they take me a walk: though tired and stiff, to climb the heights i madly agree; and, after a tumble or so from the cliff, they kindly suggest the sea. i try the rocks, and i think it cool that they laugh with such an excess of glee, as i heavily slip into every pool that skirts the cold cold sea. [illustration] y{e} carpette knyghte. i have a horse--a ryghte goode horse-- ne doe i envye those who scoure y{e} playne yn headye course tyll soddayne on theyre nose they lyghte wyth unexpected force-- yt ys--a horse of clothes. i have a saddel--"say'st thou soe? wyth styrruppes, knyghte, to boote?" i sayde not that--i answere "noe"-- yt lacketh such, i woote: yt ys a mutton-saddel, loe! parte of y{e} fleecye brute. i have a bytte--a ryghte good bytte-- as shall bee seene yn tyme. y{e} jawe of horse yt wyll not fytte; yts use ys more sublyme. fayre syr, how deemest thou of yt? yt ys--thys bytte of rhyme. [illustration: "i have a horse"] hiawatha's photographing. [in an age of imitation, i can claim no special merit for this slight attempt at doing what is known to be so easy. any fairly practised writer, with the slightest ear for rhythm, could compose, for hours together, in the easy running metre of 'the song of hiawatha.' having, then, distinctly stated that i challenge no attention in the following little poem to its merely verbal jingle, i must beg the candid reader to confine his criticism to its treatment of the subject.] from his shoulder hiawatha took the camera of rosewood, made of sliding, folding rosewood; neatly put it all together. in its case it lay compactly, folded into nearly nothing; but he opened out the hinges, pushed and pulled the joints and hinges, till it looked all squares and oblongs, like a complicated figure in the second book of euclid. [illustration] this he perched upon a tripod-- crouched beneath its dusky cover-- stretched his hand, enforcing silence-- said "be motionless, i beg you!" mystic, awful was the process. all the family in order sat before him for their pictures: each in turn, as he was taken, volunteered his own suggestions, his ingenious suggestions. first the governor, the father: he suggested velvet curtains looped about a massy pillar; and the corner of a table, of a rosewood dining-table. he would hold a scroll of something, hold it firmly in his left-hand; he would keep his right-hand buried (like napoleon) in his waistcoat; he would contemplate the distance with a look of pensive meaning, as of ducks that die in tempests. grand, heroic was the notion: yet the picture failed entirely: failed, because he moved a little, moved, because he couldn't help it. next, his better half took courage; she would have her picture taken. _she_ came dressed beyond description, dressed in jewels and in satin far too gorgeous for an empress. [illustration: "first the governor, the father"] gracefully she sat down sideways, with a simper scarcely human, holding in her hand a bouquet rather larger than a cabbage. all the while that she was sitting, still the lady chattered, chattered, like a monkey in the forest. "am i sitting still?" she asked him. "is my face enough in profile? shall i hold the bouquet higher? will it come into the picture?" and the picture failed completely. next the son, the stunning-cantab: he suggested curves of beauty, curves pervading all his figure, which the eye might follow onward, till they centered in the breast-pin, centered in the golden breast-pin. he had learnt it all from ruskin (author of 'the stones of venice,' 'seven lamps of architecture,' 'modern painters,' and some others); and perhaps he had not fully understood his author's meaning; but, whatever was the reason, all was fruitless, as the picture ended in an utter failure. [illustration: "next the son, the stunning-cantab"] next to him the eldest daughter: she suggested very little, only asked if he would take her with her look of 'passive beauty.' her idea of passive beauty was a squinting of the left-eye, was a drooping of the right-eye, was a smile that went up sideways to the corner of the nostrils. hiawatha, when she asked him, took no notice of the question, looked as if he hadn't heard it; but, when pointedly appealed to, smiled in his peculiar manner, coughed and said it 'didn't matter,' bit his lip and changed the subject. nor in this was he mistaken, as the picture failed completely. so in turn the other sisters. [illustration: "next to him the eldest daughter"] last, the youngest son was taken: very rough and thick his hair was, very round and red his face was, very dusty was his jacket, very fidgety his manner. and his overbearing sisters called him names he disapproved of: called him johnny, 'daddy's darling,' called him jacky, 'scrubby school-boy.' and, so awful was the picture, in comparison the others seemed, to one's bewildered fancy, to have partially succeeded. finally my hiawatha tumbled all the tribe together, ('grouped' is not the right expression), and, as happy chance would have it, did at last obtain a picture where the faces all succeeded: each came out a perfect likeness. [illustration: "last, the youngest son was taken"] then they joined and all abused it, unrestrainedly abused it, as the worst and ugliest picture they could possibly have dreamed of. giving one such strange expressions-- sullen, stupid, pert expressions. really any one would take us (any one that did not know us) for the most unpleasant people!' (hiawatha seemed to think so, seemed to think it not unlikely). all together rang their voices, angry, loud, discordant voices, as of dogs that howl in concert, as of cats that wail in chorus. but my hiawatha's patience, his politeness and his patience, unaccountably had vanished, and he left that happy party. neither did he leave them slowly, with the calm deliberation, the intense deliberation of a photographic artist: but he left them in a hurry, left them in a mighty hurry, stating that he would not stand it, stating in emphatic language what he'd be before he'd stand it. hurriedly he packed his boxes: hurriedly the porter trundled on a barrow all his boxes: hurriedly he took his ticket: hurriedly the train received him: thus departed hiawatha. [illustration] melancholetta. with saddest music all day long she soothed her secret sorrow: at night she sighed "i fear 'twas wrong such cheerful words to borrow. dearest, a sweeter, sadder song i'll sing to thee to-morrow." i thanked her, but i could not say that i was glad to hear it: i left the house at break of day, and did not venture near it till time, i hoped, had worn away her grief, for nought could cheer it! [illustration: "at night she sighed"] my dismal sister! couldst thou know the wretched home thou keepest! thy brother, drowned in daily woe, is thankful when thou sleepest; for if i laugh, however low, when thou'rt awake, thou weepest! i took my sister t'other day (excuse the slang expression) to sadler's wells to see the play, in hopes the new impression might in her thoughts, from grave to gay effect some slight digression. i asked three gay young dogs from town to join us in our folly, whose mirth, i thought, might serve to drown my sister's melancholy: the lively jones, the sportive brown, and robinson the jolly. the maid announced the meal in tones that i myself had taught her, meant to allay my sister's moans like oil on troubled water: i rushed to jones, the lively jones, and begged him to escort her. vainly he strove, with ready wit, to joke about the weather-- to ventilate the last '_on dit_'-- to quote the price of leather-- she groaned "here i and sorrow sit: let us lament together!" i urged "you're wasting time, you know: delay will spoil the venison." "my heart is wasted with my woe! there is no rest--in venice, on the bridge of sighs!" she quoted low from byron and from tennyson. i need not tell of soup and fish in solemn silence swallowed, the sobs that ushered in each dish, and its departure followed, nor yet my suicidal wish to _be_ the cheese i hollowed. some desperate attempts were made to start a conversation; "madam," the sportive brown essayed, "which kind of recreation, hunting or fishing, have you made your special occupation?" her lips curved downwards instantly, as if of india-rubber. "hounds _in full cry_ i like," said she: (oh how i longed to snub her!) "of fish, a whale's the one for me, _it is so full of blubber_!" the night's performance was "king john." "it's dull," she wept, "and so-so!" a while i let her tears flow on, she said they soothed her woe so! at length the curtain rose upon 'bombastes furioso.' in vain we roared; in vain we tried to rouse her into laughter: her pensive glances wandered wide from orchestra to rafter-- "_tier upon tier!_" she said, and sighed; and silence followed after. [illustration] a valentine. [sent to a friend who had complained that i was glad enough to see him when he came, but didn't seem to miss him if he stayed away.] and cannot pleasures, while they last, be actual unless, when past, they leave us shuddering and aghast, with anguish smarting? and cannot friends be firm and fast, and yet bear parting? and must i then, at friendship's call, calmly resign the little all (trifling, i grant, it is and small) i have of gladness, and lend my being to the thrall of gloom and sadness? and think you that i should be dumb, and full _dolorum omnium_, excepting when _you_ choose to come and share my dinner? at other times be sour and glum and daily thinner? must he then only live to weep, who'd prove his friendship true and deep? by day a lonely shadow creep, at night-time languish, oft raising in his broken sleep the moan of anguish? the lover, if for certain days his fair one be denied his gaze, sinks not in grief and wild amaze, but, wiser wooer, he spends the time in writing lays, and posts them to her. and if the verse flow free and fast, till even the poet is aghast, a touching valentine at last the post shall carry, when thirteen days are gone and past of february. farewell, dear friend, and when we meet, in desert waste or crowded street, perhaps before this week shall fleet, perhaps to-morrow, i trust to find _your_ heart the seat of wasting sorrow. the three voices. the first voice. [illustration] he trilled a carol fresh and free: he laughed aloud for very glee: there came a breeze from off the sea: it passed athwart the glooming flat-- it fanned his forehead as he sat-- it lightly bore away his hat, all to the feet of one who stood like maid enchanted in a wood, frowning as darkly as she could. with huge umbrella, lank and brown, unerringly she pinned it down, right through the centre of the crown. then, with an aspect cold and grim, regardless of its battered rim, she took it up and gave it him. a while like one in dreams he stood, then faltered forth his gratitude in words just short of being rude: for it had lost its shape and shine, and it had cost him four-and-nine, and he was going out to dine. [illustration: "unerringly she pinned it down."] "to dine!" she sneered in acid tone. "to bend thy being to a bone clothed in a radiance not its own!" the tear-drop trickled to his chin: there was a meaning in her grin that made him feel on fire within. "term it not 'radiance,'" said he: "'tis solid nutriment to me. dinner is dinner: tea is tea." and she "yea so? yet wherefore cease? let thy scant knowledge find increase. say 'men are men, and geese are geese.'" he moaned: he knew not what to say. the thought "that i could get away!" strove with the thought "but i must stay." "to dine!" she shrieked in dragon-wrath. "to swallow wines all foam and froth! to simper at a table-cloth! "say, can thy noble spirit stoop to join the gormandising troop who find a solace in the soup? "canst thou desire or pie or puff? thy well-bred manners were enough, without such gross material stuff." "yet well-bred men," he faintly said, "are not unwilling to be fed: nor are they well without the bread." her visage scorched him ere she spoke: "there are," she said, "a kind of folk who have no horror of a joke. "such wretches live: they take their share of common earth and common air: we come across them here and there: "we grant them--there is no escape-- a sort of semi-human shape suggestive of the man-like ape." "in all such theories," said he, "one fixed exception there must be. that is, the present company." baffled, she gave a wolfish bark: he, aiming blindly in the dark, with random shaft had pierced the mark. she felt that her defeat was plain, yet madly strove with might and main to get the upper hand again. fixing her eyes upon the beach, as though unconscious of his speech, she said "each gives to more than each." he could not answer yea or nay: he faltered "gifts may pass away." yet knew not what he meant to say. "if that be so," she straight replied, "each heart with each doth coincide. what boots it? for the world is wide." [illustration: "he faltered 'gifts may pass away.'"] "the world is but a thought," said he: "the vast unfathomable sea is but a notion--unto me." and darkly fell her answer dread upon his unresisting head, like half a hundredweight of lead. "the good and great must ever shun that reckless and abandoned one who stoops to perpetrate a pun. "the man that smokes--that reads the _times_-- that goes to christmas pantomimes-- is capable of _any_ crimes!" he felt it was his turn to speak, and, with a shamed and crimson cheek, moaned "this is harder than bezique!" but when she asked him "wherefore so?" he felt his very whiskers glow, and frankly owned "i do not know." [illustration: "this is harder than bezique!"] while, like broad waves of golden grain, or sunlit hues on cloistered pane, his colour came and went again. pitying his obvious distress, yet with a tinge of bitterness, she said "the more exceeds the less." "a truth of such undoubted weight," he urged, "and so extreme in date, it were superfluous to state." roused into sudden passion, she in tone of cold malignity: "to others, yea: but not to thee." but when she saw him quail and quake, and when he urged "for pity's sake!" once more in gentle tone she spake. "thought in the mind doth still abide: that is by intellect supplied, and within that idea doth hide: "and he, that yearns the truth to know, still further inwardly may go, and find idea from notion flow: "and thus the chain, that sages sought, is to a glorious circle wrought, for notion hath its source in thought." so passed they on with even pace: yet gradually one might trace a shadow growing on his face. [illustration] the second voice. [illustration] they walked beside the wave-worn beach; her tongue was very apt to teach, and now and then he did beseech she would abate her dulcet tone, because the talk was all her own, and he was dull as any drone. she urged "no cheese is made of chalk": and ceaseless flowed her dreary talk, tuned to the footfall of a walk. her voice was very full and rich, and, when at length she asked him "which?" it mounted to its highest pitch. he a bewildered answer gave, drowned in the sullen moaning wave, lost in the echoes of the cave. he answered her he knew not what: like shaft from bow at random shot, he spoke, but she regarded not. she waited not for his reply, but with a downward leaden eye went on as if he were not by: sound argument and grave defence, strange questions raised on "why?" and "whence?" and wildly tangled evidence. when he, with racked and whirling brain, feebly implored her to explain, she simply said it all again. wrenched with an agony intense, he spake, neglecting sound and sense, and careless of all consequence: "mind--i believe--is essence--ent-- abstract--that is--an accident-- which we--that is to say--i meant--" when, with quick breath and cheeks all flushed, at length his speech was somewhat hushed, she looked at him, and he was crushed. it needed not her calm reply: she fixed him with a stony eye, and he could neither fight nor fly, while she dissected, word by word, his speech, half guessed at and half heard, as might a cat a little bird. [illustration: "he spake, neglecting sound and sense."] then, having wholly overthrown his views, and stripped them to the bone, proceeded to unfold her own. "shall man be man? and shall he miss of other thoughts no thought but this, harmonious dews of sober bliss? "what boots it? shall his fevered eye through towering nothingness descry the grisly phantom hurry by? "and hear dumb shrieks that fill the air; see mouths that gape, and eyes that stare and redden in the dusky glare? "the meadows breathing amber light, the darkness toppling from the height, the feathery train of granite night? "shall he, grown gray among his peers, through the thick curtain of his tears catch glimpses of his earlier years, [illustration: "shall man be man?"] "and hear the sounds he knew of yore, old shufflings on the sanded floor, old knuckles tapping at the door? "yet still before him as he flies one pallid form shall ever rise, and, bodying forth in glassy eyes "the vision of a vanished good, low peering through the tangled wood, shall freeze the current of his blood." still from each fact, with skill uncouth and savage rapture, like a tooth she wrenched some slow reluctant truth. till, like a silent water-mill, when summer suns have dried the rill, she reached a full stop, and was still. dead calm succeeded to the fuss, as when the loaded omnibus has reached the railway terminus: when, for the tumult of the street, is heard the engine's stifled beat, the velvet tread of porters' feet. with glance that ever sought the ground, she moved her lips without a sound, and every now and then she frowned. he gazed upon the sleeping sea, and joyed in its tranquillity, and in that silence dead, but she to muse a little space did seem, then, like the echo of a dream, harped back upon her threadbare theme. still an attentive ear he lent but could not fathom what she meant: she was not deep, nor eloquent. he marked the ripple on the sand: the even swaying of her hand was all that he could understand. he saw in dreams a drawing-room, where thirteen wretches sat in gloom, waiting--he thought he knew for whom: he saw them drooping here and there, each feebly huddled on a chair, in attitudes of blank despair: oysters were not more mute than they, for all their brains were pumped away, and they had nothing more to say-- save one, who groaned "three hours are gone!" who shrieked "we'll wait no longer, john! tell them to set the dinner on!" the vision passed: the ghosts were fled: he saw once more that woman dread: he heard once more the words she said. he left her, and he turned aside: he sat and watched the coming tide across the shores so newly dried. [illustration: "he sat and watched the coming tide"] he wondered at the waters clear, the breeze that whispered in his ear, the billows heaving far and near, and why he had so long preferred to hang upon her every word: "in truth," he said, "it was absurd." [illustration] the third voice. [illustration] not long this transport held its place: within a little moment's space quick tears were raining down his face. his heart stood still, aghast with fear; a wordless voice, nor far nor near, he seemed to hear and not to hear. "tears kindle not the doubtful spark. if so, why not? of this remark the bearings are profoundly dark." "her speech," he said, "hath caused this pain. easier i count it to explain the jargon of the howling main, "or, stretched beside some babbling brook, to con, with inexpressive look, an unintelligible book." low spake the voice within his head, in words imagined more than said, soundless as ghost's intended tread: "if thou art duller than before, why quittedst thou the voice of lore? why not endure, expecting more?" "rather than that," he groaned aghast, "i'd writhe in depths of cavern vast, some loathly vampire's rich repast." [illustration: "he groaned aghast"] "'twere hard," it answered, "themes immense to coop within the narrow fence that rings _thy_ scant intelligence." "not so," he urged, "nor once alone: but there was something in her tone that chilled me to the very bone. "her style was anything but clear, and most unpleasantly severe; her epithets were very queer. "and yet, so grand were her replies, i could not choose but deem her wise; i did not dare to criticise; "nor did i leave her, till she went so deep in tangled argument that all my powers of thought were spent." a little whisper inly slid, "yet truth is truth: you know you did." a little wink beneath the lid. and, sickened with excess of dread, prone to the dust he bent his head, and lay like one three-quarters dead. the whisper left him--like a breeze lost in the depths of leafy trees-- left him by no means at his ease. once more he weltered in despair, with hands, through denser-matted hair, more tightly clenched than then they were. when, bathed in dawn of living red, majestic frowned the mountain head, "tell me my fault," was all he said. when, at high noon, the blazing sky scorched in his head each haggard eye, then keenest rose his weary cry. and when at eve the unpitying sun smiled grimly on the solemn fun, "alack," he sighed, "what _have_ i done?" [illustration: "tortured, unaided, and alone"] but saddest, darkest was the sight, when the cold grasp of leaden night dashed him to earth, and held him tight. tortured, unaided, and alone, thunders were silence to his groan, bagpipes sweet music to its tone: "what? ever thus, in dismal round, shall pain and mystery profound pursue me like a sleepless hound, "with crimson-dashed and eager jaws, me, still in ignorance of the cause, unknowing what i broke of laws?" the whisper to his ear did seem like echoed flow of silent stream, or shadow of forgotten dream, the whisper trembling in the wind: "her fate with thine was intertwined," so spake it in his inner mind: [illustration: "a scared dullard, gibbering low"] "each orbed on each a baleful star: each proved the other's blight and bar: each unto each were best, most far: "yea, each to each was worse than foe: thou, a scared dullard, gibbering low, and she, an avalanche of woe!" tÈma con variaziÓni. [why is it that poetry has never yet been subjected to that process of dilution which has proved so advantageous to her sister-art music? the diluter gives us first a few notes of some well-known air, then a dozen bars of his own, then a few more notes of the air, and so on alternately: thus saving the listener, if not from all risk of recognising the melody at all, at least from the too-exciting transports which it might produce in a more concentrated form. the process is termed "setting" by composers, and any one, that has ever experienced the emotion of being unexpectedly set down in a heap of mortar, will recognise the truthfulness of this happy phrase. for truly, just as the genuine epicure lingers lovingly over a morsel of supreme venison--whose every fibre seems to murmur "excelsior!"--yet swallows, ere returning to the toothsome dainty, great mouthfuls of oatmeal-porridge and winkles: and just as the perfect connoisseur in claret permits himself but one delicate sip, and then tosses off a pint or more of boarding-school beer: so also---- i never loved a dear gazelle-- _nor anything that cost me much: high prices profit those who sell, but why should i be fond of such?_ to glad me with his soft black eye _my son comes trotting home from school; he's had a fight, but can't tell why-- he always was a little fool!_ but, when he came to know me well, _he kicked me out, her testy sire: and when i stained my hair, that belle, might note the change, and thus admire_ and love me, it was sure to dye _a muddy green or staring blue: whilst one might trace, with half an eye, the still triumphant carrot through_. a game of fives. [illustration] five little girls, of five, four, three, two, one: rolling on the hearthrug, full of tricks and fun. five rosy girls, in years from ten to six: sitting down to lessons--no more time for tricks. five growing girls, from fifteen to eleven: music, drawing, languages, and food enough for seven! five winsome girls, from twenty to sixteen: each young man that calls, i say "now tell me which you _mean_!" [illustration: "now tell me which you _mean_!"] five dashing girls, the youngest twenty-one: but, if nobody proposes, what is there to be done? five showy girls--but thirty is an age when girls may be _engaging_, but they somehow don't _engage_. five dressy girls, of thirty-one or more: so gracious to the shy young men they snubbed so much before! * * * * * five _passé_ girls--their age? well, never mind! we jog along together, like the rest of human kind: but the quondam "careless bachelor" begins to think he knows the answer to that ancient problem "how the money goes"! poeta fit, non nascitur. [illustration] "how shall i be a poet? how shall i write in rhyme? you told me once 'the very wish partook of the sublime.' then tell me how! don't put me off with your 'another time'!" the old man smiled to see him, to hear his sudden sally; he liked the lad to speak his mind enthusiastically; and thought "there's no hum-drum in him, nor any shilly-shally." "and would you be a poet before you've been to school? ah, well! i hardly thought you so absolute a fool. first learn to be spasmodic-- a very simple rule. "for first you write a sentence, and then you chop it small; then mix the bits, and sort them out just as they chance to fall: the order of the phrases makes no difference at all. "then, if you'd be impressive, remember what i say, that abstract qualities begin with capitals alway: the true, the good, the beautiful-- those are the things that pay! "next, when you are describing a shape, or sound, or tint; don't state the matter plainly, but put it in a hint; and learn to look at all things with a sort of mental squint." "for instance, if i wished, sir, of mutton-pies to tell, should i say 'dreams of fleecy flocks pent in a wheaten cell'?" "why, yes," the old man said: "that phrase would answer very well. "then fourthly, there are epithets that suit with any word-- as well as harvey's reading sauce with fish, or flesh, or bird-- of these, 'wild,' 'lonely,' 'weary,' 'strange,' are much to be preferred." "and will it do, o will it do to take them in a lump-- as 'the wild man went his weary way to a strange and lonely pump'?" "nay, nay! you must not hastily to such conclusions jump. "such epithets, like pepper, give zest to what you write; and, if you strew them sparely, they whet the appetite: but if you lay them on too thick, you spoil the matter quite! [illustration: "the wild man went his weary way"] "last, as to the arrangement: your reader, you should show him, must take what information he can get, and look for no im- mature disclosure of the drift and purpose of your poem. "therefore, to test his patience-- how much he can endure-- mention no places, names, or dates, and evermore be sure throughout the poem to be found consistently obscure. "first fix upon the limit to which it shall extend: then fill it up with 'padding' (beg some of any friend): your great sensation-stanza you place towards the end." "and what is a sensation, grandfather, tell me, pray? i think i never heard the word so used before to-day: be kind enough to mention one '_exempli gratiâ_.'" and the old man, looking sadly across the garden-lawn, where here and there a dew-drop yet glittered in the dawn, said "go to the adelphi, and see the 'colleen bawn.' "the word is due to boucicault-- the theory is his, where life becomes a spasm, and history a whiz: if that is not sensation, i don't know what it is. "now try your hand, ere fancy have lost its present glow--" "and then," his grandson added, "we'll publish it, you know: green cloth--gold-lettered at the back-- in duodecimo!" then proudly smiled that old man to see the eager lad rush madly for his pen and ink and for his blotting-pad-- but, when he thought of _publishing_, his face grew stern and sad. [illustration] the hunting of the snark, an agony in eight fits. preface. if--and the thing is wildly possible--the charge of writing nonsense were ever brought against the author of this brief but instructive poem, it would be based, i feel convinced, on the line (in p. ) "then the bowsprit got mixed with the rudder sometimes:" in view of this painful possibility, i will not (as i might) appeal indignantly to my other writings as a proof that i am incapable of such a deed: i will not (as i might) point to the strong moral purpose of this poem itself, to the arithmetical principles so cautiously inculcated in it, or to its noble teachings in natural history--i will take the more prosaic course of simply explaining how it happened. the bellman, who was almost morbidly sensitive about appearances, used to have the bowsprit unshipped once or twice a week to be revarnished; and it more than once happened, when the time came for replacing it, that no one on board could remember which end of the ship it belonged to. they knew it was not of the slightest use to appeal to the bellman about it--he would only refer to his naval code, and read out in pathetic tones admiralty instructions which none of them had ever been able to understand--so it generally ended in its being fastened on, anyhow, across the rudder. the helmsman[ ] used to stand by with tears in his eyes: _he_ knew it was all wrong, but alas! rule of the code, "_no one shall speak to the man at the helm_," had been completed by the bellman himself with the words "_and the man at the helm shall speak to no one_." so remonstrance was impossible, and no steering could be done till the next varnishing day. during these bewildering intervals the ship usually sailed backwards. as this poem is to some extent connected with the lay of the jabberwock, let me take this opportunity of answering a question that has often been asked me, how to pronounce "slithy toves." the "i" in "slithy" is long, as in "writhe"; and "toves" is pronounced so as to rhyme with "groves." again, the first "o" in "borogoves" is pronounced like the "o" in "borrow." i have heard people try to give it the sound of the "o" in "worry." such is human perversity. [ ] this office was usually undertaken by the boots, who found in it a refuge from the baker's constant complaints about the insufficient blacking of his three pair of boots. this also seems a fitting occasion to notice the other hard words in that poem. humpty-dumpty's theory, of two meanings packed into one word like a portmanteau, seems to me the right explanation for all. for instance, take the two words "fuming" and "furious." make up your mind that you will say both words, but leave it unsettled which you will say first. now open your mouth and speak. if your thoughts incline ever so little towards "fuming," you will say "fuming-furious"; if they turn, by even a hair's breadth towards "furious," you will say "furious-fuming"; but if you have that rarest of gifts, a perfectly balanced mind, you will say "frumious." supposing that, when pistol uttered the well-known words-- "under which king, bezonian? speak or die!" justice shallow had felt certain that it was either william or richard, but had not been able to settle which, so that he could not possibly say either name before the other, can it be doubted that, rather than die, he would have gasped out "rilchiam!" fit the first. _the landing._ "just the place for a snark!" the bellman cried, as he landed his crew with care; supporting each man on the top of the tide by a finger entwined in his hair. "just the place for a snark! i have said it twice: that alone should encourage the crew. just the place for a snark! i have said it thrice: what i tell you three times is true." the crew was complete: it included a boots-- a maker of bonnets and hoods-- a barrister, brought to arrange their disputes-- and a broker, to value their goods. [illustration: "supporting each man on the top of the tide"] a billiard-marker, whose skill was immense, might perhaps have won more than his share-- but a banker, engaged at enormous expense, had the whole of their cash in his care. there was also a beaver, that paced on the deck, or would sit making lace in the bow: and had often (the bellman said) saved them from wreck, though none of the sailors knew how. there was one who was famed for the number of things he forgot when he entered the ship: his umbrella, his watch, all his jewels and rings, and the clothes he had bought for the trip. he had forty-two boxes, all carefully packed, with his name painted clearly on each: but since he omitted to mention the fact, they were all left behind on the beach. the loss of his clothes hardly mattered, because he had seven coats on when he came, with three pair of boots--but the worst of it was he had wholly forgotten his name. [illustration: "he had wholly forgotten his name"] he would answer to "hi!" or to any loud cry, such as "fry me!" or "fritter my wig!" to "what-you-may-call-um!" or "what-was-his-name!" but especially "thing-um-a jig!" while, for those who preferred a more forcible word, he had different names from these: his intimate friends called him "candle-ends," and his enemies "toasted-cheese." "his form is ungainly--his intellect small--" (so the bellman would often remark)-- "but his courage is perfect! and that, after all, is the thing that one needs with a snark." he would joke with hyænas, returning their stare with an impudent wag of the head: and he once went a walk, paw-in-paw, with a bear, "just to keep up its spirits," he said. he came as a baker: but owned, when too late-- and it drove the poor bellman half-mad-- he could only bake bride-cake--for which, i may state, no materials were to be had. the last of the crew needs especial remark, though he looked an incredible dunce: he had just one idea--but, that one being "snark," the good bellman engaged him at once. he came as a butcher: but gravely declared, when the ship had been sailing a week, he could only kill beavers. the bellman looked scared, and was almost too frightened to speak: but at length he explained, in a tremulous tone, there was only one beaver on board; and that was a tame one he had of his own, whose death would be deeply deplored. the beaver, who happened to hear the remark, protested, with tears in its eyes, that not even the rapture of hunting the snark could atone for that dismal surprise! it strongly advised that the butcher should be conveyed in a separate ship: but the bellman declared that would never agree with the plans he had made for the trip: [illustration: "the beaver kept looking the opposite way"] navigation was always a difficult art, though with only one ship and one bell: and he feared he must really decline, for his part, undertaking another as well. the beaver's best course was, no doubt, to procure a second-hand dagger-proof coat-- so the baker advised it--and next, to insure its life in some office of note: this the banker suggested, and offered for hire (on moderate terms), or for sale, two excellent policies, one against fire, and one against damage from hail. yet still, ever after that sorrowful day, whenever the butcher was by, the beaver kept looking the opposite way, and appeared unaccountably shy. fit the second. _the bellman's speech._ the bellman himself they all praised to the skies-- such a carriage, such ease and such grace! such solemnity, too! one could see he was wise, the moment one looked in his face! he had bought a large map representing the sea, without the least vestige of land: and the crew were much pleased when they found it to be a map they could all understand. "what's the good of mercator's north poles and equators, tropics, zones, and meridian lines?" so the bellman would cry: and the crew would reply "they are merely conventional signs! [illustration: ocean-chart.] "other maps are such shapes, with their islands and capes! but we've got our brave captain to thank" (so the crew would protest) "that he's bought _us_ the best-- a perfect and absolute blank!" this was charming, no doubt: but they shortly found out that the captain they trusted so well had only one notion for crossing the ocean, and that was to tingle his bell. he was thoughtful and grave--but the orders he gave were enough to bewilder a crew. when he cried "steer to starboard, but keep her head larboard!" what on earth was the helmsman to do? then the bowsprit got mixed with the rudder sometimes: a thing, as the bellman remarked, that frequently happens in tropical climes, when a vessel is, so to speak, "snarked." but the principal failing occurred in the sailing, and the bellman, perplexed and distressed, said he _had_ hoped, at least, when the wind blew due east, that the ship would _not_ travel due west! but the danger was past--they had landed at last, with their boxes, portmanteaus, and bags: yet at first sight the crew were not pleased with the view which consisted of chasms and crags. the bellman perceived that their spirits were low, and repeated in musical tone some jokes he had kept for a season of woe-- but the crew would do nothing but groan. he served out some grog with a liberal hand, and bade them sit down on the beach: and they could not but own that their captain looked grand, as he stood and delivered his speech. "friends, romans, and countrymen, lend me your ears! (they were all of them fond of quotations: so they drank to his health, and they gave him three cheers while he served out additional rations). "we have sailed many months, we have sailed many weeks, (four weeks to the month you may mark), but never as yet ('tis your captain who speaks) have we caught the least glimpse of a snark! "we have sailed many weeks, we have sailed many days, (seven days to the week i allow), but a snark, on the which we might lovingly gaze, we have never beheld till now! "come, listen, my men, while i tell you again the five unmistakable marks by which you may know, wheresoever you go, the warranted genuine snarks. "let us take them in order. the first is the taste, which is meagre and hollow, but crisp: like a coat that is rather too tight in the waist, with a flavour of will-o-the wisp. "its habit of getting up late you'll agree that it carries too far, when i say that it frequently breakfasts at five o'clock tea, and dines on the following day. "the third is its slowness in taking a jest. should you happen to venture on one, it will sigh like a thing that is deeply distressed: and it always looks grave at a pun. "the fourth is its fondness for bathing-machines, which it constantly carries about, and believes that they add to the beauty of scenes-- a sentiment open to doubt. "the fifth is ambition. it next will be right to describe each particular batch: distinguishing those that have feathers, and bite, from those that have whiskers, and scratch. "for, although common snarks do no manner of harm, yet i feel it my duty to say some are boojums--" the bellman broke off in alarm, for the baker had fainted away. fit the third. _the baker's tale._ they roused him with muffins--they roused him with ice-- they roused him with mustard and cress-- they roused him with jam and judicious advice-- they set him conundrums to guess. when at length he sat up and was able to speak, his sad story he offered to tell; and the bellman cried "silence! not even a shriek!" and excitedly tingled his bell. there was silence supreme! not a shriek, not a scream; scarcely even a howl or a groan, as the man they called "ho!" told his story of woe in an antediluvian tone. "my father and mother were honest, though poor--" "skip all that!" cried the bellman in haste. "if it once becomes dark, there's no chance of a snark-- we have hardly a minute to waste!" "i skip forty years," said the baker, in tears, "and proceed without further remark to the day when you took me aboard of your ship to help you in hunting the snark. "a dear uncle of mine (after whom i was named) remarked, when i bade him farewell--" "oh, skip your dear uncle!" the bellman exclaimed, as he angrily tingled his bell. "he remarked to me then," said that mildest of men, "'if your snark be a snark, that is right: fetch it home by all means--you may serve it with greens and it's handy for striking a light. "'you may seek it with thimbles--and seek it with care; you may hunt it with forks and hope; you may threaten its life with a railway-share; you may charm it with smiles and soap--'" ("that's exactly the method," the bellman bold in a hasty parenthesis cried, "that's exactly the way i have always been told that the capture of snarks should be tried!") "'but oh, beamish nephew, beware of the day, if your snark be a boojum! for then you will softly and suddenly vanish away, and never be met with again!' "it is this, it is this that oppresses my soul, when i think of my uncle's last words: and my heart is like nothing so much as a bowl brimming over with quivering curds! "it is this, it is this--" "we have had that before!" the bellman indignantly said. and the baker replied "let me say it once more. it is this, it is this that i dread! "i engage with the snark--every night after dark-- in a dreamy delirious fight: i serve it with greens in those shadowy scenes, and i use it for striking a light: [illustration: "but oh, beamish nephew, beware of the day"] "but if ever i meet with a boojum, that day, in a moment (of this i am sure), i shall softly and suddenly vanish away-- and the notion i cannot endure!" fit the fourth. _the hunting._ the bellman looked uffish, and wrinkled his brow. "if only you'd spoken before! it's excessively awkward to mention it now, with the snark, so to speak, at the door! "we should all of us grieve, as you well may believe, if you never were met with again-- but surely, my man, when the voyage began, you might have suggested it then? "it's excessively awkward to mention it now-- as i think i've already remarked." and the man they called "hi!" replied, with a sigh, "i informed you the day we embarked. "you may charge me with murder--or want of sense-- (we are all of us weak at times): but the slightest approach to a false pretence was never among my crimes! "i said it in hebrew--i said it in dutch-- i said it in german and greek: but i wholly forgot (and it vexes me much) that english is what you speak!" "'tis a pitiful tale," said the bellman, whose face had grown longer at every word: "but, now that you've stated the whole of your case, more debate would be simply absurd. "the rest of my speech" (he explained to his men) "you shall hear when i've leisure to speak it. but the snark is at hand, let me tell you again! 'tis your glorious duty to seek it! "to seek it with thimbles, to seek it with care; to pursue it with forks and hope; to threaten its life with a railway-share; to charm it with smiles and soap! [illustration: "to pursue it with forks and hope."] "for the snark's a peculiar creature, that won't be caught in a commonplace way. do all that you know, and try all that you don't: not a chance must be wasted to-day! "for england expects--i forbear to proceed: 'tis a maxim tremendous, but trite: and you'd best be unpacking the things that you need to rig yourselves out for the fight." then the banker endorsed a blank cheque (which he crossed), and changed his loose silver for notes: the baker with care combed his whiskers and hair, and shook the dust out of his coats: the boots and the broker were sharpening a spade-- each working the grindstone in turn: but the beaver went on making lace, and displayed no interest in the concern: though the barrister tried to appeal to its pride, and vainly proceeded to cite a number of cases, in which making laces had been proved an infringement of right. the maker of bonnets ferociously planned a novel arrangement of bows: while the billiard-marker with quivering hand was chalking the tip of his nose. but the butcher turned nervous, and dressed himself fine, with yellow kid gloves and a ruff-- said he felt it exactly like going to dine, which the bellman declared was all "stuff." "introduce me, now there's a good fellow," he said, "if we happen to meet it together!" and the bellman, sagaciously nodding his head, said "that must depend on the weather." the beaver went simply galumphing about, at seeing the butcher so shy: and even the baker, though stupid and stout, made an effort to wink with one eye. "be a man!" cried the bellman in wrath, as he heard the butcher beginning to sob. "should we meet with a jubjub, that desperate bird, we shall need all our strength for the job!" fit the fifth. _the beaver's lesson._ they sought it with thimbles, they sought it with care they pursued it with forks and hope; they threatened its life with a railway-share; they charmed it with smiles and soap. then the butcher contrived an ingenious plan for making a separate sally; and had fixed on a spot unfrequented by man, a dismal and desolate valley. but the very same plan to the beaver occurred: it had chosen the very same place: yet neither betrayed, by a sign or a word, the disgust that appeared in his face. each thought he was thinking of nothing but "snark" and the glorious work of the day; and each tried to pretend that he did not remark that the other was going that way. but the valley grew narrow and narrower still, and the evening got darker and colder, till (merely from nervousness, not from good will) they marched along shoulder to shoulder. then a scream, shrill and high, rent the shuddering sky, and they knew that some danger was near: the beaver turned pale to the tip of its tail, and even the butcher felt queer. he thought of his childhood, left far far behind-- that blissful and innocent state-- the sound so exactly recalled to his mind a pencil that squeaks on a slate! "'tis the voice of the jubjub!" he suddenly cried. (this man, that they used to call "dunce.") "as the bellman would tell you," he added with pride, "i have uttered that sentiment once." "'tis the note of the jubjub! keep count, i entreat; you will find i have told it you twice. 'tis the song of the jubjub! the proof is complete, if only i've stated it thrice." the beaver had counted with scrupulous care, attending to every word: but it fairly lost heart, and outgrabe in despair, when the third repetition occurred. it felt that, in spite of all possible pains, it had somehow contrived to lose count, and the only thing now was to rack its poor brains by reckoning up the amount. "two added to one--if that could but be done," it said, "with one's fingers and thumbs!" recollecting with tears how, in earlier years, it had taken no pains with its sums. "the thing can be done," said the butcher, "i think. the thing must be done, i am sure. the thing shall be done! bring me paper and ink, the best there is time to procure." the beaver brought paper, portfolio, pens, and ink in unfailing supplies: while strange creepy creatures came out of their dens, and watched them with wondering eyes. so engrossed was the butcher, he heeded them not, as he wrote with a pen in each hand, and explained all the while in a popular style which the beaver could well understand. "taking three as the subject to reason about-- a convenient number to state-- we add seven, and ten, and then multiply out by one thousand diminished by eight. "the result we proceed to divide, as you see, by nine-hundred-and-ninety-and-two: then subtract seventeen, and the answer must be exactly and perfectly true. "the method employed i would gladly explain, while i have it so clear in my head, if i had but the time and you had but the brain-- but much yet remains to be said. [illustration: "the beaver brought paper, portfolio, pens"] "in one moment i've seen what has hitherto been enveloped in absolute mystery, and without extra charge i will give you at large a lesson in natural history." in his genial way he proceeded to say (forgetting all laws of propriety, and that giving instruction, without introduction, would have caused quite a thrill in society), "as to temper the jubjub's a desperate bird, since it lives in perpetual passion: its taste in costume is entirely absurd-- it is ages ahead of the fashion: "but it knows any friend it has met once before: it never will look at a bribe: and in charity-meetings it stands at the door, and collects--though it does not subscribe. "its flavour when cooked is more exquisite far than mutton, or oysters, or eggs: (some think it keeps best in an ivory jar, and some, in mahogany kegs:) "you boil it in sawdust: you salt it in glue: you condense it with locusts and tape: still keeping one principal object in view-- to preserve its symmetrical shape." the butcher would gladly have talked till next day, but he felt that the lesson must end, and he wept with delight in attempting to say he considered the beaver his friend: while the beaver confessed, with affectionate looks more eloquent even than tears, it had learned in ten minutes far more than all books would have taught it in seventy years. they returned hand-in-hand, and the bellman, unmanned (for a moment) with noble emotion, said "this amply repays all the wearisome days we have spent on the billowy ocean!" such friends, as the beaver and butcher became, have seldom if ever been known; in winter or summer, 'twas always the same-- you could never meet either alone. and when quarrels arose--as one frequently finds quarrels will, spite of every endeavour-- the song of the jubjub recurred to their minds, and cemented their friendship for ever! fit the sixth. _the barrister's dream._ they sought it with thimbles, they sought it with care; they pursued it with forks and hope; they threatened its life with a railway-share; they charmed it with smiles and soap. but the barrister, weary of proving in vain that the beaver's lace-making was wrong, fell asleep, and in dreams saw the creature quite plain that his fancy had dwelt on so long. he dreamed that he stood in a shadowy court, where the snark, with a glass in its eye, dressed in gown, bands, and wig, was defending a pig on the charge of deserting its sty. [illustration: "'you must know--' said the judge: but the snark exclaimed 'fudge!'"] the witnesses proved, without error or flaw, that the sty was deserted when found: and the judge kept explaining the state of the law in a soft under-current of sound. the indictment had never been clearly expressed, and it seemed that the snark had begun, and had spoken three hours, before any one guessed what the pig was supposed to have done. the jury had each formed a different view (long before the indictment was read), and they all spoke at once, so that none of them knew one word that the others had said. "you must know--" said the judge: but the snark exclaimed "fudge! that statute is obsolete quite! let me tell you, my friends, the whole question depends on an ancient manorial right. "in the matter of treason the pig would appear to have aided, but scarcely abetted: while the charge of insolvency fails, it is clear, if you grant the plea 'never indebted.' "the fact of desertion i will not dispute: but its guilt, as i trust, is removed (so far as relates to the costs of this suit) by the alibi which has been proved. "my poor client's fate now depends on your votes." here the speaker sat down in his place, and directed the judge to refer to his notes and briefly to sum up the case. but the judge said he never had summed up before; so the snark undertook it instead, and summed it so well that it came to far more than the witnesses ever had said! when the verdict was called for, the jury declined, as the word was so puzzling to spell; but they ventured to hope that the snark wouldn't mind undertaking that duty as well. so the snark found the verdict, although, as it owned, it was spent with the toils of the day: when it said the word "guilty!" the jury all groaned and some of them fainted away. then the snark pronounced sentence, the judge being quite too nervous to utter a word: when it rose to its feet, there was silence like night, and the fall of a pin might be heard. "transportation for life" was the sentence it gave, "and _then_ to be fined forty pound." the jury all cheered, though the judge said he feared that the phrase was not legally sound. but their wild exultation was suddenly checked when the jailer informed them, with tears, such a sentence would have not the slightest effect, as the pig had been dead for some years. the judge left the court, looking deeply disgusted: but the snark, though a little aghast, as the lawyer to whom the defence was intrusted, went bellowing on to the last. thus the barrister dreamed, while the bellowing seemed to grow every moment more clear: till he woke to the knell of a furious bell, which the bellman rang close at his ear. fit the seventh. _the banker's fate._ they sought it with thimbles, they sought it with care; they pursued it with forks and hope; they threatened its life with a railway-share; they charmed it with smiles and soap. and the banker, inspired with a courage so new it was matter for general remark, rushed madly ahead and was lost to their view in his zeal to discover the snark. but while he was seeking with thimbles and care, a bandersnatch swiftly drew nigh and grabbed at the banker, who shrieked in despair, for he knew it was useless to fly. he offered large discount--he offered a cheque (drawn "to bearer") for seven-pounds-ten: but the bandersnatch merely extended its neck and grabbed at the banker again. without rest or pause--while those frumious jaws went savagely snapping around-- he skipped and he hopped, and he floundered and flopped, till fainting he fell to the ground. the bandersnatch fled as the others appeared led on by that fear-stricken yell: and the bellman remarked "it is just as i feared!" and solemnly tolled on his bell. he was black in the face, and they scarcely could trace the least likeness to what he had been: while so great was his fright that his waistcoat turned white-- a wonderful thing to be seen! [illustration: "so great was his fright that his waistcoat turned white."] to the horror of all who were present that day, he uprose in full evening dress, and with senseless grimaces endeavoured to say what his tongue could no longer express. down he sank in a chair--ran his hands through his hair-- and chanted in mimsiest tones words whose utter inanity proved his insanity, while he rattled a couple of bones. "leave him here to his fate--it is getting so late!" the bellman exclaimed in a fright. "we have lost half the day. any further delay, and we sha'n't catch a snark before night!" fit the eighth. _the vanishing._ they sought it with thimbles, they sought it with care; they pursued it with forks and hope; they threatened its life with a railway-share; they charmed it with smiles and soap. they shuddered to think that the chase might fail, and the beaver, excited at last, went bounding along on the tip of its tail, for the daylight was nearly past. "there is thingumbob shouting!" the bellman said. "he is shouting like mad, only hark! he is waving his hands, he is wagging his head, he has certainly found a snark!" they gazed in delight, while the butcher exclaimed "he was always a desperate wag!" they beheld him--their baker--their hero unnamed-- on the top of a neighbouring crag, erect and sublime, for one moment of time. in the next, that wild figure they saw (as if stung by a spasm) plunge into a chasm, while they waited and listened in awe. "it's a snark!" was the sound that first came to their ears, and seemed almost too good to be true. then followed a torrent of laughter and cheers: then the ominous words "it's a boo--" then, silence. some fancied they heard in the air a weary and wandering sigh that sounded like "--jum!" but the others declare it was only a breeze that went by. [illustration: "then, silence"] they hunted till darkness came on, but they found not a button, or feather, or mark, by which they could tell that they stood on the ground where the baker had met with the snark. in the midst of the word he was trying to say, in the midst of his laughter and glee, he had softly and suddenly vanished away-- for the snark _was_ a boojum, you see. size and tears. [illustration] when on the sandy shore i sit, beside the salt sea-wave, and fall into a weeping fit because i dare not shave-- a little whisper at my ear enquires the reason of my fear. i answer "if that ruffian jones should recognise me here, he'd bellow out my name in tones offensive to the ear: he chaffs me so on being stout (a thing that always puts me out)." ah me! i see him on the cliff! farewell, farewell to hope, if he should look this way, and if he's got his telescope! to whatsoever place i flee, my odious rival follows me! for every night, and everywhere, i meet him out at dinner; and when i've found some charming fair, and vowed to die or win her, the wretch (he's thin and i am stout) is sure to come and cut me out! [illustration: "he's thin and i am stout"] the girls (just like them!) all agree to praise j. jones, esquire: i ask them what on earth they see about him to admire? they cry "he is so sleek and slim, it's quite a treat to look at him!" they vanish in tobacco smoke, those visionary maids-- i feel a sharp and sudden poke between the shoulder-blades-- "why, brown, my boy! you're growing stout!" (i told you he would find me out!) "my growth is not _your_ business, sir!" "no more it is, my boy! but if it's _yours_, as i infer, why, brown, i give you joy! a man, whose business prospers so, is just the sort of man to know! "it's hardly safe, though, talking here-- i'd best get out of reach: for such a weight as yours, i fear, must shortly sink the beach!"-- insult me thus because i'm stout! i vow i'll go and call him out! [illustration] atalanta in camden-town. ay, 'twas here, on this spot, in that summer of yore, atalanta did not vote my presence a bore, nor reply to my tenderest talk "she had heard all that nonsense before." she'd the brooch i had bought and the necklace and sash on, and her heart, as i thought, was alive to my passion; and she'd done up her hair in the style that the empress had brought into fashion. [illustration] i had been to the play with my pearl of a peri-- but, for all i could say, she declared she was weary, that "the place was so crowded and hot, and she couldn't abide that dundreary." then i thought "'tis for me that she whines and she whimpers!" and it soothed me to see those sensational simpers, and i said "this is scrumptious!"--a phrase i had learned from the devonshire shrimpers. and i vowed "'twill be said i'm a fortunate fellow, when the breakfast is spread, when the topers are mellow, when the foam of the bride-cake is white, and the fierce orange-blossoms are yellow!" o that languishing yawn! o those eloquent eyes! i was drunk with the dawn of a splendid surmise-- i was stung by a look, i was slain by a tear, by a tempest of sighs. and i whispered "'tis time! is not love at its deepest? shall we squander life's prime, while thou waitest and weepest? let us settle it, license or banns?--though undoubtedly banns are the cheapest." "ah, my hero," said i, "let me be thy leander!" but i lost her reply-- something ending with "gander"-- for the omnibus rattled so loud that no mortal could quite understand her. the lang coortin'. the ladye she stood at her lattice high, wi' her doggie at her feet; thorough the lattice she can spy the passers in the street. "there's one that standeth at the door, and tirleth at the pin: now speak and say, my popinjay, if i sall let him in." then up and spake the popinjay that flew abune her head: "gae let him in that tirls the pin: he cometh thee to wed." o when he cam' the parlour in, a woeful man was he! "and dinna ye ken your lover agen, sae well that loveth thee?" [illustration] "and how wad i ken ye loved me, sir, that have been sae lang away? and how wad i ken ye loved me, sir? ye never telled me sae." said--"ladye dear," and the salt, salt tear cam' rinnin' doon his cheek, "i have sent thee tokens of my love this many and many a week. "o didna ye get the rings, ladye, the rings o' the gowd sae fine? i wot that i have sent to thee four score, four score and nine." "they cam' to me," said that fair ladye. "wow, they were flimsie things!" said--"that chain o' gowd, my doggie to howd, it is made o' thae self-same rings." "and didna ye get the locks, the locks, the locks o' my ain black hair, whilk i sent by post, whilk i sent by box, whilk i sent by the carrier?" "they cam' to me," said that fair ladye; "and i prithee send nae mair!" said--"that cushion sae red, for my doggie's head, it is stuffed wi' thae locks o' hair." "and didna ye get the letter, ladye, tied wi' a silken string, whilk i sent to thee frae the far countrie, a message of love to bring?" "it cam' to me frae the far countrie wi' its silken string and a'; but it wasna prepaid," said that high-born maid, "sae i gar'd them tak' it awa'." "o ever alack that ye sent it back, it was written sae clerkly and well! now the message it brought, and the boon that it sought, i must even say it mysel'." then up and spake the popinjay, sae wisely counselled he. "now say it in the proper way: gae doon upon thy knee!" the lover he turned baith red and pale, went doon upon his knee: "o ladye, hear the waesome tale that must be told to thee! "for five lang years, and five lang years, i coorted thee by looks; by nods and winks, by smiles and tears, as i had read in books. "for ten lang years, o weary hours! i coorted thee by signs; by sending game, by sending flowers, by sending valentines. "for five lang years, and five lang years, i have dwelt in the far countrie, till that thy mind should be inclined mair tenderly to me. "now thirty years are gane and past, i am come frae a foreign land: i am come to tell thee my love at last-- o ladye, gie me thy hand!" the ladye she turned not pale nor red, but she smiled a pitiful smile: "sic' a coortin' as yours, my man," she said "takes a lang and a weary while!" and out and laughed the popinjay, a laugh of bitter scorn: "a coortin' done in sic' a way, it ought not to be borne!" [illustration: "and out and laughed the popinjay"] wi' that the doggie barked aloud, and up and doon he ran, and tugged and strained his chain o' gowd, all for to bite the man. "o hush thee, gentle popinjay! o hush thee, doggie dear! there is a word i fain wad say, it needeth he should hear!" aye louder screamed that ladye fair to drown her doggie's bark: ever the lover shouted mair to make that ladye hark: shrill and more shrill the popinjay upraised his angry squall: i trow the doggie's voice that day was louder than them all! the serving-men and serving-maids sat by the kitchen fire: they heard sic' a din the parlour within as made them much admire. [illustration: "o hush thee, gentle popinjay!"] out spake the boy in buttons (i ween he wasna thin), "now wha will tae the parlour gae, and stay this deadlie din?" and they have taen a kerchief, casted their kevils in, for wha should tae the parlour gae, and stay that deadlie din. when on that boy the kevil fell to stay the fearsome noise, "gae in," they cried, "whate'er betide, thou prince of button-boys!" syne, he has taen a supple cane to swinge that dog sae fat: the doggie yowled, the doggie howled the louder aye for that. syne, he has taen a mutton-bane-- the doggie ceased his noise, and followed doon the kitchen stair that prince of button-boys! [illustration: "the doggie ceased his noise"] then sadly spake that ladye fair, wi' a frown upon her brow: "o dearer to me is my sma' doggie than a dozen sic' as thou! "nae use, nae use for sighs and tears: nae use at all to fret: sin' ye've bided sae well for thirty years, ye may bide a wee langer yet!" sadly, sadly he crossed the floor and tirlëd at the pin: sadly went he through the door where sadly he cam' in. "o gin i had a popinjay to fly abune my head, to tell me what i ought to say, i had by this been wed. "o gin i find anither ladye," he said wi' sighs and tears, "i wot my coortin' sall not be anither thirty years: "for gin i find a ladye gay, exactly to my taste, i'll pop the question, aye or nay, in twenty years at maist." [illustration] four riddles. [these consist of two double acrostics and two charades. no. i. was written at the request of some young friends, who had gone to a ball at an oxford commemoration--and also as a specimen of what might be done by making the double acrostic _a connected poem_ instead of what it has hitherto been, a string of disjointed stanzas, on every conceivable subject, and about as interesting to read straight through as a page of a cyclopædia. the first two stanzas describe the two main words, and each subsequent stanza one of the cross "lights." no. ii. was written after seeing miss ellen terry perform in the play of "hamlet." in this case the first stanza describes the two main words. no. iii. was written after seeing miss marion terry perform in mr. gilbert's play of "pygmalion and galatea." the three stanzas respectively describe "my first," "my second," and "my whole."] i. there was an ancient city, stricken down with a strange frenzy, and for many a day they paced from morn to eve the crowded town, and danced the night away. i asked the cause: the aged man grew sad: they pointed to a building gray and tall, and hoarsely answered "step inside, my lad, and then you'll see it all." * * * * * yet what are all such gaieties to me whose thoughts are full of indices and surds? x{ } + x + = / . but something whispered "it will soon be done: bands cannot always play, nor ladies smile: endure with patience the distasteful fun for just a little while!" a change came o'er my vision--it was night: we clove a pathway through a frantic throng: the steeds, wild-plunging, filled us with affright: the chariots whirled along. within a marble hall a river ran-- a living tide, half muslin and half cloth: and here one mourned a broken wreath or fan, yet swallowed down her wrath; and here one offered to a thirsty fair (his words half-drowned amid those thunders tuneful) some frozen viand (there were many there), a tooth-ache in each spoonful. there comes a happy pause, for human strength will not endure to dance without cessation; and every one must reach the point at length of absolute prostration. at such a moment ladies learn to give, to partners who would urge them over-much, a flat and yet decided negative-- photographers love such. there comes a welcome summons--hope revives, and fading eyes grow bright, and pulses quicken: incessant pop the corks, and busy knives dispense the tongue and chicken. flushed with new life, the crowd flows back again: and all is tangled talk and mazy motion-- much like a waving field of golden grain, or a tempestuous ocean. and thus they give the time, that nature meant for peaceful sleep and meditative snores, to ceaseless din and mindless merriment and waste of shoes and floors. and one (we name him not) that flies the flowers, that dreads the dances, and that shuns the salads, they doom to pass in solitude the hours, writing acrostic-ballads. how late it grows! the hour is surely past that should have warned us with its double-knock? the twilight wanes, and morning comes at last-- "oh, uncle, what's o'clock?" the uncle gravely nods, and wisely winks. it _may_ mean much, but how is one to know? he opes his mouth--yet out of it, methinks, no words of wisdom flow. ii. empress of art, for thee i twine this wreath with all too slender skill. forgive my muse each halting line, and for the deed accept the will! * * * * * o day of tears! whence comes this spectre grim, parting, like death's cold river, souls that love? is not he bound to thee, as thou to him, by vows, unwhispered here, yet heard above? and still it lives, that keen and heavenward flame, lives in his eye, and trembles in his tone: and these wild words of fury but proclaim a heart that beats for thee, for thee alone! but all is lost: that mighty mind o'erthrown, like sweet bells jangled, piteous sight to see! "doubt that the stars are fire," so runs his moan, "doubt truth herself, but not my love for thee!" a sadder vision yet: thine aged sire shaming his hoary locks with treacherous wile! and dost thou now doubt truth to be a liar? and wilt thou die, that hast forgot to smile? nay, get thee hence! leave all thy winsome ways and the faint fragrance of thy scattered flowers: in holy silence wait the appointed days, and weep away the leaden-footed hours. iii. the air is bright with hues of light and rich with laughter and with singing: young hearts beat high in ecstasy, and banners wave, and bells are ringing: but silence falls with fading day, and there's an end to mirth and play. ah, well-a-day! rest your old bones, ye wrinkled crones! the kettle sings, the firelight dances. deep be it quaffed, the magic draught that fills the soul with golden fancies! for youth and pleasance will not stay, and ye are withered, worn, and gray. ah, well-a-day! o fair cold face! o form of grace, for human passion madly yearning! o weary air of dumb despair, from marble won, to marble turning! "leave us not thus!" we fondly pray. "we cannot let thee pass away!" ah, well-a-day! iv. my first is singular at best: more plural is my second: my third is far the pluralest-- so plural-plural, i protest it scarcely can be reckoned! my first is followed by a bird: my second by believers in magic art: my simple third follows, too often, hopes absurd and plausible deceivers. my first to get at wisdom tries-- a failure melancholy! my second men revered as wise: my third from heights of wisdom flies to depths of frantic folly. my first is ageing day by day: my second's age is ended: my third enjoys an age, they say, that never seems to fade away, through centuries extended. my whole? i need a poet's pen to paint her myriad phases: the monarch, and the slave, of men-- a mountain-summit, and a den of dark and deadly mazes-- a flashing light--a fleeting shade-- beginning, end, and middle of all that human art hath made or wit devised! go, seek _her_ aid, if you would read my riddle! fame's penny-trumpet. [affectionately dedicated to all "original researchers" who pant for "endowment."] blow, blow your trumpets till they crack, ye little men of little souls! and bid them huddle at your back-- gold-sucking leeches, shoals on shoals! fill all the air with hungry wails-- "reward us, ere we think or write! without your gold mere knowledge fails to sate the swinish appetite!" and, where great plato paced serene, or newton paused with wistful eye, rush to the chace with hoofs unclean and babel-clamour of the sty! be yours the pay: be theirs the praise: we will not rob them of their due, nor vex the ghosts of other days by naming them along with you. they sought and found undying fame: they toiled not for reward nor thanks: their cheeks are hot with honest shame for you, the modern mountebanks! who preach of justice--plead with tears that love and mercy should abound-- while marking with complacent ears the moaning of some tortured hound: who prate of wisdom--nay, forbear, lest wisdom turn on you in wrath, trampling, with heel that will not spare, the vermin that beset her path! go, throng each other's drawing-rooms, ye idols of a petty clique: strut your brief hour in borrowed plumes, and make your penny-trumpets squeak: [illustration: "go, throng each other's drawing-rooms"] deck your dull talk with pilfered shreds of learning from a nobler time, and oil each other's little heads with mutual flattery's golden slime: and when the topmost height ye gain, and stand in glory's ether clear, and grasp the prize of all your pain-- so many hundred pounds a year-- then let fame's banner be unfurled! sing pæans for a victory won! ye tapers, that would light the world, and cast a shadow on the sun-- who still shall pour his rays sublime, one crystal flood, from east to west, when ye have burned your little time and feebly flickered into rest! the end. [turn over. works by lewis carroll. alice's adventures in wonderland. with forty-two illustrations by tenniel. crown vo, cloth, gilt edges, price _s._ seventy-first thousand. translations of the same--into french, by henri buÉ--into german, by antonie zimmermann--and into italian, by t. pietrocÒla rossetti--with tenniel's illustrations. crown vo, cloth, gilt edges, price _s._ each. through the looking-glass, and what alice found there. with fifty illustrations by tenniel. crown vo, cloth, gilt edges, price _s._ fifty-second thousand. rhyme? and reason? with sixty-five illustrations by arthur b. frost, and nine by henry holiday. (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 vo, cloth, coloured edges, price _s._ n.b. in selling the above-mentioned books to the trade, messrs. macmillan and co. will abate _d._ in the shilling (no odd copies), and allow per cent. discount for payment within six months, and per cent. for cash. in selling them to the public (for cash only) they will allow per cent. discount. mr. lewis carroll, having been requested to allow "an easter greeting" (a leaflet, addressed to children, and frequently given with his books) to be sold separately, has arranged with messrs. harrison, of , pall mall, who will supply a single copy for _d._, or for _d._, or for _s._ macmillan & co., london. london: r. clay, sons, and taylor, printers. transcriber's notes: passages in italics are indicated by _italics_. superscripted characters are indicated by {superscript}. generously made available by the internet archive.) feeding the mind uniform with the present volume. _ s. net each; leather, s. net each._ prayers written at vailima. by r. l. stevenson. a christmas sermon. by r. l. stevenson. london: chatto & windus. feeding the mind by lewis carroll with a prefatory note by william h. draper london chatto & windus [_all rights reserved_] note _the history of this little sparkle from the pen of lewis carroll may soon be told. it was in october of the year that he came on a visit to a certain vicarage in derbyshire, where he had promised, on the score of friendship, to do what was for him a most unusual favour--to give a lecture before a public audience._ _the writer well remembers his nervous, highly-strung manner as he stood before the little room full of simple people, few of whom had any idea of the world-wide reputation of that shy, slight figure before them._ _when the lecture was over, he handed the manuscript to me, saying: 'do what you like with it.'_ _the one for whose sake he did this kindness was not long after called_ 'into the silent land.' _so the beautifully-written ms., in his customary violet ink, has been treasured for more than twenty years, only now and then being read over at christmastime to a friend or two by the study fire, always to meet with the same welcome and glad acknowledgment that here was a genuine, though little flame that could not have belonged to any other source but that which all the world knew in_ alice in wonderland _and_ through the looking-glass. _there may be, perhaps, many others who, gathering round a winter fire, will be glad to read words, however few, from that bright source, and whose memories will respond to the fresh touch of that cherished name._ _it remains to add but one or two more associations that cling to it and make the remembrance more vivid still. while lewis carroll was staying in the house, there came to call a certain genial and by no means shy dean, who, without realizing what he was doing, proceeded, in the presence of other callers, to make some remark identifying mr. dodgson as the author of his books._ _there followed an immense explosion immediately on the visitor's departure, with a pathetic and serious request that, if there were any risk of a repetition of the call, due warning might be given, and the retreat secured._ _probably not many readers of the immortal alice have ever seen the curious little whimsical paper called_ eight or nine wise words about letter-writing _which their author had printed and used to send to his acquaintance, accompanied by a small case for postage-stamps._ _it consists of forty pages, and is published by emberlin and son, oxford; and these are the contents:_ page on stamp-cases, how to begin a letter, how to go on with a letter, how to end a letter, on registering correspondence, _in this little script, also, there are the same sparkles of wit which betoken that nimble pen, as, for example, under_ 'how to begin a letter': '"and never, never, dear madam" (n.b.--this remark is addressed to ladies _only_. no _man_ would ever do such a thing), "put 'wednesday' simply as the date! "_that way madness lies!_"' _from section _: 'how to go on with a letter.'--'a great deal of the bad writing in the world comes simply from writing too _quickly_. of course you reply, "i do it to save _time_." a very good object, no doubt, but what right have you to do it at your friend's expense? isn't _his_ time as valuable as yours? years ago i used to receive letters from a friend--and very interesting letters too--written in one of the most atrocious hands ever invented. it generally took me about a _week_ to read one of his letters! i used to carry it about in my pocket and take it out at leisure times, to puzzle over the riddles which composed it--holding it in different positions and at different distances, till at last the meaning of some hopeless scrawl would flash upon me, when i at once wrote down the english under it. and when several had been thus guessed the context would help one with the others, till at last the whole series of hieroglyphics was deciphered. if _all_ one's friends wrote like that, life would be entirely spent in reading their letters!' _rule for correspondence that has, unfortunately, become_ controversial. '_don't repeat yourself._--when once you have had your say fully and clearly on a certain point, and have failed to convince your friend, _drop that subject_. to repeat your arguments all over again, will simply lead to his doing the same, and so you will go on like a circulating decimal. _did you ever know a circulating decimal come to an end?_' * * * * * _rule ._--'if your friend makes a severe remark, either leave it unnoticed, or make your reply distinctly less severe; and if he makes a friendly remark, tending towards making up the little difference that has arisen between you, let your reply be distinctly _more_ friendly. * * * * * 'if, in picking a quarrel, each party declined to go more than _three-eighths_ of the way, and if in making friends, each was ready to go _five-eighths_ of the way--why, there would be more reconciliations than quarrels! which is like the irishman's remonstrance to his gad-about daughter: "shure, you're _always_ goin' out! you go out three times for _wanst_ that you come in!"' * * * * * _rule ._--'don't try to get the last word.... (n.b.--if you are a gentleman and your friend a lady, this rule is superfluous: _you won't get the last word!_)' * * * * * _let the last word to-day be part of another rule, which gives a glimpse into that gentle heart:_ 'when you have written a letter that you feel may possibly irritate your friend, however necessary you may have felt it to so express yourself, _put it aside till the next day_. then read it over again, and fancy it addressed to yourself. this will often lead to your writing it all over again, taking out a lot of the vinegar and pepper and putting in honey instead, and thus making a _much_ more palatable dish of it!' 'quis desiderio sit pudor aut modus tam cari capitis?' w. h. d. _november ._ feeding the mind breakfast, dinner, tea; in extreme cases, breakfast, luncheon, dinner, tea, supper, and a glass of something hot at bedtime. what care we take about feeding the lucky body! which of us does as much for his mind? and what causes the difference? is the body so much the more important of the two? by no means: but life depends on the body being fed, whereas we can continue to exist as animals (scarcely as men) though the mind be utterly starved and neglected. therefore nature provides that, in case of serious neglect of the body, such terrible consequences of discomfort and pain shall ensue, as will soon bring us back to a sense of our duty: and some of the functions necessary to life she does for us altogether, leaving us no choice in the matter. it would fare but ill with many of us if we were left to superintend our own digestion and circulation. 'bless me!' one would cry, 'i forgot to wind up my heart this morning! to think that it has been standing still for the last three hours!' 'i can't walk with you this afternoon,' a friend would say, 'as i have no less than eleven dinners to digest. i had to let them stand over from last week, being so busy, and my doctor says he will not answer for the consequences if i wait any longer!' well, it is, i say, for us that the consequences of neglecting the body can be clearly seen and felt; and it might be well for some if the mind were equally visible and tangible--if we could take it, say, to the doctor, and have its pulse felt. 'why, what have you been doing with this mind lately? how have you fed it? it looks pale, and the pulse is very slow.' 'well, doctor, it has not had much regular food lately. i gave it a lot of sugar-plums yesterday.' 'sugar-plums! what kind?' 'well, they were a parcel of conundrums, sir.' 'ah, i thought so. now just mind this: if you go on playing tricks like that, you'll spoil all its teeth, and get laid up with mental indigestion. you must have nothing but the plainest reading for the next few days. take care now! no novels on any account!' * * * * * considering the amount of painful experience many of us have had in feeding and dosing the body, it would, i think, be quite worth our while to try and translate some of the rules into corresponding ones for the mind. first, then, we should set ourselves to provide for our mind its _proper kind_ of food. we very soon learn what will, and what will not, agree with the body, and find little difficulty in refusing a piece of the tempting pudding or pie which is associated in our memory with that terrible attack of indigestion, and whose very name irresistibly recalls rhubarb and magnesia; but it takes a great many lessons to convince us how indigestible some of our favourite lines of reading are, and again and again we make a meal of the unwholesome novel, sure to be followed by its usual train of low spirits, unwillingness to work, weariness of existence--in fact, by mental nightmare. then we should be careful to provide this wholesome food in _proper amount_. mental gluttony, or over-reading, is a dangerous propensity, tending to weakness of digestive power, and in some cases to loss of appetite: we know that bread is a good and wholesome food, but who would like to try the experiment of eating two or three loaves at a sitting? i have heard a physician telling his patient--whose complaint was merely gluttony and want of exercise--that 'the earliest symptom of hyper-nutrition is a deposition of adipose tissue,' and no doubt the fine long words greatly consoled the poor man under his increasing load of fat. i wonder if there is such a thing in nature as a fat mind? i really think i have met with one or two: minds which could not keep up with the slowest trot in conversation; could not jump over a logical fence, to save their lives; always got stuck fast in a narrow argument; and, in short, were fit for nothing but to waddle helplessly through the world. * * * * * then, again, though the food be wholesome and in proper amount, we know that we must not consume _too many kinds at once_. take the thirsty a quart of beer, or a quart of cider, or even a quart of cold tea, and he will probably thank you (though not so heartily in the last case!). but what think you his feelings would be if you offered him a tray containing a little mug of beer, a little mug of cider, another of cold tea, one of hot tea, one of coffee, one of cocoa, and corresponding vessels of milk, water, brandy-and-water, and butter-milk? the sum total might be a quart, but would it be the same thing to the haymaker? * * * * * having settled the proper kind, amount, and variety of our mental food, it remains that we should be careful to allow _proper intervals_ between meal and meal, and not swallow the food hastily without mastication, so that it may be thoroughly digested; both which rules, for the body, are also applicable at once to the mind. first, as to the intervals: these are as really necessary as they are for the body, with this difference only, that while the body requires three or four hours' rest before it is ready for another meal, the mind will in many cases do with three or four minutes. i believe that the interval required is much shorter than is generally supposed, and from personal experience, i would recommend anyone, who has to devote several hours together to one subject of thought, to try the effect of such a break, say once an hour, leaving off for five minutes only each time, but taking care to throw the mind absolutely 'out of gear' for those five minutes, and to turn it entirely to other subjects. it is astonishing what an amount of impetus and elasticity the mind recovers during those short periods of rest. and then, as to the mastication of the food, the mental process answering to this is simply _thinking over_ what we read. this is a very much greater exertion of mind than the mere passive taking in the contents of our author. so much greater an exertion is it, that, as coleridge says, the mind often 'angrily refuses' to put itself to such trouble--so much greater, that we are far too apt to neglect it altogether, and go on pouring in fresh food on the top of the undigested masses already lying there, till the unfortunate mind is fairly swamped under the flood. but the greater the exertion the more valuable, we may be sure, is the effect. one hour of steady thinking over a subject (a solitary walk is as good an opportunity for the process as any other) is worth two or three of reading only. and just consider another effect of this thorough digestion of the books we read; i mean the arranging and 'ticketing,' so to speak, of the subjects in our minds, so that we can readily refer to them when we want them. sam slick tells us that he has learnt several languages in his life, but somehow 'couldn't keep the parcels sorted' in his mind. and many a mind that hurries through book after book, without waiting to digest or arrange anything, gets into that sort of condition, and the unfortunate owner finds himself far from fit really to support the character all his friends give him. 'a thoroughly well-read man. just you try him in any subject, now. you can't puzzle him.' you turn to the thoroughly well-read man. you ask him a question, say, in english history (he is understood to have just finished reading macaulay). he smiles good-naturedly, tries to look as if he knew all about it, and proceeds to dive into his mind for the answer. up comes a handful of very promising facts, but on examination they turn out to belong to the wrong century, and are pitched in again. a second haul brings up a fact much more like the real thing, but, unfortunately, along with it comes a tangle of other things--a fact in political economy, a rule in arithmetic, the ages of his brother's children, and a stanza of gray's 'elegy,' and among all these, the fact he wants has got hopelessly twisted up and entangled. meanwhile, every one is waiting for his reply, and, as the silence is getting more and more awkward, our well-read friend has to stammer out some half-answer at last, not nearly so clear or so satisfactory as an ordinary schoolboy would have given. and all this for want of making up his knowledge into proper bundles and ticketing them. do you know the unfortunate victim of ill-judged mental feeding when you see him? can you doubt him? look at him drearily wandering round a reading-room, tasting dish after dish--we beg his pardon, book after book--keeping to none. first a mouthful of novel; but no, faugh! he has had nothing but that to eat for the last week, and is quite tired of the taste. then a slice of science; but you know at once what the result of that will be--ah, of course, much too tough for _his_ teeth. and so on through the whole weary round, which he tried (and failed in) yesterday, and will probably try and fail in to-morrow. mr. oliver wendell holmes, in his very amusing book, 'the professor at the breakfast table,' gives the following rule for knowing whether a human being is young or old: 'the crucial experiment is this--offer a bulky bun to the suspected individual just ten minutes before dinner. if this is easily accepted and devoured, the fact of youth is established.' he tells us that a human being, 'if young, will eat anything at any hour of the day or night.' to ascertain the healthiness of the _mental_ appetite of a human animal, place in its hands a short, well-written, but not exciting treatise on some popular subject--a mental _bun_, in fact. if it is read with eager interest and perfect attention, _and if the reader can answer questions on the subject afterwards_, the mind is in first-rate working order. if it be politely laid down again, or perhaps lounged over for a few minutes, and then, 'i can't read this stupid book! would you hand me the second volume of "the mysterious murder"?' you may be equally sure that there is something wrong in the mental digestion. if this paper has given you any useful hints on the important subject of reading, and made you see that it is one's duty no less than one's interest to 'read, mark, learn, and inwardly digest' the good books that fall in your way, its purpose will be fulfilled. billing and sons, ltd., printers, guildford generously made available by the internet archive/american libraries.) eight or nine wise words about letter-writing by _lewis carroll_ emberlin and son , magdalen street oxford first published . contents. page. _on stamp-cases_ _how to begin a letter_ _how to go on with a letter_ _how to end a letter_ _on registering correspondence_ § . _on stamp-cases._ some american writer has said "the snakes in this district may be divided into one species--the venomous." the same principle applies here. postage-stamp-cases may be divided into one species, the "wonderland." imitations of it will soon appear, no doubt: but they cannot include the two pictorial surprises, which are copyright. you don't see why i call them 'surprises'? well, take the case in your left-hand, and regard it attentively. you see alice nursing the duchess's baby? (an entirely new combination, by the way: it doesn't occur in the book.) now, with your right thumb and forefinger, lay hold of the little book, and suddenly pull it out. _the baby has turned into a pig!_ if _that_ doesn't surprise you, why, i suppose you wouldn't be surprised if your own mother-in-law suddenly turned into a gyroscope! this case is _not_ intended to carry about in your pocket. far from it. people seldom want any other stamps, on an emergency, than penny-stamps for letters, sixpenny-stamps for telegrams, and a bit of stamp-edging for cut fingers (it makes capital sticking-plaster, and will stand three or four washings, cautiously conducted): and all these are easily carried in a purse or pocketbook. no, _this_ is meant to haunt your envelope-case, or wherever you keep your writing-materials. what made me invent it was the constantly wanting stamps of other values, for foreign letters, parcel post, &c., and finding it very bothersome to get at the kind i wanted in a hurry. since i have possessed a "wonderland stamp case", life has been bright and peaceful, and i have used no other. i believe the queen's laundress uses no other. each of the pockets will hold stamps, comfortably. i would recommend you to arrange the , before putting them in, something like a _bouquet_, making them lean to the right and to the left alternately: thus there will always be a free _corner_ to get hold of, so as to take them out, quickly and easily, one by one: otherwise you will find them apt to come out two or three at a time. according to _my_ experience, the _d._, _d._, and _s._ stamps are hardly ever wanted, though i have constantly to replenish all the other pockets. if your experience agrees with mine, you may find it convenient to keep only a couple (say) of each of these kinds, in the _s._ pocket, and to fill the other pockets with extra _d._ stamps. § . _how to begin a letter._ if the letter is to be in answer to another, begin by getting out that other letter and reading it through, in order to refresh your memory, as to what it is you have to answer, and as to your correspondent's _present address_ (otherwise you will be sending your letter to his regular address in _london_, though he has been careful in writing to give you his _torquay_ address in full). next, address and stamp the envelope. "what! before writing the _letter_?" most certainly. and i'll tell you what will happen if you don't. you will go on writing till the last moment, and just in the middle of the last sentence, you will become aware that 'time's up!' then comes the hurried wind-up--the wildly-scrawled signature--the hastily-fastened envelope, which comes open in the post--the address, a mere hieroglyphic--the horrible discovery that you've forgotten to replenish your stamp-case--the frantic appeal, to every one in the house, to lend you a stamp--the headlong rush to the post office, arriving, hot and gasping, just after the box has closed--and finally, a week afterwards, the return of the letter, from the dead-letter office, marked "address illegible"! next, put your own address, _in full_, at the top of the note-sheet. it is an aggravating thing----i speak from bitter experience----when a friend, staying at some new address, heads his letter "dover," simply, assuming that you can get the rest of the address from his previous letter, which perhaps you have destroyed. next, put the date _in full_. it is another aggravating thing, when you wish, years afterwards, to arrange a series of letters, to find them dated "feb. ", "aug. ", without any year to guide you as to which comes first. and never, never, dear madam (n.b. this remark is addressed to ladies _only_: no _man_ would ever do such a thing), put "wednesday", simply, as the date! "_that way madness lies._" § . _how to go on with a letter._ here is a golden rule to begin with. _write legibly._ the average temper of the human race would be perceptibly sweetened, if everybody obeyed this rule! a great deal of the bad writing in the world comes simply from writing _too quickly_. of course you reply, "i do it to save _time_". a very good object, no doubt: but what right have you to do it at your friend's expense? isn't _his_ time as valuable as yours? years ago, i used to receive letters from a friend----and very interesting letters too----written in one of the most atrocious hands ever invented. it generally took me about a _week_ to read one of his letters! i used to carry it about in my pocket, and take it out at leisure times, to puzzle over the riddles which composed it----holding it in different positions, and at different distances, till at last the meaning of some hopeless scrawl would flash upon me, when i at once wrote down the english under it; and, when several had been thus guessed, the context would help one with the others, till at last the whole series of hieroglyphics was deciphered. if _all_ one's friends wrote like that, life would be entirely spent in reading their letters! this rule applies, specially, to names of people or places----and _most_ specially to _foreign names_. i got a letter once, containing some russian names, written in the same hasty scramble in which people often write "yours sincerely". the _context_, of course, didn't help in the least: and one spelling was just as likely as another, so far as _i_ knew: it was necessary to write and tell my friend that i couldn't read any of them! my second rule is, don't fill _more_ than a page and a half with apologies for not having written sooner! the best subject, to _begin_ with, is your friend's last letter. write with the letter open before you. answer his questions, and make any remarks his letter suggests. _then_ go on to what you want to say yourself. this arrangement is more courteous, and pleasanter for the reader, than to fill the letter with your own invaluable remarks, and then hastily answer your friend's questions in a postscript. your friend is much more likely to enjoy your wit, _after_ his own anxiety for information has been satisfied. in referring to anything your friend has said in his letter, it is best to _quote the exact words_, and not to give a summary of them in _your_ words. _a's_ impression, of what _b_ has said, expressed in _a's_ words, will never convey to _b_ the meaning of his own words. this is specially necessary when some point has arisen as to which the two correspondents do not quite agree. there ought to be no opening for such writing as "you are quite mistaken in thinking i said so-and-so. it was not in the least my meaning, &c., &c.", which tends to make a correspondence last for a lifetime. a few more rules may fitly be given here, for correspondence that has unfortunately become _controversial_. one is, _don't repeat yourself_. when once you have said your say, fully and clearly, on a certain point, and have failed to convince your friend, _drop that subject_: to repeat your arguments, all over again, will simply lead to his doing the same; and so you will go on, like a circulating decimal. _did you ever know a circulating decimal come to an end?_ another rule is, when you have written a letter that you feel may possibly irritate your friend, however necessary you may have felt it to so express yourself, _put it aside till the next day_. then read it over again, and fancy it addressed to yourself. this will often lead to your writing it all over again, taking out a lot of the vinegar and pepper, and putting in honey instead, and thus making a _much_ more palatable dish of it! if, when you have done your best to write inoffensively, you still feel that it will probably lead to further controversy, _keep a copy of it_. there is very little use, months afterwards, in pleading "i am almost sure i never expressed myself as you say: to the best of my recollection i said so-and-so". _far_ better to be able to write "i did _not_ express myself so: these are the words i used." my fifth rule is, if your friend makes a severe remark, either leave it unnoticed, or make your reply distinctly _less_ severe: and if he makes a friendly remark, tending towards 'making up' the little difference that has arisen between you, let your reply be distinctly _more_ friendly. if, in picking a quarrel, each party declined to go more than _three-eighths_ of the way, and if, in making friends, each was ready to go _five-eighths_ of the way--why, there would be more reconciliations than quarrels! which is like the irishman's remonstrance to his gad-about daughter--"shure, you're _always_ goin' out! you go out _three_ times, for _wanst_ that you come in!" my sixth rule (and my last remark about controversial correspondence) is, _don't try to have the last word_! how many a controversy would be nipped in the bud, if each was anxious to let the _other_ have the last word! never mind how telling a rejoinder you leave unuttered: never mind your friend's supposing that you are silent from lack of anything to say: let the thing drop, as soon as it is possible without discourtesy: remember 'speech is silvern, but silence is golden'! (n.b.--if you are a gentleman, and your friend a lady, this rule is superfluous: _you won't get the last word_!) my seventh rule is, if it should ever occur to you to write, jestingly, in _dispraise_ of your friend, be sure you exaggerate enough to make the jesting _obvious_: a word spoken in _jest_, but taken as _earnest_, may lead to very serious consequences. i have known it to lead to the breaking-off of a friendship. suppose, for instance, you wish to remind your friend of a sovereign you have lent him, which he has forgotten to repay--you might quite _mean_ the words "i mention it, as you seem to have a conveniently bad memory for debts", in jest: yet there would be nothing to wonder at if he took offence at that way of putting it. but, suppose you wrote "long observation of your career, as a pickpocket and a burglar, has convinced me that my one lingering hope, for recovering that sovereign i lent you, is to say 'pay up, or i'll summons yer!'" he would indeed be a matter-of-fact friend if he took _that_ as seriously meant! my eighth rule. when you say, in your letter, "i enclose cheque for £ ", or "i enclose john's letter for you to see", leave off writing for a moment--go and get the document referred to--and _put it into the envelope_. otherwise, you are pretty certain to find it lying about, _after the post has gone_! my ninth rule. when you get to the end of a note-sheet, and find you have more to say, take another piece of paper--a whole sheet, or a scrap, as the case may demand: but, whatever you do, _don't cross_! remember the old proverb '_cross-writing makes cross reading_'. "the _old_ proverb?" you say, enquiringly. "_how_ old?" well, not so _very_ ancient, i must confess. in fact, i'm afraid i invented it while writing this paragraph! still, you know, 'old' is a _comparative_ term. i think you would be _quite_ justified in addressing a chicken, just out of the shell, as "old boy!", _when compared_ with another chicken, that was only half-out! § . _how to end a letter._ if doubtful whether to end with 'yours faithfully', or 'yours truly', or 'yours most truly', &c. (there are at least a dozen varieties, before you reach 'yours affectionately'), refer to your correspondent's last letter, and make your winding-up _at least as friendly as his_; in fact, even if a shade _more_ friendly, it will do no harm! a postscript is a very useful invention: but it is _not_ meant (as so many ladies suppose) to contain the real _gist_ of the letter: it serves rather to throw into the shade any little matter we do _not_ wish to make a fuss about. for example, your friend had promised to execute a commission for you in town, but forgot it, thereby putting you to great inconvenience: and he now writes to apologize for his negligence. it would be cruel, and needlessly crushing, to make it the main subject of your reply. how much more gracefully it comes in thus! "p.s. don't distress yourself any more about having omitted that little matter in town. i won't deny that it _did_ put my plans out a little, at the time: but it's all right now. i often forget things, myself: and 'those who live in glass-houses, mustn't throw stones', you know!" when you take your letters to the post, _carry them in your hand_. if you put them in your pocket you will take a long country-walk (i speak from experience), passing the post-office _twice_, going and returning, and, when you get home, will find them _still_ in your pocket. § . _on registering correspondence._ let me recommend you to keep a record of letters received and sent. i have kept one for many years, and have found it of the greatest possible service, in many ways: it secures my _answering_ letters, however long they have to wait; it enables me to refer, for my own guidance, to the details of previous correspondence, though the actual letters may have been destroyed long ago; and, most valuable feature of all, if any difficulty arises, years afterwards, in connection with a half-forgotten correspondence, it enables me to say, with confidence, "i did _not_ tell you that he was 'an _invaluable_ servant in _every_ way', and that you _couldn't_ 'trust him too much'. i have a _précis_ of my letter. what i said was 'he is a _valuable_ servant in _many_ ways, but _don't_ trust him too much'. so, if he's cheated you, you really must not hold _me_ responsible for it!" i will now give you a few simple rules for making, and keeping, a letter-register. get a blank book, containing (say) leaves, about inches wide and high. it should be _well_ fastened into its cover, as it will have to be opened and shut hundreds of times. have a line ruled, in red ink, down each margin of every page, an inch off the edge (the margin should be wide enough to contain a number of digits, easily: _i_ manage with a / inch margin: but, unless you write very small you will find an inch more comfortable). write a _précis_ of each letter, received or sent, in chronological order. let the entry of a 'received' letter reach from the left-hand edge to the right-hand marginal line; and the entry of a 'sent' letter from the left-hand marginal line to the right-hand edge. thus the two kinds will be quite distinct, and you can easily hunt through the 'received' letters by themselves, without being bothered with the 'sent' letters; and _vice versâ_. use the _right-hand_ pages only: and, when you come to the end of the book, turn it upside-down, and begin at the other end, still using right-hand pages. you will find this much more comfortable than using left-hand pages. you will find it convenient to write, at the top of every sheet of a 'received' letter, its register-number in full. i will now give a few (ideal) specimen pages of my letter-register, and make a few remarks on them: after which i think you will find it easy enough to manage one for yourself. | / . || -------+ || ( ) |ap. (tu.) _jones, mrs._ am || sendg, |as present from self and mr. || j., a |white elephant. || -------+----------------------------------|| ( ) |do. _wilkins & co._ bill, for|| grand |piano, £ _s._ _d._ [pd|| , -------+----------------------------------|| ( ) |do. _scareham, h._ [writes from|| 'grand | hotel, monte carlo'] asking || to borr|ow £ for a few weeks (!) ||[symbol] -------+----------------------------------+-------- [symbol]||( ) do. _scareham, h._ would| like to ||know _object_, for wh loan is | asked, ||and _security_ offered. | ||----------------------------------+-------- ||( ) ap. . _wilkins & co._ ||in pre- ||vious letter, now before me, || you ||undertook to supply one for ||£ : ||decling to pay more. || ||----------------------------------+-------- ||( ) do. _cheetham & sharp._ | have ||written --enclosing previo|us let- ||ter--is law on my side? | [ ------++----------------------------------++------- ( ) ||ap. . _manager, goods statn_,|| _g. n.||r._ white elephant arrived, ad- || dresse||d to you--send for it at once-- || 'very ||savage'. || -------+----------------------------------+-------- | | | | | / . | ------++ | ||( ) ap. . (f) _jones, mrs._ th||anks, ||but no room for it at present, am||send- ||ing it to zoological gardens. || ||----------------------------------++------- ||( ) do. _manager, goods sta||tn, g._ ||_n. r._ please deliver, to bearer||of this ||note, case containg white ele-||phant ||addressed to me. || ||----------------------------------+-------- ||( ) do. _director zool. garde |ns._ (en- ||closing above note to r. w. ma|nager) ||call for valuable animal, prese|nted to ||gardens. | -------+----------------------------------+-------- ( ) |ap. . _cheetham & sharp._ you|| misquo|te enclosed letter, limit named || is £ | . || -------+----------------------------------||------- ( ) |ap. . _director, zoo. gardens._|| case de|livered to us contained doz.|| port--|consumed at directors' ban-|| quet--|many thanks. || -------+----------------------------------+-------- ||( ) do. t _jones, mrs._ why | call a [symbol]||doz. of port a 'white elephant'? | -------+----------------------------------+-------- ( ) |do. t _jones, mrs._ 'it was a ||[symbol] joke'. | || -------+----------------------------------+-------- | | | | | / . | -------+ | ||( ) ap. . (th) _page & co._|orderg ||macaulay's essays and "jane |eyre" ||(cheap edtn). | -------+----------------------------------+-------- ( ) |do. _aunt jemima_--invitg for || or |days after the th. [ || -------+----------------------------------|| ( ) |do. _lon. and west. bk._ have || recevd |£ , pd to yr acct fm parkins || & co. |calcutta [en || -------+----------------------------------+-------- ||( ) do. _aunt jemima_--can|not ||possibly come this month, will|write ||when able. | [ ||----------------------------------+-------- ||( ) ap. . _cheetham and |co._ re- ||turn letter enclosed to you. | [× ||----------------------------------+-------- ||( ) do. _morton, philip._ co|uld you ||lend me browning's 'dramati|s per- ||sonæ' for a day or ? | -------+----------------------------------+-------- ( ) |ap. . _aunt jemima_, leav- || ing ho|use at end of month : address || ' , |royal avenue, bath.' [ || -------+----------------------------------|| ( ) |ap. . _cheetham and co._, || returng|letter as reqd, bill / / . [ || -------+----------------------------------+-------- | | | | | / . | -------+ | ( ) |ap. . (tu) _page & co._ bill ||} for boo|ks, as ordered, / [ ||} -------+----------------------------------||} ( ) |do. ¶ _do._ books ||} -------+----------------------------------+-------- ||( ) do. _cheetham and co._ c|an un- ||derstand the / --what is £ |for? -------+----------------------------------+-------- ( ) |ap. . ¶ _morton, p._ 'dra- || matis |personæ', as asked for. [retd || -------+----------------------------------+-------- ||( ) do. _wilkins and co._ w|ith ||bill, / / , and ch. for do.| [en ||----------------------------------+-------- ||( ) do. _page and co._ bill,| / , ||postal [symbol] for /- and| stps. -------+----------------------------------+-------- ( ) |ap. . _cheetham and co._ it || was a |'clerical error' (!) || -------+----------------------------------+-------- ||( ) ap. . _morton, p._ retu|rng ||browning with many thanks. | -------+----------------------------------+-------- ( ) |do. _wilkins and co._ receptd || bill. | || -------+----------------------------------+-------- | | | | i begin each page by putting, at the top left-hand corner, the next entry-number i am going to use, _in full_ (the last digits of each entry-number are enough afterwards); and i put the date of the year, at the top, in the centre. i begin each entry with the last digits of the entry-number, enclosed in an oval (this is difficult to reproduce in print, so i have put round-parentheses here). then, for the _first_ entry in each page, i put the day of the month and the day of the week: afterwards, 'do.' is enough for the month-day, till it changes: i do not repeat the week-day. next, if the entry is _not_ a letter, i put a symbol for 'parcel' (see nos. , ) or 'telegram' (see nos. , ) as the case may be. next, the name of the person, underlined (indicated here by italics). if an entry needs special further attention, i put [____ at the end: and, when it has been attended to, i fill in the appropriate symbol, e.g. in no. , it showed that the bill had to be _paid_; in no. , that an answer was really _needed_ (the '×' means 'attended to'); in no. , that i owed the old lady a visit; in no. , that the item had to be entered in my account book; in no. , that i must not forget to write; in no. , that the address had to be entered in my address-book; in no. , that the book had to be returned. i give each entry the space of lines, whether it fills them or not, in order to have room for references. and, at the foot of each page i leave or lines _blank_ (often useful afterwards for entering omitted letters) and miss one or numbers before i begin the next page. at any odd moments of leisure, i 'make up' the entry-book, in various ways, as follows:-- ( ) i draw a _second_ line, at the right-hand end of the 'received' entries, and at the left-hand end of the 'sent' entries. this i usually do pretty well 'up to date'. in my register the first line is _red_, the second _blue_: here i distinguish them by making the first thin, and the second _thick_. ( ) beginning with the last entry, and going backwards, i read over the names till i recognise one as having occurred already: i then link the two entries together, by giving the one, that comes first in chronological order, a 'foot-reference' (see nos. , ). i do not keep this 'up-to-date', but leave it till there are or pages to be done. i work back till i come among entries that are all supplied with 'foot-references', when i once more glance through the last few pages, to see if there are any entries not yet supplied with head-references: _their_ predecessors may need a special search. if an entry is connected, in subject, with another under a different name, i link them by cross-references, distinguished from the head- and foot-references by being written _further from the marginal line_ (see no. ). when consecutive entries have the same name, and are both of the same kind (i.e. both 'received' or both 'sent') i bracket them (see nos. , ); if of different kinds, i link them with the symbol used for nos. , . ( ) beginning at the earliest entry not yet done with, and going forwards, i cross out every entry that has got a head- and foot-reference, and is done with, by continuing the extra line _through_ it (see nos. , , ). thus, wherever a _break_ occurs in this extra line, it shows there is some matter still needing attention. i do not keep this anything like 'up to date', but leave it till there are or pages to look through at a time. when the first page in the volume is thus completely crossed out, i put a mark at the foot of the page to indicate this; and so with pages , , &c. hence, whenever i do this part of the 'making up', i need not begin at the beginning of the volume, but only at the _earliest page that has not got this mark_. all this looks very complicated, when stated at full length: but you will find it perfectly simple, when you have had a little practice, and will come to regard the 'making-up' as a pleasant occupation for a rainy day, or at any time that you feel disinclined for more severe mental work. in the game of whist, hoyle gives us one golden rule, "when in doubt, win the trick"--i find that rule admirable for real life: when in doubt what to do, i 'make-up' my letter-register! the end. works by lewis carroll. published by macmillan & co., ltd., london. alice's adventures in wonderland. with forty-two illustrations by tenniel. (first published in .) crown vo, cloth, gilt edges, price _s._ net. ninetieth thousand. the same; people's edition. (first published in .) crown vo, cloth, price _s._ _d._ net. one hundred and forty-third thousand. the same; illustrated pocket classics for the young. fcap. vo, cloth, with full gilt back and gilt top, _s._ net. limp leather, with full gilt back and gilt edges, _s._ net. the same. vo, sewed, _d._; cloth, _s._ the same; miniature edition. pott vo, _s._ net. the same; little folks' edition. square mo. with coloured illustrations. _s._ net. aventures d'alice au pays des merveilles. traduit de l'anglais par henry bue. ouvrage illustré de vignettes par john tenniel. (first published in .) crown vo, cloth, gilt edges, price _s._ net. second thousand. le avventure d'alice nel paese delle meraviglie. tradotte dall' inglese da t. pietrocola-rossetti. con vignette di giovanni tenniel. (first published in .) crown vo, cloth, gilt edges, price _s._ net. alice's adventures under ground. being a facsimile of the original ms. book, which was afterwards developed into "alice's adventures in wonderland." with thirty-seven illustrations by the author. (begun, july, ; finished, feb., ; first published, in facsimile, in .) crown vo, cloth, gilt edges, price _s._ net. fourth thousand. through the looking-glass; and what alice found there. with fifty illustrations by tenniel. (first published in .) crown vo, cloth, gilt edges, price _s._ net. sixty-third thousand. the same; people's edition. (first published in .) crown vo, cloth, price _s._ _d._ net. eighty-fourth thousand. the same; illustrated pocket classics for the young. fcap. vo, cloth, with full gilt back and gilt top, _s._ net. limp leather, with full gilt back and gilt edges, _s._ net. the same. vo, sewed, _d._; cloth _s._ the same; little folks' edition. square mo. with coloured illustrations. _s._ _d._ net. alice's adventures in wonderland; and through the looking-glass; people's editions. both books together in one volume. (first published in .) crown vo, cloth, price _s._ _d._ net. the hunting of the snark. an agony in eight fits. with nine illustrations, and two large gilt designs on cover, by henry holiday. (first published in .) crown vo, cloth, gilt edges, price _s._ _d._ net. twenty-third thousand. rhyme? and reason? with sixty-five illustrations by arthur b. frost, and nine by henry holiday. (first published in , being a reprint, with a few additions, of the comic portions of "phantasmagoria, and other poems," published in , and of "the hunting of the snark," published in .) crown vo, cloth, gilt edges, price _s._ net. eighth thousand. sylvie and bruno concluded. with forty-six illustrations by harry furniss. (first published in .) fifth thousand. crown vo, cloth, gilt edges, price _s._ _d._ net. people's edition, _s._ _d._ net. n.b.--this book contains pages. the story of sylvie and bruno, in one volume. with illustrations by harry furniss. crown vo, _s._ _d._ net. three sunsets, and other poems. with twelve illustrations by e. gertrude thomson. fcap. to, cloth, gilt edges, price _s._ net. n.b.--this is a reprint, with a few additions, of the serious portion of "phantasmagoria, and other poems," published in . works by lewis carroll. published by chatto & windus, st. martin's lane, london, w.c. price _s._ net, boards; _s._ net, bound in leather. feeding the mind. a lecture delivered in . with preface by william h. draper. always in stock at emberlin & son, oxford. postage one penny. advice to writers. buy "the wonderland case for postage-stamps," invented by lewis carroll, october , , size inches by , containing separate pockets for stamps of different values, coloured pictorial surprises taken from _alice in wonderland_, and or wise words about letter-writing. it is published by messrs. emberlin & son, magdalen street, oxford. price _s._ n.b.--if ordered by post, an additional payment will be required, to cover cost of postage, as follows:-- one, two, three, or four copies, _d._ five to fourteen do., _d._ each subsequent fourteen or fraction thereof, _d._ the wonderland [illustration] postage-stamp case published by emberlin and son, , magdalen street, oxford. [illustration] (post free, d.) price one shilling [illustration] [illustration] [illustration] invented by [illustration] lewis carroll mdccclxxxix transcriber's notes: passages in italics are indicated by _italics_. the original text includes an intention blank space that is represented in this text version as ____.