The Phantom Tollbooth

  “Well, why didn’t you say so?” said the Mathemagician, who was busily measuring the edge of a raindrop. “What’s the greatest number you can think of?”

  “Nine trillion, nine hundred ninety-nine billion, nine hundred ninety-nine million, nine hundred ninety-nine thousand, nine hundred ninety-nine,” recited Milo breathlessly.

  “Very good,” said the Mathemagician. “Now add one to it. Now add one again,” he repeated when Milo had added the previous one. “Now add one again. Now add one again. Now add one again. Now add one again. Now add one again. Now add one again. Now add——”

  “But when can I stop?” pleaded Milo.

  “Never,” said the Mathemagician with a little smile, “for the number you want is always at least one more than the number you’ve got, and it’s so large that if you started saying it yesterday you wouldn’t finish tomorrow.”

  “Where could you ever find a number so big?” scoffed the Humbug.

  “In the same place they have the smallest number there is,” he answered helpfully; “and you know what that is.”

  “Certainly,” said the bug, suddenly remembering something to do at the other end of the room.

  “One one-millionth?” asked Milo, trying to think of the smallest fraction possible.

  “Almost,” said the Mathemagician. “Now divide it in half. Now divide it in half again. Now divide it in half again. Now divide it in half again. Now divide it in half again. Now divide it in half again. Now divide——”

  “Oh dear,” shouted Milo, holding his hands to his ears, “doesn’t that ever stop either?”

  “How can it,” said the Mathemagician, “when you can always take half of whatever you have left until it’s so small that if you started to say it right now you’d finish even before you began?”

  “Where could you keep anything so tiny?” Milo asked, trying very hard to imagine such a thing.

  The Mathemagician stopped what he was doing and explained simply, “Why, in a box that’s so small you can’t see it—and that’s kept in a drawer that’s so small you can’t see it, in a dresser that’s so small you can’t see it, in a house that’s so small you can’t see it, on a street that’s so small you can’t see it, in a city that’s so small you can’t see it, which is part of a country that’s so small you can’t see it, in a world that’s so small you can’t see it.”

  Then he sat down, fanned himself with a handkerchief, and continued. “Then, of course, we keep the whole thing in another box that’s so small you can’t see it—and, if you follow me, I’ll show you where to find it.”

  They walked to one of the small windows and there, tied to the sill, was one end of a line that stretched along the ground and into the distance until completely out of sight.

  “Just follow that line forever,” said the Mathemagician, “and when you reach the end, turn left. There you’ll find the land of Infinity, where the tallest, the shortest, the biggest, the smallest, and the most and the least of everything are kept.”

  “I really don’t have that much time,” said Milo anxiously. “Isn’t there a quicker way?”

  “Well, you might try this flight of stairs,” he suggested, opening another door and pointing up. “It goes there, too.”

  Milo bounded across the room and started up the stairs two at a time. “Wait for me, please,” he shouted to Tock and the Humbug. “I’ll be gone just a few minutes.”

  16. A Very Dirty Bird

  Up he went—very quickly at first—then more slowly—then in a little while even more slowly than that—and finally, after many minutes of climbing up the endless stairway, one weary foot was barely able to follow the other. Milo suddenly realized that with all his effort he was no closer to the top than when he began, and not a great deal further from the bottom. But he struggled on for a while longer, until at last, completely exhausted, he collapsed onto one of the steps.

  “I should have known it,” he mumbled, resting his tired legs and filling his lungs with air. “This is just like the line that goes on forever, and I’ll never get there.”

  “You wouldn’t like it much anyway,” someone replied gently. “Infinity is a dreadfully poor place. They can never manage to make ends meet.”

  Milo looked up, with his head still resting heavily in his hand; he was becoming quite accustomed to being addressed at the oddest times, in the oddest places, by the oddest people—and this time he was not at all disappointed. Standing next to him on the step was exactly one half of a small child who had been divided neatly from top to bottom.

  “Pardon me for staring,” said Milo, after he had been staring for some time, “but I’ve never seen half a child before.”

  “It’s .58 to be precise,” replied the child from the left side of his mouth (which happened to be the only side of his mouth).

  “I beg your pardon?” said Milo.

  “It’s .58,” he repeated; “it’s a little bit more than a half.”

  “Have you always been that way?” asked Milo impatiently, for he felt that that was a needlessly fine distinction.

  “My goodness, no,” the child assured him. “A few years ago I was just .42 and, believe me, that was terribly inconvenient.”

  “What is the rest of your family like?” said Milo, this time a bit more sympathetically.

  “Oh, we’re just the average family,” he said thoughtfully; “mother, father, and 2.58 children—and, as I explained, I’m the .58.”

  “It must be rather odd being only part of a person,” Milo remarked.

  “Not at all,” said the child. “Every average family has 2.58 children, so I always have someone to play with. Besides, each family also has an average of 1.3 automobiles, and since I’m the only one who can drive three tenths of a car, I get to use it all the time.”

  “But averages aren’t real,” objected Milo; “they’re just imaginary.”

  “That may be so,” he agreed, “but they’re also very useful at times. For instance, if you didn’t have any money at all, but you happened to be with four other people who had ten dollars apiece, then you’d each have an average of eight dollars. Isn’t that right?”

  “I guess so,” said Milo weakly.

  “Well, think how much better off you’d be, just because of averages,” he explained convincingly. “And think of the poor farmer when it doesn’t rain all year: if there wasn’t an average yearly rainfall of 37 inches in this part of the country, all his crops would wither and die.”

  It all sounded terribly confusing to Milo, for he had always had trouble in school with just this subject.

  “There are still other advantages,” continued the child. “For instance, if one rat were cornered by nine cats, then, on the average, each cat would be 10 per cent rat and the rat would be 90 per cent cat. If you happened to be a rat, you can see how much nicer it would make things.”

  “But that can never be,” said Milo, jumping to his feet.

  “Don’t be too sure,” said the child patiently, “for one of the nicest things about mathematics, or anything else you might care to learn, is that many of the things which can never be, often are. You see,” he went on, “it’s very much like your trying to reach Infinity. You know that it’s there, but you just don’t know where—but just because you can never reach it doesn’t mean that it’s not worth looking for.”

  “I hadn’t thought of it that way,” said Milo, starting down the stairs. “I think I’ll go back now.”

  “A wise decision,” the child agreed; “but try again someday—perhaps you’ll get much closer.” And, as Milo waved good-by, he smiled warmly, which he usually did on the average of 47 times a day.

  “Everyone here knows so much more than I do,” thought Milo as he leaped from step to step. “I’ll have to do a lot better if I’m going to rescue the princesses.”

  In a few moments he’d reached the bottom again and burst into the workshop, where Tock and the Humbug were eagerly watching the Mathemagician perform.

  �
�Ah, back already,” he cried, greeting him with a friendly wave. “I hope you found what you were looking for.”

  “I’m afraid not,” admitted Milo. And then he added in a very discouraged tone, “Everything in Digitopolis is much too difficult for me.”

  The Mathemagician nodded knowingly and stroked his chin several times. “You’ll find,” he remarked gently, “that the only thing you can do easily is be wrong, and that’s hardly worth the effort.”

  Milo tried very hard to understand all the things he’d been told, and all the things he’d seen, and, as he spoke, one curious thing still bothered him. “Why is it,” he said quietly, “that quite often even the things which are correct just don’t seem to be right?”

  A look of deep melancholy crossed the Mathemagician’s face and his eyes grew moist with sadness. Everything was silent, and it was several minutes before he was able to reply at all.

  “How very true,” he sobbed, supporting himself on the staff. “It has been that way since Rhyme and Reason were banished.”

  “Quite so,” began the Humbug. “I personally feel that——”

  “AND ALL BECAUSE OF THAT STUBBORN WRETCH AZAZ,” roared the Mathemagician, completely overwhelming the bug, for now his sadness had changed to fury and he stalked about the room adding up anger and multiplying wrath. “IT’S ALL HIS FAULT.”

  “Perhaps if you discussed it with him——” Milo started to say, but never had time to finish.

  “He’s much too unreasonable,” interrupted the Mathemagician again. “Why, just last month I sent him a very friendly letter, which he never had the courtesy to answer. See for yourself.”

  He handed Milo a copy of the letter, which read:

  “But maybe he doesn’t understand numbers,” said Milo, who found it a little difficult to read himself.

  “NONSENSE!” bellowed the Mathemagician. “Everyone understands numbers. No matter what language you speak, they always mean the same thing. A seven is a seven anywhere in the world.”

  “My goodness,” thought Milo, “everybody is so terribly sensitive about the things they know best.”

  “With your permission,” said Tock, changing the subject, “we’d like to rescue Rhyme and Reason.”

  “Has Azaz agreed to it?” the Mathemagician inquired.

  “Yes, sir,” the dog assured him.

  “THEN I DON’T,” he thundered again, “for since they’ve been banished, we’ve never agreed on anything—and we never will.” He emphasized his last remark with a dark and ominous look.

  “Never?” asked Milo, with the slightest touch of disbelief in his voice.

  “NEVER!” he repeated. “And if you can prove otherwise, you have my permission to go.”

  “Well,” said Milo, who had thought about this problem very carefully ever since leaving Dictionopolis. “Then with whatever Azaz agrees, you disagree.”

  “Correct,” said the Mathemagician with a tolerant smile.

  “And with whatever Azaz disagrees, you agree.”

  “Also correct,” yawned the Mathemagician, nonchalantly cleaning his fingernails with the point of his staff.

  “Then each of you agrees that he will disagree with whatever each of you agrees with,” said Milo triumphantly; “and if you both disagree with the same thing, then aren’t you really in agreement?”

  “I’VE BEEN TRICKED!” cried the Mathemagician helplessly, for no matter how he figured, it still came out just that way.

  “Splendid effort,” commented the Humbug jovially; “exactly the way I would have done it myself.”

  “And now may we go?” added Tock.

  The Mathemagician accepted his defeat with grace, nodded weakly, and then drew the three travelers to his side.

  “It’s a long and dangerous journey,” he began softly, and a furrow of concern creased his forehead. “Long before you find them, the demons will know you’re there. Watch for them well,” he emphasized, “for when they appear, it might be too late.”

  The Humbug shuddered down to his shoes, and Milo felt the tips of his fingers suddenly grow cold.

  “But there is one problem even more serious than that,” he whispered ominously.

  “What is it?” gasped Milo, who was not sure he really wanted to know.

  “I’m afraid I can tell you only when you return. Come along,” said the Mathemagician, “and I’ll show you the way.” And, simply by carrying the three, he transported them all to the very edge of Digitopolis. Behind them lay all the kingdoms of Wisdom, and up ahead a narrow rutted path led toward the mountains and darkness.

  “We’ll never get the car up that,” said Milo unhappily.

  “True enough,” replied the Mathemagician, “but you can be in Ignorance quick enough without riding all the way; and if you’re to be successful, it will have to be step by step.”

  “But I would like to take my gifts,” Milo insisted.

  “So you shall,” announced the Dodecahedron, who appeared from nowhere with his arms full. “Here are your sights, here are your sounds, and here,” he said, handing Milo the last of them disdainfully, “are your words.”

  “And, most important of all,” added the Mathemagician, “here is your own magic staff. Use it well and there is nothing it cannot do for you.”

  He placed in Milo’s breast pocket a small gleaming pencil which, except for the size, was much like his own. Then, with a last word of encouragement, he and the Dodecahedron (who was simultaneously sobbing, frowning, pining, and sighing from four of his saddest faces) made their farewells and watched as the three tiny figures disappeared into the forbidding Mountains of Ignorance.

  Almost immediately the light began to fade as the difficult path wandered aimlessly upward, inching forward almost as reluctantly as the trembling Humbug. Tock as usual led the way, sniffing ahead for danger, and Milo, his bag of precious possessions slung over one shoulder, followed silently and resolutely behind.

  “Perhaps someone should stay back to guard the way,” said the unhappy bug, offering his services; but, since his suggestion was met with silence, he followed glumly along.

  The higher they went, the darker it became, though it wasn’t the darkness of night, but rather more like a mixture of lurking shadows and evil intentions which oozed from the slimy moss-covered cliffs and blotted out the light. A cruel wind shrieked through the rocks and the air was thick and heavy, as if it had been used several times before.

  On they went, higher and higher up the dizzying trail, on one side the sheer stone walls and brutal peaks towering above them, and on the other an endless, limitless, bottomless nothing.

  “I can hardly see a thing,” said Milo, taking hold of Tock’s tail as a sticky mist engulfed the moon. “Perhaps we should wait until morning.”

  “They’ll be mourning for you soon enough,” came a reply from directly above, and this was followed by a hideous cackling laugh very much like someone choking on a fishbone.

  Clinging to one of the greasy rocks and blending almost perfectly with it was a large, unkempt, and exceedingly soiled bird who looked more like a dirty floor mop than anything else. He had a sharp, dangerous beak, and the one eye he chose to open stared down maliciously.

  “I don’t think you understand,” said Milo timidly as the watchdog growled a warning. “We’re looking for a place to spend the night.”

  “It’s not yours to spend,” the bird shrieked again, and followed it with the same horrible laugh.

  “That doesn’t make any sense, you see——” he started to explain.

  “Dollars or cents, it’s still not yours to spend,” the bird replied haughtily.

  “But I didn’t mean——” insisted Milo.

  “Of course you’re mean,” interrupted the bird, closing the eye that had been open and opening the one that had been closed. “Anyone who’d spend a night that doesn’t belong to him is very mean.”

  “Well, I thought that by——” he tried again desperately.

  “That’s a diffe
rent story,” interjected the bird a bit more amiably. “If you want to buy, I’m sure I can arrange to sell, but with what you’re doing you’ll probably end up in a cell anyway.”

  “That doesn’t seem right,” said Milo helplessly, for, with the bird taking everything the wrong way, he hardly knew what he was saying.

  “Agreed,” said the bird, with a sharp click of his beak, “but neither is it left, although if I were you I would have left a long time ago.”

  “Let me try once more,” Milo said in an effort to explain. “In other words——”

  “You mean you have other words?” cried the bird happily. “Well, by all means, use them. You’re certainly not doing very well with the ones you have now.”

  “Must you always interrupt like that?” said Tock irritably, for even he was becoming impatient.

  “Naturally,” the bird cackled; “it’s my job. I take the words right out of your mouth. Haven’t we met before? I’m the Everpresent Wordsnatcher, and I’m sure I know your friend the bug.” And then he leaned all the way forward and gave a terrible knowing smile.

  The Humbug, who was too big to hide and too frightened to move, denied everything.

  “Is everyone who lives in Ignorance like you?” asked Milo.

  “Much worse,” he said longingly. “But I don’t live here. I’m from a place very far away called Context.”

  “Don’t you think you should be getting back?” suggested the bug, holding one arm up in front of him.

  “What a horrible thought.” The bird shuddered. “It’s such an unpleasant place that I spend almost all my time out of it. Besides, what could be nicer than these grimy mountains?”

  “Almost anything,” thought Milo as he pulled his collar up. And then he asked the bird, “Are you a demon?”

  “I’m afraid not,” he replied sadly, as several filthy tears rolled down his beak. “I’ve tried, but the best I can manage to be is a nuisance,” and, before Milo could reply, he flapped his dingy wings and flew off in a cascade of dust and dirt and fuzz.