Summary
Chapter 13: Unfortunate Conclusions
Milo runs back to the people, writes what he has done, and aims his lips to the cannon. The little “but” flies toward the wall and a moment later, the walls come tumbling down. All the sounds ever made hurtle out – speeches, babies crying, cannons, crying, singing, and many more. After all the confusion, things settle down and are normal once more.
The Soundkeeper is sitting on a pile of the walls’ rubble. She is very upset, and says it will take years to collect all the sounds again. She knows, though, that you can’t improve sound with just silence; you have to use each at the proper time. She wishes Rhyme and Reason were there, and is happy to hear that Milo and friends are going to rescue them. She gives Milo a bag of sounds he might be happy to hear in lonely places. Finally, she tells them to take the road to the sea and turn left and they will be in Digitopolis.
The three travelers begin driving along the sea and see a lovely island in the distance. Tock is happy that they have plenty of time, the Humbug thinks nothing can go wrong, and Milo is pleased it couldn’t be a nicer day.
Suddenly the three of them find themselves on the little island with a bunch of other people. They meet a man Milo calls “Canby,” for he boasts that he is as tall/short/stupid/graceful etc. as he can be. Canby tells them that this is the Island of Conclusions, and people get there by jumping. As he says this, more arrive. It is difficult for people to leave but easy for them to come. The only way is to swim, which Milo and friends grudgingly decide to do.
After a long time, they finally make it to shore, and Milo gasps that he will do his best in the future not to jump to conclusions. They get back in the car and head onward toward Digitopolis.
Chapter 14: The Dodecahedron Leads the Way
They approach a sign at a place where the road divides into three; the sign gives the miles, yards, feet, inches, etc. to the city. The travelers debate which is better. Suddenly a man appears – he is comprised of a lot of angles with many different sides. He wears a beret and introduces himself as the Dodecahedron, a shape with twelve faces. He uses different faces for different occasions and thinks it is odd Milo only has one.
He also comments that everything in Digitopolis is precise and things are named for what they are; thus, “Milo” is a strange name. Milo asks for help with getting to the city and the Dodecahedron presents him with complicated math problems. Tock figures out that the roads will all arrive at the same place at the same time, and gleefully the Dodecahedron commends him and says he will take them.
They approach the land of numbers, which is a lot bleaker than the land of words. Milo asks if numbers are made and the Dodecahedron replies that they are dug up. When he asks why Milo doesn’t know much about numbers, Milo replies that he does not think they are that important. The Dodecahedron becomes very angry. He praises numbers and their beauty and value. The Dodecahedron then leads them into a mountain. They can hear digging and scraping sounds; they are in a numbers mine.
A man strides toward them. He is wearing a long, flowing robe with mathematical equations on it and carries a staff in his left hand. The staff resembles a gigantic pencil. It is the Mathemagician, and Milo says hello and asks if there are precious stones in the mine. The Mathemagician shows them the beautiful numbers that rest in the carts after being dug up, and explains that they are sent all over the world. When Milo drops a number and it shatters, the Mathemagician tells him not to worry because they are used for fractions.
When the Humbug asks if they have any diamonds or rubies or emeralds, the Mathemagician leads him to a massive pile of precious stones. It is an unfathomable amount of wealth, and the Mathemagician simply says they cannot figure out what to do with these. The irony of the comment stuns the Humbug into silence.
Chapter 15: The Way to Infinity
Eight strong miners enter the cavern with a steaming cauldron, which the Mathemagician graciously offers to Milo, Tock, and the Humbug. They discover, however, that the more they eat the hungrier they become. He tells them it is subtraction stew and in this city they eat when they are full and eat until they are hungry. This way, he says, the more you want, the less you get and the less you get, the more you have.
The Mathemagician makes the surroundings vanish and suddenly they are all in his workshop. It is a circular room with tiny arched windows corresponding to the sixteen points of the compass, labels on every piece of furniture proclaiming its measurements, an artist’s easel, and a collection of scales, rulers, and weights.
After the Mathemagician shows them how he can multiply using himself, Milo asks if he can make things disappear. The Mathemagician assents. Milo asks for the biggest number there is and the Mathemagician takes out a huge 3; Milo asks for the longest number there is and he takes out an incredibly long 8. Milo is frustrated because this is not what he means, but Tock clarifies that they want the number of the greatest possible magnitude.
The Mathemagician asks Milo what the biggest number he can think if is, and then asks him to add 1 to it. This could go on forever; it is the same with dividing numbers. He tells them he keeps the smallest number there is in a box that is so small one cannot see it. He tells them all these tall, big, short, small, most, and least numbers are kept in the land of infinity. Milo asks if there is a quick way to get there and the Mathemagician points him up a staircase into the sky.
Chapter 16: A Very Dirty Bird
Milo climbs and climbs but never seems to make any progress. When he complains, a voice tells him Infinity is a place where they can never make ends meet. Milo notices a small child, but he looks like he is only half a child. Milo remarks that he has never seen half a child before, but the boy tells him he is .58 to be exact. He is from an average family, and since the average family has 2.58 children, he is the .58. When Milo comments that averages are made up, the boy rebuts him – what about average rainfall? Ultimately, the boy says “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, are” (197).
Milo thinks to himself he better learn more in order to rescue the princesses. He heads back down the staircase, giving up his goal of reaching Infinity, and is met by his friends and the Mathemagician. When he says ruefully that things are correct but never seem to be right, the Mathemagician agrees that it has been that way since Rhyme and Reason left. He sobs, and then angrily says it is all Azaz’s fault. He fervently claims numbers are better, but Tock interjects that they want to rescue the princesses.
The Mathemagician asks if Azaz agreed and when Tock says yes, he roars that he will not. Milo uses logic to get him to see that they are both in agreement, and the Mathemagician consents ruefully. He tells them it is a dangerous journey and he has to be aware of the demons; also, he has something else to tell them but can only do so when they return.
He transports them to the edge of the city, right in front of the treacherous ascent into the Mountains of Ignorance. They cannot take their car, so they will have to walk. The Dodecahedron appears and gives Milo all his accrued gifts, and the Mathemagician gives him a gift as well – a little staff that looks just like a pencil. He and the Dodecahedron vanish.
Before them is a narrow, difficult path. The sun goes down and the night seems full of shadows and evil intentions. A harsh wind blows as they go higher and higher into the brutal peaks.
When Milo wonders if they should wait until morning, a voice remarks that people will be mourning them soon enough. The travelers see a large, soiled, and haggard bird. Milo tells him they just want a place to spend the night, and the bird cackles that the night is not theirs to spend. The bird is confusing to Milo and he tries again. The bird interrupts, and introduces himself as the Everpresent Wordsnatcher.
Milo asks if everyone on the Mountains of Ignorance is like the bird, and he says they are worse. He is actually from a place called Context, though, and he spends most of his time out of it. Milo asks if he is a demon and the bird sadly says he is only a nuisance. The bird flies off.
The travelers then see a finely dressed man with a completely blank face leaning against a dead tree. He is very friendly and asks after all of them. They all think he is the nicest person they have met.
The man asks them for help with a few small jobs, and they happily agree. Milo is tasked with moving a pile of fine sand from one spot to another only with a pair of scissors. Tock has to empty a well and fill another with only a dropper. The Humbug has to dig a hole through the cliff with only a needle.
They all begin to work and hour after hour after hour passes.
Analysis
In these chapters Milo, Tock, and the Humbug meet several of the more interesting and sinister characters in the novel. In the former category, we have the Dodecahedron and Mathemagician, the .58 child, and Canby. In the latter, we have the bird and the Terrible Trivium (the latter to be discussed in the next analysis). Milo’s hero’s journey is still in the “journey” phase as he learns more about the quest and himself, encounters friends who have advice and gifts, and faces enemies who would waylay him if they were successful. Specifically, he learns not to jump to conclusions, not to interrupt ("snatch words"), and not to be waylaid by untrustworthy individuals or those who are too obsessively blindsided by their own pursuits.
The Mathemagician is the counterpart (which makes sense, as they are brothers) to Azaz. He is impressive and wise, and even bears a similar style of cloak, albeit one with numbers instead of letters. He is fiercely devoted to numbers and touts their importance above words. The Dodecahedron also undertakes this task, yelling at Milo, “Why numbers are the most beautiful and valuable things in the world!” (177). Their arguments, along with those of .58, are persuasive. They cause Milo to think about how numbers are used to know how long traveling somewhere takes, how things are organized and measured by numbers, and how even averages and fractions are valuable. The Mathemagician also helps Milo understand the consequences and characteristics of infinity, which Juster helps make concrete for the reader as well through Milo’s staircase foray.
Like Azaz, the Mathemagician has more than just a dollop of pride and stubbornness. He is also extremely precise, ironically demonstrating for Milo the importance of words when he “misunderstands” Milo’s query about the longest and biggest number. His commitment solely to numbers and his inability to accept the importance of words, again, mirrors Azaz’s intransigence and emphasizes the necessity of Milo’s quest.
The Mathemagician is the source for one of the most appealing metaphors in the book. When the Humbug dismisses his staff as merely a pencil, the Mathemagician replies, “True enough… but once you learn to use it, there’s no end to what you can do” (188). When he bids goodbye to Milo, he gives him his own pencil and makes a similar comment: “Use it well and there is nothing it cannot do for you” (201). Literally, the pencil can write out words and numbers. Metaphorically, though, the Mathemagician is stressing that writing is a gateway to ideas, creativity, power, fulfillment, and more. Milo’s pencil will take him into drawings, equations, and brainstorms; it is indeed a powerful tool.
One of the intellectual arguments of the mid-20th century can be applied to The Phantom Tollbooth, and is one Juster has mentioned in interviews as being something he was cognizant of, though he did not deliberately set out to flesh the argument out. In 1959, C.P. Snow, a British scientist and novelist, posited the thesis that the intellectual life of the Western world is irrevocably split between science and the humanities (see Tom Stoppard’s Arcadia for another fictional exploration of this idea). First given as a talk, Snow published it in book form as The Two Cultures and the Scientific Revolution. While Snow argues that it is primarily the literary types that exacerbate the distance between the two and that scientists promote real prosperity and security, he does in fact argue for education not being too specialized. Without going deeper into his work, we can already see the similarities with Juster’s novel. The gulf between Digitopolis and Dictionopolis is irrational and harmful; it takes both numbers and letters to make life both livable and meaningful. Extremity in the defense of one thing – numbers, letters, beautiful sounds – is a vice. The Mathemagician needs to embrace words, and Azaz needs to embrace numbers. The Soundkeeper needs to embrace the whole spectrum of sounds. The Doldrums need to allow laughter and thinking. Faintly Macabre has to let people use the words they want. Milo learns all of this on his journey, and will take it back with him to his real life.
