Once upon a prime, p.12
Once Upon a Prime, page 12
What can explain the hold that the number 3 has on our psyche? I propose that the mathematics of triangles and trichotomies enables the triumph of the triple. Three, in geometry, is very special. First, it’s the smallest number of points that can define a two-dimensional shape. If you have only two points, then you get just a line. Three points (as long as they don’t all lie on the same line) give you a triangle. But it’s better than that. Imagine trying to make a rigid, stable structure out of sticks or rods. With two rods, you can’t do anything. You can join two ends together, but the other two ends just flop about uselessly. But if I start with three rods of any lengths I like, I can fit them together in exactly one way to make a triangle. If you also have three rods of the same length and do the same thing, our two triangles will look the same. That’s the second special thing about the number 3. It is not true for any higher number. With four rods, there are infinitely many quadrilaterals (four-sided shapes) you can make. Even in the superspecial case in which I want all four side-lengths to be the same, there are infinitely many possibilities. You can make a square, sure, but then you can squeeze it at the corners to make a series of ever-thinner diamonds. The triangle is the only straight-line shape that can’t be deformed in this way. That’s why in structures made with steel rods, such as geodesic domes, the basic shape is the triangle. It is the strongest shape.
The third (of course) special geometric property of the number 3 is that three is the largest number of points you can have in a plane that are all the same distance from one another. The three points of an equilateral triangle are mutually equidistant. It’s impossible to draw four or more points on a piece of paper all of which are the same distance from each of the others. (You can do it for four points if you go to the third dimension, with what’s called a tetrahedron, but even then, this is a shape made by joining four equilateral triangles together.) These geometric properties of the triangle could be one reason why sets of three things give us a sense of strength and completeness, and also often of equitability. All for one and one for all, as the Three Musketeers say. With two, it’s just the up or down, left or right, or north or south of a line. With three, suddenly we can encompass a whole space.
The final mathematical aspect of 3 is the trichotomy. Imagine the whole number line laid out, and stick a pin in it at a point x. Every other number has a relation to x, and there are precisely three possibilities (a trichotomy). It is less than x, or it equals x, or it is greater than x. This kind of trichotomy is all over the place in mathematics. Every angle is either acute (less than 90 degrees), a right angle (equal to 90 degrees), or obtuse (greater than 90 degrees). Numbers are negative, positive, or zero. Time can be past, present, or future. In statistics, a data point can be higher than the mean, lower than the mean, or bang-on average.
A variant of this idea is the set of three we obtain from the two extremes plus the middle. Smallest, biggest, and everything in between. Sunrise, daytime, sunset. Birth, life, death. Trichotomies like this happen regularly, in both the language and the structure of narrative. We have three layers for adjectives: good, better, best; bad, worse, worst; brave, braver, bravest. The youngest of three fairy-tale brothers is invariably the wisest; the youngest sister is always the prettiest; the third billy goat gruff is the biggest and defeats the troll. And what better example of trichotomy than the threefold verdicts given by everyone’s favorite housebreaker, Goldilocks? Daddy Bear’s porridge is too hot. Mummy Bear’s porridge is too cold. Baby Bear’s porridge is just right. Goldilocks was clearly familiar with Aristotle’s doctrine of the mean. He says that every ethical virtue is a golden mean (just right) between two vices—one an excess, the other a deficiency. Courage is a virtue, an excess of courage is the vice of recklessness, and a deficiency of courage is the vice of cowardice. When it comes to money, liberality is a virtue, an excess of liberality is profligacy, and a deficiency is miserliness. And when it comes to beds, Daddy Bear’s bed is too hard, Mummy Bear’s is too soft, and Baby Bear’s epitomizes the Aristotelian mean—it’s just right.
Stories themselves have a beginning, middle, and end. The most common multivolume set is the trilogy. These often are trilogies only in hindsight; a common structure is a self-contained initial volume, followed by a Book 2 that ends on a cliff-hanger, or at least with matters unresolved, and then a concluding Book 3 wrapping up all the loose ends, so that the trilogy is a larger-scale version of beginning, middle, end. Consider also the three-act play, in which each scene itself must also have a beginning, middle, and end. The book you are holding in your hands itself has three parts.
The appearance of magic numbers in fiction may be the most obvious manifestation of mathematics in literature, but that’s just the beginning. Later, I’ll show you the ways in which much more sophisticated mathematical ideas, from geometry to algebra and even calculus, have made their appearance in great works of literature, from Moby-Dick to War and Peace. Numbers are such a crucial part of human thought that they are even hidden inside words themselves, sometimes in the most unexpected places. Think of the fateful bowl of punch that dashes Becky Sharp’s hopes for a proposal from Jos Sedley in Vanity Fair. No numbers there, right? Except that the word “punch” derives from the Sanskrit word for “five,” panca, because the drink originated with an Indian concoction that had five ingredients. Numbers really are part of the fabric of language, in (to use the ancient Greek word for “ten thousand”) myriad ways.
6
Ahab’s Arithmetic
Mathematical Metaphors in Fiction
I mentioned in the introduction that the seeds for this book were planted when I heard a mathematician mention that Moby-Dick contains a reference to an interesting curve called a cycloid. Curiously enough, when I emailed my friend Tony (the mathematician in question) a couple of years back to thank him for his recommendation, he replied saying I’d been the one to recommend it to him—so I guess we’ll never know the truth. At any rate, one morning I sat down on the train, opened the book, started reading, and within a few minutes encountered a brilliant description with a definite mathematical tinge to it. Ishmael spends the night at the Spouter-Inn, whose landlord is somewhat stingy with his drinks: “Abominable are the tumblers into which he pours his poison. Though true cylinders without—within, the villainous green goggling glasses deceitfully tapered downwards to a cheating bottom. Parallel meridians rudely pecked into the glass, surround these footpads’ goblets.” It’s a great image, and there’s an undeniably geometrical air to the true cylinder marked with parallel meridians. It piqued my interest.
As I read on, I kept encountering mathematical allusions, so many in fact that it became clear to me that Melville obviously relished mathematical ideas—they were bound to escape from his mind onto the page, and when he reached for a metaphor, more often than not something mathematical would present itself. In praising the loyalty of his cabin boy, Captain Ahab says, “True art thou, lad, as the circumference to its centre.” And indeed this is exactly right—the points on the circumference of a circle steadfastly remain exactly the same distance from the center, all the way around.
The world of mathematics is a glorious source of metaphors. Some of these have become clichés in everyday speech, like “squaring the circle”—a reference to the ancient Greek problem of constructing a square with the same area as a given circle. Few people using this phrase know that the mathematical proof of its impossibility took more than two millennia to find. But sometimes you’ll come across authors who, like Melville, clearly have an affinity for mathematics and cannot help but use mathematical metaphors in their writing. In this chapter, I’ll give you a guided tour of some of the loveliest mathematical allusions in the work of classic writers like Melville, George Eliot, Leo Tolstoy, and James Joyce. Understanding these references adds another layer to our enjoyment of great literature, and it’ll give you a totally new perspective on some much-loved books, as well as on their authors.
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Before I show you any more of Melville’s mathematical metaphors, I want to tell you a little about Melville himself and how he came to write what D. H. Lawrence described as “a surpassingly beautiful book … a great book, a very great book, the greatest book of the sea ever written. It moves awe in the soul.” Melville tried various professions (teacher, engineer, deckhand on a whaling ship) before writing his first novel, Typee, a fictionalized account of his time with the Polynesian tribe of that name. This, and the follow-up, Omoo (the Polynesian word for “wanderer”), were well received, and he wrote three more seafaring stories over the next few years. I am focusing on Moby-Dick, his sixth book, because it’s my favorite of Melville’s books and it’s the best known.1 But Melville’s love of mathematics seeps into everything he does. In his earlier novel Mardi, he has a character cry out, “Oh Man, Man, Man! Thou art harder to solve, than the Integral Calculus.” His publisher was clearly concerned that discussing philosophy and mathematics might not be as profitable as writing about scantily clad young Polynesian ladies, and Melville reassured him that the next book would contain “no metaphysics, no conic-sections, nothing but cakes & ale.” Fortunately for literature, he comprehensively failed to keep that promise.
Moby-Dick was written in 1850 and published in 1851. Reviews were … mixed. Harper’s New Monthly Magazine loved it: “The genius of the author for moral analysis is scarcely surpassed by his wizard power of description.” But a reviewer for the London Athenaeum felt that “Mr. Melville has to thank himself only if his horrors and his heroics are flung aside by the general reader, as so much trash belonging to the worst school of Bedlam literature.” It’s amazing to think that the author of Moby-Dick more or less gave up writing within a few years of its publication; he spent the last two decades of his life working for the US customs service and died in obscurity in 1891. His lifetime earnings from perhaps the most influential American novel of the nineteenth century amounted to $556.37. We don’t know much about Melville the man—as an indication of how carefully he guarded his privacy, he used to hang a towel over the doorknob of his study so that nobody could look through the keyhole. Few of his letters survive, and all that his close friend Nathaniel Hawthorne could find to say was that, although he was a gentleman, he was “a little heterodox in the matter of clean linen.” But listen: if you’ve not done so already, please overlook the dirty laundry and read Moby-Dick. It’s like no other book.
Our narrator, Ishmael, goes to work as a deckhand on a whaling ship, the Pequod, with its captain, Ahab; first mate, Starbuck (of coffee shop fame); and second mate, Stubb. Gradually it becomes clear that Ahab is obsessed with hunting and killing the great white whale Moby Dick—a previous encounter with whom led to Ahab’s losing his leg. (By the way, the book is titled Moby-Dick, but the whale is named Moby Dick. If you are angry about the inconsistency, please take it up with Ishmael.) Ultimately, Ahab’s hubristic and monomaniacal pursuit of the white whale drives him to insanity, endangering the whole crew, and let’s just say it doesn’t end well for Ahab.
This is no ordinary adventure story. There are “excerpts” discussing mentions of whales and whaling from a dizzying array of sources, including Shakespeare, the Bible, and books on natural history and navigation. There’s a whole chapter on the meaning of Moby Dick’s whiteness, and many philosophical musings from Ishmael and others. Ishmael explains that the book must necessarily have a huge compass because its subject, Leviathan, is so vast. “Give me a condor’s quill!” he says. “Give me Vesuvius’s crater for an inkstand!”
If I asked you to predict where mathematics might appear in a nineteenth-century sea story, you might think of quadrants and sextants and quite rightly suggest that it could be involved in descriptions of navigation. We do indeed hear about Ahab’s doing mathematical calculations “on the upper part of his ivory leg,” and Ishmael talks of “studying the mathematics aloft there” in the crow’s nest where he perches, scanning the sea for signs of whales. But Melville goes a lot deeper. The almost magical powers of mathematics, for those initiates who can decipher its “cabalistic contrivances,” are spoken of by the crew with a mixture of awe and suspicion: “I have heard devils can be raised with Daboll’s arithmetic,” says the second mate, Stubb. Generations of American schoolchildren would have been familiar with Daboll’s Arithmetic, the most widely used textbook in US schools for the first half of the nineteenth century. (The full title is Daboll’s Schoolmaster’s Assistant: Being a Plain, Practical System of Arithmetic, Adapted to the United States.) The author, Nathan Daboll, was a Connecticut mathematics teacher, and we know that Melville used Daboll’s Arithmetic as a pupil and probably as a teacher. You would have Daboll for arithmetic and Euclid for geometry.
Looking at the book with modern eyes, it’s not at all surprising that Stubb compared it to some sort of alchemy. It provides techniques, which are to be learned by rote, for all manner of calculations, from the basics of arithmetic to currency conversion and rules for calculating interest, annuities, profit and loss, and ship tonnage. Methods are even given for the extraction of square roots and cube roots by hand. The rules given are often presented almost as magic formulae. For instance, to convert from South Carolina dollars to Maryland dollars, “multiply the given sum by 45, and divide the product by 28.” Or there’s the mysterious “Rule of three direct,” which teaches, “By having three numbers given to find a fourth, which shall have the same proportion to the third, as the second has to the first.” Here is the rule for finding the circumference of a circle, given its diameter: “As 7 is to 22, so is the given diameter to the circumference. Or, more exactly, as 115 is to 355; the diameter is found inversely.” The circumference of a circle is its diameter d multiplied by but amazingly there is no mention of here, or the fact that these rules work because and are approximations to They are just magic numbers to be deployed.
For Stubb, mathematics is mysterious, even malign. But for Ishmael, mathematics, and symmetry in particular, symbolize virtue. The sperm whale has a “pervading dignity” because of the “mathematical symmetry” of its head. In describing this head, Ishmael even claims to define a new mathematical concept. He explains, “Regarding the Sperm Whale’s head as a solid oblong, you may, on an inclined plane, sideways divide it into two quoins, whereof the lower is the bony structure, forming the cranium and jaws, and the upper an unctuous mass wholly free from bones.” In a footnote, he explains, “Quoin is not a Euclidean term. It belongs to the pure nautical mathematics. I know not that it has been defined before. A quoin is a solid which differs from a wedge in having its sharp end formed by the steep inclination of one side, instead of the mutual tapering of both sides.” This could have come straight out of a geometry book!
You could argue that it’s fair enough to get a bit geometrical when describing a shape (though it does indicate at least comfort and facility with these terms), but Euclid gets name-checked in several other places too. When explaining that the whale’s eyes, being on opposite sides of its head, present its brain with two completely distinct views that must be processed simultaneously, Ishmael says that if the whale can really do this, then “it is as marvelous a thing in him, as if a man were able simultaneously to go through the demonstrations of two distinct problems in Euclid.” The best mathematical moments in Moby-Dick are in places like this, where Melville throws in a mathematical allusion just for the fun of it.
It takes a geometer’s eye, for instance, to connect a whale’s fin to the gnomon of a sundial, as in Ishmael’s observation here:
When the sea is moderately calm, and slightly marked with spherical ripples, and this gnomon-like fin stands up and casts shadows upon the wrinkled surface, it may well be supposed that the watery circle surrounding it somewhat resembles a dial, with its style and wavy hour-lines graved on it. On that Ahaz-dial the shadow often goes back.
Pleasingly enough, that mention of Ahaz recalls what is now thought to be the earliest written reference to sundials, in the Old Testament book of Isaiah. God causes the shadow on a sundial to move miraculously backward ten degrees, as a sign that he will cure the sickness of Hezekiah, son of King Ahaz of Judah.
But perhaps the most fascinating bit of geometry in Moby-Dick involves cycloids, those mathematical curves I mentioned at the start of this chapter. Ishmael thinks about them while he is cleaning the great try-pots on the deck of the Pequod. Try-pots are huge metal vats—think massive cauldrons—where the whale blubber is rendered to produce oil:
Sometimes they are polished with soapstone and sand, till they shine within like silver punchbowls.… While employed in polishing them—one man in each pot, side by side—many confidential communications are carried on, over the iron lips. It is a place also for profound mathematical meditation. It was in the left hand try-pot of the Pequod, with the soapstone diligently circling round me, that I was first indirectly struck by the remarkable fact, that in geometry all bodies gliding along the cycloid, my soapstone for example, will descend from any point in precisely the same time.
