by Ellen Kaplan
We hear Loomis exclaiming, “One moment, sir! What role has trigonometry played here?” For the proof no more than relabeled (in terms of angle functions) the ratios we knew from the Pythagorean proof in Chapter Three, then withdrew to let those ratios play their familiar game. This seems an instance of tautology yielding nothing new under the central sun of Mind.
A wholly—and profoundly—trigonometric proof blossoms, however, when a great swath of mathematics is rethought as grounded in infinite series (a point of view that—although it puts formalism before intuition—unifies much, solves delicate problems, and opens fresh vistas). For sine and cosine, thought of as functions of their angles, are defined independently of the Pythagorean relation, and their derivatives—the functions that measure their slopes—can likewise be found without it. These lead (via ingenious series devised by an eighteenth-century Englishman, Brook Taylor, in response to a conversation in a coffeehouse) to the wonderful expressions which better and better approximate them, and are at their limit the infinite series which are identical to the sine and cosine functions themselves:
Squaring these seriesf and adding the results yields
This proof (which lived on the dark side of Loomis’s moon) also reminds us of the broader nature of math, by showing that, while it may take a considerable yet finite effort of bookkeeping to keep track of how terms cancel out or vanish, it needs time’s infinite expansion to conclude.
Stepping back just an ergo from this flurry of proving, you see that Loomis spoke from an excluded middle, being neither right nor wrong. Proofs of the Pythagorean Theorem via trigonometry are indeed possible—though some are no more than cosmetic changes on proofs closer to the bone, while others coil as infinite series, like stem cells in their very marrow.
If we are to find further significant proofs of the Pythagorean Theorem, we should begin by looking in its proper home, which is geometry: measuring earth. Long before he became famous for his Flashman novels, George MacDonald Fraser had fought in the jungles of Burma, and he describes in his memoirs this strange moment from 1944:
“You mentioned isosceles triangles. Will it do if I prove Pythagoras for you?”
“Jesus,” he said. “The square on the hypotenuse. I’ll bet you can’t.”
I did it with a bayonet, on the earth beside my pit—which may have been how Pythagoras himself did it originally, for all I know. I went wrong once, having forgotten where to drop the perpendicular, but in the end there it was. . . . He followed it so intently that I felt slightly worried; after all, it’s hardly normal to be utterly absorbed in triangles and circles when the surrounding night may be stiff with Japanese.1
This story not only renews the sense of our ultimate human kinship: it caters to that feeling you get on overcast days, that while the real magic of things casts no shadow, there is an occult meaning foreshadowed by everything. What if Pythagoras really had seen this theorem by scratching it in the earth? Wasn’t he after all a traveler through chthonic doorways? So Faust, millennia later, called Mephistopheles up through similar diagrams on the floor, and romantic musings find the irrational caged in all such symbols. And at these times, when significance seems to shine only against a dark background and the gnomon’s shadow to tell more than the light that cast it, you find yourself in sympathy once again with the Hearers and the Name Worshippers and the tribe of astrologers and readers of entrails and of Bible Codes, and all references are covert and everything comes to stand for something else.
Let’s return to our anthology of proofs, and relish a few of the many flowers that grow in these Pythagorean fields—some as exotic as orchids, some daisy-simple, with a damask rose next to the latest hybrid tea. The Greeks thought that the rare mortal might touch just once, and then but briefly, the bronze sky, through a superlative act of courage, strength, or thought—such as grasping an immortal truth. These proofs of the Pythagorean Theorem, brewing up, who can say how, out of our disorderly daily affairs, may, unrecognized, have marked the acme of their inventors’ lives.
Here, for example, is one put together by Thabit ibn Qurra, who was born in 836 in Mesopotamia—two millennia after those Babylonians who did their devising in Chapter Two. Thabit had been a money changer in Harran, but was brought by a chance meeting into Baghdad’s House of Wisdom, where he remained to translate ancient Greek texts into Syriac, and to become an expert in medicine, astronomy, astrology, linguistics, magic, philosophy, mechanics, physics, music, geography, botany, natural history, agriculture, meteorology—and mathematics. Think of him when next you look at a sundial: he and his grandson Ibrahim studied the curves that went into its making (such as those you see here in Ptolemy’s ‘analemma’).
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Thabit begins his magical proof by picturing our triangle ABC and its attendant squares, with the small square on side a, and then carry ing off those squares and placing them side by side, as squares HBCD and FGDE. Thus ED = b and DC = a.
Now he locates a point A on ED so that EA = a. This means that AD = b − a, so ADC = b − a + a = b. Drawing AB, he has his original triangle back again.
He now draws FA. Let’s number the pieces as here:
Rotating piece 1 counterclockwise by 90°, it rests on FG as piece 6, with the new vertex K. Of course 6 . 1, so their areas are the same.
Now rotating ACB clockwise by 90°, it rests on BH, as part 7. 7 4 + 5, again with the same areas.
This means that the square on the hypotenuse of ACB is made up of the pieces 2 + 3 + 6 + 7 2 + 3 + 1 + 4 + 5, which is the sum of the squares on the legs. Done!
Or should we have said done—again? Has Thabit’s proof cast a shadow not his, but Guido’s, with two of the triangles differently moved and interpreted? For (lettering as below) triangles P and Q remain as they were, outlining the square on the hypotenuse, but S and R have each been rotated 180° counterclockwise, and no longer outline the square but have become part of it!
Which do you prefer, the simple or the subtle?
And which do you prefer, a proof like Thabit’s that involves cutting up, moving, and adding pieces of polygons, or one that uses no more than comparing—with the added fillip of needing the whole Euclidean plane to do so? But if we’re not to dissect, then what will we be comparing? Take the two squares with side-lengths a and b that Thabit began with, set next to each other, and tile the whole plane with them, in every direction, as if this were the Turkish Bathhouse of the Gods. Mathematics is nothing if not extreme.
Pause now and pull a transparent layer over this flooring, and on it trace a second pattern—this one just of the hypotenuse squares, but tilted, as in Thabit’s diagram:
The side of each such square (we’ll call its side-length 1) runs, as a hypotenuse should, from the top of an a side to the opposite end of the paired b side:
Now pick out a small portion T (for top) of this hypotenuse tiling—here it is with the floor underneath and the hypotenuse square pattern superimposed on it. To see the bigger (but not too big) picture, let’s choose to have n = 3 hypotenuse squares on a side, so this piece is made of n2 = 9 tilted hypotenuse squares, and it looks as if there were n2 = 9 large and n2 = 9 small square tiles on the floor below it.
Were this exactly so, we would have proven the Pythagorean Theorem: one hypotenuse square would take up as much area as the squares on the two legs. But is it exact? As in our little sample, parts of some of the tiles below stick out of T, and parts of other tiles intrude. Let’s not stoop to measure, but call the underneath tiling of n2 large and small tiles, in our extract, U (yes, for “underneath”), and denote its area—as we did in Chapter Four—by | U |. We want to show that | T | = | U |, but do this by looking, like gods, from afar—since gods don’t sweat the small stuff (even in a Turkish bath).
If the area of our 3 × 3 patch, T, fell short of that of the small and large squares associated with it below in U, it would certainly still fall short in a 4 × 4 or 5 × 5 or any n × n extract. Hence it would fall short i
f we covered the whole plane in this way, taking the limit as n goes to infinity (in mathematical shorthand, lim as )g:
(We’ve written rather than <, to hold out hope for the equality we want.)
But if the 3 × 3 area of T exceeded that of U, it would still do so at the limit:
That means there must be some constant c, which can do double duty: were T’s side-length, n, increased by this c to n + c, we and the gods could make a new square that would comfortably enclose U; but also were n shrunk by c to n − c, we’d have a square smaller than T which would be enclosed by U. That is:
Divideh this in e quality through by n2, and take the limit as n goes to infinity once more—which means letting T become an ever larger square of these 1 × 1 hypotenuses, while U increases with it, until the entire plane is covered by U and this pulsing T above. So (expanding those squared parentheses and dividing, as we promised, by n2)
then, expanding the numerators and carry ing out the division by n2,
(but strangely!) proven the Pythagorean Theorem.2
This proof, so different from any we’ve seen, takes up not only more room but more thought than Thabit’s. We’ve practiced a new sort of trapeze-work here: setting up a less and a greater which turn out to be the same—thanks to some ancient arithmetic and some eighteenth-century taking of limits. You’d be right to feel dizzied at first by this Cirque under a distant Soleil. Yet it would have been a relief to Thabit, since, as you saw, nothing in it moved, save the focal point: out to infinity and back again. Thabit was a firm believer in the Greek view that the objects of mathematics are always at rest, so he must have been troubled by the scrambling around in what he called his “reduction to triangles and rearrangement by juxtaposition”.3
The style of a proof reflects the character of its maker. Our fictitious Guido’s was swift, Thabit’s deft, this latest wildly imaginative—but da Vinci’s is cryptic. Construct, he says, the squares on a right triangle’s three sides, as lettered here, and then draw EF, making CEF CAB.
Now construct JKH CAB on KH, and last, lines CJ, DC, and CG—an unpromising beginning. At least DCG is a straight line, since its two parts, DC and CG, bisect equal opposite angles.
Leonardo, like a master of misdirection, now asks us to look at four quadrilaterals. After a moment’s thought you will see that DEFG DABG, and CAHJ JKBC (in each case a shared side and equal vertex angles). It will take more than a moment to see that DEFG and CAHJ are congruent too, so that all four of them are (and are therefore equal in area).
Making a six-sided figure out of the first pair and another out of the second, we therefore have their areas equal: | ADEFGB | = | CAHJKB |.
Begin now to peel equal parts of their areas equally away:
the Pythagorean Theorem.
What ambiguous angel signaled this vision to him?
Arthur Colburn, with his 108 proofs, wasn’t the only Washingtonian to busy himself with the Pythagorean Theorem. In 1876, while James A. Garfield was still in the House of Representatives (four years before he won the Republican nomination on the thirty-sixth ballot, and became the country’s twentieth president—only to be assassinated four months later), he tells us that he was discussing some mathematical amusements with other members of Congress, when he came up with this elegant proof (and remarked, “We think it something on which the members of both houses can unite without distinction of party.”)
Since a trapezoid’s area is its height times half the sum of its bases, the area of this one is
On the other hand, this trapezoid is made up of two right triangles with legs a and b, and half of the square on side c—so its area can also be thought of as
Obvious once you see it—like so much math; but (again, like so much math) not at all obvious before. Yet might you not indeed have met it in another existence—or at least in a previous chapter? Isn’t it our oldest diagram, sliced slantwise in half?
Could some shade of that Greek or Babylonian figure have fallen on Garfield’s past, during his 1840s schooldays in Ohio’s Orange Township, or at the Western Reserve Eclectic Institute, or while he was at Williams College in Massachusetts? Or is it so deeply lodged in our mathematical genes that an hour spent with convivial colleagues in Washington, or scratching in the Italian dirt, will call it up?
This dizzying variety of proofs must make you want to take stock and classify. The last two examples built outward from the triangle, the Pythagorean inward, and the trigonometric ignored it altogether. We have proofs bounded in a nutshell and others needing infinite space. Would it help to set up kingdoms of algebra and geometry, then phyla of the intuitive and the formal, classes of analytic and synthetic, say, orders of simple and subtle, and families that look to which squares are folded over or folded back? How would we fit in distinctions between proofs without words and proofs without pictures? Should we stick with our anthologist’s plan to no more than sample these flowers, or does the Linnaeus in us demand a perfected plenum, which our inner Darwin would then turn sideways and send through time?i Yet classifying rigidifies, and might make us miss the world cavorting beyond our ken.
Would either shelving or browsing explain why there are so many proofs? Is it the fad of an adolescence we pass through or abide in, a beacon of awe that beckons our growing strength? Is it a desire to make the impersonal one’s own, or an expression of the astonishment Hobbes felt on first seeing Euclid I.47: “By God! This is impossible!” A reasonable exclamation, since, as we noticed, we haven’t the instinct for area that we have for length. This drove him to follow its ancestry back to ever more venerable forebears, as it might us to prior convictions that bear our own likeness on them. While we may sketch these on paper, Hobbes was given to scribbling geometrical diagrams on his thigh (perhaps this is why Pythagoras’s was golden). Proofs by proportion, proofs by dissection—perhaps you feel, “Seen one, seen ’em all”—but what if you couldn’t see?
THE BLIND GIRL’S PROOF
Loomis credits his fourteenth geometric proof “to Miss E. A. Coolidge, a blind girl.” He then gives us no more than a reference to the journal he found it in. What!? Were his curiosity, his imagination, his compassion, not stirred? Did his compulsion to move on to the next, and the next, and the next leave him no time to wonder at the visions of the blind? Not being in such a hurry, we hunted out the Journal of Education, volume 28, 1888, through the stacks in Harvard’s Gutman Library, where it had—who knows how many years before—been mis-shelved, giving the diligent librarian a dusty two hours before she cornered it down a stack receding to infinity. “That made my day!” she said. And there, was Miss Coolidge’s proof, across from the “Notes and Queries” (“What is the difference between bell fast in Chicago, Belfast in Maine and Belfast in Ireland?” “What is the Agynnian sect?”). Her proof, however, was not as it appears in Loomis, who had clearly exercised his editorial powers over it (he was somewhat high in manner). Here is her proof rescued from the journal, with our reconstruction of what reasons weren’t given there:
In the square on its hypotenuse, copy ABC four times as shown, calling these copies 1, 2, 3, and 4. This leaves square 5, of side-length a − b, in the middle. Call the square on side b, 6.
Copy ABC twice again in the square on the long side a, calling those copies 1 and 2. Divide the rectangle that remains in this square into a square, 8, congruent to 5, and rectangle 7.
Letting the numbers now stand for the areas of the figures they denote, Miss Coolidge needs to show that
i.e., that
Since 8 was constructed so that 8 . 5, it remains only to show that 6 + 7 = 3 + 4. But 3 and 4 each have area , hence 3 + 4 = ab; 6 = b2 and 7 has sides b and a − b, hence area b(a − b) = ab − b2.
So 6 + 7 = b2 + ab − b2 = ab, as desired.
What is particularly fascinating about this proof is that Miss Coolidge begins it geometrically—making and comparing shapes—but then turns to calculating with letters when, perhaps, she can’t quite figure out what’s left and what’s wanting.
You might think that, being blind, Miss Coolidge would have resorted to abstraction as early as possible, but in fact she delays it as long as she can, trying to stay true to the spirit of geometry. She needs to account for square 5, so constructs the square 8 congruent to it in what remains of the square on a (after 1 and 2 are removed); and only then turns to a formula for calculating the area of that remaining rectangle, 7.
Had she not bothered to make 8, then the rectangle left in the square on a, after triangles 1 and 2 are removed, has area a (a − b) = a2 − ab, so that this rectangle, with 6, is a2 − ab + b2; and that’s just what areas 3 + 4 + 5 add up to.
In fact, had she been wholly comfortable with symbolic manipulations, she could have avoided constructions in the square on a altogether,
Having followed her proof, if we now follow her we may better understand the play between abstraction and different sorts of sensory information. But how find the woman behind Loomis’s brief “Miss E. A. Coolidge”? It struck us that Coolidge is a good Boston name, and the journal in which her proof first appeared was published in Boston. Might she not then have been a student at the renowned Perkins School for the Blind? We e-mailed Jan Seymour-Ford, their research librarian, who answered: “That was inspired guesswork! Emma A. Coolidge was a Perkins student. She was born August 4, 1857, in Stur-bridge, MA.”