by Ellen Kaplan
1.47:
In ABC, right-angled at C, the sum of the areas of the squares on the two legs (AHKC and CLMB) equals the area of the square on the hypotenuse (ABEG).
PROOF:
Part I.
1. Drop the perpendicular from C to AB, meeting it at D, and extend it to meet GE at F (there’s the legacy from the Pythagorean proof by proportions).
2. Consider AHB. The altitude to its base HA from B is equal in length to AC (think of HA extended, making a parallel to KCB. The altitude from B is then the perpendicular, equal in length to AC, between these two parallels).
3. That gives the area of AHB, half the base times the height to that base, as HA · AC.
4. Hence AHB has half the area of square AHKC.
5. Consider ACG. Using ‘’ to mean what its two parts imply—not only similar but equal—in length, in angle measure, or in shape, AC AH, AG AB, and CAG HAB (each is a right angle plus CAB), so ACG AHB, by side-angle-side, hence their areas are equal.
6. But reasoning as in step 2, the area of ACG is AG· AD.
7. Hence (by transitivity of areas), the area of AHKC equals that of rectangle AGFD.
Part II.
Proceed exactly as in Part I, considering now ABM and CEB:
1. As in the steps 2–4 of Part I, ABM has half the area of square CLMB.
2. As in step 5, ABM CEB.
3. As in step 6, the area of CEB is EB· BD.
4. As in step 7, the area of CLMB therefore equals that of rectangle BEFD.
Since the rectangles AGFD and BEFD together make the area of ABEG, adding the results of Parts I and II gives us the sum of the areas of the squares on the two legs (AHKC and CLMB) equaling the area of the square on the hypotenuse (ABEG), as desired.
Of course the diagram with which Euclid confronts his reader presents both parts at once, and with HB, CG, AM, and CE (which, being “auxiliary lines”, we dotted in) shown as solidly as the rest; so that the first thing the reader sees is this cat’s-cradle, with its pairs of obtuse triangles, reminiscent of Guido’s four right triangles, caught as if in a double exposure:
No wonder Schopenhauer, mixing his metaphors in the throes of a warp-spasm, called this “a mouse-trap proof” and “a proof walking on stilts, in fact, a mean, underhanded proof” (the German hinterlistiger sounds even more villainous).2
Worse: don’t those sharp angles suggest a dangerous and barely caged creature? If you’ve ever wondered why we speak of the “horns of a dilemma”, your answer is here. In his 1570 edition of Euclid, Sir Henry Billingsley (who later became a Haberdasher, then Sheriff, then Lord Mayor of London) wrote that I.47 “hath been commonly called of barbarous writers of latter time Dulcarnon”, from the Persian du-, two, and karn-, horned3 : the dilemma that impales whoever would tackle this proof.
“I am, til God me bettre minde sende,
At dulcarnon, right at my wittes ende.”
That’s Criseyde, speaking two centuries before in Chaucer’s Troilus and Criseyde. Yet those who conquered this proof came to venerate it. Proclus, writing in the fifth century A.D., says: “If we listen to those who wish to recount ancient history, we may find some of them referring this theorem to Pythagoras and saying that he sacrificed an ox in honor of its discovery. But for my part, while I admire those who first observed the truth of this theorem, I marvel more at the writer of the Elements . . . because he made it fast by a most lucid demonstration.” 4
“Made it fast”: Proclus uses the same image of tethering that Plato puts in Socrates’s mouth when kidding Meno about the supposed statues in Athens made by Daedalus, which he pretends would run away were they not tied down (by way of explaining to Meno that true opinions only become knowledge when tethered by recollection). This tells us something important about the Greek view of diagrams, constructions, and proofs, in terms of their attitude toward what is versus what we know: that distinction between ontology and epistemology which has ever since split the Western mind as surely as the Great Longitudinal Fissure divides the brain’s two hemi spheres.
For the image of tethering “the truth of this theorem” (i.e., making it more than just a true opinion) by a demonstration shows that these mathematicians were as aware as we are that it takes constructive work to grasp what lies in being, rather than becoming. Euclid’s constructions (he wrote three books, now lost, about them)5 are mean proportionals between being and thinking: scaff oldings to be removed once the finished work is revealed.b Kant, more than a millennium later, rethought this as the synthetic and the a priori, the made and the found, recognizing them as two sides of the same coin. At the end of a demonstration Euclid presents us with everything, constituent and auxiliary lines alike, on a par. The end of his work is the beginning of ours, who must, like readers of Plato, rebuild his artifices and then, with him, remove them.c
We can now understand why those awkward obtuse triangles came to give this theorem its other, gentler, medieval nickname: the Bride’s Chair. Do you see the maiden luxuriating there as an Eastern bearer carries her palanquin on his back to the wedding? Heath discovered her once again in a Great War issue of the magazine La Vie Parisienne, where Euclid’s diagram was the framework for a French poilu shouldering his bride and all his house hold belongings.6
This jolly misogynist version went back to an inn sign painted by Hogarth, The Man Loaded with Mischief. Heath caught a glimpse too of such a sign somewhere in the fen-land near his own Cambridge.7 Those days are perhaps farther from ours than are Euclid’s.
It isn’t quite true that Euclid proves the Pythagorean Theorem twice: when he returns to it in Book VI he actually proves a strikingly broader statement.c For in developing Eudoxus’s theory of proportions, he shows that not just the areas of squares but of any similar polygons on a right triangle’s three sides will be in the relation ad + b2 = c2, where a, b, and c are the side-lengths of these fitted polygons.
As you’ll soon see, Euclid’s proof will glide as smoothly as a swan—but the vigorous paddling beneath the surface will both deepen the story of intermediate constructions and shift Euclid from a name on a staid classic to an almost companionable fellow being, by showing us the cast of his thought if not the cut of his jib. For he makes what we’ll call Euclid’s Converter by breaking up similar polygons, as we would, into networks of similar triangles, and then showing (see the appendix to this chapter) that the areas of similar triangles are to each other as the squares of their corresponding sides—hence so must be the areas of their sums. But it is just here that we have the advantage of him, being the proud possessors of algebra, the ultimate in generic reasoning.
In algebraic style we would have no diagrams whatsoever but call our similar triangles 1 and 2, and denote their areas by | 1 | and | 2 |.
Since we know that a triangle’s area is half the base times the height, .
If 1 ~ 2, with a constant of proportionality k (so that b2 = kb1 and h2 = kh1), then
Since Euclid had general figures, not abstract ones, he could reach this result only by some clever geometric manipulations—and we know from I.47 which way his mind inclined: toward making an intermediate triangle that would allow transitivity of areas!
once remarked that there is a “sort of mathematics that no gentleman does in public.”8 You might well think Euclid’s machinations are of this sort, so lest you run screaming at the sight, or at that of the apparatus in his mind’s gymnasium, we’ll hide them in the appendix at the end of this chapter.
Now at last we’re ready for Euclid’s enhanced Pythagorean Theorem, in modern notation:
THEOREM: In ABC, right-angled at C, if the polygons Pa and Pb on the two legs are each similar and similarly situated to that on the hypotenuse, Pc, then the sum of their areas equals the area of that on the hypotenuse: | Pa| + | Pb| = | Pc|.
PROOF:
[representing any such triple of similar polygons by similar parallelograms, and using “” to mean “implies”]:
This proof legitimizes the first of our theor
em’s many progeny, waiting in what heaven for their human parents to be born?
APPENDIX
We spoke of Euclid’s gymnastics in proving that the areas of similar triangles are to one another as the squares of their corresponding sides. Here are the workout machines, which he needs in the proof of the generalized Pythagorean Theorem.
(A) [This is the modern version of his porism to Proposition 19 of Book V]:
Euclid now establishes that the areas of two triangles are to one another as the squares of their corresponding sides. He first proves that
PROOF:
(It helps to draw these triangles as here, and then to draw BD).
Now he can prove
PROOF:
Exercise machines? If you look at these and at the rest of his proofs in Book V, they may remind you more of a carpenter’s shop before the invention of lathes. There are the beautiful saws and draw knives and files to shape and smooth a chair leg: the work of contemplative hours, with the end-product marked by the craftsmanship of risk. We want the chairs, of course, and many of them, and now—but at the cost of an almost sensuous feeling for form?
CHAPTER FIVE
Touching the Bronze Sky
Some people collect Ketchikan beer coasters, some Sturmey Archer three-speed hubs, others wives or ailments. Jury Whipper collected proofs of the Pythagorean Theorem. He wasn’t the first: fifty-nine years before him, in 1821, Johann Joseph Ignatius Hoffmann published more than thirty; in 1778 a Frenchman named Fourrey included thirty-eight among his Curiosités Géometriques, and a Herr Graap had translated others from a Russian anthologist. Nor was Jury Whipper the last: Professors Yanney and Calderhead, from the Universities of Wooster in Ohio and Curry in Pennsylvania, gathered together some hundred proofs between 1896 and 1899. A lawyer at the District of Columbia Bar, named Arthur Colburn, published 108 of his own, starting in 1910; perhaps the currents of litigation ran more slowly in the days before air conditioning.
On the shoulders of these giants still stands Elisha Scott Loomis: the boy born in a log cabin in 1852, who rose to be a 32nd degree Freemason and, he tells us, “plowed habit-formation grooves in the plastic brains of over 4000 boys and girls and young men and women.” He tells us that of all the honors conferred upon him, he prized the title of Teacher more than any other, “either educational, social, or secret.”
What might some of those secret titles have been? And was he as leonine as his portrait shows him, florid moustache and wing collar? Or should we believe a penciled note in the Harvard Library’s copye of his book, The Pythagorean Proposition: Its Proofs Analyzed and Classified and Bibliography of Sources for Data of the Four Kinds of Proofs: “He was somewhat high in manner, but was in reality a good sport. I never met him.” Then how did you know, G. W. Evans?
Pythagorean investigations breed mystery. Here’s another. How many proofs are there in Loomis’s book? Be a Babylonian, you say, and count them. But counting, as even a Babylonian knew, is one of the hardest of human tasks. Loomis claims to have 367 proofs in his second edition, though some are circular, some defective, some no more than variations on or parts of others. Are the algebraic proofs, which he says readily follow from this or that geometric demonstration, to be thought of as different from them? He asks about a possible proof here, “can’t calculate the number” of others there, and speaks of “several”, “a number of ”, and “countless” different proofs from those he gives. 9,728 proofs, for example, derive from his figures for Algebraic Proofs Six and Seven, he tells us, and 65,780 more from his Figure Eight. When he writes, as in his note to Geometric Proof 110, of more cases extant, does he mean more than he has given? Are two proofs really different if a square in one has no more than slid sideways from that in another (as in his Geometric Proofs 111 and 112), or if a grid of lines is differently parsed (proofs 119 and 124)?
We conclude that his book contains 367 proofs minus a few, plus several, increased by a number derivable but not in fact derived, to which are added those that are “other” and “different”, resulting in many plus a multitude, increased by what he describes as an unlimited horde of the likely, and a quantity indefinitely great that will be discovered by “the ingenious resources and ideas of the mathematical investigator”, giving us as an approximate total more than we should, or could, or may, or want to, count. This amplitude is consonant with the generous spirit of brotherhoods, Pythagorean or secret, and is an image of life itself: in the earth below each tree of spreading order the mice of Somewhat gnaw, while chaos in its foliage is made by the insects of Et Cetera.
The Babylonian urge makes us want to number Loomis’s proofs; the Egyptian urge, to picture them. His is an anthology of insights had by generals and artists and lawyers and children and principals of normal schools and presidents of free schools; by people from as far away as ancient India and Renaissance Italy and modern England, and others as close to home as Independence, Texas, Hudson, New York, and both Whitwater and Black Hawk, Wisconsin. Among their variety nestle fifty-five proofs by Loomis himself (or is it ninety-four?).
He lets us look a little down the reversed telescope of time to March 28, 1926, which was a particularly good day for him. At ten in the morning on that Saturday he invented a snappy geometric demonstration, and at three in the afternoon another; and then, before he went to sleep, came up with a third, at nine thirty in the evening. And here, twenty-six years before, is Loomis at work on August 1, 1900, during a summer vacation from plowing grooves in the plastic brains of his students at West High School in Cleveland, Ohio. What was a sleepy Wednesday for some meant enough leisure for him to contrive five proofs—and four more for that matter, in the next eight days, after three in the week before (we shall do our best to keep from reading any Pythagorean significance into those numbers). His thoughts were running on figures where the square on a right triangle’s short side is folded over onto part of the triangle itself, and onto part of the square built on its hypotenuse. In 1926 it was the square on the hypotenuse that he saw folded back over the triangle. He urges every teacher of geometry to use paper-folding proofs, although the one he cites requires folds we are unable to see.
Might these numerous proofs not reduce in fact to variations on Guido’s, or the Pythagorean proof by proportions, with the occasional crossover product such as Euclid’s? Does the stark, surprising relation these proofs aim to establish not constrict the flow of invention toward it into a few narrow channels—and at the other end, aren’t the sources of our play as challengingly limited as a child’s: a stick and a string?
Were you to turn on the news and hear “Now for some baseball scores. Four to three, one to nothing, eight to five . . .”, you’d think the world had gone comically mad. This is the madness within whose shadow algebra lives. If some insubstantial part of us, which we variously call instinct and intuition, inclines us this way or that when devising a proof, algebra is the machine in this ghost, churning out, like the mad scientist’s Igor, a succession of means without any care for what end they might serve, or whom they should benefit.
Take those 75,508 or so algebraic proofs that Loomis cites. You could tie the triangle up with ever fancier ribbons and then unwrap it, before a familiar construction inside is smothered to death. You could fence it about with palisades and ankle-breakers, keeping ratios in and rationality out.
Proofs, more proofs, Master! Were you an aficionado of trig, you could surround it: since a circle’s tangent squared equals the product of a secant through it with its external segment, a2 = (c + b)(c − b) = c2 − b2c2 = a2 + b2.
Or, if you were Loomis, and it was the twenty-third of February in 1926, you could prove it without a square or a mean proportional in sight:
You see in proofs like these a miniature history of the baroque becoming the rococo. Some of Loomis’s are tortuous, some teasing; some pert, some monstrous. Yet while most of mental space, like cyberspace, is empty, its convolutions have no room for the convoluted: math worthy of the name is as simple a
s possible—but not, as Einstein pointed out, simpler.
If algebra pours out such a vacuous plenty, wouldn’t trigonometry follow with ampler waves? No, Loomis thunders: “There are no trigonometric proofs, because all the fundamental formulae of trigonometry are themselves based upon the truth of the Pythagorean Theorem; because of this theorem we say sin2 A + cos2 A = 1, etc. Trigonometry is because the Pythagorean Theorem is.” Here, however, is a trigonometric proof deciphered from something sketched on the back flyleaf in Harvard’s copy of Loomis (by the same Mr. Evans who told us that Loomis was a good sport):