by Ellen Kaplan
Emma had lost her sight from whooping cough when she was a year old—she could detect only light and shadow, and none of the remedies her parents tried (having her wear gold earrings, putting talcum powder into her eyes, blistering her temple with poisoned flies) helped. After graduating from Perkins, she studied at Wellesley for a year, then returned as a teacher to Perkins, where one of her students is said to have been Annie Sullivan, later Helen Keller’s tutor. Emma married, had a daughter, wrote children’s books, taught music in her New Hampshire village school, sewed, knitted, would kill and dress a fowl for dinner if her husband was away, and boldly went out alone, tying white rags to doorways so she could find her way from place to place.4 Isn’t this what we see in her proof: catching the chiaroscuro of prominent shapes, but navigating otherwise by those abstract relations with which practice and memory furnish the mind?
What sort of imagination this involves may matter here. The blind mathematician Louis Antoine was led by the famous analyst Henri Lebesgue to study two-and three-dimensional topology, because “in such a study the eyes of the spirit and the habit of concentration will replace the lost vision.” His equally outstanding compatriot, Bernard Morin, who has been completely blind since age six, was asked how he knew the correct sign in a long and difficult computation. “By feeling the weight of the thing”, he said. More tellingly, Morin distinguishes between what he calls time-like and space-like mathematical imagination, and surprisingly says that he excels at the latter. A problem with picturing geometrical objects is that we tend to see only their outsides, which hide what might be complicated within. Morin, who works with extremely intricate objects in three dimensions, has taught himself how to pass from outside to inside (from one “room” to another). “Our spatial imagination”, he says, “is framed by manipulating objects. You act on objects with your hands, not with your eyes. So being outside or inside is something that is really connected with your actions on objects.”5 Think of Emma sewing and knitting, or killing and dressing fowl.
Might Emma’s extraneous calculations have come from an intrusion of the time-like into her sounder space-like imagination—and is the tactile yet one more intermediate between Being and Becoming?
THE HAWK AND THE RAT
You may not have dwelt on how Emma Coolidge packed her hypotenuse square—a way which could, we showed, by itself have proven the Pythagorean Theorem. In fact, unknown to her, it had—some twelve centuries before in India, by an astronomer and mathematician named Bhaskara. He speaks only of a 3-4-5 right triangle, and after a contorted verbal description deigns to say: “A field is sketched in order to convince the dull-minded”, with this diagram:
The problems he gives show that Bhaskara knew the Theorem in broader terms, although this pastoral example is still in the 3-4-5 family:
A hawk was resting upon a wall whose height was twelve hastas. A departing rat was seen by that hawk at a distance of twenty-four hastas from the foot of the wall; and the hawk was seen by the rat [a feeling here for Nemesis as well as mathematics]. There, because of his fear, the rat ran with increasing speed towards his own residence which was in the wall. On the way he was killed by the hawk moving along the hypotenuse. In this case I wish to know what is the distance not attained by the rat, and what is the distance crossed by the hawk.j
Life is not only stranger than we think, it may (to paraphrase J. B. S. Haldane) be stranger than we can think. Five hundred years later, another and much more famous Indian astronomer and mathematician named Bhaskara (no relation)6 published the same figure (but flipped around the vertical axis and with the numbers and outside box removed) to accompany a proof of the general theorem.
In the intervening centuries, however, the dull-minded had also been flipped into the sagacious, for this later Bhaskara explains the proof briefly, then says that just seeing the figure suffices. (In fact, after a terse summary, he writes: “And otherwise, when one has set down those parts of the figure there, seeing.” Over the years this has been whittled down to the story that his proof consisted of no more than the figure and the single word, “Behold!”)7
When you do, you see Guido’s figure, built not on a square of side-length a + b but of c, the hypotenuse. A negligible change, you think, until you try Guido-like rearrangements. You may find this one,
but it isn’t nearly as pleasing as Guido’s, since it needs a new line imagined or drawn in, others removed, and some mental arithmetic, to see the squares on the sides. And yet there is that powerful proof of the
Theorem without any further drawing: the square’s area, c2, is made up of four right triangles, each of area , and that middle square, whose side-length is b − a. So
as desired.
To achieve this result, we moved not shapes but symbols around: we construed rather than constructed—and from this slim difference opens the gap between geometry and algebra, between visual and abstract math.
Not that this is yet algebra, since its hallmark x, the unknown, has still to make its appearance. To understand, however, what a deep chasm has just been leapt, we need to sharpen our seeing of a split that is usually ignored. On the one side is thinking that uses placeholders—what was once accurately but misleadingly called “literal arithmetic” (so in x + 3 = 5, x really means ‘What?’; it holds the place for the answer ‘2’, which immediately springs to mind). On the other side is algebra proper, with its unknown that itself later metamorphoses into a variable (in x + 3 = y, x and y vary together: if y were 5, x would be 2, but were y 13, x would be 10—that placeholder has become a variable). You see this distinction again in language, between names and nouns: to pass from bodies out there to names for them (‘Adele’ and ‘Barbara’ for the two girls whispering across the room, ‘3’ for those sticks, | / ), is already a full turn of abstraction up our spiraling architectural instinct. To enmesh these names as nouns in a grammatical matrix is a further turn, freeing them now to move together into unguessed configurations. A third, algebraic, turn will invite us to solve for what is still missing, just by grammatical rearrangements of our nouns (3 + x2 = 19—ah, x2 = 16, so x is 4 or −4). How do these changes happen—how do we think our way from 32 + 42 = 52 and 52 + 122 = 132 to the universal a2 + b2 = c2? Take the simpler case of area. We replace specific numbers by generic names (4 becomes the triangle’s ‘base’ and 3 its ‘height’) and describe its area no longer as but as ‘half the base times the height’.
Then, by what seems a trivial act of abbreviation but in fact resounds with consequences, we write: ‘A’, ‘b’, and ‘h’. We next clamp these symbols (somewhere between nouns and pronouns) together in a ‘formula’—a little form:
Only now can we hear and respond one further turn up to algebra’s call: if you knew that A was 6 and b was 3, what must h be? Grammar ignores the objects (in this case shapes) its nouns stand for and allows us to rearrange, wholly within this new linguistic context:
Haven’t we just found the proper context in which to see, as Parmenides did, that motion is an illusion and only Being is? Euclid constructed temporary scaffoldings. Later (in the proof that involved tiling the endless bathhouse floor), invoking infinity allowed us to keep the objects of mathematics still, while godlike thought hurried over them. Here those objects are if anything yet more at rest as abstraction grows, shape yields to form, and movement is confined to symbols in our language (or in our thought, if language faithfully mirrors it—or in both, as is the way with reflection).
Bhaskara’s diagram needed none of this third-storey work, but the grammar of these symbols (aptly called rules rather than laws of multiplication, subtraction, and distribution) sufficed on the second level to rewrite (b − a)2 as b 2 − 2ab + a2, which, along with grammatical steps on the same level, produced the Pythagorean Theorem from this orderly dance of signs. We are centuries past diagrams in the dust.
Are these what you said were no more than tautologies, Russell? Meaning is the aftermath of form. We want to see the hang of things, and abstraction’s up
ward path leads us, like Dante’s, to it. Yes, but at what cost? What have you relished more in this book so far, the glimpses of Hippasus and Miss Coolidge, of Loomis and Thabit, or the growing clarity of a once hidden harmonious structure? We can have both. The signs that decorate this structure pick out doors opening inward on the vast tower of time. Behind each one are singular people, odd events, histories of invention that led to discovery, and the marks that personalities have left on impersonal truth—which give even it a singular character too.
What would you do were you carry ing a precious platter, and tripped? Were you frivolous and French, and a footman in the service of the Sun King, Louis XIV, you would sweep the shards aside while an even more glorious platter, bearing an even more sumptuous bird, was brought in by the footman waiting in the kitchen for this planned accident of conspicuous consumption. Were you Greek, serious, and talking with Socrates, you would agree that the One had thus lightly become the Many.8 But were you Chinese and living a millennium ago, you would (the legend runs) have gathered up the fragments in a panic and tried to reassemble them into their original square shape—and failing, would have instead devised the thousand patterns of the Tangram: the puzzle that once rapt people away as fully as Sudoku does now.
The pieces, called tans, have sedately settled into seven fixed shapes: a pair of small and a pair of large isosceles right triangles; one more, of middle size; a square; and a disconcerting parallelogram—all made to pack into an attractive square box.
The game is to rearrange them all to make the countless different figures whose outlines only are in the booklet that came with your puzzle: a cat, say,
or a swan:
Shall we see if the Pythagorean Theorem is among them?
Here are the pieces in their box, making the square on the hypotenuse, and (if our given right triangle is isosceles), taken out of the box and arranged to make the squares on the two sides:
You may feel that the arbitrary number and shapes of a Tangram make this not so much a problem as a puzzle (is this another criterion to add to our classification of proofs—two genera that will each branch out from the families, with the hint of more finely twigged species in between?). And you may find it still more disappointing that we can’t generalize with tangrams to other right triangles. Well, were we not bound by historical convention, we could make these instead be our seven pieces:
For now they will miraculously do just what we had wished (once again, showing them packed and unpacked around the unnumbered right triangle):
Even more of a miracle—Behold! Isn’t this just Bhaskara’s figure? But with those two cuts giving us seven pieces from his original five, we can now do what he couldn’t and make the small and larger square without either invisible lines or mental arithmetic (we’ve shown here, in one picture, the seven pieces arranged to make the square on the hypotenuse, and, differently arranged, to make the squares on the two sides of the unnumbered right triangle).
This idea came, twenty-five years ago, from the Orientalist Donald Wagner,9 who later, however, found out that he had been preceded by a young German mathematician, Benjir von Gutheil, killed in the trenches of France in 1914.
If we look past the legend of the Tangram’s origin, and its slightly more probable source in furniture sets of the Song Dynasty, we may come on Liu Hui’s third-century A.D. commentary on a text two or three centuries earlier, The Nine Chapters on Mathematical Art. The Pythagorean Theorem, for a 3-4-5 right triangle, plays a large part in it, and although none of the diagrams that were probably in Liu Hui’s commentary have survived, here is a playful attempt to reconstruct them.10 Like one of those antique inlaid puzzle boxes, it is made with sliding panels and hidden pressure-points.
Given that 3 and 4 were important numbers in ancient China, since 3 was taken to be the circumference of a unit circle while 4 was the perimeter of a unit square,11 it seemed reasonable to look for a circle and square inside a 3-4-5 right triangle—and this is what D. G. Rogers did. The Tangram-like interpretation of Liu Hui that he came up with in fact generalizes to any right triangle.
In right triangle ABC, bisect angles A and B, and let these bisectors meet at J (the center of the triangle’s incircle—the circle, that is, tangent to the triangle’s three sides—our hidden pressure-point).
Drop perpendiculars from J to the sides, meeting BC at E, AB at F, and AC at G. Number the resulting triangles as shown.
1 2, 3 4 (by shared sides, right angles and the equal, corresponding, bisected angles). Hence JG = JF = JE = r, the incircle’s radius. Let 2r = d, this circle’s diameter.
You see that JG = JE makes 5 a square.
Taking 1, 2, 3, 4, and 5 as our tans, slide these panels like this:
Make three more copies of this tangram and put all four together:
Note the lengths a, b, and c, as shown; call the height of this rectangle d.
The area of this inlaid rectangle is d (a + b + c), but since each of the four blocks composing it has the area of the original triangle, namely , we have
We want to unlock this box, so, sliding the panels in the lower left rectangle only, we have
Comparing the lower edges of our two four-block formations, we see that
Equating (1) and (2),
Are you giddy with all this prestidigitation, or is that a sudden feeling of déjà vu? Didn’t we see something very like this back in Loomis’s proof of February 23, 1926? There he used only the bisector of A and the perpendicular to AB, from the intersection of the bisector with BC, and finished his proof with no more than three tans and a fourth of the steps. And we thought that was baroque!
But if you want déjà vu all over again, isn’t this also Guido’s proof, with an intaglio of extra lines? Rogers himself rearranges his twenty tans in those tried and true ways:
We may not have to push our invention quite this hard in order to reconstruct what the Chinese mathematicians of the third century A.D. had in mind. A 1213 edition has survived of a commentary by Liu Hui’s contemporary Zhao Shuang on The Gnomon of Zhou (a book even earlier than The Nine Chapters), with this “Figure of the Hypotenuse”,12 its four vermilion triangles surrounding the solitary central yellow square. Look familiar?
With all sorts of communication over the centuries having been possible between the Far East and India, would you want us now to revise our attribution of this figure from Bhaskara to The Gnomon of Zhou? There is the slight problem that the Chinese stuck resolutely to 3-4-5 triangles, while Bhaskara, by avoiding specific numbers, made his proof work for any right triangle at all. More to the point, this question of chest-thumping priority detracts from what counts in mathematics as much here as it did among the Pythagoreans and Greeks. Every attempt to shore up a proof by invoking remote authority weakens the innate validity of its impersonal argument. Add, besides, the evidence of Miss Coolidge having given parthenogenetic birth to this figure: it can arise independently, and doubtless has, time and again, being one of the characters in the abstract repertory company that performs our inner commedia dell’arte.
Einstein, after all, came up as a boy with a proof via that other mask, Pythagorean proportions:
I remember that an uncle told me the Pythagorean theorem before the holy geometry booklet had come into my hands. After much effort I succeeded in “proving” this theorem on the basis of the similarity of triangles . . . for anyone who experiences [these feelings] for the first time, it is marvelous enough that man is capable at all to reach such a degree of certainty and purity in pure thinking as the Greeks showed us for the first time to be possible in geometry.13
Does it take anything away from his discovery that others had made it before him? Georg Christoph Lichtenberg once wrote: “What you have been obliged to discover by yourself leaves a path in your mind which you can use again when the need arises.” As, for Einstein, it did, and he did.
Of much greater significance in this Chinese “figure of the hypotenuse” is what it tells us about the risks with which we use e
xamples. When we point, we want others to look where we’re pointing, not at the end of our finger. One problem is the nervous viewer’s eagerness to take us exactly at our word—but another is the speaker’s failure to imagine how many types of ambiguity he is open to. Drawing these 3-4-5 right triangles in a 7 × 7 grid might have been meant to stand for any right triangles, with the grid no more than a concession to artistic taste, architectural convention, and printers’ convenience.
What comes across from text and commentary, however, is that the grid was essential, and that the diagram meant exactly what it shows: the Pythagorean relation belongs to this single case, which is also singular, given the mathematical importance of 3 and 4.k This is no mere example but The Exemplar. It isn’t meant to conjure up differently proportioned right triangles—only perhaps to remind us that 3, 4, and 5 model the order of the world, and that balance within each family, in all the vermilion land, is guaranteed by the Emperor, in his central Yellow Palace, having the mandate of heaven.
Our common humanity makes it hard to imagine that another society might have weighted its emphases so differently from ours as to think that looking at an isolated instance, the 3-4-5 right triangle, showed what was true for all—and that this “all” wasn’t other instances but algorithms based on manipulations on and with it. The diagram instantiated the algorithms. “The key fact”, says the historian Karine Chemla, “is that these figures form the basis for the whole development in the following sense: all the algorithms placed after the figures derive from them, in that the reasons for their correctness are drawn from these figures and only from them.”14l