by Ellen Kaplan
How to embody the general accurately yet suggestively: that was the great leap from the specific diagrams of the Babylonians to the unmarked figures of the Greeks. While exemplary thought may promote a spread of relations from those given, it is justified in broader contexts only by the validity of these relations in the narrow original.
We turned away from Liu Hui because Zhao Shuang’s commentary seemed to give us more of an insight into the thought of the time—but it would be wrong to sell short a scholar of Liu’s intelligence. He worked within a tradition so respectful of its ancients that correcting them wasn’t acceptable. Liu may well have understood that the structure of this proof supported its generalization to all right triangles, but would have had to express himself in written or spoken asides.15
Loomis interrupts the march of his crabbed diagrams suddenly, after his Geometric Proof 52, which bristles like a hedgehog:
He writes:
This figure and proof is taken from the following work, now in my library, the title page of which is:
Euclides Elementorum Geometricorum
Libros Tredecim
Isidorum et Hypsiclem
& Recentiores de Corporibus Regularibus, &
Proclii
Propositiones Geometricas
The work from which the above is taken is a book of 620 pages, 8 inches by 12 inches, bound in vellum, and, though printed in A.D. 1645, is well preserved. It once had a place in the Sunderland Library, Blenheim Palace, England, as the book plate shows—on the book plate is printed—“From the Sunderland Library, Blenheim Palace, purchased, April, 1882.” I found the book in a second-hand book store in Toronto, Canada, and on July 15, 1891, I purchased it. E. S. Loomis.
The work has 408 diagrams, or geometric figures, is entirely in Latin, and highly embellished.
Ah, Loomis, Loomis.
Take that tumble of right triangles you saw in Chapter Three. What if each had been scaled up from the first (which has sides a, b, and c)—one of them by a, one by b, and one by c:
Since the first and the second now each have a side ab, paste those sides together to form a new right triangle with legs ac and bc, and hypotenuse a2 + b2. But the third right triangle, scaled up by c, also has legs ac and bc, with hypotenuse c2. Since these two triangles are congruent (by S.A.S.), their hypotenuses are equal, so a2 + b2 = c2.
A miraculous proof.16 Once again, construing a mute figure in two different ways makes it speak. For all the translucency of Guido’s and the old Pythagorean inward construction, is this outward expansion instead the most transparent of all?
By contrast, we came on a proof in a professorial note titled “Pythagoras Made Difficult”,17 with the writer’s reflection that “If you are feeling particularly hostile toward the whole universe and you want to do something evil, show this argument to your calculus students and tell them they need to learn differential equations to understand how to prove the Pythagorean theorem. With any luck, half of them will believe you.”
Whether or not you were Professor Hardy’s student, if you are no friend of differential equations, heed the epitaph W. B. Yeats wrote for himself: Horseman, Pass By!
Otherwise, consider this diagram with the familiar beginning of the alphabet replaced by the variables of calculus, x, y, and z.
Look at a very small triangle similar—with a flip—to the whole, tucked into its left-hand corner (this triangle is cut off with a stroke perpendicular to y):
Its side-lengths, in calculus-speak, are dx and dy: increments that will tellingly dwindle to nothing. The similarity of the small to the large triangle gives us (short/long):
Using the powerful technique called “separation of variables” (miracled up in the seventeenth century by Guillaume François Antoine, le Marquis de l’Hôpital), we can rewrite this equation as
and integrate:
where K is the “constant of integration”. Being a constant, its value doesn’t change as those of x and y shrink, so that, at the limit when x = 0, y2 = K.
But look at the diagram: as the side x goes to 0, y collapses onto z—that is, when x = 0, y = z, hence y2 = z2.
This means that the constant K is now and always has been z2, so
y2 = x2 + z2.
“There now, wasn’t that easy?” as the Contortionist said to the Fat Lady.
HILBERT IN PARIS
What we gain in new proofs we lose in attempts to classify them, trawling with anti-torpedo nets for minnows. Perhaps, though, we needn’t carry rigor as far as mortis in order to satisfy our legal longings and understand better what we want of a proof. Tying an impersonal truth to the axioms of our making?m Certainly. Doing this gracefully, and with the sudden surprises that mark revelation? Yes—so that to Bhaskara’s “Behold!” the Miss Coolidge we shelter answers, “I see!”
In 1900 David Hilbert announced twenty-three problems at the International Congress of Mathematicians in Paris. These picked out the boundaries of mathematical inquiry for the century ahead (ten have since been solved, five not; the rest swelter in various kinds of limbo). A twenty-fourth, which never made it past his journal, asks for a way of determining the simplest proof of a theorem (he cites the Pythagorean)—as by counting the number of steps needed, once the chain of reasoning has been methodically reduced.18
By asking for the simplest, and asking in these terms, Hilbert confronted the formal and the intuitive. But are simplest and shortest really the same? Hasn’t many a shining explication of complexity emerged from mazy wanderings, and elegance, as in music, from prolonging a single note into a trill? Would we have noticed the profundity of 32 + 42 = 52 had it dawned on us as negligently as 3 + 4 = 7?
The question of simplicity isn’t itself simple. It is inextricably bound up with the vertiginous issues of consistency and completeness, and of judging whether the truth of a mathematical statement can be decided mechanically. These are fraught issues, since the axiomatic foundations of the rigorous formal methods for laying bare the armature of such statements can, ironically, themselves be no more than intuitively convincing (or should their fecundity, say, rather than their clarity, be the criterion for choosing these axioms?). Should every step along the way have a finite character, or must “ideal” statements (i.e., statements about infinite collections) be added to them? Simplicity is the very devil.n
Categorizing may cause what is ultimately a single proof, pure as a diamond, to seem like a multitude, as we are dazzled by the shimmer reflecting off its facets. Not only does this hide from us a One behind the Many, it keeps us from seeing the endless subtle issues at play in the background, such as (with the Pythagorean Theorem) the nature of area and the architecture of motions on the Euclidean plane, and in general the opposite pulls of abstraction and embodiment.
So if we are not to categorize proofs of the Pythagorean Theorem by some scheme or other, nor dare to arrange them by degrees of simplicity, aren’t we led to cultivating our taste for them in terms of what light they shed on their locales, whose shadowed depths in turn develop them?
The Greeks, we said, believed that a rare mortal might touch the bronze sky for a brief moment. Twenty-five hundred years ago the Cretan runner Ergoteles won the long race at the Olympic Games, and was the glory of his time. In 1924 Mr. Ericson, the math teacher at Milwaukee’s Washington High School, told his junior class that if any of them could work out a new proof for the Pythagorean Theorem, he would see to it “that the matter would be properly advertised in the local newspaper.” After a time Alvin Knoerr came forward.
“Although I never expected to be able to develop a new proof,” Knoerr later wrote,19
I tried a little experimenting a few days later just to pass time. This proved so interesting that I continued working out all kinds of ideas. The usual result of fifteen or twenty minutes of work, however, was that the solutions would boil down to an identity, thus bringing me back to the original starting point.
After experimenting in this manner for about a month I began to realize tha
t my efforts were useless. Finally I struck upon a plan which seemed to me to be far more logical than others because instead of starting from the beginning I started from the end and took the fact that the square on the hypotenuse is equal to the sum of the squares on the other two sides for granted.
Then I broke this equation up into several other equations and finally ended up in a proportion. We had just finished similar triangles so the proportion which I had developed suggested the usage of similar triangles. Then after I had the triangles constructed it was a matter of only an hour or so to work out the proof.
Here is what Alvin Knoerr came up with, simplified a bit from his Rube Goldberg original.
Given: ACB has a right angle at C.
AB is extended to E, so that CB BE;
point D is placed so DB CB BE.
Alvin Knoerr’s reminiscence is the only record we have found of the creative work behind a proof of the Pythagorean Theorem. How representative of others was the movement of his thought back from conclusion to premises, then forward again—the method often called “analysis and synthesis”? Removing the scaffolding before unveiling the building heightens the jolt that joins the true to the beautiful, but loses us the contemplative pleasures of watching the mind shape and mortar its bricks.
What became of Ergoteles after he won his town the race? He was granted citizenship in the Sicilian city of Himera, where he had fled after an uprising in Cretan Knossos, and later gained the coveted right to own land. And Alvin Knoerr? We catch a glimpse of him in 1936, tramping out a gigantic M on a snowy hillside in Wisconsin, to prove that the projected monument to the School of Mines, where he was a student, would be visible from distant Platteville. Is that him, prospecting for atomic minerals after the war? He edited the Engineering and Mining Journal in the ’sixties, writing that most research “is nothing more than searching for ideas that have already been set down in print.” He died, eighty-eight years old, in 1995, in Queens, New York. His hometown historian could find no traces of him. He has even been deleted from Wikipedia.o
Here is a proof with a hypermodern flavor, outflanking the Theorem. It, too, cunningly flirts with, then slips past, trigonometry.
A right triangle is uniquely determined by the length of one side and the size of an acute angle. Let’s choose c as our side and the smallest
angle, ø. We know this triangle’s area is
Let’s look at areas again:
What happened? The pea must have been under one of those shells, and now it’s gone!
This is an instance of ‘dimensional analysis’, where we needn’t inquire into the nature of the function f, just glean the desired result from this way of looking, which dates back to the physicist Percy Bridgman in the 1920s.20 It allows you to solve a problem by attending to no more than the dimensions of its variables (thus finding easily, for example, that the drop in pressure of a fluid flowing through a pipe is proportional to Q/R4, where R is the pipe’s radius and Q the volume of flow in a unit of time). You might think of this proof as a streamlined version of the Pythagorean, sped up by removing all the finicking Euclidean constrictions.
Having crossed the divide from mathematics to physics, let’s linger a moment to relish a proof that depends on the impossibility of perpetual motion. Make yourself a box with parallel congruent right triangles ABC for top and bottom, and with side height h. Fill it with gas at pressure p, seal it closed, then hinge it at corner A to a vertical pole, around which it can pivot freely.
We define torque, T, as the twisting ability of a force F on a lever of length k about a point A: T = F · k. These levers will be the faces of our box. As good Newtonians, we make these assumptions:
1. There is no perpetual motion.
2. A gas pushes equally on all the faces of its container.
3. The force F with which the gas pushes on a face is concentrated at the midpoint of that face.
4. This force is the product of the gas pressure, p, times the area of that face.
Let’s now calculate these forces. We’ll ignore those acting on the top and bottom faces (those right triangles), since they are equal and opposite, hence cancel each other out.
The force acting on the AB face, with area h · AB, is
As you see from the diagram, this force will act to rotate the box clockwise.
The forces acting on faces AC and BC try to rotate the box counterclockwise.
Finally, since C is a right angle, we can in our minds slide BC down along AC to A without changing its angle, so that the force FCB, acting on CB’s midpoint, indeed gives us the torque TCB around A:
Now here is the crux of the argument. Because we assume that there can be no perpetual motion, the clockwise and counter clockwise torques must balance:
That is,
and simplifying, AB2 = AC2 + CB2, as we had hoped.
Toto, where are we? Very far from the fruited plains of mathematics. All that talk of circular motion may stir up fears of a deeper circularity: not that the Pythagorean Theorem might already lie in the handling of vectors here, but that the axioms of physics invoked may reflect an underlying geometry, as if we were deriving this proof not from axioms at all but from their consequences. That would make the result not so much a proof as an illustration—a drawing-down of abstract structure into the visible. And what exactly is the turn-and-turnabout relation between physics and mathematics: do the insights of each propel the other? Are they hopelessly or hopefully entangled? May an anti-cyclone whirl us home again.21
LE VASISTDAS
A curious French name for a small window that can be cranked open, like a transom, is le vasistdas, from the German Was ist das?, “What is
that?” (claimed to be what provincial German soldiers said when they first saw these transoms, in 1870s Paris). You might want to call this diagram a vasistdas:
What does it look out on, with its strangely leaded lights? Well, take this square by its top edge and bend it down and around into a cylinder, gluing top to bottom:
Next take what were its left-and right-hand sides—now the left and right circles at the ends of the cylinder—and glue them together, having bent the cylinder around into a doughnut (which has the
glorified name of ‘torus’). What have you now? That trapezoid and those triangles have come together into two squares, so that the initial square (which you could take as having been that on the hypotenuse) is fully covered by the squares on the two legs! This is the other world that window disclosed, the farthest fetch of a proof, it seems, by way of mapping the parts onto a torus made of their whole.22 When the Cheshire cat disappeared, at least it left its smile behind. Here the original right triangle has vanished into the doughnut’s hole.
We’ve sailed past so many islanded proofs—stopping to dine on some and waving at others from afar—that we begin to feel as legendary as Odysseus. Is that Samos over there, where Pythagoras was born? Kepler thought that the soul of Pythagoras might have migrated into his own. But are all these proofs, traveling abroad from the inland sea, not his true reincarnations?
CHAPTER SIX
Exuberant Life
The Pythagorean Theorem is an ancient oak in the landscape of thought. We’ve traced many of its roots down to where they clutch the rocks of our intuition—those proofs that in fact make it a theorem. But trees have their branches in the air, growing more luxuriantly in the direction of light and open spaces, more densely toward neighboring woods, creating the local chaos and changing symmetry that mark living things.
The tree of man was never quiet, nor are the trees in the forest of his certainty, taking the least opportunity to broaden and so conquer more of thinking’s space. Here we’ll follow some striking ways that the Pythagorean Theorem has spread out. First, though, since generalizing may precede as well as follow, let’s look at how the Pythagorean relation can be drawn from one yet broader.
Its source is Ptolemy, whom you met in Chapter Two with his wonderful method of approximating square roots by gnomon
s. He was a Roman citizen in the second century A.D., a member of Alexandria’s lively Greek community, steeped in Babylonian data, and called in later Arabic sources “the Upper Egyptian”—more a confluence than a person. His Almagest (the Arabic rendering of the Greek Syntaxis as “The Great Work”) established a geocentric model for astronomy on the basis of detailed mathematical research—which included what came to be called Ptolemy’s Theorem.
For Ptolemy saw that if a quadrilateral ABCD were to be inscribed in a circle,
along with its diagonals, then the product of (the lengths of) these diagonals would equal the sum of the products of the opposite sides:
This hardly leaps off the page at you, so that just seeing it is already worth a niche in the mathematical Pantheon. Much less obvious is how to prove this—which Ptolemy did with astonishing economy (and this leads the sun to shine on his bust). He simply drew a line from A to BD, meeting it at E, making BAE CAD.