Numbers of Sides
'Many cheerful facts about the square of the hypotenuse,' and beyond.
Feb 25, 2008, Vol. 13, No. 23 • By DAVID GUASPARI
The Pythagorean Theorem
The Pythagorean Theorem is perhaps the one mathematical fact an Average Joe might be able to name. It is ancient. Evidence of the Pythagorean Theorem can be found on Babylonian clay tablets from 1800 B.C.; versions exist in manuscripts from India circa 600 B.C. and from the Han Dynasty. The first rigorous proof is ascribed to Greeks of the school of Pythagoras, in the mid-6th century B.C.
Eli Maor says that the Pythagorean Theorem is "arguably the most frequently used theorem in all of mathematics" and makes that the premise, or McGuffin, for touring a swath of mathematical history. He aims at the general reader, wishing to provide both an intellectual adventure, complete with proofs, and a genial ramble. (An appended chronology notes that soon after "Einstein publishes his general theory of relativity . . . Stanley Jashemski, age nineteen, of Youngstown, Ohio, proposes possibly the shortest known proof of PT.")
He begins, regrettably, with a sin of anachronism--miscasting the original meaning of the Theorem into modern terms. Euclid's famous treatise on geometry presents us with a fact about area: If we draw a square on each side of a right triangle, the area of the square on the hypotenuse is the total of the areas of the squares on the other sides. Nowadays we are inclined to express this as an equation--a2+b2=c2--in which a, b, and c are numbers representing the lengths of the triangle's sides. Maor treats these as interchangeable formulations, and from the modern point of view they are.
But Pythagoras and Euclid would find the modern version unintelligible, for reasons interesting and deep. They distinguished numbers, which are "multitudes" (that can be counted), from lengths, areas, and volumes, which are continuously varying "magnitudes." Multitudes differ essentially from magnitudes. And magnitudes themselves come in different kinds. We may meaningfully compare one line segment to another line segment (is it greater?) but not to a different kind of magnitude, such as a circle or a cube.
It makes sense to total the magnitudes of two squares, but not to total a square with a line. It makes sense to multiply numbers, obtaining another number as a result--three groups of four things amount to 12 things altogether. But it seems merely confused to speak of multiplying one line segment by another, of multiplying by something that is not a multiplicity.
Presented with these careful distinctions, and the rigorous and brilliant Greek science that respected them, a reader might suffer a profitable moment of uncertainty and discomfort, wondering how he could have thought in any other way--uncertain, at least for that moment, how there could be any coherent sense in (or any use for) some mongrel notion of "number" and practice of "algebra" that embraced the counting numbers and magnitudes of all kinds. Yet the massive triumphs of mathematical physics, for one thing, assure us that there can be.
We can't solve that problem here--to begin with, a rigorous mathematical account of the modern notion of number is highly technical--but it is illuminating to consider a simple strategy that holds out hope of dissolving it: In ordinary speech we don't say that the length of a line is "three"--we say that it's "three feet" or "three furlongs" or some such thing. We choose a unit and measure the line as some multiple of the unit--at least, when it comes out exactly.
That suggests a way to unify the distinct quantitative ideas of multitude and magnitude case-by-case: Given a right triangle, for example, choose a unit of which all three sides are exact multiples. That assigns a number to each side and those numbers will satisfy a2+b2=c2.
This strategy fails, for an astonishing reason: The innocent assumption that we can always find such a unit is false. For example, there is no unit of which both the sides and the diagonal of a square are exact multiples. The Pythagoreans not only discovered that but proved it. Here shines one particular brilliance of Greek mathematics: that its results are established by proof. And so far as we know, the notion of mathematical proof--of developing an entire body of knowledge by rigorous deduction from a set of first principles--has emerged only once in human history.
For the Pythagoras cult, this had a tragic aspect. Scholars dispute about the precise beliefs of Pythagoras and his followers, but agree that they included a mystical conviction that numbers (multitudes) are, in some sense, the fundamental constituents of the world.
This seems to have received powerful support from the discovery, attributed to Pythagoras, that basic musical intervals are "rational." Stop a tensed violin string somewhere in the middle and consider the difference between the pitches produced by plucking its two parts. If the ratio between the lengths of those parts is two to one, the pitch difference is an octave; if two to three, it's a perfect fifth; and so on.
Note that the ratio of two lengths is the same as the ratio of two counting numbers precisely when there is a unit of which both lengths are exact multiples. An irrefutable proof that the sides and diagonal of a square are, in this sense, "irrational"--and that irrationality is an essential feature of the mathematical world--can only have been a metaphysical blow.
The first section of Maor's book stretches from the Babylonians to Archimedes, the greatest ancient mathematician, and one of the greatest ever. Insofar as it has a single theme, this section asks which societies knew of the Pythagorean Theorem, and in what form and in what way they knew it. The next thousand or so years get brief treatment as an interlude, an era of "translators and commentators"--illustrated by episodes from Chinese, Hindu, and Arabic, as well as Western, mathematics.
The final section begins in the mid-16th century with François Viète--often regarded as the first modern mathematician--and tells two related stories. One is the introduction of infinite methods and infinities into mathematics--controversial but successful innovations that had to wait 300 years for a rigorous basis. The other develops the "non-Euclidean" geometry that plays a central role in modern physics as the mathematical setting for Einstein's theory of general relativity.
The Pythagorean Theorem holds for figures drawn on a flat surface--that is, for the objects of Euclidean geometry. In other settings--figures drawn on the surface of a sphere, for example--it fails. The Theorem is a characteristic of flatness, hence its ubiquity: The calculations of trigonometry, of the lengths of lines (straight or curved), etc., are all intimately tied to it. The converse insight, that the geometry of a surface can be captured by describing the ways in which it deviates from the Pythagorean Theorem, makes it possible to represent and reason about unvisualizable geometries such as the "curved space-time" of Einstein's theory.
Maor ventilates these stories with frequent digressions. One (rather dull) chapter called "The Pythagorean Theorem in Art, Poetry, and Prose" provides a laundry list. There is the patter song from The Pirates of Penzance in which the modern major general boasts of his acquaintance "with many cheerful facts about the square of the hypotenuse." There are encomia to Pythagoras from Johannes Kepler and (descending from the sublime) Jacob Bronowski. There is the famous story of Thomas Hobbes's exclamation, on first seeing the Pythagorean Theorem: "By God, this is impossible!"
Another chapter presents excerpts from the curious life work of Elisha Scott Loomis, who undertook to gather all known proofs of the Pythagorean Theorem--of which he found 371, including one by President James Garfield (before his election). Also included are brain teasers, mathematical curiosities, and a short essay on the possibility of composing a message that would be understood by intelligent life in distant galaxies.
Bits of potted history serve as glue. Not all of this is reliable, as when Plato's contribution to geometry is described as "his recognition of its importance to learning in general, to logical thinking, and, ultimately, to a healthy democracy." It is not likely that Plato was fond of diseased democracy, but safe to say that promoting democracy was not one of his concerns. Plato's interest in geometry was metaphysical: The relation between ideal geometric figures (grasped by reason) and the imperfect copies that we draw or otherwise encounter through our senses prefigures the relation between a Platonic form, such as Goodness, and its imperfect realizations in the world of ordinary experience.
It is hard to predict who would be charmed by The Pythagorean Theorem, but all will recognize the author's enthusiasm for his subject and his respect for the reader: Three cheers for including those proofs!
David Guaspari is a writer in Ithaca, New York.