Numbers of Sides
GUASPARI, DAVID
Numbers of Sides ‘Many cheerful facts about the square of the hypotenuse,’ and beyond. BY DAVID GUASPARI The Pythagorean Theorem is perhaps the one mathematical fact an Average Joe might be...
...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...
...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...
...He begins, regrettably, with a sin of anachronism—miscasting the original meaning of the Theorem into modern terms...
...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...
...Maor ventilates these stories with frequent digressions...
...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...
...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...
...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...
...BY DAVID GUASPARI The Pythagorean Theorem is perhaps the one mathematical fact an Average Joe might be able to name...
...Yet the massive triumphs of mathematical physics, for one thing, assure us that there can be...
...minating 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...
...That assigns a number to each side and those numbers will satisfy a2+b2=c2...
...There is the famous story of Thomas Hobbes’s exclamation, on fi rst seeing the Pythagorean Theorem: “By God, this is impossible...
...and so on...
...We may meaningfully compare one line segment to another line segment (is it greater...
...and from the Han Dynasty...
...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...
...We choose a unit and measure the line as some multiple of the unit—at least, when it comes out exactly...
...Maor treats these as interchangeable formulations, and from the modern point of view they are...
...If the ratio between the lengths of those parts is two to one, the pitch difference is an octave...
...Evidence of the Pythagorean Theorem can be found on Babylonian clay tablets from 1800 B.C...
...Bits of potted history serve as glue...
...This strategy fails, for an astonishing reason: The innocent assumption that we can always fi nd such a unit is false...
...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 Garfi eld (before his election...
...In other settings—fi gures drawn on the surface of a sphere, for example—it fails...
...It is ancient...
...But it seems merely confused to speak of multiplying one line segment by another, of multiplying by something that is not a multiplicity...
...The fi rst section of Maor’s book stretches from the Babylonians to Archimedes, the greatest ancient mathematician, and one of the greatest ever...
...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...
...It makes sense to multiply numbers, obtaining another number as a result—three groups of four things amount to 12 things altogether...
...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...
...Multitudes differ essentially from magnitudes...
...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 fi rst principles—has emerged only once in human history...
...They distinguished numbers, which are “multitudes” (that can be counted), from lengths, areas, and volumes, which are continuously varying “magnitudes...
...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...
...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 illuDavid Guaspari is a writer in Ithaca, New York...
...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...
...The fi rst rigorous proof is ascribed to Greeks of the school of Pythagoras, in the mid-6th century B.C...
...The Pythagorean Theorem holds for fi gures drawn on a fl at surface—that is, for the objects of Euclidean geometry...
...One (rather dull) chapter called “The Pythagorean Theorem in Art, Poetry, and Prose” provides a laundry list...
...Presented with these careful distinctions, and the rigorous and brilliant Greek science that respected them, a reader might suffer a profi table 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...
...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 spacetime” of Einstein’s theory...
...Plato’s interest in geometry was metaphysical: The relation between ideal geometric fi gures (grasped by reason) and the imperfect copies that we draw or otherwise encounter through our senses prefi gures the relation between a Platonic form, such as Goodness, and its imperfect realizations in the world of ordinary experience...
...The Pythagoreans not only discovered that but proved it...
...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...
...Here shines one particular brilliance of Greek mathematics: that its results are established by proof...
...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...
...The fi nal section begins in the mid16th century with Fran?ois Vi?te—often regarded as the fi rst modern mathematician—and tells two related stories...
...But Pythagoras and Euclid would fi nd the modern version unintelligible, for reasons interesting and deep...
...There are encomia to Pythagoras from Johannes Kepler and (descending from the sublime) Jacob Bronowski...
...It is not likely that Plato was fond of diseased democracy, but safe to say that promoting democracy was not one of his concerns...
...This seems to have received powerful support from the discovery, attributed to Pythagoras, that basic musical intervals are “rational...
...versions exist in manuscripts from India circa 600 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 McGuffi n, for touring a swath of mathematical history...
...if two to three, it’s a perfect fi fth...
...One is the introduction of infi nite methods and infi nities into mathematics—controversial but successful innovations that had to wait 300 years for a rigorous basis...
...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...
...And magnitudes themselves come in different kinds...
...For example, there is no unit of which both the sides and the diagonal of a square are exact multiples...
...Stop a tensed violin string somewhere in the middle and consider the difference between the pitches produced by plucking its two parts...
...He aims at the general reader, wishing to provide both an intellectual adventure, complete with proofs, and a genial ramble...
...For the Pythagoras cult, this had a tragic aspect...
...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 Theorem is a characteristic of fl atness, hence its ubiquity: The calculations of trigonometry, of the lengths of lines (straight or curved), etc., are all intimately tied to it...
Vol. 13 • February 2008 • No. 23