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Euclid in a Nutshell

By: Eva Brann
March 28, 2014

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A review of The King of Infinite Space: Euclid and His Elements, by David Berlinski

There are hosannas due in heaven—or better iai iais on Olympus—for a book in praise of Euclid, especially one as accessible, amusing, and enthusiastic as David Berlinski’s. “Euclid is universally acclaimed great,” he begins, and thus he continues in his chapter headings: “The Greater Euclid,” “Euclid the Great.” Just so students at St. John’s (where I teach), who study the 13 books of Euclid’s Elements for much of their freshman year, tend to begin their first papers; these are often on the beautiful narrative told by the propositions leading to the “Pythagorean theorem,” which is the climax of the first book: “In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle.” They get chided in their paper conferences for indulging in so vapid—because unearned—a judgment. Berlinski, a senior fellow of the Discovery Institute’s Center for Science and Culture, however, is a mathematician and knows whereof he speaks. Euclid wrote, he says, “by far the most successful of mathematical text books,” most of which rightly “have a short and ignominious life.”

Textbooks are usually pre-cooked, boiled-down presentations of other people’s fresh inventions—desiccated originals. Not so Euclid’s. While much of the greatest mathematics in the Elements, including the very notion of “element” (basic, as distinct from “elementary,” simple), is well-known to have been discovered by Euclid’s predecessors (though some of its wonderful individual propositions are probably his), he is undoubtedly the author of that great composition as a whole. The title “Elements” betokens: choice of beginnings, logical sequence, economy of theorems, dramatic culmination. What those terms describe is a system: Euclid, Berlinski points out, shows us that the mathematical insights he draws into his book are interconnected, that there is “mathematics,” a tight-woven realm of related “learnables,” as the Greek word, mathematika, signifies, a world for which Euclid has found deft beginnings and dramatic directions.

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Such system-making is anything but a grinding of the logical crank; it requires artful choice. As Berlinski puts it, the Pythagorean theorem (“in right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle”), which is the climax of the first book, is “the same” as the Fifth Postulate, the most notorious of Euclid’s beginnings. Thus Euclid might have, it seems, begun with the theorem and deduced the postulate.

He divides his unproved, presumably unprovable, beginnings into “postulates,” meaning demands or requirements, and “common notions,” also called “axioms,” meaning “what is worthy” of belief. These two sets of fundamentals are often collapsed into one set of axioms, but Euclid’s distinction matters. The common notions—for example, that “equals added to equals are equal”—are generally plain to everyone, except perhaps the fifth and last one: “The whole is greater than its parts.” Our students, on easy terms with “actual,” that is, realized, accessible infinity, will quickly point out that the whole of the counting-numbers is in fact equal to each of its halves, the odd and the even part. But thereby hangs a tale, to be taken up in a moment.

The postulates, however, are not so easily accepted, and so Euclid demands belief for them, especially for that notorious fifth which says, in essence, that if two lines fall on a straight line, so that the inside angles together are less than a right angle, then they’ll meet on that side (and thus form a triangle). In fact, this postulate is so opaque, not, to be sure, as a geometric insight but as an unprovable truth, that it is at the center of one of the most remarkable episodes in the modern development of mathematics. The intended disproof of the Fifth Postulate, which was meant to fail and in failing became a proof of its unprovableness, unwittingly produced propositions of a first non-Euclidean geometry, based on the denial of the postulate.

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But imagine Euclid beginning with the Pythagorean theorem as an axiom: if the Parallel Postulate is hard to swallow, what unperspicuous unsightliness would have ensued, while the student unpacked all the theorems leading up to it! So when Berlinski says that the beginning and the end of Elements I are the same, that’s a little over-stated; they aren’t even equivalent, since the theorem is so much overtly richer than the postulate, whose powers are purely implicit. That’s Euclid’s art precisely: to demand of the student acceptance of a postulate that is obvious visually, puzzling logically, and astoundingly potent. For it enables a whole series of propositions of which the Pythagorean theorem is but the crown.

Berlinski translates this climactic Pythagorean theorem into algebra: When a, b, and h are the measured lengths of the sides of a right triangle, then a2 + b2 = h2, where h measures the hypotenuse; a numerical example is 32 + 42 = 52. When 3, 4, 5 are numbers measuring the lengths of three lines, the latter will come together as a right triangle; this is the converse of the Pythagorean theorem.

Now as Berlinski points out, “The apparatus of modern algebra was not available to Euclid.” The interesting fact here is that, although Euclid gives clear evidence of thinking of magnitudes in general, that is, of relations that hold for geometric quantities as well as for numeral multitudes, there is not even the unborn ghost of a general magnitude in evidence within the Elements. General magnitudes are the unspecified quantities that the “algebraic apparatus” of x, y, z works with—abstracted, secondary notions. In fact, the translation of Euclidean theorems into algebra is a latter-day imposition that fudges one of the most amazing developments of mathematics, the modern melding of geometry and arithmetic in symbolic algebra, whose letters stand indistinguishably for magnitudes and multitudes. Euclid’s generality, however, covers, under one theory of proportions, two fundamentally distinct kinds of amount: geometric and arithmetical. Ancient geometry and arithmetic thus preserve a vestigial concreteness that is not so much primitive as determinedly immediate to sensory life.

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Here is an oddity of Berlinski’s book, which shows up in three respects. First, the 13 books of the Elements do not only tell stories in parts, book by book, but as a whole: The first book is essentially the tale of the transformation of areas; the Pythagorean case is just the most gloriously ingenious of such transformations—the addition of two square-shaped areas into a third similar shape. The first of these area transformations (Theorem I 35) is so counterintuitive that the ancients thought of it as paradoxical. It proves that two parallelograms that stand on the same base, and whose tops lie on one and the same line parallel with the base, are equal in area: Thus, astoundingly, the area of two figures with vastly unequal perimeters can be equal (figure 1).

Figure 1:

Euclid, wisely, gives no definition of area, a terminally mystifying, transformable something—some strange geometric stuff underlying our figural geometric imagination. You can see that the theorem demands the existence of parallel lines as Euclid defines them: as lines that, when produced indefinitely in either direction, do not meet. Euclid does not mean, as we tend to do, that if you leap in thought to infinity, there and only there they’ll meet. He rather means that you may project your imagination as far as you please, and they still won’t meet. Here he is an Aristotelian, for Aristotle proscribes “actual infinity” as inaccessible to sound thinking; we can imagine infinity only in terms of an indefinite approach, a potential completion. Moreover, I 35 requires the use of the Fifth Postulate. So Elements I depends on a feature of the mathematical imagination which Aristotle calls a “mathematical material,” a kind of fungible field, a transformable stuff—that, together with parallelism, accounts for the counter-intuitive fact that vastly different-looking rectilinear shapes can have a provably equal area. This is one of the wonders of elementary geometry.

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Second, the center of the Elements, its fifth and sixth books, is devoted to the theory of proportions—discovered by Euclid’s great predecessor Eudoxos. The famous Fifth of the Definitions introducing these books is, however, evidently Euclid’s. It sets down the test for, and with it the conceptual explication of, proportionality—what it means for pairs of magnitudes to be in the same ratio (logos). Proportions turn out to be the avatars of our algebraic equations, but their immediate application to geometry is in the sixth book, my favorite. It might well be called the human center of the Elements. For here are treated similar shapes, those that are the same in form but different in size—as in the first book we had shapes different in form but equal in size. Their possibility, once again, depends on the Parallel Postulate. Here is formally founded, what is, to my mind, our most intimately human capability, another feature of our imagination: image-making, be it of internal, mental images spread on that mental stuff called area, or external pictures, plans, geometric diagrams, crafted onto an external medium.

This image-making capacity is lost with the denial of the Fifth, the Parallel Postulate. Playfair’s Axiom—that through a given point not on a given line only one line can be drawn parallel to that base line—is almost immediately inferrable from this postulate. Now there are two non-Euclidean geometries that arise when this axiom is denied: In one, the first to be discovered, there are two parallel lines through that point—one in each direction. In the other, there are none—no two lines are ever parallel. And in both of these latter geometries there is no similarity. For similarity of (rectilinear) figures requires that their angles remain the same as their sizes change. But it can be shown that in the first non-Euclidean geometry, as the size of a triangle increases, its angle sum decreases, while in the second geometry as the size increases, the angle sum increases. Hence in the non-Euclidean geometries there are no images in the human sense, no similarities, no scaled copies. Euclidean geometry is uniquely the geometry of our local, image-ridden, human world.

We have no way of representing these non-Euclidean figures on our blackboards without grossly fudging the drawing, for example, putting curved lines for straight. Or we can model the geometry on a Euclidean figure and rename the parts. For example, imagine a sphere and give the name straight line to each of its great circles (the lines, like meridians, that connect any two points on a sphere by the shortest distance). You’ll see that on that sphere no two so-called straight lines can be parallel—any two lines intersect, in fact, twice (and re-enter).

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Third, the last book, the thirteenth, gives the construction of the five so-called Platonic (regular) solids and ends by proving that there are only these five: pyramid, cube, octahedron, icosahedron, dodecahedron. Now these geometric jewels played a great role in the analysis both of the material elements of bodies and the mathematical constitution of the heavens—even for Kepler, the founder of modern astronomy, an astronomy that accounts not only for orbital shapes (as could the ancients with reasonable accuracy) but for the physical causes of motion (that were beyond them). Hence Euclid’s Elements were thought of as being a preparation for the study of both heaven and earth. Indeed the Definitions of its first book, which Berlinski finds “moving” by reason of their tentative imperfection, can be seen as perfect for their purpose—if they are taken as levers from the material to the mathematical realm. Thus the often derided First Definition of the whole work, “A point is that which has no part,” might be read as an invitation to our imagination to engage in an abstractive shrinking of a physical body to focus on the bare “here”: A point is a nothing which is somewhere. And from it is generated an equally abstracted structure of lines (breadthless lengths) and surfaces, which have “length and breadth only.” (Solids come in the later books.) Thus physicality is sloughed off, and its ghost, Euclidean lines, planes, solids are built up. So seen the definitions constitute a pedagogy for inducing a mathematical imagination that matches the physical world, a world not so much foregone as etherialized.

So now back to the above mentioned oddity: These Euclidean highpoints, which reveal the Elements to be not only the elements of “geometry,” that is, “earth measurement,” and of artifact-construction, but also of proportions, arithmetic, and astronomy, are barely alluded to in Berlinski’s encomium. Why Berlinski does not extend his praise to these praiseworthy elements of the Elements is not clear. One must stop somewhere, and he does certainly find plenty to praise in the first book. But is it Euclid he is praising? Perhaps he omits these later highpoints because the first book contains the elementary, that is, the simple, carpenter’s geometry, with which most people might be familiar. And indeed he finds plenty in this first book to praise—with ever well-hedged enthusiasm. So, as I mentioned, to extract the full efficacy of that Pythagorean proposition, Berlinski turns it into analytic geometry. (I interchange “proposition” and “theorem,” as is common—actually a theorem demonstrates the truth of an enunciated proposition.) For, he says, so understood this proposition brings the concept of (measured) distance under control. But surely not for Euclid, for whom lines or plane figures are relatively equal or unequal, but never absolutely measurable, and for whom no separate figure has a definite distance to any other. For they are seen only as in their own place. Berlinski translates the proposition from Euclidian place into Cartesian space, which extends infinitely in all directions, can be given an arbitrary “origin,” and plays host to a grid centered on that origin, numbered O. In that form, the theorem can be applied (figure 2).

Figure 2:

Since x2 + y2 = h2, here a2 + b2 = h2, the square root of h2 gives the numerical distance d of P from O. But is that still “Euclid the Great”? I think not.

The point that “has no parts” but only sheer thereness inspires a fruitful puzzlement: where on earth, or beyond, is it? One answer might be that the Euclidean point has a position, either as underlying, and so directing, lines, or off them, but it has in fact no definite location, no articulable address. In Cartesian space, on the other hand, once we put ourselves at the center of space at O, P is suddenly somewhere, in a given direction and at a definite distance from us. But for Euclid there is no such origin and, as I said, no such distance: Euclid’s figures are not in space—if by (plane) space we mean an infinitely extended field whose center is up to us and whose every location is definite in relation to that origin and measurable from it, either by general algebraic magnitudes (x, y) or intended constants (a, b).

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For Euclid’s figures are strictly in place. Place, as Aristotle defines it, is the shape-envelope into which a figure is received by its environment as in a finite, local, outside-in container. Straight lines that don’t enclose a figure can, to be sure, be extended “indefinitely,” further and further from wherever you are, but they are, as I said, never thought of as going all the way to infinity. Not only is any one figure in its own place (and we hover over it), not only is there absent any distance-connection to all the other figures, but any figure’s size relation to any other is obtainable only by the highly dubious, implicitly postulated, operation of superposition.

Berlinski knows, or at least senses, all that, even as he draws Euclid into modernity. So what’s his point in praising so antediluvian a mathematical world? The epigraph by Michael Atiyah, heading Berlinski’s chapter “The Devil’s Offer,” accounts for part of the enthusiasm. Atiyah says that “Algebra is the offer made by the devil to the mathematicians.” It promises a “powerful machine” in exchange for his soul. Vision-based geometry, the intelligible beauties contemplated by our soul’s organ for such sightliness, the imagination, is turned in for a great access in problem-solving power offered by abstract calculation. Berlinski feels that sensory loss—without wishing to resign his modern mathematical advantage.

I’d like to supplement that essentially aesthetic sentiment. To enter the Elements on Euclid’s own terms is, I think, to enter into a universe not so much antique or naïve as ever-young and immediate, a realm in which notions, that would later be defiantly acquired and are by now unreflectively inherited, were not yet on the scene, a world that is still close to first awakenings: figures that take their own containing place rather than spaces that supply connectable loci; extensions that we keep in our sights, going locally step by step, rather than limits that we leap into by a super-stretch of the intellect; numbers that are collections of concrete units rather than symbols that signify abstract quantities. Nothing is better for a fresh view of ourselves than, from time to time, to inhabit such a counterworld, so as to have the experience of being sometimes more than merely our own contemporaries. To study Euclid is thus a return to our origins, a work of intellectual recovery, an exercise in human conservation. It grounds us.

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When Berlinski entitles his little venture into that world The King of Infinite Space, he is quoting a part of Hamlet’s speech (II.ii.260): “I could be bounded in a nutshell and count myself a king of infinite space....” The part unquoted in the title (italicized by me) is, as it happens, truer of Euclid’s finitude than the part cited, and therein lies much of this book’s interest: the deep questions it allusively raises about the nature and development of mathematics and, above all, mathematics’ progressive abstraction from the immediate imagination, that are inexplicitly, allusively raised.

In his spirited way, Berlinski draws attention to many more Euclidean accomplishments than I have noted. Besides the system-building, there is Euclid’s theorem-forming; he seems to have, if not devised, at least established, the standard form of a demonstration, which captures, I think, the way that seeing the mathematical truth in a diagram is related to articulating its logic in a proof: each proof is pictorially particular and logically general. As Berlinski observes, Euclid readily uses different sorts of proof, among them the reductio ad absurdum, the negative proof, regarded with suspicion by some mathematicians. Euclid’s willing use, Berlinski would probably agree, may well be a token of his familiarity with the philosophers—the sense that he gives the receptive student of having deep, sometimes meta-mathematical reasons for what he does and avoids doing. Thus, it seems, the maligned negative proof is the recognizably formal counterpart of the powerfully dialectical “Socratic refutation” set out in the Platonic dialogues: to treat a false opinion as if it were true, and then to bring out, through questioning conversation, the fact that the consequences are untenable and so to insinuate the opposite opinion.

I should conclude by mentioning one of the charms of Berlinski’s Euclid-appreciation: his many lively metaphors and similes, meant to vivify the imaginative delights to be found among mathematical objects and their relations. Once, however, he seems to me to collude with a label to go over the top: For the sheer fun of it, he depicts a so-called cardioid: “its pencilled heart emerging from the billet-doux of (x2 + y2 + ax)2 = a2 (x2 + y2).” To me, the equation doesn’t bill and coo, and the diagram looks like a striated lima bean.