The King of Infinite Space

by David Berlinski

  Proposition forty-seven is the first theorem in which a masterful Euclid imposes on his readers the distinction between strategy and tactics. His tactics involve two sets of congruent triangles. They play the role of proxies. They are congruent, those proxies, and so the same. But as Euclid will show, they are also equal to squares or parts of squares. The strategy of his proof thus involves a feint toward incidental objects, followed by a sweep toward the theorem’s essential identities, the axis of his attack curving like a scythe.

  IN ORDER TO obtain the first of his proxies, Euclid drops a line from A to L, one parallel to either BD or CE. He then joins the lines AD and FC. BAC and BAG are right angles. CA and AG constitute a single straight line. Euclid’s fourth axiom is in play, and so is his tenth definition (which the reader is invited to rescue from its obscurity).

  But BA is also a straight line with respect to AH. And what is more, the angles DBC and FBA are equal because they are right angles.

  Euclid now adds the angle ABC both to DBC and FBA. It follows that the whole angle DBA is equal to the whole angle FBC. Euclid’s second common notion is now in the game and doing useful work at last.

  But look: BD is equal to BC. They are sides of the same square. And FB is equal to BA, for the same reason. The triangles ABD and FBC are thus congruent by Euclid’s fourth proposition.

  Euclid has finished feinting; and the first half of his proof is complete.

  EUCLID NOW ESTABLISHES a conclusion about figures that are not triangles at all. The parallelogram BDL, he argues, is twice the triangle ABD. They have the same base BD and are in the parallels BD and AL.

  In justifying this assertion—the only recondite assertion in his proof—Euclid appeals to his forty-first theorem: “If a parallelogram have the same base with a triangle, and be in the same parallels, the parallelogram is double of the triangle.”1

  By the same reasoning, the square GB is double the triangle FBC. They have the same base in FB and are in the same parallel lines FB and GC. The double of equals is equal to one another. It follows that the parallelogram BDL is equal to the square GB.

  Euclid now repeats his reasoning. If AE and BK are joined, he asserts, two new congruent triangles appear in HBK and AEC. The parallelogram CL must be equal to square HC.

  A farewell to those fabulous feints is in order. They have done their work. Euclid’s attack is now direct; it is straightforward; and his proof proceeds by the solid, time-tested tactic of putting two and two together. The square BDEC is equal to its parts in EL and CL. But CL is equal to the square GL; and EL to the square AK. When reassembled, the square BDEC, having been divided for purposes of proof, is equal to GL and AK.

  The square on the side BC is equal to the squares on the sides BA and AC.

  Done.

  See that, Son?

  Yes, Sir.

  Attaboy.

  1. The translation of this proposition would be easier to understand were the words “in the same parallels” replaced by the words “are lying within the same parallels.” The parallelogram and the triangle, in other words, are bounded by the same parallel lines.

  Chapter VII

  VISIBLE AND INVISIBLE PROOF

  Reductio ad absurdum, which Euclid loved so much, is one of a mathematician’s finest weapons. It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.

  —G. H. HARDY

  Some paint comes across directly onto the nervous system and other paint tells you the story in a long diatribe through the brain.

  —FRANCIS BACON

  IN ANTOINE WATTEAU’S wonderful painting Jupiter and Antiope, a tense and muscular Jupiter has withdrawn Antiope’s silken robe from her sleeping body, and, of course, bound by durable pigments, he does nothing more, the poor brute forever locked where Watteau left him, lost in longing and fuming with impatience.

  This is the great limitation of the Western pictorial tradition. The plane is static. Nothing moves. It is a limitation aching to be violated. In the eighteenth and nineteenth centuries, inventions appeared in which a series of stiff paper scraps would, when rapidly flipped, create a fragile illusion of real life. Adults were enchanted, children amused. There was the zoetrope, the magic lantern, the praxinoscope, the thaumatrope, the phenakistoscope, and the flip book. Praxinoscopic visionaries could see that a series of mounted scraps might one day depict Jupiter in all his massive muscular force doing something more than crouching in impatience. Whatever the unfolding that the cinema reveals, it is one prefigured in the experience that the sophisticated imagination brings to the pictorial plane itself. A great painting invites its own analytical continuation, an arrangement of two-dimensional shapes allowed during a moment of aesthetic fantasy to shed its confinement and enter into the future or the past. In commenting on John Ruskin, the art historian Kenneth Clark appealed to a superiority that allowed him “to conjure images vividly in the mind’s eye.” Whatever Ruskin’s superiority, it embodies a power that in part we all share, the ability in looking at a painting to wriggle out of the present and slip into the stream of time.

  The analytic continuation of a great painting very often controls its aesthetic properties and so its natural critical vocabulary. Watteau’s Jupiter and Antiope is filled with tension—the obvious word—and its arrangement of shapes unstable, if for no better reason than the discomfort Jupiter is shortly to feel in his right arm. Johannes Vermeer’s View of Delft is, by way of contrast, serene. Projected into the future or recovered from the past, it hardly changes, the river passing into an open canal, the clouds, the reflections on the water’s surface, the sand—not so much timeless as indifferent, a flow, a ripeness, too.

  THE ELEMENTS IS unusual as a mathematical treatise in that it is meticulously illustrated. For every theorem, there is a picture; and with rare exceptions, the pictures are marvels, the Elements providing its readers with a series of ingenious geometrical tableaus: triangles, circles, squares, rectangles, lines crossed or in parallel, the stable and familiar shapes of art and architecture, each presented in isolation, a pedagogical handmaiden to the text, the work of a masterful teacher who knew just when the confidence of his students was about to sag. It may be possible to acquire the Elements without once attending to its illustrations, but no one has done so.

  Like Watteau’s Jupiter and Antiope or Vermeer’s View of Delft, there is nothing in the Elements that corresponds to what in life is a fluid ever-changing succession—those images that yet, fresh images beget. The illustrations are essential because they are a beginning. “Who could dispense with the figure of the triangle, [or] the circle with its centre?” David Hilbert asked in 1900. The axioms have nothing to do with it. “We do not habitually follow the chain of reasoning back to the axioms,” Hilbert observed. “On the contrary we apply, especially in first attacking a problem, a rapid, unconscious, not absolutely sure combination, trusting to a certain arithmetical feeling for the behavior of the arithmetical symbols, which we could dispense with as little in arithmetic as with the geometrical imagination in geometry.”

  It is in the rich and fascinating interplay between the logical structure of his theorems and their brilliantly contrived illustrations that Euclid’s art comes most alive.

  Yes, alive; yes, art.

  Do we know whether Euclid composed his own illustrations? We do not. The manuscript trail goes cold in the Middle Ages, no more than scraps found earlier, shuffling antiquarians fingering them absently in Cairo or Baghdad and then consigning them to cedar cases. And this is another aspect of the Elements, the enigma of the book, the identity of its author.

  EUCLID’S TWENTY-SEVENTH proposition affirms that if a straight line EF falling on two straight lines AB and CD makes the alternate angles AEF and EFD equal, then AB is parallel to CD (Figure VII.1).1 Some stripping of the theorem is required. A straight line EF falls on two straight lines AB and CD. Said once, it need not be said again. The three straight li
nes are like Somerset Maugham’s three fat women of Antibes: they are there. The stripped-down theorem: if AEF is equal to EFD, then AB is parallel to CD.

  FIGURE VII.1. Proposition twenty-seven

  Figure VII.1 tells us no more than this: angles are equal; lines are parallel. It is monotonous. If the figure is understood as a temporal slice—one depicting a figure frozen at one moment—might extending it into the future reveal something more of its inner tensions, the balance of forces leading to the theorem itself? Crouching in Watteau’s oil, Jupiter might moments later be imagined pouncing, but in projecting Figure VII.1 into the next moment, we see nothing that we have not seen before. The thing is as it was. Its angles are equal, its lines parallel.

  Just where is that “rapid, unconscious, not entirely sure” visual intuition when it is needed? Needless to say, Figure VII.1 is not the diagram Euclid used.

  EUCLID’S TWENTY-SEVENTH proposition says that if AEF is equal to EFD, then AB is parallel to CD. What has been stripped down may now be stripped bare: if P, then Q. It is only when Euclid goes bone deep that he is able to observe the logical space in which his arguments and illustrations fuse completely.

  Hypothetical propositions contain two propositions in P and Q; there is correspondingly a fourfold region of logical space which they may cohabit. If P, then Q is front to back, and if Q, then P is back to front. One is the converse of the other. A proposition and its converse are logically independent; they are free to go their separate ways. If ~P, then ~Q is again front to back (with ~P meaning “not P” and ~Q meaning “not Q”) and is called the inverse of if P, then Q. The converse and inverse of a given proposition are logically equivalent. There is no distance between them; they say the same thing. And finally, there is if ~Q, then ~P, the contrapositive of if P, then Q, the coupling of conversion and inversion (rather a desperate description, now that I think about it). A proposition and its contrapositive are logically equivalent.

  Although Euclid begins his proof with proposition twenty-seven taken straight-up, his argument depends on its contrapositive: if AB is not parallel to CD, then AEF and EFD are not equal.

  In order to demonstrate this proposition, Euclid undertakes a maneuver that is common throughout mathematics and therefore throughout geometry. By dividing his mind, he assigns to one half the position he wishes to rebut, and to the other half, the ensuing right of ridicule. The technique is known as reductio ad absurdum, or proof by contradiction. Euclid’s strategy is to prove that a proposition is true by assuming that it is false, and then demonstrating what a mess that makes.

  From the assumption that the contrapositive to his twenty-seventh proposition is false, Euclid will show that AEF and EFD are equal and that they are not.

  This is the mess.

  EUCLID’S PROOF IS not self-contained. If it were, there would be no reason to place twenty-six proofs before it. Proposition twenty-seven employs Euclid’s sixteenth proposition and his nineteenth and twenty-third definitions. The sixteenth proposition says that in any triangle, if one of the sides is extended, the exterior angles must be greater than either of the interior and opposite angles. Figure VII.2 carries an enviable air of visual authority: angle ACD is greater than angles CBA or BAC. The twenty-third definition has something to say about parallel lines—among other things, if two lines are not parallel, then sooner or later they must meet at a point.2 His nineteenth definition offers the obvious and expected account of just which figures are triangles.

  FIGURE VII.2. Proposition sixteen

  Euclid is ready now to argue. His claim is that if AB is not parallel to CD, then the angles AEF and EFD shown in Figure VII.1 are not equal. So begin with this. And suppose the proposition false. If false, its antecedent must be true: AB is not parallel to CD. And if false, its consequent must be false: AEF and EFD are equal.

  If AB is not parallel to CD, then these lines must meet at a point G. It is here, and only here, that Euclid’s own diagram plays a role. Figure VII.3 does one thing. It illustrates the supposition that AB is not parallel to CD, the antecedent—and only the antecedent—of the proposition Euclid means to reject. The figure is as spare and unforgiving as a sneer. It does not show lines AB and CD converging toward G. Were Euclid to have been interrogated on this point, he might have said with perfect aplomb, Why should I have bothered?

  FIGURE VII.3. Proposition sixteen, Euclid’s diagram

  Why should he have bothered?

  Given that AB and CD meet at G, Euclid next constructs the triangle GEF. He requires his nineteenth definition and his first axiom, the latter to connect the dots and the former to say what they mean. Euclid’s sixteenth proposition affirms that if in any triangle one of the sides is extended, then the exterior angles must be greater than either of the interior and opposite angles. The denial of Euclid’s twenty-seventh proposition has encountered Euclid’s antecedent proof of proposition sixteen. The turning point of the proof has arrived.

  And while Euclid provides no diagrams to illustrate this point, a diagram is easy to contrive (Figure VII.4). The balance of Euclid’s proof is now a matter of stating the obvious. From his sixteenth proposition, Euclid has concluded that AEF and EFD cannot be equal. But from his assumption that his twenty-seventh proposition is false, Euclid has also concluded that AEF and EFD must be equal. The result is the foreseen shambles: AEF and EFD are both equal and unequal.

  FIGURE VII.4. Contrapositive to proposition twenty-seven

  We may allow Euclid to dissolve the distinctions in his mind between back and forth. The denial of proposition twenty-seven had led to a contradiction. Therefore the proposition must be true. Euclid has persuaded himself of himself.

  EUCLID’S ELEMENTS REMAINS locked in the world before our own, and it thus demands the kind of richly imagined embedding still demanded by Watteau. The painting or the Euclidean diagram is a slice of a temporal continuum, the mind placing in proper perspective the slices that have gone before and those that go afterward.

  The purpose of a proof is to compel belief, and to compel his reader’s belief, Euclid has invested his twenty-seventh proposition with pictorial life. His diagram and the logical structure of the theorem it expresses undergo fusion. Far more than the axiomatic method alone, it is this fusion that is at the heart of Euclid’s method.

  In Figure VII.1, lines drawn parallel strike off for the infinite, always together, and by the same distance, too, but always apart, doomed thus to travel in companionable isolation throughout the whole of space. Figure VII.3, by contrast, expresses the contrapositive kernel of Euclid’s argument. It offers a lucid and balanced view into the future, but a view narrowed to a single pictorial and geometrical point. The lines are not parallel; therefore, they meet at a point.

  Euclid’s single diagram leads backward and forward by means of a sequence that Euclid expects the reader to create and then complete, a night series of shapes, fluid and fantastic, the day’s unrelenting logic dissolved. Parallel lines bend toward one another, drooping in defiance of the facts, and deflected from parallel position, they converge toward some point G, and converging toward some point G, they register an effect at a distance, and an effect registered, angles change, and the angles changing, “Well, you know or don’t you kennet or haven’t I told you every telling has a taling and that’s the he and the she of it, Look, the dusk is growing. My branches are taking root.”1

  Figures VII.1 and VII.3 depict worlds in counterfactual collision. Lines that are parallel in Figure VII.1 are not parallel in Figure VII.3. But worlds in collision on the level of the image represent worlds in collusion on the level of the theorem. Having grasped his proof, you will, Euclid is persuaded, understand its illustration, for with his exquisite power to unify the logical structure of his proof and the diagrams by which the proof is conveyed, he has illustrated a temporal flow.

  As far as Euclid the Magician is concerned, nothing more need be done. He has gotten you to do something rapid, unconscious, and not entirely certain.

 
; As far as Euclid the Logician is concerned, nothing more need be added. He has gotten you to see a sequence of propositions hurtling toward a contradiction.

  As far as Euclid the Geometer is concerned, nothing more need be done or added.

  THE PROPOSITIONS THAT Euclid demonstrated in the Elements ascend by number, and the numbers are a reasonable guide to their difficulty. Euclid’s twenty-seventh proposition retains something of the obvious. It encourages the student (or the reader) to a concurrent grunt of affirmation. The theorem is dramatic nevertheless in its reach and power. It draws a connection between a pair of equal angles on the one hand and a pair of parallel lines on the other.

  A look is enough to gauge the character of an angle, but no look, however lingering, does much to determine the character of parallel lines. Straight lines are parallel if they never meet. Within the Euclidean plane, never goes on and on. How is the geometer to establish that lines that never met will never meet? Once they have passed the last point of public inspection, lines that seem parallel might willfully undertake an unexpected decision to draw close after all.

  But equal angles are equal locally, visible in the here and now. By checking the angles made by certain straight lines, the geometer may determine their parallel character once and for all. There is no need to track them to the back of beyond.

  This theorem is interesting without in any way being extraordinary. What is extraordinary is what is so often hidden in the Elements: the rich ensemble of instruments that Euclid has employed to serve his ends. The proof of proposition twenty-seven is entirely a matter of the scant few lines needed to move with logical assurance from Euclid’s premises to his conclusions. But like an army, every one of Euclid’s theorems carries a long logistical tail: its apparatus of propositions, axioms, definitions, common notions, and rules of inference. And its illustrations, those diagrams that provide an intuition that is “rapid, unconscious, and not entirely certain.”