Why Beauty is Truth Page 4

  Euclid . . . put together the Elements, arranging in order many of Eudoxus’s theorems, perfecting many of Theaetetus’s, and also bringing to irrefutable demonstration the things which had been only loosely proved by his predecessors. This man lived in the time of the first Ptolemy; for Archimedes, who followed closely upon the first Ptolemy, makes mention of Euclid, and further they say that Ptolemy once asked him if there were a shorter way to study geometry than the Elements, to which he replied that there was no royal road to geometry. He is therefore younger than Plato’s circle, but older than Eratosthenes and Archimedes; for these were contemporaries, as Eratosthenes somewhere says. In his aim he was a Platonist, being in sympathy with this philosophy, whence he made the end of the whole Elements the construction of the so-called Platonic figures.

  The treatment of some topics in the Elements provides indirect but compelling evidence that Euclid must at some point have been a student at Plato’s Academy in Athens. Only there, for example, could he have learned about the geometry of Eudoxus and Theaetetus. As for his character, all we have are some fragments from Pappus, who described him as “most fair and well disposed towards all who were able in any measure to advance mathematics, careful in no way to give offence, and although an exact scholar, not vaunting himself.” A few anecdotes survive, such as one told by Stobaeus. One of Euclid’s students asked him what he would get through an understanding of geometry. Euclid called his slave and said, “Give him a coin, since he must make a profit from what he learns.”

  The Greek attitude to mathematics was very different from that of the Babylonians or the Egyptians. Those cultures saw mathematics largely in practical terms—although “practical” could mean aligning shafts through a pyramid so that the ka of the dead pharaoh could be launched in the direction of Sirius. For some Greek mathematicians, numbers were not tools occasionally employed in support of mystical beliefs, but the very core of those beliefs.

  Aristotle and Plato tell of a cult, centered on Pythagoras, that flourished around 550 BCE and that viewed mathematics, especially number, as the basis of the whole of creation. They developed mystical ideas about the harmony of the universe, based in part on the discovery that harmonious notes on a stringed instrument are related to simple mathematical patterns. If a string produces a certain note, a string of half the length produces a note one octave higher—the most harmonious of all intervals. They investigated various number patterns, in particular polygonal numbers, formed by arranging objects in polygonal patterns. For instance, the “triangular numbers” 1, 3, 6, and 10 are formed from triangles, and the “square numbers” 1, 4, 9, and 16 are formed from squares:

  Triangular and square numbers.

  Pythagoreanism embraced some nutty numerology—it considered 2 to be male and 3 female, for example—but the view that the deep structure of nature is mathematical survives today as the basis of most theoretical science. Although later Greek geometry was less mystical, the Greeks generally saw mathematics as an end in itself, more a branch of philosophy than a tool.

  There are reasons to believe that this does not tell the whole story. It is well established that Archimedes, who may have been a pupil of Euclid’s, employed his mathematical abilities to design powerful machines and engines of war. There survive a tiny number of intricate Greek mechanisms whose cunning design and precision manufacture hint at a well-developed tradition of craftsmanship—an ancient version of “applied mathematics.” Perhaps the best-known example is a machine found in the sea near the small island of Antikythera that appears to be a calculating device for astronomical phenomena built from a complex tangle of interlocking cogwheels.

  Euclid’s Elements certainly fits this rarefied view of Greek mathematics—possibly because that view is largely based on the Elements. The book’s main emphasis is on logic and proof, and there is no hint of practical applications. The most important feature of the Elements, for our story, is not what it contains but what it does not.

  Euclid made two great innovations. The first is the concept of proof. Euclid refuses to accept any mathematical statement as being true unless it is supported by a sequence of logical steps that deduces it from statements already known to be true. The second innovation recognizes that the proof process must start somewhere, and that these initial statements cannot be proved. So Euclid states up front five basic assumptions on which all his later deductions rest. Four of these are simple and straightforward: two points may be joined by a line; any finite line can be extended; a circle can be drawn with any center and any radius; all right angles are equal.

  But the fifth postulate is very different. It is long and complicated, and what it asserts is not nearly so reasonable and obvious. Its main implication is the existence of parallel lines—straight lines that never meet, but run forever in the same direction, always the same distance apart, like two sidewalks on either side of an infinitely long, perfectly straight road. What Euclid actually states is that whenever two lines cross a third, the first two lines must meet on whichever side creates two angles that add up to less than two right angles. It turns out that this assumption is logically equivalent to the existence of exactly one line parallel to a given line and passing through a given point (not on the given line).

  Euclid’s fifth postulate.

  For centuries the fifth postulate was viewed as a blemish—something to be removed by deducing it from the other four, or to be replaced by something simpler and just as obvious as the others. By the nineteenth century, mathematicians understood that Euclid was absolutely right to include his fifth postulate, because they could prove that it can’t be deduced from his other assumptions.

  To Euclid, logical proofs were an essential feature of geometry, and proof remains fundamental to the mathematical enterprise. A statement that lacks a proof is viewed with suspicion, however much circumstantial evidence seems to favor it and however important its implications may be. Physicists, engineers, and astronomers tend to view proofs with disdain, as a kind of pedantic appendage, because they have an effective substitute: observation.

  For instance, imagine an astronomer trying to calculate the movements of the Moon. He will write down mathematical equations that determine the Moon’s motion, and promptly get stuck because there seems to be no way to solve the equations exactly. So the astronomer may tinker with the equations, introducing various simplifying approximations. A mathematician will worry that these approximations might have a serious effect on the answer, and will want to prove that they do not cause trouble. The astronomer has a different way to check that what he has done makes sense. He can see whether the motion of the Moon fits his calculations. If it does, that simultaneously justifies the method (because it got the right answer) and verifies the theory (for the same reason). The logic here is not circular because if the method is mathematically invalid, then it will almost certainly fail to predict the Moon’s motion.

  Without the luxury of observations or experiments, mathematicians have to verify their work by its internal logic. The more important the implications of some statement are, the more important it is to make sure that the statement is true. So proof becomes even more crucial when the statement is something that everyone wants to be true, or that would have enormous implications if it were true.

  Proofs cannot rest on thin air, and they cannot trace logical antecedents back forever. They have to start somewhere, and where they start will by definition be things that have not been—and will not be—proved. Today we call these unproved starting assumptions axioms. The axioms for a piece of mathematics are the rules of the game.

  Anyone who objects to the axioms can change them if they wish: however, the result will be a different game. Mathematics does not assert that some statement is true: it asserts that if we make various assumptions, then the statement concerned must be a logical consequence. This does not imply that the axioms are unchallengeable. Mathematicians may debate whether a given axiomatic system is better than another for some purpose, or whether
the system has any intrinsic merit or interest. But these discussions are not about the internal logic of any particular axiomatic game. They are about which games are worthwhile, interesting, or fun.

  The consequences of Euclid’s axioms—his long, carefully selected chains of logical deductions—are extraordinarily far-reaching. For example, he proves—with logic that in his day was considered impeccable—that once you agree to his axioms you necessarily must conclude that:

  • The square on the hypotenuse of a right triangle is equal to the sum of the squares on the other two sides.

  • There exist infinitely many prime numbers.

  • There exist irrational numbers—not expressible as an exact fraction. An example is the square root of two.

  • There are precisely five regular solids: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron.

  • Any angle can be divided exactly into two equal parts using only straightedge and compass.

  • Regular polygons with 3, 4, 5, 6, 8, 10, and 12 sides can be constructed exactly using only straightedge and compass.

  I have expressed these “theorems,” as we call any mathematical statement that has a proof, in modern terms. Euclid’s point of view was rather different: he did not work directly with numbers. Everything that we would interpret as properties of numbers is stated in terms of lengths, areas, and volumes.

  The content of the Elements divides into two main categories. There are theorems, which tell you that something is true. And there are constructions, which tell you how to do something.

  A typical and justly famous theorem is Proposition 47 of Book I of the Elements, usually known as the Pythagorean theorem. This tells us that the longest side of a right triangle bears a particular relationship to the other two sides. But it does not, without further effort or interpretation, provide a method for achieving any goal.

  The Pythagorean theorem.

  A construction that will be important in our story is Proposition 9 of Book I, where Euclid solves the “bisection problem” for angles. Euclid’s method of bisecting an angle is simple but clever, given the limited techniques available at this early stage of the development.

  How to bisect an angle with straightedge and compass.

  Given (1) an angle between two line segments, (2) place your compass tip where the segments meet, and draw a circle, which crosses the segments at two points, one on each (dark blobs). Now (3) draw two circles of equal radius, one centered at each of the new points. They meet in two points (only one is marked), and (4) the required bisector (dotted) runs through both of these.

  By repeating this construction, you can divide an angle into four equal pieces, or eight, or sixteen—the number doubles at each step, so we obtain the powers of 2, which are 2, 4, 8, 16, 32, 64, and so on.

  As I mentioned, the main aspect of The Elements that affects our story is not what it contains but what it doesn’t. Euclid did not provide any method for:

  • Dividing an angle into three exactly equal parts (“trisecting the angle”).

  • Constructing a regular 7-sided polygon.

  • Constructing a line whose length is equal to the circumference of a given circle (“rectifying the circle”).

  • Constructing a square whose area is equal to that of a given circle (“squaring the circle”).

  • Constructing a cube whose volume is exactly twice that of a given cube (“duplicating the cube”).

  It is sometimes said that the Greeks themselves saw these omissions as flaws in Euclid’s monumental work and devoted a great deal of effort to repairing them. Historians of mathematics have found very little evidence to back up these claims. In fact, the Greeks could solve all of the above problems, but they had to use methods that were not available within the Euclidean framework. All of Euclid’s constructions were done with an unmarked straightedge and compass. Greek geometers could trisect angles using special curves called conic sections; they could square the circle using another special curve called a quadratrix. On the other hand, they do not seem to have realized that if you can trisect angles, you can construct a regular 7-gon. (I do mean 7-gon. There is an easy construction for a 9-gon, but there is also a very clever one for a 7-gon.) In fact, they apparently did not follow up the consequences of trisection at all. Their hearts seem not to have been in it.

  Later mathematicians viewed Euclid’s omissions in a rather different light. Instead of seeking new tools to solve these problems, they began to wonder what could be achieved with the limited tools Euclid used: straightedge and compass. (And no cheating with marks on the ruler: the Greeks knew that “neusis constructions” with sliding rulers and alignment of marks could trisect the angle effectively and accurately. One such method was devised by Archimedes.) Finding out what could or could not be done, and proving it, took a long time. By the late 1800s we finally knew that none of the above problems can be solved using straightedge and compass alone.

  How Archimedes trisected an angle.

  This was a remarkable development. Instead of proving that a particular method solved a particular problem, mathematicians were learning to prove the opposite, in a very strong form: no method of this-and-that kind is capable of solving such-and-such problem. Mathematicians began to learn the inherent limitations of their subject. With the fascinating twist that even as they were stating these limitations, they could prove that they genuinely were limitations.

  In the hope of avoiding misconceptions, I want to point out some important aspects of the trisection question.

  What is required is an exact construction. This is a very strict condition within the idealized Greek formulation of geometry, where lines are infinitely thin and points have zero size. It requires cutting the angle into three exactly equal parts. Not just the same to ten decimal places, or a hundred or a billion—the construction must be infinitely precise. In the same spirit, however, we are allowed to place the compass point with infinite precision on any point that is given to us or is later constructed; we can set the radius of the compass, with infinite precision, to equal the distance between any two such points; and we can draw a straight line that passes exactly through any two such points.

  This is not what happens in messy reality. So is Euclid’s geometry useless in the real world? No. For instance, if you do what Euclid prescribes in Proposition 9, with a real compass on real paper, you get a pretty good bisector. In the days before computer graphics, this is how draftsmen bisected angles in technical drawings. Idealization is not a flaw: it is the main reason mathematics works at all. Within the idealized model, it is possible to reason logically, because we know exactly what properties our objects have. The messy real world isn’t like that.

  But idealizations also have limitations that sometimes make the model inappropriate. Infinitely thin lines do not, for example, work well as painted lane markers on roads. The model has to be tailored to an appropriate context. Euclid’s model was tailored to help us work out the logical dependencies among geometrical statements. As a bonus, it may also help us understand the real world, but that certainly wasn’t central to Euclid’s thinking.

  The next comment is related, but it points in a rather different direction. There is no problem finding constructions for trisecting angles approximately. If you want to be accurate to one percent or one thousandth of a percent, it can be done. When the error is one thousandth of the thickness of your pencil line, it really doesn’t matter for technical drawing. The mathematical problem is about ideal trisections. Can an arbitrary angle be trisected exactly? And the answer is “no.”

  It is often said that “you can’t prove a negative.” Mathematicians know this to be rubbish. Moreover, negatives have their own fascination, especially when new methods are needed to prove them. Those methods are often more powerful, and more interesting, than a positive solution. When someone invents a powerful new method to characterize those things that can be constructed using straightedge and compass, and distinguish them from those that cannot,
then you have an entirely new way of thinking. And with that come new thoughts, new problems, new solutions—and new mathematical theories and tools.

  No one can use a tool that hasn’t been built. You can’t call a friend on your cell phone if cell phones don’t exist. You can’t eat a spinach soufflé if no one has invented agriculture or discovered fire. So tool-building can be at least as important as problem-solving.

  The ability to divide angles into equal parts is closely related to something much prettier: constructing regular polygons.

  A polygon (Greek for “many angles”) is a closed shape formed from straight lines. Triangles, squares, rectangles, diamonds like this ◊, all are polygons. A circle is not a polygon, because its “side” is a curve, not a series of straight lines. A polygon is regular if all of its sides have the same length and if each pair of consecutive sides meet at the same angle. Here are regular polygons with 3, 4, 5, 6, 7, and 8 sides:

  Regular polygons.

  Their technical names are equilateral triangle, square, (regular) pentagon, hexagon, heptagon, and octagon. Less elegantly, they are referred to as the regular 3-gon, 4-gon, 5-gon, 6-gon, 7-gon, and 8-gon. This terminology may seem ugly, but when it becomes necessary to refer to the regular 17-sided polygon—as it shortly will—then the term “17-gon” is far more practical than “heptadecagon” or “heptakaidecagon.” As for the 65,537-gon (yes, that too!)—well, you get the picture.

  Euclid and his predecessors must have thought a great deal about which regular polygons can be constructed, because he offers constructions for many of them. This turned out to be a fascinating, and decidedly tricky, question. The Greeks knew how to construct regular polygons when the number of sides is