This is page 1 Printer: Opaque this CHAPTER 1 Geometry: The Parallel Postulate 1.1 Introduction The first half of the nineteenth century was a time of tremendous change and upheavals all over the world. First the American and then the French revolution had eroded old power structures and political and philosophi- cal belief systems, making way for new paradigms of social organization. The Industrial Revolution drastically changed the lives of most people in Europe and the recently formed United States of America, with the newly perfected steam locomotive as its most visible symbol of progress. The mod- ern era began to take shape during this time (Exercise 1.1). No wonder that mathematics experienced a major revolution of its own, which also laid the foundations for the modern mathematical era. For twenty centuries one distinguished mathematician after another attempted to prove that the ge- ometry laid out by Euclid around 300 b.c.e. in his Elements was the “true” and only one, and provided a description of the physical universe we live in. Not until the end of the eighteenth century did it occur to somebody that the reason for two-thousand years’ worth of spectacular failure might be that it was simply not true. After the admission of this possibility, proof of its reality was not long in coming. However, in the end this “negative” answer left mathematics a much richer subject. Instead of one geometry, there now was a rich variety of possible geometries, which found applica- tions in many different areas and ultimately provided the mathematical language for Einstein’s relativity theory. To understand what is meant by this statement we need to begin by taking a look at the structure and content of Euclid’s Elements. Just like other authors before him, Euclid had produced a compendium of geometric results known at the time. What made his Elements different from those of 2 1. Geometry: The Parallel Postulate his predecessors was a much higher standard of mathematical rigor, not to be surpassed until a few centuries ago, and the logical structure of the work. Beginning with a list of postulates, which we might consider as fundamental truths accepted without proof, Euclid builds up his geometrical edifice as a very beautiful and economical succession of theorems and proofs, each de- pending on the previous ones, with little that is superfluous. This structure was greatly influenced by the teachings of Aristotle. Naturally, much de- pends on one’s choice of “fundamental truths” that one is willing to accept without demonstration as foundation of the whole theory. Among the ten postulates, or axioms, as they would be called today, the five most important ones are of two types [51, vol. I, pp. 195 ff]. The first three postulates assert the possibility of certain geometric constructions. 1. To draw a straight line from any point to any point. 2. To produce a finite straight line continuously in a straight line. 3. To describe a circle with any center and radius. The next one states that 4. All right angles are equal to one another. Thus, a right angle is a determinate magnitude, by which other angles can be measured. A rather subtle consequence of this postulate is that space must be homogeneous, so that no distortion occurs as we move a right angle around to match it with other right angles. We will have more to say about this later. Finally, the last and most important postulate concerns parallel lines. Again, faithful to Aristotelian doctrine, Euclid precedes his postulates by definitions of the concepts to be used. He defines two parallel straight lines to be “straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction” [51, vol. I, p. 190]. The fifth, or “parallel,” postulate, as it is known, states: 5. If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, will meet on that side on which are the angles less than two right angles. It is a witness to Euclid’s genius that he chose this particular statement as the basis of his geometry and viewed it as undemonstrable, as history was to show. (For more information on Euclid and his works see the chapter on number theory. A detailed description of the Elements can be found in [42].) Much of what we know about the role of the Elements in antiquity comes from an extensive commentary by the philosopher Proclus (410–485), head of the Platonic Academy in Athens and one of the last representatives of classical Greek thought. According to him, the parallel postulate was questioned from the very beginning, and attempts were made either to 1.1 Introduction 3 prove it using the other postulates or to replace it by a more fundamental truth, possibly based on a different definition of parallelism. Proclus himself says: This ought even to be struck out of the Postulates altogether; for it is a theorem involving many difficulties, which Ptolemy, in a certain book, set himself to solve, and it requires for the demonstration of it a number of definitions as well as theorems. And the converse of it is actually proved by Euclid himself as a theorem. It may be that some would be deceived and would think it proper to place even the assumption in question among the postulates as affording, in the lessening of the two right angles, ground for an instantaneous belief that the straight lines converge and meet. To such as these Geminus1 correctly replied that we have learned from the very pioneers of this science not to have any regard to mere plausible imaginings when it is a question of the reasonings to be included in our geometrical doctrine. For Aristotle says that it is as justifiable to ask scientific proofs of a rhetorician as to accept mere plausibilities from a ge- ometer; and Simmias is made by Plato to say that he recognizes as quacks those who fashion for themselves proofs from probabilities. So in this case the fact that, when the right angles are lessened, the straight lines converge is true and necessary; but the statement that, since they converge more and more as they are produced, they will sometime meet is plausible but not necessary, in the absence of some argument showing that this is true in the case of straight lines. For the fact that some lines exist which approach indefinitely, but yet re- main non-secant, although it seems improbable and paradoxical, is nevertheless true and fully ascertained with regard to other species of lines. May not then the same thing be possible in the case of straight lines which happens in the case of the lines referred to? Indeed, until the statement in the Postulate is clinched by proof, the facts shown in the case of other lines may direct our imagination the opposite way. And, though the controversial arguments against the meeting of the straight lines should contain much that is surprising, is there not all the more reason why we should expel from our body of doctrine this merely plausible and unreasoned hypothesis? [51, pp. 202 f.] This objection to the parallel postulate, so aptly described here by Pro- clus, was shared by mathematicians for the next two thousand years and produced a vast amount of literature filled with attempts to furnish the proof Proclus calls for. His description also gives a glimpse of the age-old debate about what constitutes mathematical rigor, which was to play an important role in the subsequent history of the problem. 1Greek mathematician, approx. 70 b.c.e. 4 1. Geometry: The Parallel Postulate FIGURE 1.1. Transport of angles. Why would Euclid choose to include such an odd and unintuitive state- ment among his postulates? Parallels play a central role in Euclidean geometry, because they allow us to transport angles around, the central tool for proving even the most basic facts. Thus, given an angle with sides l and l0, we want to draw the same angle with side l0, through a point P not on l. In order to do this, we need to be able to draw a line through P that is parallel to l (Figure 1.1). This raises the question whether such parallels always exist. But to be useful for constructions there must also be a unique parallel to l through P. Euclid proves the existence of paral- lels in Proposition 31 of Book I, without the use of the parallel postulate. Uniqueness follows from Proposition 30 [51, Vol. I, p. 316], for which the parallel postulate is necessary (Exercise 1.2). The first source in this chapter is Proposition 32 from Book I of the Elements, which asserts that the angle sum in a triangle is equal to two right angles. Book I is arranged in such a way that the first 28 propositions can all be proved without using the parallel postulate. It is used, however, in Euclid’s proof of Proposition 32. Much subsequent effort was focused on understanding the precise relationship between this result and the parallel postulate, as we will see later in the chapter. The other central consequence of the parallel postulate in Euclidean geometry is the Pythagorean Theorem,2 perhaps the best-known math- ematical result in the world. Babylonian civilizations knew and used it at least 1,000 years before Pythagoras, and from 800–600 b.c.e. the Sul- basutras of Indian Vedic mathematics and religion told how to use it to construct perfect religious altars [91, pp.