Non-Euclidean Geometry Glenn Mason-Riseborough (17/10/1997) Introduction. In more than 2000 years since Euclid wrote the Elements (300 BCE), his fifth (parallel) postulate seems to have caused more debate than any of his other propositions, and perhaps more than any other statement in the history of mathematics. The argument was that it was not self-evident and should thus not be included as a postulate, but as a theorem that can be proven using the other axioms and postulates. Alternatively, it was suggested that a more self-evident postulate would be found which would then necessarily prove the parallel postulate, again relegating it to a theorem. It is unknown why Euclid included this postulate in the fashion that he did, we can only guess that he understood the significance of it as a postulate, when so many people who wrote commentaries on it later did not. Over the centuries numerous people developed different forms of Euclid’s parallel postulate, thinking that they led to a proof of the postulate. Some of these have seemed more self-evident than others, but all of them have been shown to be equivalent statements to the parallel postulate. Eventually it was realised that by negating the parallel postulate one did not arrive at a contradiction and hence alternative geometries were discovered. It is the aim of this essay to discuss the theory of parallels as given by Euclid and the alternative theories that were proposed over the centuries, leading to the realisation of non-Euclidean geometries. With this in mind, we will focus on the attempts made by mathematicians in the eighteenth and nineteenth centuries to prove the parallel postulate by reductio ad absurdum. This essay will then briefly discuss some alternative geometries that do not assume Euclid’s fifth postulate, and the ways in which they can be realised on a surface. Proclus’ commentary of the parallel postulate. Euclid’s fifth postulate is unlike any of the other previous postulates in that not only is not self-evident, but it is also a lot longer and more complex to understand. It also lacks the generality of the other postulates. Coxeter (1961) states the postulate as follows: That, if a straight line meets two other straight lines so as to make the two interior angles on one side of it less than two right angles, the other straight lines, if extended indefinitely, will meet on that side on which are the angles less than the two right angles. Because of this lack of generality, it was argued by many people over the centuries that it was more of a theorem than a postulate. The first commentary of the Elements that has been preserved (the commentaries by Heron of Alexandria have been lost) was by Proclus (c410-485) (Lanczos, 1970). Proclus was born in the latter years of the Hellenistic Age, and as such had access to the wide variety of important works and accomplishments of his predecessors. Early in his career he grew into the habit of writing commentaries on his extensive readings, and one such commentary was on Euclid’s Elements. In this commentary on the first book of the Elements, Proclus is very scathing of the fifth postulate. His first comments relating to it are ‘This ought to be struck from the postulates altogether. For it is a theorem ...’ (Proclus, 1970, p150). Proclus then proceeds to say that it is a provable theorem of which Ptolemy even claims to have demonstrated. He demonstrates that Ptolemy’s (85-165) proof begs the question and then attempts a proof of his own (which is also circular). Pogorelov (1966) gives the following ‘proof’ given by Proclus, for the theorem of parallels (see Fig. 1): Given ? + ? < 180?, it is required to prove that the lines g’ and g’’ intersect at some point C. The line g’’’ is drawn through A parallel to g’, and a perpendicular from g’’’ is drawn to point B on g’’. As the point B recedes from A its distance from g’’’ increases indefinitely , but the distance between the parallel lines g’ and g’’’ is constant, so a point C can be found which is the intersection between g’’ and g’’’. The properties of parallel lines used in this proof are not contained explicitly in the remaining postulates and consequently it cannot be deduced from them. Proclus also asserted that Euclid proves the converse of the postulate later in the Elements, and that surely if the converse of a proposition can be demonstrated then so too can the proposition itself (Proclus, 1970). Equivalent propositions to the parallel postulate. Proclus gave a number of other alternatives to Euclid’s fifth postulate. One of the more well known of these became known as Playfair’s Axiom. This proposition states that through a point not on a given line it is possible to draw no more than one line which is parallel to the given line (Pogorelov, 1966). Another equivalent proposition given by Proclus was: ‘any straight line which cuts one of two parallels, cuts also the other’ (Lanczos, 1970). An alternative proposition given by A. M. Legendre (1752-1833) and later by Farkas Bolyai (1775-1856) was: ‘given any three points not on a straight line, a circle can be found which passes through all three of them.’ Legendre also gave another form of the postulate: ‘if the sum of the angles of one single triangle is a straight angle, then the same is true for all triangles’ (Lanczos, 1970). John Wallis (1616-1703) suggested that the parallel postulate should be replaced by a characteristic of our space perception and gave the following alternative: ‘if the sides of a triangle are changed in the same ratio, the angles of the triangle remain unchanged’ (Lanczos, 1970). Other alternatives given by many different people up until the nineteenth century included: ? All the perpendiculars to one side of an acute angle intersect the other side of it. ? There exists similar triangles which are not congruent. ? There exists triangles having arbitrarily large areas. ? There exists triangles with the sum of the angles equal to two right angles. ? If two straight lines are parallel to a third straight line, they are also parallel to each other. Negating the fifth postulate. In the eighteenth century it was hoped that by assuming the negation of the fifth postulate, a contradiction could be arrived at, thus finally proving the truth of the postulate (by reductio ad absurdum). In 1733 Gerolamo Saccheri (1667-1733) considered a quadrilateral with two right base angles and equal lateral sides (see Fig. 2). He stated that it was clear that the other two angles are equal, but regarding their angle sizes, there were three alternative hypotheses—they were either right, obtuse or acute (Pogorelov, 1966). Saccheri then proved that the statement that both angles are right is equivalent to the fifth postulate—if we assume one, this leads to a proof of the other and vice versa. When Saccheri postulated the obtuse angle hypothesis he arrived at a contradiction, and when he postulated the acute angle hypothesis his results were absurd from the viewpoint of Euclidean geometry. These results suggested that either parallel lines had one common perpendicular and then diverged indefinitely, or that they approached one another asymptotically in one direction and diverged in the other. Saccheri did not accept this and searched for a contradiction, finally finding one which later was discovered to be simply a computational error. In 1766 Johann Lambert (1738-1777) similarly considered a quadrilateral in the hopes of finding a proof of the parallel postulate (Pogorelov, 1966). This quadrilateral had three right angles (see Fig. 3) and Lambert considered the alternatives for the fourth angle. Just as Saccheri did before him, Lambert considered that the angle could be either right, obtuse or acute. Lambert proved that the right angle hypothesis is equivalent to the parallel postulate and that the obtuse angle hypothesis is impossible. In the same way as Saccheri, Lambert arrived at counter-intuitive results when he considered the acute angle hypothesis. Because he did not arrive at a contradiction, Lambert continued to develop a system from this hypothesis and soon discovered an analogy of this system with the geometry on a sphere. Lanczos (1970) states that Lambert also considered the equivalent statement of whether the angle sum of a triangle is equal to, greater than, or smaller than 180?. In this case, the equal to 180? hypothesis is equivalent to the parallel postulate, the greater than 180? hypothesis is equivalent to the obtuse angle hypothesis, and the less than 180? hypothesis is equivalent to the acute angle hypothesis. Equivalent results to the quadrilateral argument were proved in Lambert’s book, Theory of Parallels. Lanczos (1970) considers this a genuine example of non- Euclidean geometry, but admits that Lambert did not realise the significance of his findings. Legendre also used the angle sum of triangle hypothesis in the same way to that of Lambert to attempt to prove the fifth postulate (Pogorelov, 1966). Again he considered the three alternatives (equal to, greater than, or smaller than 180?) and found the first is equivalent to the fifth postulate and the second is impossible. For the third hypothesis, Legendre found a contradiction by implicitly making use of the fifth postulate from one of its equivalents. Lobachevski, J. Bolyai and Gauss. Within the space of a few years and completely independently of each other, these three men all developed new geometries in which Euclid’s parallel postulate does not hold. Pogorelov (1966) considers the honour of discovering a new geometry (1) went to Nikolai Lobachevski (1793-1856) for his Lobachevskian geometry (Lobachevski called it Pangeometry). The results of which were published in German in 1829-30 in his book titled On the Principles of Geometry (Lanczos, 1970). Lobachevski replaced the parallel postulate by the statement: ‘Through a point not on a line in the plane there pass two lines which do not intersect the given line’ (Pogorelov, 1966, p 16). In the hope of finding a contradiction, Lobachevski then developed this geometry to a similar extent as Euclid’s Elements. From this basis Lobachevski came to the conclusion in 1826 that the system that he had developed was as equally valid as the Euclidean system. He realised that from the point of view of logical consistency, his geometry was in a similar position as the Euclidean geometry. There was no guarantee that by developing either system further, there would be no eventual contradiction. In fact, it was later shown that the logical consistency of one geometry depends on the logical consistency of the other (Pogorelov, 1966). The question of which geometry better represents our world thus becomes an empirical question. In 1832 Farkas Bolyai published a basic textbook on geometry (usually referred by the first word of its name—Tentamen) in which his son, János Bolyai (1802-1860) included an appendix on what he called the ‘science of absolute geometry’ (Lanczos, 1970). In this appendix János Bolyai discussed non-Euclidean geometries in which the first four Euclidean postulates were assumed, but the fifth was not. Bolyai’s theories were similar to Lobachevski’s, but he did not develop them to the same extent. Farkas Bolyai (who was a friend of Carl Gauss (1777-1855) as a student at the University of Goettingen) showed his son’s work to Gauss, who, although recognising the brilliance of the work, remarked that he had known of the results for years (Lanczos, 1970). János Bolyai was hurt by this comment by such a great mathematician and did not pursue his research in the area of non- Euclidean geometries. In fact, Gauss had been working on the problem associated with the parallel postulate for many years previously and his solutions were far more comprehensive than the geometrical approach of either Lobachevski or Bolyai (Lanczos, 1970). Gauss never published these results—he published very few papers, preferring to write ‘few, but mature’ papers (Lanczos, 1970, p 88). Much of his work, however can be dated from his notebooks and letters to friends and colleagues. In a letter written in 1824 to his friend Taurinus, Gauss writes of his significant accomplishments in the area of non-Euclidean geometry (Lanczos, 1970). Lobachevskian geometry. Lobachevski’s and Bolyai’s geometries and results were very similar to each other in many respects. Both of them initially eliminated the possibility that the angle sum of triangle is greater than 180? as it is ‘contradicted by the uniqueness and infinity of a straight line’ (Lanczos, 1970, p 65). They both then progressed to the possibility that the angle sum of triangle is less than 180? and found no contradiction. Figure 4 shows the geometrical results of Lobachevski’s argument which follows. A perpendicular PP’ is dropped from a point P to a given straight line, l. A straight line, p is drawn through P perpendicular to PP’. Line p cannot intersect line l because (due to symmetry) it would then have to intersect on both sides and thus contradict the uniqueness of a straight line between two points. Hence line p is parallel to line l. By assuming the parallel postulate, line p is the only parallel to l through P. However by dropping this, we get a different result. In this case, we may get an infinite number of parallel lines through P, because none of the lines within a small but finite angle ?? between p and p’ will intersect l. Even though this result may intuitively seem implausible, this reluctance to accept it is based on our intuitive use of the law of similarity and our implicit assumption of Euclidean geometry, both of which are not logically necessary. Acceptance of non-Euclidean geometries. Despite three separate results showing the existence of valid geometries that did not include Euclid’s parallel postulate, non- Euclidean geometries were not widely accepted amongst mathematicians for a number of years. It was not until the work of a number of geometers after Lobachevski, that there was general acceptance of these new geometries. One of these geometers was E. Beltrami (1835-1900), who in 1862 proved that plane Lobachevskian geometry holds on a surface of constant negative curvature (Pogorelov, 1966). In this geometry, Lobachevskian lines are thought of as geodesic curves. At the time this result was considered as proof of the consistency of Lobachevskian geometry, and a contradiction in this would also mean a contradiction in the theory of surfaces in Euclidean space, i.e., a contradiction in Euclidean geometry. David Hilbert (1862-1943) showed later that in Euclidean space there does not exist a complete surface with constant negative curvature without singularities, and thus on this surface, only part of the geometry of the Lobachevskian plane can be interpreted. The interpretations of F. Klein (1849-1925) and H. Poincaré (1854-1912) eliminated this problem. Klein showed that Lobachevskian geometry can be realised on the interior of a disc in the Euclidean plane (Pogorelov, 1966). In this interpretation, a line in Lobachevskian geometry corresponds to a chord on a disc in the Euclidean plane. The consistency of this interpretation of Lobachevskian geometry was proved, and in doing so this proved the independence of Euclid’s fifth postulate from the other four postulates. In this way it was finally proved that the parallel postulate is a necessary part of Euclidean geometry, but that its negation leads to equally valid geometries. Conclusions. The theory of parallel lines has a long history in geometry that started with Euclid’s fifth postulate in his Elements. For the next 2000 years many mathematicians and writers of commentaries questioned the inclusion of that postulate. Its lack of generality made it seem as if it should be a theorem that was provable by other more self-evident postulates. It was not until the 18th century that, in the hope of finding a proof by reductio ad absurdum, it was discovered that negating the parallel postulate did not lead to a contradiction. Saccheri, Lambert, and Legendre separately attempted proofs by this method and were all unable to arrive at (valid) contradictions for all possible cases. Unfortunately, none of them were aware of the importance of their findings and it was not until Gauss, Lobachevski and János Bolyai attempted proofs in a similar manner that the significance was realised. Within a few years of each other, they all separately realised that negating the parallel postulate did not lead to a contradiction. Hence, it was realised that by modifying the parallel postulate, additional geometries could be developed which were self-consistent. These new geometries, while seeming strange and counterintuitive, were nonetheless perfectly valid systems. Numerous mathematicians such as Beltrami, Klein, Poincaré and Hilbert then assisted in showing the validity and consistency of these new geometries. Since then, many additional geometries have been developed from different initial assumptions, and today these are used in many various areas of mathematics, physics and in other sciences and arts. References: Coxeter, H. S. M. (1961). Introduction to geometry. New York: Wiley & Sons. Daniels, N. (1974). Thomas Reid’s Inquiry: The geometry of visibles and the case for realism. New York: Franklin. Hamilton, W. B. (1863). The works of Thomas Reid, D. D. Edinburgh: MacLachlan & Stewart. Lanczos, C. (1970). Space through the ages: The evolution of geometrical ideas from Pythagoras to Hilbert and Einstein. London: Academic Press. Proclus (1970). A commentary on the first book of Euclid’s Elements (G. R. Morrow, Trans.). Princeton: Princeton University. Pogorelov, A. V. (1966). Lectures on the foundations of geometry (L. F. Boron, Trans.). Groningen, Netherlands: Noordhoff. Endnotes: 1 It is of interest to note that Daniels (1964) states that he considers Thomas Reid’s ‘geometry of visibles’ (first published in 1764) a clear example of non-Euclidean geometry. However, it must be realised that Reid did not consider the theory of parallels when he developed his geometry. He merely stated that ‘if two lines be parallel —that is, everywhere equally distant from each other—they cannot both be straight’ (Hamilton, 1863, p 148). Daniels (1964) also states that spherical geometries was well known during the eighteenth century and often appended to editions of Euclid’s Elements (e.g., an edition of 1781 included an appendix by Dr. John Keil).