Euclidean Geometry Glenn Mason-Riseborough (29/8/1997) Introduction Today we know of many different consistent geometrical systems, but for much of our written history mathematicians were aware of only one geometrical system—Euclidean geometry. Euclidean geometry is the name given to the system of geometry devised by Euclid and enunciated in his thirteen books collectively referred to as the Elements (written circa 300 BCE) (Lanczos 1970). This essay will discuss the logical structure of Euclidean geometry, with specific reference to Euclid’s Elements. It will start with a brief overview of Greek mathematics at the time of Euclid and the impact Euclid’s system has had on Western history up to the present day. This essay will then discuss the general structure of the Elements—the way Euclid initially started with definitions, postulates, and axioms before moving into solving problems and proving theorems. We will then look specifically at the definitions, postulates and axioms—what is the difference between these terms and why were certain propositions chosen above others? Finally, we will look at the specific theorems and problems contained in the Elements and discuss the logical structure of some individual proofs. Historical Overview of Euclidean Geometry Any system that lays claim to being consistent must conform to some logical criteria. However, the specifics of this logical structure may differ in different societies and what was once accepted may not be considered rigorous practice today. For example the mathematics of all societies prior to the Greeks (or at least those whose writings we have discovered) consisted of using ‘trial-and-error’ empirical evidence to come to general conclusions (Eves, 1963). This was still a logical approach (although not formally so) because it gave reasoned answers to problems based on practical experience and experimentation. Arguably, one of the most important developments made by the Greeks was the insistence on deductive proofs as evidence for geometric and mathematical facts. This insistence on deductively valid arguments is based on the logical structure of the proof. We say that an argument is deductively valid if the truth of the premises require that the conclusion is also true. Thus by presupposing the truth of certain axioms, we can then necessarily show the truth of consequent theorems. Many Greek mathematicians such as Thales of Miletus (624-547 BCE), Pythagoras (572-490 BCE), Eudoxus (408-355 BCE) and members of the Pythagorean school developed deductive proofs for a wide range of different problems (Lanczos 1970). Attempts were made to string these proofs together, and people such as Hyppocrates of Chios (c450 BCE), Leon, and Theudius had partial success at producing chains of propositions based on a few initial definitions and assumptions (Eves, 1963). It is the remarkable work of Euclid however that has withstood the test of time. Not only did his Elements link together 465 propositions covering many topics of mathematics, but it was done in such a way that for over 2000 years it was considered the model of mathematical reasoning (Eves 1963). It is no detraction from Euclid that much of his work was a compilation of his predecessors’ work. It is in fact extremely fortunate that Euclid followed so many of the great Greek geometers because his work is consequently such a complete account of Greek geometrical knowledge that otherwise would have been lost. It is unfortunate that no original copies of the Elements has been found. Modern editions of the Elements are based on Theon of Alexander’s revision made circa 400 CE. Early in the 19th century an older copy was discovered in the Vatican library. This copy shows minor differences from Theon’s revision, specifically in the initial definitions, axioms and postulates, however the proofs have stayed largely unchanged (Eves 1963). Overview and Structure of the Elements The title, ‘Elements’ was used to mean foundations or basic principles, in the sense that it built a system based on a number of elementary assumptions or foundations (Lanczos 1970). Euclid’s Elements is remarkable in that it used a surprisingly small number of basic assumptions and yet was able to create an extremely large and complex system. What is even more remarkable is that these assumptions were ‘correctly’ chosen from a large universe of possible assumptions (the following section of this essay will discuss this selection in more depth). The propositions which followed were then proved deductively from the assumptions and previously proved theorems. The Elements was divided into thirteen books which covered a wide variety of topics in geometry (both two dimensional and three dimensional) and number theory. Books I to VI are on two dimensional magnitudes. Book I starts with some initial definitions, postulates and axioms before proving 48 propositions regarding the properties of triangles, parallel lines, parallelograms and squares. Book II contains the transformation of areas and book III is on circles, chords and tangents. Book IV gives methods of construction with straightedge and compass of regular polygons of three, four, five, six, and fifteen sides. Book V contains the Eudoxian theory of proportion and book VI applies this theory to plane geometry (Eves 1963). Books VII to IX contain 102 propositions on number theory. Book VII discusses proportion, book VIII is concerned with sequences or geometric progressions and book IX deals with prime numbers and perfect numbers. Book X is the longest book of the Elements; it discusses irrational numbers (Eves 1963). Books XI to XIII discuss solid geometry. Specifically, book XI contains definitions and theorems about lines and planes in space and parallelepipeds. Volumes of pyramids, cones and spheres are discussed in book XII. Finally, book XIII shows constructions of the five regular polyhedra (cube, tetrahedron, octahedron, dodecahedron, and icosahedron) and proves that these are the only possible regular polyhedra (Eves 1963). Definitions, Axioms and Postulates It is to the credit of the Greeks that they realised that in order to create a comprehensive system they had to first start with some basic assumptions. If this is not done, they would be left with an infinite regress in which each proposition must be backed up by other propositions, which must be backed up by other propositions ad infinitum. Thus, the axiomatic system was devised, and Euclid’s Elements is a remarkable example of this, despite a number of flaws. Definitions The first statements Euclid makes in the Elements are definitions of the terms he wishes to use in the postulates, axioms and theorems that follow. Many people have attacked these definitions on the grounds that from them we cannot get the proper concepts of many of the terms (Lanczos 1970). The German mathematician Moritz Pasch recognised the importance of distinguishing between explicit and implicit definitions. Pasch saw explicit definitions as definitions in which the new term was described in terms of previously defined terms. On the other hand, implicitly defined terms are defined through the medium of the context in which they occur (Eves 1963). Pasch stated that deductive science must be independent of meaning, that is, axiomatic systems must be completely devoid of any semantics (Eves 1963). Having made this distinction, we can now see that despite Euclid’s attempt at explicit definitions for terms such as point, line, and plane, his definitions are for the most part implicit. For example, the definition of point assumes knowledge of the meaning of terms such as ‘parts’ or ‘magnitude’. Thus, while Euclid based many of his assumptions on visual imagery (and thus made many hidden assumptions), Pasch was able to keep the geometry purely formal by considering it as a purely hypothetico-deductive system (Eves 1963). In 1889 the Italian mathematician Guiseppi Peano symbolised the primitive terms of Pasch’s version of Euclidean geometry and was thus able to protect himself from the trap that Euclid fell into (Eves 1963). During the winter term of 1898- 1899 the German mathematician David Hilbert gave a series of lectures at the University of Gottingen on the foundations of Euclidean geometry. Hilbert’s reformation showed the hypothetico-deductive nature of the geometry based on 21 axioms and using six primitive terms (Eves 1963). Axioms (or Common Notions) and Postulates The axioms and postulates were intended to be self-evident truths that anyone could clearly see. Today we do not distinguish between axioms and postulates—we use the words interchangeably to mean assumptions on which our system is founded. The Greeks however, chose to distinguish between the two by defining axioms as assumptions that are common to all areas of study, whereas postulates are assumptions which are specific to the subject under discussion (Eves 1963). The Elements thus contains both axioms and postulates explicitly formulated and numbered separately. In many later editions the line became blurred and propositions which were originally classed as postulates are numbered amongst the axioms or vice versa (e.g. Todhunter (1894) includes the fifth postulate as the twelfth axiom). The generally accepted axioms and postulates as given by Euclid are explicitly listed as follows (Tuller 1967): The Postulates 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 centre and distance. 4. That all right angles are equal to one another. 5. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. The Axioms 1. Things which are equal to the same thing are also equal to one another. 2. If equals be added to equals, the wholes are equal. 3. If equals be subtracted from equals, the remainders are equal. 4. Things which coincide with one another are equal to one another. 5. The whole is greater than the parts. It can be seen from this list of propositions that the fourth axiom should in fact be a postulate. Lanczos (1970) notes that the proposition is of a geometrical nature because the equality of coinciding figures is geometrical rather than logical. Lanczos (1970) also argues that the first postulate is incomplete. Not only is it necessary to be able to draw a straight line between any two points, but we must also demand that there be one and only one such line. One of the conditions of axioms and postulates is that they are all necessary—that is, no axiom or postulate can be deduced from the others. Because of the lack of intuitiveness and elegance of the fifth postulate, many people argued that it should be able to be deduced from the other postulates and axioms and hence be excluded. However it is one of the greatest achievements of Euclid that he realised the importance of this proposition on the foundations of Euclidean geometry and included it as a postulate. It was not until the eighteenth century that people such as J. H. Lambert (1728-1777) realised that modifying the fifth postulate would result in a different but consistent system (Lanczos 1970). Another condition of axioms and postulates is that they must be consistent. The premises should be chosen such that they do not lead (using deductive logic) to contradictory propositions (e.g. p and ~p) and violate the law of non-contradiction. Again the Elements is remarkable in that Euclid was able to devise a complex system that is also consistent. Theorems The bulk of Euclid’s Elements consists of theorems that are proved from axioms, postulates and previous theorems using standard logical rules of inference. For the most part these proofs are considered perfectly valid, however there are a small number of cases in which there are logical deficiencies in the proof. In this section, this essay will initially discuss some techniques Euclid uses in his proofs before showing some of the few shortcomings in an otherwise well structured, logical sequence of arguments. A type of proof which Euclid often used is a form of argument called reductio ad absurdum (RAA) (or proof by contradiction). As the name suggests, this form of argument assumes the opposite of what we are trying to prove then proceeds to show that a contradiction necessarily follows. The form of this proof is initially used in the proof of theorem I.6 (Todhunter 1894). This theorem is the converse of the previous theorem (I.5), it proves that two sides of triangle are equal to each other if the opposite angles are equal to each other. Theorem I.6 could have been proved using a method similar to that used in I.5, but it is probable that Euclid wished to introduce the reader to the technique of RAA early in the Elements. The first theorem of the Elements to assume the fifth postulate as part of its proof is theorem I.27 (Todhunter 1894); it is also the first of Euclid’s theorems which discusses parallel lines. This theorem is an integral theorem of book I as every other consequent theorem in the book assumes its truth as part of their proof. As stated above in the previous section, Euclid sometimes used visual imagery as part of his proofs. Today we do not find this rigorous practice and we do not consider proofs of theorems that use this technique as being valid. An example of this is theorem I.4 (Todhunter 1894) in which Euclid uses the principle of superposition—assuming that moving a triangle will not change its internal structure. This theorem states that if a triangle has two sides equal to the corresponding sides of another triangle, and the angle contained by the sides is also equal then the triangles will be equivalent (the other two angles and the side are equal). The proof uses a technique such that the points of one triangle are applied to the points of the other (effectively sliding one triangle on top of the other) and we consequently see that since the equivalent sides and angle are equal then the other side and angles match directly one over top of the other. Today we take this as an axiom of “the rigidity of a triangle with a tail” (Coxeter 1969). Another two examples which Prenowitz and Jordan (1965) use to show the deficiencies of using assumptions based on visual evidence are the “theorems” ‘every triangle is isosceles,’ and ‘there exists a triangle with two right angles.’ Due to space constraints this essay will not discuss the first of these “theorems,” however the second “proof” is as follows: Two circles meet at points A and B (see Fig. 1). Lines AC and AD are their respective diameters from A, and CD meets the two circles at E and F respectively. is a right angle since it is inscribed in semicircle AEC. Similarly, is also a right angle since it is inscribed in semicircle AFD. Thus, has two right angles, namely and . From a diagram it may seem that CD passes through B, and hence B, E and F are the same point, however we cannot know this simply from constructing a diagram. Conclusions Euclid’s Elements is a tremendous body of work that tied together the geometrical knowledge of many of the greatest mathematicians of antiquity. The axiomatic structure and deductive validity of Euclidean geometry defined the way we saw the world for the next 2000 years. Euclid showed that it is possible to create a complex but consistent system with only a small number of basic assumptions. The logical way in which this was accomplished—starting from the simple and progressing to the complex, became a benchmark which many scholars of later generations aimed at. The close scrutiny by modern mathematicians has uncovered a number of inconsistencies in the structure and methodology of the system, for example we no longer consider visual evidence as sufficient for a deductive proof. We also realise that Euclidean geometry is not the only possible geometry and have developed numerous other geometries using different initial assumptions. Despite this, Euclidean geometry as described by the Elements is still one of the most important and influential works to date. References: Coxeter, H. S. M. (1969) Introduction to Geometry. New York: Wiley & Sons Eves, H. (1963) A survey of Geometry, Volume One. Boston: Allyn and Bacon. Lanczos, C. (1970) Space through the ages, the Evolution of Geometrical Ideas from Pythagoras to Hilbert and Einstein. London: Academic Press Prenowitz, W & Jordan, M. (1965) Basic Concepts of Geometry. Toronto: Xerox Todhunter, I. (1894) The Elements of Euclid. London: MacMillan and Co. Tuller, A. (1967) A Modern Introduction to Geometries. Princeton: D. Van Nostrand Co.