Euclid and The Elements C R Pranesachar When a crime is committed and the culprit responsible for the crime is nabbed, the job of the investigators is not over. To convince the judge, they have to prove beyond doubt that it was the accused who committed the crime. Likewise, mathematical statements require proofs for their veracity to be established. Further, to make these statements precise we need to have a set of proper definitions. It was Euclid, who more than two thousand years ago, started this tradition when he bequeathed to the world his book entitled The Elements. This tradition has continued since then. Definitions, proofs besides axioms have become an inalienable part of mathematical lore. We owe all this to Euclid, one of the greatest pioneers of rigour and reasoning. In this article we discuss the work of Euclid. C R Pranesachar is at Mathematical Olympiad Cell, HBCSE, TIFR at the Department of Mathematics, Indian Institute of Science, Bangalore. His PhD is in combinatorics. His interests are enumeration and triangle geometry. Introduction There is no royal road to Geometry Euclid Geometry is one branch of Mathematics that abounds in beautiful and striking theorems at a very elementary level. Here, by geometry, we mean the geometry of triangles and circles. Till the 1960 s, pure geometry was taught not only in schools (upto the plus two level) but also at the undergraduate level. Students of the present generation are not generally familiar with some of these interesting results in pure geometry as it has given way to other more important topics in the curriculum. Let us have a look at some of the significant but almost forgotten results: 1. The orthocentre H, the centroid G and the circumcentre O of Keywords Definitions, postulates, axioms, theorems, proofs, prime numbers, gcd. 19
Figure 1 (left). K, L are midpoints of BC, CA. D, E are the feet of altitudes. Figure 2. (right). K, L, M are midpoints of BC, CA, AB. D, E, F are the feet of altitudes. any triangle lie on a straight line. Further HG : GO = 2:1. This line is called the Euler line of the triangle. (See Figure 1.) 2. In a triangle the midpoints of its sides, the feet of altitudes on the sides and the midpoints of the joins of the orthocenter to the vertices (in all, nine points) lie on a circle called the nine-point circle and its centre N is the midpoint of OH. Thus N also lies on the Euler line! (See Figure 2.) 3. The incircle and excircles of a triangle touch the nine-point circle internally and externally, respectively. This is called Feuerbach s Theorem. The concept of proof was alien to many other civilizations although several mathematical truths had been empirically verified. There are many more such succinct statements and quite a few have several proofs each. These proofs have been made possible by the diligence of one man named Euclid, although he himself did not prove the above theorems. Euclid was a Greek mathematician who lived between circa 325 BC and circa 265 BC. Unfortunately, not much is known about his life except that he came from a rich family and thus was able to go to an Advanced School to study and that for most of his life he lived in Alexandria, Egypt. He was the first to see that mathematical truths were based on certain definitions, postulates and axioms and all results (theorems) had to be proved. The concept of proof was alien to many other civilizations although several mathematical truths had been empirically verified. For Euclid, proof meant much. After all, one had to substantiate what one said. This meant that one had to use a series of statements each based on the previous statements, 20
definitions, postulates, etc., and logically derive the conclusion and then only one could end the argument with the letters QED! It was Euclid who systematically went on his business of writing about his findings (some compiled and some his own). He started with definitions, postulates, common notions and built a huge imperishable edifice on them, which he named The Elements. This treatise, on Geometry and Number Theory, has 13 volumes. Although the original book was lost, its translations existed, especially in Arabic and Latin and now everything is put together in its place. Surprisingly it is said to be devoid of any preface, epilogue or witty comments! In Geometry, Euclid proved results on triangles (scalene, isosceles and equilateral), collinearity, concurrency, parallelism, similarity, congruency, tangency and concyclicity. The famous Pythagoras Theorem on right-angled triangles is proved in the first volume. In Number Theory he defined divisibility, prime numbers, greatest common divisor and proved several properties of these notions. They include the famed theorem on the infinitude of prime numbers and the Euclidean algorithm for finding the gcd of two numbers using repeated division. This latter notion is relevant to different algebraic structures in modern Algebra, especially Rings. In Solid Geometry he proved the existence of five regular polyhedra 1 : the tetrahedron, the cube, the octahedron, the dodecahedron and the icosahedron. The different topics covered in The Elements are as follows: Volumes I to VI Geometry; Volumes VII to X Number Theory; Volumes XI to XIII Solid Geometry. Each volume begins with definitions, and then propositions are proved based on the previous material. The first volume has, in addition, five postulates and five axioms. Some of his definitions are disputable. For example, he defines a point as that which has no part. He also says the ends of the straight line are points. He In Geometry, Euclid proved results on triangles (scalene, isosceles and equilateral), collinearity, concurrency, parallelism, similarity, congruency, tangency and concyclicity. The famous Pythagoras Theorem on rightangled triangles is proved in the first volume. 1 A polyhedron is a 3-dimensional object bounded by identical regular polygons as faces with the same number of edges meeting at each vertex. Each volume begins with definitions, and then propositions are proved based on the previous material. 21
Almost from the time of its writing and lasting almost to the present, The Elements has exerted continuous and major influence on human affairs. B L Van der Waerden defines a straight line as that which evenly lies with points on itself. Yet with all its imperfections, The Elements has stood the test of time and was a prized textbook for several centuries. It has seen several editions and has been translated into several languages. B L Van der Waerden remarked that Almost from the time of its writing and lasting almost to the present, The Elements has exerted continuous and major influence on human affairs. No wonder the methodology of Euclid continues to hold its grip on us even now. Euclid was not a first-rate mathematician, but he was the first to realize that proof and reasoning were as important as the statement of the theorem. Euclid also made use of the powerful method of Reductio-ad-absurdum to prove some of his results: that is, you negate the conclusion and arrive at an absurd (or contradictory) statement(s) by using hypothesis or otherwise. There is even a far-fetched hypothesis that Euclid did not exist and a team of mathematicians who gave themselves this name compiled the 13 volumes which they named The Elements. Incidentally, in Greek mythology, water, earth, fire and air constitute the Elements. They are supposed to constitute the whole matter in the world. The word Theorem is perhaps related to God. Towards the end of the 19th century Hilbert and others developed an axiomatic theory of geometry and laid two dimensional and three dimensional geometry on firmer foundations. In this context, the reader may refer to a series of articles by K Paranjape published in Resonance [2]. Volume I of The Elements In the first volume there are 5 common notions, 5 postulates and 23 definitions. We give below all the common notions and postulates, along with a few definitions. Besides these, there are 48 propositions. Common Notions 1. Things which equal the same thing also equal one another. [If a = c and b = c, then a = b.] 22
2. If equals are added to equals, then the wholes are equal. [If a = b and c = d, then a + c = b + d.] 3. If equals are subtracted from equals then the remainders are equal. [If a = b and c = d, then a c = b d.] 4. Things which coincide with one another equal one another. 5. The whole is greater than part. Some Definitions 1. A point is that which has no part. 2. A line is breadthless length. 3. The ends of a line are points. 4. A straight line is a line that lies evenly with the points on itself. 8. A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line. 23. Parallel straight lines which, being in the same plane and produced indefinitely in both directions, do not meet one another in either direction. Postulates 1. To draw a straight line from any point to any point. (That is, any two points may be joined by a straight line). 2. To produce a finite straight line continuously in a straight line. 3. To describe a circle with any centre and radius. 4. That all right angles equal one another. 5. That, if a straight line falling on two straight lines makes the The last postulate, called Euclid s 5th Postulate became a highly debated topic in the 19th century. Abandoning it or altering it resulted in the so-called non- Euclidean Geometry. 23
interior angles on the same side less than two right angles, then the two straight lines, if produced indefinitely meet on that side on which the sum of the angles is less than two right angles. 2 See Renuka Ravindran s article on pp.26 36. The last postulate, called Euclid s 5th Postulate 2 became a highly debated topic in the 19th century. Abandoning it or altering it resulted in the so-called non-euclidean Geometry. It may also be mentioned that the Pythagoras theorem that the square on the hypotenuse of a right-angled triangle equals the sum of the squares on the other two sides is proved in the 47th proposition in the first volume, while its converse in the last and 48th proposition. An Overview of the XIII Volumes Table 1 gives the subject-wise list of the number of definitions, theorems, problems, porisms (corollaries), lemmas, postulates, and axioms. We describe briefly some of the more important theorems below. Here, Roman numerals refer to the volume number, and the Arabic numbers to the corresponding proposition in it. IX.20 There are infinitely many prime numbers. Table 1. Proof: Suppose there are finitely many prime numbers, say p 1, p 2, p 3,..., p k. Consider the number M = p 1.p 2.p 3..... p k + 1. Either M itself is a prime number or else has a prime factor q which cannot be any of p 1, p 2,, p k as none of these divide M. Hence, in either case, there is a new prime number different from the original set p 1, p 2,..., p k, yielding a contradiction. QED Subject Definitions Theorems Problems Porisms Lemmas Postulates Axioms Geometry 64 125 48 8 0 5 5 Number Theory 38 185 32 7 11 0 0 Solid Geometry 28 62 13 4 6 0 0 Total 120 372 93 19 17 5 5 24
I.20 In any triangle the sum of any two sides is greater than the third. I.4,8,26 These three theorems prove the congruence of two triangles with equal side-angle-side, side-side-side, angle-sideangle components. I.47,48 Pythagoras theorem and its converse. III.35 If two chords of a circle intersect at a point inside the circle then the rectangle contained by the segments of one chord equals the rectangle contained by the segments of the other. Suggested Reading [1] Florian Cajori, A History of Mathematics, AMS Chelsea Publishing, 1999. [2] Kapil Paranjape, Geometry, Resonance, Vol.1, Nos. 1 6, January June, 1996. [3] John Stilwell, Mathematics and its History, Springer, 2002. III.36 Tangent Secant Theorem. If P is a point outside a circle and a tangent from P to the circle touches it at T, and another line (called secant) through P cuts the circle at two points A and B, then PT 2 = PA. PB. IV.11 This theorem describes the construction of a regular pentagon. VI.19 Areas of similar triangles are proportional to the squares of any pair of corresponding sides. XII.10 The volume of a cone is one-third of the volume of the cylinder having the same base radius and height. XIII.13-18 These theorems describe the existence of the five polyhedra (already mentioned on p.21) and prove that these are the only five. To conclude, Euclid was one of the finest systematisers. He paved the way to his successors in the art of writing rigorous mathematics. Schools the world over in general and in India in particular may be losing out a great deal of good mathematics if Euclid is removed from the curriculum. Address for Correspondence C R Pranesachar Mathematical Olympiad Cell Department of Mathematics Indian Institute of Science Bangalore 560 012, India. Email:pran@math.iisc.ernet.in 25