Mathematical Analysis in Ancient Greece

WDS'08 Proceedings of Contributed Papers, Part I, 27–31, 2008. ISBN 978-80-7378-065-4 © MATFYZPRESS Mathematical Analysis in Ancient Greece K. Cernˇ eková Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic. Abstract. The article deals with ideas of mathematical analysis which appeared in the ancient Greece. Our goal is to present the method, that gave us the modern perspective on the history of these ideas. We start with the theory of exhaustion used in the twelfth book of Euclid’s Elements (ca. 300 B.C.), which bears the idea of correct integration techniques based on Proposition X,1. However, Euclid did not invented this theory, nor referred to the author. Later on, Archimedes mentioned in his works that earlier geometers used this lemma to prove important theorems of plane geometry, and Eudoxus (4th century B.C.) was the first one, who proved some of them. This indicates, and is nowadays believed, that Eudoxus proved Proposition X,1 and many other theorems from the twelfth book of Euclid’s Elements. On the other hand Simplicius stated that one of those important geometrical theorems was already proved by Hippocrates of Chios in the 5th century B.C. But he did not mention, how Hippocrates proved it. This yields us a dilemma whether to call a founder somebody who discovered the idea and intuitively used it in one case, or someone who firmly formulated and proved the basic idea and explained its usage in many problems. Ideas of Mathematical Analysis The theory of mathematical analysis from a modern point of view was established in the 17th century, but we can trace the ideas throughout history. They can be divided into four areas: concepts of infinity and continuity • infinite series • lengths of curves, areas of figures (of revolution), volumes of solids (of revolution); i.e. • integration techniques tangents to curves; i.e. differentiation techniques • Ancient Greece The first attempts to use methods of mathematical analysis date back to ancient Greeks, who were interested only in two parts of mathematical analysis. Ancient philosophers developed the concept of infinity and found out a method for calculating the area of some plane figures and the volume of some solids. Archimedes also used infinite series for his computations and calculated a tangent to the Archimedean spiral. However, these results are exceptional and no one else was concerned with them and no general methods were established for calculating infinite series or tangents to curves. This article will focus on the development of integration techniques. I would like to deal with the questions about the foundations and founders of integration techniques. Euclid1 is undoubtedly one of the most important Greek mathematicians. His Elements are said to be a collection of the all known mathematics at that time (except for the theory of 1Euclid of Alexandria, was a Greek mathematician flourishing in the Library of Alexandria (Egypt), during 27 CERNEKˇ OVA:´ MATHEMATICAL ANALYSIS IN ANCIENT GREECE conic sections), organized and structured in an axiomatic system still used today. Around the year 300 B.C. he created an excellent textbook which was used for two thousand years. The importance of this book increases due to the fact that no older mathematical text is preserved (or maybe Euclid was the reason that no other text survived, because they were no longer needed since the Elements was written). The oldest Greek mathematics has to be reconstructed from the comments and references of younger scholars. Elements contains theorems and theories from various mathematical disciplines including number theory, theory of proportions, plane geometry, and stereometry. From the point of view of mathematical analysis Proposition X, 1 is very useful: Two unequal magnitudes being set out, if from the greater there be subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and if this process be repeated continually, there will be left some magnitude which will be less than the lesser magnitude set out. [Euclid 1956, III, 14] It bears the flavor of continuity and enabled to avoid the limits. Euclid used this lemma in his twelfth book in many propositions to calculate ratios between areas of different figures and ratios between volumes of different solids2. I would like to present a slightly modified proof of one of the propositions to show how infinitesimal techniques emerged. Proposition XII, 2 says that circles are to one another as the squares on the diameters [Euclid 1956, III, 371]. This means that the ratio of areas of two circles is equal to the ratio of the areas of the squares above their corresponding diameters. M F E N P G H O Figure 1. Denote the circles K and k, let their corresponding areas be A and a, and let their corresponding diameters be D and d. We will proceed by the contradiction. Assume that a : A = d2 : D2 is not true, then let a′ : A = d2 : D2, where a′ is the area of a circle k′, which is different from the circle k. the reign of Ptolemy I (323 B.C.–283 B.C.). He may have also studied at Plato’s Academy in Greece. This little bibliographical information comes from commentaries by Proclus and Pappus of Alexandria; nothing more is known about Euclid’s life. He is the author of the Elements, in which the principles of mathematics are deduced from a small set of axioms. Euclid’s method of proving mathematical theorems by logical deduction from accepted principles remains the backbone of all mathematics, imbuing that field with its characteristic rigor. Euclid also wrote works on perspective, conic sections, spherical geometry, and possibly quadric surfaces: Data; On Divisions of Figures; Optics; Phaenomena; Catoptrics, which is of doubtful authenticity, being perhaps written by Theon of Alexandria. All of these works follow the basic logical structure of the Elements, containing definitions and proved propositions. There are four works credibly attributed to Euclid which have been lost: Conics; Porisms; Pseudaria or Book of Fallacies; Surface Loci [Gillispie 1981, 4, 414–437]. 2Greek mathematicians were not interested in the quantitative calculations of areas or volumes, they always consider the unknown area in ratio with some known area (preferably square). 28 CERNEKˇ OVA:´ MATHEMATICAL ANALYSIS IN ANCIENT GREECE First, let a′ < a, then k′ is smaller than k. Then into the circle k we can inscribe a polygon of area p, such that a′ < p < a. To prove this we use Proposition X, 1. We subtract the inscribed square EF GH (Fig. 1) from the circle k (notice that the area of square EF GH is obviously more than half of the area of k). Then from the remaining part we subtract the triangles EMF , FNG, GOH, HPE (notice that the sum of areas of triangles EMF , FNG, GOH, HPE is obviously more than half of reminded part of the circle). Continuing in the same way, we can by Proposition X, 1 “exhaust” the circle k such that the area of the remaining part will be smaller than a a′. Now let P be the− area of the similar polygon inscribed into the circle K. Then we know (Proposition XII, 1) that p : P = d2 : D2 = a′ : A. But since p > a′, it follows that P > A, which is not possible, because the polygon of the area P is inscribed into the circle K. So the assumption a′ < a was not true. We can show in similar way (and Euclid did that) that the assumption a′ > a leads to a contradiction as well. So the original proposition has to be true. An application of Proposition X, 1 enabled Euclid to avoid the statement that the area of a circle is the limit of areas of inscribed polygons. His proof is based on the exhaustion3 of circle, i.e. inscribing the polygon, which approximates circle as much as needed. Greeks never thought that there are literally infinitely many steps in this procedure. For them there was always a tiny piece, which was not exhausted, even if this piece could be made arbitrarily small. Thus the Greeks did not have to deal with actual infinity, this procedure requires only its potential existence. It is very similar to how we compute limits today. In our ε δ definition of limit we do not use any infinitesimals, we always compute with the finite quantities− ε and δ. From this it follows that at the turn of the 3rd century B.C., there was an established theory based on the ideas of mathematical analysis. It is believed that Euclid himself did not think up most of Elements’ mathematics, but unfortunately he did not refer to his ancestors. Euclid did not mention who discovered the method of exhaustion. The only reference about this method coming from ancient Greece is in the Preface of Archimedes’s treatise Quadrature of the Parabola: Earlier geometers have also used this lemma; for, by using this lemma, they proved that circles are to one another in the duplicate ratio of their diameters, and that the spheres are one to another in the triplicate ratio of their diameters, and also that any pyramid is a third part of the prism having the same base as the pyramid and equal height; and, further, by assuming a lemma similar to that aforesaid, they proved that any cone is a third part of the cylinder having the same base as the cone and equal height. [Ivor 1957, II, 231] In 3rd century B.C. Archimedes4 himself used Proposition X, 1 (or a very similar one) to find the areas and volumes of different figures and solids.

Load more