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Home , Data (Euclid), Dinostratus, Euclid, Eudoxus of Cnidus, Hippocrates of Chios, Mathematical analysis

WDS'08 Proceedings of Contributed Papers, Part I, 27–31, 2008. ISBN 978-80-7378-065-4 © MATFYZPRESS

Mathematical in Ancient

K. Cernˇ ekov´a Charles University, Faculty of and , Prague, Czech .

Abstract. The article deals with ideas of which appeared in the . Our goal is to present the method, that gave us the modern on the history of these ideas. We start with the theory of exhaustion used in the twelfth book of ’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, mentioned in his works that earlier geometers used this lemma to prove important of , 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 of 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 of view was established in the 17th century, but we can trace the ideas throughout history. They can be divided into four : concepts of infinity and continuity • infinite • of , areas of figures (of revolution), of solids (of revolution); i.e. • integration techniques 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 of some plane figures and the of some solids. Archimedes also used infinite series for his computations and calculated a 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 . His Elements are said to be a collection of the all known mathematics at that time (except for the theory of

1Euclid of , was a Greek flourishing in the Library (Egypt), during

27 CERNEKˇ OVA:´ MATHEMATICAL ANALYSIS IN ANCIENT GREECE conic sections), organized and structured in an 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 that no other text survived, because they were no longer needed since the Elements was written). The oldest has to be reconstructed from the comments and references of younger scholars. Elements contains theorems and theories from various mathematical disciplines including 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 out, if from the greater there be subtracted a mag- nitude 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 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 are to one another as the squares on the [Euclid 1956, III, 371]. This that the 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 k′, which is different from the circle k. the reign of I (323 B.C.–283 B.C.). He may have also studied at ’s in Greece. This little bibliographical information comes from commentaries by and ; 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 . 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, , and possibly quadric surfaces: ; On Divisions of Figures; ; Phaenomena; , which is of doubtful authenticity, being perhaps written by . 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; ; 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 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 is the 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 − 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 . The only reference about this method coming from ancient Greece is in the Preface of Archimedes’s 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 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. In another two Prefaces of his On the and Cylinder and Method, respectively, Archimedes specified some parts of the previous announcement:

3The name “method of exhaustion” got this procedure in the 17th century. 4Archimedes was born in Syracuse ca. 287 B.C. and died in Syracuse in 212 B.C. He is famous as mathemati- cian, physicist, engineer, inventor, and . Only a few details of his life are known, because a biography written by his friend Heracleides has not survived. Archimedes’s father was astronomer Phidias and Archimedes himself was probably kinsman of the ruler of Syracuse, King Heron II. He studied for some time in Alexandria. It is believed that he invented various mechanical devices and discovered the law of lever and Archimedes’s law of hydrostatic named after him. Archimedes also explained how to construct a planetarium in the book On Sphere Making that is lost. His mathematical works can be divided into three groups: (1) proof of theorems related to the areas and volumes of figures bounded by curved lines and surfaces: On the Sphere and Cylinder, On the Measurement of the Circle, On the Conoids and Spheroids, , On the Quadrature of Parabola; (2) geometrical analysis of statical and hydrostatical problems, the use of statics in geometry: On the Equilib- rium of Planes, On Floating Bodies, On the Method of Mechanical Theorems; (3) others: The Sandreckoner, The Cattle-Problem, Stomachion [Gillispie 1981, I, 213–231].

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For this reason I cannot feel any hesitation in setting these [theorems] side by side both with the investigations of other geometers and with those of the theorems of Eudoxus on solids which seem to stand out pre-eminently, namely, that any pyramid is a third part of the prism having the same base as the pyramid and equal height, and that any cone is a third part of the cylinder having the same base as the cone and equal height; for though these properties were naturally inherent in these figures all along, yet they were in fact unknown to the many competent geometers who lived before Eudoxus and had not been noticed by anyone. [Ivor 1957, I, 409–410] This is a reason why, in the case of those theorems the cone and the pyramid of which Eudoxus first discovered the proof, the theorems that the cone is a third part of the cylinder, and the pyramid of the prism, having the same base and equal height, no small share of the credit should be given to , who was the first to make the assertion with regard to the said figure, though without proof. [Ivor 1957, I, 229–230] To summarize these passages one can see that Archimedes ascribed Proposition X, 1 to the geometers, who proved that (i) the circles are one to another in the “duplicate ratio” of their diameters, or that (ii) the spheres are one to another in the “triplicate ratio” of their diameters, or that (iii) the volume of pyramid is one third of the volume of circumscribed prism, or that (iv) the volume of cone is one third of the volume of circumscribed cylinder. Then he claimed that the first proofs of (iii) and (iv) and other geometrical theorems were demonstrated by Eudoxus5. According to this account it is generally believed that Proposition X, 1 and probably most of the propositions in book XII of Elements was proved and used by in the 4th century B.C., i.e. about 70 years before Euclid. It is also very likely that Euclid in his book XII rewrote Eudoxus’s proofs, because according to Archimedes, Eudoxus used the lemma to prove his geometrical theorems and we can hardly imagine essentially different proofs using the same method. On the other hand Simplicius6 stated that (i) was proved by Hippocrates of Chios7 in the

5Eudoxus of Cnidus, son of the of Cnidus, was according to Apollodorus in his prime between 368– 365 B.C. and since we are said that he lived for 53 years, we can assume that he lived ca. 408–355 B.C. Eudoxus first traveled to Tarentum () to study with , from whom he learned mathematics. Then he studied medicine with Philiston at . When he was 23, he left for with physician Theomedon and attended Plato’s lectures for two months. He walked seven miles each direction, each day, because he was too poor that he could only afford an apartment at . From Athens he returned to Cnidus and then around 381–380 he went to Egypt with a letter for king Nectabenus II, which was sent by Agesilaus II from . He lived there 16 months and studied mathematics and . From Egypt, he traveled north to , at the south shore of the Sea of Marmara, and the Propontis. Then he traveled south to the court of Maussolus. During his travels he gathered many students of his own. He came to Athens with his students in 368 or little later. Eudoxus returned to his native Cnidus, where he served in the city assembly. He was interested in astronomy and mathematics, according to ’s and his commentators, Eudoxus found the routes for all 7 known celestial solids composed from concentrical spheres with the center in Earth and different radii. Aristotle’s estimation of the Earth’s radius is based probably on Eudoxus’s calculations, because Archimedes mentioned, that Eudoxus calculated the relative of Earth from Sun. According to the anonymous commentator of Euclid’s Elements (probably Proclus), Eudoxus discovered the content of the fifth book of Elements, which treats the theory of proportions. All Eudoxus’s treatises are lost. His pupils were brothers Meneachmus and [Gillispie 1981, 4, 465–467]. 6Simplicius of Cilicia, a disciple of Ammonius and of , was one of the last Neoplatonists. When, in 529 A.D., the school of at Athens was disendowed and the teaching of philosophy forbidden, the scholars Damascius, Simplicius, Priscianus Lydius and four others resolved in 531 or 532 to seek the protection of Khosrau I, king of Persia, but they returned to Greece within two years. After his return from Persia Simplicius wrote commentaries upon Aristotle’s De coelo, Physica, De anima and Categoriae, which, with a commentary upon the Enchiridion of , have survived. His commentaries are very valuable, as they contain many fragments of the older philosophers as well as of his immediate predecessors [Gillispie 1981, 12, 440]. 7Hippocrates of Chios was, according to ’s Commentary on Aristotle’s Physics, a merchant. After he was robbed by pirates, he went to Athens, possibly for litigation. Hippocrates stayed in Athens between 450–430 B.C. and studied geometry. He was one the first, who taught for money. We are told, that he squared the lunes, which are now named by him [Gillispie 1981, 6, 410–411].

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5th century B.C. In his Commentary on Aristotle’s Physics Simplicius claimed that according to the written by Eudemus of Rhodes8 (which is lost):

He [Hippocrates] made his starting-point, and set out as the first of the theorems useful to this purpose, that the similar segments of circles have the same ratios as the bases in square. And this he proved by showing that the squares on the diameters have the same ratio as the circles. [Ivor 1957, I, 239]

Since Eudemus is considered to be a reliable source, it is admitted that Hippocrates made some kind of demonstration of presented . But we cannot be sure whether he used Proposition X, 1 or not. I would like to emphasize that it is not the problem itself, which is representative of mathe- matical analysis, but the method of exhaustion, by which it could be solved. However we know nothing about how Hippocrates “proved” his statement. On the other hand Eudoxus in the 4th century B.C. proved many geometrical theorems using correct integration techniques based on Proposition X,1. He was renowned mathematician and physicist, so he could stand the task of establishing new theory of exhaustion. To answer the introductory questions about founding and founders of the first integration techniques one should consider following questions: Did Hippocrates in the 5th century B.C. formulated and proved Proposition X, 1? Is founder somebody who thought the idea and intuitively used it in one case, or someone who firmly formulated and proved the basic idea and explain its use in many problems?

References Boyer, C. B., The History of the and its Conceptual Development, Dover Publications, Inc., New York, 1959. Edwards, C. H., The Historical Development of the Calculus, Springer-Verlag New York, Inc., 1979. Euclid, The Thirteen Books of the Elements. Translated with introduction and commentary by Sir Thomas L. Heath, Volume I–III, Courier Dover Publications, New York, 1956. Gillispie, C. C. (ed.), Dictionary of Scientific Biography, Volume 1–12, Scribner, New York, 1981. Ivor, T., Selections illustrating the history of Greek Mathematics with an English Translation by Ivor Thomas, Volume I–II, Cambridge, Massachusetts, Harvard University Press, London, 1957.

8Eudemus of was an philosopher, who lived from ca. 370 BC until ca. 300 BC. Eudemus was born on the isle of Rhodes, but spent a large part of his life in Athens, where he studied philosophy at Aristotle’s . After Aristotle’s death, Eudemus returned to Rhodes, where he founded his own philosophical school, continued his own philosophical research, and went on editing Aristotle’s work. He was the first historian of and one of Aristotle’s most important pupils, editing his teacher’s work and making it more accessible. Eudemus’s collaboration with Aristotle was long-lasting and close, and he was generally considered to be one of Aristotle’s most brilliant pupils; he and Theophrastus of Lesbos were regularly called not Aristotle’s “disciples,” but his “companions.” Eudemus wrote Physics, Analytics, Categories, On discourse and On the , where he expounded and interpret Aristotle’s ideas. From his historical works are known titles and some fragments of History of , History of Geometry and . Two other historical works, History of and History of Lindos (Lindos is a port on Rhodes), are attributed to Eudemus, but here his authorship is not certain [Gillispie 1981, 4, 460–463].

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