Laura Winters Writing Sample SCHOOLS OF GREEK MATHEMATICAL PRACTICE: GEOMETRY The following writing sample is adapted from parts of the introduction, first chapter, and conclusion of my dissertation, in order to give an overall sense of the thesis and its scope. Pages 1-4 introduce the terminology and methods I use in my research; pages 5-11 give a brief analysis of the methodological and stylistic conventions that characterize the schools, with special attention to Euclid’s Elements; pages 11-20 survey the social and philosophical factors that contributed to the schools’ formations and that shaped the intellectual history of Greek geometry. INTRODUCTION Greek geometry, it is often claimed, was unlike any contemporary tradition of geometry: it neither followed the conventions of Babylonian mathematics nor used the methods of Egypt, despite the claims of Greeks themselves (including Herodotus) that the Egyptians were their first teachers in the art.1 Already, in its earliest extant texts Greek geometry appears as an advanced, highly systematic, theoretical study practiced by members of the philosophical schools. This has led many scholars to ascribe an almost mystical genius to the Greeks, and to see in the abstractness and complexity of their mathematics seeds of the modern habit of scientific inquiry.2 But by “Greek geometry” here scholars usually mean only geometry as Euclid did it. The particular conventions and standards of rigor practiced by Euclid have so thoroughly informed modern scholarship, that many still ascribe the characteristics of his work to Greek geometry in general. The textual legacy of Greek geometry, however, shows a diversity of methods, of which the style of Euclid is only a single strain. As impressive as the achievements of this strain are, their prominence in the historical record has eclipsed authors who worked with other methods.3 1 Herodotus, Histories II.109. Aristotle makes a similar claim in Metaphysics 981b21, but he uses the general µαθηµατικαί instead of Herodotus’ more specific γεωµετρίη. See also Proclus, In Primum Euclidis Elementorum Librum Commentarii, ed. Gottfried Friedlein (Leipzig: Teubner, 1873), 64.18-19. 2 Paul Tannery, La Géométrie Grecque: Comment Son Histoire nous Est Parvenue et ce Que nous en Savons: Essai Critique par Paul Tannery. 1. Ptie. Historie Génêrale de la Géométrie Élémentaire (Paris: Gauthier-Villars, 1887), 4. 3 For example, D. H. Fowler, The Mathematics of Plato’s Academy : A New Reconstruction, 2nd ed. (Oxford: Oxford University Press, 1999), 10: “My first characteristic of early Greek mathematics is negative: it seems to be 1 Laura Winters Writing Sample Most modern scholarship on Greek mathematics, therefore, focuses on the history of particular concepts, or on the similarities and differences between ancient and modern methods. But I mean to suggest for the first time that there were discernible schools of Greek mathematicians, which were recognized in antiquity but have not been handed down to modernity. I call one school the systematists. This is the group of which Euclid and Apollonius of Perga are members, and it is the style of geometry usually thought of as “Euclidean”4 and quintessentially Greek. The other school, that of Archimedes and Heron of Alexandria, I call the heurists. They represent perhaps a much larger group of ancient mathematicians,5 but they have been much more obscured by posterity. I shall argue that the Greek authors who wrote and worked in the tradition of Euclid were a small and specialized group, and that the others worked with very different methods and in independent networks; and furthermore that systematist geometry has its origins in the circle of Plato, which influenced its distinctive character. Preliminary Comments Several points of terminological and methodological clarity are in order. First, I have divided the Greek texts and authors into groups called “systematist” and “heurist.” These terms are original, as is the taxonomy they represent. They are meant to highlight the attitudes with which each group acquires, investigates, and presents mathematical knowledge. They do not, completely non-arithmetised.” or Morris Kline, Mathematical Thought from Ancient to Modern Times (New York: Oxford University Press, 1990), 49: “Classical Greek science, as we shall see, was qualitative.” 4 I use the term “Euclidean” here to describe the style and methods used by Euclid himself, not its meaning for present-day mathematics. In modern mathematics, “Euclidean geometry” refers to geometry which assumes a certain postulate about the nature of space, specifically Euclid’s fifth postulate, that parallel lines do not ever meet when produced. Both systematist and heurist geometry are “Euclidean” according to this modern usage. “Non- Euclidean geometry” explores what can be logically claimed about space and spatial objects if the parallel postulate is not necessarily true. The validity of Euclid’s parallel postulate has been a topic of debate since antiquity, but non- Euclidean geometry was only developed in the nineteenth century by authors such as Bolyai, Lobachevsky, and Riemann. 5 Many of the heurists’ methods bear similarities to the geometry of ancient Babylon and Egypt (see note 145), which may indicate a wider mathematical diaspora and an older tradition. 2 Laura Winters Writing Sample however, express the full extent of the differences between the two schools. For example, the heurists treat geometrical objects as having physical presence, whereas the systematists idealize them. No single term can cover all such differences, because they are a historical accident rather than the necessary result of any mathematical theory or practice. Next, I shall avoid modern algebraic notation. Since the time of Descartes, geometry has used algebraic notation as its principal means of problem-solving. The innovation of Descartes was the division of all space into a three-dimensional numbered grid, so that any spatial object could be described by the set of coordinates that fulfill a given equation. No ancient Greek work of either school used these methods. Any text that uses modern algebraic notation to express Greek geometry is a translation, with all of the risks and benefits that such entails. But in this case it is a translation not only of language but of the methods themselves, and so it raises the question of how far the means of expressing and “doing” mathematics can actually differ before the substance of the text itself is lost. A scholarly debate over whether or not algebraic notation is an appropriate translation choice for the ancient texts has continued for nearly a hundred years.6 Both sides have their merits, but since my purpose is to examine different methods used by Greeks to solve mathematical problems, I shall keep strictly to their idiom, including the use of lettered diagrams and a good deal of specialized vocabulary, which I shall gloss whenever appropriate. 6 For a survey of the debate, see Viktor Blåsjö, “In Defence of Geometrical Algebra,” Archive for History of Exact Sciences 70, no. 3 (May 1, 2016): 325–59. The major works include: H. G. Zeuthen, Die Lehre von den Kegelschnitten im Altertum., trans. R. J. D. von Fischer-Benzon (Kopenhagen: A.F. Höst & Sohn, 1886); Bernard Vitrac, “Peut-on Parler d’algèbre Dans Les Mathématiques Grecques Anciennes ?,” Mirror of Heritage (Ayene-Ne Miras) 3 (2005): 1–44; Sabetai Unguru, “On the Need to Rewrite the History of Greek Mathematics,” Archive for History of Exact Sciences 15, no. 1 (1975): 67–114; Szabó, Anfänge Der Griechischen Mathematik. (München, Wien: Oldenbourg, 1969); and Paul Tannery, De la Solution géométrique des problèmes du second degré avant Euclide. (Bordeaux: Impr. de G. Gounouilhou, 1884). 3 Laura Winters Writing Sample A final point of terminological clarification: when I say that Greek geometers used different methods, I am not talking about the difference between what we call theoretical and applied mathematics. Many scholars have already observed that some Greeks practiced purely theoretical mathematics and others did not. 7 But when scholars use the terms “theoretical” and “applied,” they are referring to how mathematical information is used, not how it is acquired. I argue that the primary difference in method between one group of geometers and another has little to do with the modern categories.8 Authors outside the systematist group wrote very theoretical works (such as Archimedes’ Spirals), and Euclid himself wrote a work on astronomy, which most modern mathematicians would classify as applied. The primary difference between these traditions of Greek geometry is not between theory and application, but rather has to do with how mathematical objects (like lines and figures) are conceptualized and how knowledge about them is generated. THE SYSTEMATIST SCHOOL Euclid 7 e.g. Kline, Mathematical Thought from Ancient to Modern Times, 49-50, “Educated people did not concern themselves with practical problems […] Mathematical thought was thus separated from practical needs […] When the barrier between the cultured and slave classes was breached in the Alexandrian period (300 B.C. to about A.D. 600) and educated men interested themselves in practical affairs, the emphasis shifted to quantitative knowledge […].” See also S. Cuomo, Ancient Mathematics (London ; New York: Routledge, 2001), 79-105 for a comparison between Euclid and some of the geometers I have called “heurist.” 8 Ancient and Late Ancient authors recognized a difference between the mathematical