Ptolemy's mathematical approach

Description

The study is an examination of the mathematical methods of Ptolemy and his predecessors. It attempts, so far as possible, to situate this work in the context of what we know about the rest of Greek mathematics and the exact sciences, with little or no reference to current scientific and mathematical knowledge.Each of these chapters describes a domain of Greek mathematical practice that is not witnessed in the theoretical texts and is generally left out of discussions of Greek mathematics. Moreover, in each case, I help the reader develop a sense for the methods and practices of the ancients instead of focusing simply on their results.The third chapter is an examination of all of the evidence we have for the so-called Menelaus Theorem, the fundamental theorem of ancient spherical trigonometry. It studies the texts of Ptolemy, his predecessors and his commentators and shows that the line of transmission cannot have been as straightforward as has previously been assumed. This is followed by an investigation of Ptolemy's practices in applying the fundamental theorem. This study of Ptolemy's spherical astronomy acts as a case study which gives us insight into the deductive structure of Ptolemy's exact science. This investigation allows us to develop a sense for how the ancient mathematical astronomer used these methods to produce new results.The final chapter is an exegesis of ancient methods of projecting the sphere onto the plane. It explores the texts of Ptolemy and his predecessors which are concerned with projecting the sphere either for the purpose of drawing maps or in order to model the sphere and solve for arc lengths. This leads to discussions of two important ancient methods of doing spherical geometry.The second chapter is a study of the first and most crucial application of these methods: the development of the chord table and its application to trigonometric problems. It also examines the trigonometric methods of the Hellenistic mathematical astronomers and shows how these fundamentally differed from Ptolemy's practice. It develops a general picture of the mathematical practices used in the trigonometry by means of chord tables.After a brief discussion of Ptolemy's philosophy of mathematics, the first chapter gives a classification of types of mathematical text found in Ptolemy and the Greek applied mathematical tradition in general. This is followed by sections that deal with the use of ratio and tables in Ptolemy's work. In order to apply metrical methods to geometrical problems, Ptolemy uses proportions as equations and develops tables to model continuous functions. Both of these practices, although natural to us, are unusual in the context of Greek mathematics. I examine the implicit assumptions and explain how these methods serve the applied mathematician.

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