Tag: Ptolemy

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The Almagest – Book I: Corollaries to Ptolemy’s Theorem

If you’ve been following the Almagest posts, you’ll recall that we’ve done some work to derive the chord lengths of various angles. But Ptolemy’s goal is to derive the chord length for every angle between 0-180º in 12º intervals. To do that, we’re going to have to develop some new tools using Ptolemy’s theorem on the angles we already know in order to add, subtract, and divide them. These new tools are referred to as corollaries since they come from applications of Ptolemy’s theorem.

The first one comes from the following diagram:

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The Almagest – Book I: Ptolemy’s Theorem

We shall next show how the remaining individual chords can be derived from the above, first of all setting out a theorem which is extremely useful for the matter at hand.

Having derived a handful of special angle-chord relationships, Ptolemy next set out to derive a more general theorem to get the rest. So Ptolemy constructs a new diagram from which to start his calculations.

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The Almagest – Book I: Introduction

Ptolemy begins by drawing a distinction between practical and theoretical philosophy. The distinction is not defined here although Ptolemy gives “moral virtues” as an example of the former and “understanding of the universe” as an example of the latter. Likely, he is referring to Aristotle’s work, Metaphysics (Ptolemy almost never cited sources and Aristotle is the one name which is later cited which indicates how much Ptolemy drew from him), which defined three types of philosophy: The theoretical (knowledge for its own sake), the practical (morality), and the productive (works of utility and beauty). Continue reading “The Almagest – Book I: Introduction”