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Introduction to Lunules
The Methods of Eudemus and Alexander Compared
Hippocrates of Chios lived in the middle of the 5th. cent. BCE. He is reported to have associated with Anaxagoras and to have written a book of mathematical 'elements', presumably a treatise collecting basic propositions. Aristotle also reports views of his and his student Aeschylus on the nature of comets. However, he is particularly associated with two problems, one is the reduction of the problem of doubling a cube to the discovery of the solution to a double mean proportion (a : b = b : c = c : 2a). The other is the discovery of a crescent or lunule, i.e., curvilinear figure, that can be squared. This may have been regarded as a contribution to the problem of squaring a circle.
In his enormous Commentary on Aristotle's Physics, Simplicius (6th cent. CE), our only source for the mathematical details of the problems, cites two sources, Alexander of Aphrodisias (late 2nd cent. CE) and Eudemus of Rhodes (4th cent. BCE). The methods of Alexander and Eudemus are very different, while Alexander only provides the first and fourth of Eudemus' examples, which also happen to be easier than the other two. Alexander's method is conceptually simpler and happens to be more general than the method used by Eudemus. Indeed, it encompasses the class of all lunules whose construction does not require the squaring of a circle. This generality may be incidental; the simplicity of the method may indicate nothing more than that the method of Alexander was developed as a school presentation. Although Eudemus' account is the older, it may be impossible to determine which, if either, is original to Hippocrates, or even whether Hippocrates discovered all four lunules discussed by Eudemus.
This module consists of a mathematical analysis of these two descriptions of Hippocrates' work on lunules in four sections. There is general introduction to the mathematics of lunules. This discussion does not concern Hippocrates per se and is anachronistic. Then there is an analysis of Alexander's method . This analysis assumes that the method may be generalizable, and so uses a lunule that is not in any ancient account. The analysis of Eudemus' method likewise uses a lunule not in any ancient account. Finally, the two methods, of Alexander and Eudemus, are compared.