Eudoxus system of concentric spheres

The Babylonians were concerned with predicting the time at which a particular phenomenon (e.g. a planetary opposition) would occur, since it was the date that was ominous. Their astronomy was thus concerned with the analysis of discrete processes. On the other hand, as we shall see, the Greeks in their astronomy focused on predicting where a celestial body would be at a given time, and they were thus concerned with modelling a continuous process, which naturally leads to the use of geometrical methods. The first person to produce a geometrical model of celestial motions was Eudoxus, who was born in Cnidus on the western coast of Asia Minor in about 400 BC. According to Diogenes Laertius' Lives of Philosophers (written in AD third century) he was taught geometry by Archytas of Tarentum, one of the leading Pythagorean philosophers, and studied with Plato, who was about 30 years older than him. None of Eudoxus' works have survived. Most of the information we have about his system of concentric spheres comes from a brief contemporary account in Aristotle's Metaphysics and a more substantial description due to Aristotle's influential commentator Simplicius (AD sixth century).

1 Simplicius based his work on that of the philosopher Sosigenes (AD second century - not to be confused with the Sosigenes who helped Julius Caesar reform the calendar in the first century BC), and ultimately on the lost history of astronomy by Eudemus, who lived only a generation after Eudoxus. It is important to recognize that Simplicius was writing 900 years after the event and he may well have been putting his own interpretation on Eudoxus' work. By itself, the information supplied by Aristotle and Simplicius does not provide a clear picture of Eudoxus' model, but in a classic paper in 1877, the Italian Giovanni Schiaparelli produced a reconstruction of the Eudoxan system which has remained the accepted version ever since. Recently, however, Yavetz (1998, 2001) has argued that Schiaparelli's interpretation is not the only one that can be put on the original sources which is consistent with what we know about Greek astronomy of the time, and that there is no reason to believe one rather than the other. Here we will stick to the standard interpretation.

Eudoxus had a considerable influence in many areas of mathematics. In fact, he was the inspiration behind two of the most profound mathematical advances of the fourth century BC. The first of these was the theory of proportion, which formed the basis of much of Euclid's Elements (see p. 36), and the second was the method of exhaustion, which was used extensively in the same work and represents the beginnings of the subject now known as the 'integral calculus'. Both of these contributions were of fundamental significance for the development of mathematics, but it would appear that, as far as other scholars from antiquity were concerned, his main claims to fame were his astronomical dis-

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