aspect of nature but rather a moral and psychological dilemma. In short, Bernal's argument serves, if anything, to strengthen the case for recognizing a revolutionary approach to the study of nature in classical Greece. Whether, in this connection, one wishes to talk about a "Greek miracle" is perhaps just a matter of taste. (I myself abhor the phrase and would be quite pleased never to hear it again.) 88
It may be helpful to add a summary of my main claims in this essay:
The Egyptians never invented a mathematical astronomy in the sense that the Babylonians and the Greeks did. Such an astronomy must include both (i) some form of systematic--though not necessarily very precise--observation of sun, moon, and planets, and (2) some observationally based procedures for computing new celestial observations from known ones. Whether the Egyptians ever made any celestial observations on which computations could have been based may be open to dispute; but it should be noted that mere recognition of particular stars or constellations is hardly enough. There is, however, no evidence that the Egyptians developed, on their own, the requisite computational procedures--though, of course, by Hellenistic times Egyptian astronomers were using easily identifiable Babylonian and Greek procedures. Was Greek astronomy, then, entirely uninfluenced by Egyptian ideas? Well, almost entirely, the exception being the adoption of the Egyptian calendar by later Greek astronomers, including Ptolemy.
Egyptian mathematics never approached the depth of understanding revealed in the most advanced Babylonian mathematics, which was in turn far surpassed by the Greeks, so that it is difficult to see how the peak Egyptian achievements--even generously extrapolated to hypothetical results now lost to us--could ever have led to Greek mathematics with its clear conception of rigorous demonstration and its characteristic methods of formulating and solving problems. Specifically, the remarkable Babylonian result on Pythagorean triples implies a deeper understanding of number-theoretical properties (such as the property of being what Neugebauer calls a "regular" number, that is, a number whose reciprocal is expressible as a finite sexagesimal fraction) than anything implicit in the Egyptians' numerical formulas for volumes, areas, or π; yet the failure of the Babylonians to grasp the irrationality of(for which they had worked out a good approximation) indicates how far they fell short of the Greeks. But the surpassing Greek achievements in mathematics stemmed from the development of logical procedures for mathematical proofs (such as indirect or reductio ad absurdum proof) and the formulation of powerful general principles (such as a general theory of proportion and a general technique for finding areas and volumes by the method of exhaustion). In what ways, then, could the Greeks have been stimulated by Egyptian mathematics? Perhaps--and this is only speculation--as a chal-