*Elements*), geometrical models of the heavens with quantitative predictive power

The ancient Greeks traced the origin of many aspects of their

The upshot of this work has been to change the terms of the debate: the burden of proof is now on those who would deny that contact with neighboring civilizations contributed in a significant way to the Greeks’ distinctive cultural achievements. Yet it also raises a challenge to define more precisely the modalities of cultural influence, which have too often been conceived of as a matter of simple

*First-order and second-order knowledge.*

*Modes of diffusion.* I distinguish between diffusion through

My main argument is twofold: first, insofar as

Two preliminary points are crucial. First I do not claim that the Greeks *invent-**ed* *only* forms of such knowledge that were influential.

With these points in mind I now turn to a brief examination of four related areas of

## 8.1 Cosmology

By

First, these Greek cosmologies offer a certain kind of explanation of the universe. They typically seek to reduce the diversity of observable phenomena to the interaction of a small number of factors, which behave in consistent ways in a wide variety of contexts. And they are “rational” in the sense that they are supported by explicit reasons and arguments. For example, Anaximenes explains all physical transformations by condensation and rarefaction, and offers evidence (the behavior of breath exhaled from the mouth) that heating and cooling can be reduced to those processes.

Second, early Greek cosmologies typically envision the large-scale structure of the universe in terms of geometrical models with a high degree of symmetry. For example, Anaximander conceives of the sun, moon and stars as apertures in a set of concentric rings, which are supposed to explain phenomena such as eclipses and the phases of the moon.

Finally, analogies with various crafts are an important source of both the particular explanations and the geometrical models characteristic of this tradition. Thus Anaximander’s cosmic rings are likened to wheels, while Anaximenes likens condensation and rarefaction to the production of felt from wool.

The earliest Greek cosmologies are an example of

Mesopotamian cosmologies are reflected in texts whose goals were assuredly not to construct a definitive cosmic picture to serve as the framework for inquiry about natural phenomena. Rochberg 1993, 51

That, of course, is precisely what the Greeks *were* doing. Yet it is surely no accident that this particular tradition in Greek thought begins in Miletus, at the heart of the cultural crossroads that was Asia Minor in the sixth century BCE. Through Parmenides’ critique in the early fifth century BCE Greek cosmological thought becomes second order, as explicit standards for the validity of cosmological accounts and arguments are developed and articulated. But this critique of course presupposes the existence of the earlier systems.

## 8.2 Mathematics

A distinctive achievement of early Greek *Metrica* (probably first century CE) testify to another type of Greek mathematical knowledge, one concerned with practical problems of calculation and mensuration rather than deductive proof. Such texts may reflect the diffusion of much older techniques from the near East Neugebauer 1957; Høyrup 1996. While there is very little direct evidence, knowledge of basic

My main concern here is with the notion of mathematical proof as we find it in Euclid’s *Elements*, which developed between the beginning of the sixth and end of the fourth centuries BCE. Since we have almost no Greek mathematical texts from this period, any reconstruction of these developments is unavoidably speculative. What we can do is compare the *Elements* itself with the extraordinarily rich sources for Babylonian

The case of the *a fortiori* to the claim, widespread in mid-twentieth century scholarship, that the Babylonians developed a kind of *Elements*). Recent work by Jens Høyrup has shown that much of Old Bablyonian

The factors that drove the development of the notion of proof in the Greek context seem to have been quite different from those which stimulated critical reflection in the

the development of new mathematical concepts including incommensurability, which both reflected and called for analysis of the conditions under which they held of mathematical objects;

the rapid increase in the number of mathematical results discovered by the investigation of such concepts;

the development of a new kind of notation in which letters of the

the possible impact of an emerging concern with

An additional factor *may* have been familiarity with some of the results of Babylonian *Meno* (82–86) where Socrates leads a slave boy to recognize a special case of the

## 8.3 Medicine

*On Diseases 2,* for example, consists of a catalog of diseases indicating the signs by which they can be recognized and the appropriate treatment. There are affinities between the therapies mentioned in Greek texts and earlier material Goltz 1974; Geller 2010. Further affinities have been noted between *Odyssey*. There is thus no reason to reject the notion that the

But we can also identify in the Hippocratic texts a concern with methodological reflection that is not present in the material from the neighboring cultures. In particular, the conception of medicine as a form of expertise (*technê*) that has a basis in explanatory theory is developed by some (though by no means all) of the Hippocratic writers. This development was a result of several interacting factors. The impulse toward highly reductive explanation that can be traced in early cosmological thought had its impact on medicine, as the cosmological theorists tended to speculate on the construction of the human body or the *Epidemics*, which contain *Epidemics* testify to an ongoing engagement with the problem of relating general rules to particular cases, for as well as individual case histories they also contain an extensive body of prognostic and therapeutic generalizations that are closely related to the material in what is perhaps the most representative and influential Hippocratic text of all, the *Aphorisms*. The geographical range of the Greek doctors also prompted reflection on the conditions under which generalizations such as those expressed in the *Aphorisms* could be considered valid, for a rule that held under one set of climatic or geographical conditions might not hold elsewhere. The important early treatise *Airs, Waters, Places,* for example, sets out a general theory of the effects of environmental factors on human beings, and incorporates it into a wide-ranging ethnographic discussion of foreign lands and peoples. Similarly, the treatise *Prognostic* ends by saying that the prognostic “signs” it sets forth will be valid everywhere, not just in certain locales. Reflection on geographical variation and individual differences stimulated the development of general theories of the working of humoral factors such as phlegm and bile, which were supposed to explain the effects of the environment on all individuals *wherever* they might be located, and *whatever* their constitution might be.

Thus in *why* they worked.

## 8.4 Astronomy

Mathematical modeling of planetary phenomena with the goal of exact prediction arose first in Babylon, and reached the pinnacle of its development during the Seleucid period Neugebauer 1957. Instead of constructing geometrical models of the cosmos, the Babylonians used combinations of *a fortiori* the general development of these methods) must be laboriously reconstructed. The scribes do not seem to have committed their methods to writing; nor did they record whatever ideas they may have had about the meaning or general significance of the periodicities that their work so accurately represented.

The spread of these Babylonian methods across the Greek-speaking Hellenistic world is the most well-documented and extensive case of the transmission of scientific knowledge in the ancient Mediterranean world. That Hipparchus in the second century BCE and Ptolemy in the second century CE used Babylonian parameters in constructing their geometrical models has long been recognized. But it is not just a question of adopting Babylonian parameters: Babylonian methods also spread across the Greek-speaking world to an extent that has only recently become clear Swerdlow 1998. Not only Hipparchus himself, but also pre-Ptolemaic writers such as Hypsicles and Geminus make use of Babylonian methods, often without drawing attention to their provenance Evans 1998; Berggren and Evans 2006. A number of papyri from Greco-Roman Egypt indicate that

*Almagest* on the inadequacy of all earlier attempts to offer theories of planetary motion are as much a criticism of the Babylonian or “Greco-Babylonian” approaches as they are a claim to his own original achievement (*Almagest* 9.2; cf. Neugebauer 1945). Ptolemy made use of Babylonian parameters, which were the essential foundation of a system with quantitative predictive power, but the diffusion of Babylonian methods stimulated him to develop his own strictly geometrical approach. The end result was the Ptolemaic system, a powerful fusion of highly accurate prediction with an overarching cosmological framework which, despite internal tensions, dominated

## 8.5 Conclusions

I hope that this survey, however brief and speculative, has at least succeeded in showing the usefulness of the concepts I have introduced for understanding the impact of Egypt and the Near East on the development of

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