Argument

In this paper I offer some reflections on the thirty-second proposition of Book I of Euclid’s Elements, the assertion that the three interior angles of a triangle are equal to two right angles, reflections relating to the character of the theorem and the reasoning involved in it, and especially on its historical background. I reject a common view according to which there was at some time a petitio principii in the theory of parallels and argue that certain kinds of skepticism about Proclus’ historical reports are excessive. The evidence we have makes it reasonable to suppose that the so-called common notions were made explicit in the earlier fourth century BCE and the postulates, including the parallel postulate, somewhat later than that. On the other hand, it seems clear that some proof of I,32 was available by the mid-fifth century. I attempt to describe what such a proof would have been like and reflect on its significance for our understanding of early Greek geometry.

Argument

My purpose in this paper is to venture an explanation for how the peculiar Greek idiom “application of area” came into being, and why Greek mathematicians preferred this idiom to one which seems more straightforward and is undoubtedly older. To those conversant with the Greek theory of , especially Elements VI.28–29, there is little new in my exposition, and they may want to skip sections 1 and 2. I present them nevertheless, partly to initiate newcomers, partly in order to show that Elements II.5 works as a maximum condition (analogous to VI.27) besides being, like II.6, a useful lemma in the solution of the problem. My main point is the shifting of focus from one rectangle to another in problems 3a and 4a, which may have given rise to the esoteric – and more general – wording of problems 3b and 4b.

Argument

The past five years have seen revisions of what had been regarded as the single definitive reconstruction of Eudoxus’ homocentric sphere theory. Crucial to these revisions, which present alternative constructions for the moon (Mendell 1998a, 1998b) and the five planets (Yavetz 1998), is a careful assessment of Simplicius’ testimony regarding the Eudoxan theory. The purpose of this paper is to show that even though we can account for lunar motion by means of two distinct homocentric models, both models involve forced and ambiguous readings of Simplicius’ account. Both reconstructions adequately accommodate the lunar month and the 18.5-year precession of the lunar nodes in a homocentric model that fits the general spirit of Eudoxus’ approach to the problem. Both are also in full accord with Aristotle’s short, general account of the Eudoxan system. However, since neither of them can be naturally reconciled with Simplicius’ text, and since we know of no other appropriate alternatives, it seems to follow that Simplicius’ account may suffer from serious technical inadequacies, and that only limited historical weight can be given to his testimony regarding the details of the system.

Argument

This paper proposes an alternative view to Becker’s reconstruction of pre-Eudoxean theory of proportion. No extant document explicitly demonstrates the alleged alternation of theories of proportion before Book V of the Elements. Books V and VI of the Elements are not so complete as a theory of proportion in the abstract, and can be interpreted better as a collection of propositions useful in geometry. It follows then that older theories, if they existed at all, must have been less complete. Prevailing interest in geometry on the part of Greek mathematicians is also visible in some expressions in Book VI for specific ratio and proportion used only in definite geometric context. It is therefore more fruitful to consider the “theory” of proportion before Eudoxus as an aggregate of techniques about proportion that are useful in geometry, rather than as the result of a conscious effort to build a logically consistent set of propositions based on a definition.

Argument

Apollonius of Perga’s Conica, like almost all Greek mathematical works, relies heavily on the use of proportion, of analogia. Analogy as the assertion of a resemblance, however, also plays a role in the Conica. The homologue, which Apollonius introduces in Book VII, is a striking example of analogia in both senses. On the one hand, the homologue is defined by means of a precise proportion relating the diameter and latus rectum of a conic section to fixed segments along the diameter. On the other hand, Apollonius’ use of the homologue in Book VII makes it clear that he meant it truly to evoke the latus rectum in the reader’s mind; in this way, Apollonius treats the homologue as an image of the latus rectum. The example of the homologue thus suggests the possibility that, in Greek mathematics, proportion was not only a vital manipulatory tool but also a means of making images.

Argument

The extant sources for ancient Egyptian mathematics are extremely limited. It is therefore necessary to read the few sources carefully and use additional information from further Egyptian sources in order to achieve the most detailed picture possible. Traditional approaches to Egyptian mathematics have provided only a superficial account of mathematical practices and almost no information about the role of mathematics within Egyptian culture. To enlarge our knowledge it is crucial to use a different methodological approach in the analysis of ancient mathematical techniques. In addition, it is indispensable to contextualize the mathematical problems with sources that are not specifically mathematical per se. In this article I discuss several possibilities for these additional sources, such as administrative texts, reliefs found in tombs, and other archaeological evidence. I exemplify the use of these sources with two problems from the Moscow mathematical papyrus.

Argument

In this article, I compare Sabetai Unguru’s and Wilbur Knorr’s views on the historiography of ancient Greek mathematics. Although they share the same concern for avoiding anachronisms, they take very different stands on the role mathematical readings should have in the interpretation of ancient mathematics. While Unguru refuses any intrusion of mathematical practice into history, Knorr believes this practice to be a key tool for understanding the ancient tradition of geometry. Thus modern historians have to find their way between these opposing views while avoiding an unsatisfactory compromise. One approach to this, I propose, is to take ancient rhetoric into account. I illustrate this proposal by showing how rhetorical categories can help us to analyze mathematical texts. I finally show that such an approach accommodates Knorr’s concern about ancient mathematical practice as well as the standards for modern historical research set by Unguru 25 years ago.

Argument

Abstraction is commonly valued as being essential to mathematics or even consubstantial with it. In relation to this belief, mathematical texts from Antiquity – be they from Babylon, Egypt, or China –, which are composed of seemingly concrete problems and algorithms solving them, have been considered to be practice-oriented and deprived of theory. This paper offers an alternative view on both issues. Relying on evidence given by third-century commentaries on The Nine Chapters on Mathematical Procedures, a Chinese treatise composed around the beginning of the Common Era,this paper argues that, in the scholarly mathematical tradition of ancient China, generality was more valued than abstraction. In this respect, problems must be interpreted as paradigms, in the sense grammarians use the term. One of the goals of theoretical endeavor was to exhibit the most general operations possible, and this purpose can be read in quite a few specific features of mathematical practice and concepts. Moreover, it is shown that abstraction is not absent from The Nine Chapters, but that it entertains with generality a relationship that requires analysis and can by no means be taken for granted. These ancient sources hence constitute an invitation to develop a critical approach toward the relation of mathematics to abstraction and generality taken separately, as well as the relation of these two values with each other.