Archimedes’ Sand-Reckoner presents a system for naming extraordinarily large numbers, larger than the number of grains of sand that would fill the cosmos. Curiously, Archimedes addresses the treatise not to another specialist but to King Gelon II of Syracuse. While the treatise has thus been seen as evidence for the dynamics of patronage, difficulties in both Archimedes’ treatment of Gelon and his discussion of astronomical models make it fit incongruously within contemporary court and scientific contexts. This article offers a new (...) reading of the Sand-Reckoner based on a reconsideration of the relationship between author and addressee: deferring assumptions about the historicity of that relationship, it analyzes Gelon’s role in the treatise with respect to both the stylistic features of Archimedes’ prose and a broader tradition of narratives about a variety of cultural actors who engage with kings, speaking not so much truth as wit to power. Such a reading resolves the social and scientific difficulties of the treatise, and develops the literary-experimental qualities of Hellenistic science. In turn, the article proposes a revised approach, sensitive to broader patterns of authorship, to understanding ancient scientific authors’ relationship to royal authority. It concludes, finally, that the royal patronage seemingly exemplified by the Sand-Reckoner had greater significance as a cultural trope than as a social institution. (shrink)

This article examines the structures of the epico-Purāṇic divisions of time (yugas/sandhyās/kalpas) and asks what is joined by the Purāṇic ages known as yugas or joinings. It concludes that these structures reflect a combining of three systems of number—Greek acrophonic, Babylonian sexagesimal and Hindu decimal— represented as divisions of time. Since most interpretations of these structures, particularly yugas, focus on questions of dharma and its decline over the various ages rather than on number, it asks in conclusion if there is (...) any necessary relationship between number and dharma. (shrink)

The Latin edition of the Mathematicae Collectiones was published in print in 1588, thirteen years after Federico Commandino’s demise. For his Latin version of Pappus’s work, Comandino used two Greek codices, formerly identified by Treweek. In this article, another Greek manuscript, revised and annotated by Commandino, is revealed. Two letters from Commandino to Ettore Ausonio shed new light on the edition of Pappus’s Collectio and show the partnership between the two mathematicians in elaborating supplementary proofs to include in the comments. (...) Using these letters, we can date the first draft of the Latin version in the late 1560s. The posthumous edition of the Mathematicae Collectiones involved Commandino’s disciples and, in particular, Guidobaldo del Monte. The comparison between the manuscripts and the printed edition reveals an important role played by the disciple in revising the Latin translation of his master. (shrink)

I present a systematic interpretation of the foundational purpose of constructions in ancient Greek geometry. I argue that Greek geometers were committed to an operationalist foundational program, according to which all of mathematics—including its entire ontology and epistemology—is based entirely on concrete physical constructions. On this reading, key foundational aspects of Greek geometry are analogous to core tenets of 20th-century operationalist/positivist/constructivist/intuitionist philosophy of science and mathematics. Operationalism provides coherent answers to a range of traditional philosophical problems regarding classical mathematics, such (...) as the epistemic warrant and generality of diagrammatic reasoning, superposition, and the relation between constructivism and proof by contradiction. Alleged logical flaws in Euclid can be interpreted as sound operationalist reasoning. Operationalism also provides a compelling philosophical motivation for the otherwise inexplicable Greek obsession with cube duplication, angle trisection, and circle quadrature. Operationalism makes coherent sense of numerous specific choices made in this tradition, and suggests new interpretations of several solutions to these problems. In particular, I argue that: Archytas’s cube duplication was originally a single-motion machine; Diocles’s cissoid was originally traced by a linkage device; Greek conic section theory was thoroughly constructive, based on the conic compass; in a few cases, string-based constructions of conic sections were used instead; pointwise constructions of curves were rejected in foundational contexts by Greek mathematicians, with good reason. Operationalism enables us to view the classical geometrical tradition as a more unified and philosophically aware enterprise than has hitherto been recognised. (shrink)

A discussion of Lloyd's Tarner Lectures at Trinity College. The importance of Lloyd's previous scholarship is characterized and these sweeping, erudite lectures are placed in the context of that scholarship. In the broadest terms, the lectures are a call to culturally and historically comparative study of human reasoning. At their heart is a comparative history of scientific theorizing from the ancients through to modern science. Lloyd rejects the positivist picture, and the view of modern and ancient science as discontinuous; he (...) urges scholars to undertake comparative work on the ancient sciences in different traditions. This critical notice evaluates Lloyd's view and raises several questions for further reflection. (shrink)