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Notes linguistiques et critiques sur le livre II des Coniques d'Apollonius de Perge (2e partie)
Abstract
Follow-up and end of the article the first part of which was published in the Revue des Études Grecques, t. 112 (1999/2), 409-443. This second part is devoted to examining propositions 31-53 in Book II of the Conies. As in the first part, it can also be used to serve as a complement to the Dictionnaire historique de la terminologie géométrique des Grecs by Ch. Mugler. — It ends with a Supplement to the critical notes on Book I of the Conies.
The present study aims to analyze the evolution of methods used in the publication of mathematical works from Greek antiquity. The progress made in the history of texts offers the instruments necessary for measuring what can separate the author’s text in its original form from the various editions produced in the course of time. The evolution of book production techniques, the influence exerted by the circles that took over the transmission of scientific knowledge, and the conceptual bases that have influenced the intervention of publishers are all elements to be taken into account by the historian of science in judging the exact nature of the transmitted text and evaluating what its present form owes to the path taken to reach us.
This paper is a contribution to our understanding of the technical concept of given in Greek mathematical texts. By working through mathematical arguments by Menaechmus, Euclid, Apollonius, Heron and Ptolemy, I elucidate the meaning of given in various mathematical practices. I next show how the concept of given is related to the terms discussed by Marinus in his philosophical discussion of Euclid’s Data. I will argue that what is given does not simply exist, but can be unproblematically assumed or produced through some effective procedure. Arguments by givens are shown to be general claims about constructibility and computability. The claim that an object is given is related to our concept of an assignment—what is given is available in some uniquely determined, or determinable, way for future mathematical work.
The aim of this article is to present and discuss the language of the «givens», a typical stylistic resource of Greek mathematics
and one of the major features of the proof format of analysis and synthesis. I shall analyze its expressive function and its
peculiarities, as well as its general role as a deductive tool, explaining at the same time its particular applications in
subgenres of a geometrical proposition like the locus theorems and the so-called «porisms». The main interpretative theses of this study are the following: the language of the
«givens» (1) is the standard idiom in which “existence and uniqueness” of a mathematical object was proved, (2) was conceived
as an unified framework reducing to a strictly deductive format disparate argumentative steps such as deductions, constructions,
and calculations.
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