In 1813, J.-V. Poncelet discovered that if there exists a polygon of n-sides, which is inscribed in a given conic and circumscribed about another conic, then infinitely many such polygons exist. This theorem became known as Poncelet’s porism, and the related polygons were called Poncelet’s polygons. In this article, we trace the history of the research about the existence of such polygons, from the “prehistorical” work of W. Chapple, of the middle of the eighteenth century, to the modern approach of (...) P. Griffiths in the late 1970s, and beyond. For reasons of space, the article has been divided into two parts, the second of which will appear in the next issue of this journal. (shrink)

The first part of the article appeared in the previous issue of this journal. It deals with the history of the research on Poncelet’s porism, and related subjects, which were developed from the middle of the eighteenth century until the end of the nineteenth century. In this second part, we take the research developed in the twentieth century. We also offer a comparison of the main works on the subject, and we draw some conclusions.

Of Euclid’s lost manuscripts, few have elicited as much scholarly attention as the Porisms, of which a couple of brief summaries by late-Antiquity commentators are extant. Despite the lack of textual sources, attempts at restoring the content of this absent volume became numerous in early-modern Europe, following the diffusion of ancient mathematical manuscripts preserved in the Arabic world. Later, one similar attempt was that of French geometer Michel Chasles (1793–1880). This paper investigates the historiographical tenets and practices involved in (...) Chasles’ restoration of the porisms, as well as the philosophical and mathematical claims tentatively buttressed therewith. Echoes of the Quarrel of the Ancients and the Moderns, and of a long-standing debate on the authority and usefulness of the past, are shown to have decisively shaped Chasles’ enterprise—and, with it, his integration of mathematical and historical research. (shrink)

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. (shrink)