Book 1 of Euclid's Elements opens with a set of unproved assumptions: definitions, postulates, and ‘common notions’. The common notions are general rules validating deductions that involve the relations of equality and congruence. The attested postulates are five in number, even if a part of the manuscript tradition adds a sixth, almost surely spurious, that in some manuscripts features as the ninth, and last, common notion. The postulates are called αἰτήματα both in the manuscripts of the Elements and in the (...) ancient exegetic tradition. It is not said, however, that this denomination is original, as it coincides with the nomen rei actae associated with the verbal form introducing the postulates themselves. Since antiquity it has been recognized that the postulates naturally split into two groups of quite different character: the first three are rules licensing basic constructions, the fourth and the fifth are assertions stating properties of particular geometric objects. I transcribe postulates 1–3 in the text established by Heiberg : ᾐτήσθω ἀπὸ παντὸς σημείου ἐπὶ πᾶν σημεῖον εὐθεῖαν γραμμὴν ἀγαγεῖνκαὶ πεπερασμένην εὐθεῖαν κατὰ τὸ συνεχὲς ἐπ᾽ εὐθείας ἐκβαλεῖνκαὶ παντὶ κέντρῳ καὶ διαστήματι κύκλον γράϕεσθαι. (shrink)

ArgumentThis article presents ancient documents on the subject of homeomeric lines. On the basis of such documents, the article reconstructs a definition of the notion as well as a proof of the result, which is left unproved in extant sources, that there are only three homeomeric lines: the straight line, the circumference, and the cylindrical helix. A point of particular historiographic interest is that homeomeric lines were the only class of lines defined directly as the extension of a mathematical property, (...) a move that is unparalleled in Greek mathematics. The far-reaching connections between mathematical homeomery and key issues in the ancient cosmological debate are extensively discussed here. An analysis of its relevance as a foundational theme will be presented in a companion paper in a future issue of Science in Context. (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)

ArgumentThis article is the sequel to an article published in the previous issue ofScience in Contextthat dealt with homeomeric lines (Acerbi 2010). The present article deals with foundational issues in Greek mathematics. It considers two key characters in the study of mathematical homeomery, namely, Apollonius and Geminus, and analyzes in detail their approaches to foundational themes as they are attested in ancient sources. The main historiographical result of this paper is to show thatthere wasa well-establishedmathematicalfield of discourse in “foundations of (...) mathematics,” a fact that is by no means obvious. The paper argues that the authors involved in this field of discourse set up a variety of philosophical, scholarly, and mathematical tools that they used in developing their investigations. (shrink)

Aristotle claims that the necessity of the syllogisms in the first figure is evident, and calls them ‘perfect’ on this basis. The perfection of such syllogisms, most notably barbara, appears to be correlated with the actual disposition of the middle term. G. Patzig strengthened the correlation to an explanation, claiming that in virtue of that disposition the transitivity of the relation ‘belongs to all’ between the terms becomes manifest. The present article shows that the modern scheme of transitivity, namely, with (...) contiguous middle term, is not the same as the one employed with remarkable consistency in the ancient mathematical texts. On these grounds, I argue that Patzig’s calling into play the notion of transitivity cannot count as an explanation of first figure syllogisms being perfect, not even of those in barbara. Keywords: Aristotle; logic; syllogism; perfect syllogism; first figure; barbara; proportion; transitivity. (shrink)

The article analyzes in detail the assumptions and the proofs typical of the research field of the geometry of burning mirrors. It emphasizes the role of two propositions of the Archimedean Quadratura parabolae, never brought to bear on this subject, and of a complex system of projections reducing a sumptōma of a parabola to some specific linear lemmas. On the grounds of this case-study, the much-debated problem of the heuristic role of analysis is also discussed.

Three problems in book I of Diophantus’ Arithmetica contain the adjective plasmatikon, that appears to qualify an implicit reference to some theorems in Elements, book II. The translation and meaning of the adjective sparked a long-lasting controversy that has become a nonnegligible aspect of the debate about the possibility of interpreting Diophantus’ approach and, more generally, Greek mathematics in algebraic terms. The correct interpretation of the word, a technical term in the Greek rhetorical tradition that perfectly fits the context in (...) which it is inserted in the Arithmetica, entails that Diophantus’ text contained no (implicit) reference to Euclid’s Elements. The clause containing the adjective turns out to be a later interpolation, that cannot be used to support any algebraic interpretation of the Arithmetica. (shrink)

The identification of a “new” handwriting of Manuel Chrysoloras allows to assign to him a number of hitherto unpublished witnesses, all preceding his teaching activity in Florence from 1397. This new dossier illustrates Chrysoloras’ youth in Constantinople, providing precious and so far unknown information on his family, his father John, his studies in the anti-Palamite milieux with Isaac Argyros and Demetrios Kydones, and, of course, his books, his graphic education, and his readings. Among these, a special place is occupied by (...) Aristotle’s logic: particular attention is therefore paid to a mock-up of a manuscript containing Aristotle’s Organon with a marginal commentary, here attributed to Chrysoloras and realized in a well-known manuscript by his master Argyros. (shrink)

The passage has been an object of scholarly debate: the lack of independent sources on the mathematical construction described by Aristotle, the terseness of the formulation and the resulting syntactical ambiguities make the exact interpretation of the text quite difficult, as already noted by Philoponus. What does it mean that the gnomons are ‘placed round the one and without’ (περὶ τὸ ἓν καὶ χωρίς)? And in what sense is this an indication of the even being ‘cut off, enclosed (ἐναπολαμβανόμενον), and (...) limited (περαινόμενον) by the odd’? The aim of the present paper is to discuss the available interpretations of the passage and to propose a new one. (shrink)