There are a number of ways in which Greek mathematics can be seen to be radically original. First, at the level of mathematical contents: many objects and results were first discovered by Greek mathematicians. Second, Greek mathematics was original at the level of logical form: it is arguable that no form of mathematics was ever axiomatic independently of the influence of Greek mathematics. Finally, third, Greek mathematics was original at the level of form, of presentation: Greek mathematics is written in (...) its own specific, original style. This style may vary from author to author, as well as within the works of a single author, but it is still always recognizable as the Greek mathematical style. This style is characterized by the use of the lettered diagram, a specific technical terminology, and a system of short phrases. I believe this third aspect of the originality—the style—was responsible, indirectly, for the two other aspects of the originality. The style was a tool, with which Greek mathematicians were able to produce results of a given kind, and to produce them in a special, compelling way. This tool, I claim, emerged organically, and reflected the communication-situation in which Greek mathematics was conducted. For all this I have argued elsewhere. (shrink)

Les Sphériques ont été écrites par Ménélaos sous la forme d’un traité d’apparence purement mathématique. Cependant, la matière qui y est développée dans les deuxième et troisième livres est étroitement liée à des problèmes rencontrés en astronomie: calcul des coordonnées équatoriales du Soleil, établissement de tables d’ascensions, étude du mouvement du Soleil dans la sphère oblique, levers simultanés. Ce lien, qui demeure implicite dans le texte, a été mis en évidence par deux mathématiciens et astronomes arabo-musulmans, qui ont exposé la (...) signification astronomique de certains théorèmes des Sphériques. Nous présenterons, commenterons et complèterons cette explicitation en classant les théorèmes concernés en trois groupes: sens de variation de certaines grandeurs sur la sphère, propositions de trigonométrie sphérique et applications, sens de variation de rapports entre certaines grandeurs sur la sphère. The Spherics were written by Menelaos in the form of a purely mathematical treatise. However, the material developed in the second and third book is closely linked to problems met in astronomy: computation of equatorial coordinates of the Sun, setting up of rising-time tables, study of the motion of the Sun in the sphaera obliqua, simultaneous risings. This link, which remains implicit in the text, was clearly displayed by two arabo-islamic mathematicians and astronomers, who expounded the astronomical meaning of some theorems of the Spherics. We describe, comment and complement their explanations, by classifying the implications of the theorems in three groups: direction of variation of some quantities on the sphere, spherical trigonometry and applications, direction of variation of ratios of some quantities on the sphere. (shrink)

The phrase τά ένί διαστηματι γ ραφόμενα occurs in that part of Pappus' Collection Book VIII which deals with instrumental solutions to problems more practical than purely geometrical. In the preceding section an instrumental solution for the problem of doubling the cube has been propounded, which is dependent on the use of a ruler passing through a point about which it is turned in the generation of the locus of points known as the cissoid, and in the subsequent section a (...) solution is propounded for the problem of finding the diameter of a cylinder of which only an incomplete portion of the side remains, leaving neither base intact. (shrink)

The phrase τά ένί διαστηματι γ ραφόμενα occurs in that part of Pappus' Collection Book VIII which deals with instrumental solutions to problems more practical than purely geometrical. In the preceding section an instrumental solution for the problem of doubling the cube has been propounded, which is dependent on the use of a ruler passing through a point about which it is turned in the generation of the locus of points known as the cissoid, and in the subsequent section a (...) solution is propounded for the problem of finding the diameter of a cylinder of which only an incomplete portion of the side remains, leaving neither base intact. (shrink)

This paper examines the widely accepted contention that geometrical constructions serve in Greek mathematics as proofs of the existence of the constructed figures. In particular, I consider the following two questions: first, whether the evidence taken from Aristotle's philosophy does support the modern existential interpretation of geometrical constructions; and second, whether Euclid's Elements presupposes Aristotle's concept of being. With regard to the first question, I argue that Aristotle's ontology cannot serve as evidence to support the existential interpretation, since Aristotle's ontological (...) discussions address the question of the relation between the whole and its parts, while the modern discussions of mathematical existence consider the question of the validity of a concept. In considering the second question, I analyze two syllogistic reformulations of Euclidean proofs. This analysis leads to two conclusions: first, it discloses the discrepancy between Aristotle's view of mathematical objects and Euclid's practice, whereby it will cast doubt on the historical and theoretical adequacy of the existential interpretation. Second, it sets the conceptual background for an alternative interpretation of geometrical constructions. I argue, on the basis of this analysis that geometrical constructions do not serve in the Elements as a means of ascertaining the existence of geometrical objects, but rather as a means of exhibiting spatial relations between geometrical figures. (shrink)