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Ancient Geometry and Plato's Philosophy on the Base of Pappus' 'Comment on the Xth Book of Elements of Euclid'.

100%

Krol Z.

Kwartalnik Historii Nauki i Techniki

2006

vol. 51

issue 3-4

169-203

The article is treating of a new interpretation of ancient geometry (part I) and is willing to explain several mathematical and historical conceptions that were presented in Pappus' 'Comment on the Xth Book of 'Elements' of Euclid' (part II). Euclid's 'Elements' were a kind of 'intuitive model', quite different from the contemporary one, divested of the 'infinite space' notion. Reconstruction of the hermeneutic horizon of the ancient mathematics allows us to explain the structure and mathematics presented in the columns of the Xth book of 'Elements'. The following subjects were handled: (1) reasons for elimination of the Euclid's 'infinite space' notion and substituting it for Plato's Diad in ancient times, (2) basing geometry and searches over the incommensurable magnitudes on one distinguished line together with mathematical consequences, (3) differences in the way of thinking of ancient and contemporary mathematician. Scientific studies allow to qualify from the historical point of view the share in development of the incommensurable magnitudes theories presented by Theaetetus of Athens, Apollonius of Perga, Euclid and Eudoxus. In the article a reconstruction of the mathematical contents of the lost Apollonius' treatise on incommensurable magnitudes is also presented A traditionally established pattern of the development of geometry, according to which Euclidean geometry used to extend as theory basing on relatively unalterable outfit of the fundamental intuition as, for instance, Euclid's infinite space, continuum intuitions and metric intuitions (what is important, the first revolutionary change was a discovery of non–Euclidean geometry in the 19th century) cannot be sustained.

Introduction to Ancient Theories of Proportion

88%

Krol Z.

Kwartalnik Historii Nauki i Techniki

2007

vol. 52

issue 1

77-96

Some of the things that are nowadays taken for granted in mathematics, namely that line segments of a certain length can be well ordered, and 'Euclidean' space is characterized by continuity and metricity, were problematic in antiquity. The main problem of ancient mathematics consisted in attempts to formulate anew a single mathematic theory after its disintegration into arithmetic and geometry caused by the discovery of incommensurability. Successive theories aimed at the metrization of geometric concepts and encompassed an ever increasing variety of mathematical objects. The paper proposes a new scheme of the development of ancient theories of proportion, which includes: 1. Early theories of proportion (P_1), among which two phases of development and two further subtypes have been distinguished in phase two: P_1a - early theories of numerical proportions and P_1b - early theories of geometrical proportions. 2. Theories of numerical proportions motivated by studies of irrational magnitudes: the theory of Archytas (P_2) and the theory of Theaetetus (P_3). 3. Theories of purely geometrical proportions P_4 (mainly book IV of Euclid's Elements) 4. The first theory of proportion that included mixed proportions, i.e. numerical and geometrical proportions (P_5). 5. Eudoxus' theory of proportions (P_6). The research of which the current paper presents the development of mathematics in a new light, and its results allow a reconstruction of the hermeneutic horizon for ancient mathematics.

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2 Kwartalnik Historii Nauki i Techniki

2 Krol Z.

1 2007

1 2006

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Ancient Geometry and Plato's Philosophy on the Base of Pappus' 'Comment on the Xth Book of Elements of Euclid'.

Introduction to Ancient Theories of Proportion