Introduction to Ancient Theories of Proportion
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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