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

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.