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 X'h Book of 'Elements' of Euclid (part 11). Euclid's Elements were a kind of 'intuitive model', quite different from the contemporary one. Elements were divested of the 'infinitespace' notion. Reconstruction of the hermeneutic horizon of the ancient mathematics lets us explain structure and mathematics presented in the columns ofthe Xth book of Elements. The following subjects were handled: 1. reasons for elimination of the Euclid's ' infinite space' notion and substitutin.s 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 let 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 there is also presented a reconstruction of the mathematical contents of the lost Apollonius' treatise on incommensu- able magnitudes. 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 important, the first revolutionary change was a discovery of non -Euclidean geometry in the XIXth century) -cannot be sustained.