In the development of mathematical thinking two fundamental phenomena can be indentified, i.e., quantity and form. The mathematical disciplines that are related to them are algebra and geometry. In the course of the development they shared the leadership. In the phylogeny of mathematical thinking each period of taking the lead brought semantical shifts. Among the most notable semantical transformations in the development of mathematics we rank Pythagorean and Descartean ones. In Pythagorean semantical transformation, algebraic thinking was changed to geometrical one, whereas in the Descartean semantical transformation geometrical thinking was changed to algebraic one. In the paper the above transformations are commented thoroughly. In the next part of the paper, there is an illustration of the semantical transformation accomplished in the present time. In this illustration geometrical situations are algebraized to form an idempotent, medial and commutative quasi group and then all is modelled in the algebraic language. The algebraic structure that has been built takes the lead from geometry and modifies the known geometric situations.