In its fourth volume, the Elements of Mathematics collection covers the most important application of the so-called: self-consistent geometries, non-Euclidean geometry. The author maintains the objective of the series as he is concerned with the reader’s cultural background in mathematics, since non-Euclidean geometry, in the 20th century’s theoretical physics, models the construction of space-time geometry, which introduces modern physics with the theory of relativity. The book also addresses basic concepts of modern physics for undergraduate students in mathematics in order to promote professional competence by connecting everyday knowledge with academic knowledge.