Among the problems of Hilbert, formulated at the turn of the XIX and XX centuries, a special place is occupied by the third problem - the only one related to the method of teaching elementary mathematics. In it, Hilbert raises the question of whether it is possible to abandon the limit transition in the derivation of the triangular pyramid volume formula and limit oneself only to the method of equivalence. This problem has led to a large number of works (M.) Dan, who gave a negative solution to Hilbert's problem, B. F. Kagan, mathematics of the Swiss school, etc.. ). The book introduces the reader to the current state of the theory of equivalence, which in recent years has been enriched by a number of new results. It is intended for researchers, university teachers, peduniaries, schools, students of mathematics and all readers who are seriously interested in mathematics.