In 1880 he published the translation of Beginnings of Euclid with a big introduction where were viewed the main principles of the hyperbolic geometry.

In 1871, Mikhail Vaschenko-Zakharchenko married Vera Nikolayevna Mel'nickaya (1840—1895) from Tver noble family.

In 1862, for the first time systematically gave a lecture on the operational calculation and applied it to solve the differential equations.

In 1866, he defended his doctor dissertation Riemann's theory of compound variable functions.

He is an author of more than dozen textbooks on the analytical geometry, projection geometry, algebra, calculus of variations, and including an important work on the history of mathematics in which he discussed the history of mathematics up to the 15th century.