How the foundations of new mathematics were developed in the first half of the 17th century, the book tells on the example of the work of Descartes, Fermat, Torricelli and Roberval. These scientists participated in the creation of differential and integral calculus, finalized and completed later by Newton and Leibniz. The development of basic mathematical ideas from antiquity to the 17th century is shown, as well as the continuity of these ideas in the works of the creators of modern science.