History of Science and Engineering Annotation << Back Isaac Newton's Method of Regular Polygons and the Development of the Tangent Method Balamirzoev A.G., Rizaev M.K. The development of differentiation rules in the works of Euler (1748, 1755), Lagrange (1770), C.F. Lacroix, O.L. Cauchy, and the development of two methods for fi nding approximate solutions to Newton's algebraic equation are described – the polygon method, used, in particular, to fi nd the root, and the tangent method, while expanding the concept of the number y Newton's law. It is noted that Newton did not analyze the convergence of his methods. This article shows how the analysis of an algebraic equation led to the formulation of two fundamental propositions of the theory of functions – the root gap theorem and the derivative root theorem – and to the creation of the mean value theorem. Keywords: polygon method, equation roots, derivative, convergence. DOI: 10.25791/intstg.4.2026.1612 Pp. 16-23.