Fagnano's method for solving algebraic equations: its historical overview and development

In 1750, Giulio Carlo Fagnano dei Toschi's treatise “Produzioni matematiche” was published in two volumes. The second volume of the treatise contains a work in which Fagnano proposed a uniform method for solving algebraic equations up to the fourth degree. Fagnano's method, as we call it in this article, is based on a comparison of algebraic equations with identities arising in the representation of an expression of the form $ (a_1 + \cdots + a_m)^n $ in terms of lesser powers of $ a_1 + \cdots + a_m. $ Here we explore the results related to the use of certain types of algebraic identities in solving a number of families of algebraic equations that are solvable by radicals. Various connections of these identities with linear recurrent sequences and trigonometric identities are considered. A historical survey devoted to the use of Fagnano's method for solving algebraic equations is presented in section 1. Section 2 is devoted to algebraic equations that are solvable by radicals closely related to Chebyshev polynomials and whose solution by radicals is based on the use of the Kummer identity. In section 3, Fagnano's method is used to study some families of algebraic equations that are solvable by radicals. In sections 4 and 5, the connections of the identities considered in the previous sections with the well-known linear recurrent sequences are analyzed. In section 6, a connection is established between the identities covered in this article and the group determinants.