On some new formulæ for the numerical calculation of the mutual induction of coaxial circles

The importance of realising rapid and accurate methods of calculating the elliptic integrals, now denoted by F (ϕ,k ) = ∫ϕ 0 d ϕ/ ∆ (ϕ,k ), E (ϕ,k = ∫ϕ 0 ∆ (ϕ,K )d ϕ, where ∆ (ϕ,K ) = √(1 -k 2 sin2 ϕ), was first remarked by Euler (1766), although it was not until several years later that Landen (1775) discovered in geometrical form the transformation which forms the basis of existing methods of numerical calculation of the elliptic integrals. A method of successive transformations for the ultimate reduction of the algebraic forms of the elliptic integrals to elementary forms was published by Lagrange in 1784-5. This memoir contains an exposition of the scales of arithmetico-geometrical means and discusses their use in calculating the elliptic integrals of the first and second kinds in a manner practically identical with that formulated about the same time by Legendre.