Landen’s theorem

Landen's theorem, named after mathematician John Landen, relates the arc lengths of ellipses and hyperbolas. Specifically, it expresses the arc length of a hyperbola in terms of the arc lengths of two ellipses. Landen, a British mathematician trained as a surveyor, made significant contributions to the study of elliptic integrals. His theorem was published in 1775 and later included in his Mathematical Memoirs. Landen also worked on other mathematical and scientific topics, including astronomy and physics, and made early contributions to the study of rotary motion.1