Previously, between the mid-1700s to the mid-1800s, Leonhard Euler, Adrien-Marie Legendre, and Carl Gustav Jacob Jacobi were able to establish necessary but insufficient conditions for the existence of a strong relative minimum. In 1879, Karl Weierstrass added a fourth that does indeed guarantee such a quantity exists. Carathéodory constructed his method for deriving sufficient conditions based on the use of the Hamilton–Jacobi equation to construct a field of extremals. The ideas are closely related to light propagation in optics. The method became known as "Carathéodory's method of equivalent variational problems" or "the royal road to the calculus of variations". A key advantage of Carathéodory's work on this topic is that it illuminates the relation between the calculus of variations and partial differential equations. It allows for quick and elegant derivations of conditions of sufficiency in the calculus of variations and leads directly to the Euler-Lagrange equation and the Weierstrass condition. He published his "Variationsrechnung und Partielle Differentialgleichungen Erster Ordnung" (Calculus of Variations and First-order Partial Differential Equations) in 1935.