Euler

It is not quite certain when Euler first became seriously interested in the calculus of variations. Carathéodory, who edited Euler’s magnificent 1744 opus, The Method of Finding Plane Curves that Show Some Property of Maximum or Minimum … , believed it unlikely that it occurred during his period in Basel with John Bernoulli.1 However, we should note that Euler considered in 1732 and 1736 problems more or less arising out of James Bernoulli’s isoperimetric problems; and even as early as the end of 1728 or early in 1729, as Eneström showed, he wrote “On finding the equation of geodesic curves.” In effect, Euler in 1744, following John Bernoulli, examined the question of end-curves that cut a family of geodesics so that they have equal length. He showed that the end-curves must be orthogonal to the geodesics. This is the precursor of the so-called envelope theorem of Chapter 7 below and plays a key part in Carathéodory’s work.