1.1. The brachistochrone 3 (1697), Isaac Newton (1695–7, 1697), John Bernoulli (1697a), James (or Jakob) Bernoulli (1697), and Guillaume l’Hˆ opital (1697). Leibniz provided a geometrical solution. He derived the differen- tial equation for the brachistochrone but did not specify the result- ing curve (Goldstine, 1980). Leibniz also suggested that the brachis- tochrone be called the tachystoptotam (from ταχιστoς, swiftest, and πιπτιν, to fall). Mercifully, this suggestion was ignored. Newton’s anonymous solution was published in the Philosophical Transactions it was then reprinted in the Acta Eruditorum. Newton provided the correct answer but gave no clue to his method. Despite Newton’s anonymity, John Bernoulli recognized that the work was “ex ungue Leonem” (from the claw of the Lion) and the Acta Eruditorum listed Newton in its index of authors. John Bernoulli provided two solutions. The first solution relied on an analogy between the mechanical brachistochrone and light. Bernoulli (1697a) was quite taken with Fermat’s principle of least time for light and argued that the brachistochrone “is the curve that a light ray would follow on its way through a medium whose density is inversely proportional to the velocity that a heavy body acquires during its fall.” He broke up the optical medium into thin horizontal layers, chose an appropriate index of refraction, and used Snell’s law of refraction and calculus to determine the shape of the brachistochrone. John Bernoulli (1718) described his second solution many years later. This second solution received little attention at the time but is now viewed as the first suﬃciency proof in the calculus of variations. James Bernoulli’s solution was not as elegant as that of his young- er brother, but it contained the key idea of varying only one value of the solution curve at a time. This idea provided the basis for further work in the calculus of variations. James Bernoulli called his solution an oligochrone (from oλιγoς, little, and χρoνoς, time). We shall see that the brachistochrone is the inverted cycloid x(φ) = a + R(φ − sin φ) , y(φ) = ya − R (1 − cos φ) , (1.6) where the parameter R is uniquely determined by the initial and terminal points. This cycloid is the curve traced by a point on the

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