2 1. Introduction a b ya yb A B x y M Figure 1.1. Curve of descent energy and potential energy remains constant. If our particle starts from rest, we may thus write 1 2 mv2 + mgy = mgya . (1.2) The particle’s speed is then v = 2g(ya − y) . (1.3) We now wish to find the brachistochrone (from βραχιστoς, short- est, and χρoνoς, time John Bernoulli originally, but erroneously, wrote brachystochrone). That is, we wish to find the curve y = y(x) ≤ ya (1.4) that minimizes the integral T = 1 √ 2g b a 1 + y 2 ya − y dx . (1.5) Several famous mathematicians responded to John Bernoulli’s challenge. Solutions were submitted by Gottfried Wilhelm Leibniz

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2014 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.