Bernoulli brothers, 17th century Swiss mathematicians. At right is Jean (Johann) Benoulli (1667-1748). At left is Jacques (Jacob) Bernoulli (1654-1705). Jean wrote on differential equations and curves. Jacques studied infinite series, curves and differential calculus, inventing the term 'integral'. The brothers often quarreled and are seen debating a mathematical problem in 1696. The brachistochrone problem was the derivation of the curve along which a particle will most quickly move from one point to another which is not directly below it. Jacques devised a correct proof leading to a cycloid curve. Using Jacques' idea, Jean used a cycloid curve but substituted an incorrect proof.