Bernoulli brothers. Engraving of Jean (Johann) and Jacques (Jacob) Bernoulli, Swiss mathematicians. Jean (1667-1748), sitting, wrote on differential equations & curves. Jacques (1654- 1705) studied infinite series, curves and differential calculus; he invented the term 'integral'. The brothers often quarrelled 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.