Josiah Willard Gibbs (February 11, 1839 – April 28, 1903) (generally known as J. Willard Gibbs) was a preeminent American mathematical-engineer, theoretical physicist, and chemist noted for his contributions to physical chemistry and statistical mechanics. He introduced the "phase rule," which predicts the number of phases that a substance can manifest at a particular temperature and pressure. He devised much of the theoretical foundation for chemical thermodynamics as well as physical chemistry. As a mathematician, he was an inventor of vector analysis. The American Mathematical Society has described him as one of the greatest scientists America has ever produced.