In his last memoir on mathematical physics, Henri Poincaré presented one of the most profound and compelling proofs of the hypothesis of quanta. This highly original proof, which is actually three separate proofs, is based on first principles and is full of physical insight, mathematical rigor, and elegant simplicity. The memoir is refreshingly uncluttered by some of the conventional, and more abstract concepts, such as temperature and entropy, that Planck and others relied on in their work. Poincaré’s analysis is based on an ingenious physical model consisting of long‐period resonators interacting with short‐period resonators. A unique formulation of statistical mechanics, based on the calculus of probabilities, Fourier’s integral, and complex analysis, logically unfolds throughout the memoir. Poincaré invents an ‘‘inverse statistical mechanics’’ that allows him to prove a crucial result that no one had proved before: The hypothesis of quanta is both a sufficient and anecessarycondition to account for Planck’s law of radiation. In a separate, more universal proof, Poincaré proves that the existence of a discontinuity in the motion of a resonator is necessary to explainanyobserved law of radiation. Given the significant impact of Poincaré’s memoir on quantum theory and statistical physics, it is surprising that most physicists are not aware of its valuable mathematical and physical ideas. Poincaré’s tour de force proofs are presented here in a form suitable for use in a standard course in quantum mechanics, statistical mechanics, or mathematical physics.