to Henri Poincaré, writing in 1904, even though Maxwell and Boltzmann had previously explained the irreversibility of macroscopic physical processes in probabilistic terms, "the one who has seen it most clearly, in a book too little read because it is a little difficult to read, is Gibbs, in his Elementary Principles of Statistical Mechanics." Gibbs's analysis of irreversibility, and his formulation of Boltzmann's H-theorem and of the ergodic hypothesis, were major influences on the mathematical physics of the 20th century. Gibbs was well aware that the application of the equipartition theorem to large systems of classical particles failed to explain the measurements of the specific heats of both solids and gases, and he argued that this was evidence of the danger of basing thermodynamics on "hypotheses about the constitution of matter". Gibbs's own framework for statistical mechanics, based on ensembles of macroscopically indistinguishable microstates, could be carried over almost intact after the discovery that the microscopic laws of nature obey quantum rules, rather than the classical laws known to Gibbs and to his contemporaries. His resolution of the so-called "Gibbs paradox", about the entropy of the mixing of gases, is now often cited as a prefiguration of the indistinguishability of particles required by quantum physics. Vector analysis British scientists, including Maxwell, had relied on Hamilton's quaternions in order to express the dynamics of physical quantities, like the electric and magnetic fields, having both a magnitude and a direction in three-dimensional space. Following W. K. Clifford in his Elements of Dynamic (1888), Gibbs noted that the product of quaternions could be separated into two parts: a one-dimensional (scalar) quantity and a three-dimensional vector, so that the use of quaternions involved mathematical complications and redundancies that could be avoided in the interest of simplicity and to facilitate teaching. In his Yale classroom notes he defined distinct dot and cross products for pairs of vectors and introduced the now common notation for them. Through the 1901 textbook Vector Analysis prepared by E. B. Wilson from Gibbs notes, he was largely responsible for the development of the vector calculus techniques still used today in electrodynamics and fluid mechanics. While he was working on vector analysis in the late 1870s, Gibbs discovered that his approach was similar to the one that Grassmann had taken in his "multiple algebra". Gibbs then sought to publicize Grassmann's work, stressing that it was both more general and historically prior