to Hamilton's quaternionic algebra. To establish priority of Grassmann's ideas, Gibbs convinced Grassmann's heirs to seek the publication in Germany of the essay "Theorie der Ebbe und Flut" on tides that Grassmann had submitted in 1840 to the faculty at the University of Berlin, in which he had first introduced the notion of what would later be called a vector space (linear space). As Gibbs had advocated in the 1880s and 1890s, quaternions were eventually all but abandoned by physicists in favor of the vectorial approach developed by him and, independently, by Oliver Heaviside. Gibbs applied his vector methods to the determination of planetary and comet orbits. He also developed the concept of mutually reciprocal triads of vectors that later proved to be of importance in crystallography. Physical optics Though Gibbs's research on physical optics is less well known today than his other work, it made a significant contribution to classical electromagnetism by applying Maxwell's equations to the theory of optical processes such as birefringence, dispersion, and optical activity. In that work, Gibbs showed that those processes could be accounted for by Maxwell's equations without any special assumptions about the microscopic structure of matter or about the nature of the medium in which electromagnetic waves were supposed to propagate (the so-called luminiferous ether). Gibbs also stressed that the absence of a longitudinal electromagnetic wave, which is needed to account for the observed properties of light, is automatically guaranteed by Maxwell's equations (by virtue of what is now called their "gauge invariance"), whereas in mechanical theories of light, such as Lord Kelvin's, it must be imposed as an ad hoc condition on the properties of the aether. In his last paper on physical optics, Gibbs concluded that Shortly afterwards, the electromagnetic nature of light was demonstrated by the experiments of Heinrich Hertz in Germany. Scientific recognition Gibbs worked at a time when there was little tradition of rigorous theoretical science in the United States. His research was not easily understandable to his students or his colleagues, and he made no effort to popularize his ideas or to simplify their exposition to make them more accessible. His seminal work on thermodynamics was published mostly in the Transactions of the Connecticut Academy, a journal edited by his librarian brother-in-law, which was little read in the US and even less so in Europe. When Gibbs submitted his long paper on the equilibrium of heterogeneous