By far the greater portion of extant knowl edge of quatermons, both of the theory and of its applications, is due directly to Hamilton and is published in the
Hamilton laid the foundations of the new subject during the years 1833-43. In No vember of the latter year his first paper on
Meanwhile, in 1844, Grassmann published his now celebrated work, (Die Ausdehnungslehre,) which occupies common ground with quater nion theory, but plans a much wider range of investigation than was contemplated by Hamilton in his system. The two systems have a common body of fundamental prin ciples, but they have different points of view, different notations and in part different pur poses. Both form the foundations upon which
all subsequent work of the kind must stand.
Hamilton's of Quaternions,' which still remains the storehouse of knowledge for its subject, was a posthumous work, published in 1866. A new edition has recently appeared in two volumes.
After the work of Hamilton and Grassmann had been finished the next most important forward step in the creation of new algebras was taken by Benjamin Peirce in his essay on (Linear Associative Algebra,' presented to the National Academy of Sciences at Washington in 1870, subsequently published in the Amer ican Journal of Mathematics (1881). In this work the author enumerates the types of linear associative algebras, and classifies them by a set of criteria which takes account of the num ber and the assumed laws of combination of their irreducible elements (extraordinaries, or yids). These laws are exhibited in the form of a series of multiplication tables.
Some of the more recent attempts to apply the quaternion analysis and the Ausdeh nungslehre to physical problems have appeared under the title of Analysis,' sometimes with modified, sometimes with wholly changed notations. Of these the contributions of Wil lard Gibbs and of Heaviside should be men tioned as of special importance. See VECTOR