a. Two events and are hyperbolic-orthogonal when Canonical events and are hyperbolic orthogonal and lie on the axes of a frame of reference in which the events simultaneous with the origin are proportional to . In 1933 Max Zorn was using the split-octonions and noted the composition algebra property. He realized that the Cayley–Dickson construction, used to generate division algebras, could be modified (with a factor gamma, ) to construct other composition algebras including the split-octonions. His innovation was perpetuated by Adrian Albert, Richard D. Schafer, and others. The gamma factor, with as base field, builds split-complex numbers as a composition algebra. Reviewing Albert for Mathematical Reviews, N. H. McCoy wrote that there was an "introduction of some new algebras of order 2e over F generalizing Cayley–Dickson algebras." Taking and corresponds to the algebra of this article. In 1935 J.C. Vignaux and A. Durañona y Vedia developed the split-complex geometric algebra and function theory in four articles in Contribución a las Ciencias Físicas y Matemáticas, National University of La Plata, República Argentina (in Spanish). These expository and pedagogical essays presented the subject for broad appreciation. In 1941 E.F. Allen used the split-complex geometric arithmetic to establish the nine-point hyperbola of a triangle inscribed in . In 1956 Mieczyslaw Warmus published "Calculus of Approximations" in Bulletin de l’Académie polonaise des sciences (see link in References). He developed two algebraic systems, each of which he called "approximate numbers", the second of which forms a real algebra. D. H. Lehmer reviewed the article in Mathematical Reviews and observed that this second system was isomorphic to the "hyperbolic complex" numbers, the subject of this article. In 1961 Warmus continued his exposition, referring to the components of an approximate number as midpoint and radius of the interval denoted. Synonyms Different authors have used a great variety of names for the split-complex numbers. Some of these include: (real) tessarines, James Cockle (1848) (algebraic) motors, W.K. Clifford (1882) hyperbolic complex numbers, J.C. Vignaux (1935) bireal numbers, U. Bencivenga (1946) approximate numbers, Warmus (1956), for use in interval analysis double numbers, I.M. Yaglom (1968), Kantor and Solodovnikov (1989), Hazewinkel (1990), Rooney (2014) anormal-complex numbers, W. Benz (1973) perplex numbers, P. Fjelstad (1986) and Poodiack & LeClair (2009) countercomplex or hyperbolic, Carmody (1988) Lorentz numbers, F.R. Harvey (1990) hyperbolic numbers, G. Sobczyk (1995) paracomplex numbers, Cruceanu, Fortuny & Gadea (1996) semi-complex numbers, F. Antonuccio (1994) split binarions, K. McCrimmon