in 1908, wrote articles on hypercomplex numbers for German and French Encyclopedias of Mathematics, respectively, where they use these arithmetics with linear fractional transformations in imitation of those of Möbius. In 1902 Theodore Vahlen contributed a short but well-referenced paper exploring some linear fractional transformations of a Clifford algebra. The ring of dual numbers D gave Josef Grünwald opportunity to exhibit P(D) in 1906. Corrado Segre (1912) continued the development with that ring. Arthur Conway, one of the early adopters of relativity via biquaternion transformations, considered the quaternion-multiplicative-inverse transformation in his 1911 relativity study. In 1947 some elements of inversive quaternion geometry were described by P.G. Gormley in Ireland. In 1968 Isaak Yaglom's Complex Numbers in Geometry appeared in English, translated from Russian. There he uses P(D) to describe line geometry in the Euclidean plane and P(M) to describe it for Lobachevski's plane. Yaglom's text A Simple Non-Euclidean Geometry appeared in English in 1979. There in pages 174 to 200 he develops Minkowskian geometry and describes P(M) as the "inversive Minkowski plane". The Russian original of Yaglom's text was published in 1969. Between the two editions, Walter Benz (1973) published his book which included the homogeneous coordinates taken from M. See also Euclid's orchard Notes and references Sky Brewer (2012) "Projective Cross-ratio on Hypercomplex Numbers", Advances in Applied Clifford Algebras, DOI 10.1007/s00006-12-0335-7. I. M. Yaglom (1968) Complex Numbers in Geometry. Further reading G. Ancochea (1941) "Le théorèm de von Staudt en géométrie projective quaternionienne", Journal für Mathematik, Band 184, Heft 4, SS. 193–8. N. B. Limaye (1972) "Cross-ratios and Projectivities of a line", Mathematische Zeitschrift 129: 49–53, . B.V. Limaye & N.B. Limaye (1977) "The Fundamental Theorem for the Projective Line over Commutative Rings", Aequationes Mathematica 16:275–81. . B.V. Limaye & N.B. Limaye (1977) "The Fundamental Theorem for the Projective Line over Non-Commutative Local Rings", Archiv der Mathematik 28(1):102–9 . Marcel Wild (2006) "The Fundamental Theorem of Projective Geometry for an Arbitrary Length Two Module", Rocky Mountain Journal of Mathematics 36(6):2075–80. External links Mitod Saniga (2006) Projective Lines over Finite Rings (pdf) from Astronomical Institute of the Slovak Academy of Sciences Algebraic geometry Ring theory Projective geometry