Editing Projective line over a ring (section)

== History ==
[[August Ferdinand Möbius]] investigated the [[Möbius transformation]]s between his book ''Barycentric Calculus'' (1827) and his 1855 paper "Theorie der Kreisverwandtschaft in rein geometrischer Darstellung". [[Karl Wilhelm Feuerbach]] and [[Julius Plücker]] are also credited with originating the use of homogeneous coordinates. [[Eduard Study]] in 1898, and [[Élie Cartan]] in 1908, wrote articles on [[hypercomplex numbers]] for German and French ''Encyclopedias of Mathematics'', respectively, where they use these arithmetics with [[linear fractional transformation]]s 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]].<ref>{{citation |last=Vahlen |first=Theodore |authorlink=Theodore Vahlen |date=1902 |title=Über Bewegungen und complexe Zahlen |journal=[[Mathematische Annalen]] |volume=55 |issue=4 |pages=585–593 |doi=10.1007/BF01450354 }}</ref> The ring of [[dual numbers]] ''D'' gave Josef Grünwald opportunity to exhibit P<sup>1</sup>(''D'') in 1906.<ref name="Grünwald"/> [[Corrado Segre]] (1912) continued the development with that ring.<ref name=CS/>

[[Arthur W. Conway|Arthur Conway]], one of the early adopters of relativity via [[biquaternion]] transformations, considered the quaternion-multiplicative-inverse transformation in his 1911 relativity study.<ref>{{citation |last=Conway |first=Arthur |authorlink=Arthur W. Conway |date=1911 |title=On the application of quaternions to some recent developments of electrical theory |journal=[[Proceedings of the Royal Irish Academy]] |volume=29 |pages=1–9, particularly page 9 }}</ref> In 1947 some elements of inversive quaternion geometry were described by P.G. Gormley in Ireland.<ref>{{citation |first=P.G. |last=Gormley |date=1947 |title=Stereographic projection and the linear fractional group of transformations of quaternions |journal=[[Proceedings of the Royal Irish Academy]], Section A |volume=51 |pages=67–85 }}</ref> In 1968 [[Isaak Yaglom]]'s ''Complex Numbers in Geometry'' appeared in English, translated from Russian. There he uses P<sup>1</sup>(''D'') to describe [[line coordinates#With complex numbers|line geometry]] in the Euclidean plane and P<sup>1</sup>(''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<sup>1</sup>(''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,<ref name=Benz73/> which included the homogeneous coordinates taken from&nbsp;''M''.