in projective geometry, ratio that is of fundamental importance in characterizing
projections. In a projection of one line onto another from a central point
(see Figure), the double ratio of lengths on the first line (AC/AD)/(BC/BD)
is equal to the corresponding ratio on the other line. Such a ratio is
significant because projections distort most metric relationships (i.e.,
those involving the measured quantities of length and angle), while the
study of projective geometry centres on finding those properties that remain
invariant. Although the cross ratio was used extensively by early 19th-century
projective geometers in formulating theorems, it was felt to be a somewhat
unsatisfactory concept because its definition depended upon the Euclidean
concept of length, a concept from which projective geometers wanted to
free the subject altogether. In 1847 the German mathematician Karl G.C.
von Staudt showed how to effect this separation by defining the cross ratio
without reference to length. In 1873 the German mathematician Felix Klein
showed how the basic concepts in Euclidean geometry of length and angle
magnitude could be defined solely in terms of von Staudt's abstract cross
ratio, bringing the two geometries together again, this time with projective
geometry occupying the more basic position.
