> From what I read and learned I got the impression, that Hirst > simply generalised the classical 'inversion': Given a center > C and a (regular or singular) polarity \pi in a projective > space, then the (C,\pi)-inverse X' to a point X (not= C) is > defined by > (HI-1) C, X, X' are collinear, > (HI-2) X, X' are \pi-conjugate. > > One will call X' the Hirst inverse to X with respect to the > inversion center C and the polarity \pi.
This transformation has been referred to as "conic inversion". Giusto Bellavitis (1803-1880) seems to have been the first to study it as early as 1838. I seem to recall that Franz Seydewitz (1807-1852) was also interested in this transformation (but not earlier than 1840). I may look some pointers up, if necessary. Thomas Archer Hirst (1830-1892) came later (c. 1865). By the way, there is a curious six-part series on Hirst's life published in the Monthly (1993). And, just for the record, let me point out that the MacTutor History of Mathematics Archive (St Andrews, Scotland) at http://www-history.mcs.st-and.ac.uk/history/Mathematicians/Hirst.html has Hirst's death-date wrong -- in fact, TAH died on February 16, 1892.