Date: Jan 8, 2000 11:25 AM
Author: J.G.Fauvel (John Fauvel)
Subject: Re: [HM] Name needed for a point: Hirst?

1. Picking up Gunter Weiss's helpful suggestion about H P Hudson's 'Cremona
transformations', I consulted its bibliography which lists three works by
Hirst on the subject:

'On the quadric inversion of plane curves', Proc Roy Soc London 14 (1865),
(This also appeared in Italian, in Rome -- Ann di M 7 (1865), 49-65; and in
Naples -- Battaglini G 4 (1866), 278-293)

'Sur la transformation quadrique', Nouv. Ann (2) 5 (1866), 213-218

'On quadric transformation', Quart J 17 (1881), 301-311
(Hudson remarks that an abstract of this paper appeared in the British
Association report for 1865 (Birmingham))

Gunter Weiss's citation of the 1864 (Bath) meeting of the British
Association refers to another short report by Hirst, 'On a generalization of
the method of geometrical inversion', which is not included in Hudson's

2. H P Hudson is Hilda Hudson (1881-1965). For people who don't know it,
her book is a fine achievement, whose final chapter (pp. 388-395) is a
history of Cremona transformations from Apollonius onwards. Her discussion
of Hirst (this is getting towards answering Clark Kimberling's query) is
contained in the following sentence (which illustrates her compact
historiographical style):

"Inversion, as a transformation of points, is treated by Plucker, 1831-4,
Bellavitis, who extends it to space and generalizes it into conical
inversion, Thomson, who was led to it in his electro-static researches,
Liouville, who elaborates Thomson's method and introduces the phrase
'reciprocal radii', Mobius, whose Kreisverwandtschaft became widely known,
Hirst, after whom this form of involution is often named, and many other
writers of the period."

(Hudson p.389, omitting bibliographical reference numbers.)

3. Gunter made a very helpful and interesting final remark about the value
and possible use of Hirst inversion in high school classes. If any
school-teachers would take up the challenge and share their lesson plans for
such a classroom discussion, it would be a great service to those concerned
about the uses of history in the mathematics classroom and would be worth
sharing more widely for those concerned about geometry education.

John Fauvel