John Casey, professor of mathematics at the Catholic university of Dublin, has given elementary demonstrations founded on the theory of similitude and coaxal circles which are reproduced in his Sequel to Euclid; an analytical solution by Gergonne is given in Salmon's Conic Sections.
2 Gergonne, Annales de math.
Gergonne had shown that when a number of the intersections of two curves of the (p+q)th degree lie on a curve of the pth degree the rest lie on a curve of the qth degree.
Plucker finally (Gergonne Ann., 1828-1829) showed how many points must be taken on a curve of any degree so that curves of the same degree (infinite in number) may be drawn through them, and proved that all the points, beyond the given ones, in which these curves intersect the given one are fixed by the original choice.
The term " pole " was first used by Francois Joseph Servois, and " polar " by Joseph Diez Gergonne (Gerg.
And iv., 1828, 1829) subsequent to the memoir by Gergonne, Considerations philosophiques sur les elemens de la science de l'etendue (Gerg.
In this memoir by Gergonne, the theory of duality is very clearly and explicitly stated; for instance, we find " da p s la geometrie plane, a chaque theoreme ii en repond necessairement un autre qui s'en deduit en echangeant simplement entre eux les deux mots points et droites; tandis que dans la geometrie de l'espace ce sont les mots points et plans qu'il faut echanger entre eux pour passer d'un theoreme a son correlatif "; and the plan is introduced of printing correlative theorems, opposite to each other, in two columns.
There was a reclamation as to priority by Poncelet in the Bulletin universel reprinted with remarks by Gergonne (Gerg.
Xix., 1827), and followed by a short paper by Gergonne, Rectifications de quelques theoremes, &c., which is important as first introducing the word class.