466 
CURVE. 
[785 
The intersections which lie on the radical axis are two out of the four inter 
sections of the two circles. The question as to the remaining two intersections did 
not present itself to Gaultier, but it is answered in Poncelet’s Traité des propriétés 
projectives (1822), where we find (p. 49) the statement, “deux circles places arbitraire 
ment sur un plan...ont idéalement deux points imaginaires communs à l’infini”; that 
is, a circle qua curve of the second order is met by the line infinity in two points ; 
but, more than this, they are the same two points for any circle whatever. The 
points in question have since been called (it is believed first by Dr Salmon) the 
circular points at infinity, or they may be called the circular points; these are also 
frequently spoken of as the points I, J ; and we have thus the circle characterized 
as a conic which passes through the two circular points at infinity ; the number of 
conditions thus imposed upon the conic is = 2, and there remain three arbitrary con 
stants, which is the right number for the circle. Poncelet throughout his work makes 
continual use of the foregoing theories of imaginaries and infinity, and also of the 
before-mentioned theory of reciprocal polars. 
Poncelet’s two memoirs “ Sur les centres des moyennes harmoniques,” and “ Sur la 
théorie générale des polaires réciproques,” although presented to the Paris Academy in 
1824 were only published (Crelle, t. in. and iv., 1828, 1829), subsequent to the memoir 
by Gergonne, “ Considerations philosophiques sur les élémens de la science de l’étendue ” 
(Gerg., t. XVI., 1825—26). In this memoir by Gergonne, the theory of duality is very 
clearly and explicitly stated ; for instance, we find “ dans la géométrie plane, à chaque 
théorème il en répond nécessairement un autre qui s’en déduit en échangeant simple 
ment entre eux les deux mots points et droites; tandis que dans la géométrie de 
l’espace ce sont les mots points et plans qu’il faut échanger entre eux pour passer d’un 
théorème à son corrélatif” ; 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., t. xix., 
1827), and followed by a short paper by Gergonne, “Rectifications de quelques théorèmes, 
&c.,” which is important as first introducing the word class. We find in it explicitly 
the two correlative definitions :—“ a plane curve is said to be of the mth degree (order) 
when it has with a line m real or ideal intersections,” and “ a plane curve is said to 
be of the mth class when from any point of its plane there can be drawn to it m real 
or ideal tangents.” 
It may be remarked that in Poncelet’s memoir on reciprocal polars, above referred 
to, we have the theorem that the number of tangents from a point to a curve of 
the order m, or say the class of the curve, is in general and at most = m (m — 1), 
and that he mentions that this number is subject to reduction when the curve has 
double points or cusps. 
The theorem of duality as regards plane figures may be thus stated :—two figures 
may correspond to each other in such manner that to each point and line in either 
figure there corresponds in the other figure a line and point respectively. It is to 
be understood that the theorem extends to all points or lines, drawn or not drawn ; 
thus if in the first figure there are any number of points on a line drawn or not 
drawn, the corresponding lines in the second figure, produced if necessary, must meet