CURVE. 
465 
The Géométrie Descriptive by Monge was written in the year 1794 or 1795 
(7th edition, Paris, 1847), and in it we find stated, in piano with regard to the circle, 
and in three dimensions with regard to a surface of the second order, the fundamental 
theorem of reciprocal polars, viz. “ Given a surface of the second order and a circum 
scribed conic surface which touches it ... . then if the conic surface moves so that its 
summit is always in the same plane, the plane of the curve of contact passes always 
through the same point.” The theorem is here referred to partly on account of its 
bearing on the theory of imaginaries in geometry. It is, in Brianchon’s memoir “ Sur 
les surfaces du second degré” {Jour. Polyt., t. vi., 1806), shown how for any given 
position of the summit the plane of contact is determined, or reciprocally ; say the 
plane XY is determined when the point P is given, or reciprocally ; and it is noticed 
that when P is situate in the interior of the surface the plane XY does not cut 
the surface; that is, we have a real plane XY intersecting the surface in the imaginary 
curve of contact of the imaginary circumscribed cone having for its summit a given 
real point P inside the surface. 
Stating the theorem in regard to a conic, we have a real point P (called the 
pole) and a real line XY (called the polar), the line joining the two (real or imaginary) 
points of contact of the (real or imaginary) tangents drawn from the point to the conic ; 
and the theorem is that when the point describes a line the line passes through a 
point, this line and point being polar and pole to each other. The term “ pole ” was 
first used by Servois, and “polar” by Gergonne {Gerg., t. I. and III., 1810—13); and 
from the theorem we have the method of reciprocal polars for the transformation of 
geometrical theorems, used already by Brianchon (in the memoir above referred to) for 
the demonstration of the theorem called by his name, and in a similar manner by 
various writers in the earlier volumes of Gergonne. We are here concerned with the 
method less in itself than as leading to the general notion of duality. And, bearing 
in a somewhat similar manner also on the theory of imaginaries in geometry (but the 
notion presents itself in a more explicit form), there is the memoir by Gaultier, on 
the graphical construction of circles and spheres {Jour. Polyt., t. ix., 1813). The well- 
known theorem as to radical axes may be stated as follows. Consider two circles 
partially drawn so that it does not appear whether the circles, if completed, would or 
would not intersect in real points, say two ares of circles ; then we can, by means of 
a third circle drawn so as to intersect in two real points each of the two arcs, 
determine a right line, which, if the complete circles intersect in two real points, passes 
through the points, and which is on this account regarded as a line passing through 
two (real or imaginary) points of intersection of the two circles. The construction in 
fact is, join the two points in which the third circle meets the first arc, and join 
also the two points in which the third circle meets the second arc, and from the 
point of intersection of the two joining lines, let fall a perpendicular on the line 
joining the centre of the two circles ; this perpendicular (considered as an indefinite 
line) is what Gaultier terms the “ radical axis of the two circles ” ; it is a line 
determined by a real construction and itself always real ; and by what precedes it is 
the line joining two (real or imaginary, as the case may be) intersections of the given 
circles. 
C. XI. 
59