M. B. PORTER—SPHERICS. 
49 
monic ratio. The word projective is used to denote that the two rows 
are in some position perspective, or are projective in the above sense 
to other projective rows. 
Two pencils are projective when they can be arranged so that all the 
corresponding rays intersect on a geodesic. 
3 ° . By this definition four points fixed and one variable determine, 
in the case of the reciprocal curve, a system of projective pencils; 
while on the ellipse four fixed tangents and one variable determine a 
system of projective ranges. 
The Theorems of Pascal and Brianchon readily follow from either of 
these relations. 
11. Thus figure 5. M. FNCB/\C. FEDBaA. FE 
DBAM. ALDB. therefore M. FNCB/\M. ALDB, 
and since three corresponding elements coincide, 
the fourth must also, L, M, and N are collinear. 
The line L M N determines the two 'points P, P 
such that A. P'DEF/\D.P'ABC and A. PDEF/\D. PABC. Pascal’s 
line, thus, determines a seventh point, which taken along with any 
three of six given points is perspective to the remaining three and 
itself taken in a corresponding order. The problem has in general 
two solutions, but only one when the Pascal’s line is tangent, and the 
problem is impossible when the line is without the curve entirely. 
** 12. In figure 6, suppose (FC), (EB), and 
(DA) all to pass through the same point P; 
draw a line through DE to meet OMN (Pas¬ 
cal’s line) at L. The ranges PBSE and PA 
RD are projective, while L. PARDAL. PBS 
E; so that D, E and L are on one straight 
line. Thus, if a secant turn about P the join 
(EC) (BF) is always on the line OL. This 
^ is called De la Hire’s theorem of the pole 
and polar, P being called the pole, and OL the polar. 
Cor. I. D. ABCC' A A. DEFF'. 
Cor. II. This, together with Pascal’s theorem, affords an elegant 
method for drawing a tangent to a curve of this kind, on piano , with a 
ruler only. 
POLES AND POLARS. 
13. From the method of generating the polar, it will be seen that 
the polars of all the points on any line pass through a certain point, 
the pole of the line and conversely; so that to every line there corre¬ 
sponds a point, and to every point a line. This is only a more gen¬ 
eral aspect of the principle of Duality, as hitherto employed (quad- 
