M. B. PORTER—SPHERICS. 
55 
that the error is of the second order, which, however, presents little 
difficulty. 
To reduce theorems in spherics to corresponding ones in -piano , it is 
only necessary to consider the sphere to grow indefinitely large. It is 
often expedient, however, to select some particular point, as the center 
of the planar universe, and it is the quadrantal polar of this point that 
becomes the line at infinity. It will be noted in the reduction to the 
plane, that only the hemisphere is dealt with and the portion of the 
sphere beyond the line chosen to be line at infinity is disregarded. 
These considerations show why it is admissible to regard the line at 
infinity as having no direction, or rather any direction, as being the 
locus of the intersection of all parallel lines; why on y one parallel can 
be drawn to a line through a point. 
Many of the results obtained for spherical may be immediately stated 
for the flat conics by changing the word cyclic line to asymptote. But 
in 24 it will be found convenient take the polar of C to become the line 
at infinity; 24 then becomes an expression for the difference between 
the hyperbolic arc from the vertex to the point at infinity, and the 
asymptote between the center and the point at infinity. In 16, 1 ° let 
the line along which the quadrant is displaced be taken as the line at 
infinity, and it becomes the well known theorem: The locus of the 
vertex of a right angle, whose sides touch a conic, is a concentric cir¬ 
cle. When AA'= a quadrant, the conic is the locus of a point equally 
distant from F and the quadrantal polar of F r ; so that th e parabola is 
said to be tangent to the line at infinity; the tangent can thus coincide 
with the line at infinity, and the lines joining its intersections with two 
fixed tangents to the focus would be parallel to the two fixed tangents 
respectively, and make a constant angle with each other; so that the in 
tersections of two fixed tangents, and a variable third, are concyclic 
with the focus. Conversely, if a series of parabolas touch three lines, 
the locus ol their foci is a circle passing through the three points of in¬ 
tersection of the tangent, a theorem due to Lambert. 
The property of the circle, in piano, that two points on a circle 
subtend a constant angle at any third point on the same circle, may 
be deduced in the same way. Considering the foci (10.2 ° ) to coin¬ 
cide, the intercept on the polar of the center will be constant, and tak¬ 
ing this line as the line at infinity, the theorem is evident. All graph¬ 
ical relations are, of course, quite independent of the selection of the 
line at infinity , so that Pascal’s, Brianchon’s, and the polar properties 
are proved at the same time for the sphere and plane. It is possible 
to pass by an inverse process from plane to the sphere, and, by sup¬ 
posing the plane to become uniformly bent in the third dimension, 
deduce from the properties of circles, in piano , theorems concerning 
conics on the sphere. The theorem of the circle, the vertex of any 
