the triangles _POA_ and _POC_ are equal, as also are the triangles _P’OB_
and _P’OC_. Therefore the angle _POP’_ is constant, being equal to half
the constant angle _AOC + COB_. This being true, if we take any four
harmonic points, _P__1_, _P__2_, _P__3_, _P__4_, on the line _u_, they
will project to _O_ in four harmonic lines, and the tangents to the circle
from these four points will meet _u’_ in four harmonic points, _P’__1_,
_P’__2_, _P’__3_, _P’__4_, because the lines from these points to _O_
inclose the same angles as the lines from the points _P__1_, _P__2_,
_P__3_, _P__4_ on _u_. The point-row on _u_ is therefore projective to the
point-row on _u’_. Thus the tangents to a circle are seen to join
corresponding points on two projective point-rows, and so, according to
the definition, form a pencil of rays of the second order.




*81. Tangents to a conic.* If now this figure be projected to a point
outside the plane of the circle, and any section of the resulting cone be
made by a plane, we can easily see that the system of rays tangent to any
conic section is a pencil of rays of the second order. The converse is
also true, as we shall see later, and a pencil of rays of the second order
is also a set of lines tangent to a conic section.




*82.* The point-rows _u_ and _u’_ are, themselves, lines of the system,
for to the common point of the two point-rows, considered as a point of
_u_, must correspond some point of _u’_, and the line joining these two
corresponding points is clearly _u’_ itself. Similarly for the line _u_.