FIG. 10




*48.* We may also give an illustration of a case where two superposed
projective point-rows have no self-corresponding points at all. Thus we
may take two lines revolving about a fixed point _S_ and always making the
same angle a with each other (Fig. 10). They will cut out on any line _u_
in the plane two point-rows which are easily seen to be projective. For,
given any four rays _SP_ which are harmonic, the four corresponding rays
_SP’_ must also be harmonic, since they make the same angles with each
other. Four harmonic points _P_ correspond, therefore, to four harmonic
points _P’_. It is clear, however, that no point _P_ can coincide with its
corresponding point _P’_, for in that case the lines _PS_ and _P’S_ would
coincide, which is impossible if the angle between them is to be constant.




*49. Fundamental theorem. Postulate of continuity.* We have thus shown
that two projective point-rows, superposed one on the other, may have two
points, one point, or no point at all corresponding to themselves. We
proceed to show that

_If two projective point-rows, superposed upon the same straight line,
have more than two self-corresponding points, they must have an infinite
number, and every point corresponds to itself; that is, the two point-rows
are not essentially distinct._