In projective geometry, non-intersecting lines do not exist (they all meet at infinity). In hyperbolic geometry on the other hand, through any point not on the line $\ell$, there are more than one line that does not intersect $\ell$, so at least some non-intersecting lines do exist. In all, they rather look incompatible together. A geometry cannot be both projective and hyperbolic. But still, it was Cayley who said it.
So how is hyperbolic geometry a projective geometry? I guess that it has to do with transformations/reflections and that ilk, but how does that relate to hyperbolic geometry in particular?