Having adjoined the ideal elements to the visible universe, von Staudt proves the usual theorems of projective geometry, noting the different cases that arise from different arrangements of the elements. As the only means of comparing figures is by projection, all the proofs depend on this; in many cases on the theorem that a given cast (Wurf) is unaltered when any two elements are interchanged, provided the other two are interchanged also (G. 59), which can be written symbolically

while if the cast is harmonic, viz. AG harmonic with respect to BD, (ABCD) = (CBAD) = (ADCB).