Sentence:  If a point O on "I" is related to A by a <WSD>prospectivity</WSD>, then all <WSD>prospectivities</WSD>, which (1) have the same double point U, and (2) relate O to A, give the same correspondent (Q, in figure) to any point P on the line "I"; in fact they are all the same <WSD>prospectivity</WSD>, however "m", "n", S, and S' may have been varied subject to these conditions. 
Meaning of the word marked with <WSD> tags:  A projective relationship between a line and itself by projecting onto an intersecting line from a point on another line that intersects at the same point, and then back on to the first line from a different point on the first intersecting line.