More about special circles and spheres

Abstract

In these Notes (III 287) I devoted a note to this subject, in which I limited myself to only a complete derivation of the geometric images related here. The same conclusion, of course, could be produced in other ways, which should lead to identical results. All conoprims, for example, can be derived by a collinear transformation from circles, but also by notching two projective prime rays, and from elementary manuals it can be seen that the last example, if not more correct, is at least more visual, and in this sense simpler (see article). From the note mentioned at the beginning, it is clear that the existence of special circles and spheres brings great disruption to the ideas about the circle that have been established, not even for centuries, but for millennia. As it is proven in it, the idea of a center and equal radii must be completely eliminated from this concept."