Systems of circles on a sphere
Abstract
Any aggregate of circles whatsoever will not differ from the collection of circles of the previous system, but will constitute only half of the collection of that system, with the linear primas and seconds of ordinary circles remaining such also for this system; but the linear collections of vectorial circles of the previous system will no longer be such for this system, because the tangent linear primas of the previous system are no longer linear primas of this system. It is easy to prove that in this system aggregates of vectorial circles are altogether absent; indeed, one cannot even postulate them. In fact, if I postulate, for example, a right vectorial circle, then the diametrically opposite one is already a left vectorial circle; we obtain essentially two vectorial circles, by which their linear prima on the sphere is completely and uniquely determined; it is clear that in its presence one cannot postulate a third, arbitrary circle; generally, it would no longer be part of a defined linear prima.
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