34 MATHESIS. 



regarded ; in the sphere, however, the direction of these 

 positions, or of series of numbers. 



145. The doctrine of the sphere is Geometry ; for all 

 forms are contained in the sphere. All geometrical proofs 

 admit of being conducted through the sphere. Geometry 

 has originated directly from arithmetic, or is arithmetic 

 itself, with this difference, that the latter regards series 

 of numbers as individualities, the former, however, as a 

 whole. Arithmetic is a geometry seriebus discretis ; 

 geometry is an arithmetic seriebus continuis, a solidified 

 arithmetic. 



146. Geometry is a science of equal value with arith- 

 metic ; it is even as certain, because it has no other pro- 

 positions ; it is equally eternal, is the same realization of 

 the primary act, the Dem geometrizans of the Pytha- 

 goreans. Everything to be certain must therefore 

 resemble geometry, must be itself a position of geometry, 

 only under other relations. 



147. Geometry is more real, more finite, therefore 

 also more apparent, and, as it were, more mate- 

 rial than arithmetic. The ideas in it have become 

 something determinate, have assumed form, while before 

 they still fluctuated formless in arithmetic ; here were 

 they mere ghosts without veils, but in geometry they 

 have received these veils. Time has received for its 

 form, its body, the line ; space, the surface ; life, the 

 globe, consequently the rotation for its form or body. 

 It is to be here remarked, that ideas always become 

 more real and more finite, always approximate nearer to 

 actual manifestation, the lower they descend or the more 

 they are considered individually. Geometry has not 

 originated later than arithmetic, but is only a more 

 indvidual view of ideas, arithmetic being more uni- 

 versal. Geometry is arithmetic with stationary numbers, 

 = points. The Divine thus approximates to manifes- 

 tation, to materiality, the more individual it becomes ; 

 and this is very natural, for it verily limits itself more 

 and obtains always more predicates. The more a thing 

 obtains predicates, by so much the more perfect is its 



