more definitely, consists of abstractions from finite properties. Mathematical content as such is essentially magnitude. Geometrical figures pertain to space, and have thus in themselves externality as their principle, since they are distinguished from real objects, and represent only the one-sided spatiality of these objects, as distinguished from their concrete filling up, through which they first became real. So number has the unit for its principle, and is the putting together of a multiplicity of units which are independent, and is thus a completely external combination. The knowledge which we have here before us can only attain its greatest perfection in this field, because that field contains only simple and definite qualities, and the dependence of these upon each other, the insight into the nature of which is proof, is thus stable, and ensures for proof the logical progress of necessity. This kind of knowledge is capable of exhausting the nature of its objects. The logical nature of the process of proof is not, however, confined to mathematical content, but enters into all departments of natural and spiritual material; but we may sum up what is logical in knowledge in connection with proof by saying that it depends on the rules of inference; the proofs of the existence of God are therefore essentially inferences. The express investigation of these forms belongs, however, partly to logic, and for the rest the nature of the fundamental defect must be ascertained in the course of the examination of these proofs which is about to be taken in hand. For the present it is enough to remark further, in connection with what has been said, that the rules of inference have a kind of foundation which is of the nature of mathematical calculation. The connection of propositions which are requisite to constitute a syllogistic conclusion depends on the relations of the sphere which each of them occupies as regards the other, and which is quite properly regarded as greater or smaller. The definite extent of such
