For the purpose of this book it is not necessary to expound in detail why modern mathematical thinking has been led to look for thought-forms other than those of classical geometry. It is enough to remark that for quite a long time there had been an awareness of the fact that the consistency of Euclid's definitions and proofs fails as soon as one has no longer to do with finite geometrical entities, but with figures which extend into infinity, as for instance when the properties of parallel straight lines come into question. For the concept of infinity was foreign to classical geometrical thinking. Problems of the kind which had defeated Euclidean thinking became soluble directly human thinking was able to handle the concept of infinity.