[Footnote 2: In his book, _On the Limits of Philosophy_, Mr. W. Tobias maintains that axioms of a kind which I formerly enunciated are a misunderstanding of Kant’s opinion. But Kant specially adduces the axioms, that the straight line is the shortest (_Kritik der reinen Vernunft_, Introduction, v. 2nd ed. p. 16); that space has three dimensions (_Ibid._ part i. sect. i. § 3, p. 41); that only one straight line is possible between two points (_Ibid._ part ii. sect. i. ‘On the Axioms of Intuition’), as axioms which express _a priori_ the conditions of intuition by the senses. It is not here the question, whether these axioms were originally given as intuition of space, or whether they are only the starting-points from which the understanding can develop such axioms _a priori_ on which my critic insists.] It is precisely this relation of geometry to the theory of cognition which emboldens me to speak to you on geometrical subjects in an assembly of those who for the most part have limited their mathematical studies to the ordinary instruction in schools. Fortunately, the amount of geometry taught in our gymnasia will enable you to follow, at any rate the tendency, of the principles I am about to discuss. I intend to give you an account of a series of recent and closely connected mathematical researches which are concerned with the geometrical axioms, their relations to experience, with the question whether it is logically possible to replace them by others. Seeing that the researches in question are more immediately designed to furnish proofs for experts in a region which, more than almost any other, requires a higher power of abstraction, and that they are virtually inaccessible to the non-mathematician, I will endeavour to explain to such a one the question at issue. I need scarcely remark that my explanation will give no proof of the correctness of the new views. He who seeks this proof must take the trouble to study the original researches. Anyone who has entered the gates of the first elementary axioms of geometry, that is, the mathematical doctrine of space, finds on his path that unbroken chain of conclusions of which I just spoke, by which the ever more varied and more complicated figures are brought within the domain of law. But even in their first elements certain principles are laid down, with respect to which geometry confesses that she cannot prove them, and can only assume that anyone who understands the essence of these principles will at once admit their correctness. These are the so-called axioms.