Although I agree with Helmholtz in thinking the distinction between Euclidean and non-Euclidean spaces empirical, I cannot think his argument on the "imaginability" of the latter a very happy one. The validity of any proof must turn, obviously, on the definition of imaginability. The definition which Helmholtz gives in his answer to Land is as follows: Imaginability requires "die vollständige Vorstellbarkeit derjenigen Sinneseindrücke, welche das betreffende Object in uns nach den bekannten Gesetzen unserer Sinnesorgane unter allen denkbaren Bedingungen der Beobachtung erregen, und wodurch es sich von anderen ähnlichen Objecten unterscheiden würde" (Wiss. Abh. II. p. 644). This definition is not very clear, owing to the ambiguity of the word "_Vorstellbarkeit_." The following definition seems less ambiguous: "Wenn die Reihe der Sinneseindrücke vollständig und eindeutig angegeben werden kann, muss man m. E. die Sache für _anschaulich vorstellbar_ erklären" (Vorträge und Reden, II. p. 234). This makes clear, what also appears from his manner of proof, that he regards things as imaginable which can be _described_ in conceptual terms. Such, as Land remarks (Mind, Vol. II. p. 45), "is not the sense required for argumentation in this case." That Land's criticism is just, is shown by Helmholtz's proof for non-Euclidean spaces, for it consists only in an analogy to the volume inside a sphere, which is mathematically obtained thus: We take the symbols representing magnitudes in "pseudo-spherical" (hyperbolic) space, and give them a new Euclidean meaning; thus all our symbolic propositions become capable of two interpretations, one for pseudo-spherical space, and one for the volume inside a sphere. It is, however, sufficiently obvious that this procedure, though it enables us to _describe_ our new space, does not enable us to _imagine_ it, in the sense of calling up images of the way things would look in it. We really derive, from this analogy, no more knowledge than a man born blind may derive, as to light, from an analogy with heat. The dictum "Nihil est in intellectu quod non fuerit ante in sensu," would unquestionably be true, if for _intellect_ we were to substitute _imagination_; it is vain, therefore, _if_ our actual space be Euclidean, to hope for a power of _imagining_ a non-Euclidean space. What Helmholtz might, I believe with perfect truth, have urged against Land, is that the image we actually have of space is not sufficiently accurate to exclude, in the actual space we know, all possibility of a slight departure from the Euclidean type. But in maintaining that we cannot imagine, though we can conceive and describe, a space different from that we actually have, Land is, in my opinion, unquestionably in the right. For a pure Kantian, who maintains, with Land, that none of the axioms can be proved, this question is of great importance. But if, as I have maintained, some of the axioms are susceptible of a transcendental