[2024]. On the supposition that Euclidean geometry is not valid, it is easy to show that similar figures do not exist; in that case the angles of an equilateral triangle vary with the side in which I see no absurdity at all. The angle is a function of the side and the sides are functions of the angle, a function which, of course, at the same time involves a constant length. It seems somewhat of a paradox to say that a constant length could be given a priori as it were, but in this again I see nothing inconsistent. Indeed, it would be desirable that Euclidean geometry were not valid, for then we should possess a general a priori standard of measure.—Gauss.
Letter to Gerling (1816); Werke, Bd. 8 (Göttingen, 1900), p. 169.
[2025]. I am convinced more and more that the necessary truth of our geometry cannot be demonstrated, at least not by the human intellect to the human understanding. Perhaps in another world we may gain other insights into the nature of space which at present are unattainable to us. Until then we must consider geometry as of equal rank not with arithmetic, which is purely a priori, but with mechanics.—Gauss.
Letter to Olbers (1817); Werke, Bd. 8 (Göttingen, 1900), p. 177.
[2026]. There is no doubt that it can be rigorously established that the sum of the angles of a rectilinear triangle cannot exceed 180°. But it is otherwise with the statement that the sum of the angles cannot be less than 180°; this is the real Gordian knot, the rocks which cause the wreck of all.... I have been occupied with the problem over thirty years and I doubt if anyone has given it more serious attention, though I have never published anything concerning it. The assumption that the angle sum is less than 180° leads to a peculiar geometry, entirely different from the Euclidean, but throughout consistent with itself. I have developed this geometry to my own satisfaction so that I can solve every problem that arises in it with the exception of the determination of a certain constant which cannot be determined a priori. The larger one assumes this constant the more nearly one approaches the Euclidean geometry, an infinitely large value makes the two coincide. The theorems of this geometry seem in part paradoxical, and to the unpracticed absurd; but on a closer and calm reflection it is found that in themselves they contain nothing impossible.... All my efforts to discover some contradiction, some inconsistency in this Non-Euclidean geometry have been fruitless, the one thing in it that seems contrary to reason is that space would have to contain a definitely determinate (though to us unknown) linear magnitude. However, it seems to me that notwithstanding the meaningless word-wisdom of the metaphysicians we know really too little, or nothing, concerning the true nature of space to confound what appears unnatural with the absolutely impossible. Should Non-Euclidean geometry be true, and this constant bear some relation to magnitudes which come within the domain of terrestrial or celestial measurement, it could be determined a posteriori.—Gauss.
Letter to Taurinus (1824); Werke, Bd. 8 (Göttingen, 1900), p. 187.
[2027]. There is also another subject, which with me is nearly forty years old, to which I have again given some thought during leisure hours, I mean the foundations of geometry.... Here, too, I have consolidated many things, and my conviction has, if possible become more firm that geometry cannot be completely established on a priori grounds. In the mean time I shall probably not for a long time yet put my very extended investigations concerning this matter in shape for publication, possibly not while I live, for I fear the cry of the Bœotians which would arise should I express my whole view on this matter.—It is curious too, that besides the known gap in Euclid’s geometry, to fill which all efforts till now have been in vain, and which will never be filled, there exists another defect, which to my knowledge no one thus far has criticised and which (though possible) it is by no means easy to remove. This is the definition of a plane as a surface which wholly contains the line joining any two points. This definition contains more than is necessary to the determination of the surface, and tacitly involves a theorem which demands proof.—Gauss.
Letter to Bessel (1829); Werke, Bd. 8 (Göttingen, 1900), p. 200.
[2028]. I will add that I have recently received from Hungary a little paper on Non-Euclidean geometry, in which I rediscover all my own ideas and results worked out with great elegance,.... The writer is a very young Austrian officer, the son of one of my early friends, with whom I often discussed the subject in 1798, although my ideas were at that time far removed from the development and maturity which they have received through the original reflections of this young man. I consider the young geometer v. Bolyai a genius of the first rank.—Gauss.
Letter to Gerling (1832); Werke, Bd. 8 (Göttingen, 1900), p. 221.