[2019]. The most suggestive and notable achievement of the last century is the discovery of Non-Euclidean geometry.—Hilbert, D.
Quoted by G. D. Fitch in Manning’s “The Fourth Dimension Simply Explained,” (New York, 1910), p. 58.
[2020]. Non-Euclidean geometry—primate among the emancipators of the human intellect....—Keyser, C. J.
The Foundations of Mathematics; Science History of the Universe, Vol. 8 (New York, 1909), p. 192.
[2021]. Every high school teacher [Gymnasial-lehrer] must of necessity know something about non-euclidean geometry, because it is one of the few branches of mathematics which, by means of certain catch-phrases, has become known in wider circles, and concerning which any teacher is consequently liable to be asked at any time. In physics there are many such matters—almost every new discovery is of this kind—which, through certain catch-words have become topics of common conversation, and about which therefore every teacher must of course be informed. Think of a teacher of physics who knows nothing of Roentgen rays or of radium; no better impression would be made by a mathematician who is unable to give information concerning non-euclidean geometry.—Klein, F.
Elementarmathematik vom höheren Standpunkte aus (Leipzig, 1909), Bd. 2, p. 378.
[2022]. What Vesalius was to Galen, what Copernicus was to Ptolemy, that was Lobatchewsky to Euclid. There is, indeed, a somewhat instructive parallel between the last two cases. Copernicus and Lobatchewsky were both of Slavic origin. Each of them has brought about a revolution in scientific ideas so great that it can only be compared with that wrought by the other. And the reason of the transcendent importance of these two changes is that they are changes in the conception of the Cosmos.... And in virtue of these two revolutions the idea of the Universe, the Macrocosm, the All, as subject of human knowledge, and therefore of human interest, has fallen to pieces.—Clifford, W. K.
Lectures and Essays (New York, 1901), Vol. 1, pp. 356, 358.
[2023]. I am exceedingly sorry that I have failed to avail myself of our former greater proximity to learn more of your work on the foundations of geometry; it surely would have saved me much useless effort and given me more peace, than one of my disposition can enjoy so long as so much is left to consider in a matter of this kind. I have myself made much progress in this matter (though my other heterogeneous occupations have left me but little time for this purpose); though the course which I have pursued does not lead as much to the desired end, which you assure me you have reached, as to the questioning of the truth of geometry. It is true that I have found much which many would accept as proof, but which in my estimation proves nothing, for instance, if it could be shown that a rectilinear triangle is possible, whose area is greater than that of any given surface, then I could rigorously establish the whole of geometry. Now most people, no doubt, would grant this as an axiom, but not I; it is conceivable that, however distant apart the vertices of the triangle might be chosen, its area might yet always be below a certain limit. I have found several other such theorems, but none of them satisfies me.—Gauss.
Letter to Bolyai (1799); Werke, Bd. 8 (Göttingen, 1900), p. 159.