On Nikolai I Lobachevsky

"The analytical geometry of Descartes and the calculus of Newton and Leibniz have expanded into the marvelous mathematical method - more daring than anything that the history of philosophy records - of Lobachevsky and Riemann, Gauss and Sylvester. Indeed, mathematics, the indispensable tool of the sciences, defying the senses to follow its splendid flights, is demonstrating today, as it never has been demonstrated before, the supremacy of the pure reason." (Nicholas M Butler,"What Knowledge is of Most Worth?", [Presidential address to the National Education Association] 1895)

"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. [...] Now the enormous effect of the Copernican system, and of the astronomical discoveries that have followed it, is […] the change effected by Copernicus in the idea of the universe. But there was left another to be made. For the laws of space and motion. […] So, you see, there is a real parallel between the work of Copernicus and […] the work of Lobatchewsky." (William K Clifford, "The Postulates of Time And Space", [in "Lectures and Essays" Vol. 1] 1901) 

"The full impact of the Lobatchewskian method of challenging axioms has probably yet to be felt. It is no exaggeration to call Lobatchewsky the Copernicus of Geometry,* for geometry is only a part of the vaster domain which he renovated; it might even be just to designate him as a Copernicus of all thought." (Eric T Bell, "Men of Mathematics", 1937)

"C. F. Gauss and N. I. Lobachevski were so deeply convinced of the consistency of the new geometry that the whole issue was of little concern to them. Bolyai, on the other hand, was greatly exercised by this question; his deep insight into the whole complex of problems connected with hyperbolic geometry is truly astounding. Bolyai tried very hard to prove rigorously that hyperbolic geometry is consistent, but he failed, because his mathematical training was far inferior to that of Gauss and Lobachevski." (Isaak Yaglom, "Geometric Transformations",  1973)

"Historically, hyperbolic geometry was first developed on an axiomatic basis. It arose as a result of efforts to prove the axiom of parallels from the other axioms. Doubt persisted for a long time as to whether this axiom could be deduced from the remaining axioms of Euclidean geometry. In their attempts to prove this axiom, mathematicians used the method of 'proof by contradiction' i.e., they assumed that the axiom of parallels was false and tried, on the basis of this assumption, to obtain a contradiction. All of these attempts were fruitless. True, the theorems obtained by negating the axiom of parallels appeared strange, but they did not contradict one another. The issue was resolved when C. F. Gauss, N. I. Lobachevski and J. Bolyai first stated explicitly that by negating the axiom of parallels one arrives at a new geometry, just as consistent as the usual (Euclidean) geometry." (Isaak Yaglom, "Geometric Transformations", 1973)

"The great merit of C. F. Gauss, N. I. Lobachevski and J. Bolyai is that they were the first to destroy the notion that Euclidean geometry was unique and irreplaceable. While it is true that Gauss, Lobachevski and Bolyai developed their geometry quite extensively without encountering contradictions, they nevertheless left unanswered the question whether it was, in principle, free of contradictions." (Isaak Yaglom, "Geometric Transformations",  1973)

"The mathematicians who originally conceived of non-euclidean geometry, Bolyai, Lobachevsky and to some extent Gauss, seem all to have conceived of the theory as one which is potentially applicable to physical space. [...] The original BL [Bolyai-Lobachevsky, or hyperbolic] geometers saw their results as holding for the case of a single parallel or for the case of multiple parallels. Because of this, the issue facing the pioneers of BL geometry was not strictly speaking consistency but truth. It was the question of whether the possibilities they envisioned of multiple non-intersecting lines were ever realized." (Michael J Scanlan, "Beltrami’s model and the independence of the parallel postulate", History and Philosophy of Logic 9, 1988)

"Mathematical theories have sometimes been used to predict phenomena that were not confirmed until years later. For example, Maxwell’s equations, named after physicist James Clerk Maxwell, predicted radio waves. Einstein’s field equations suggested that gravity would bend light and that the universe is expanding. Physicist Paul Dirac once noted that the abstract mathematics we study now gives us a glimpse of physics in the future. In fact, his equations predicted the existence of antimatter, which was subsequently discovered. Similarly, mathematician Nikolai Lobachevsky said that “there is no branch of mathematics, however abstract, which may not someday be applied to the phenomena of the real world." (Clifford A Pickover, "The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics", 2009)

"At any rate, long before the curvature of space was first detected, Beltrami’s construction of the hyperbolic plane showed that more than one kind of geometry is possible. Beltrami assumed that Euclidean space exists, and constructed a non-Euclidean plane inside it, with nonstandard definitions of 'line' and 'distance' (namely, line segments in the unit disk and pseudodistance). This shows that the geometry of Bolyai and Lobachevsky is logically as valid as the geometry of Euclid: if there is a space in which 'lines' and 'distance' behave as Euclid thought they do, then there is also a surface in which 'lines' and 'distance' behave as Bolyai and Lobachevsky thought they might." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics", 2018)