On János Bolyai

"If I commenced by saying that I am unable to praise this work [by J:1oos], you would certainly be surprised for a moment. But I cannot say otherwise. To praise it, would be to praise myself. Indeed the whole contents of the work, the path taken by your son, the results to which he is led, coincide almost entirely with my meditations, which have occupied my mind partly for the last thirty or thirty-five years. [...] of which up till now I have put little on paper; my intention was not to let it be published during my lifetime. [...] On the other hand it was my idea to write down all this later so that at least it should not perish with me. It is therefore a pleasant surprise for me that I am spared this trouble, and I am very glad that it is just the son of myoid friend, who takes the precedence of me in such a remarkable manner." (Carl Friedrich Gauss, [letter to Farkas Bolyai] 1832

"Bolyai [Janos] projected a universal language for speech as we have it for music and mathematics." (George B Halsted,  "János Bolyai, Science Absolute of Space", 1896)

"The profound mathematical ability of Bolyai János showed itself physically not only in his handling of the violin, where he was a master, but also of arms, where he was unapproachable." (George B Halsted,  "János Bolyai, Science Absolute of Space", 1896)

"The scheme of laws of nature so largely due to Newton is merely one of an infinite number of conceivable rational schemes for helping us master and make experience; it is commode, convenient; but perhaps another may be vastly more advantageous. The old conception of true has been revised. The first expression of the new idea occurs on the title page of John Bolyai's marvelous Science Absolute of Space, in the phrase 'haud unquam a priori decidenda'." (George B Halsted, 1913) 

"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)

"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)