Great minds think alike: how did Bolyai and Lobachevsky come to revolutionise geometry simultaneously yet independently?

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The two mathematicians credited with the discovery of non-Euclidean geometry, János Bolyai and Nikolai Lobachevsky, knew nothing of each other’s work at the time, yet many of their theorems and even their diagrams are virtually identical. The obvious explanation for two great minds coinciding in this way is that both drew inspiration from the successes and failures of their predecessors. In fact the failure to prove Euclid’s parallel postulate led Bolyai and Lobachevsky to develop a geometry in which the parallel postulate is replaced by the hyperbolic postulate. Some of the curious objects to be found in this geometry, such as asymptotic parallels, horocycles and horospheres, are the natural consequences of this new postulate. So it is no surprise that the theorems deriving these objects should appear in similar form in the work of both mathematicians. But other theorems, such as the derivation of the fundamental identity of hyperbolic trigonometry, are by no means obvious and were proved by Bolyai and Lobachevsky in entirely different ways. Given the complexity of the arguments deployed, it is hard to believe that they arrived at their proofs without having deduced the result in advance. So how did they know the right answer? In the case of the fundamental identity I hope to suggest a possible explanation.