Lobachevsky and Bolyai, the pioneers of non-Euclidean geometry explored the logical consequences of the hyperbolic axiom which we introduced in our first post.
It is the newer one replacing the fifth axiom of Euclidean Geometry – the parallel postulate. This new geometry was bizarrely different from that of Euclid, as commented earlier also.
Before them, many others, most notably Saccheri (in 1733) and Lambert (in 1766), had discovered some of the consequences of the hyperbolic axiom.
However, their aim in exploring these consequences had been to find a contradiction, which they believed would finally prove that Euclidean Geometry was the One True Geometry.
Certainly, Saccheri believed he had found a clear contradiction when he published ‘Euclid Freed of Every Flaw’. But Lambert is a much more perplexing case, and he is perhaps an unsung hero in this story. The results of Johann Heinrich Lambert penetrated so deeply into this new geometry that is seems impossible that he did not at times believe in the reality of what he was doing (even though his primary objective was to find a contradiction in the hyperbolic axiom)! Regardless of his motivations and beliefs, Lambert was certainly the first to discover a remarkable fact about the angle sum of a triangle under the hyperbolic axiom, and his result will be central to much that follows in a latter chapter of our discussion.
Nevertheless, Lobachevsky and Bolyai richly deserve their fame for having been the first to recognize and fully embrace the idea that they had discovered an entirely new and consistent non-Euclidean Geometry. But what this new geometry really meant, and what it might be useful for, even they could not say (as the nature of this discovery was heavily axiomatic and the concrete part of it was yet to be uncovered).
Lobachevsky did in fact put this geometry to use to evaluate previously unknown integrals, but this particular application must be viewed as relatively minor, at least in hindsight.
Remarkably and surprisingly, it was the ‘Differential Geometry of curves surfaces‘ that ultimately resolved these questions. In 1868, the Italian mathematician Eugenio Beltrami finally succeeded in giving Hyperbolic Geometry a concrete interpretation, setting it upon a firm and intuitive foundation from which it has since grown and flourished. Sadly, neither Lobachevsky nor Bolyai lived to see this, they died in 1856 and 1860 respectively.
It turns out that this non-Euclidean Geometry had in fact already manifested itself in various branches of mathematics throughout history, but always in disguise! Henri Poincaré (beginning around 1882) was the first not only to strip it of its camouflage, but also to recognize and exploit its power in such diverse areas as Complex Analysis, Differential Equations, Number Theory, and Topology.
Its continued vitality and centrality in the mathematics of the 20th and 21st centuries is demonstrated by Thurston’s work on 3-manifolds. Wiles’s proof of Fermat’s Last Theorem, and Perelman’s proof of the Poincaré conjecture (as a special case of Thurston’s Geometrization Conjecture), to name but three examples.
Later in our discussion, we shall describe Beltrami ‘s breakthrough, as well as the nature of Hyperbolic Geometry, but before that we will be exploring a different, simpler kind of non-Euclidean Geometry, one that was already known to the Ancients. Guess what? It is none other than the Spherical Geometry!
Our next post in this series will be focused on understanding Spherical Geometry and we will see later that analogizing and comparing with this geometry gives us a better insight into the study of Hyperbolic Geometry and that is the sole purpose to discuss about spherical geometry in the next post.
Leave a Reply