[PDF] 1830-1930: A Century of Geometry: Epistemology, History and Mathematics (English and French Edition) Luciano Boi, Dominique Flament, Jean-Michel Salanskis

These harmless little articles are not terribly useful, but I was prompted to make some remarks on Gauss. Houzel writes on “The Birth of Non-Euclidean Geometry” and summarises the facts. Basically, in Gauss’s correspondence and Nachlass one can find evidence of both conceptual and technical insights on non-Euclidean geometry. Perhaps the clearest technical result is the formula for the circumference of a circle, k(pi/2)(e^(r/k)-e^(-r/k)). This is one instance of the marked analogy with spherical geometry, where circles scale as the sine of the radius, whereas here in hyperbolic geometry they scale as the hyperbolic sine. Even so, one must confess that there is no evidence of Gauss having attacked non-Euclidean geometry on the basis of differential geometry and curvature, although obviously “it is difficult to think that Gauss had not seen the relation”. When it comes to assessing Gauss’s claims, after the publications of Bolyai and Lobachevsky, that this was known to him already, one should perhaps remember that he made similar claims regarding elliptic functions—saying that Abel had only a third of his results and so on—and that in this case there is more compelling evidence that he was essentially right. Gauss shows up again in Volkert’s article on “Mathematical Progress as Synthesis of Intuition and Calculus”. Although his thesis is trivially correct, Volkert gets the Gauss stuff all wrong. The discussion concerns Gauss’s 1799 doctoral dissertation on the fundamental theorem of algebra. Supposedly, the problem with Gauss’s proof, which is supposed to exemplify “an advancement of intuition in relation to calculus” is that “the continuity of the plane … wasn’t exactified”. Of course, anyone with the slightest understanding of mathematics will know that “the continuity of the plane” is no more an issue in this proof of Gauss that in Euclid’s proposition 1 or any other geometrical work whatsoever during the two thousand years between them. The real issue in Gauss’s proof is the nature of algebraic curves, as of course Gauss himself knew. One wonders if Volkert even bothered to read the paper since he claims that “the existance of the point of intersection is treated by Gauss as something absolutely clear; he says nothing about it”, which is plainly false. Gauss says a lot about it (properly understood) in a long footnote that shows that he recognised the problem and, I would argue, recognised that his proof was incomplete.