1830-1930: A Century of Geometry: Epistemology, History and by Luciano Boi, Dominique Flament, Jean-Michel Salanskis

By Luciano Boi, Dominique Flament, Jean-Michel Salanskis

Those innocuous little articles usually are not extraordinarily precious, yet i used to be caused to make a few feedback on Gauss. Houzel writes on "The start of Non-Euclidean Geometry" and summarises the proof. primarily, in Gauss's correspondence and Nachlass you'll discover proof of either conceptual and technical insights on non-Euclidean geometry. possibly the clearest technical result's the formulation for the circumference of a circle, k(pi/2)(e^(r/k)-e^(-r/k)). this is often one example of the marked analogy with round geometry, the place circles scale because the sine of the radius, while the following in hyperbolic geometry they scale because the hyperbolic sine. in spite of this, one needs to confess that there's no proof of Gauss having attacked non-Euclidean geometry at the foundation of differential geometry and curvature, even supposing evidently "it is hard to imagine that Gauss had no longer visible the relation". in terms of assessing Gauss's claims, after the courses of Bolyai and Lobachevsky, that this used to be identified to him already, one should still probably keep in mind that he made related claims concerning elliptic functions---saying that Abel had just a 3rd of his effects and so on---and that during this situation there's extra compelling facts that he was once primarily correct. Gauss exhibits up back in Volkert's article on "Mathematical development as Synthesis of instinct and Calculus". even supposing his thesis is trivially right, Volkert will get the Gauss stuff all flawed. The dialogue issues Gauss's 1799 doctoral dissertation at the basic theorem of algebra. Supposedly, the matter with Gauss's facts, that's alleged to exemplify "an development of instinct in terms of calculus" is that "the continuity of the aircraft ... wasn't exactified". in fact, someone with the slightest realizing of arithmetic will understand that "the continuity of the airplane" isn't any extra a subject during this evidence of Gauss that during Euclid's proposition 1 or the other geometrical paintings whatever in the course of the thousand years among them. the true factor in Gauss's evidence is the character of algebraic curves, as in fact Gauss himself knew. One wonders if Volkert even stricken to learn the paper for the reason that he claims that "the existance of the purpose of intersection is handled through Gauss as whatever totally transparent; he says not anything approximately it", that's it appears that evidently fake. Gauss says much approximately it (properly understood) in an extended footnote that indicates that he regarded the matter and, i'd argue, regarded that his facts used to be incomplete.

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Extra info for 1830-1930: A Century of Geometry: Epistemology, History and Mathematics (English and French Edition)

Example text

Eine rationale Zahl ~ mit teilerfremden Zahlen a E Z und b b E 1':1, fur die gelten wiirde a3 a --3--1=0 <=> a 3 =b 3 +3ab 2 =b 2 (b+3a). 3 b b Jeder Primteiler von b ware auch ein Teiler von a, was wegen der Teilerfremdheit ein Widerspruch ist. Daher mtisste die Zahl b = 1 sein. Daraus wtirde aber folgen, dass jeder Primteiler von a auch 1 teilen wiirde, was zu a = 1 oder a = -1, also auch einem Widerspruch fiihren wiirde. Es folgt, dass g(y) und damit f(x) irreduzibel tiber ()l sind, und a = 60 0 lasst sich nicht mit Zirkel und Lineal dritteln.

537 als Prirnzahlen. Erst Leonard Euler (1707 - 1783) fand 1732 die Zerlegung F 5 = 641 . 297 durch einen genialen zahlentheoretischen Trick. Es gilt Fs = 2 25 +1 = 232+1. 228 == (_1)4 == 1 mod 641. 2 28 = _2 32 == 1 mod 641, woraus wie behauptet 641 I F5 folgt.

27 £)! Also ist dann der Koordinatenk6rper K der Ebene nicht nur nach b. ein Schieik6rper, sondem sogar ein K6rper, und nach c. gilt der Satz von Pappus! Fur diese rein geometrische Aussage ist meines Wissens kein rein geometrischer Beweis bekannt. 11 auch die Umkehrung P {:= D. 12: Dieser Beweis ist im Detail umfangreich, es seien hier nur einige Schritte angedeutet bzw. ausgeflihrt. Die Struktur der Translationen spiegelt sich in der additiven Struktur des Koordinatenk6rpers K wider, entsprechend die Struktur der Streckungs-Fixgruppen DF in der multiplikativen Struktur von K, woraus jeweils die zweiten Aquivalenzpfeile folgen.