User:Dioskorides/sandbox

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 * The problem of largest ellipse inscribed in a quadrilateral is quite difficult (even the analogous problem for a triangle is difficult (see Steiner inellipse), and for a quadrilateral it is even more difficult). I cannot state what is essentially the new idea here (I myself don't know how to solve this problem), but it does not seem to contain a revolutionary idea. This is why I included it in the "minor contributions" section.
 * The new idea in "rotation in space" is very clear, it is an anticipation of the revolutionary idea of quaternions, which is the first example of non-commutative algebra and a highlight of modern algebra. Initially I even thought to include it in a more significant section and not in the "minor contributions", but since it is still a small part of Gauss's work, I eventually decided to write it here.
 * Gauss's result on the area of planar pentagons is a curiosity of recreational math; it is discused in the article "geometry of pentagons: from Gauss to Robbins". The surprising point of Gauss's result is that for a pentagon one does not have to use triangulation to compute its area, but rather use only the vertex triangles.
 * I agree that Gauss's solution to Apollonius problem is not very significant, it is one and half page long and not so interesting. This was not an edition of mine, if I recall correctly.
 * About the "Pentagramma myrificum" I cannot give a comprehensive answer, but I am pretty sure it was one of his more significant minor investigations.
 * 2. About the "Riemann sphere"- if you will look at Gauss's fragment that Schlesinger refers to on p.198, you will see that Gauss used the stereographic projection there (in order to identify the extended complex plane with the sphere). The same idea was mentioned in one of Riemann's lectures, and that is why several decades later Riemann's name became attached to this idea. By the way, I personally discovered this aspect of Gauss's work several years ago when reading John Stillwell's book "mathematics and its history". עשו (talk) 10:32, 19 June 2024 (UTC)

-- In the lifetime of Gauss a vivid discussion on the Parallel postulate in Euclidean geometry was going on. Numerous efforts were made to prove it in the frame of the Euclidean axioms, whereas some mathematicians discussed the possibility of geometrical systems without it. Gauss thought about the basics of geometry since the 1790s years, but in the 1810s years he realized that a non-Euclidean geometry without the parallel postulate could solve the problem. In a letter to Franz Taurinus of 1824, he presented a short comprehensible outline of what he himself named a "non-Euclidean geometry", but he strongly forbed Taurinus to make any use of it.

The first publications on non-Euclidean geometry in the history of mathematics were authored by Nikolai Lobachevsky in 1829 and Janos Bolyai in 1832. In the following years, Gauss wrote his ideas on the topic but did not publish them, thus avoiding influencing the contemporary scientific discussion. Gauss commended the ideas of Janos Bolyai in a letter to his father and university friend Farkas Bolyai claiming that these were congruent to his own thoughts since some decades. However, it is not quite clear to what extent he preceded Lobachevsky and Bolyai, for he his letter remarks are only vague and obscure.

Sartorius mentioned Gauss's work on non-Euclidean geometry firstly in 1856, but only the edition of left papers in Volume VIII of the Collected Works (1900) showed Gauss's own ideas on that matter, at a time when non-Euclidean geometry had yet grown out of controversial discussion.