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Carl Friedrich Gauss is a former featured article. Please see the links under Article milestones below for its original nomination page (for older articles, check the nomination archive) and why it was removed.

This article appeared on Wikipedia's Main Page as Today's featured article on August 4, 2005.

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June 27, 2005	Peer review	Reviewed
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Non-Euclidean Geometry

I think the present text will cause some misconceptions on Gauss' role in the scientific dfevelopment of this branch of maths. I have modified the text, focussing his interaction with the scientific community, what only causes the progress. Dioskorides (talk) 00:20, 9 January 2024 (UTC)[reply]

References needed

I think the present version is lacking some references.

1. In sub-chapter "Further Investigations" we read in the third section: „…his two papers on biquadratic residues … are considered second in importance …“ Who considers? Such an arbitrary statement needs reference.

2. The same case in the third section of sub-chapter "Analysis": „Perhaps the most remarkables…“

3. Both the first two sections of "Analysis" are completely lacking references. Dioskorides (talk) 18:51, 31 January 2024 (UTC)[reply]

1. I remember I Read this statement somewhere, but I do not remember where. Anyway, relative importance is in the eye of the observer, so this is not such an important sentence. Most sources mention his article on biquadratic residues, so it is definitely one of his most important publications in number theory.

2. For the unique aspects of Gauss's tesselation in Gauss's work I can only give my private research on the topic, this Mathoverflow post - https://mathoverflow.net/questions/370190/how-did-gauss-characterize-the-metrical-relations-in-the-uniform-4-4-4-tiling. In particular, it is shown here that the drawing and Gauss's statements next to it can be derived with the help of Poincare disk model, so definitely Gauss had a significant anticipation of later ideas. This tessellation is also mentioned in John Stillwell's book "Mathematics and its history", but without the derivation of Gauss's statements.

3. Can you tell in more details which parts of the first two sections of "Analysis" deserve references? עשו (talk) 19:41, 31 January 2024 (UTC)[reply]

Where can we find it?

The article provides a large overview of Gauss’s own writings, and the text often refers to it (e.g. "Gauss (1809)". But in some remarks it is not clear tio which of Gauss’s writings it may concern.

1. Sub-chapter "Analysis", first section: "One of Gauss’s first independent discoveries…" Where published?

2. Sub-chapter "Analysis", second section: "…works which culminates with his discovery in 1808 of the very general Jacobi triple product identity…" Where can I find it?

3. Sub-chapter "Analysis", fifth section: "In 1822 Gauss published his prize winning essay on conformal mappings…" Title? Where published?

And: "In addition, in unpublished fragments from the years 1834-1839…" Which ones? The Collected Works contain all the unpublished things, which of them are concerned?

4. Sub-chapter "Numerical analysis", first section: "In 1815, he published an article on numeric integration…" Title? Where? Dioskorides (talk) 18:52, 31 January 2024 (UTC)[reply]

1. It is published in the Schlesinger's essay mentioned in the list of sources, where it is claimed that Gauss discovered the elementary properties of the AGM already in 1794. The exact year cannot be determined, but since already in Gauss's notebooks from the years 1797-1799 he arrived at more advanced results (not only "elementary" results) on diverse topics such as the lemniscate elliptic functions, Landen's transformation, it is quite probable that the dating to 1794 is accurate. Anyway, it was indeed one of his first independent discoveries.

2. For the claim in this sentence just look at the references in Rajan Roy's book.

3. (i) His prize winning essay (1822) is entitled "General solution to the task of depicting the parts of a given surface on another given surface in such a way that the image becomes similar to the smallest parts of what is depicted" (original german: Allgemeine Auflösung der Aufgabe die Theile einer gegebenen Fläche auf einer andern gegebnen Fläche so abzubilden, dass die Abbildung dem Abgebildeten in den kleinstenTheiIen ähnlich wird). You can find it in volume 4 of his collected works.

(ii) These unpublished fragments are in p.311-320 of volume 10-1 of his collected works. It is true that these fragments have not been discussed in depth anywhere (at least as far as I could find) except in Schlesinger's essay, but neverthless thay do represent a significant part of his work.

4. Gauss's article on numerical integration is entitled "Methodus nova integralium valores per approximationem inveniendi" and can be found in volume 3 of of his collected work.

Glad to help. עשו (talk) 19:26, 31 January 2024 (UTC)[reply]

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-== Chapter: "Religious views" ==
-The chapter "Religious views", do we need it really in this extensive form? There is only little knowledge of Gauss' religious views, some disseminated remarks in his correspondence with relation to God, and the report of Sartorius. But Sartorius himself was careful, and mentioned the danger, that for his presentation of Gauss' religious views he "would inevitably be charged with having confused my own (= Sartorius) with his (= Gauss)". (Sartorius 2006, English version, p. 83)
-The chapter is, to a great extent, based on Dunnigton, who does not quote his sources exactly or does not quote at all. The long citation has the character more of a sermon than of a scientific text. I worry that the danger Sartorius had feared is applicable on Dunnington excessively. And this holds for Rudolf Wagner, the second man telling about his talks with Gauss, too.
-I think we can give the main facts of this subject in a few sentences, as part of "Personality", and the excessive quotation is not needed. [[User:Dioskorides|Dioskorides]] ([[User talk:Dioskorides|talk]]) 13:47, 24 August 2023 (UTC)
-:Done. --[[User:Dioskorides|Dioskorides]] ([[User talk:Dioskorides|talk]]) 23:00, 28 August 2023 (UTC)
-== Suggestion for a new division into chapters ==
-Hello!
-Since number theory is the field of pure mathematics in which Gauss's contributions were most significant, I think it will be better to separate the first subchapter of "Scientific work" into two titles:
-* Algebra and number theory
-* analysis.
-This will enable wikipedians to write more comprehensively on Gauss's contributions to number theory - the deep and important results communicated in the two papers on biquadratic residues are not mentioned at all right now.
-In addition, the chapter on analysis in itself should also be written more comprehensively. Gauss's contributions to analysis were mostly unpublished during his lifetime, but should occupy some place here because it completes the the view of his achievements. His accomplishments here include his work on lemniscatic functions and elliptic functions in general, on Jacobi theta functions and Jacobi triple product identity (discovered by Gauss already in 1808), on the complex AGM algorithm and its relation to modular functions, and more. There is even a drawing in his Nachlass of a tessellation of the unit disk with curved triangles ("hyperbolic triangles"), and his results on the metrical relations in this network suggest he may had anticipated the formulas related to Poincare disk model.
-I mentioned all these facts just to argue in favor of separation of the first chapter into two. Thanks in advance! [[User:עשו|עשו]] ([[User talk:עשו|talk]]) 12:14, 2 September 2023 (UTC)
-:OK, I agree with the division of this subchapter. I would encourage you to do it. I think the "mathematics" are the most difficult subchapter of this article. Of course, further enlargement can be done, but we must keep in mind, that this Gauss article can give only an overview on the subjects. The vast information must be added to the special articles. Those, however, mostly lack a short historical part; at least, it should give the the original paper or book nof Gauss, where the things were published first. --[[User:Dioskorides|Dioskorides]] ([[User talk:Dioskorides|talk]]) 17:30, 3 September 2023 (UTC)
-:The statements should always be connected with a short reference to the writings (e.g. Gauss (1818)), so that the reader knows where Gauss dealt with the matter. --[[User:Dioskorides|Dioskorides]] ([[User talk:Dioskorides|talk]]) 22:31, 3 September 2023 (UTC)
-:To be published or not to be published, that's always the question. I think it is reasonable, when we – especially in Gauss' mathematics – always make clear, whether the work was published by him in time, published posthumously from the Nachlass, published implicitely by Gauss without a special publisation on it, or only taken from his diary or letters without a following publication. Otherwise we cannot see, what he provided for his contemporaries. They could not take any advantage from unpublished work, of course. --[[User:Dioskorides|Dioskorides]] ([[User talk:Dioskorides|talk]]) 22:31, 3 September 2023 (UTC)
-::I myself will finish my work on Gauss, I have worked on the lemma since half a year now, I have no more ideas. I worry, if I would continue it would be over-detailed. But of course, I can do some editorial work. Greetings --[[User:Dioskorides|Dioskorides]] ([[User talk:Dioskorides|talk]]) 22:36, 3 September 2023 (UTC)
 == Non-Euclidean Geometry ==