User talk:עשו

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September 2023

Hello, I'm TechnoSquirrel69. I noticed that you added or changed content in an article, Carl Friedrich Gauss, but you didn't provide a reliable source. It's been removed and archived in the page history for now, but if you'd like to include a citation and re-add it, please do so. You can have a look at referencing for beginners. If you think I made a mistake, you can leave me a message on my talk page. Thank you. —TechnoSquirrel69 (sigh) 19:11, 1 September 2023 (UTC)[reply]

Gauss

Hello, just for your information:

I have finished my work on the Gauss article and nominated it as Good article on March 3th. Today another user has taken it from the list because lack of 43 references, marked with tag "citation needed". 23 of them are in the mathematical chapters "Algebra", "Analysis", "Theory of errors", Differential geometry", "Early topology", and "Minor mathematical accomplishments". Can you please supply the references, I am not suited to the mathematical part, I will add the other missing ones. Many thanks for your constructive collaboration. Greetings. Dioskorides (talk) 18:55, 6 March 2024 (UTC) In addition to that: the last three sections of "Electromagnetism" are created by you and still need some reference. --Dioskorides (talk) 21:08, 6 March 2024 (UTC)[reply]

Hey!

After checking the list of statements with tag "citation needed" I saw that at least half of them are not really neccesary - the statements are either contained in other wikipedia articles or in the references listed, or are implied by other statements (which are sourced) in the article. I will try to supply the references for the statements I believe truly deserve references. עשו (talk) 08:04, 7 March 2024 (UTC)[reply]

Dioskorides, I added many references to the statements I believe need better sourcing. Many of the statements with "citation needed" tag do not really need additional sourcing - I will give a few examples:

In the "Disquisitiones Arithmeticae" subchapter - this tag is added right after a passage that describes the contents of this book; however, this is obviously correct by just reading the article on the book Disquisitiones Arithmeticae.
In the chapter "Further investigations" that are 4 statements with this tag, of which 2-3 do not really deserve it. For example, the information establishing the statement with the second tag can be found in wikipedia article Gaussian integers (in the "historical background" chapter).
In the chapter "Theory of errors", this tag appears after the first passage, but by reading wikipedia article on Gauss–Markov theorem, one is asserted about the correctness of the information given in this passage.
In the chapter "Differential geometry" this tag appears 3 times, but I believe it is nowhere justified. The significance of the Theorema Egregium and its relavence to carthography are well covered in many other wikipedia articles, such as Differential geometry of surfaces. עשו (talk) 10:41, 7 March 2024 (UTC)[reply]

Hello,

of course, I have been convinced that this article is quite complete, and that there were enough references, when I started the GA-nomination. I agrree with you, that some of the citation tags are incomprehensible. I have read the text once more and tried to find the particular reference problems. And I try to understand the remarks of User:David Eppstein in that way, that we cannot refer the Gauss article with references in Gauss-related articles. We cannot take for granted that readers of the Gauss article have read the Gauss-related articles, or even that the readers of the more general Gauss article are able to understand the more spezialized Gauss-related articles. The Gauss article is really blessed with related articles, but it's a pity that they very often give no historical background, or if so, it's often hard to find. And furthermore, all evaluating statements, they may be right or wrong, need a reference. We must present an origin for these statements, otherwise we would arouse suspicion that these were "self-made" valuations by Wikipedians.

Now some remarks to your examples, I marked the problematic statements, in my opinion, of course.

In the "Disquisitiones Arithmeticae" subchapter - this tag is added right after a passage that describes the contents of this book; however, this is obviously correct by just reading the article on the book Disquisitiones Arithmeticae.

The entries in Gauss' Mathematical diary indicate that he was busy with the subject of number theory at least since 1796. A detailed study of previous researches showed him that some of his findings had been already done by other scholars. In the years 1798 and 1799 Gauss wrote a voluminous compilation of all these results in the famous Disquisitiones Arithmeticae, published in 1801, that was fundamental in consolidating number theory as a discipline and covered both elementary and algebraic number theory. Therein he introduces, among other things, the triple bar symbol ( $\equiv$ ) for congruence and uses it in a clean presentation of modular arithmetic. It deals with the unique factorization theorem and primitive roots modulo n. In the main chapters, Gauss presents the first two proofs of the law of quadratic reciprocity, which allows mathematicians to determine the solvability of any quadratic equation in modular arithmetic, and develops the theories of binary and ternary quadratic forms.^{[citation needed]}

Bold Types: evaluation

In the chapter "Further investigations" that are 4 statements with this tag, of which 2-3 do not really deserve it. For example, the information establishing the statement with the second tag can be found in wikipedia article Gaussian integers (in the "historical background" chapter).

Gauss proved Fermat's Last Theorem for n = 3 and sketchingly proved it for n = 5 in his unpublished writings. The particular case of n = 3 was proved much earlier by Leonhard Euler, but Gauss developed a more streamlined proof which made use of Eisenstein integers; though more general, the proof was simpler than in the real integers case.^{[citation needed]}

In his two important papers on biquadratic residues (published in 1828 and 1832) Gauss introduces the ring of Gaussian integers $\mathbb {Z} [i]$ , and shows that this ring is a unique factorization domain. He generalizes into this ring many key arithmetic concepts, such as Fermat's little theorem and Gauss's lemma. The main objective of introducing this ring was to formulate the law of biquadratic reciprocity – as Gauss discovered, rings of complex integers are the natural setting for such higher reciprocity laws.^{[citation needed]}

I cannot find the name Fermat or Gauss' lemma in article Gaussian integers, nor the 'complex imntegers".

In the second paper, he states the general law of biquadratic reciprocity and proves several special cases of it, but proof of the general theorem is lacking, despite Gauss's statements that he found such a proof around 1814. He promised a third paper with a general proof, which has never appeared. In an earlier publication from 1818 containing his fifth and sixth proofs of quadratic reciprocity, he claims the techniques of these proofs (Gauss sums) can be applied to prove higher reciprocity laws.^{[citation needed]}

Gauss's publications on biquadratic residues opened the way for boundless enlargement of the theory of numbers, and are memorable for the wealth of investigations in "higher arithmetic" that they led to.^{[citation needed]}

In the chapter "Theory of errors", this tag appears after the first passage, but by reading wikipedia article on Gauss–Markov theorem, one is asserted about the correctness of the information given in this passage.

It is likely that Gauss used the method of least squares for calculating the orbit of Ceres to minimize the impact of measurement error. The method was published first by Adrien-Marie Legendre in 1805, but Gauss claimed in Theoria motus (1809) that he had been using it since 1794 or 1795. In the history of statistics, this disagreement is called the "priority dispute over the discovery of the method of least squares". Gauss proved the method under the assumption of normally distributed errors (Gauss–Markov theorem) in his paper Theoria combinationis observationum erroribus minimis obnoxiae from 1821.^{[citation needed]}

May we take the papaer of R.L. Plackett as reference to it?

In this paper, which was relatively little known in the English speaking world in the first century after its publication, he stated and proved Gauss's inequality (a Chebyshev-type inequality) for unimodal distributions, and stated without proof another inequality for moments of the fourth order (a special case of Gauss-Winckler inequality). He derived lower and upper bounds for the variance of sample variance. In a supplement to this paper Gauss described recursive least squares methods that went unnoticed until 1950, when his work was rediscovered as a consequence of the growing demand of quick estimation for various new technologies. Gauss's work on the theory of errors was extended in several directions by the geodesist Friedrich Robert Helmert, and the Gauss-Helmert theory is considered today as the "classical" theory of errors.^{[citation needed]}

In the chapter "Differential geometry" this tag appears 3 times, but I believe it is nowhere justified. The significance of the Theorema Egregium and its relavence to carthography are well covered in many other wikipedia articles, such as Differential geometry of surfaces. עשו (talk) 10:41, 7 March 2024 (UTC)[reply]

The geodetic survey of Hanover fueled Gauss' interest in differential geometry and topology, fields of mathematics dealing with curves and surfaces. This led him in 1828 to the publication of a memoir that marks the birth of modern differential geometry of surfaces, as it departed from the traditional ways of treating surfaces as cartesian graphs of functions of two variables, and instead pioneered a revolutionary approach that initiated the exploration of surfaces from the "inner" point of view of a two-dimensional being constrained to move on it. Its crowning result, the Theorema Egregium (remarkable theorem), established a property of the notion of Gaussian curvature. Informally, the theorem says that the curvature of a surface can be determined entirely by measuring angles and distances on the surface. That is, curvature does not depend on how the surface might be embedded in 3-dimensional space or 2-dimensional space.^{[citation needed]}

The Theorema Egregium leads to the abstraction of surfaces as doubly-extended manifolds - it makes clear the distinction between the intrinsic properties of the manifold (the metric) and its physical realization (the embedding) in ambient space. A consequence is the impossibility of an isometric transformation between surfaces of different Gaussian curvature. This means practically that a sphere or an ellipsoid cannot be transformed to a plane without distortion, what causes a fundamental problem in designing projections for geographical maps.^{[citation needed]}

An additional significant portion of his essay is dedicated to a profound study of geodesics. In particular, Gauss proves the local Gauss-Bonnet theorem on geodesic triangles, and generalizes Legendre's theorem on spherical triangles to geodesic triangles on arbitrary surfaces with continuous curvature; he found that the angles of a "sufficiently small" geodesic triangle deviate from that of a planar triangle of the same sides in a way that depends only on the values of the surface curvature at the vertices of the triangle - regardless of the behaviour of the surface in the triangle interior.^{[citation needed]}

We cannot give a reference "Gauss (1828), Art. 26-28", as given in Legendre's theorem on spherical triangles, that is the way of refering in original research. We need a reference from the secondary literature.

I have supplied some further references, too, but I have no idea for the remaining ones. Greetings. --Dioskorides (talk) 21:43, 12 March 2024 (UTC)[reply]

Hello,

thank you for adding some of the claimed references. I have also added some references, and worked a little bit on the style. But I have still some questions, some facts are not quite clear, or not sufficiently referenced.

In "Analysis", 3rd section: "His unpublished writings include several drawings that show he was quite aware of the geometric side of the theory; in the context of his work on the complex AGM he recognized and made a sketch of the key concept of fundamental domain for the modular group." has no reference, only note m, but the note is not referenced either.
In "Astronomy" last section: "Gauss presents the method of evaluating the gravitational attraction of such an elliptic ring, which includes several complicated steps; one of them involves a direct application of the arithmetic-geometric mean (AGM) algorithm to calculate an elliptic integral." is still without reference.
In "Theory of errors", 2nd section: "In this paper, which was relatively little known in the English speaking world in the first century after its publication, he stated and proved Gauss's inequality (a Chebyshev-type inequality) for unimodal distributions, and stated without proof another inequality for moments of the fourth order (a special case of Gauss-Winckler inequality)." We don't have a lemma on "Gauss-Winckler inequality" in the List of things named after Carl Friedrich Gauss, and it seems to be a seldom-used concept. The reader cannot get any information of it in Wikipedia. I for myself find this mathematician "Winckler" neither in English nor in German Wikipedia.

Note s as appendix of the next sentence is without reference. Do we need this information? Was it a severe mistake of Gauss?

In "Differential geometry", last section: "Based on this evidence and the announcement in the memoir of further investigations on the curvature integral, it is very likely that he knew the more general version of the Gauss-Bonnet theorem proved by Pierre Ossian Bonnet in 1848, which is closer in spirit to the global version of this theorem." I find this difficult to understand. The reference "Minimal surfaces (2010) p.50" is not really usuable. And I think, there is no clarification with the Note w as appendix, what is Bolza's "essay on Gauss's contributions to calculus of variations? If I look at Oskar Bolza, I find his "Lectures On The Calculus of Variations" in four versions (English, German), do you mean that? And if so, to which edition refer the given page numbers?

It would be fine, if we can clarify these points before a renewed nomination. Greetings --Dioskorides (talk) 18:37, 31 March 2024 (UTC)[reply]

Hey!

I added sources for three claims (out of the four claims you mentioned) for which you requested credible source:

In "Analysis" - I cited the relavent pages in Schlesinger's essay on the fundamental domain of the modular group. You can also find less comprehensive discussions of Gauss's work on these mathematical themes in different books and articles.
In "Astronomy" - I cited the relavent pages from Schlesinger essay. The point here is that Gauss arrived (after several steps) at a certain integral, which he evaluated using the AGM algorithm.
In "Theory of errors" - you are right that the term "Gauss-Winckler inequality" is seldom used; I used this term just to help the reader distinguish it from the better known Gauss's inequality - these are not the same result, as Gauss stated the former result as a kind of extension for the latter (just look at his original publication). Therefore, I do not think this claim needs better sourcing. Regarding the mistake of Gauss mentioned in note s - what is really important is that by a comprhensive procedure Gauss derived these bounds (and his upper bound was correct), and certainly this was not a severe mistake of him. But I have to warn that I am far from being familiar with his work on errors theory, so my opinion may be wrong.
In "Differential geometry" - Bolza's essay is in volume 10-2 of Gauss's Werke, just like Schlesinger's and Stackel's essays (and several others). I added it in the list of sources.

עשו (talk) 22:11, 31 March 2024 (UTC)[reply]

Hello,

thank you for your last edits, esp. for Bolza. As I am no mathematician, I have no idea about the variation calculus, and so I have overloooked the Bolza paper.

I think, the next week I will try once more to start a Good-Article nomination. I will keep you informed about what's going on. Greetings. --Dioskorides (talk) 12:09, 5 April 2024 (UTC)[reply]

Hello,

User:David Eppstein (who is a mathematician with own WP-article and an experienced Wikipedian) mentioned still two non-referenced statements in the chapter "Analysis". They can easily be found at the end of the 2nd and 4th section, where references are missing. I am not experienced in this subject.

So I may ask: Can we put one of the previous references to the section ends, e.g. the refs. no. 130 resp. 133, or do we need special references for it?

Additionally from me: In the 3rd section, I miss a reference for footnote p. May we take Schlesinger, p. 102 (with his footnote no.140)? It's the only site on pp. 101–106, where he mentions the Disquisitiones.

And in the chapter "Differential geometry", I don't understand ref. no. 191, I find it incomplete.

Another question: For me, the sections 1, 3, and 4 seem to be connected by the subjects of AGM and elliptic functions; what about joining them together with another sequence of sections? I propose changing the sequence (present numbers) to 1,3,4,2,5 .

Greetings. --Dioskorides (talk) 20:52, 24 April 2024 (UTC)[reply]

1.(a) I added a reference for the statement at the end of the 2th section of "Analysis" (on Gauss's 1811 article about the sign of qudratic Gauss sums); this claim really needed reference.

(b) I don't understand why the statement at the end of the 4th section needs reference - Gauss continued fraction appears explicitly in his published work (in the "Disquisitiones generales circa series infinitam..." (1813)) and not in his unpublished work, so the required reference is simply this paper of Gauss.

2. I added another reference for footnote p, this reference repeats on Schlesinger's statement but is clearer than him. If you are willing to understand this footnote from the mathematical point of view, it is harder to explain, but I will say that there is a bijection between binary quadratic forms (a,b,c) and points of the upper half plane, and replacing the variables x,y of the form by new variable x',y' that are related to x,y by a linear substitution corresponds to the action of the modular group on the upper half plane. So the fundamental domain (which contains exactly one representative from each orbit) for the modular group (which has significance in analysis) is actually the fundamental domain for binary quadratic forms (of importance in number theory).

3. I think Ref. no.191 is the only place where Stackel discusses Gauss's generalization of Legendre's theorem on spherical triangles, but he does not make a comprehensive discussion of this result, so perhaps this claim really needs better sourcing. What exactly do you wish to further understand? I can give another source (PETER DOMBROWSKI, 150 years after Gauss’ « disquisitiones generales circa superficies curvas ») which on p.114-120 also mentions this result and describes Gauss's ideas of its proof. Don't ask me to explain the ideas of its proof because I myself don't know it, but I do understand the theorem (but not its proof).

4. I tried to describe the development of Gauss's analytic ideas in a chronological order - first he discovered the AGM and lemniscatic elliptic functions (section 1), than he turned to theta functions (section 2) and later to the foundations of the general theory of modular forms (section 3). After that his interests shifted to the study of hypergeometric series (section 4) and from the 1820s and on to conformal mappings (section 5). I think that if there is a need for change, it is to the order 1,3,2,4,5, but I still feel the current sequence is best. Anyway, you may take advice from other wikipedians regerding the best order, I will not resist to change if other will think it is better otherwise. עשו (talk) 13:36, 25 April 2024 (UTC)[reply]

Hello,

Many thanks for your edits and your instructive comments. I fear I have occupied you too much.

Firstly, I want to avoid a possible misunderstanding. I have not claimed for further information on content, neither did David Eppstein, as far as I can see. It's only a formal matter to finish a section with a source remark from the secondary literature, otherwise it looks like unsourced, and only very trivial content may be without reference. I think David Eppstein only made a short overview of the article, and thereby noticed these unsourced sections, regardless of their content.

For example your point 1.b: The last sentence of "Analysis", 4th section, is not trivial and needs reference. How shall the reader know that he can find the "continued fractions" in the Disquisitiones generales, neither the title nor the non-existing table of content tell it. And the Disq. generales itself is a primary source, but we need secondary sources. I have found it now in Schlesinger, and added it. I have put a remark to the table of writings, too. But, of course, we addditionally can give the Gauss text as service to the reader.

Thanks for "Abel", I think he should be mentioned in the text, because he finally published all these important ideas, no one has use any benefit from unpublished stuff.

I will try now to shorten some of the sections resp. notes and then present it on WP:GAN once more. I am curious to get a reviewer's comment, and I fear he or she will say: "too much".:) Greetings--Dioskorides (talk) 22:30, 27 April 2024 (UTC)[reply]

Dioskorides, no problem! you may keep occuping me, although I have to admit I do prefer to discuss the contents of this article rather than giving better references/sources. I think this article has a very large number of references and sources to be considered very reliable. By the way, after adding a new passage to the "Analysis" chapter (now there are six passages), I made a small change in the order of the passages, as you suggested (now it is 1,2,4,3,5,6).

I have to take advice from you on a possible addition to this article. I noticed that Gauss's publication "Theoria attractionis..." (1813) is not mentioned at all in this article. It contained the first (or one of the first, I am not sure of that) closed-form analytic solutions for the gravitational attraction of homogeneous ellipsoids. Since it had a closed form, this solution avoided convergence issues of previous infinite series solutions. It also contained Gauss's theorem in vector analysis, one of his more known theorems.

Therefore, I think this publication should be mentioned, but I have no idea in which chapter to add it without harming the presentation of things. So can you recommend where to mention this publication? עשו (talk) 20:45, 28 April 2024 (UTC)[reply]

Thanks for your attention. I have just inserted a short comment into the chapter "Selected writings". I am with you, we should write more on the obvious published results of the lemma person than on all the stuff he left hidden in the drawer of his desk.

Today I have enlarged the article of Ludwig Adolf Sohncke, whom you have mentioned as one of the guys who found the solutions for the transformations of the modular forms. You write that he had discovered the 7th and some higher orders, and I am sure you have a valuable source for it. I have linked the available papers of Sohncke. In his paper from 1834 (page 178), he deals with the 11th, 13th, and 17th order, but I cannot find the 7th order. But, if you leaf backwards to page 172, you will find the previous article that in fact deals with the 7th order. This article is not from Sohncke, but from the mathematician C. Gützlaff, and Sohncke, who knew him, referred to this text. The full title is: Aequatio modularis pro transformatione functionum ellipticarum septimi ordinis (pages 172–177) So it might be possible that this had been overlooked in the historical literature, and all the content of these two articles might be attributed to the more prominent professor Sohncke than to Carl Eduard Gützlaff, who was a high school teacher at the gymnasium (cf. pp. 22 and 23) of Marienwerder (today Kwidzyn in Poland). Gützlaff also refers to Jacobi as the discoverer of the 3th and the 5th order. This only to your information, we should not discuss all these details in the Gauss text. Greetings --Dioskorides (talk) 21:49, 28 April 2024 (UTC)[reply]

Hello,

thank you for your new useful edit. You write: " I am not sure this passage fits well in the mechanics chapter." I have an idea to solve this problem. We create a new chapter "Potential theory" from your latest edit together with the 2nd half of the 4th section of "Geomagnetism", because this shows the same problem, and place the new chapter between "Magnetism" and "Optics". If you agree, I will do it in this way. Greetings --Dioskorides (talk) 21:51, 29 April 2024 (UTC)[reply]

I agree. עשו (talk) 22:00, 29 April 2024 (UTC)[reply]

Happy birthday

to CF Gauss! I think, this is just the right time to finish the work for the moment, and declare it complete for Wikipedia. I think when a reviewer will have given his comment, the work will continue anyway. Thank you for your constructive collaboration. Greetings --Dioskorides (talk) 14:11, 30 April 2024 (UTC) The article is now listed in Wikipedia:Good article nominations#Mathematics. --Dioskorides (talk) 15:08, 30 April 2024 (UTC)[reply]

I want to inform you that the review of the Gauss article has started yesterday, see here. --Dioskorides (talk) 22:57, 2 June 2024 (UTC)[reply]

Hello,

the review of the Gauss article is going on, and I think it is going successful. But I have still a problem with the chapter "Minor mathematical accomplishments". User:Broc has complained about some wording, but I have another problem.

In the first sentence we read "...he helped spread the new mathematical ideas..." That's right, but which ideas? The new idea is the concept of complex number, the examples in the second section are related to it. But as far as I can see, the proof of the orthocenter in the fourth section and the pentagramma mirificium were solved with complex numbers, too. So I would like to add them both to the second section.

Now, what is the matter with the other examples: the problem of Apollonius, the area of the ellipse, the "Rotation in space", the area of the pentagon, the Poncelet pentagon, and Napiers's spherical pentagon? Which "new idea(s)" did Gauss apply to solve these problems?

Last problem: In the second section I see ref. 235 (= Schlesinger p.198) only as reference for the last part of the sentence. I don't find "Riemann sphere" or stereographic projection on this page or pages around. If we don't find a suitable reference, I think we can omit such an example for the moment. Greetings --Dioskorides (talk) 22:38, 18 June 2024 (UTC)[reply]

@Dioskorides: 1.

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)[reply]

Hello,

now the article on Carl Friedrich Gauss belongs to the "Good Articles". Many thanks for your contributions and your help, without your edits this result would not have been achieved. And it was your work, that prompted me to enlarge the articlesw on Seeber and the Sohncke family. Greetings --Dioskorides (talk) 20:44, 8 July 2024 (UTC)[reply]