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21

A good account is Weinstein, Max Born, Albert Einstein and Hermann Minkowski's Space-Time Formalism of Special Relativity. They did no have much of a relationship, what it was is well-summarized by Sommerfeld: "Strangely enough no personal contacts resulted between his teacher of mathematics, Hermann Minkowski, and Einstein. When, later on, Minkowski ...


12

Poincare refers to the Lie's solution of the so-called problem of space, a.k.a. the Helmholtz , or Riemann-Helmholtz, or Helmholtz-Lie problem of space, which amounts to characterizing all manifolds (originally, only 3-dimensional) with free mobility of figures (roughly, homogeneity and isotropy). In modern terms, free mobility amounts to constant Riemannian ...


11

Technically, the first was Lobachevski (published in 1829-30). Bolyai was independent (published in 1832). Gauss discovered it independently of both (not published). A more complicated research is needed to find out when it was actually discovered by each person, and we can never be 100% sure of the result. Perhaps Gauss was the first. But the usual way to ...


11

One must firmly protest these orgies of a formalism with which even technicians are getting harassed today. (Literally: of the formalism, with which one is beginning to harass even technicians today.)To your questions: 1) Techniker is for technicians, engineers, graduates of the Technische Hochschule where Weyl gave these lectures, as opposed to scientists ...


9

Riemann discussed a "unified field theory", including light, electromagnetism and gravity, in the unpublished paper Neue Mathematische Principien der Naturphilosophie (New Mathematical Principles of Natural Philosophy, 1853, the title obviously alludes to Newton's), and in Gravitation und Licht (Gravity and Light), the last section of his Fragmente on ...


7

You are guessing correctly. Riemann did not use $g$ for the metric tensor, he writes things like $ds^2$ or $\sum dx^2$ instead, see his 1854 lecture "On the Hypotheses which lie at the Bases of Geometry" (1854). Originally, "g" was for gravity. Einstein and Marcel Grossman, his mathematician friend who introduced him to tensor calculus and collaborated ...


7

See : Detlef Laugwitz, Bernhard Riemann 1826-1866 : Turning Points in the Conception of Mathematics (1996 - German ed.1996), page 182; we have : Riemann's habilitation paper "Uber die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe" (W.227-265) ("On the representability of a function by means of a trigonometric series") was completed ...


6

The Clifford torus was introduced by Clifford in 1873, not as embedded into $\mathbb{R}^4$ or $\mathbb{C}^2$, but first projectively and then intrinsically, by identifying the opposite sides of a flat parallelogram, a commonplace method in modern topology textbooks. The embedding into $\mathbb{R}^4$ first appeared in Killing's Die Nichteuklidischen ...


4

Usually this is attributed to, in alphabetical order, Bolyai, Gauss, and Lobachevski all working at about the same time in the first third of the 19th century. This certainly predates Riemann. A colorful Harvard mathematician and guitarist Tom Lehrer seems to suggest in one of his songs that Lobachevski was guilty of plagiarism. However, In the liner ...


4

History does not often develop in the order of textbook expositions. Today the exponential map is introduced early in both Riemannian geometry and Lie group theory, but many results it is used to derive were originally derived without it. There is no "exponential map" in Gauss's General Investigations of Curved Surfaces (1825,27) or Riemann's On the ...


3

One has to object thoroughly to the orgies of formalism, with which recently even technicians have been bothered. Man muß ... protestieren -> One has to object to (in the context of legislative elements like mandatory paperwork) nachdrücklich -> thoroughly (or intensely, not just a little) man heute ... zu belästigen beginnt -> with which recently ... have ...


3

Elliptic geometry is not equivalent to geometry on the sphere because there is non-unique line through antipodal points on the sphere, contrary to one of the axioms. One needs to identify the antipodal points, which gives the real projective plane. With induced metric it will be a model. According to Kline, Riemann introduced manifolds of constant ...


3

Kharkiv University (Ukraine) subscribed to all models made M. Schilling, who probably was a student of Klein, and who run a company making and selling these models. Currently they photograph them and place on internet: http://touch-geometry.karazin.ua/list This page also has Schilling's catalog of his production. Recently I learned that they have some ...


3

Regarding first developments in non- euclidean geometry, Eugenio Beltrami considered Lobachevsky-Bolyai geometry as nothing else but euclidean geometry on a space with (constant) negative curvature. In 2 memoirs published in 1868 he proposed some realizations of such geometries: one by using geodesics on a pseudosphere and others, including what is now known ...


3

Regarding : Morris Kline, Mathematics: The Loss of Certainty (1982) the answer is : yes; Ch.IV The First Debacle : The Withering of Truth is dedicated to non-Eucliidean geometry, and it is worth to be read. More details into : BA Rosenfeld, A history of non-euclidean geometry (1988) Jeremy Gray, Worlds Out of Nothing: A Course in the History of Geometry ...


2

Have you taken a look at Richard Trudeau's The non-Euclidean revolution (with an introduction by H. S. M. Coxeter. Birkhäuser Boston, Inc., Boston, MA, 1987. xiv+269 pp. ISBN: 0-8176-3311-1)? The first two paragraphs of the review of this book, which K. Strubecker contributed to MathSciNet, read thus: Starting from a very detailed, critical overview of ...


2

http://www.nytimes.com/2008/06/19/science/19gromoll.html: In the soul theorem, published in 1972, Dr. Gromoll and Dr. Cheeger were studying the properties of certain surfaces that could have flat regions or curves like the outside of a sphere but not regions shaped liked saddles. They found that the properties of such surfaces, infinite in extent and ...


1

There is a monolithic Einstein myth consolidated by scores of popular books which is a great inconvenience for anybody trying to make some sense of the history. Cornifold's answer and refs already point at the main isuue: at first Einstein did not understand and avoided mentioning Minkowski's approach which nevertheless turned to be crucial for his later ...


1

Heney L. Beose's translation (last sentence of ch. 1, § 6, p. 54) says: An emphatic protest must be entered against these orgies of formalism which are threatening the peace of even the technical scientist.


1

"Man muß gegen die Orgien des Formalismus, mit dem man heute sogar die Techniker zu belästigen beginnt, nachdrücklich protestieren." Both of the translations you got are 'decent'. Technician is indeed the literal translation of the German "Techniker", and engineer (Ingeneur) following. Technical scientists are beyond my knowledge, but I would expect ...


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