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8

"Libration" loosely refers to longitudinal oscillations due to orbital resonances. The tidal lock and the trojans, the only situations that Wikipedia mentions, are just special cases. The existence of resonances does not require one body to revolve around the other, and Jupiter exerts a major pull on the asteroid belt just as a planet would on its moons. ...


6

While you already accepted an answer, it seems not superfluous to add another one, in particular since you are implicitly asking for a better translation/understanding of the passage you quoted. I will give you a translation here, and some comments of mine. Beforehand, though, I think it is fair to say hat you seem to have been misled when you wrote ...


5

zy_: There are two parts to Biermann's hypothesis on the meaning of "Vicimus GEGAN": i) GEGAN should be read in inverse order as standing for (vicimus) N[exum medii] A[rithmetico-] G[eometricum] E[xpectationibus] G[eneralibus]; ii) With "Vicimus GEGAN" the great Gauß was alluding to his discoveries on the connection between the arithmetic-geometric ...


5

Gauss was probably afraid of attacks of some mathematicians and philosophers: the idea looked too radical for that time. The history shows that Gauss was right. When non-Euclidean geometry was published (by Lobachevski), it was had a poor reception in the beginning. Even among some mathematicians (for example Ostrogradskii). Later when Riemann published his ...


3

Fortunately (and surprisingly) i found the answer very quickly - at the website of Springer they allow the readers to see the first 2 pages of each chapter, and to see the complete list of references for this book. Since the desired reference is reference 279, the relevant pages of Gauss's Nachlass are p.56-57 of volume X,1, which are entitled "Hauptmomente ...


2

According to Klein (Lectures on history of mathematics in 19 century), he absolute invariant $J$ was introduced by Gauss in his manuscript "On summatory function", which is reproduced on p. 386 of volume III of Gauss collected works.


2

I asked a mathematician who is an expert to abstract algebra and he showed me that Gauss's congruence was actually correct. To prove Gauss congruence let's introduce the following notation: $$x = a+bi,y = c+ di, u = \alpha + \beta i, v = \gamma + \delta i, X = A + Bi, Y = C + Di$$. First of all, one has to understand that Gauss's notation is different from ...


2

"Curtius's problem" is Schumacher's personal name repeated only by Dickson. By scaling, one can reduce the problem to that about triangles with rational sides and the circumradius, which is equivalent to the more commonly considered problem about triangles whose sides and the area are rationals, since $r=abc/4A$. These, or the version where the sides and the ...


1

The "midpoint equation" is another name for the "equation of the centre", related to kepler's equation. What Gauss wanted to determine in this investigation is the exact mathematical expression for the coefficients of the trigonometric series (Fourier series) for the difference between true anomaly $v$ and mean anomaly $M$: $$v - M = \sum_{n=1}^\infty \frac{...


1

Would have been easier if spelled right... sigh. From an online dictionary dull; obtuse; without cultural refinement. In this case, a cultural slur on the residents of Boeotia, and presumably anyone else that Gauss disapproved of.


1

As for the figures on p.621, i think (but i'm not sure) that the problem which Gauss tries to solve in it is (in modern terms): "to find the self-inductance of a single circular loop". This is a very difficult problem in mathematical physics that is often omitted in textbooks on electromagnetism - who usually adhere to the simple problem of calculating the ...


1

I'm quite not an electricity engineer, but it seems that the figure on p. 603 (title 2) - which actually represents an interesting electric circuit with a current source "a" (to use Gauss's designation) connected with a configuration of resistors that includes a "Y" shaped collection of 3 resistors (c,e,f in Gauss's notation) positioned in the centre of a ...


1

As for the third question (on "sympathetic vibrations"), p. 127-136 of Schaefer's treatise are the best source on this. According to these pages, much of Gauss's experimental investigations on magnetism was concerned with the determination of the magnetic moment of needles by measuring their period of oscillation; his main objective was to use such precise ...


1

@ José Hdz. Stgo.: Thank you for this great answer. I think I should add a few comments on the related materials I collected from different books. (Since I am neither a native speaker of German language nor well-versed in it, sometimes I need google translation or deepL translation for a better understanding on all these mysteries) I believe Biermann used ...


1

(Edited). My German is not too good, but the dot clearly stands for multiplication, and $i=\sqrt{−1}$ so $$\frac{\mathrm{tg}i.24}{i}$$ is what we call the hyperbolic tangent today, $\tanh(24)$. Symbol probably $\partial$ means the differential. If you can read German, read the comments below the note: Stackel explains what is going on there. In particular ...


1

No, this surface is actually less interesting than the Moebius strip, it is just a torus (if you do not know what this means, think of the surface of a doughnut) with a (rather large) hole in it. However, it does suggest a nontrivial (at the time of writing, but quite standard now) and useful mathematical idea, namely, constructing surfaces by identifying ...


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