Antonio Lao said:
Turbo-1,
Thanks. But did he called it magnetic? I think I need to get Epple M 1998 Math. Intelligencer 20 45 as mentioned by Berger.
I have this issue of the Mathematical Intelligencer (Winter 1998), and Moriz Epple's article is entitled
Orbits of Asteroids, a Braid, and the First Link Invariant. Epple quotes an integral formula for the intertwinings of two curves in three dimensional space, leading to a number that counts the number of intertwinings and is insensitive to interchange of the curves. This statement, without proof, was from a fragment found among Gauss' paper after his death, and his editor published it with his electromagnetic researches, which he did with Weber.
Epple is concerned to uncover Gauss' path to this result, and he first reviews its understanding as electromagnetic in the nineteenth century by such figures as Moebius, Maxwell and Tait. Epple concedes the relevance of the invariant to electromagnetism but suggests a second way that Gauss might have been led to it; the orbits of asteroids. Gauss had been the one who calculated the orbit of the first discovered asteroid, Ceres, in 1801. In 1804 he published a paper on the determination of the part of the sky in which a newly discovered planet or asteroid might be expected to appear. Epple says; "Published in an astronomical journal, the treatise addressed, at the same time, issues of practical astronomy, such as recent observational data, and the making of star maps, and mathematical topics in geometry, differential equations, and
geometria situs{now called topology}". Gauss says that if the orbit of the body is inside or outside the Earth's orbit, you get a definite region of the celestial sphere where the body is restricted to appear. But if the orbits are linked, the body might appear anywhere in the celestial sphere. Epple works through the math in this paper and shows that you can interpolate the linking integral into an argument where Gauss, being Gauss, declines to show his work. So that he may have come upon the link invariant in discussing the paths of heavenly bodies through the sky.
