Abstract
Johannes Kepler discovered the very elegant elliptical path of planets with the Sun in one focus of that ellipse in 1605. Kepler inspired generations of researchers to study properties hidden in those elliptical paths. The visible elliptical paths belong to the Aristotelian World. On the other side there are invisible mathematical objects in the Plato´s Realm that might describe the mechanism behind those elliptical paths. One such curve belonging to the Plato´s Realm discovered Isaac Newton in 1687 - the locus of radii of curvature of that ellipse (the evolute of the ellipse). Are there more curves in the Plato´s Realm that could reveal to us additional information about Kepler´s ellipse? W.R. Hamilton in 1847 discovered the hodograph of the Kepler´s ellipse using the pedal curve with pedal points in both foci (the auxiliary circle of that ellipse). This hodograph depicts the moment of the tangent momentum of orbiting planets. Inspired by the hodograph model we propose newly to use two contrapedal curves of the Kepler´s ellipse with contrapedal points in both the Kepler´s occupied and Ptolemy´s empty foci. Observers travelling along those contrapedal curves might bring new valuable experimental data about the orbital angular velocity of planets and a new version of the Kepler´s area law. Based on these contrapedal curves we have defined the moment of the normal momentum. The first derivation of the moment of the normal momentum reveals the torque of the ellipse. This torque of ellipse should contribute to the precession of the Kepler´s ellipse. In the Library of forgotten works of Old Masters we have re-discovered the Horrebow´s circle (1717) and the Colwell´s anomaly H (1993) that might serve as an intermediate step in the solving of the Kepler´s Equation (KE). Have we found the Arriadne´s Thread leading out of the Labyrinth or are we still lost in the Labyrinth?
Highlights
The famous quote of Heraclitus “Nature loves to hide” was described in details by Pierre Hadot in 2008
We propose to use the very-well known Antikythera Mechanism as an analogy for the visible Keplers ellipse - a part of our Aristotelian World - connected deeply with invisible curves from the Platos Realm - Newtons evolute (1687), Horrebows circle (1717) and Colwells anomaly (1993), Hamiltons pedal curve (1847), two contrapedal curves (2018), there are two more curves describing the moment of the normal momentum and the torque of the Keplers ellipse (2018)
Are there some more hidden curves in the Platos Realm connected to the Keplers ellipse?
Summary
The famous quote of Heraclitus “Nature loves to hide” was described in details by Pierre Hadot in 2008. W.R. Hamilton in 1847 discovered a very elegant model of the hodograph using the pedal curve with pedal points located in both foci (the auxiliary circle). Hamilton in 1847 discovered a very elegant model of the hodograph using the pedal curve with pedal points located in both foci (the auxiliary circle) This classical model of the Keplers ellipse could not properly explain the precession of the planets and Albert Einstein in 1915 replaced this classical model with his concept of the elastic spacetime. In this contribution we have been working with these mathematical objects from the Platos Realm: 1) Ellipse properties discovered by Appolonius of Perga - the Great Geometer and many his scholars. (We are aware of the famous quote of Richard Feynman from the year 1962: “Theres certain irrationality to any work in gravitation, so it is hard to explain why you do any of it.”)
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