**What are the plausibility and long-term effects of elliptic orbits being developed in ancient times?**

Regarding the first part, I became interested in whether elliptic orbits could be accessible to the ancients given their observations and conceptual tools. Other than perhaps requiring some implausibly lucky inspirations, it does not seem to be at all impossible for the mathematics and observations of the times to have achieved this. To facilitate this, I will posit two mathematical developments, first a relatively minor one to link ellipses and Greek conception of circular motion as natural, so that they might widen their perspectives to the possibilities, and then a more direct construction of the ellipse linked to Keplerian orbits. The actual point of divergence could occur in multiple ways; ideally, it before the invention of the equant (ca. 150 CE historically), since the following scheme is extremely similar.

To be clear, I don't necessarily mean Keplerian orbits, as the the spatial shape of the orbit does not by itself determine the correct speed of outside a physical theory. Although whether Keplerian orbits can be developed early is also an interesting issue; if elliptical orbits are used, eventual correction to Keplerian is conceptually smaller. Historically, Aristarkhos of Samos was an early proponent of heliocentrism.

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The first use of epicycles is typically credited to Apollōnios of Perga, ca. 200 BCE, from whom we also get the modern definitions and names for the conic sections (though the names might predate him). Given the predilections of Greek physics of considering circular motion as natural, circles upon circles is a sensible compromise. However, in a direct sense, ellipses are some of the simplest of possible epicycles: a point on a smaller rotating circle that is itself rotating upon a larger one, at the same angular frequency but in oppose directions, traces out an ellipse.

Analytically, z = ½(a+b)exp(iωt) + ½(a-b)exp(-iωt) traces an ellipse of semi-major axis a and semi-minor axis b. If one writes R = a/(1+k) = b/(1-k), the front coefficients become R and Rk, respectively, which is a hypotrochoid formed by a circle of radius R rolling inside a circle of radius 2R with the linked tracing point a distance Rk from the inner circle's centre. Equivalently, one can think of selected points on arbitrary concentric rotating circles, take the vector from the inner rotating point to the outer one, and translate it to the centre. If the circles rotate in the same direction, the result is an epitrochoid, and if in opposite direction, a hypotrochoid.

This could conceivably be discovered very early, e.g. as a slight generalisation of a construction originally developed by Nasir al-Din Tusi (1247). In the Tusi couple, a circle rolls without slipping inside a larger one of twice the size, and a point on the inner circle traces out a line segment, a diameter of the larger. If instead a point inside the rolling circle is picked, the result is an ellipse.

To try to fix this, one can instead characterise an ellipse and the corresponding orbit in one of two equivalent ways:

(A) Given a line segment AB and points O,F on it such that AO=FB, the locus of points P' such that OP'+FP'=AB is an ellipse with major axis AB and foci O,F.

The planetary model would have the Sun at F and a uniform rotation about O, which keeps the sum of the distances to the Sun and this shadow focus constant. This is now the common ‘tack-and-string’ definition, though I'm not sure how old the construction actually is; however, that every ellipse has this property is Apollōnios' *Conics*III.52. The term ‘focus’ (‘hearth’) is anachronistic, being introduced by Kepler in

*Optics*(1604), but Apollōnios did construct those two points for ellipses, calling them ‘the points arising out of the application’, the application essentially being the taking away an appropriate amount from both sides of the major axis.

Another equivalent characterisation of ellipses is the following, which is nicer for eventual correction to Keplerian orbits:

(M) Given a circle with centre O and a radius OP, pick a point F in the interior. Find P' as the intersection of OP and the perpendicular bisector of FP. As P is varied along the circle, the locus of the resulting points P' is an ellipse with foci at O,F and major axis equal to OP.

Here the trick is to show FP'=PP', so FP'+OP'=OP is constant; the perpendicular bisector of FP is tangent to the ellipse and bisects the angle FP'P, which is related to *Conics*III.48. Therefore, this construction would not be unnatural to ancient geometers. Under (M), orbit could be thought of as generated from the uniformly rotating circle directly or as being driven by it, with the ellipse as a counter-rotating epicycle as before.

(M) θ˙/n = (1±e)/(1∓e) = 1 ± 2e + 2e² + O(e³),

(K) θ˙/n = (1±e)²/√(1-e²)³ = 1 ± 2e + (5/2)e² + O(e³),

thus having relative error 1-½e²+O(e⁴). Under this model, the maximum and minimum angular speeds are inversions of each other, θ˙₊θ˙₋ = n², while for Kepler, the spatial speeds are instead, v₊²v₋² = (an)². Before the equal-area law is realised, the Tusi couple could perform a finer speed adjustment, which was its original purpose, should observations require it.(K) θ˙/n = (1±e)²/√(1-e²)³ = 1 ± 2e + (5/2)e² + O(e³),

*Matter and Motion*(1877) and one of Feynman's lectures (1964); Feynman credited it as inspired by Ugo Fano's work on Rutherford scattering. Rather than a uniform rotation in space, as is used here for simplicity and plausibility of what the ancients might have tried first, what's actually going on in Kepler orbits is that hodograph is a circle, i.e. in velocity space, the orbit is a circle with ratio of centre offset to radius equal to e = c/a.

Interestingly, the hodographs of constant specific energy are a pencil of circles discovered by Apollōnios. The elliptic pencil is in in red; the common points of intersection are |v| = an = √(-2ε). The circles in the blue pencil are formed as the locus of points with a constant ratio of distances from the previous two points, as is according to the Apollōnios' characterisation of a circle; they are related to hyperbolic orbits instead and are everywhere orthogonal to the red pencil.

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