Ellipses
In the previous post, we’ve seen how naturally deferents and epicycles arise from a simple heliocentric model: just switch to a geocentric frame.
But Kepler tells us that the orbit of each planet (around the Sun) is not a circle, but an ellipse. And the Sun is at one focus, not at the center. And the speed of the planet is not constant, but governed by the area law.
Finally, the orbits of the planets lie in different planes. We table this final difficulty for awhile; that’s not a bad approximation since all the planetary orbits do lie close to the plane of the ecliptic.
Even with these complications, the steps in the previous post still work just fine. The deferents and epicycles become ellipses, but so what. The exact shape of the orbit doesn’t matter: we can write 
From now on, we will adopt the heliocentric frame. In this section, we will see how Kepler’s 1st law left its mark on the Ptolemaic system.
First we review ellipse geometry. Look at this figure:
Its semi-major axis is denoted a, its semi-minor axis b. The distance between the foci is 2c. As we can see from the figure, b2+c2=a2. The eccentricity is the ratio e=c/a. In other words, if we scale the ellipse so a=1, then 2e is the distance between the foci.
Circumscribe a circle of radius a around the ellipse, like so:
Let us use this circle as an approximation to the elliptical orbit. Keep the Sun at the focus. So we ignore one part of Kepler’s 1st law (the shape) but adopt the other (the eccentric position of the Sun).
We measure the closeness of this approximation with another quantity. Define the ellipticity of the ellipse to be the ratio η=(a–b)/a. In other words, if we scale the ellipse so a=1, then η is the difference between the semi-major and semi-minor axes. Clearly η measures how “squashed” the ellipse is. For a circle, η=0. In the figure, e≈0.4 and η≈0.1.
It’s easy to express η in terms of e:
Using the approximation 
η ≈ 1-(1-½ e2) = ½ e2
So the eccentricity is a first-order effect, but the ellipticity is second-order. For example, Mercury has the greatest eccentricity of any of the known planets, about 1/5, so its ellipticity is about 1/50. Kepler discovered his 1st law in his “battles with Mars” in the Astronomia nova; Mars has an eccentricity of about 1/10 and an ellipticity of about 1/200. Earth’s eccentricity is about 1/60 and its ellipticity is about 1/7200. Pluto is no longer a planet, and wasn’t known to Ptolemy, but its eccentricity is about 1/4 and its ellipticity is about 1/32.
We have our first approximation to Kepler’s system: the planets travel in so-called eccentric orbits, that is, circles with the Sun displaced from the center. In the next post, we’ll look at the speed law.
Diagonal Argument 