Showing Kepler's first law using the Triangulation method

Question

Historically, Kepler came up with his first law using the triangulation method on Mars's orbit (here is a video of Terence Tao referencing the same: https://www.youtube.com/watch?v=7ne0GArfeMs&t=2367s).

Mars has an orbital period of 687 days. Suppose that the geocentric longitude (angle between fixed reference and Mars as viewed from Earth) and heliocentric longitude (angle between fixed reference and Earth as viewed from Sun) are measured at a particular time. Then, 687 days later, Mars would return to the same position but Earth wouldn't (it would be 43 days behind its original position). Now, we can plot two different lines from Earth to Mars. Their intersection is the position of Mars.

This process can be repeated several times, to obtain different positions of Mars throughout a year. To try to show this, I used the following data I found online:

Date	Heliocentric longitude of Earth ($\theta$)	Geocentric longitude of Mars ($\phi$)
Feb 17th 1585	159	135
Jan 5th 1587	115	182
Sept 19th 1591	005	284
Aug 6th 1593	323	346
Dec 7th 1593	086	003
Oct 25th 1595	042	050
Mar 28th 1587	197	168
Feb 12th 1589	154	219
Mar 10th 1585	180	132
Jan 26th 1587	136	185

My assumptions are that Earth has a perfectly circular orbit around the Sun and that the orbit of Earth is on the same plane as that of Mars. I use the first row. Now, for Feb 17th 1585 and Jan 5th 1587, the two lines are:

$$y - sin(159) = tan(135)(x - cos(159))$$ $$y - sin(115) = tan(182)(x - cos(115))$$

The gradient of the line from Earth to Mars is $tan(\phi)$. Additionally, a point on the line is the position of Earth, which is $(cos(\theta),sin(\theta))$ (radius = 1 AU). Solving the above equations gives the single point (-1.45,0.87).

Doing this for all five pairs of lines yields the following graph on Desmos:

Clearly, this looks nothing like an ellipse (unless you argue that I need significantly more points and only then will it start to resemble an ellipse). So, I have the following questions after doing all of these calculations:

(1) Is my method of calculation correct

(2) Is the data/method of calculation too imprecise to give an ellipse?

(3) If the answer to (2) is yes, then how exactly did Kepler deduce the elliptical nature of Mars's orbit? What numerical calculations did he do to come to this conclusion? Can these calculations be replicated?

Answer 1

I may be able to say a couple of things.

Incidentally, I noticed that there is a 1992 translation of Kepler's book Astronomia Nova. Translator: William H. Donahue

I noticed that the historian of mathematics Ann Elizabeth Leighton Davis has published much about Kepler's methods, but it would appear that all of that material is behind paywall.

I came across some blog posts that draw on Davis' works.

Kepler's Ellipse

Kepler's equal areas law

Source:
blog site: Meditations on Mathematics, articles tagged 'physics'
Author: James Stevenson

About stages in Kepler's progress:

Kepler was from a very early point relying on (precursors of) Kepler's second law.

With the very limited data points that Kepler had: what he needed was a two-fold hypothesis:

• a hypothesis for the shape of the orbit
• a hypothesis for the speed of the planet at any point of its orbit

(With the 'circles and epi-circles' approach the velocity along each circle is assumed to be uniform. Without the 'circles and epi-circles' simplification the problem is impossible.)

It was clear that one way or another a non-uniform angular velocity had to be implemented. But once a supposition of uniform angular velocity is abandoned the question is: what mathematical relation to replace it with?

One possibility is a supposition that planets move faster the closer they are to the Sun, and Kepler tried that one. It is clear that at an early stage Kepler came to prefer an area law.

There was the choice of shape, and the location of that shape, and the orientation of that shape. For each permutation that he tried Kepler applied the area hypothesis to calculate a velocity profile.

For instance, among the shapes that Kepler tried was an egg shape. What that illustrates, it would appear, is that Kepler had very little clue as to the shape of the orbit.

The problem is of course: once the supposition of circular motion is abandoned the possibilities are infinite.

It would appear that Kepler sought to limit the range of possibilities by assuming that the actual shape, while not a perfect circle, is a mathematically elegant entity.

It may well be that Kepler's reasons for embracing the area law were actually wrong reasons. Be it as it may, it proved a most fortunate choice. As we know, a Kepler orbit satisfies the area law exactly.

With the combination of a shape and its velocity profile according to the area law Kepler could see whether the orbit matched the sparse observations.

As you point out, the simplified story that is everywhere doesn't cut it. Kepler was up against a far, far more difficult problem than the simplified story suggests. Kepler's achievement is astounding.

A list of articles by A. E. L. Davis is available in the references section of the Stanford Encyclopedia of Philosophy entry for Kepler

I haven't read the following source, I have only glanced at the 'Chapters' section:
PhD Thesis by A. E. L Davis:
A mathematical elucidation of the bases of Kepler's laws