Ptolemy’s cosmos, to scale

Diagrams of Ptolemy’s epicycles and equants rarely show their correct relative scale. So the “Whole cosmos” and “Inner planets” pages of this site show accurate animations of his system, with parameters derived from the real-world motion of the planets:

Here are some notes about the diagrams:

They’re built by me, Brandon Rhodes.
You can find the Python scripts that generate the parameters and SVG diagrams on GitHub.
The Earth is at each diagram’s center.
The dotted circles are all exactly concentric, and are all centered on the Earth. They each show the circle that their planet would travel if its orbit had zero eccentricity and was perfectly centered on the Earth.
But planet orbits are eccentric — the circles are not all centered on the Earth. So instead of moving along its dotted circle, each planet moves along the corresponding solid circle, which is called the planet’s deferent. The deferent is closest to Earth where it touches the dotted circle (the point is marked with a short tick mark) and furthest from Earth on the opposite side.
Motion along the deferent is not uniform if viewed from the Earth: each planet moves faster when close to the Earth and slower when far away. The motion is only uniform when viewed from the planet’s equant, an invisible point over on the opposite side of the deferent’s center from the Earth. (Neither the deferents’ centers nor the equants are shown in the diagram, as they would create a small crowded forest of dots down around the Earth that would be visually difficult to associate with the correct planets.)
The Sun’s motion is simplest. It travels directly along its deferent.
The motion of every other planet is complicated by an epicycle: a circle whose center travels along the deferent. The planet itself revolves around the circumference of the epicycle at a simple uniform speed.
Ptolemy used ancient and contemporary observations to determine the eccentricity of each equant and the size of each planet’s epicycle compared to its deferent. The parameters in these diagrams are instead generated by taking a modern NASA ephemeris of planetary positions and running a SciPy optimizer against them to find a best fit for each parameter.
One crucial parameter for each orbit is still undetermined even after all the other numbers have been determined: how big is the orbit? The Greeks could measure no planetary parallax against the fixed stars and so lacked any sense of their scale. Even if, for example, you doubled the size of Saturn’s deferent, you could still make Saturn trace the right path in the sky by doubling the size of its epicycle to match. So Ptolemy suggested several restrictions that together provide a possible scale for the cosmos:
1. The distance to the Moon’s orbit was known: the Greeks estimated it to be 59.7 Earth radii.
2. Ptolemy chose an order in which to place the planets above the Moon. First he listed Mercury and Venus, that never stray far from the Sun; then the Sun itself; and then the outer planets from fastest to slowest: Mars, Jupiter, and Saturn.
3. Next, Ptolemy reasoned that the orbits must be far enough apart that the epicycles never collide — for example, Saturn’s orbit must be large enough to keep its epicycle from ever overlapping with that of Jupiter.
4. Suspecting that the cosmos would not include large yawning gulfs that served no purpose, he finished by suggesting that the planet orbits would otherwise be stacked as tightly as possible.
The diagrams here impose exactly these conditions, resulting in a maximum height for the outermost planet Saturn of 16,551 Earth radii — above which must stand the outer edge of the cosmos, the sphere of the fixed stars. Ptolemy’s own estimate was that the cosmos was 20,000 Earth radii across, so the parameters computed here have permitted a slightly tighter packing than those of Ptolemy.
As you watch the planets move, you will notice three coincidences which proved momentous for the future of Western science.
1. Coincidence #1: The center of Mercury’s epicycle and Venus’s epicycle are always roughly aligned with the Sun. If the Earth is the center of planetary motion, why aren’t Mercury and Venus free to move independently of the Sun, like the outer planets?
2. Coincidence #2: The epicycles of Mars, Jupiter, and Saturn all have exactly the same period, one solar year. Why aren’t their epicycles free to have different periods?
3. Coincidence #3: The epicycles of Mars, Jupiter, and Saturn always point in nearly the same direction as a line connecting the Earth and Sun. If the Earth is the only center of planetary motion, why is every outer planet copying the Sun’s motion?
More than a thousand years later, these coincidences drove a Polish scientist named Copernicus to famously suggest an alternative to Ptolemy’s system, as described in my 2013 Keynote talk at DjangoCon Europe in Warsaw.
The model illustrated in these diagrams is very slightly simpler than Ptolemy’s real model:
1. The diagram ignores inclination and pretends that all the planets orbit in exactly the same plane. This seemed reasonable for a two-dimensional diagram which would have trouble depicting inclination angles anyway.
2. The diagram makes the Sun’s deferent symmetric with those of the other planets by giving it an equant. Ptolemy instead deferred to the model of his predecessor Hipparchus by giving the Sun no equant and instead putting the Earth twice as far from the center of the Sun’s deferent.
3. The orbit of Mercury shown here uses only a single epicycle, whereas Ptolemy gave it two epicycles to better model its motion over the long centuries between his own era and the observations of the ancient Babylonian astronomers.
I am unlikely to fiddle with this diagram further, but if anyone else is interested in tackling the second and third inaccuracies listed above, I welcome discussion at the GitHub repository where the code for this diagram lives.
In early drafts of this page I kept referring to the model as “Ptolemy’s Solar System” but it’s not a solar system: the Sun isn’t at its center.
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