Ptolemy's Epicycles
This is the machine Ptolemy used to predict the planets, drawn from the numbers in his Almagest. A large circle, the deferent, carries a small one, the epicycle, and the planet rides the small circle. The loops it traces are his explanation of why planets sometimes seem to move backwards. The blue line is where the real planet went, so you can see how close the old model comes.
| Mars's epicycle | 39;30 on a deferent of 60 |
|---|---|
| Mars loops | 7 in 15 years |
| Largest gap, 2001 to 2100 | Mars 8.5°, Jupiter under 1° |
| Apogees carried forward | About 26° since AD 137 |
What you are looking at
The Earth sits near the middle, a little off the centre of the big circle, and the zodiac runs around the edge with 0 degrees Aries on the left, as on a birth chart. While the picture builds you see the working parts: the deferent, the epicycle riding on it, the planet on the epicycle, and a dashed gold line from a point called the equant to the epicycle's centre. The thin line from the Earth out to the wheel is where the model says the planet appears in the sky.
As the years go by the planet traces a loop every time the epicycle carries it back toward the Earth. Those loops are retrogrades. When the picture is finished the circles fade, a faint epicycle is left at the centre of every loop, and the whole trace stands on its own. The blue line underneath is the real planet's path, computed from its actual positions and drawn to the same scale.
Pick another planet to see its own circles. Mars has a big epicycle and seven loops in 15 years. Saturn has a small one and draws a ring of 29 little loops in 30 years. Venus draws five loops in eight years, the same five-fold pattern as the Rose of Venus.
The math
Ptolemy measured every circle in parts of a deferent of radius 60. His epicycles are 39;30 for Mars, 11;30 for Jupiter, 6;30 for Saturn, 43;10 for Venus and 22;30 for Mercury, written in his base-60 notation, so 39;30 means 39 and a half. In modern terms the epicycle of an outer planet is the Earth's own orbit seen from that planet: 39;30 out of 60 is 0.658, and the Earth's orbit is 0.656 of the size of Mars's. For Venus it is the other way round: 43;10 out of 60 is 0.719, and Venus's orbit is 0.723 of the Earth's.
The Earth is not at the centre of the deferent. The centre sits a small distance away (6 parts for Mars, 2;45 for Jupiter, 3;25 for Saturn, 1;15 for Venus), and the equant sits the same distance again beyond it. The epicycle's centre moves around the deferent at a speed that looks steady from the equant, not from the centre and not from the Earth. That trick, which Copernicus later objected to, does much of the work Kepler's ellipses would do nearly 1,500 years later.
For Mars, Jupiter and Saturn, the line from the epicycle's centre to the planet always stays parallel to the line from the Earth to the mean Sun, which is why each loop happens when the planet is opposite the Sun. For Venus the epicycle's centre follows the mean Sun itself. Mercury has an extra crank: its deferent's centre circles a point 6 parts from the Earth, so Mercury comes closest, 55;34 from the Earth, about 120 degrees either side of its apogee, and is farthest, 69, at the apogee itself.
How it is computed
The circles, the eccentricities, the equants and the apogees are Ptolemy's. His apogees were fixed among the stars (Mars at Cancer 25;30 in his own time), so they are carried forward by the real precession of the equinoxes since AD 137, about 26 degrees. The rates and the starting positions are modern: each circle turns at the planet's modern average rate and starts where the planet's average position really was on the start date, taken from NASA JPL's mean orbital elements. Ptolemy's own tables would add nearly 1,900 years of drift in his year length and his precession, which says nothing about his geometry; this way the gap between the model and the sky is the geometry alone.
The real path comes from the astronomy-engine library, the same planetary theory the app's other engines use, sampled every one to four days in a fixed frame and compared with the model degree by degree. Over 2001 to 2100 the model strays up to about 8.5 degrees from the real Mars and about 9 from Mercury, but stays within 4 degrees for Venus, 2 for Saturn and 1 for Jupiter. The corner of the finished picture gives the largest gap for the span on screen.
Questions
What is an epicycle?
A small circle whose centre rides around a larger circle, the deferent. A planet on the small circle sometimes moves with the big circle's motion and sometimes against it, so seen from the Earth it speeds up, stops, runs backwards and carries on. Ptolemy used it to explain retrograde motion with nothing but circles.
What is the equant?
A point on the far side of the deferent's centre from the Earth, the same distance away. The epicycle moves at a speed that looks steady from the equant, so it runs faster when it is near the Earth and slower when it is far. Copernicus disliked it because it broke the rule of steady motion around a circle's centre, but it is a close stand-in for Kepler's second law.
How accurate was Ptolemy's model?
With his circles set going at modern rates, it stays within a degree of the real Jupiter, within 2 degrees of Saturn and within 4 of Venus over this century. Mars and Mercury are harder: up to about 8.5 and 9 degrees. Mars was the planet circles could never quite capture, and working on it is what led Kepler to the ellipse.
Is this the path the planet really draws?
The gold or coloured line is Ptolemy's model. The blue line is the real planet, from its actual positions. They lie almost on top of each other for Jupiter and Saturn and separate visibly for Mars and Mercury, which is the point of the comparison.