Earth, planets and orbits
c. AD 150
Circles on circles
Claudius Ptolemy
Why would a planet stop, go backwards, and then carry on?
Planets sometimes seem to stop and drift backwards across the sky. Ptolemy explained this with small circles riding on bigger circles, all centred near the Earth. His Almagest predicted planet positions well enough to be used for 1,400 years.
First, make a guess
How many spinning circles does it take to trace Mars’s looping path exactly, as seen from Earth (with circular orbits)?
- Circles used
- 2 of 256
- Shape captured
- 97.9%
- Note
- Two circles are exact here: Ptolemy’s deferent and epicycle are Earth’s and Mars’s orbits
Draw any closed shape on the canvas. It is split into spinning circles (a Fourier series): each circle turns at a whole-number speed, and the biggest ones come first. Watch how few circles you need before the shape appears.
How they figured it out
Greek tradition said heavenly motion must be made of perfect circles turned at steady speeds. But Mars sometimes drifts backwards against the stars for weeks. Ptolemy’s answer: put each planet on a small circle (an epicycle) whose centre rides around a big circle (the deferent).
To match the sky better he added a stranger trick, the equant, a point off the centre about which the motion only looks steady. It worked: his predictions were good to about a degree, and his Almagest was used for 1,400 years. Copernicus disliked the equant so much that getting rid of it was one of his motives.
The equation
Add a small spinning circle to a big spinning circle and the point on the end traces loops. (Written here in modern notation Ptolemy never had.)
See it in real life
The fairground teacups
On a teacup ride your cup spins on a platform that is itself turning. Filmed from above, your path is a chain of loops: an epicycle. A spirograph draws the same curves.
Where engineers use it
Signal analysis
Circles on circles are a Fourier series: the maths behind audio compression and radio.
Mechanisms
Planetary gearboxes in cars and wind turbines are literal epicycles.
Where it came from
- The first trigonometry table c. 150 BC
Built on
Ptolemy’s model of the heavens ran on Hipparchus’s chord tables.
What it led to
- Everything is waves 1822
Analogy only
Circles riding on circles can trace almost any repeating path, just as stacked waves can build almost any shape.
Where the comparison stops: Ptolemy fitted circles by hand to match the sky, with no theory of why or when it works. Fourier’s mathematics says exactly when such sums succeed.
- The Sun in the centre 1543
Led to
Copernicus kept the circles but moved the Sun to the centre.