Ptolemy's equant makes Mars predictable
To fit Mars, Ptolemy let it sweep its circle at a steady rate around a point that was not the centre — a cheat that broke Greek doctrine and worked for 1,400 years.
Around 150 CE in Alexandria, Claudius Ptolemy broke the central rule of his own tradition in order to make Mars behave. Greek astronomy demanded uniform motion on perfect circles. Mars would not do it — it speeds up and slows down by a margin too large to hide. So Ptolemy kept the circle and moved the reference point: he placed an equant off-centre, as far from the circle's centre on one side as the Earth sits on the other, and declared that Mars sweeps out equal angles in equal times as seen from there. Steady from the equant. Not steady from anywhere real.
The Greeks had been watching the planet for centuries before that. They called it Pyroeis, the fiery one, and the star of Ares; Rome swapped in its own war god and gave us Mars. They also produced one hard empirical result. In On the Heavens, Aristotle records watching a half-lit Moon pass across Mars, which vanished behind the shadowed limb and came out on the bright side. Kepler put the date at 4 April 357 BCE. Recomputation leaves one candidate that fits the description: the evening of 4 May 357 BCE. Kepler was a month out, and on his date no occultation was visible from Athens at all. From that single sighting Aristotle drew the only thing anyone then knew for certain about the planet's distance: Mars is further away than the Moon.
Mars was the hard case for a reason. It is the nearest of the outer planets, so its retrograde loop is the largest, and of the planets anyone could follow across the sky its orbit is the most lopsided — eccentricity 0.093, against Jupiter's 0.049 and Venus's 0.007. Only Mercury, which never leaves the twilight, is worse. The result is loops that differ every time round: different width, different duration, different place in the zodiac. One circle at one speed cannot produce that.
The equant worked because it is a decent first-order stand-in for what we now call Kepler's second law. Offsetting the centre gets the changing distance roughly right; offsetting the equant gets the changing speed roughly right. Together they held Ptolemy's Mars longitudes to within a degree or two — enough for thirteen books of the Almagest, a catalogue of over a thousand stars, and tables astronomers were still using in the sixteenth century. The error only became intolerable when Tycho Brahe started measuring to arc minutes.
What the model could not do was say how far away anything was. It is scale-invariant: only ratios appear, so nothing in it fixes Mars in miles or Earth-radii. And one of those ratios is quietly extraordinary. Ptolemy's Mars epicycle is about two-thirds the size of its deferent. Invert it and you get roughly 1.5 — Mars's real distance from the Sun in units of Earth's. The number was sitting inside the geocentric model the whole time. It took fourteen centuries for anyone to read it that way.