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Let's go to Mars

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.

A coloured engraving. At the centre, a terrestrial globe lettered Europa, Africa, India and Mar di; around it seven concentric orbit circles drawn in perspective, labelled in Latin Luna, Mercurius, Venus, Sol, Mars, Iupiter and Saturnus and each marked with a planet symbol, with a wide zodiac band crossing them at an angle and a graduated great circle enclosing the whole. Winged figures and putti hold two title banners across the top corners; below, scholars sit with globes and an open book, two men sight along cross-staffs beside a church tower, and a ruined building stands at the right.
Plate 2 of Andreas Cellarius's Harmonia Macrocosmica, published in Amsterdam in 1660: the Ptolemaic cosmos, with the Earth fixed at the centre and the Moon, Mercury, Venus, the Sun, Mars, Jupiter and Saturn each carried on a circle around it. Mars runs outside the Sun's circle and inside Jupiter's, and the plate draws all seven circles concentric on the Earth, without the off-centre equant Ptolemy used to fit Mars. · Image: Andreas Cellarius, Harmonia Macrocosmica, Amsterdam, 1660, Plate 2

Around 150 CE in Alexandria, Claudius Ptolemy introduced the equant. Greek astronomy required uniform motion on circles; Mars speeds up and slows down too much for that. Ptolemy kept the circle: the equant sits off-centre, as far from the circle's centre on one side as the Earth sits on the other, and Mars's epicycle centre sweeps equal angles in equal times as seen from it.

Greek observers called it Pyroeis, the fiery one, and the star of Ares; Rome substituted its own war god, Mars. In On the Heavens, Aristotle records a half-lit Moon passing across Mars: the planet vanished behind the shadowed limb and reappeared on the bright side. Kepler dated it to 4 April 357 BCE. Recomputation leaves one candidate, the evening of 4 May 357 BCE; from Athens no occultation was visible on Kepler's date. Aristotle concluded that Mars lies further away than the Moon.

Mars is the nearest of the outer planets, so its retrograde loop is the largest, and its orbit is the most eccentric of the planets anyone could follow: 0.093, against Jupiter's 0.049 and Venus's 0.007. Only Mercury, which never leaves the twilight, is worse. The loops differ every time round in width, duration and place in the zodiac, which one circle at one speed cannot produce.

The equant is a first-order stand-in for Kepler's second law. Offsetting the centre approximates the changing distance, offsetting the equant the changing speed. Together they usually held Ptolemy's Mars longitudes to within a degree, several degrees near close oppositions — enough for thirteen books of the Almagest, a catalogue of over a thousand stars, and tables still in use in the sixteenth century. The error became intolerable when Tycho Brahe began measuring to arc minutes.

The model is scale-invariant: nothing fixes Mars in miles or Earth-radii. Ptolemy's Mars epicycle is about two-thirds the size of its deferent; inverting that ratio gives roughly 1.5, Mars's real distance from the Sun in units of Earth's.