Two skies, one motion

What you are looking at

Nothing in the picture moves differently when you press one of the three buttons. The bodies keep the positions they had; only the point the camera holds still changes, and the trails are redrawn from where you are now standing. Hold Sun-centred and every planet runs a circle. Switch to Earth-centred and Mars begins to double back on itself — the retrograde loop, which is not a motion of Mars at all but of the earth overtaking it on the inside.

That is the whole content of the Copernican change, and it is worth being clear about what it did and did not settle. It did not catch the ancients in an error about where the planets appear; the Ptolemaic tables were good, and the Rambam's are good, as the readout below the diagram shows. What it did was replace a set of circles that had to be tuned to reproduce the loops with a frame in which the loops do not need reproducing, because they are not there.

The nine galgalim

The third view is not a redrawing of the second. It is a construction — the cosmos as the Rambam sets it out in the third chapter of Hilchot Yesodei haTorah, with the earth at rest and nine spheres around it, each carrying its body.

Once that view settles, the nine thin out to two. The moon's sphere and the sun's are the only ones he builds a machine for, so the rest step back and the five planets leave the picture entirely, and what is left is the working of chapters 12 to 15 laid out as a figure: every point he names marked in his own Hebrew, the points his steps imply but he never names marked in English only, and every angle he tells you to add or subtract drawn as an arc with its value on it. Theמסלול is there as a real angle turned about the centre of the sun's circle; the מנת המסלול is the wedge between the mean direction and the true one; theהמרחק הכפול is swept from the far point of the moon's large circle round to its mean. Zoom in and the finer marks appear, down to the two apogees of the small sphere: the one facing the earth, from which the המסלול הנכון is counted, and the one facing the point opposite the large circle's centre, from which the אמצע המסלול is — and the angle between them is exactly what 15:3 tells you to add.

Watch the transition into that view carefully. Each planet slides off its true distance and onto its shell, but it never changes thedirection it is in. That is the honest summary of what a sphere model does and does not claim: the spheres fix an order and a set of radii, but what they are fitted to is the angle, because the angle is the only thing anyone could measure.

The sun's circle is not centred on us

Turn on Construction in the Rambam's view and a thin gold circle appears just off-centre where the fourth sphere was. The sun travels that circle at a perfectly even rate; the earth sits a little to one side of its centre. That offset is the entire reason the sun runs fast in January and slow in July, and the Rambam gives you the machinery to compute it: a mean position that advances 59′ 8″ a day, an apogee that creeps a second and a half every ten years, and a table converting the angle between them into a correction.

The correction table peaks at 1° 59′. In an eccentric-circle model the largest possible correction is the arcsine of the offset, so that single number fixes the geometry: the earth stands about one part in twenty-nine of the radius away from the centre. The gold circle on screen is drawn at exactly that offset, and the sun is placed on it by the construction rather than by the table — the two agree to within about half an arc-minute, which is the coarseness of the table itself and not a fault in the drawing.

The moon needs a second circle

The moon will not submit to one circle. The Rambam gives it two: a large one around the earth, carrying a small one that the moon itself rides.

This is the epicycle, named plainly and without apology. Its table peaks at 5° 8′, which by the same arcsine argument makes the small sphere about a ninth of the large one — and the peak falls at a course of 100°, not 90°, which is exactly where an epicycle puts it and where a simple offset circle could not. The Rambam's own table, in other words, carries the fingerprint of the construction it came from.

One further correction sits on top: before entering the table you adjust the moon's place on its small sphere by an amount depending on thedouble elongation, twice its distance from the sun. The readout shows this quantity live. It only matters near the new moon, which is the only time anyone needed it — the whole apparatus exists to answer one question, whether the crescent will be visible tonight.

What gives the moon its height

Switch Off-centre orbit off under the diagram, so that the large circle is drawn about the earth, and the moon's distance from the earth depends on one thing only: where it stands on its small sphere. The large circle then contributes nothing; the epicycle carries the moon a ninth of that radius out and back. At a course of nothing the moon sits at the far point of the small sphere and is as high as it ever gets; at 180° it is at the near point and as low.

That is checkable against the sky, and it checks out. Taking the real moon's perigees through 1178–79 and asking what course this model gives at each, the answers cluster around 180° — 192°, 189°, 168°, 167°, 172°, 180°, 188°, 193° — and over four years the model's predicted height and the moon's true distance correlate at r = 0.954. The epicycle, its centre and its starting point are all correctly phased. The one place it overreaches is the size of the swing: the construction runs from 0.911 to 1.089, a ratio of 1.197, where the real moon manages only 1.141. Switch the off-centre orbit back on and the large circle's wandering centre carries the small sphere from 1.21 radii at new and full moon to 0.79 at the quarters — Ptolemy's figure, and Ptolemy's famous overreach, since no moon ever swung by anything like that. The Rambam's equation table quietly declines to follow it, which is the subject of the next section.

The other apogee, and what it really meets

There is a second apogee in this model, and confusing the two is the easiest mistake to make here. The Rambam says of every one of the seven spheres that "although it encompasses the world, the earth is not at its centre" (11:13), and the commentaries give the moon's orbit an apogee accordingly, saying the double elongation measures the distance from it. The arithmetic is exact: the moon's mean runs east at 13° 10′ 35″ a day and that apogee runs west at 11° 12′ 19″, separating at 24° 22′ 54″ a day, precisely twice the 12° 11′ 27″ by which the moon pulls away from the sun. It is drawn in the diagram as the far point of the large circle, with a dashed line out from the earth to it and the double elongation swept as an arc from it round to the moon's mean.

At conjunction that dashed line swings into coincidence with the radius carrying the small sphere — because the double elongation is zero there, by definition. Slow the clock and watch it happen. Exactly zero atmean conjunction, when the two mean positions meet; at the new moon you actually see, which is true conjunction, it comes within a dozen degrees and no closer, because the true positions differ from the mean ones by the equations. But note what meets what. The orbit's apogee meets thecentre of the small sphere — the moon's mean position. It does not meet the moon. The moon may be anywhere on its small sphere at that moment, and usually is.

Both centres are marked in the diagram, and neither is the earth. The sun's circle is centred on a gold cross set off to one side of the earth and fixed there, and that offset is what makes its apogee a real high point. The moon's circle is centred on a silver cross that is not fixed at all: it runs round the earth once a month, always standing where it keeps the circle's far point a double elongation behind the moon's mean. How far from the earth? He never says, and gives the point no name. But his bands of 15:3 are, to the degree, what an offset of a fifth of the radius produces — Ptolemy's own figure, 10;19 in 49;41 — so that is where the cross is drawn, and the diagram labels it in English only, as it does every point his steps imply but his text does not name.

Here is the seam in the model. An off-centre large circle brings the small sphere nearer at some times than others, and a nearer sphere subtends a wider angle — so the size of the equation ought to vary with it. The Rambam's table does not vary: one course, one angle, always. His double-elongation correction moves where the moon is on the small sphere, never how far the small sphere can displace it. The diagram follows him rather than Ptolemy on this: the small sphere is drawn at whatever size subtends the angle his table gives it, so it visibly shrinks as its centre is carried nearer. One luminary he gives geometry you can point at; the other he gives a table, and part of the geometry stays behind the curtain.

The two are on quite different clocks. The small sphere's circuit takes 27.55 days against the 29.53 of a lunation, so the moon stands about 26° further round it at each new moon than at the last, returning to where it began only after some fourteen months. The moon is therefore notat the same height at every conjunction — and should not be. That mismatch is real sky: it is why some new moons are closer than others, and why the largest full moons arrive on their own slow cycle.

The clock hidden in the model

There is a step here that is easy to miss and that gives the whole calculation away. Everything else is reckoned from the start of the night, but the moon is not looked for at the start of the night — it is looked for about twenty minutes after sunset, and sunset is only at six o'clock near the equinoxes. So before anything else is done to it, the moon's mean is shifted by up to half a degree according to where the sun is standing.

Eight bands, running 0, +15′, +30′, +15′, 0, −15′, −30′, −15′ around the year, symmetric about the equinoxes: a table of the length of the day, written as a correction to the moon. It is the seam where this piece of abstract geometry is stitched to a particular hillside in Eretz Yisrael at dusk. Without it the worked example in 15:8 comes out fifteen minutes of arc wrong — which is exactly how I found that I had left it out.

How well it holds

The readout prints the Rambam's sun and moon beside a modern ephemeris for the same instant. Jump to his epoch — the eve of Thursday, 3 Nisan 4938, which is 22 March 1178 — and then to today, eight and a half centuries later. The sun holds to about a degree and the moon to a degree and a half, and neither drifts as the centuries pass. That is a fair test, because his figures were never re-fitted in between.

Note the word eve. The epoch is an evening, not a midnight: day zero begins at nightfall on the Wednesday. Anchor it to the start of the civil day instead and the moon comes out 9.6° ahead of the sky — two thirds of an hour's worth of lunar motion for every hour you are wrong by. The sun barely notices, which is what makes the mistake so easy to keep: it is the moon, moving thirteen times faster, that tells you your clock is off.

It only holds, though, if you take his mean motions from the long-period tables rather than the daily one. The daily rate is stated as 59′ 8″, and 59′ 8″ taken literally throws the sun 28° off in the time since. His figure for ten thousand days — 136° 28′ 20″ — implies 59′ 8.33″, and that rate is still good today. The precision was always there; it is the rounding, not the model, that fails.

What is drawn faithfully, and what is not

• The sun and the moon are computed entirely from Hilchot Kiddush HaChodesh 12–15 — mean motions, apogee, both correction tables, the double-elongation adjustment, and the sighting-time correction of 14:5. Both of the Rambam's own worked examples, the sun at 13:9 and the moon at 15:8–9, come back out of the code to the arc-minute he states them in.
• The five planets are not, and in his view they are not shown at all. He gives no model for them, so rather than hang borrowed positions on his spheres the bodies are put away when that view arrives, leaving the two luminaries he does account for. The spheres themselves stay, and stay named — those are his. Switch to either of the other two frames to get the planets back.
• Distances are compressed by default, as the square root of the semi-major axis, so that Mercury and Saturn can share a frame. Switch Distances to True under the diagram for the real proportions — Saturn stays put and everything inside Mars collapses into the middle, which is why almost no diagram of the solar system is drawn that way. Angles are exact under either setting.
• Latitude is ignored. Everything is flattened into the plane of the ecliptic; the inclined circles and the regressing nodes of chapters 16–17 are a separate diagram.
• The tables are read his way. He says exactly what to do off the ten-degree grid — "calculate the average increase per degree and add the proportionate amount to the lower figure" (13:7) — and what to do past 180°: subtract from 360 and read the same angle, now added rather than subtracted (13:5–6, 15:7). Both are followed as written. The double-elongation table of 15:3 is a different animal: it is stated as bands of whole degrees, and it is applied as the staircase it is, not interpolated.
• The moon's large circle is drawn off-centre by a fifth of its radius, its centre running round the earth. He states no such figure; it is the one that reproduces his bands of 15:3 to the degree, and it is Ptolemy's. The small sphere is drawn at whatever size subtends the angle his equation table gives it, since that table takes no account of distance. Switch Off-centre orbit off for the circle drawn about the earth.
• Two extensions past where he stops. The double elongation never leaves 5°–63° on a night the moon is sighted (15:2), so that is all he tabulates — but an orrery runs right round the month. Past 180° the correction is subtracted rather than added, the construction being symmetric about the line of syzygies; and between 63° and 180° the geometry his own bands come from is followed on round — it climbs to about 13° and returns to zero at 180°, where the correction must vanish. Neither is his, both are marked as such in the code, and no sighting calculation ever touches them.
• The course is not rounded. He discards the minutes of the course before entering the table (13:9, 15:8), which is what you want when computing by hand. Here it is interpolated continuously — a refinement in the direction of precision, and the only place the arithmetic departs from his.

Sources: Hilchot Yesodei haTorah 3; Hilchot Kiddush HaChodesh 11–15. Translation adapted from the Moznaim edition.