An instrument that computes with circles

Angles survive and areas do not proved the two properties of stereographic projection that matter most: it takes every circle on the sphere to a circle or a straight line, and it preserves every angle. The sphere that complex numbers live on showed that the sphere’s rotations become, in the projected plane, the maps written as one linear expression divided by another. Twice those essays named a debt: the instrument that was built on the projection two thousand years before anyone proved why it worked.

The astrolabe is that instrument. In its common planispheric form it is a brass disc a hand’s breadth across, with a plate engraved for one latitude and a pierced star map, the rete, that turns over it. Astronomers, navigators and astrologers used it from late antiquity through the seventeenth century to tell the time from the stars, find when the sun would rise, measure heights, and cast horoscopes. Every one of those uses is a computation in spherical astronomy, and the astrolabe performs each by lining up circles. This essay rebuilds the instrument from the projection and checks what it computes against the spherical trigonometry it replaces.

The sky projected from its south pole #

Picture the stars on a sphere centred on the observer, the celestial sphere, turning once a day about the axis through its north and south poles. A star’s position is given by its declination, its angle north or south of the celestial equator, and by its angle round the axis. Project the sphere from its south pole onto the plane of its equator, as a sphere is a plane plus one point did for the globe: each star lands on the line from the south pole through it, where that line meets the plane.

A star at declination $\delta$ lands at distance $R\tan\big((90° - \delta)/2\big)$ from the centre, where $R$ is the radius of the equator’s image. The north celestial pole lands at the centre. The equator lands on itself, a circle of radius $R$. The tropic of Cancer, the sun’s path at midsummer, at declination $+23.44°$, lands inside it at radius $0.656R$; the tropic of Capricorn, at $-23.44°$, lands outside at $1.524R$. Everything south of Capricorn lands farther out still, and the south pole would land at infinity, so the instrument stops at Capricorn: its plate is the disc inside the tropic of Capricorn, and the southern sky beyond that is simply not on it. For an observer in the northern hemisphere that sky never rises far, and nothing useful is lost.

On the plate are engraved the circles that depend on where the observer stands: the horizon, the circles of equal altitude above the horizon — the almucantars, from the Arabic — and the circles of equal direction, which pass through the zenith, the point straight overhead. On the sphere every one of these is a circle: the horizon is a great circle, each almucantar a small circle parallel to it, each direction circle a great circle through the zenith. So on the plate every one of them is a circle, and the hero figure checks it numerically: sixteen curves computed point by point from the sphere, each compared with the circle through three of its points, and the largest departure is $9 \times 10^{-15}$ of the equator’s radius — the rounding of double-precision arithmetic.

That is the practical meaning of the circle-preserving property. A plate needed perhaps forty circles, and every one could be drawn with compasses once its centre and radius were known. A projection that sent circles to other curves would have needed every line plotted point by point and joined by hand.

The maker’s two numbers per circle #

Because each line is a circle, an instrument maker needed only its centre and its radius, and both have closed forms. An almucantar at altitude $a$ on the plate for latitude $\varphi$ crosses the meridian at two points whose distances from the pole follow from the projection formula, and the circle through them has its centre on the meridian, towards the zenith, at a distance $c$ from the pole, and a radius $r$:

$$c = \frac{R\cos\varphi}{\sin\varphi + \sin a}, \qquad r = \frac{R\cos a}{\sin\varphi + \sin a}.$$

For the horizon, $a = 0$, these simplify to a centre at $R\cot\varphi$ and a radius of $R/\sin\varphi$, and the horizon passes through the two points where the equator meets the east and west — as it must, since the celestial equator crosses every horizon due east and due west. At latitude 52° the horizon’s radius is $1.269R$; at 30° it is exactly $2R$, since $\sin 30° = \tfrac12$. The hero figure’s sixteen circles were fitted from points rather than from these formulas, and the formulas agree with the fits to within $2 \times 10^{-16}$. A medieval maker worked from tables of exactly these quantities, computed with the spherical trigonometry that the triangle a globe gets wrong set out, and once they were tabulated the engraving needed nothing but compasses and a ruler.

The construction is the circular version of a fact met in a different setting. Every triple on one circle projected a circle from one of its own points onto a line, and found that rational points of the circle go to rational points of the line; that is the same projection one dimension down, and its circle-to-line behaviour is what makes the plane picture of a sphere exact.

Turning the rete is turning the sky #

Above the plate lies the rete, a cut-away brass map carrying the bright stars, each marked by the tip of a pointer, and a ring for the ecliptic, the sun’s path through the stars over a year. The ecliptic is a great circle inclined at $23.44°$ to the equator, so it projects to a circle touching both tropics, off-centre. The rete turns about the central pin.

The sky turns about the pole, and the pole sits at the plate’s centre, so turning the sky is turning the rete about the pin — a rotation of the plane. In the language of the companion essay it is the simplest Möbius transformation, multiplication of the complex coordinate by a number of size one, which multiplying is turning introduced. The plate does not turn, because the horizon and the zenith are fixed to the observer. So the instrument’s one moving part implements the one motion of the sky, and every relation between a star and the local horizon at any moment can be read by setting the rete to the moment and looking.

The figure turns it to three sidereal times — the time measured by the stars rather than the sun — and asks which of twelve stars are up. On the plate the question is whether each star’s pointer lies inside the horizon circle. In spherical trigonometry it is whether the star’s altitude, $\arcsin(\sin\varphi\sin\delta + \cos\varphi\cos\delta\cos H)$ at latitude $\varphi$, declination $\delta$ and hour angle $H$, is positive. The two answers agree for every star at every setting, as they must: the plate’s horizon is the projection of the sky’s horizon, and a point is above a circle on the sphere exactly when its image is inside the image circle.

The commonest use ran the same relation backwards. To find the time at night, the user hung the astrolabe from its ring, sighted a known star through the rule on the back, and read its altitude from the scale round the rim — say 35°. Then the rete was turned until that star’s pointer lay on the 35° almucantar, on the correct side of the meridian, and the time was read from the rete’s position against the hour scale round the plate. In trigonometry the same step solves $\sin 35° = \sin\varphi\sin\delta + \cos\varphi\cos\delta\cos H$ for the hour angle $H$, an arccosine of a combination of five sines and cosines; on the instrument it is a turn of the wrist until a dot touches a circle, and the answer is exact up to the user’s eyesight. Because the circles are exact, for questions of this kind the instrument has no approximation error at all; its error is entirely in reading.

Sunrise is where two circles cross #

The question an astrolabe answered most often was the time, and the next most often when the sun would rise.

On a given day the sun has a fixed declination, so its path across the sky is a circle round the pole, which projects to a circle round the plate’s centre. The sun is up while that circle is inside the horizon, and it rises and sets where the two circles cross. A user found the sun’s position on the ecliptic ring for the date, turned the rete until it touched the horizon on the eastern side, and read the time from the scale round the plate’s edge.

Done with geometry rather than brass, the figure intersects the day circle with the horizon circle and measures the angle from the meridian to the crossing. At midsummer it is $123.71°$, so the sun is up for $2 \times 123.71°/15° = 16.49$ hours; at the equinox exactly $90°$, twelve hours; at midwinter $56.29°$, seven and a half hours. Spherical trigonometry gives the same angle as $\arccos(-\tan\varphi\tan\delta)$, and the plate and the formula agree to within a millionth of a degree. The figures are for the sun’s centre with no allowance for the bending of light by the air, which lifts the sun’s image by about half a degree at the horizon and lengthens each day by several minutes; astrolabe users allowed for it, or not, by eye.

A plate for every latitude #

The rete is the same everywhere on Earth; the sky is the same sky. The plate is not.

The horizon depends on where the observer stands, so a plate is engraved for one latitude, and an astrolabe came with a stack of them, one for each city its owner might use it in, stored in the hollow of the main body, the mater. At the equator the horizon passes through both celestial poles, including the south pole the projection is taken from, and a circle through the centre of projection projects to a straight line — the one case of the circle property where the image is a line rather than a circle, exactly as the companion essay’s theorem allows. As the latitude rises the zenith moves from the edge towards the centre, reaching the pole at the pole, and the horizon contracts round it.

Some instruments carried a universal plate instead, the saphea of al-Zarqali in eleventh-century Toledo, which projects from a point on the equator rather than from the pole, so that one plate serves every latitude, at the price of losing the rete’s simple rotation about the centre. The choice between the two is the choice of which point of the sky to project from, and so of which circles come out as lines, and it is the same freedom that one chart is never enough used in choosing where to put the missing point.

The one set of lines that is not a circle #

Below the horizon, many astrolabes carry curves that are not circles on the sphere at all.

Before mechanical clocks, the day and the night were each divided into twelve unequal hours, so that a summer day’s hour was long and a winter day’s hour short. To read them, astrolabes engraved eleven lines below the horizon: the $k$-th line joins, for each declination, the point where the sun stands $k$ twelfths of the way from sunset to sunrise. These points are not defined by any circle on the sphere. They are defined by dividing an arc whose length varies with the declination, and the curve they trace is not a circle, so the projection owes it nothing.

Makers drew each as the circular arc through three points: where the sun stands on the tropic of Capricorn, on the equator and on the tropic of Cancer at that hour. The figure computes the true lines and the arcs and measures the gap. The largest miss is $0.36\%$ of the plate’s radius, on the third hour and its mirror image the ninth, and on a plate of radius a hundred millimetres that is $0.36$ millimetres, about the width of an engraved line. On the sixth line, midnight, there is no miss at all: the sun is at the bottom of its daily circle, on the meridian below the pole, and that line is the straight lower half of the meridian. The approximation was good enough that it is invisible on the instruments, and it is the one place in the whole construction where the makers were approximating rather than exact.

What the instrument computes, and what it does not #

The plate and rete solve one class of problem: anything that depends on the position of a point of the sky relative to the local horizon at a given moment. Rising and setting times, the altitude of a star at a given time and the time at which a star reaches a given altitude, the direction of sunrise, the length of twilight, the sidereal time from a measured altitude. Each is a relation between a circle on the rete and a circle on the plate, and the projection reduces it to lining up two circles, which the eye does to a fraction of a degree. The figures here confirm the instrument’s answers against trigonometry at full precision, which a brass instrument never reaches; real astrolabes were accurate to a few minutes of time.

The plate does not compute angles between arbitrary points on the sphere, because stereographic projection preserves angles at a point but not distances along a curve. A distance on the sky would need the projected chord turned back into an arc, which the instrument cannot do. Users who wanted it measured it directly with the sighting rule on the back.

Older than its proof #

The projection is generally credited to Hipparchus in the second century BC, and Ptolemy’s Planisphaerium describes it for star maps. The oldest surviving astrolabes come from the Islamic world in the tenth century, the oldest dated one signed by Nastulus around 927, and the instrument reached Europe through Spain; Geoffrey Chaucer wrote a treatise on its use for his son Lewis around 1391. The circle property was used throughout; the angle property was proved by Thomas Harriot around 1590, unpublished, and by Edmond Halley in the 1690s. For roughly fifteen centuries the instrument worked because circles stay circles, before anyone could say in general why they did.

Still open: how the projection was first found #

Nothing is open about the mathematics of the astrolabe, which is entirely classical. What is unsettled is the history: how much of the projection Hipparchus knew, whether the instrument Ptolemy calls an astrolabe was the planispheric one or an armillary sphere, and when the rete and plate took their familiar form, are argued from scanty texts, and the earliest instruments have not survived. The Planisphaerium itself survives only in Arabic translation and a Latin version made from that, so even the text from which the projection’s history starts has passed through two languages.

Compasses instead of trigonometry #

An astrolabe is a theorem made usable, and the same theorem later made the three-sphere drawable, when the circles that fill a three-sphere were carried into ordinary space by projection from one point. The projection sends the sky’s circles to circles, so the plate can be engraved with compasses; it turns the sky’s rotation into a rotation of the plane, so a single pin carries the whole sky’s motion; and it puts circles on the sphere into circles on the plate, so above and below the horizon become inside and outside a circle. The figures check the instrument against spherical trigonometry and find agreement to rounding error, except on the one family of lines that was never a circle on the sphere, where the makers’ approximation is out by about the width of the line they engraved.