Astrolabe
The planispheric astrolabe is an analogue computing instrument, and shows the stereographic projection of the celestial sphere onto the plane of the equator, seen from outside and pinned on the elevated pole. This page builds it one layer at a time.
The top row of the menu holds the five pieces of the instrument: the complete astrolabe, the mater, the plate, the alidade and the rete. Pressing one shows the finished piece; underneath appear the stages that build it.
The construction of the plate begins with the orthographic meridian projection of the celestial sphere. R is the radius PnO of the sphere, in arbitrary units. The Z knob changes the observer's latitude (which is also the altitude of the elevated pole, the angle PnON), and its value can also be typed exactly in the φ field below.
Stereographic projection is a special kind of projection, widely used to represent the celestial sphere. It has a point of projection which, for a northern astrolabe, is the south celestial pole, and a plane of projection which is the plane of the equator. The line joining a star to the point of projection meets the plane of the equator in a point that represents its stereographic position. Although it preserves neither distances nor areas, stereographic projection has two decisive advantages for astronomical instruments: it preserves angles, and the circles of the celestial sphere project as circles. With a simple pair of compasses it is therefore easy to draw the circles of the celestial sphere on the plane, once the projections of two diametrically opposite points — and hence their midpoint, the centre — have been found.
The drawing alongside is in two parts: the plane of the meridian projection above, and the plane of the stereographic projection below. On the stereographic projection one first carries over the circle of the equator, of radius R, identical to the circle of the meridian in the drawing above. The circle of the meridian projects onto the stereographic plane as a horizontal straight line. In this example the zenith Z and the circle of the horizon are projected, starting from the projections of N and S, which are two diametrically opposite points of the horizon. Once the points N″ and S″ have been found, their midpoint C is where the compasses are set to draw the stereographic projection of the horizon. The two points where the equator meets the horizon are the cardinal points E and W. Note that the direction EW passes through the north celestial pole Pn and is perpendicular to the meridian line NS. In actual construction the two drawings — the meridian projection and the stereographic projection below it — are superimposed: the circle of the meridian falls on the circle of the equator. Here they are kept apart only for clarity.
Drawing a circle of declination is quite simple. The A handle on the meridian projection picks out a circle of declination, which appears here as a horizontal chord. The point A″ in the stereographic projection is enough to draw that circle of declination, whose radius is necessarily Pn″A″. A precise value of the declination δ can be typed in to obtain the radius of one particular circle of declination.
Of all the circles of declination that could be drawn we keep only the most important ones: the tropic of Cancer ♋ and the tropic of Capricorn ♑. Their size is set by R alone. On a northern astrolabe the tropic of Capricorn also fixes the diameter of the plate: the plate is limited to the size of the tropic of Capricorn, whose radius is 1.52 R. There is no point in drawing further circles of declination: they would crowd the plate, and, as we shall see, a stereographic ruler will be enough to measure the declination of the stars.
Here is the construction of a single almucantar, that is, of a circle formed by the points of equal altitude above the horizon. The A handle is added, tied to an almucantar AB which, in the meridian projection, appears as a segment. The two diametrically opposite points A and B project onto the stereographic plane as the points A″ and B″. Their midpoint C is the centre on which to set the compasses to draw the almucantar.
Eight almucantars are drawn, from the one at altitude h = 10° up to h = 80°. They are obtained by repeating the construction of the previous stage. They nest around Z″, where h = 90°. The outermost almucantar of the family, the one with h = 0°, is the horizon itself.
Now we learn how to add the stereographic projection of the vertical circles. A vertical circle passes through the zenith Z and through the nadir Zi. We begin with the east prime vertical and the west prime vertical. We mark the projection of the nadir Zi″ and the midpoint C between Z″ and Zi″, which we use to draw the whole circle. We then pick out the useful arcs of that circle: the east vertical (from Z″ to E″) and the west vertical (from Z″ to W″). To construct the other verticals we shall need the axis of the centres, perpendicular to the line NS and passing through C (see the next stage).
Drawing any vertical circle rests on a simple rule of geometry about chords: every circle through the ends of a chord has its centre on the axis of that chord. All the vertical circles pass through the zenith Z″ and the nadir Zi″, so their centres necessarily lie on the axis of the centres we have drawn. The K handle stands for one of those centres. Since stereographic projection preserves angles — and preserves them at the point where the circles meet, that is at Z″ — it is the angle CZ″K that settles the azimuth: the ring centred on Z″ reads it off directly as A₁, with 90° at C, which is the prime vertical. Placing K therefore constructs the pair of verticals of azimuth A₁ and A₂ = A₁ + 180°, the two halves of one and the same circle, parted by Z″ and Zi″.
This is what the plate looks like, complete with all the vertical circles at every 10° of azimuth, set in its proper place inside the mater. The meridian line towards the north coincides with 24ʰ (mean solar time) and with 180° of hour angle; towards the south it coincides with 12ʰ of solar time and with 0° of hour angle.
The first disc is called the mater; it has a cylindrical hollow that will take the second disc, the plate, and it is surrounded by a graduated limb. The limb carries the ring of the solar hours, running from 0ʰ to 24ʰ. On a northern astrolabe the hours increase clockwise, like the apparent motion of the celestial sphere seen from the north pole. If the hour angle is engraved on the ring instead, the degrees appear from 0° to 360°, increasing clockwise as well, with their zero at the position of 12ʰ solar time.
The plate sits in the hollow of the mater. On it are engraved the local references: for the horizontal system, the outline of the horizon, the zenith, the almucantars and the vertical circles; for the equatorial system, the celestial equator, the tropic of Cancer and the tropic of Capricorn. The last of these lies on the outer edge, against the limb of the mater. The construction of the plate begins with the method of stereographic projection.
The celestial equatorial system, or uranographic system, brings in right ascension. The astrolabe reads off the declination δ and the right ascension α by means of the stereographic ruler drawn on the alidade, pinned at the elevated pole, and of a graduated ring on the third disc, which is called the rete. Both the alidade and the rete can turn about the elevated pole independently of the plate and of the mater. The next page shows how the ruler of declination is built.
Construction of the stereographic ruler of declinations on the alidade. The δ knob runs over a scale of useful declinations (from 90° to −30°). For every position of δ the corresponding point D is found on the edge of the alidade, held fixed in the direction of hour angle 0°.
This is the instrument as it stands at this point of the construction. The alidade, with its scale of stereographic declinations, and the ring of right ascension will be used to complete the rete, adding the ecliptic, the ring of ecliptic longitude and the principal stars.
On the meridian projection the ecliptic can be set in various positions, according to the time of year. To make the drawing easier we choose the December solstice and put that solstitial point S_d towards the south, on the right, with the June solstitial point S_g opposite to it, towards the north, on the left. Care must be taken to place the zero of right ascension of the rete — the first point of Aries γ — in its proper position: here it falls at the top, in the direction of east. Halfway between the two solstitial points lies the centre C, on which the compasses are set to draw the ecliptic on the rete. We take the opportunity to draw also the stereographic projections of the poles of the ecliptic, π_n and π_s, which will be needed to build the scale of ecliptic longitude.
The divisions of the ecliptic longitude scale are found by drawing the ecliptic meridians. These necessarily pass through the poles of the ecliptic, so, following the same reasoning that led us to draw the vertical circles, one draws the axis of the centres through the midpoint C_m between the poles of the ecliptic. The K handle opens an angle λ_1 = C_mπ_nK. The circle of the ecliptic meridian, having centre K and radius Kπ_n, meets the ecliptic in two points, and these mark two opposite divisions of the longitude scale: λ_1 and λ_2 = 180 + λ_1.
The ecliptic meridians found in the construction of the previous stage are not to be drawn: they serve only to mark the points of the longitude scale all along the ecliptic. The divisions are struck with the alidade: they are pieces of hour circles, all directed toward the centre of the astrolabe, not toward the centre of the ecliptic.
Drawing the ecliptic meridians is rather laborious. The construction of the longitude scale can be simplified by using the formula tan α = sin λ · cos ε / cos λ, which for every value of longitude yields the corresponding right ascension. The alidade is used to mark that correspondence. The table below may help.
The alidade and the scale of right ascension let us plot on the rete, and quickly, the principal stars of the northern hemisphere and some of the southern one, provided their declination is greater than −23.4°. The table below lists 24 of them with their equatorial coordinates for the epoch asked for. The choice of epoch can be useful in checking the correctness of ancient astrolabes.
The rete is a transparent disc — in the old instruments a pierced, openwork one — carrying the stars and the uranographic references: right ascension, and the ecliptic with its ring of ecliptic longitude. Being transparent (or pierced), the rete lets one read the plate beneath and the limb with their local references: the meridian line, the cardinal directions, the horizon, the vertical circles and the almucantars. Seen through it are also the fundamental celestial parallels — the tropic of Cancer, the tropic of Capricorn and the celestial equator — which belong to the uranographic and to the local references alike.