Stereographic projection and astrolabe

We will only discuss the astrolabe in brief. It is a simple medieval astrometric instrument based on stereographic projection. PhDr. Alena Hadravová, CSc., from the Institute of Contemporary History of the Academy of Sciences wrote in her article published on @muni.cz: “The astronomical instrument astrolabe, which was used for maybe two thousand years thanks to its versatility, was a little scientific miracle, a sophisticated and yet beautiful object. The fact that it was the tractate written by the Czech university professor of the 15th century, Křišťan of Prachatice, which was, among the great number of the then existing texts on this instrument, printed as the first essay on the astrolabe ever is, in my opinion, a nice achievement of Czech medieval science.”

The astronomical dial of the Prague Astronomical Clock is also an astrolabe. Today, we would consider it as a mechanical analog computer using a special representation of the space – celestial sphere – in the plane of the dial. This representation is called stereographic projection. The first to use the projection of the celestial sphere to a plane was probably Hipparchus from Bithynia around 180–125 BC. Thanks to this projection, movements of celestial objects can be simulated in a plane and thus perform “calculations of spherical astronomy” in quite a simple way. Stereographic projection has an important quality that each spatial circle is represented by another circle in the plane.

Construction of the main circles according to Křišťan

The dial of the Prague Astronomical Clock uses a projection from the North Pole. The centre of the dial represents the South Pole. In deference to tradition, we will use the procedure of construction of an astrolabe according to Křišťan of Prachatice. The construction is carried out directly on the dial; it begins from the largest circle and thus minimizes construction deviations. We can find almost the same (only rotated) construction in the copy of Táborský’s report on the astronomical clock from the 17th century on page 71. (Figure on the left.) This reassures us that this procedure was used with the dial of the astronomical clock.

The trick of the medieval construction consists in the fact that the projection is mostly made to the plane passing through the equator from the point G. Both tropics are equally distant from the equator so the construction is performed twice, using the similarity of the triangles N E B and L E G.

The construction (figure on the right) begins by arbitrary plotting of the largest circle depending on the assumed size of the dial. If projecting from the North Pole, this is the circle of the tropic of Cancer. We continue by plotting the angle A E N from the centre; this represents the inclination of the Earth’s axis. According to Křišťan, the angle shall be 24°, according to the copy of the report 23.5°. (The inclination of the Earth’s axis changes in the course of time and currently it is 23.439°). Here we adhered to the value from the report because the dial must be approached to as a monument as well. The line passing through the points B and N intersects the vertical axis in the point F, through which we pass the equator circle from the same centre. We obtain the point M by means of the line G L. Through this point, we pass the circle of the tropic of Capricorn because it is the stereographic projection of the tropic of Capricorn when projection from the point G to the equator plane. The ecliptic would be an inscribed circle of the Cancer circle and circumscribed circle of the Capricorn circle. Its diameter is, therefore, the sum of radii of the tropic circles.

It is a very simple construction that cannot practically go wrong. Or could it? It seems that even the authors of the great reconstruction of the dial 1865 were not so familiar with the stereographic projection, as they made the ecliptic ring the same size as the equator. On the sphere, the ecliptic and equator have the same diameter, but their stereographic projections are different. This leads to later “extension of the ring” but, surprisingly, not to correction of the positions of the Sun and the Moon.

Construction of the night

The basic circles of the tropics, equator and ecliptic are not dependent on the geographical position of the clock. On the contrary, the position of the dusk circle (horizon) and the circle of the astronomical night are in connection with the latitude. On the following interactive construction, you can observe this dependence. The construction deals with the fixed part of the dial, that is why the ecliptic circle is not displayed.

Let us remark one more thing: for construction of astronomical clocks, the above mentioned projections from the North Pole as well as from the South Pole were used. The projection from the North Pole has the advantage of the tropic of Cancer being the outward circle so that the Sun moves along the largest arc in the summer. What is an advantage for the Sun is not convenient for stars. An astrolabe with the projection from the North Pole represents the southern sky. All preserved portable planispheric astrolabes are intended for observation of stars and that is why they use the projection from the South Pole. It was probably for their growing popularity and effort for unification why this projection was used for astronomical clocks as well. The first astronomical clock of this kind was probably built in Mantua in 1473. Modern astronomical clocks returned back to the more comprehensible northern projection. (Litomyšl, Prostějov, Stará Bystrica, Tokyo, Soul, ...).

You can tell a clock dial projected from the South Pole by swapped tropics and opposite arching of the dusk circle. You can get a rough idea on our model of dusk by setting the angle to 130 °C and dragging the line segment representing the astronomical night over the centre to the same distance.