Astronomical Tables

Astronomical tables were designed to facilitate the calculation of planetary positions, lunar phases, eclipses and calendrical information. They often included explanations of astronomical instruments also. The various ways in which such tables are set up are an important indication of the purpose and motivation of astronomical studies in past societies. The wide-spread diffusion of this type of work is evidence of the active use to which astronomy was put throughout history.

Ptolemy's Almagest provided the theoretical explanation and tables needed for every-day calculations, mainly for astrological purposes. The arrangement of the Almagest, however, was not necessarily the most convenient for those practitioners, and thus the Handy Tables were compiled, with explanations at the beginning.

The Arabic z-îj, meaning a complete set of tables, varied considerably in form and content: some were arithmetical or trigonometric aids; some for converting calendar dates, others were for calculating setting and rising of the sun and the moon, monthly or daily positions of planets, lunar or solar eclipses, or the date of the first visibility of the crescent moon, which had a religious significance. Many included tables to factor in precessional change, or to adjust the figures according to geographic co-ordinates. Ibn Yunus' al-Zij al-Hakimi al-kabir is a particularly fine example of a medieval Islamic table. In the second half of the eleventh century, Muslim astronomers gathered in Toledo developed and compiled the Toledan Tables from disparate elements Ü some parts derived from the work of al-Battani, and other parts from al-Khawarizmi and Ptolemy. These Toledan Tables became highly popular alongside al-Khawarizmi's z-îj and were translated into Latin in the twelfth century.

In the thirteenth century, Alfonso X of Leon and Castile, also known as Alfonso the Wise, encouraged many Arabic works to be translated into Castilian. Although possibly of later origin, the Alfonsine Tables take the eve of his coronation, 31 May 1252, as the starting point. Along with the canon (derived from the Arabic word 'qanun', meaning 'thread' or 'model') or introductory instructions of John of Saxony, the Alfonsine Tables became a highly influential set of astronomical tables in Europe. Copernicus learnt how to use the Alfonsine Tables at the University of Cracow. By following the rules of calculation, based on periods of planetary motions, in principle the user could derive from the base Alfonsine year the planetary position for any given time or any given place.

The canons of John of Saxony explained how, in order to derive the planetary positions (longitudes) for any given time or any place, the user had to calculate the length of time between the basic year and the year in question, divide them by the mean figures of planetary orbits, adde or subtracted values to adjust for the hours and minutes, and make corrections for the difference in latitude of the location in question. To expedite calculations, the table was often accompanied by a table of sexagesimal multiplication.

Users of the table, however, often seem to have been confused over whether to add or subtract corrections from certain points. Thus the Alfonsine Tables were repeatedly transformed to reduce the amount of computation needed. Such tables were often called the 'Resolved Tables', and usually tabulated planetary positions for certain years or certain latitudes. These tables were designed for calendrical or astrological purposes. A 'Resolved Table' by Johannes Schöner (Tabulae astronomicae, quas vulgo, quia omni difficultate et obscuritate carent resolutas vocant, Schöner, Nuremberg 1536), for instance, included a famous defence of astrology by Melanchthon hailing these tables as worthier than the wooden doves or automata of his time. Another tendency was to dispense with tables altogether. The solution was to use paper discs, called volvelles, which functioned like 'analogue computers'. The most extraordinary and deluxe publication in this genre was Peter Apian's Astronomicum Caesarium, made for the Emperor and his relatives.

Astronomical Tables A page from Peurbach's Tabulae Eclypsium.

Large image (161K).
Very Large image (732K).

With the advent of movable-type printing, ephemerides and calendars derived from these tables began to be printed in large numbers. Ephemerides (or almanacs) normally provided pre-calculated positions of planets for several years to come. Johannes Regiomontanus' Ephemerides gave for the first time daily planetary positions (instead of every 5-10 days), which meant that only a minimum calculation was now required for horoscopic purposes. He also composed the Table of directions (1467), giving the progressions of planetary conjunctions and devising a new way of dividing the sky into twelve houses for drawing up horoscopes. Regiomontanus' method radically simplified the task of casting horoscopes.

In the sixteenth century, ephemerides or almanacs giving daily planetary positions for the coming several years became common. Many of them, like Johannes Stöffler's Almanach nova, explained the basic principles of astrology, such as planetary conjunctions and their meanings. For the year 1524, Stöffler forecast a world-wide major commotion of unprecedented scale in February when there would be twenty different conjunctions of planets, all of which would occupy a watery sign. Stöffler's forecast only anticipated a spectacular commotion of a general description, but, as is well-known, it gave rise to an explosion of vernacular pamphlets on Prognostication speculating on the possible impending disaster, from major floods to the end of the world.

It seems that even Copernicus's famous De revolutionibus, was eagerly awaited by astronomers for its improved and more accurate tables. In reality, however, the tables in the De revolutionibus were not exhaustive and not terribly useful. Thus, Erasmus Reinhold set out to re-calculate afresh, from Copernicus's basic parameters, a new set of astronomical tables. This was the Prussian Tables (1551), dedicated to Albert, Duke of Prussia. Throughout his explanatory canons, Reinhold used as his paradigm the position of position of Saturn at the birth of the Duke, on 17 May 1490. With these tables, Reinhold intended to replace the Alfonsine Tables; he added redundant tables to his new tables so that compilers of almanacs familiar with the older Alfonsine Tables could perform all the steps in an analogous manner.

Several tables based on the Alfonsine Tables were published after the publication of the Prussian Tables. Copernicus's heliocentric claims did not, then, win over the hearts of all European astronomers overnight. Rather, the Prussian Tables became popular in German speaking countries for nationalistic and confessional reasons, it seems, and it is through these tables that Copernicus's reputation was established as a skilled mathematician or an astronomer on a par with Ptolemy.

In 1627, the Rudolphine Tables, originally Tycho Brahe's project for Rudolf II but completed by Johannes Kepler, was published. In it, Kepler rehearsed the canons of astrological interpretation by using Rudolf IIÍs horoscope as an example, and discussed also the merits of Regiomontanus's division of astrological houses. Compared to earlier tables, the Rudolphine Tables, based on Kepler's model of elliptical orbits, yielded significantly improved predictions of planetary positions. The method of finding the longitude of a given planet at a given time was based on Kepler's equation and he exploited logarithms for this tabulation. Despite Tycho Brahe's wish, the tables expressed Kepler's belief in the heliocentric system in that the precise geocentric positions were worked out from calculating the heliocentric positions of the planets and the earth separately, and then combined to give the planetary positions as seen on earth.

Recommended Reading

Owen Gingerich, 'The role of Erasmus Reinhold and the Prutenic Tables in the dissemination of Copernican Theory', Studia Copernicana 6 (1973), 43-62; 'The accuracy of ephemerides 1500-1800' Vistas in Astronomy 28 (1985), 339-42; 'The Alphonsine tables in the age of printing', in . M. Comes et al. (eds) De astronomia Alphonsi Regis, Barceolona 1987, 89-95

J. D. North, The Fontana History of Astronomy, London 1994,180f., 207-13, 217-23, 259

Full Bibliography