Astronomical Tables: Applications and Improvements During the Middle Ages

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Astronomical Tables: Applications and Improvements During the Middle Ages

Overview

Medieval astronomers were frequently called upon to resolve practical questions pertaining to social or religious matters. This was especially true in the Islamic world, where the motions of heavenly bodies were, and still are, closely tied to religious law. Astronomers also had to respond to the technical demands of astrologers who occupied an important place in Islamic society. Efforts throughout the Middle Ages to address these and related needs adequately led to improvements in existing astronomical tables and produced important theoretical developments that had applications far beyond the specific problems they were intended solve. The culmination of this work was the Alfonsine Tables, introduced in Paris around 1320.

Background

Ever since the time of the Babylonians, theoretical models have been developed for predicting the time of occurrence and location of celestial phenomena. Unfortunately, without calculators or computers, performing even the simplest calculations with these models was cumbersome and time-consuming. Astronomical tables were constructed to simplify the procedure.

Medieval astronomical tables were based almost exclusively on Ptolemy's (100?-170?) geocentric models. Ptolemy developed his geometrical models in the Almagest. Computing planetary positions based on his theory required knowing the numbers that specified the actual geometry of the models. Seven parameters were necessary, five of which were independent for each planet. Knowing these parameters and the geometry of the model, it was possible to find the celestial longitude of any planet.

In the Almagest, Ptolemy showed how to construct tables that made this procedure much more tractable. The initial starting position, or radix, for each celestial body was determined for some fixed time and displayed in tabular form. The mean motion of each body was then derived from the underlying theory and tabulated. In addition, tables of correction factors were provided. Also included were specialized tables for conjunctions of the Sun and Moon, eclipses, parallaxes, and other phenomena.

The procedure for using the tables was straightforward. To find the location of a planet at some time in the future, the mean motion tables were used in conjunction with the planet's radix to calculate how far it had moved from its initial position. However, this only indicated the approximate location of the planet. Since the motions of the Sun, Moon, and planets are not uniform, a series of corrections was necessary to fix the position more accurately. In the case of a planet, retrograde motion needed to be accounted for. The necessary adjustments for this could be made by simply locating the appropriate correction factor in the relevant table and applying it.

To further facilitate their use, Ptolemy abstracted the tables of the Almagest from their underlying theory. Several of the parameters and procedures were then modified in accordance with the results obtained in his Planetary Hypotheses. He then assembled these improved tables as the Handy Tables (c. 150). This served as the basis for Islamic astronomical tables, and the specific tabular form that Ptolemy devised persisted well into the seventeenth century.

Impact

Applied astronomy was very important in Islam for many reasons. Since the Islamic calendar is lunar, the beginning of the holy month of Ramadan and various religious festivals are regulated by the appearance of the lunar crescent. The times of the five daily prayers, required of all Muslims, are determined through the observation of celestial bodies. Other religious prohibitions and obligations are similarly tied to astronomical phenomena. Furthermore, the prayer wall in mosques (Muslim places of worship) is always aligned along the qibla, the direction of Islam's most sacred site—the Kaaba in Mecca. The Koran also encouraged Muslims to use the stars for guidance—a remark largely responsible for Islam's ceaseless parade of astrologers, who required astronomical details to ply their trade properly. Judicious decisions pertaining to agriculture, geography, navigation, and the like also relied on the results of the astronomers.

Before the founding of Islam in 622, the Bedouins of the Arabian peninsula had developed an extensive body of atheoretical knowledge about the motions of the Sun, Moon, and planets, as well as information on the seasons and fixed stars. In the early years of Islam, this folk astronomy was applied by legal scholars to address the religious practices just mentioned. As Muslim scientists became increasingly familiar with Hellenistic, Hindu, and Persian astronomical lore, more sophisticated methods were developed to deal with these secular and religious needs.

The works of Ptolemy were first translated into Arabic during the early ninth century, and Islamic astronomers quickly adopted the Ptolemaic geocentric view. Their own observations though—made at a different latitude and some 700 years after the construction of Ptolemy's Handy Tables—revealed significant deviations from Ptolemy's predictions. Thabit ibn Qurra (836?-901), al-Battani (858?-929), and others isolated and corrected erroneous parameters. New estimates were made of the speed of precessional movement, the obliquity of the ecliptic, the solar eccentricity, and the position of solar apogee. Based on these revised values, new tables were computed for various meridians and radices given for the date of the hijrah (the prophet Muhammad's emigration to Medina) in 622.

These tables were collected into handbooks known as zijes. Along with the tables for mean motions, radices, and correction factors, they included accurate calendric, prayer, and qibla tables. They also often included various trigonometric tables, indispensable for solving problems of spherical astronomy, as well as star catalogs. The introduction to a zij would usually discuss the underlying model and specific parameter values used to construct the tables along with canons for their use.

Many of the problems Islamic astronomers were called upon to solve required new mathematical methods. For instance, determining the qibla involved complex problems in spherical trigonometry. Al-Khwarizmi (800?-847?), in Zij al-sindhind, and al-Battani, in Zij-i Djadid Sultani, advanced astronomical theory by providing tables of sine functions to assist in solving such problems. Al-Battani's Zij also contained sophisticated tables of special trigonometric functions for solving problems involving spherical triangles. De motu stellarum, the Latin version of this work, printed in Nuremberg (1537), was important in the development of European astronomy.

The determination of the exact time prayer was to begin also involved spherical geometry. Ibn Yunus (940?-1009), in al-Zij al-Hakimi, made impressive strides in this direction. He compiled useful timekeeping tables that were widely imitated. They also helped establish the timekeeping institution of the muwaqqit, which was later to be associated with mosques and madrasas (Koranic schools). The timekeeping tables of Shams al-Din al-Khalili (1320?-1380?) represent the crowning achievement of medieval Islamic solutions to problems in spherical astronomy. Some of his tables for regulating prayer times were used well into the nineteenth century.

The Toledan Tables were compiled in the eleventh century under the direction of Cadi Ibn Sa'id. These tables are usually attributed to al-Zarqali (1028-1087), who participated in the project. The tables, however, have no unified underlying astronomical theory. Disparate methods and incompatible parameter values were used to compute the various tables. For instance, the tables of differences of ascension and the tables of right ascension are calculated using different values for the obliquity of the ecliptic—the former using Ptolemy's value from the Handy Tables, the latter using al-Battani's value from Zij-i Djadid Sultani. Also, the equations of the Sun, Moon, and planets follow al-Khwarizmi's tables, while the theory of trepidation follows Thabit ibn Qurra's tables. Nevertheless, the Toledan Tables were immensely popular and adapted to many locations in Europe. They also influenced the production of almanacs, which were designed not to provide the means for calculating planetary positions, but rather to give those positions explicitly.

The Toledan Tables provided the model for King Alfonso X of Castile's (1221-1284) tables. Compiled during his reign, these tables were composed in Castellan Spanish. New observations were made to establish a consistent set of parameters and the tables recomputed on the Toledo meridian. Completed in about 1272, Alfonso's tables closely followed al-Zarqali's, but Alfonso's Spanish version exerted no influence on astronomy outside of Spain. Nasir al-Din al-Tusi's (1201-1274) Zij-i ilkhani, based on 12 years of observations at the Maragha observatory, also appeared in 1272. It, unlike Alfonso's work, was very influential, particularly in the East.

The Castellan Alfonsine tables were hardly circulated and only the canons for their use are extant, the actual tables having been lost. This stands in stark contrast to the hundreds of manuscript copies and thousands of printed versions of what is known as the Latin Alfonsine Tables. This work has long been incorrectly attributed to Alfonso. Evidence now strongly suggests that this is a completely independent work compiled by a group of Parisian astronomers at least as early as 1327.

A number of important innovations appear in these tables. They were formulated with a sliding sexagesimal system that allowed the mean motion tables to be used with different sets of radices. Thus, it was possible to make predictions using any calendar. In fact, the tables came with no fewer than 10 different sets of radices taken from various calendars, including the Islamic lunar calendar and the Christian Julian calendar. This flexibility goes a long way toward explaining the universal appeal these tables were to have. That which most clearly differentiated this work from its predecessors was its advanced precessional theory, which took into account sidereal motion. Consequently, the Alfonsine Tables generated planetary positions in tropical coordinates.

The Alfonsine Tables became the most influential handbook of practical astronomy in Europe. It was the basis for almost every almanac and ephemeris until superseded by the Tabulae Prutenicae (1551) of Erasmus Reinhold (1511-1553). The Tabulae Prutenicae was the first practical set of planetary tables based on Nicolaus Copernicus's (1473-1543) heliocentric theory. Reinhold cast his tables in essentially the same format as the Alfonsine Tables, so its users would have to make no commitment to Copernican heliocentrism.

STEPHEN D. NORTON