Mathematical Techniques in Astronomy

Mathematics is and always has been of central importance to astronomy. As soon as observations became quantified the possibility for calculation and prediction based on observations was open to astronomers. Mathematical developments were both applied to and motivated by astronomical calculations, and many of the most famous astronomers were also mathematicians and vice versa. Although techniques have become increasingly complex, the majority of mathematical astronomical techniques are concerned with positioning and calculation of relative distances of heavenly bodies. The basis of this is spherical trigonometry, which allows calculations on the celestial sphere based on observations taken from an observer on earth. The projection of the celestial sphere onto a flat surface allowed the construction of instruments such as the astrolabe and the mapping of the heavens. Techniques for increasingly accurate calculation were crucial to the development of astronomy as an exact science. It must be borne in mind, however, that not everyone studying or using astronomy was aware of or capable of applying the latest mathematical techniques. For example, there is evidence of a monk in northern France in the twelfth century positioning stars relative to architectural landmarks in his monastery, such as the windows along the dormitory wall.

The first developments of mathematical astronomy came during the Mesopotamian and Babylonian civilisations, especially during the Seleucid Kingdom (ca. 320BC to ca. 620AD). Techniques were developed for prediction of eclipses and positions of the heavenly bodies, in terms of degrees of latitude and longitude and measured relative to the sun's apparent motion. Tables were calculated and written for reference, based on arithmetic methods. These tables were available to the Greeks, who adopted many elements of the approach taken by the Babylonians (the sexagesimal system of calculation remained in use in astronomy right up to the Early Modern period) in areas of maths. Many of the Egyptian methods developed for surveying could also be applied to mathematical problems in astronomy.

Among the techniques developed and improved by the Greeks were geometric solutions of triangulation problems, including their application to three dimensions. Systems based on combinations of uniform circular motions were proposed to explain and predict the motion of the heavenly bodies, Eudoxus being among the first to suggest a model based on the rotations of concentric spheres. This kind of model wasn't very accurate at predicting positions, but generated a new type of curve, the hippopede, which provided a new area of research for geometry. Other popular types of model were based on epicycles (planets orbit along a circular path whose centre is at or near the earth) or eccentrics (the planets rotate around the sun, which in turn rotates around the earth). The development of ever more complex models of the celestial sphere required more complex calculations, and more sophisticated geometry to back them up. Textbooks on the sphere were written, consolidating mathematical techniques for astronomy these were called spherics.

Mathematicians and astronomers including Hipparchus developed techniques for the measurement of angles, and tables for calculations with these angles. Archimedes and Aristarchus studied the numerical ratios in triangles, and sophisticated theories and treatises on the application of these new theories to astronomy were published. These texts were the precursors of spherical trigonometry, which became vital to astronomy. Ptolemy's Almagest summarised and advanced these techniques and Hipparchus and Menelaus of Alexandria produced tables of what would today be called values of the sine function.

The learning of the Greeks was transmitted to Arabic areas, who in turn added Indian and Chinese mathematical and astronomical texts to the corpus of works available. The Arab scholars improved and combined the methods they read about, predictive astronomy being central to many aspects of Islam. Important advances in mathematical techniques included al-Khwarizmi's predictions of the times of visibility of the time of the new moon, and calculations of the qibla or direction of Mecca, in which to pray, from astronomical observations.

The Arabs worked with the sexagesimal system inherited from the Babylonians via the Greeks, but often converted numbers to the decimal system for complex calculations since it was easier. They called base-60 numbers the arithmetic of the astronomers. They also incorporated elements of the Indian system including adding zero to the number system.

Thabit and Ibrahim developed geometric methods for sundials, including the solution of conic sections and the application of this to the construction of sundials. In the late 10th century Abu al-Wafa and Abu Nasr Mansur proved theories of plane and spherical trigonometry and derived the laws of sines and tangents. Highly accurate tables and techniques for the calculation of trigonometric problems were produced. Abu Nasr's pupil, al-Biruni (973-1050) applied these techniques with great success to geographic and astronomical problems.

The Greeks had developed the astrolabe, but the Arabs applied their new techniques to its improvement and the development of universal astrolabes that did not require separate plates for each degree of latitude. They perfected techniques for the projection of the celestial sphere onto a flat plane, described by the Greeks, and the marking of scales and lines to enable calculations of positions on the celestial sphere to be carried out on a flat surface.

From the end of the 10th century West European scholars became increasingly interested in the writings of the Greeks and Arabs, and translations were made of important texts. Astronomy was part of the quadrivium (arithmetic, geometry, astronomy and music) of mathematical subjects which were taught to students in church educational institutions. With the founding of the universities came and increased study of Greek and Arab texts, including the mathematics of astronomy. The techniques of spherical trigonometry and other important applied geometry techniques were studied, commented on, and used to calculate astronomical tables for West European latitudes. For the next few centuries the majority of work on mathematical astronomy concentrated on consolidation and improvement of existing techniques.

By the seventeenth century mathematics was becoming more institutionalised, with increasingly efficient means of communication between mathematicians and their colleagues. This enabled advances in maths to become widely known and applied more quickly. The publication in 1614 of John Napier's work on logarithms was quickly adopted as a way of simplifying mathematical calculations in astronomy, and new logarithmic, trigonometric and astronomical tables followed. These included Kepler's Rodolphine Tables, which made great use of the new techniques and was based on elliptic orbits about the sun. The accuracy of tables and techniques increased quickly, as did the accuracy with which the heavenly bodies could be observed.

The development of calculus in the seventeenth century allowed calculation of changing quantities with greater accuracy and ease, including quantities like the speed a body is moving at. Developments in the representation of geometric quantities by algebraic expressions facilitated the further refinement of astronomical models. Increased understanding of the forces at work in the universe enabled calculations and predictions to take account of why things behaved as they did more effectively, and build this into the mathematical models used for calculation.

The development of mathematical techniques for astronomy did not stop at the end of the seventeenth century, although much of the groundwork had been laid. In the following centuries more sophisticated mathematical methods were developed, building on the fundamentals of trigonometry and calculus and were applied to astronomy. The principles of spherical trigonometry underpin the calculations of modern astronomy, although the calculations are now carried out by computers rather than slide rules.

Recommended Reading

Grattan-Guinness, I. The Fontana History of the Mathematical Sciences, London 1997

The Encyclopaedia Britannica on CD-Rom (1998) has some great introductory articles on history of maths and astronomy and so on, and explanations of mathematical techniques.

Full Bibliography