«I d '~ DO-TH 98/18 BYZANTINE ASTRONOMY FROM A.D. 1300 '~ EMMANUEL A. PASCHOS ¢ Department of Physics, University of.I2QrtmJJndl. 44221 Dortmund, ...»
FROM A.D. 1300
'~ EMMANUEL A. PASCHOS ¢
Department of Physics, University of..I2QrtmJJndl
44221 Dortmund, Germany
A Byzantine article from the 13th century contains advanced astronomical ideas and pre-Copernican diagrams. The models are geocentric but contain improve I"'" ments on the trajectories of the Moon and Mercury. This talk presents several '-I, A models and compares them briefly with the Astronomy of Ptolemy, Arabic Astro nomies of that time and the heliocentric system.
1. Introduction It is an important historical fact that Byzantium preserved the traditions and scientific knowledge of the ancient world. The Byzantines considered the traditions of ancient Greece and Rome to be their own heritage and preserved them for many centuries. Numerous studies have been written on the fields of literature, art, philop sophy, law, etc., but there are fewer studies on the scientific developments during the Byzantine period.
Among the valuable material delivered to us are scientific writings from ancient Greece. Historians of science state that "the majority of manuscripts on which our knowledge of Greek science is based are Byzantine codices, written between 500 and
--~- 1500 years after the lifetime of their authors".1 Thus "while the Greek scientific he ritage was [to a large extent] lost in Western Europe between the collapse of the Roman Empire in the fifth century and the translation movement of the twelfth and thirteenth centuries" 2, it remained intact in the Eastern Roman Empire (Byzantium) in manuscripts attributed to the ancient authors and at the same time it was modified in the articles and commentaries of Byzantine scholars. In contrast to Western Eu ropeans "the Arabs had virtually full access to that [Greek] heritage from the eighth century onward. This occured because of a momentous translation effort whereby the great works of Greece and other cultures were translated in Arabic". 2 Later on (12th and 13th centuries) the classical knowledge was transmitted to Western Europe through Byzantine and Arabic sources and Irish monks who travelled across Europe founding monasteries and scriptoria. 3 It is now interesting to ask, "As the Byzantines were copying the ancient texts for almost a thousand years, did they also study their contents?" We know that the texts were taught almost continuously at the University of Constantinople and the Patriarchal School, and, in addition, professors wrote commentaries and books (lecture notes) on these subjects. In addition recent studies of mathematical and astronomical texts 4 show that from the 11th to the 13th centuries Byzantine scholars began to question the ancient writings and started introducing their own improve ments. Deviations from ancient theories have been established in astronomical texts, where the improvements in theorems and models are unambiguous.
Several studies of the past thirty years mention 5,6 that a short Byzantine article contains pre-Copernican figures and ideas. The article is of purely scientific nature and contains numerical parameters and 12 pages of diagrams which make possible the reconstruction of the models. For this reason it provides a unique opportunity for comparisons with the Astronomy of Ptolemy, Arabic Astronomies of this period and the heliocentric system developed later by Copernicus, Kepler and Galilei.
The article under discussion survives in three manuscripts. Two of them are in the Vatican Library and one in the Laurentiana Library of Florence. 5 It was written around A.D. 1280, and it is unsigned. David Pingree from Brown University ma de comparative studies and attributes the article to Gregory Chioniades, 7 who was born in Constantinople between 1240 and 1250 and died in Trabizond about 1320.
Chioniades travelled extensively, first to Trabizond (Black Sea) and then to Tabriz (Ira:q) and became familiar with Persian and Arabic Astronomy. Since the article contains a complete astronomy of that time and deviates on several points from the classical tradition we prepared the edition and translation of the text together with an analysis of its contents. Our study appears in a book published together with Prof.
P. Sotiroudis with the title "The Schemata of the Stars".8 I shall frequently refer to the Byzantine article as "The Schemata of the Stars".
In this talk I will cover a few topics from the book trying to indicate the level of
Astronomy at that time. Among them I will discuss:
1. Values for the obliquity and the precession of the equinoxes, which indicate the observational accuracy, 2. the shape of the earth, whether it is spherical or flat, and
3. models for the sun, the moon and the five planets. The models are very in teresting because they apply a geometrical theorem of Arabic origin. For the comparison of the epicyclic with the Newtonian trajectories we need an analy tic formalism for writing the epicyclic models. Several authors, including us, found it useful to visualize each circular motion as a rotating vector of the ra dius and then add up the rotating radii. 9 We discovered, in our studies, that the description of rotating vectors and the calculation of the resultant positions and velocities simplify tremendously when we write each vector as a complex function. The method will be used in the article and is briefly described in Appendix A.
2. Observational Accuracy
I will begin this section with several definitions. As the earth moves around the sun, it defines a plane: the ecliptic. In addition, the earth rotates every 24 hours around its axis passing through the north pole. The extension of this axis intersects the celestial sphere at a point called the celestial north pole. The equator is a plane perpendicular to the earth's axis of rotation. The two planes of the equator and the ecliptic do not coincide but form an angle of 23° 27'. This angle is called the obliquity.
In addition, the earth rotates like a "top" and its axis of rotation is not fixed but precesses in a conical motion; that is the north pole is not fixed but precesses on a circle and completes a revolution in 26,000 years. Consequently, the celestial north pole coincides now with the star Polaris, but in 12,000 years it will move very close to the bright star Vegas. Another way of describing the precession of the celestial north pole is in terms of the equinoxes. The extension of the earth's equator intersects the celestial sphere on a circle: the celestial equator. The two circles - the ecliptic and celestial equator - intersect at two points the equinoxes. The precession of the celestial north pole can be described as a precession of the equinoxial points.
In a geocentric system the definitions are similar. In this system, the ecliptic is defined as the plane of the apparent motion of the sun. The definition of the various quantities in the geocentric system is illustrated in figure 1. The obliquity and the
precession of the equinoxes are responsible for two periodic events. The obliquity is responsible for the seasons and the precession of the equinoxes is important for the definition of the length of a year. In fact, the tropical year is defined as the time interval between two vernal equinoxes, being equal to 365.2422 days.
The Byzantine model is geocentric and has nine spheres. To explain the apparent motion of the fixed stars and the sun, Chioniades introduces three spheres: two for the fixed stars and one for the sun. The remaining six spheres are used for the moon and the five planets. The outer sphere of the universe is the ninth sphere, which rotates once every 24 hours. It carries with it the eight inner spheres with their stars and is responsible for day and night. All fixed stars, including the signs of the zodiac, are located on the eighth sphere which rotates very slowly and accounts for the precession of the equinoxes. Its axis of rotation is at an angle of approximately
23.5° relative to the axis of the ninth sphere. The angle between the two axes is the obliquity. Chioniades reviews these values and we give a summary of them in Table 1.
For comparison, we included the values of al-Tusi from an Arabic article known as al- Tadhkira. 10 Chioniades gives precise values for three groups of ancient observers.
The mention of al-Tusi is interesting in itself, because Chioniades refers to his precise measurement which was the best value at that time. The value attributed to al Tusi is smaller than the one written in the al- Tadhkira. There is an explanation for this. The smaller value quoted here was obtained by al-Tusi after the writing of the al- Tadhkira.
For the precession of the equinoxes Chioniades states that "according to the an cients it is 1° in 100 years; according to later scholars 1° in 66 years and completes a revolution in 24,000 years". In Table 2 we give a summary of values quoted by various authors. One degree in 100 years is the value adopted by Ptolemy and one degree in 66 years was found by astronomers working for Caliph al-Mamoun. The contemporary value is lOin 72 years. This shows an accuracy of 9%. A change in the value for the precession implies a modification for the length of the year. This necessitates the reform of the Julian calendar which Nikephoros Gregoras recognized and proposed in the 14th century, but was not adopted for fear of religious unrest.
The new calendar was finally introduced by Pope Gregory XIII in 1582.
In the Middle Ages, a great effort was invested in improving angular measure ments. The best values were achieved by Arab astronomers and al-Tusi states that "a difference in position of less than 10' is undetectable."ll Measurements of the distances to the planets were much worse and observers could measure accurately only the parallax of the moon. Other distances were much less accurate and were obtained by indirect methods.
3. The Shape of the Earth
It is generally believed that everybody who read Aristotle knew that the earth is spherical. The shape of the earth is not mentioned explicitly in "The Schemata of the Stars", but in five diagrams the earth is drawn as a spherical globe. In figure 2 I show two of the diagrams for solar and lunar eclipses where the earth is spherical.
The article was written some 200 years before the voyage of Columbus to America and here we have one more evidence that the Byzantines considered the earth to be spherical. Arabic astronomy also considered the earth to be spherical.
An exception to this rule is an article by Cosmas Indicopleustes who wrote the "Christian Topography" around A.D. 550. There he describes the earth to be flat with an inclination relative to the sun, which explained the divisions of day and night. This simple view was not taken very seriously and it was not thought worthy of mention by medieval commentators.12
4. The Motion of the Sun
As mentioned already, the models for the sun and the planets are geocentric. For each celestial body we shall introduce a system of spheres whose axes and rates of rotation are at our disposal. They must be chosen appropriately, so that the resultant motion reproduces the correct longitudes, angular velocities as well as stations and retrogrations. These epicyclic models are an approximation to the elliptic motions through a superposition of uniform circular motions.
I begin with the model for the sun. First there is the fireball of the sun. This is the sphere where the sun is located. Next, the sphere of the sun is located inside the epicycle and the surface of the sun touches the concave surface of the epicyle. Finally the epicyle is tangent to the concave side of the major sphere, Le. the deferent. The system for the circles of the sun is shown in figure 3. As the deferent rotates through the signs of the zodiac, it carries with it the epicycle. The epicycle rotates with the same angular velocity as the deferent but in opposite sense. The net effect is that the sun rotates on a circle which is eccentric relative to the earth. This model describes the apparent movement of the sun and it was adequate for the observational accuracy of the Middle Ages. It is an ancient model invented by Hipparchus, which was known to Ptolemy. It has been used by astronomers in Byzantium and the Arab World.
I describe the model with complex functions because it is easy to generalize it for models with many epicycles. All the motions of the planets lie on planes. Planar motions are easily described in polar coordinates or with complex functions. The location of the point E of the major circle is given by
This is shown explicitly in figure 4 where after the first rotation the radius of the epicycle is in direction of EA'. After the second rotation W2 the radius of the epicycle
--+ is restored to the direction EA. The position of the point A is given by
which is the equation of an eccentric circle.
5. Comparison with the Newtonian Trajectory The eccentric trajectory is an approximation to the elliptic orbit. It is worthwhile to ask how accurate is the eccentric trajectory. According to Newtonian mechanics
We see that the two trajectories and angular velocities are periodic, repeating them selves every time Breaches 211". The maximum difference in the angular velocities at any value of B is 0.05' / day. One should note that such a small difference was undetectable with the experimental accuracy of that time (1300 A.D.).
For the accumulating effect over a year we must integrate eq. (13) over time.
What is the corresponding equation for the Newtonian trajectory? Most books give the radius r as a function of the angle, but we need the angle as a function of time.