Reaching for the Stars
The nightly celestial sphere that surrounds our earth, with all its multitude of stars, galaxies, planets, and the like has always been a source of fascination for people since the most ancient times. There is barely an ancient civilization that did not have some records of star watching. Some did not keep written records, but they watched nevertheless and recorded their observations by glyphs on ancient dwellings or at times even erected real monuments for what they wished to observe. The field of archeoastronomy is filled with examples of the latter type. Others kept very elaborate records such as the cuneiform tablets of ancient Mesopotamia.
By the time the Islamic civilization appeared on the stage of human history, much had been already said about the stars in records that they could read, or listen to, when they were orally recited, or they could have translated, as they did very early on. Even the dwellers of pre-Islamic cities and desert stretches of Arabia were very well acquainted with the night skies, and could navigate their ways through the various star configurations that they could imagine. The sheer beauty of those imagined configurations moved some people to deploy the beautiful imagery in composing poems. Others developed folk tales about other imaginary events that took place in the sky, along lines not too dissimilar from the stories of the gods who inhabited the skies of Greek mythology. Old pre-Islamic anecdotes spoke, for example, of the celestial lion that lifted its tail one day, and slapped the ground with it, thus frightening a few desert gazelles that happened to be nearby. The frightened gazelles then jumped to the north and left behind them a trace of three couplets of stars that are still called al-ulā, al-thāniya and al-thālitha (the first, the second, and the third) till this day. In our modern constellation iconography those couplets fall on three of the feet of the Big Bear. The Big Bear itself had a different name in ancient Arabian lore, and its most distinctive four stars that formed a figure very close to a square were thought of as the four pole bearers of a deceased person, while the three stars that followed were called banāt na'sh, the (daughters of [the deceased in] the coffin).
There is no systematic reconstruction of the constellations of ancient Arabia, and there is no iconographic evidence to highlight their contours in the night skies. All that is known about them comes from anecdotes, similar to the ones just cited, that are preserved in classical literary sources, or in sources of a special genre called anwa' literature. The latter genre of writings is devoted to the description of star groups whose risings and settings are a necessary consequence of their relationship to the position of the sun during the solar year. The groups that are close to the sun will necessarily become invisible, on account of the overpowering solar rays, while the others that are directly opposite to the solar position will become visible as soon as the sun sets. And since the passage of the sun in the sky is only a reflection of the earth's yearly journey around the sun, the visible nightly sky can be used as a solar clock marking the various periods of the solar year. So when the sun reaches the constellation of Scorpio, for example, the stars of the constellation of Taurus, that are diametrically across from the sign of Scorpio, begin to rise over the eastern horizon as the sun sets together with the stars of Scorpio. No wonder then that the evening rising of the Pleiades, a cluster of stars located over the hump of Taurus, began to be considered as the announcer of the beginning of the rainy season in ancient Arabia, and its morning rising, just before sunrise, as the announcer of the hot season. The ancient folk medicine of Arabia was also tied to such events, and in this particular instance the dictum said: Assure me of good times between the rising and the setting of the Pleiades, I assure you of the rest of the year. Meaning if the summer heat and the lack of rain does not kill the agricultural produce and cause widespread famine and disease then I can assure you of the opulence of the year.
Such observations, especially when noted in connection with the agricultural calendar, or with the migration calendar of various Bedouin tribes looking for grazing land for their flocks, were in fact very well tuned and highly developed in the literary traditions of pre-Islamic Arabia. Names were given to these special groups of stars and the sky was divided into 28 regions, corresponding to the lunar month, where the moon was supposed to reside each night in one of those regions, called mansions, in its monthly path around the earth. Those mansions too were tied to the agricultural calendar as well. And so were the lunar changes, as the moon went through its monthly phases, and the influence the moon had on the various human and agricultural cycles, a phenomenon that is still observed in farmer's almanacs or their equivalents of various cultures.
With the importation of the Greek astronomical tradition during the eighth and the ninth centuries, the Greek star lore began to compete with that of ancient Arabia in many circles of the newly emerging society. Of course there were those who profited from translating the Greek astronomical tradition into Arabic, and who could also see that the mathematical character of that tradition allowed for a much higher precision in locating stars in the skies, and thus allowed for much more developed techniques for computing the positions of the planets that seemed to wander amongst the stars. The benefits derivable from such precision and such calculations were way too tempting and thus very difficult to abandon. And then there were others who saw the Islamic astronomical tradition as defined by a civilization that was first and foremost dependent on Arabic, the language of its holy text, and thus sought to revive the ancient Arabian traditions and systematize them in order to compete with the incoming "foreign" Greek tradition.
This tense polarity between the two traditions, which was itself a reflection of the tension that the imported Greek sciences and philosophy created as they were being assimilated by the newly-emerging Islamic tradition, became the hallmark of the literature that dealt with the celestial imagery. No one tradition was completely successful in eradicating the other, and thus it seems that both traditions survived next to each other, but in some sense at the expense of the old Arabian tradition. As astronomers and astrologers required higher and higher precision in order to compete amongst themselves for government jobs, they required more and more astronomical star lore that was more and more dependent on Greek models of star descriptions that were mostly derived from the classical second-century Greek astronomical text, the Almagest of Claudius Ptolemy (circa 150 AD).
But as other sectors of the society, who concerned themselves with religious and literary concerns amongst others, strove hard to retain their own cultural identity by depending on the ever changing character of the Arabic language itself, and on their hierarchical position in terms of their access to the moral ranking within that society on account of their mastery of that language, the reconstruction of the classical Arabic star lore was always assured a position in that society. Add to that, the clumsy manner with which the classical Greek tradition described the individual stars in the night sky, with such phrases as "the star on the place where the tail joins the body" in the constellation of the Big Bear, in contrast to the much simpler old single Arabic word maghraz which literally meant the place of "the implant" or the place where the tail was so to speak "embedded" into the body. When the two traditions competed in this domain, the old Arabic tradition seems to have won, as evidenced by the retention of the single-word transliterated Arabic names of the stars in modern star maps instead of the phrase-long designations of the Greek tradition.
By the tenth century, it was no longer only the names of the stars that needed to be harmonized. Their locations in the night sky were also contested. The single most important work to resolve all these problems with one stroke was the work of the famous astronomer 'Abd al-Rahman al-Sufi (died 986) who in the year 964 AD composed Kitāb ṣuwar al-kawākib al-thābita (Book of the constellations of the fixed stars).
It was this fame that earned Sufi, in the Latinized form of his name Azophi, a crater on the modern maps of the moon. In his Kitāb ṣuwar al-kawākib al-thābita, he did not only superimpose the Greek tradition on top of the old Arabian one, but decided to re-observe all the stars that were already tabulated in the Greek Almagest of Ptolemy. While superimposing the two traditions on top of each other, he would give the description of each constellation, as he knew it from the Arabic translations of the Almagest, and then he would append it to the old Arabian lore concerning the various stars or groups of stars of that constellation. In the case of updating and re-observing the stars that were already observed by Ptolemy, he would often find himself in disagreement with the Greek text, either with respect to the longitude of the star, or its latitude or even its magnitude. As a result, his book ended up being dotted with such expressions as "both the longitude and the latitude are in error" as he did when he gave the new coordinates for the 8th star in the constellation of the Big Bear, or such and such a star was judged by Ptolemy to be of such and such magnitude, but I find it to be of another magnitude, giving his own measurements, or such and such a star or group of stars was not even mentioned by Ptolemy. In one particular instance, that is in the case of the Andromeda galaxy, Sufi was the first to note its existence and referred to it as al-la ṭkha al-sa ḥābīya (the small cloudy patch), which was also represented in some of the manuscripts by a group of small dots on the mouth of the Big Fish that was usually drawn across the waist of the woman Andromeda.
Although Sufi's book on the fixed stars was translated into Persian towards the middle of the thirteenth century, by the equally famous astronomer Nasir al-Din al-Tusi (died 1274), apparently that translation did not add anything new to the contents of the text.
And thus it remained unchallenged, and no one could supersede it until modern times.
About two hundred years after Tusi's translation, Sufi's book was taken up again by the most enlightened grandson and successor of Tamerlane, the famous Ulugh Beg (died 1447), who was also a distinguished astronomer in his own right. This time, Ulugh Beg did not seem to have gone back to the original Arabic of the text, but satisfied himself with using Tusi's Persian translation to incorporate the tabulated coordinates of the stars in his own astronomical handbook, the Zīj-i Sulta ni Gūrga nī, after, of course, updating the star longitudes by the appropriate amount of precision. In a roundabout way at least the nomenclature of the stars which was used in the tabular part of the Persian translation of Sufi's text, was re-incarnated within the text of Ulugh Beg. That text, in turn, was itself a subject of study by seventeenth-century English astronomers, and was again reprinted in Washington, DC towards the beginning of the twentieth century.
The migration of Sufi's text to Europe and its widespread reception there was brilliantly studied by Paul Kunitzsch in several articles which were published over the last twenty years or so. Noteworthy amongst his works is his most intriguing account of the use of Sufi's text by such famous Renaissance figures like Peter Apianus (died 1524), and the later celestial atlas makers such as Johannes Bayer (died 1617), who both quoted Sufi, as well as others. The intriguing questions raised by such quotations, especially when we have no evidence that such Renaissance scientists knew any Arabic, highlight once more the urgent need to study the routes by which Islamic science managed to reach the learned scientific circles of Renaissance Europe. Kunitzsch's studies clearly reveal the importance of Sufi's text for Renaissance scholars, not only because it contained an elaborate iconographic record of the constellations, a record that was completely lacking from Ptolemy's Almagest, the other possible source for Renaissance scientists knowledge of the stars, but because it contained freshly observed coordinates that were deemed more reliable than the much older Greek ones. The second reason alone may have been responsible for the wide circulation of Ulugh Beg's star catalogue after it was introduced into the Latin world during the latter part of the seventeenth century, and through the commentary of the Oxford astronomer Hyde (died 1703) may have even contributed to the retention of many Arabic star names in the modern stellar maps till this day.
The thirteenth century belletrist al-Qazwini (died 1283) who composed his own book regarding the Wonders of Creation (Ajā’ib al-makhlūqāt), also followed Sufi's work in his own cosmological section of the book. Qazwini's work was in turn translated into German towards the beginning of the nineteenth-century and through that translation many of the star names found their way into the modern star maps, such as the one that was produced by the National Geographic Society in the 1970's.
Sufi's fame and the obvious lusciousness of the illustrated manuscripts of his text, with all the mystique of the celestial constellations, attracted the attention of such Renaissance artists as Albrecht Dürer (died 1528) who painted an imaginary portrait of "Azophi Arabus" in the lower right-hand corner of his star map, while reserving the other corners for Aratus, Ptolemy, and Manilius. At least some of the iconography in Dürer's constellations doubtlessly took their inspiration from the representations that were beautifully displayed in the various manuscripts of Sufi's text that were, and still are, preserved in various European libraries. And when the time came for the first magisterial creation of a star atlas by Hevellius in 1690, he too created imaginary portraits of Ulugh Beg and even of Albategnius (al-Battani (died 929) in the two frontispieces of his magnificent star Atlas.
Man's Place in the Cosmos: Observational Astronomy
The most important Greek astronomical work, Ptolemy's Almagest, was already more than 700 years old when it was translated into Arabic, in Baghdad, during the early part of the ninth century. The same work also contained the results of a host of observations that were either conducted by Ptolemy himself, or were reported by him on the authority of more ancient Greek and Babylonian sources.
These two facts alone, and especially the mere passage of time, can easily explain the reason why a small observational error, or a minute approximation that was either intentionally or unwittingly allowed by Ptolemy, would become 700 years later, if not more, easily noticeable to ninth-century Baghdad astronomers. Add to that the fact that those same astronomers were also working in a society whose major factions were not at all too eager to accept a foreign science that they either did not understand or it contradicted and threatened to replace the traditions that they knew very well. Thus any mistake in the original Greek texts that could be noticed by a ninth century observer would immediately threaten the validity of that text and could easily endanger other texts associated with it. It would also threaten the persons who were importing and adopting those texts.
Some of the errors were easy to notice, while others were subtler than that, and required good scientific training to detect. In the first instance, there were prescribed mathematical operations in the original Greek texts that could be easily double-checked and their results verified. In fact, one such mistake, dealing with the length of the synodic lunar month, appeared to have been incorporated in the Greek text, and was silently corrected by the famous Arabic translator al-Hajjaj b. Matar (flourished circa 830).
Other equally important values could not be so easily corrected. For example, the measuring unit that was used in the Greek texts to calculate the size of the earth was systematically given in the usual Greek unit of stadion. And there were two such famous measurements in the Greek legacy: that of Ptolemy which gave the earth's circumference as being 180,000 stadions, and that of Eratosthenes, some four centuries before him, which gave the circumference as being 252,000 stadions. Obviously, there must have been two types of stadion, or that the measure of a stadion must have changed over time.
For a ninth-century Baghdad astronomer these measurements in this particular unit were not only confusing, but the stadion unit itself was essentially meaningless. It had to be "translated" into local units if any hope of making sense of this data was to be achieved, a matter that was not so simple. For how could one translate one system of units into another if one did not have a common reference measure to compare them to? No such measure existed then, and the only recourse the Baghdad astronomers had was to measure the same physical object, in this case the length of one degree of the earth's circumference, in local units of measure.
Such a project was indeed undertaken during the reign of the Abbasid caliph al-Ma'mun (813-833 AD). The sources speak of a team of astronomers and mathematicians that were dispatched at the time to the flat desert stretch in what is nowadays northern Syria. The team was supposed to split into two groups: one group was supposed to march north along a straight line, and mark the ground when the height of the North Pole star increased by one degree, and the second group was supposed to march south, in the opposite direction, along the same line, and mark the ground where the height of the Pole star decreased by one degree. Incidentally, everyone concerned knew that the height of the Pole star over a specific geographic locality was equal to the geographic latitude of that locality. The north and south distances were then measured in the local Arab miles known at the time, and the results were averaged in order to increase their precision. The value that emerged from this measurement was 66 2/3 miles for the length of one degree of the earth's meridian circle. The earth's circumference could then be calculated as the product of 360 degree and 66 2/3 miles to yield a value of 24000 miles, which is rather close to the modern accepted value for the earth's circumference.
Other values, like the rate of precession, the inclination of the ecliptic, as well as the position of the solar apogee were subjected to similar procedures of verification. And in all instances, the traditional Greek values were found wanting. In the case of the precession of the fixed stars, that is the apparent dislocation of the fixed stars with respect to the point of the vernal equinox, the value that was determined by Ptolemy stipulated that the dislocation would be in the order of 1 degree every 100 years. The positions of all those stars were measured with respect to the fixed point of the vernal equinox (the point when reached by the sun the day will be equal to the night) along the ecliptic circle, which is the middle circle of the zodiacal belt that marks the apparent yearly path of the sun. One of the famous fixed stars, in the constellation of Leo, which was called Regulus, i.e. the royal star, or the heart of the lion, happens to be very close to the ecliptic path.
Thus measuring its position with respect to the vernal equinox was relatively easy. According to Ptolemy's value for precession of 1 degree every 100 years, this star should have been dislocated by 7 degree during the ninth century, that is, after 700 years from the time when it was observed by Ptolemy. But those observers of ninth-century Baghdad whose colleagues were measuring the size of the earth also measured the position of Regulus and found it to have been dislocated by some 11 degrees instead of 7. After repeating this measurement several times, they finally concluded that the Greek value of 1 degree every 100 years was in fact too slow, and the better value to be adopted was more like 1 degree every 70 years, a value much closer to the modern one.
Similarly, the apparent yearly path of the earth around the sun gives rise to the various seasons that all the earth's inhabitants experience. The phenomenon of seasons is caused by the inclination of the earth's axis with respect to the plane described by the earth's path. The plane of this path is the same as the plane of the ecliptic, or the plane described by the sun in its yearly path, if one were to think of the earth as being fixed and the sun moving around it, as most astronomers of the classical Greek tradition did and were followed by the Islamic astronomers till very recent times. According to the Greek tradition, the inclination of the earth's axis was determined by Ptolemy to be 23 degrees, 51 minutes, and 20 seconds (or 23;51,20°). And since ninth century astronomers were in the process of double-checking such Greek values, they also tried to re-observe this inclination, the measurement of which is a relatively easy matter. It could also be highly precise if one used very large measuring instruments. When this measurement was done, ninth-century Baghdad astronomers found that inclination to be more like 23 1/2 degrees, a value which is much closer to the modern one as it continues to be attested in most modern textbooks.
The difference between the Greek value and that which was determined in ninth-century Baghdad is in fact close to 1/3 of a degree, and one may not make much of this discrepancy. But when such small numbers are multiplied by the very large astronomical numbers, which gave the term "astronomical" its frightening meaning, the results could become dramatically erroneous.
The subtler determination of the position of the solar apogee, or the point along the zodiacal belt where the earth seems to be at its farthest point from the sun, was a little more intricate for it did not only produce a value quite at variance from the one reported in the Greek tradition but it also produced a critique of the very method of observation that was used by Ptolemy. Ptolemy had already determined that the point at which the sun appeared to be at its farthest distance from the earth, or the other way around the earth at its farthest distance from the sun if the sun were fixed, was located towards the beginning of the constellation of Gemini, and was fixed exactly at 5 1/2 degree in that zodiacal sign. Again, 700 years later, it was easy to notice that the solar apogee had indeed moved by some 11 degrees and that it was not fixed as was thought by Ptolemy. The determination of the exact location of the apogee is important as a preliminary step for the determination of other astronomical values, and thus much effort was spent in perfecting its measurement. Several questions were raised about the reasons why Ptolemy got it wrong in the first place. And after much deliberation, it was realized that Ptolemy's method for this specific measurement depended on observing the sun at four critical points of its path: at the vernal and autumnal equinoxes and at the summer and winter solstices. The determination of the time of the equinoxes is relatively easy for at those times the day will be equal to the night. But the determination of the time of the longest and shortest days of the year was not that easy. In fact it was very difficult to determine it with any high precision. The reason for the difficulty can be easily noticed even by lay observers who can surely attest that the sun will rise every day from a slightly different point along their local horizon and will set again at the opposite point in the west. From day to day, the rising sun will slowly move to the north till it reaches its northernmost point around June 21, when the day becomes the longest day of the year. At that time the rising point will reverse its direction and start moving southward till it reaches the southernmost point, around December 21 when the day becomes the shortest day. But even the least observant lay person can also notice that around June or December 21st the sun will rise and set for several days from the same points along the local horizon, and thus it would be very difficult indeed to determine at which day exactly the sun will reverse its journey.
Ninth-century Baghdad astronomers noticed this flaw in the Ptolemaic observational technique. And in their search for higher precision they decided to abandon that method altogether and to seek an alternative one. By deploying the same mathematics that was used by Ptolemy and by only shilling the observational strategy, they decided to observe the sun during the mid seasons, that is when the sun was at the 15th degree of Taurus, Leo, Scorpio and Aquarius, instead of the beginnings of the seasons as was done by Ptolemy. Their argument was that at those midpoints, the motion of the rising point of the sun along the eastern horizon, and of course the point it reached along the meridian at high noon are much easier to observe as those positions change noticeably from day to day. The new method they adopted was then called the Fuṣu l method, i.e. the Seasons' method, simply because it depended on the mid points of the seasons.
Then too, with better observational strategy and better and larger instruments, new values for the solar apogee were determined, and the apogee was found to be moving rather than fixed, and a new solar eccentricity and solar equation were also determined as a byproduct. Those values are also very close to the modem values that are still used today, while the older Greek values are now completely forgotten.
For a group of astronomers, who were working some 700 years after the Ptolemaic observations, and who were also finding such dramatic variations between their results and the results they read about in the books that were being translated from Greek into Arabic during the ninth century, the only conclusion they could draw was that the Greek astronomical tradition was deeply flawed. And if the observable results that could be double checked relatively easily were found to be so much in error, then what else was wrong with the astronomical Greek tradition that they were importing at the time? This and other questions similar to it, encouraged astronomers working in the Islamic tradition to question much thoroughly the imported Greek tradition and of course to find more and more contradictions in its very foundations. That was the point when serious research began to be conducted in order to create an alternative astronomy, and it was that very search that culminated in the total reversal of astronomical thinking during the European Renaissance in the sixteenth century.
Islamic Planetary Theories: Questioning the Foundations of Greek Astronomy
After dispensing with the easily ascertainable observational mistakes in the Greek astronomical tradition, and after finding that tradition to be defective at almost every turn, ninth-century astronomers working within the Islamic tradition felt much more empowered to tackle more sophisticated questions, at times touching the very foundations of the Greek scientific and cosmological thought itself. As the central figure in that Greek thought was none other than Aristotle, his vision of the universe dominated all Greek and later Islamic cosmological thought. In Aristotle's vision, the world was centered around the earth, which was in turn defined as the center of heaviness. The wandering planets of classical antiquity, that is, Saturn, Jupiter, Mars, Sun, Venus, Mercury and the Moon, all observable with the naked eye with varying degrees of difficulty, revolved around the earth through the motion of rigid crystalline spheres to which they were attached. Many of them had very complicated motions, and those motions were explained as resulting from the actions of several spheres that moved in specific ways to account for the strange behavior of those planets. The complicated descriptions of those motions were needed in order to account for the observable behavior of the planets, as they seemed to move against the starry background from west to east, then stop and move back from east to west, to stop again, and then resume their west to cast motion again.
In a very intuitive fashion Aristotle's vision made eminent sense to earth-bound observers who did in "fact" see the sun rise in the morning and set in the evening, instead of "seeing" the earth rotate on its axis. They also saw a stone tossed upwards inevitably come back to the earth (for it had to come back to its natural place of heaviness as Aristotle would assert). And they did in fact see flames of their camp fires ascend upwards to what was thought of as the fiery region of the sky, and so on. They also saw that in addition to the daily rising and setting of the sun, the sun also seemed to partake of a second motion as it moved in a yearly path that took it to the farthest rising point at the northeastern region of their local horizon, at the summer solstice, and to the farthest southeastern rising point at the winter solstice, and repeat that cycle. All those intuitive observations made common sense and the job of the astronomer was to account for them, and more profitably devise mathematical models that could predict their behavior.
In the case of the wandering planets, the ability to predict their behavior had an extra benefit as it helped astrologers cast horoscopes, kings to determine times of wars, patients to take their medication and all such activities whose nature and future circumstances would be highly desirable to know.
Naturally, the mathematical devices that were invented by the foremost Greek astronomer, Ptolemy, to explain the complicated paths of the planets, were themselves very complicated mathematically and were not within the easy grasp of lay people. Even sophisticated astronomers had to have good mathematical training before they could understand their workings.
In the case of the motion of the moon around the earth, for example, Ptolemy had to explain, in one mathematical model, all the variations in the observed behavior of the moon. At the time when the moon was full or new moon it seemed that its real observable position could vary by as much as 5 degrees and one minute from its expected mean position if calculated by its mean motion. These variations could be easily detected from the occurrence of eclipses as the lunar eclipses always occur when the moon is full and the solar ones occur when the moon is new. Since the timing of eclipses could be calculated from old rules of repetitive mean behavior going back in some cases to records already kept in ancient Mesopotamian libraries, a delay or a hastening of the motion of the moon by some 5 degrees could be easily noticeable and calculated.
When the moon was only 90 degrees away from the sun, that is, at quarter or three quarters moon, that is, when the moon was one week or three weeks old, the maximum variation of its real observable motion from the mean calculated motion could reach as much as 7 degrees and 40 minutes, quite a noticeable difference from the variation at the other times. These discrepancies had to be explained away by the mathematical model that should not only account for such possible variations but should also allow the astronomer, and for that matter the astrologer, to calculate the position of the moon at any time in between. Ptolemy's devised mathematical model did that remarkably well, for it allowed for all those positions to be calculated rather accurately.
However, there were at least two major problem with the proposed Ptolemaic model, and both had to do with the manner in which one needed to use the observational data of the moon. The first and very obvious problem arose from Ptolemy's model that allowed the moon to be brought very close to the earth at the time when it was 90 degrees away from the sun in order to explain its greater longitudinal variation of 7 degrees and 40 minutes as a phenomenon of perception. That is, when objects are brought closer to the observer, in Ptolemy's model the distance almost cut in half, they would appear bigger, and a variation that was only about 5 degrees at full moon could become 7 degrees if the moon was brought closer to the earth when the moon was one week or three weeks old. If one were thinking only of the amount of variation of the real motion of the moon from its mean motion, then the model worked rather well. But when objects are brought closer to our eyes they also must appear bigger. So the body of the moon itself that was brought closer by Ptolemy's model almost half way, in order to explain away the variation in its motion had to also make the moon appear almost twice as big as it would have looked when it was a full moon since now it was only half far away. Naturally that phenomenon never occurs. That is, we never see the moon, when it is one week old, looking twice as big as when it is a full moon. And that is exactly what Ibn al-Shatir of Damascus (died 1375) asserted when he confronted Ptolemy's mathematical model for the motion of the moon by saying: lam yura kadha lika (the moon was never seen like that).
The second problem with the Ptolemaic predictive model for the motion of the moon arose from a cosmological consideration: all motions had to be generated by moving spheres as Aristotle claimed and Ptolemy, after him, accepted. Spheres that carry objects either on their surfaces or in between their shells, if they are conceived of as two concentric shells of spheres, could only move those objects in circular motions that were all centered around the center of the sphere. So it was difficult to represent the complicated motions of the moon as being caused by simple sets of spheres all nested around the fixed earth as Aristotle would have wanted. Instead, Ptolemy imagined a set of at least three spheres, arranged in specific positions, and made their combined motion account for the complicated motions of the moon. This would have been fine cosmologically for Aristotle would not have objected to having more than one sphere causing motion. But the problem in Ptolemy's arrangement of the spheres is that he made one of those spheres, the deferent, move uniformly in place on an axis that did not pass through its center. And that was physically impossible. It was like imagining a baseball moving in its fixed place around a skewer that was not pierced through the center of the ball. This meant that Ptolemy's mathematical description of the motion of the moon, which accounted rather well for the actual position of the moon in the sky, could neither predict the size of the moon properly, nor could it be physically or cosmologically realizable.
Several astronomers before Ibn al-Shatir had already attempted to reform Greek astronomy by proposing their own predictive models for the moon. Most of them could improve on Ptolemy's model by solving one problem or the other. But none of them could solve both problems at the same time in one stroke: the cosmological problem that required real physical spheres to account for motion and the appearance problem that allowed for variations in the motion of the moon while preserving more or less a fixed distance for the moon from the earth so that it would continue to have close to the same size during the whole month. Only Ibn al-Shatir was able to devise a new model that could solve both problems at once.
And because of its success, the very same model was adopted later on by Copernicus (died 1543) during the European Renaissance. The only problem remaining for the modem historian is to determine with some certainty the manner in which Copernicus knew of the work of Ibn al-Shatir, since the works of the latter were never translated into Latin and since there is no reason to believe that Copernicus knew any Arabic.
Be that as it may, future research may one day uncover more fully the routes of contacts between the Islamic world and Renaissance Europe, and may then explain the mathematical details that were available for Copernicus. The fact that such contacts did indeed take place is further confirmed by evidence that is similar in nature to the mathematical representation of the lunar motion. While describing the motion of all the other planets, with the exception of the sun, Ptolemy had to assume in each and every one of them the existence of a sphere that moved uniformly in place but on an axis that did not pass through its center, a cosmological and physical impossibility. And that was the problem that most astronomers of the Islamic world attempted to resolve, some with much better success than others. Two of them stand out in that regard, not only on account of their ability to resolve the physical contradictions of the Greek astronomical tradition, but also because they were able to develop new mathematical theorems to serve that purpose.
One of those astronomers was a man by the name of Nasir al-Din al-Tusi (died 1274). When he tried to cleanse Greek astronomy of its contradictions he had to propose a new mathematical theorem that proved to be very useful in solving several other problems as well. The theorem is very simple to state: If one takes two spheres, now called the Tusi Couple in the literature, one twice the size of the other, and if the smaller sphere is placed inside the larger one in such a way that it was internally tangent to it at one point, and if one assumed that the larger sphere moved, in place, at a specific speed, in any direction, and assumed the smaller sphere to move, also in place, at twice that speed, in the opposite direction, then the original point of tangency would oscillate back and forth along the diameter of the larger sphere.
In effect, this spherical Tusi Couple, allowed for the production of a linear motion as a result of two circular motions. And that result turned out to have several applications both in planetary astronomical theory as well as in daily applications as in the case of translating the linear motions of pistons into circular motions of wheels and the like.
Tusi's Couple also proved to be very successful and seems to have influenced the same Renaissance astronomer Copernicus who also had a great need for linear motion being produced by circular motion in the construction of his own theory. In a separate chapter in his latest and greatest astronomical work, De Revolutionibus, he devoted the whole chapter [Book III, chap. 4] to the proof of the same theorem. And what the late historian of astronomy Willy Harmer noticed, already in 1973, was that Copernicus also used the same alphabetic designators of geometric points as was done by Tusi some three hundred years before him. That is, whenever Tusi used the Arabic letter "alif"' to designate a specific geometric point, Copernicus used the Latin equivalent "A" to mark the same point, and when Tusi used "ba'," Copernicus used the phonetic equivalent "B," etc.
The discovery of this correspondence, not only in the contents of the theorem but also in the alphabetic designation of the geometric points in the two proofs, left very little doubt that Copernicus was apparently using a copy of Tusi's diagram in order to construct his own slightly different proof of the same theorem. And here too, we are not certain as to the manner in which Copernicus must have known of the earlier work of Tusi.
The second astronomer was a colleague of Tusi by the name of Mu'ayyad al-Din al-ʻUrdi of Damascus (died 1266). He was also Tusi's collaborator at the Maragha observatory (one of the most distinguished observatories of medieval Islam which was directed by Tusi). ʻUrdi had his own mathematical theorem to contribute, which was also devised specifically to solve the cosmological absurdity of the Greek astronomical tradition. And the theorem was very simple as well. It says that if one took two lines of equal length, and assumed that those two lines made equal angles with a base line, either internally or externally, then the line that joined the extremities of those two equal lines will be parallel to the base line.
The application of this theorem, however, in the construction of mathematical models that depict the motion of the various planets, is by far much more reaching than its simple statement. And like all successful theorems, this one too, that is now dubbed as the ʻUrdi Lemma, had a wide application and was in fact used by several astronomers who followed ʻUrdi including the same Ibn al Shatir mentioned before and also Copernicus. But in this instance Copernicus did not devote a special section of his book for the proof of this theorem as was done by ʻUrdi some three hundred years before him. Instead Copernicus went ahead and used it in the construction of a predictive mathematical model that was technically and mathematically the same as that of ʻUrdi. It was Kepler (died 1630) who noticed, several years later, that the theorem which was embedded within the Copernican model was left without proof and asked his own Teacher Maestlin (died 1631) about it. In his response, Maestlin supplied essentially the same proof that was already stated by ʻUrdi three centuries before. Here again. there is no indication that the work of ʻUrdi was ever translated into Latin, nor is there any evidence that either Copernicus, Kepler, or Maestlin ever knew any Arabic.
Although resolving the transmission of astronomical ideas from the world of Islam to Renaissance Europe remains enigmatic at this point, the number of instances in which Copernicus used results that were already established before in the Islamic world is not limited to the examples just mentioned. Most notable among the additional instances was Copernicus's use of the same model that was used again by lbn al-Shatir to depict the motions of the planet Mercury. This model is particularly important not only because it resolves the cosmological absurdities of the Greek astronomical tradition as it was supposed to do, but because it also used the Tusi Couple to generate linear motion as a result of two circular motions. As can be imagined the model itself is rather complicated, both mathematically and in imagining its physical behavior.
But what is most surprising is that while Copernicus adopted the same model as that Ibn al-Shatir he went on to describe its behavior in a language that revealed that he did not fully comprehend the manner in which it really functioned. That could only happen when someone is indebted to somebody else’s work, which he had not yet fully digested.
The issue of Copernicus’s indebtedness to astronomers of the Islamic world, and his reliance on their results in constructing his own astronomy, even when he had not yet fully assimilated the mathematical complexities of those results, is further confirmed by another piece of evidence that comes from the same work of Copernicus's De Revolutionibus . In the chapter, which he devoted the proof of the Tusi Couple, [Book III, 4], there too he had an additional paragraph at the end of the chapter in which he speculated about the significance and implications of the theorem. But in the published version of the same work, Copernicus decided to cut that paragraph out. Luckily for us, the original manuscript of the De Revolutionibus, in which the remark occurred, has survived and has been published in facsimile. As a result we can all read what was intended to be said in that regard.
Again, this must indicate the behavior of someone who was incorporating material he did not did not yet fully understand its significance and was still toying with ideas as to what to include of it and what to exclude. At least he was still puzzled about its range of applicability.