Quiz #3

The Geminus Parapegma

(1st century AD) The parapegma appended to Geminus's Introduction to the Phenomena is one of our most important sources for reconstructing the early history of this genre among the Greeks. The Geminus parapegma is a compilation based principally on three earlier parapegmata (now lost) by Euctemon, Eudoxus, and Callippus, but it also includes a few notices drawn from other authorities. In the Geminus parapegma, the year is divided according to zodiac signs. Each sign begins with a statement of the number of days required for the Sun to traverse the sign. Then there follow, in order of time, the risings and settings of the principal stars and constellations, together with associated weather predictions and signs of the season.

Alfonsine Tables

A very influential version of this theory was built into the Alfonsine Tables (see sec. 2.11). The makers of the Alfonsine Tables introduced a steady and uniform precession that was supposed to carry the stars and the apogees of the planets all the way around the ecliptic in 49,000 years.

luni-solar cycles

All ancient luni-solar calendars were originally regulated by observation, with-out the aid of any astronomical system. In most cultures, the month began with the first visibility of the crescent Moon—in the west, just after sunset. For this reason, in Babylonian as well as Jewish practice, the day began at sunset. A few generations of experience would suffice to show that the month varied between 29 and 30 days. Therefore, if because of unfavorable weather the new crescent could not be sighted on the 3151 evening, a new month could be declared anyway.

the alexandrian calendar

As mentioned earlier, Ptolemaios III Euergetes attempted in 238 B.C. to reform the Egyptian calendar by inserting a leap day once every four years, but the new arrangement was not accepted by his subjects. However, the same reform was reintroduced more successfully by Augustus some two centuries later, after Egypt had passed under Roman control. A sixth epagomenal day was inserted at the end of the Egyptian year 23/22 B.C., and every fourth year thereafter. The modified calendar, now usually called the Alexandrian calendar, is nearly equivalent to the Julian calendar: every four-year interval contains three common years of 365 days and one leap year of 366 days. As a result, the two calendars are locked in step with one another. For example, the Alexandrian month of Thoth always begins in the Julian month of August.

Precession

Precession is a slow revolution of the whole field of stars from west to east about the poles of the ecliptic. Figure 6.1 shows the positions of three constellations (Delphinus, Auriga, and Orion) on the celestial sphere, both for the present day and for 2,000 years ago. Each star has moved along a circular arc parallel to the ecliptic. Because the motion is parallel to the ecliptic, the stars' ecliptic coordinates change in a simple way: the latitudes remain unchanged, while the longitudes increase at a steady rate. Although this rate is slow (i° in 72 years, or 50" per year) it adds up over long periods of time. In 2,00 0 years, the precession comes to some 28°, nearly a whole zodiac sign. Every star in the sky suffers this same change in longitude.

luni-solar calendar

The Roman calendar that Caesar eliminated was a luni-solar calendar, consisting of twelve months. ' There were four months of 31 days, Martius, Maius, Quintilis (= July), and October; seven of 29 days, lanuarius, Aprilis, lunius, Sextilis (= August), September, November, and December; and one of 28 days (Februarius). The length of the year was therefore 355 days, in fair agreement with the length of twelve lunar months. But as this year was some ten days shorter than the tropical or solar year, its months would not maintain a fixed relation to the seasons. Consequently, roughly every other year an intercalary month, called Intercalaris or Mercedo-nius, consisting of 27 or 28 days, was inserted after February 23, and the five remaining days of February were dropped. Thus, the year with an intercalated month consisted of 377 or 378 days. Some scholars suggest that the intercalary month alternated regularly between it s two possible lengths, so that the calendar years went through the regular four-year cycle: 355, 377, 355, 378. Four successive calendar years therefore totaled 1,465 days, and the average year amounted to 366 days, about one day longer than the tropical year. The intercalation was in the charge of the pontifices (priests of the state religion). But, through neglect, incompetence, or corruption, the necessary intercalations had not been attended to, and by 50 B.C. the calendar was some two months out of step with the seasons.

Handy Tables

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Diodorus of Sicily

A good example of these ancient manners of designating the year is provided by Diodorus of Sicily (Diodorus Siculus), a Greek historical writer who lived in Rome during the reigns of Caesar and Augustus. Diodorus completed an enormous work, of which less than half has come down to us, that treated the history of the whole known world from the time before the Trojan War down to Caesar's conquest of Gaul. Diodorus's arrangement is chronological. Each year's events are introduced by two or three equivalent designations of the year in question. For example, Diodorus begins his account of the year corresponding to 420/419 B.C. in the following way: When Astyphilos was archon at Athens, the Romans designated as consuls Lucius Quinctius and Aulus Sempronius, and the Greeks celebrated the 9Oth Olympiad, in which Hyperbios of Syracuse won the stadion. In this year, the Athenians, to abide by an oracle, restored to the Delians their island; and the Delians, who had been living at Adramyttium, returned to their homeland. . . .

Equatorial Ring

A second specialized instrument was also available for the determination of the equinoxes: the equatorial ring. This consisted of a large metal ring (probably one to two cubits in diameter) placed in the plane of the equator (fig. 5.2). The operation of this instrument is illustrated in figure 5.3. During spring and summer, the Sun is north of the celestial equator. Its daily motion carries it in a circle parallel to the equator. Consequently, the Sun shines all day on the top face of the ring and never on the bottom face. In fall and winter, the Sun is below the equator and shines all day on the bottom face of the ring. Only at the moment of equinox does the Sun come into the plane of the equator. At this moment, the shadow of the upper part of the ring (Fin fig. 5.2) will fall on the lower part (G) of the ring. Since the Sun has some angular size, the shadow of F will actually be a little thinner than the ring itself. The equinox is indicated when the shadow falls centrally on the lower part of the ring, leaving narrow illuminated strips of equal widths above and below the shadow. The advantage of the equatorial ring is that it can indicate the actual moment of equinox: one need not restrict oneself to observations taken at noon. If the Sun comes into the plane of the equator at 9 A.M., the equatorial ring will indicate it. Of course, if the equinox should happen to occur at night, it will still be necessary to interpolate between observations taken on two successive days. A major disadvantage of the equatorial ring is the difficulty of attaining and maintaining an accurate orientation.

Alternative Realities: Epicycles or Eccentrics?

As we have seen, the Greek astronomers knew two versions of the solar theory: the epicycle-plus-concentric model illustrated in figure 5.7 and the eccentric-circle model illustrated in figure 5.8. That these two models are mathematically equivalent was known from the time of Apollonius of Perga. It was remarkable that two theories that seemed physically very different should turn out to be mathematically identical. There arose a debate over which model was correct. According to Theon of Smyrna, 21Hipparchus said that it was worth the attention of the mathematicians to investigate the explanation of the same phenomena by means of hypotheses that were so different. Theon also tells us that Hipparchus expressed a preference for the epicycle theory, saying that it was probable that the heavenly bodies were placed uniformly with respect to the center of the world

parapegma

By the fifth century B.C., this lore was systematized into the parapegma, or star calendar. (The star calendar was a bit older among the Babylonians. As we have seen, the seventh-century B.C. compilation, MUL.APIN, included a star calendar.) A parapegma listed the heliacal risings and settings of the stars in chronological order. The user of the parapegma could tell the time of year by noting which stars were rising in the early morning. The parapegma served as a supplement to the chaotic civil calendars of the Greeks. Usually, but not always, the star phases were accompanied in the parapegma by weather predictions.

Callippus's parapegma

Callippus's parapegma shows a number of striking differences from those of Euctemon and Eudoxus. First, Callippus introduced a systematic use of the twelve zodiac constellations in the parapegma. Second, he was much more selective in his use of nonzodiacal stars. Indeed, besides the twelve zodiac constellations, he used only Sirius, Arcturus, the Pleiades, and Orion. Third, Callippus introduced a systematic treatment of extended constellations in their parts. Thus, he tells us when the Virgin starts to make her morning rising, when she has risen as far as her shoulders, when she has risen to her middle, when the wheat ear (Spica) has risen, and when the Virgin has finished rising. To be sure, Euctemon and Eudoxus had already done a bi t of this. But Callippus carried it much further. Finally, Callippus did not bother to record the dates of all four phases but confined himself to the morning risings and settings. The purpose of a parapegma was twofold: to tell the time of year and to foretell the weather. In Callippus's parapegma, we see a shift away from weather prediction toward more precise time reckoning. This is clear in his use of zodiac constellations, in his exclusion of nonzodiacal stars except for those with traditional importance as seasonal signs, in his breaking down of extended constellations into their parts, and even in his exclusive use of morning phases. The morning rising of the Pleaides is the first rising to be visible in the year; the evening rising is the last to be visible. Thus, the morning phases are more certain: you know when you've see the Pleiades rise for the first time, but it may take a few days to be certain whether you've seen them rise for the last time.

A Hipparchus Coin

Figure 5.11 presents a small bronze coin from Roman Bithynia. The coin was minted during the reign of Severus Alexander, who was Emperor of Rome A.D. 222—235. The obverse of the coin bears the customary portrait of the emperor himself. But the reverse (shown here) bears an image of the astronomer Hipparchus. The Greek legend around the edge reads, "Hipparchus of Nicaea." Nicaea was Hipparchus's native town in Bithynia. In Hipparchus's day (second century B.C.), Bithynia was an independent nation. It became a Roman province in 74 B.C. On the coin we see Hipparchus. He is bearded and he wears a himation, the familiar over-the-shoulder garb of ancient Greece. He sits at a tabl e that supports a celestial globe. The coin was minted several centuries after Hipparchus's death and cannot, therefore, be taken as a literal likeness. It does demonstrate, however, that Hipparchus's astronomical accomplishments were remembered, if not understood, by his countrymen. It is the oldest piece of money that carries the portrait of an astronomer.

nychthemeron

Geminus says in his Introduction to the Phenomena (VI, 1—4) that the nychthemeron (a day and night together) is not of constant length. Moreover, Geminus gives a clear explanation of one of the causes of this inequality—the one that depends on the obliquity of the ecliptic. Geminus defines the nychthemeron as the time from sunrise to sunrise, rather than as the time from local noon to local noon as we (and Ptolemy) do. Nevertheless, the essential phenomena are the same. The variability in the length of the solar day was too small to be measured directly, by means of a water clock or some other device. Rather, this variability was deduced from theory. From Geminus's remark, it is clear that the Greek astronomers knew by the first century A.D. that the nychthemeron must vary in length. 3However, the oldest surviving detailed, mathematical discussion of this subject is that of Ptolemy.

Toledan Tables

In the West, and especially in Islamic Spain, trepidation had a warmer reception. Thabit's tables for trepidation were borrowed and included as a part of the enormously influential Toledan Tables, compiled in Spain by a group of Muslim and Jewish astronomers and put into final form by al-Zarqall around the year 1080. 47The tables and their canons (directions for their use) were soon translated into Latin. The presence of trepidation tables in a manual of practical astronomy such as the Toledan Tables probably did more to popularize the theory than did Thabit's own treatise, which was much more difficult to understand and which was couched in speculative language.

On the Revolutions of the Heavenly Spheres,

In 1543, Copernicus published a book, On the Revolutions of the Heavenly Spheres, which claimed that the Earth was a planet in orbit about the Sun. But, as radical as this hypothesis was, Copernicus was quite conservative in the technical details of his astronomy. Some celestial motions were transferred by Copernicus from the sphere of the stars to the Earth, but most of the essential features of the standard astronomy were retained. For example, Copernicus let the Earth spin on its axis and revolve about the Sun to explain the daily and annual motions. And he used a gradual displacement of the Earth's axis to explain precession. But, like all his contemporaries, Copernicus believed that the precession rate was variable—he, too, was a believer in trepidation.9 Copernicus introduced a rather complicated motion of the Earth's axis to explain (in one system) both the decrease of the obliquity and the variable precession rate. So, although Copernicus now attributed these motions to the Earth, he constructed a fairly traditional system to explain them. The Alfonsine compromise of an oscillatory motion superimposed on a steady precession was basically unchanged, apart from some adjustments to the numerical param-eters.

A Stone Parapegma from Miletus

In 1902, during the excavation of the theater at Miletus conducted by the German archaeologist Wiegand, four marble fragments were found that were recognized as parts of two parapegmata. It subsequently developed that a fifth fragment, which had been found in 1899, belonged with the others. These five fragments are crucial for our understanding of the public use of star calendars in ancient times. Before the turn of the century, not a single physical parapegma was known; all investigations could be based only on the literary sources (such as Geminus). Naturally, the literary sources left some questions unanswered. For example, even the origin of the name parapegma remained obscure. Since the turn of the century, other parapegma fragments have been discovered, but none compares with the Miletus fragments in either importance r • 47 or state of preservation

The Callippic Cycle and the Callippic Calendar

In 4th century B.C., Callippus proposed a new luni-solar cycle, the Callippic Cycle (the 76 year). The Callippic Cycle is formed from four consecutive 19 year cycles, but one day is dropped. The average length of the year in Callippus's cycle is therefore exactly 365 1/4 days. Callippus' 76 year cycle served as the basis of an artificial calendar, used by some of the Greek astronomers. In this articilar calendar, the years were counted by their place in the 76 year cycle.The month names were borrowed from the Athenian calendar. But it is important to stress that Callippus's calendar had no relation to the calendar of Athens. It was a scientific calendar used by astronomers for their own purposes. This extreme step was taken because the civil calendars of the Greeks were completely unsuitable for accurate counting of the days—for all the reasons mentioned above.

Phaesis

In one of his works, however, Ptolemy did adopt the Alexandrian calendar. This was his Phaseis, which contained a parapegma, or star calendar, listing the day-by-day appearances and disappearances of the fixed stars in the course of the annual cycle (see sec. 4.11). For example, in the Phaseis, Ptolemy writes that the winter solstice occurs on the ifith of Choiak and that, for the latitude of Egypt, a Centauri "emerges" on the 6th of Choiak. (I.e., the star first becomes visible on this date as the Sun moves away from it.) It would make less sense to compose an astronomical calendar in terms of the Egyptian calendar. Neither the winter solstice nor the emergences and disappearances of the fixed stars would take place on fixed dates, since all these events advance through the months of the Egyptian calendar at the rate of one day every four years. But, in terms of the Alexandrian calendar, these annual astronomical events really do occur on about the same date every year.

The Reform

In practice, then, Easter was celebrated on a Sunday in March or April following March 21. But by the sixteenth century the date of the equinox had retrogressed to March n, so that Easter was steadily moving toward the summer. The need for reform had long been felt, but the state of astronomy in Europe had been inadequate for the task. 14In 1545, the Council of Trent authorized Pope Paul III to act, but neither Paul nor his successors were able to arrive at a solution. Work by the astronomers continued, however, and when Gregory XIII was elected to the papacy in 1572 he found several The most difficult part of the reform involved adjustments to the luni-solar ecclesiastical calendar used for calculating Easter. The details of this part of the reform need not concern us. New lunar tables were constructed to restore the ecclesiastical Moon to agreement with the true Moon. This reformed luni-solar calendar has never been accepted by the Orthodox churches, which still reckon Easter according to the tables that the Roman Church abandoned in 1582. As a result, the Orthodox Easter may coincide with the Roman Easter, or it may lag behind it by one, four, or five weeks. 1 By contrast, the reform of the solar, or Julian, calendar was simple. First, to bring the vernal equinox back to the 2ist of March, the day following October 4, 1582, was called October 15. That is, ten days were omitted. However, there was no break in the sequence of the days of the week: this sequence has therefore continued uninterrupted since its inception. Second, to correct the discrepancy between the lengths of the calendar year and the tropical year, it was decided that three leap days every 400 years were to be omitted. These were to be centennial years not evenly divisible by 400. Thus, in the old Julian calendar the years 1600, 1700, 1800, 1900, 2000, 2100, and so on, were all leap years. But under the new Gregorian calendar, 1700, 1800, 1900, and 2100 are not leap years.

Greek Civil Calendars

In the calendars of ancient Greece, the month began with the new Moon. Generally, months of 30 days, called "full" (pleres), alternated with months of 29 days, called "hollow" (koilof). Ordinarily, the civil year consisted of twelve months, but occasionally a thirteenth month was intercalated. (more on pg 183)

Gnomon

In the fifth century B.C. the gnomon was the chief, and perhaps the only, instrument available for making observations of the Sun. The solstices were determined by observing the lengths of the noon shadows. Summer solstice occurred on the day of the shortest noon shadow; the winter solstice, on the day of the longest noon shadow. Of course, the solstices do not necessarily occur at noon. The summer solstice is the moment of the Sun's greatest northward displacement from the equator, and this is just as likely to occur at the middle of the night as at noon. It is possible to interpolate between observations of noon shadows to obtain a more precise estimate of the moment of solstice. A summer solstice often cited by the ancients 1was observed at Athens by Meton and Euctemon in the archonship of Apseudes, on the list day of the Egyptian month of Phamenoth, in the morning (June 27, 432 B.C.). This time of day for the solstice—in the morning—must have been the result of interpolation between noon observations.

Thabit ibn Qurra

In the ninth century a way was found to explain both these changes (decrease in the obliquity of the ecliptic and the variability of the precession rate) in terms of a single mechanism. The theory of trepidation is usually attributed to Thabit ibn Qurra, though some scholars doubt this attribution. Thabit's doctrine of the trepidation of the equinoxes had a profound influence on medieval and early Renaissance astronomy. Indeed, one can hardly understand the medieval astronomical literature without a familiarity with Thabit's system. Al-Sabi 3al-Harranl Thabit ibn Qurra (ca. A.D. 824—901) was born in Harran, in what is now southeastern Turkey, but passed most of his professional life in Baghdad. He was not a Muslim, however, but a Sabian. The Sabians, whose religious practice included star worship, preserved aspects of Babylonian religion. Most of Thabit's scientific works were written in Arabic, but some were composed in Syriac. Thabit was a talented and well-educated man, who wrote on mathematics and made competent Arabic translations of Greek mathematical works.

Rawlin

Newton's argument convinced a number of historians. It was soon rein-forced by Dennis Rawlins. 23Borrowing an idea from Delambre, 24Rawlins noted that the southernmost stars of the catalog crossed the meridan in Alexandria about 6° above the horizon. Now, Rawlins argued that the ancient observer—whoever he was—must have tried to observe stars right down to his southern horizon. Using heavy statistical machinery, Rawlins calculated the probability that an observer, trying to observe right down to the ground, wound up, by chance, with a star catalog that stopped 6° short of the ground. Of course, the chances came out fantastically small. Like Delambre before him, Rawlins pointed out that at Rhodes, some 5° north of Alexandria, the southernmost stars of the catalog would have crossed the meridian only i° above the horizon. Applying a similar analysis to this situation, Rawlins found that the southern boundary of the catalog appeared to be statistically consistent with the catalog having been compiled at Rhodes. Since Rhodes was Hippar-chus's presumed place of observation, Rawlins argued that his results supported Hipparchus's authorship of the catalog and made Ptolemy's impossible.

Autolycus of Pitane

One could compile a list of the heliacal risings and settings of the constella-tions, simply by observations made at dawn and dusk over the course of a year. There is no need for any sort of theory. In this sense, the parapegma may be considered prescientific. But understanding the annual cycle of star phases was an important early goal of Greek scientific astronomy. Indeed, one of the oldest surviving works of Greek mathematical astronomy is devoted to this subject. This is the book (or really two books) written by Autolycus of Pitane around 320 B.C. and called On Risings and Settings. Autolycus de-fines the various kinds of heliacal risings and settings, then states and proves theorems concerning their sequence in time and the way the sequence depends on the star's position with respect to the ecliptic. No individual star is men-tioned by name. Autolycus's goal is to provide a theory for understanding the phenomena. His style is that of Euclid.

Ptolemy's parapegma,

Ptolemy's parapegma, which forms a part of his Phaseis, introduced some innovations into the tradition. First of all, Ptolemy carried to its logical conclusion the improvement in precision that had been begun by Callippus: Ptolemy did not give the dates of the heliacal risings and settings of constella-tions or parts of constellations, but only of individual stars. He included fifteen stars of the first magnitude and fifteen of the second. In this way, he eliminated the uncertainty in the first or last appearances of extended constellations such as Orion or Cygnus.

Tycho Brahe's

Testing the Argument Based on the Southern Limit of the Catalog The best case for testing Rawlins's argument about the southern limit of the star catalog is provided by Tycho Brahe's catalog of stars. Brahe's catalog was based on observations made in the 15805 and 15905 at his observatory on the island of Hven (latitude 55.9°), just north of Copenhagen. There are two different versions of the catalog. The earlier version, containing 777 stars, was printed in Brahe's Progymnasmata. The Progymnasmata was not published until 1602, after Brahe's death, but the bulk of the observations for the 777-star catalog were made before the end of 1592. In 1595, the observations of the fixed stars were resumed, bringing the total number up to 1,000, so that Brahe should not be inferior to Ptolemy. The 1000 stars were completed in great haste at the beginning of 1597, immediately before Brahe's departure from Hven. According to Dreyer, the quality of the later star places is greatly inferior to that of the original 777. Because the Progymnasmata wa s still not complete, Brahe decided to circulate a limited number of manuscript copies of the 1,000-star catalog, in 1598. The i,ooo-star catalog was first published by Kepler in the Tabulae Rudolphinae o f 1627. Both versions of the catalog are available in Dreyer's edition of Brahe's Opera.

babylonian calendar

The Babylonian year began with the new Moon of the spring month. Years contained either twelve or thirteen months. The thirteenth month was interca-lated either by adding a second month VI or a second month XII. In this list, the Babylonian month name is preceded by the Sumerian ideogram often used in Babylonian astronomical texts. Thus, the name of the spring month, Nisannu (which would require several cuneiform signs), is usually replaced by a single ideogram, BAR. (Subscripts and accent marks on some ideograms are the Assyriologists' way of distinguishing among several cunei-form signs with the same sound.) Originally, the intercalations were performed irregularly. Notices were sent in the king's name to the priestly officials at temples throughout Babylonia. This practice was still followed in the Chaldaean period. Later, during the Persian period, the announcements of intercalations came from the scribes at the temple Esangila, who sent notices to the officials at other temples through-out Babylonia. Thus, it appears that the regulation of the calendar passed into the hands of the bureaucracy. This is what made possible the eventual adoption of a regular system of intercalation.

Pope Symmarchus

The Council of Nicaea does not seem to have regularized practice regarding the Moon, for different lunar cycles continued to be used in the East and the West. Thus, Easter was sometimes celebrated on different Sundays by different sects. For example, in A.D. 501, Pope Symmachus, following the cycle then used at Rome, celebrated Easter on March 25. But his political and religious opponents at Rome, the Laurentians, followed the Greek cycle and celebrated Easter that year on April 22. Moreover, they sent a delegation to the emperor at Constantinople to accuse Symmachus of anticipating the Easter festival. 11 Uniform practice between Eas t and West was not achieved until 525, when the nineteen-year Metonic cycle was introduced at Rome. It had long been used in the East, where Greek influence predominated. Tables were prepared, based on this cycle, by means of which the date of Easter in any year could readily be determined. Again, the date of the full Moon on or next after March 21 was determined from these tables, not from astronomical observation; die Sunday following was Easter. Even after 525, other cycles continued to be used in Gaul and Britain. Feeling often ran high. The celebration of Easter on the wrong day was often deemed sufficient grounds for excommunication. 12Com-pletely uniform practice across Europe was not achieved until about A.D. 8oo.

Julian Day Calendar

The Julian calendar was instituted in Rome by Julius Caesar in the year we now call 45 B.C.. It reached its final form by A.D. 8 and continued in use without further change until A.D. 1582, when it was modified by the Gregorian reform. The Julian calendar adopts a mean length of 365 1/4 days for the year. This is in good agreement with the length of the tropical year, that is, the time from one spring equinox to the next. The Julian calendar is therefore a solar calendar and keeps good pace with the seasons. Two kinds of calendar year are distinguished: common years and leap years. Three years of every four are common years of 365 days each. One year of every four is a leap year of 366 days.

The Easter Problem

The principal motive for reform was the desire to correct the ecclesiastical calendar of the Catholic church, particularly the placement. of Easter. As Easter is the festival of the resurrection, its celebration depended on the proper dating of the crucifixion and the events around it. According to the Gospels, the last supper occurred on a Thursday evening; the trial, crucifixion, and burial of Christ on Friday. On the evening of the same Friday, the Passover was celebrated by the Jews. Finally, the resurrection occurred on the following Sunday. The Passover, around which all these events center, is celebrated for the week beginning in the evening of the I4th day of Nisan in the Jewish calendar. Now, the Jewish calendar is of the luni-solar type, and the beginning of each month corresponds closely to a ne w Moon. It follows, then, that the I4th day of Nisan was the date of a full Moon. Moreover, the month of Nisan was traditionally connected with the spring equinox: a month was intercalated before Nisan whenever necessary to ensure that Passover week did not begin before the Jewish calendrical equinox. The proper time to celebrate Easter was therefore shortly after the first full Moon of spring.

The Physics of Aristotle's Cosmos

The very idea of a constellation presupposes that these figures remain the same for long periods. By the fourth century B.C., this fact of observation had been promoted to the rank of a first principle: the heaven was changeless because it was physically impossible for it to be otherwise. According to Aristotle, the heaven is made of a fifth element, the ether, different in nature from the four elements that make up our world of growth and decay. 3The essence of this fifth element is absolute changelessness, and its natural motion is circular revolution about the center. Aristotle's method is mainly deductive and theoretical rather than empirical. But he often reinforces his physical arguments with appeals to experience. For example, after attempting to prove logically that the fifth element is neither heavy nor light, that it is ungenerated and indestructible, that it can neither grow nor diminish and is unalterable in every way, Aristotle adds that the truth of these things "is also clear from the evidence of the senses, enough at least to warrant the assent of human faith; for throughout all past time, according to the records handed down from generation to generation, we find no trace of change either in the whole of the outermost heaven or in any one of its parts." Finally, Aristotle supposes that the substance of the heavens got its name from the very changelessness of its motion: he derives aither (ether) from aei thein (always runs).

Is the Length of the Tropical Year Constant or Variable?

There seems to have been a common suspicion that the length of the year might be variable. 6The apparent variability actually resulted from errors of observation. Around 128 B.C., Hipparchus made a study of this question, as a part of his treatise On the Change of the Tropic and Equinoctial Points. This work has been lost, but portions of its contents were summarized by Ptolemy in the Almagest, so we know something of Hipparchus's procedure. Hipparchus examined the data at hand and determined that the variation in the length of the year could not be larger than about a quarter of a day. Further, he judged that the error in the observed time of an equinox or a solstice could easily amount to a quarter day. He concluded that there was no basis for accepting a variation in the length of the year: the supposed variation was no larger than the errors of observation. The most decisive evidence came from a dozen equinoxes observed carefully by Hipparchus himself over a period of nearly thirty years.

Edmund Halley,

There was no possibility of detecting shifts in the relative positions of the stars until quite recent times, when the improved status of observational astronomy and the slowly accumulating displacements of the stars themselves finally combined to give a fair chance of success. The necessary investigation was first made by Edmund Halley, then secretary of the Royal Society, and announced by him in the Philosophical Transactions for the year 1718. 57Halley had begun an investigation of precession and the decrease in the obliquity of the ecliptic in conjunction with his work as a cataloger of stars. To this purpose he compared the positions of the stars set down in the Almagest with those determined by Brahe and other more recent observers.

Tycho Brahe

Tycho Brahe (1546—1601) was a Danish nobleman who devoted the greatest part of his adult life to astronomy. King Frederick II of Denmark gave him an island, Hven, in fief and there Brahe lived like a feudal lord. The residents of the island contributed their rents to his support and were also obliged to supply labor for his construction projects. Brahe, the most famous astronomer of his generation, reflected glory on the Danish court—and also provided some practical services by casting horoscopes at the births of the royal children. But it was on Hven that the great bulk of Brahe's work was carried out. It was on Hven that Brahe constructed an observatory and place of residence that he called Uraniborg—the celestial castle. As Brahe's fame as an astronomer grew, he gathered about him a circle of assistants, students, and visiting astronomers. In many ways, Uraniborg can be considered the first modern astronomical observatory in Europe and the model for much that followed. Similar state-supported astronomical research programs had, of course, ap-peared earlier in Islamic lands, that of Ulugh Beg at Samarkand being a good example. But, because of their lack of contact with the West, they did not influence the development of the national European observatories of the seventeenth century. Brahe's Uraniborg was notable for its sustained program of observation. From the mid-i57Os until near the end of the century Brahe and his assistants engaged in regular position measurements of the planets, Moon, and stars. The solar observations, similarly carried out over many years, resulted in the best possible values for two fundamental solar parameters—the obliquity of the ecliptic and the eccentricity of the Sun's circle. Brahe also investigated the refraction of starlight by the Earth's atmosphere and prepared tables of refraction. Although others had pointed out the existence o f atmospheric refraction, Brahe was the first to take it systematically into account.

Ulugh Beg

Ulugh Beg (1394—1449), the grandson of Tamerlane, ruled Maverannakr from its capital city of Samarkand from 1409 until his death by assassination. Ulugh was a patron of the sciences and especially of astronomy. At his direction, an observatory was constructed. The astronomers in his employ undertook a program of observation and the construction of astronomical tables. This lead to a book called Zij of Ulugh Berg, which contained the tradition of the Handy Tables of Ptolemy and Theon of Alexandria. Zij is more or less a complete manual of astronomy, with trigonoetric and astronomical tables and instructions for their use (as well as a star catalog).

Meridian Quadrant

n the Almagest, Ptolemy describes several instruments for observing the Sun that represent marked improvements over the gnomon of the fifth century astronomers. The instruments described by Ptolemy were not all original with him. Similar instruments were used by Hipparchus and may have been used by astronomers of the third century B.C. The most useful of the new instruments was the quadrant set in the plane of the meridian. In Almagest I, 12, Ptolemy describes two versions of this instrument. The simplest consisted of a bloc k of wood or stone, with a smoothly dressed face set in the plane of the meridian (fig. 5.1). Near one of the upper corners, a cylindrical peg A wa s fixed at right angles to the face. This peg served as the center of a quarter-circle CDE that was divided into degrees and, if possible, into fractions of a degree. Below A was a second peg B, which served as an aid in leveling the instrument. A plumb line was suspended from A, and then splints were jammed under the block until the plumb line passed exactly over B. When, at noon, the Sun came int o the plane of the meridian, the shadow cast by peg A would indicate the altitude of the Sun. At noon the shadow becomes rather faint. Therefore, as Ptolemy says, one may place something at the edge of the graduated scale, and perpen-dicular to the face of the quadrant, to show more clearly the shadow's position. Ptolemy does not say how large his quadrant was. Judging by the descriptions of similar instruments in the writings of Pappus and Theon of Alexandria, Ptolemy's quadrant most likely had a radius of from one to two cubits (18-36 inches). One advantage of the quadrant was that it reduced the uncertainty due to the fuzziness of shadows. In using the quadrant, the observer locates the center of the shadow, rather than the edge. As the eye is able to spot the center of a narrow line of shadow very accurately, the new procedure represented an important advance over the old. The summer solstice was determined by taking several noon altitudes and interpolating to find the moment of the Sun's greatest declination.