# Astronomy and Mathematics

# Astronomy and Mathematics

**Closely Related.** With few exceptions, mathematics and astronomy were closely related in the ancient writings on science. From the first century B.C.E. and throughout the Roman Empire, learning, teaching, and practical application was exegetical, that is, dispersed through the

interpretation of texts, often through easily understood handbooks. Mathematics was not an isolated pursuit but was considered a part of what a learned person should know. Cicero (*On the Orator* 1.10, 3.127), for example, claimed that mathematics was an exact, abstruse, and obscure field, but one that an educated person could master, many scholars having obtained perfection in its study. Far from mastering mathematics, the Romans did not indulge in its study in depth, and their contributions to it were more superficial than profound.

**Three Distinct Categories.** Early Romans knew well the celestial signs—those that told the best times to plant and to harvest, that signaled changes in season, and that gave directions to travelers in unfamiliar regions. They knew the phases of the moon, the major constellations, and the positions of various stars throughout the calendar year. Indeed, roughly, they knew what a year was, although the early Roman year was imprecise compared to later adoptions of more-sophisticated calendars developed by the ancient Egyptians, Babylonians, and Greeks. When astronomy is discussed, there are three somewhat distinct areas: the primitive astronomy of the everyday Romans (as opposed to scholars); mathematical astronomy, learned primarily from the Greeks; and astrology, also learned from the Egyptians, the Babylonians, and, to a lesser degree, the Greeks.

**Roman Calendar.** Prior to learning from the Greeks, the Romans had an awkward calendar. March was the first month of the year, and there were at that time ten months in all (hence, September from *septem*, “seven,” October from *octo*, “eight,” November from *novem,* “nine,” and December from *decent*, “ten”). In 153 B.C.E. they added the two months of January and February, and moved consular elections to the first of January. Again, practicality was the reason. With Rome engaged in a protracted war in Spain, the Roman consular elections, traditionally held on 1 March, did not allow time for the consul charged with command to organize, assemble, and transport an army to Spain in time for a full campaign. By the time the consular army arrived, the rebels had encroached on Roman territory. The remainder of the season was wastefully expended in pushing them back to the point where they were at the beginning of the previous winter when the consul’s term was over. The solution was to move the elections to January so that the consul could assemble and transport an army in time for the spring—the fighting season. March, May, Quintilis (July), and October had thirty-one days, February had twenty-eight, and the remaining months had twenty-nine. The total number of days in a year was therefore 355. To adjust to the inevitable discrepancies, February was shortened to twenty-three or twenty-four days and an “intercalary” month of twenty-seven days was added. By the time of Julius Caesar, the solar calendar was off the actual solar value by three months. Finally, in 46 B.C.E. Caesar revised the calendar by making that year 445 days in length. His reform was based on the Egyptian lunar calendar, with Greek astronomical calculations that allowed a twelve-month year by inserting days into the shorter months to bring them to thirty days. February alone was the exception: every fourth year an extra day was added between 23 and 24 February (the “leap-year” addition to the end of February is a modern adaptation). Not until 1582 were subsequent calendar changes made, when Pope Gregory XIII promulgated the reform.

**Mathematical Astronomy.** The Romans fixed upon the wrong Greek writer for their earliest authority on mathematical astronomy, but his work suited the needs of the Romans because of his simplicity. Also, he was relatively concise and practical—all qualities the Romans admired. This scholar was Aratus of Soli (circa 310-circa 240 B.C.E.), who wrote in Greek a poem on astronomy and meteorology (or weather signs). Aratus was not a scientist but a person of literature; his poem *Phaenomena* was popular with the Romans, whose introduction to learned astronomy came through this work. At least four Latin translations were made. Cicero translated 769 lines when he was a young man, and he based much of his worldview on Aratus. Drusus Germanicus, nephew of Emperor Tiberius and adopted son of Augustus, translated 857 lines. In the fourth century C.E., Avienus paraphrased in Latin most of the long poem. Early in the eighth century an anonymous writer designed a poorly written translation that heavily influenced the astronomy of the Middle Ages.

## ARATUS ON THE POLES

The numerous stars, scattered in different directions, sweep all alike across the sky every day continuously forever. The axis, however, does not move even slightly from its place, but just stays forever fixed, holds the earth in the centre evenly balanced, and rotates the sky itself. Two poles terminate it at the two ends; but one is not visible, while the opposite one in the north is high above the horizon. On either side of it two Bears wheel in unison, and so they are called the Wagons. They keep their heads forever pointing to each other’s loins, and forever they move with shoulders leading, aligned towards the shoulders, but in opposite directions.

**Source:** Aratus, *Phaenomena*, 19-30, translated by Douglas Kidd (Cambridge & New York: Cambridge University Press, 1997).

**Concentric Spheres and Horse-Fetters.** Aratus based his composition on the astronomy of Eudoxus of Cnidus (circa 400-circa 347 B.C.E.), who was more of a mathematician than an astronomer, although he wrote on both subjects. Perhaps intended more as a geometrical exercise than a description of the physical world, Eudoxus proposed a universe of twenty-seven concentric spheres revolving around the earth. The perplexing motion of planets was explained through a *hippopede*, or “horse-fetter,” a figure-eight-shaped

curve, that represented the movement of a planet in latitude and retrograde wobbles. Eudoxus’s theory could not explain the obvious variations in size and brightness the planets exhibit. A later modification by Callippus (flourishing 330 B.C.E.) added to Eudoxus’s concentric spheres two more spheres for the sun and moon, and one more for the planets, but still his theoretically constructed cosmos was irreconcilable with observational data. The Greek word *planê* meant “wandering,” because, unlike stars whose progressive movement across the sky was even, planets wandered, each with its own pattern. Sometime after Callippus, a theory of unknown authorship solved the problem by using epicyclic and eccentric forms. One saw planets moving in circles in their own orbits, thereby explaining why a planet’s path appeared to have sudden rapid movements that were nearly linear, and then appear to circle back in a retrogressive manner.

**Positional Astronomy.** Aratus chose not to relate the more technical aspects of Eudoxus’s astronomy in his poem, but instead relied upon him for the celestial observations of star movements. After beginning with a hymn to Zeus, he described the northern and southern constellations. Aratus postulated the existence of a South Pole, based on earlier logical inferences by Aristotle and Eudoxus. In *De caelo* (285b) Aristotle put the South Pole on “top,” but later he altered this view. Aratus mentioned it first but, without more ado, moved to a discussion of the two Bear constellations, also called Wagons (the Big and Little Dipper). After describing them, he noted that the Greeks relied on the Great Bear for navigation, whereas the Phoenicians depended on the Little Bear. Aratus described the northern and southern constellations and planets but refrained from a discussion of their movements. Based on Eudoxus’s model, he explicated the circles of the celestial sphere before moving to a discussion of the calendar: days in a lunar month, hours of the stars as they arise and set, and the Metonic cycle, a nineteen-year lunar cycle discovered by the Athenian astronomer Meton (flourishing 432 B.C.E.). The final part—a distinct, almost separable section—concerned weather signs. Probably inspired by Hesiod’s poem on farming, *Work and Days* (which includes much important early Greek information on astronomy), Aratus’s work was read and admired by the Romans, who were introduced to astronomy through this nontechnical, easy-to-read poem that was, above all, practical.

**Astronomy, Homer, and Stoicism.** Homer was the Greek poet the Romans knew best and, Homer’s geography was readily understood: landmasses surrounded by a vast ocean. Like Aratus, Crates was a scholar of literature who wanted to interpret Homer’s universe according to Stoic principles. As the likely founder of King Eumenes II’s library in Pergamum, Crates was devoted to Stoic values based on what he regarded as Homer’s prescient theories of cosmic arrangements. He visited Rome probably in 159 B.C.E. (where he broke his leg stepping into a sewer) and may have inspired the Romans to provide public libraries and perhaps more pedestrian-friendly streets. In the history of science an important question is not only what written works were known but the order in which each came to be known. Because the Romans learned astronomy from Aratus and Crates, whose works were practical and simplistic, they tended either to disregard, to misunderstand, or, at best, to depreciate the better Hellenistic astronomers, such as Aristarchus and Hipparchus.

**Heliocentric Universe.** Some educated Romans were aware of a proposal by Aristarchus of Samos (flourishing 280 B.C.E.) that the earth was one of seven planets circling the sun. The work where he made his proposal has not survived, no doubt in part because it was disbelieved. Aristarchus has only one surviving work, called *On the Sizes and Distances of the Sun and Moon*, in which he proposed the relative distances between the earth, sun, and moon. He assumed that the moon receives its light from the sun. When the moon appeared in the sky as a half moon, he postulated that the angle formed by the sun’s rays and the visual line from the earth would be ninety degrees. He applied the geometrical theorems that (1) the square of the hypotenuse is equal to the sum of the squares of both sides and (2) the angles within a square triangle equal 180 degrees. He determined that the angle formed by the moon to the earth and by the sun to the earth was 87 degrees (whereas it is actually 89’52”). He concluded that the ratio of relative distances from the earth to the moon is more than eighteen times, but less than twenty times, the moon’s distance from the earth. Because he lacked the instruments for a more accurate measurement of the angle, his calculation, based on true values, was inaccurate, the actual ratio being about 400:1. Aristarchus proposed a heliocentric universe, possibly elaborating on a Pythagorean notion that the earth revolved around the sun, but the work in which his proposition appeared was lost. Nonetheless, the Romans and the people of the Middle Ages knew Aristarchus’s precursor theory to Copernicus in the 1540s. Like the Greeks, the Romans rejected such a radical idea that seemed to contradict common sense. Although details are lacking, one can surmise that some of the reasons for this stand were the same objections handed (much later) to Copernicus: namely, if the earth revolved around the sun and it took a day to complete a revolution, one who jumped up should land slightly to the west. Moreover, if, in explaining day and night, man observed that the earth revolved around its axis daily, then why is there not such a wind-force so strong as to level the surface of the earth? Finally, and critically, if the earth revolved around the sun and the sun was motionless like other stars, there ought to be an angle between the earth and a star relative to the sun as the earth changed position, but this is not the case. The angle should especially be apparent every six months when the earth was on the opposite side of the sun. This phenomenon is known as the stellar parallax. It was not until the 1830s that sufficiently powerful and accurate telescopes made it possible for the angles to be seen. To the Greeks, and later the Romans, the inability to make such an observation doomed Aristarchus’s theory. Not until the eighteenth century, with the work of English scientist James Bradley, was the proposition broached that the *minimum* distance to the stars was on the order of four hundred thousand times the distance to the sun. Such magnitudes defied common sense to the Romans and, besides, such information would have had no practical application.

### PLINY’S PRAISE OF HIPPARCHUS

O mighty heroes, of loftier than mortal estate, who have discovered the law of those great divinities and released the miserable mind of man from fear; mortality dreading as it did in eclipses of the stars crimes or death of some sort … [as poets describe a solar eclipse], or in the dying of the moon inferring that she was poisoned and consequently coming to her aid with a noisy clattering of cymbals… all hail your genius, ye that interpret the heavens and grasp the facts of nature, discoverers of a theory whereby you have vanquished gods and men! For who beholding these truths and the regularity of the stars’ periods of trouble… would not forgive his own destiny for the generation of mortals.

**Source:** John F. Healy, *Natural History: A Selection* (Harmondsworth, U.K.: Penguin, 1991).

**Geocentric Universe Proven Mathematically.** Ptolemy wrote that Hipparchus (circa 192-120 B.C.E.) objected to Aristarchus on the grounds of the stellar parallax. Ironically, were it not for his commentary on Eudoxus and Aratus, Hipparchus’s more important work in astronomy would have gone uncelebrated among the Romans. Hipparchus’s astronomical calculations were based on personal observations on the island of Rhodes made during the period from 141 to 127 B.C.E., and from access to the now-lost Babylonian Tables that contained centuries of celestial data. Hipparchus’s genius was relating astronomical data to practical application and, in doing so, his theoretical basis rejected Aristarchus’s heliocentric cosmos in favor of a traditional geocentric view. Employing trigonometric functions, he constructed tables for the sun and moon with the use of ingenious instruments. He could compute and predict lunar and solar eclipses, and his calculation for the average length of the lunar month comes within one second according to modern computation. In many respects it is not known precisely what Hipparchus did, because what is known of him primarily came through others, notably Ptolemy (flourishing 150 C.E.). Even before Ptolemy, the Romans celebrated Hipparchus but did not fully understand him. They even exaggerated his influence, as did the Elder Pliny, who thought that Hipparchus had swept the heavens clean of supernatural nonsense and left rationality in its place. Despite lavish

praise on one hand, Pliny saw Hipparchus as intellectually arrogant for attempting to discover that which even the gods did not know: the number of stars and their courses. Cicero accepted Hipparchus’s criticisms of Aristarchus’s heliocentric universe, although he likely read about Hipparchus in Serapion (flourishing first century B.C.E.). Serapion’s now-lost work on astronomy and geography argued that the sun was nineteen times larger than the earth.

**The Gods and the Universe.** The Romans learned and based much of their astronomy and philosophy of cosmic purpose on the Stoic philosopher Posidonius, who attempted a coherent doctrine that the Romans found intriguing. Cicero, Pompey the Great, and other Romans traveled to Rhodes to hear him lecture. Posidonius taught that God manages the universe through reason and there were three causes: matter, the soul (prime active power), and reason (the principle or directive of activity). The finite universe he regarded as a single sphere within eternal time and indefinite space or void. The cosmos was a living, sentient entity with a soul and it operated through reason. Heavenly and earthly bodies were composed of the fifth element, ether, and the four elements were of god or reason. The latter was separated from the physical world, and, at the same time, pervasive within it. The earth nourished heavenly bodies as they revolved around it in spherical orbits, a concept on which astrology was built in the Roman Empire. The sun was pure fire, around three million English miles in orbital diameter, while the smaller moon was approximately two millions stades from the earth and five hundred million from the sun. Posidonius’s calculations were accepted by the Romans and have importance in history because his figures were employed by Christopher Columbus in the fifteenth century.

**Human Limits.** The Romans incorporated Greek astronomy and mathematics within their handbook traditions for compacting knowledge in easily understood formats. The theories of the cosmos were too tedious and impractical for Roman tastes. Even the indefatigable Pliny, who absorbed learning like a sponge and whose praise of Hipparchus was lavish, thought that man’s attempt to know the deepest cosmic secrets was beyond his capacity to understand. Pliny said Hipparchus “dared to give the number of stars to posterity and to enumerate the heavenly bodies by name,” an exercise *etiam deo improbam* (“unseemly even for a god”) (Pliny, *Natural History* 2.24.95). Similarly, when he described attempts to fix the proportions of pharmaceuticals, Pliny argued that such attempts were “beyond the capability of man”(Ibid. 29.8.25). Stoic philosophers, including Pliny, increasingly saw the universe as *divina natura*, or Divine Nature, in contrast to *imbecillitas humana*, or “limited human intelligence.”

**Handbook Astronomy.** The handbook format for science in general, astronomy in particular, appealed to the Romans, although many of the handbook writers were Greeks whose works were written or known in Latin translation. Cleomedes (flourishing from the early first century B.C.E. or as late as the early second century C.E.) wrote a handbook that primarily relied on Posidonius. Heavily influenced by Stoic philosophy, Cleomedes was concerned with astronomical calculations based on a geocentric model. For the earth, he explained remarkably accurately the equator, tropics, “arctic,” and “antarctic” circles, the values for the astronomical seasons, the sun’s eccentric orbit that makes it closer to earth during certain seasons, and the sidereal day being shorter than the solar day (indirectly from Hipparchus). He described the moon’s phases, thereby dispelling many superstitions, and gave the approximately correct zodiacal periods (except for Mars) for Mercury, Venus, Jupiter, and Saturn. Fortunately, he recorded the methods that Eratosthenes and Posidonius employed for measuring the surface of the earth.

**Number Theory.** Following the handbook tradition, Nicomachus of Gerasa (flourishing 100 C.E.) presented an arithmetic that was less inclined toward astronomy than to number theory, a subject the Romans found intriguing. The Pythagoreans heavily influenced Nicomachus, especially in regard to music theory and the mathematical harmony of the scales. Nicomachus’s eclectic work combined philosophy and pure mathematics, for example, even and odd, prime (including an explanation of Eratosthenes’ “sieve” method to find prime numbers), and uncompounded numbers, as well as Euclidean problems and the derivation of superparticulars from successive multiples. His connecting the mathematical harmony of a rational universe together with the Pythagorean mystical truth was appealing to later medieval thinking.

**“No One Ignorant of Geometry.”** Above the portal of Plato’s Academy were the words: “Let no one ignorant of geometry enter.” Plato’s theory of ideas and advocacy of mathematics are best explained in his dialogue, *Timaeus.* Only the Romans who read Greek were stimulated by Plato’s emphasis on geometry, until Calcidius (fourth to fifth centuries C.E.) partially translated *Timaeus* into Latin and provided an extensive commentary. Calcidius’s discussions of geometry and astronomy were very important in the transmission of Graeco-Roman science to the Middle Ages.

**Astronomical Handbook.** Geminus (flourishing in Rhodes, circa 70 B.C.E.) wrote a handbook on positional astronomy and mathematical geography that greatly influenced the Romans. He wrote an *Introduction to Astronomy* and a larger, six-book work on the mathematical sciences, largely appealing to the Romans because of the application of his calculations for the calendar. Without originality or ingenuity, he discussed the zodiac, the moon and planetary orbits, and the constellations. He wrote about the inequalities of the solar orbit, thus accounting for seasonal variations ranging from 88.125 days to 94.5 days, the celestial equator, tropic and arctic circles, and various Greek lunar and solar calendar schemes.

**Astrology.** Babylonian priests marked the heavens according to twelve imaginary belts, thirty degrees wide (hence, 12 x 30 = 360°), based on prominent stars and constellations. They were: Aries (ram); Taurus (bull); Gemini (twins); Cancer (crab); Leo (lion); Virgo (virgin); Libra (balance-scales); Scorpio (scorpion); Sagittarius (archer); Capricornus (goat); Aquarius (water bearer); and Pisces (fishes). By around 500 B.C.E. the Babylonian priests had defined and marked these bands as the zodiac (Greek *zöidion*, “[small] living creature,” for example, probably small carved or painted figures of these signs) and observed how the planets (including the sun and moon) moved in relation to the zodiac and fixed stars. They noticed the relationship between when the signs appeared and events on earth, such as weather changes, the tides, and seasons. Postulating that heavenly objects impinge on earthly events, they developed elaborate means of forecasting the future. Horoscopes were used to determine a person’s birth and/or conception and to predict the individual’s future. Similarly for the kingdom or community, the positions of celestial bodies could indicate short-term calamities and fortuitous events. Whereas today one relegates astrology to magic and superstition, in fact the astrologers made astronomical, mathematical, and calendar computations that bordered on what is now called science. One practical example is the charting of the tides that was truly predictive. Astrology found receptive, although controversial, support in Rome as early as the first century B.C.E. In the empire, Roman emperors from Tiberius through Vespasian, and many other emperors throughout the second and third centuries, consulted with astrologers. Indeed, at times political and military decisions were influenced and determined by astrological predictions, but not without controversy. Augustus Caesar regulated astrologers who read horoscopes in private, and periodically astrologers were expelled from Rome. Between 152 and 162 C.E. Vettius Valens, a Greek, wrote a nine-book astrology, called the *Anthologies*, that had many horoscope charts. Early Christian Church Fathers excoriated astrologers. Although their influence diminished in succeeding centuries, it is fair to say that astrology and astronomy were closely linked, each influencing the other and both emphasizing mathematics for computations.

### FIRST WOMAN MATHEMATICIAN

Hypatia, daughter and pupil of Theon, enjoys the distinction of being the first known woman mathematician and the last of the ancient mathematicians. She assisted her father in the commentary on Ptolemy’s *Almagest* and the revisions to Euclid’s *Elements.* She supposedly wrote commentaries (now lost) on Diophantus’s *Arithmetica* and Apollonius’s *Conic Sections.* She is the last-known lecturer at Alexandria, where her teachings on mathematics and Neoplatonism, a philosophy she propounded, won for her the admiration of her students and the enmity of the Christians. Likely her works were among those burned in frequent pillages of libraries by Christian mobs and she was killed by one of the mobs around the year 415.

**Source:** Maria Dzielska, *Hypatia of Alexandria*, translated by F. Lyra (Cambridge, Mass.: Harvard University Press, 1995).

**Elements of Plane Geometry.** Euclid (flourishing 295 B.C.E., in Alexandria) wrote (in Greek) the *Elements*, a well-explained text that the Moslems and Christians in the Middle Ages employed. The first section on plane geometry (hence, the name “Euclidean” geometry) was little altered through the nineteenth century as a text for teaching. Even now, high-school texts on geometry embody the same problems in much the same order. Book 1 begins with the postulates that it is possible to define a point and a straight line. In addition to *Elements*, Euclid wrote *Phenomena*, devoted to spherical geometry as applied to astronomy. The Romans appear to have learned more indirectly than directly from Euclid’s texts.

Some Greek scholars produced commentaries on it. The earliest extant commentary was written by Proclus (410-485 c.e.), who refers to several predecessors who wrote on Euclid but whose works did not survive.

**Mechanics, Mathematics, and Astronomy.** The Greek Archimedes (circa 287-212 b.c.e.) was the better scholar, yet less well known to the Romans than Euclid. The Romans knew Archimedes through his clever inventions, said to have delayed the fall of Syracuse, his native city, to the Romans in the Second Punic War. Several Romans (Livy, Plutarch, and Valerius Maximus) told of his death. When Rome took the city in 212, a Roman soldier found Archimedes teaching a class and dispatched him—a teacher’s equivalent of dying with one’s boots on. Judging by Latin sources, however, the Romans rarely read, commented on, or reflected on his works. Archimedes was a first-rate mathematician and astronomer, but they knew him indirectly through Greek writers such as Hero, Pappus, and Theon. For many of his calculations one would employ integral calculus today, but he adopted the Greek method of bypassing infinitesimals. Whereas he wrote on astronomy, those works were lost, but his method for determining the sun’s diameter is incorporated in the *Sand-Reckoner.*

**Solid Geometry and Conic Sections.** What Euclid did for plane geometry, Apollonius of Perga (third to second centuries b.c.e.) did for solid geometry and conic sections. Born in Perga in Asia Minor, Apollonius studied in Pergamum and may have worked in Alexandria, although that is uncertain. The influence of his contributions to mathematics is certain and highly regarded. Even though Apollonius did not mention him by name, he appears to have built on Archimedes’ ideas. Apollonius’s enduring work was the eight-book *Conics* that developed the geometry of parabolas, ellipses, and hyperbolas. His mathematical propositions based on geometry had direct application to mathematical astronomy. His proofs were especially important to Ptolemy in establishing the stationary point of the earth in relation to orbiting bodies in epicyclic and eccentric movements.

**Stoic Views of Astronomy.** In late Republican and early imperial times, many Romans regarded the universe and the god(s) as being the same or nearly the same. Penetrating the cosmic mysteries was impractical, disrespectful, and, possibly, blasphemous. Like Pliny, Seneca (first century c.e.) regarded astronomy as instructive about Nature’s rational functions in order to free humans from superstition. Although he based his knowledge on Greek learning, primarily on Posidonius, Seneca’s views were more astrology than astronomy. Heavenly bodies cause effects on earth, and it was the earth that received Seneca’s attention. In astronomy, Seneca rejected the conventional view (derived from Aristotle and Posidonius) that comets exist between the moon and earth, thereby combining the rectilinear motion of the four earthly elements (air, earth, fire, and water) with the circular motion of ether-composed heavenly bodies. According to this view, comets travel in a combination of rectilinear and circular motions, thus, elliptical orbits. Seneca regarded comets, like other heavenly bodies, as traveling in circular orbits. Seneca’s major work on astronomy and meteorology is called *Natural Questions;* written in Latin, it was popular during the Middle Ages. The integration of philosophy and physics with a Stoic emphasis made the message appealing to medieval thinkers. The close relationships between the earth and heavenly bodies appealed to those who were inclined to accept as truth the science that made the heavens and earth, god and man, linked more closely.

**The Ancients’ Universe as Passed to Later Ages.** The work of one man, Ptolemy, was the means by which most people in the Islamic world and Christian West knew of mathematics, especially as applied to astronomy. Ptolemy supplied the theoretical basis upon which later ages founded their own observational researches down to Copernicus in the mid sixteenth century. Little is known of his life; much is known about his influence. He flourished 150 c.e. and did his work in Alexandria. His greatest work is known through its Arabic name, the *Almagest,* because no Greek to Latin translations of it were made until after the Western Europeans discovered it through an Arabic to Latin translation by Gerard of Cremona (circa 1175).

**Almagest.** The *Almagest* has been described as a synthesis of Greek astronomy and mathematics, not an original compilation; however, modern scholars see genius and innovation in his presentations. Books 1 and 2 are an introduction whereby Ptolemy describes the universe as understood by the consensus views of Eudoxus, Aristotle, and Hipparchus, *inter alia.* The universe is centered around a stationary, spherical earth with the celestial bodies moving from east to west in circles, just as Aristotle postulated, and making one revolution per day. In the observational data of how the seemingly irregular movements occur, Ptolemy employs trigonometrical calculations that he uses throughout his work. While bodies appear to have nonuniform and irregular motion, the reality, according to Ptolemy’s account, is truly uniform, perfect motion, albeit not simple. A body travels in a circle around the earth, but the earth is eccentric to the center of the orbit. This motion accounts for some bodies, such as the sun and moon, appearing closer to the earth seasonally and more distant in other seasons. On the other hand, planets wobble and, at times, appear to slip back in a retrogressive movement; at other times, they speed up, moving almost in a line. Building on Hipparchus and other previous Greek astronomers (many unknown), Ptolemy explained how planets follow in a circle around their own orbit, an epicycle, thus creating the illusion of a retrograde motion to a stationary observer on earth. The relative position of the earth to a planet would vary according to whether the planet was in its epicycle farther from the earth or closer to the earth, this variation known as a deferent.

**Planetary Orbits.** Essentially this model explained mathematically many planetary orbits. The explanation was not only rational but also predictive, an essential characteristic of science. The eccentric and epicycle-on-deferent model could not predict all planets’ orbits, thus another model was required. Some planets move around their orbits in ways that appear to be at varying speeds. Ptolemy related that these planets move to cover equal angles in equal times as measured from an equidistant point that is centered in neither the orbit’s center nor the location of the earth.

**Enduring Models.** Ptolemy’s three models (eccentric circle, epicycle-on-deferent, and equant) served as the explanation for celestial movements until Copernicus’s reversion to the heliocentric hypothesis in 1543 and Kepler’s three laws for planetary motion published in 1609. The perfection of the universe was preserved: a perfect circular motion as is required by celestial bodies composed of the fifth element, ether. Thus, the hypothesis for cosmic behavior as known to the Middle Ages was a combination of Empedocles’ elements (earth, air, fire, and water), Aristotle’s circular motion (celestial bodies composed of ether with rectilinear motion on earth of matter composed of the first four elements), and Ptolemy’s mathematical explanation for celestial motion. In the remaining books Ptolemy explained solar and lunar theory, eclipses, parallax, a catalogue of fixed stars visible at Alexandria, planetary latitudes, and a theory of planets in longitude. His genius was in the organization, description, and application of trigonometrical methods for proofs. Another work, on geography, was ascribed to Ptolemy as well.

**Late Roman Mechanics and Mathematics.** Greeks living within the Roman Empire continued the study of mathematics, again through handbook and commentary formats. Pappus of Alexandria (flourishing 320 c.e.) asserted that mechanics was a part of mathematics. Also, in separate works he wrote commentaries on Ptolemy’s *Almagest* and Apollonius’s *Conics.* Even the eighteenth-century English physicist Isaac Newton was appreciable of Pappus’s ingenuity in relating mathematics of his predecessors and making his own contributions, especially concerning loci on planes and conics.

**Commentaries on Ptolemy.** Ptolemy’s works were not known in Latin, and it was through commentaries on his works that his contributions were kept alive in the West for a long period. These commentators were themselves talented and, occasionally, original. Theon of Alexandria (flourishing 364 c.e.) wrote an extensive commentary on Ptolemy’s *Almagest* and two commentaries on his *Handy Tables* that provided a means to compute celestial positions. Along with his daughter, Hypatia, he was among the last of the researchers and teachers at the Museum in Alexandria. The only reference to his dates place him as living in the reign of Theodorius I (379-395 c.e.). As an explanation of Ptolemy, Theon’s commentary is unoriginal, being a simplistic explanation to students on an elementary level. The *Almagest* commentary is in thirteen books, corresponding to the Ptolemy work, but book eleven and part of book five are lost. Also, he wrote an edition of Euclid’s *Elements,* where he explained, altered, and, infrequently, corrected the Euclid text in such a way that the edited text enjoyed more popularity for a period than Euclid’s original. Theon’s mathematics were competent, although unoriginal, but they were well explained in a pedagogical way, thereby making his work on Ptolemy and Euclid popular for Byzantine science and, through Arabic, for Islamic science.

## Sources

Bernard R. Goldstein, *Theory and Observation in Ancient and Medieval Astronomy* (London: Variorum, 1985).

O. Neugebauer, *A History of Ancient Mathematical Astronomy,* 3 volumes (Berlin & New York: Springer-Verlag, 1975).

Liba Chaia Taub, *Ptolemy’s Universe: The Natural Philosophical and Ethical Foundations of Ptolemy’s Astronomy* (Chicago: Open Court, 1993).

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