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After Alexander's death in 323 B.C., his general Ptolemaeus, better known as Ptolemy, took control of Egypt, ruling at Alexandria with the title of satrap. He declared himself king in 305 B.C., taking the name of Soter, or Savior, beginning a reign of more than twenty years and a dynasty, the House of Ptolemy, that would last for nearly three centuries.

Alexandria soon became a great cultural center under Ptolemy I Soter (r. 305-283 B.C.), who wrote a biography of Alexander the Great. The Alexandrian renaissance he began centered on the establishment of two renowned institutions, the Museum and the Library, which he founded and which were further developed by his son and successor Ptolemy II Philadelphus (r. 283-245 B.C.).

The Museum took its name from the fact that it was dedicated to the Muses, the nine daughters of Zeus and Mnemosyne, goddess of memory, who were the patron deities of the humanities. There were Temples of the Muses elsewhere in the Greek world, including a Museum in the Platonic Academy and one founded by Theophrastus in memory of Aristotle. The Alexandrian Museum and its attached Library were meant to be a university and research center, patterned on the famous schools of philosophy in Athens, most notably the Academy and the Lyceum.

The geographer Strabo, writing in the first quarter of the first century A.D., notes that the Museum was part of the royal palace complex of the Ptolemies: “The Museum is also a part of the royal palaces; it has a public walk, an exedra with seats, and a large house, in which is the common mess-hall of the men of learning who share the Museum. This group of men not only hold property in common, but also have a priest in charge of the Museum, who formerly was appointed by the kings, but is now appointed by Caesar.”

The Museum was more like a research institute than a college, with the emphasis on science rather than the humanities. It would have included an astronomical observatory as well as rooms for anatomical dissection and physiological experiments, and around it were botanical and zoological gardens.

The scientific character of the Museum was probably due to Straton of Lampsacus, the Physicist. Straton moved to Alexandria circa 300 B.C. to serve as tutor to the future Ptolemy II Philadelphus, remaining there until he returned to Athens in 288 B.C. to succeed Theophrastus as director of the Lyceum. The prince developed a deep interest in geography and zoology through his studies with Straton, and this was reflected in his development of the Museum when he succeeded his father as king in 283 B.C.

The organization of the Library was probably due to Dimitrios of Phaleron, the former governor of Athens, who had been forced to flee from the city in 307 B.C., after which he was given refuge in Alexandria by Ptolemy I. Dimitrios, a former student at the Lyceum in Athens, is believed to have been the first chief librarian of the Library, a post he held until 284 b. c. According to Aristeas Judaeus, a Jewish scholar in the reign of Ptolemy II Philadelphus, Dimitrios “had at his disposal a large budget in order to collect, if possible, all the books in the world, and by purchases and transcriptions he, to the best of his ability, carried the king's objective into execution.”

This policy continued through the reigns of Ptolemy II Philadelphus and Ptolemy III Euergetes (r. 247-221 B.C.). Athenaeus of Naucratis, who flourished circa A.D. 200, reports that Ptolemy II bought the books of Aristotle and Theophrastus and transferred them to “the beautiful city of Alexandria.” By the time of Ptolemy III the Library was reputed to have a collection of more than half a million parchment rolls, including all the great works in Greek science and the humanities from Homer onward. This led the third Ptolemy to build a new branch of the Library within the Serapeum, the temple of Serapis. Epiphanius of Salamis, a Christian of the fourth century A.D., refers to this addition in writing of “the first library and another built in the Serapeum, smaller than the first, which was called the daughter of the first.” Most classical authors do not refer to two libraries, but to “the Royal Library,” the “great library,” or “the libraries,” occasionally mentioning “the daughter library” in the Serapeum.

Dimitrios was succeeded as chief librarian by Zenodotus of Ephesus, who held the post until 245 B.C. His principal assistant was the poet Cal-limachus of Cyrene (ca. 305-ca. 240 B.C.), who classified the 120,000 works of prose and poetry in the library according to author and subject, the first time this had ever been done. His compilation, known as the Pinakes (Tables), bore the title Tables of Persons Eminent in Every Branch of Learning Together with a List of Their Writings, and filled more than 120 books, five times the length of Homer's Iliad

The only scientist to serve as chief librarian of the Library was Eratosthenes of Cyrene (ca. 275-ca. 195 B.C.), who was appointed by Ptolemy III in or near 235 B.C. and held the post until his death. Renowned as a mathematician, astronomer, and geographer, he also made a study of Old Attic comedy and was the first to write a chronology of Greek history and literature.

Eratosthenes was the first to draw a map of the known world based on a system of meridians of longitude and parallels of latitude, which allowed him to make an accurate measurement of the earth's circumference. This measurement was done by recording simultaneous observations made at Alexandria and Syene, a distance of 5,000 stades to the south. It was observed that at the summer solstice the sun was directly overhead at noon in Syene, while on a sundial at Alexandria it cast a shadow equal to one-fiftieth of a circle. Assuming that the sun was so far away that its rays were parallel at Syene and Alexandria, Eratosthenes concluded that the north-south distance between the two places was one-fiftieth of the earth's circumference. Thus the circumference of the earth was fifty times the distance between Syene and Alexandria, or 250,000 stades The precise value used by Eratosthenes for the length of a stade is unknown, so it is not possible to evaluate the accuracy of his result, but it was certainly of the right order of magnitude.

The great school of mathematics at Alexandria was founded by Euclid (fl. ca. 295 b.c.), who is said to have taught there early in the third century B.C., but very little else is known of his life. The philosopher Pro-clus, writing in the fifth century A.D., says that Euclid “lived in the time of the first Ptolemy” and that he was “younger than the students of Plato but older than Eratosthenes and Archimedes.”

Measurement of the earth's circumference by Eratosthenes.

Euclid is renowned for his Elements of Geometry, the earliest extant textbook on the subject, still in use today. A modern historian of science notes that the Elements “has exercised an influence upon the human mind greater than that of any other work except the Bible.”

The Elements established the foundations not only of plane geometry, but also of algebra and number theory. Proclus says that Euclid composed the Elements by “collecting many of the theorems of Eudoxus, perfecting many of those of Theaetetus [a pupil of Plato's], and also bringing to irrefutable demonstration the things which are somewhat loosely proved by his predecessors.”

Aside from the mathematical content of the Elements, one of its most important qualities is the logical form and order in which the theorems are presented, for this was to be the model for all future work in Greek mathematics and mathematical physics. Equally important is the axiomatic nature of the Elements, all of geometry following as a logical deduction from a few assumptions, themselves taken as necessarily true, which when applied to physics and astronomy represented the Platonic geometrization of nature.

Euclid's extant writings include several other works on mathematics as well as a textbook on astronomy, the Phaenomena, and an elementary treatise on perspective, the Optica The Optica was the first Greek work on the subject and the only one until Claudius Ptolemaeus wrote his treatise on optics in the mid-second century A.D. One of the assumptions made by Euclid in the Optica is that vision involves light rays proceeding in straight lines from the eye to the object. This erroneous view, known as the extramission theory, was accepted by most writers on optics until the seventeenth century.

Greek mathematical physics reached its peak with the works of Archimedes (ca. 287-212 B.C.), who was born at Syracuse in Sicily. Archimedes is said to have spent some time in Egypt, and he corresponded with Eratosthenes. It is probable that he studied in Alexandria under the successors of Euclid, for he was certainly familiar with the Elements and quoted from it extensively.

Archimedes was a relative and friend of Hieron II of Syracuse and of the king's son and successor Gelon II. He worked for both Hieron and Gelon as a military engineer, constructing devices such as catapults, burning mirrors, and a system of compound pulleys for moving large ships with minimum effort. These devices were used to great advantage by the Syracusans in resisting the siege of their city in 212 B.C. by the Roman general Marcellus. But Marcellus eventually captured the city and Archimedes was killed by a Roman soldier, supposedly while he was drawing a geometrical proposition in the sand.

Archimedes was famous for his inventions, one of which was an orrery, a working model of the celestial motions. Cicero actually saw this orrery, which he claimed showed the motion of the sun and moon and demonstrated both solar and lunar eclipses. Another of his inventions, known as Archimedes’ screw, is still used in Egypt to lift water in primitive irrigation systems. Plutarch remarks that Archimedes himself did not have high regard for his inventions, regarding them as being merely the “diversions of a geometry at play.”

Plutarch writes of Archimedes’ mathematical demonstrations that “it is not possible to find in geometry more difficult and troublesome questions or proofs set out in simpler and purer terms.” One can appreciate the nature of Archimedes’ works from the titles of some of his treatises, such as On the Measurement of a Circle, On the Sphere and the Cylinder, On the Equilibrium of Planes, On Floating Bodies, and The Sand Reckoner.

In the first of these treatises Archimedes rigorously determined the surface area of a circle using the so-called method of exhaustion, whereby he obtained successively better approximations by computing the areas of regular polygons inside and outside the circle. He used this method in his other mathematical works to measure the areas and volumes of various figures, such as in his treatise On the Sphere and the Cylinder There, considering a cylinder circumscribing a sphere, he found that the ratio of their areas was 3:2, and he was so proud of this discovery that he had the figure carved on his tomb.

The treatise On the Equilibrium of Planes deals with statics, the study of mechanical systems in equilibrium. There Archimedes uses the law of the lever to find the so-called center of gravity of various figures—that is, the point at which all of their weight is in effect concentrated. The problems are idealized, neglecting friction and other extraneous factors, and the treatment is completely deductive and geometrical, patterned on Euclid's Elements Archimedes’ work on the lever gave rise to his legendary boast to King Hieron: “Give me a place to stand and I shall move the earth.”

The treatise On Floating Bodies applies the same kind of geometric analysis to hydrostatics, the study of fluids in equilibrium. The basic proposition that he uses here is the famous Archimedes’ principle, which says that a body wholly or partially immersed in a fluid is buoyed up by a force equal to the weight of fluid displaced. The first-century Roman writer Vitruvius tells the story that Archimedes discovered this principle when he entered a bathtub and noticed the increasing sense of buoyancy as he immersed himself and the water level rose. According to Vitruvius, Archimedes “without a moment's delay, and transported with joy… jumped out of the tub and rushed home naked, crying in a loud voice that he had found what he was seeking; for as he ran he shouted repeatedly in Greek, ‘Heureka!’ [I have found it].”

Vitruvius goes on to tell how Archimedes used his principle to solve a practical problem: determining whether a golden crown made for King Hieron had been adulterated with another metal. He weighed the crown in water and found that it displaced a greater volume of water than the same weight of pure gold. This showed that the crown was less dense than pure gold, and thus that it had been made with an admixture of a lighter metal. Archimedes had discovered the concept of specific gravity, the weight of a body relative to that of an equivalent volume of water.

The Sand Reckoner is dedicated to King Gelon, to whom Archimedes explains a method that he had developed for expressing extremely large numbers. This was virtually impossible with the system then used by the Greeks, where numbers were written in terms of the letters of the alphabet. As an example, Archimedes gives the number of grains of sand in “a volume equal to that of the cosmos,” that is, “the sphere whose center is the center of the earth and whose radius is the distance between the center of the sun and the center of the earth.” He then makes reference to a new theory that had been proposed by Aristarchus of Samos, an older contemporary.

Aristarchus of Samos has, however, enunciated certain hypotheses in which it results from the premises that the universe is much greater than that just mentioned. As a matter of fact, he supposes that the fixed stars and the sun do not move, but that the earth revolves in the circumference of a circle about the sun, which lies in the middle of the orbit, and that the sphere of the fixed stars, situated about the same center as the sun, is so great that the circle in which the earth is supposed to revolve has the same ratio to the distance of the fixed stars as the center of the sphere to its surface.

The last sentence is of particular significance, for it explains why there is no stellar parallax, or apparent displacement of the stars, when the earth moves in orbit around the sun in the heliocentric theory of Aristarchus. It posits that even the nearest stars are so far away, compared to the radius of the earth's orbit around the sun, that their parallax is far too small to be detected by the naked eye. In fact, this effect was not observed until the mid-nineteenth century, by which time telescopes of sufficient resolving power had been developed.

Aristarchus of Samos (ca. 310-ca. 230 B.C.) was a student of Straton the Physicist's, probably at the Lyceum in Athens. The only work of Aristarchus's that has survived is his treatise On the Sizes and Distances of the Sun and the Moon Here the radii of the sun and moon relative to that of the earth and their distances in earth radii were calculated from geometrical considerations. The first observation was that the sun and the moon appear to be the same size, indicating that their diameters must be proportional to their distances from the earth. The second was a measurement of the lunar dichotomy (i.e., the angular separation of the sun and moon at half-moon), and the third was an estimation of the breadth of the earth's shadow where the moon passes through it at the time of a lunar eclipse. The results of these measurements led Aristarchus to conclude that the sun is about 19 times farther from the earth than the moon, and that the sun is approximately 6¾ times as large and the moon about ⅓ as large as the earth. All of his values are grossly underestimated, because of the crudeness of his observations, but his geometrical methods were sound.

Stellar parallax. E1 and E2 represent two positions of the earth six months apart in its orbit around the sun S The size of the earth's orbit is greatly exaggerated. The parallax star is one that is much closer than the distant stars, so that it will be displaced with respect to them as observed from the earth around the two positions shown. The farther away the nearby star is, the smaller will be its angle of parallax.

The only ancient astronomer known to have accepted the heliocentric theory of Aristarchus was Seleucus the Babylonian, who lived in the second century B.C. One of the reasons for the lack of acceptance was that the theory conflicted with general religious belief, which had the earth as the stationary center of the cosmos. Cleanthes of Assos, who flourished in the mid-third century B.C., wrote a tract condemning the theory, in which he said that Aristarchus not only had the earth in orbit around the sun but also had it rotating on its axis. In this work, which is quoted by Plutarch, Cleanthes remarks that Aristarchus should be charged with impiety “on the ground that he was disturbing the hearth of the universe because he sought to save [the] phenomena by supposing that the heaven is at rest while the earth is revolving along the ecliptic and at the same time is rotating about its own axis.”

From Aristarchus's On the Sizes and Distances of the Sun and the Moon. Above: The lunar dichotomy. Below: A lunar eclipse diagram.

The only Hellenistic mathematician comparable to Archimedes is his younger contemporary Apollonius of Perge. Apollonius was born circa 262 B.C. in Perge, on the Mediterranean coast of Asia Minor, and in his youth he was sent to study in Alexandria, where he flourished during the reigns of Ptolemy III and Ptolemy IV Philopator (r. 221-203 b.c.). He was also an honored guest in the court of Attalos I (r. 241-197 B.C.) of Pergamum in northwestern Asia Minor, which had become a center of Greek culture, renowned for its library.

The only major work of Apollonius's that has survived is his treatise On Conics, though even there the last book is lost. This was the first comprehensive and systematic analysis of the three types of conic sections: the ellipse (of which the circle is a special case), the parabola, and the hyperbola.

Apollonius is also credited with formulating mathematical theories to explain the apparent retrograde motion of the planets. One of the theories has a planet moving around the circumference of a circle, known as the epicycle, whose center itself moves around the circumference of another circle, called the deferent, centered at the earth. The second theory has a planet moving around the circumference of an eccentric circle whose center does not coincide with the earth. He also showed that the epicycle and eccentric circle theories are equivalent, so that either model can be used to describe retrograde planetary motion.

The conic sections.

Aside from the great theoreticians of the Hellenistic era, there were also a number of gifted inventors whose works were extremely influential in the development of applied science.

Ctesibius of Alexandria, the son of a barber, who is believed to have flourished ca. 270 B.C., was renowned as an inventor of toys and devices involving pneumatics—air under pressure. He is credited with inventing a force pump, a war catapult, a fire engine, a water clock, a hydraulic organ, and a singing statue, which he made for the empress Arsinoë, the sister and wife of Ptolemy II. The most elaborate of his water clocks told the hours with a succession of figures known asparerga, such as moving puppets and whistling birds, the ancestor of the cuckoo clock. All of his written works are now lost, but his ideas and inventions were revived by his two most notable followers, Philo of Byzantium and Hero of Alexandria.

Philo of Byzantium flourished circa 250 B.C. His extant writings comprise three books from a large work on mechanics: On Catapults, On Pneumatics, and On Besieging and Defending Towns In the first of these books Philo states that he traveled to Alexandria, where people described to him the bronze-spring catapult built by Ctesibius. The second book describes a number of demonstrations almost certainly taken from Ctesibius, including several pneumatic toys. The third of the three books deals with the besieging of walled towns, including the use of catapults and other engines of war, as well as stratagems such as secret messages, cryptography, and poisons.

The epicycle theory of Apollonius; the model was used to explain the retrograde motion of a planet P.

Hero of Alexandria flourished circa A.D. 62. The longest of his extant works by far is his Pneumatica, described by one modern historian as containing “almost exclusively apparatuses for parlor magic.” One is his famous steam engine, in which a glass bulb is made to rotate by two jets of steam escaping from it in opposite directions at either end of a diameter. The first chapters, which derive largely from Philo, describe experiments to show that air is a body and that it is possible to produce a vacuum, contrary to the Aristotelian doctrine. One of the demonstrations described by Hero may derive directly from Straton the Physicist. “Thus,” he writes, “if a light vessel with a narrow mouth be taken and applied to the lips, and if the air be sucked out and discharged, the vessel will be suspended from the lips, the vacuum drawing the flesh towards it so that the evacuated vessel be filled. It is manifest from this that there was a continuous vacuum within the vessel.”

Other inventions are described in Hero's treatise On Automata-Making, most notably the thaumata, or “miracle-working” devices such as one that opened and closed the doors of a temple using air pressure. The most elaborate of the automata are two puppet shows, one of them showing Dionysus pouring out a libation in front of a temple while bacchants dance about him to the sound of trumpets and drums, another representing a naval battle in which Athena destroys the ships of Ajax with thunder and lightning. Hero also made important contributions in optics as well as applied mathematics.

Inventions of Hero's: Above: Temple doors opened by fire on an altar. Below: A steam engine.

Hipparchus, the greatest observational astronomer of antiquity, was born in Nicaea, in northwestern Asia Minor. His life span can be estimated from the dates of his earliest known observation, the autumnal equinox of 26-27 September 147 B.C., and the latest, a lunar position on 7 July 127 B.C. It is probable that he spent the latter part of his career in Rhodes, where he is known to have made observations from 141 to 127 B.C. What little is known of his life comes from the geographer Strabo, who says that Hipparchus made use of the Library at Alexandria, and from the astronomer Ptolemy, who refers frequently to his observations and often quotes him directly.

All of the writings of Hipparchus have been lost except for his first work, a commentary on the Phainomena of Aratus of Soli (ca. 310-ca. 240 B.C.), a Greek poem describing the constellations. This commentary served to popularize the names of stars and constellations, many of which have been perpetuated in the modern world. It contained a catalog of some 850 stars, for each of which Hipparchus gave the celestial coordinates, including those of a “nova,” or new star, which suddenly appeared in 134 B.C. in the constellation Scorpio. He also estimated the brightness of the stars, assigning to each of them a “magnitude,” which equaled 1 for the brightest stars and 6 for the faintest, a system still used in modern astronomy.

One of the lost writings of Hipparchus is a book on the sizes and distances of the sun and the moon, in which he apparently made a great improvement over Aristarchus. These and other measurements and theories of Hipparchus's were used by Ptolemy, who paid due credit to his predecessor.

Hipparchus is also renowned for his discovery of the precession of the equinoxes—that is, the slow movement of the celestial pole in a circle about the perpendicular to the ecliptic. The earth's precession manifests itself as a gradual advance of the spring equinox along the ecliptic, thus causing a progressive change in the celestial longitude of the stars. Hipparchus discovered this effect by comparing his star catalog with observations made 128 years earlier by the astronomer Timocharis, which led him to conclude that the celestial longitude of the star Spica in the constellation Virgo had changed by 2 degrees in that interval of time, amounting to an annual precession of 45.2 seconds of arc. This allowed him to make an accurate determination of the length of the so-called tropical year by measuring the time between two summer solstices, one observed by Aristarchus in 280 B.C. and another by himself in 135 B.C. The value that he found was equivalent to 365.2467 days, a significant improvement over the old value of 365.25 days, which did not take into account the precession of the equinoxes. The currently accepted value of the tropical year is 365.242190 days, which means that the measurement by Hipparchus was in error by somewhat less than 1 part in 10,000.

The precession of the equinoxes. Above: Precession caused by the forces of the sun and the moon on the earth's equatorial bulge. Below: The path of the north celestial pole in the celestial sphere. The celestial pole describes a radius of 23.5 degrees around the ecliptic pole.

Hipparchus is also celebrated as a mathematician, his great achievement being the development of spherical trigonometry, which he applied to problems in astronomy.

Theodosius of Bithynia, a younger contemporary of Hipparchus's, is remembered for his Sphaerica, a treatise on the application of spherical geometry to astronomy, which was translated into Arabic and later into Latin, remaining in use until the seventeenth century.

When Ptolemy XII died, in 51 B.C., his eldest daughter, Cleopatra VII, succeeded to the throne, while her younger brother Ptolemy XIII became coruler. A civil war began between Cleopatra and her brother, both of whom were captured by Caesar in 48 B.C. after his victory over Pompey who was assassinated in Egypt. Caesar's small army was attacked by a much larger Egyptian force; during the fighting a fire broke out and destroyed many buildings in the harbor quarter of Alexandria, including at least part of the Library. Seven years later Mark Antony promised Cleopatra that he would make up the loss by giving her some 200,000 volumes from the library of Pergamum. In any event, it appears that the Library of Alexandria survived the fire, for there are several mentions of it in the imperial Roman era.

The Ptolemaic dynasty came to an end in 30 B.C. when Cleopatra committed suicide in Alexandria after she and Antony had been defeated by Octavian at the Battle of Actium. Alexandria then came under the rule of Rome, whose imperial era began when Octavian became Augustus in 27 B.C.

The lifetime of the Greek geographer Strabo (63 B.c.-ca. A.D. 25) extended from the end of the Ptolemaic period through the first half century of the Roman imperial era. He was born at Amaseia in the Pon-tus, and in his youth he studied first at Nysa in Asia Minor, then in Alexandria, and later in Rome. He lived for a long time in Alexandria, where he would likely have studied the works of Eratosthenes and other Greek geographers, whom he mentions. His earlier historical work is lost, but his more important seventeen-book Geography has survived.

Strabo followed the tradition of Eratosthenes in geography, but added encyclopedic descriptions “of things on land and sea, animals, plants, fruits, and everything else to be seen in various regions.” He said that the northern limit of the inhabited world was Ireland, which he called Ierne, “the home of men who are complete savages and lead a miserable existence because of the cold.” He went on to say that “its inhabitants are more savage than the Britons, since they are man-eaters as well as herb-eaters, and since, further, they count it an honorable thing, when their fathers die, to devour them, and openly to have intercourse, not only with the other women, but with their mothers and sisters.”

Another who came to study in Alexandria was Dioscorides Pedanius (fl. A.D. 50-70), from Anazarbus in southeastern Asia Minor, who later became a physician in the Roman army during the reigns of Claudius (r. A.D. 41-54) and Nero (r. A.D. 54-68). Dioscorides is regarded as the founder of pharmacology, renowned for his De Materia Medica, a systematic description of some six hundred medicinal plants and nearly a thousand drugs.

Nicomachus of Gerasa (fl. ca. A.D. 100) is noted for his Introduction to Arithmetic This was an elementary handbook on those parts of mathematics that were needed for an understanding of Pythagorean and Platonic philosophy. The book has many errors and other shortcomings, but it was influential until the sixteenth century, giving Nicomachus the undeserved reputation of being a great mathematician.

Ancient Greek astronomy culminated with the work of Claudius Ptolemaeus (ca. A.D. 100-ca. 170), known more simply as Ptolemy. All that is known of Ptolemy's life is that he flourished in Alexandria during the successive reigns of the emperors Hadrian (r.A.D.117-38) and Antoninus Pius (r. A.D. 138-61). The most influential of his writings is his Mathematical Synthesis, better known by its Arabic name, the Almagest, the most comprehensive work on astronomy that has survived from antiquity.

The topics in the Almagest are treated in logical order through thirteen books. Book I begins with a general discussion of astronomy, including Ptolemy's view that the earth is stationary “in the middle of the heavens.” The rest of Book I and all of Book II are devoted principally to the development of the spherical trigonometry necessary for the whole work. Book III deals with the motion of the sun and Book IV with lunar motion, which is continued at a more advanced level in Book V, along with solar and lunar parallax. Book VI is about eclipses; Books VII and VIII are about the fixed stars; and Books IX through XIII are on the planets.

Ptolemy's trigonometry and catalog of stars are based on the work of Hipparchus, and his theory of epicycles and eccentrics is derived from Apollonius. The principal modification made by Ptolemy is that the center of the epicycle moves uniformly with respect to a point called the equant, which is displaced from the center of the deferent circle, a device that was to be the subject of controversy in later times.

The extant writings of Ptolemy also include other treatises on astronomy, the Handy Tables, Planetary Hypotheses, Phases of the Fixed Stars, Analemma, and Planisphaerium; a work on astrology called the Tetrabiblos; and treatises entitled Optics, Geography,and Harmonica, the latter devoted to musical theory.

Ptolemy's Tetrabiblos is the classic Greek work on astrology, the pseu-doscience of astronomical divination based on the notion that celestial bodies influence human affairs. Ptolemy was himself skeptical about some of the credulous beliefs involved in astrology, as when he states in Book III of the Tetrabiblos that “we shall dismiss the superfluous nonsense of the many, that lacks any plausibility, in favor of the primary natural causes.”

Ptolemy's concept of the equant. In the diagram the planet (Pl.) moves on an epicycle whose center describes an eccentric circle, i.e., one whose center (Ctr.) is outside the earth. The center of the epicycle moves uniformly with respect to the equant (Equt.) The equant and the earth are equidistant from the center of the eccentric circle on opposite sides of a diameter.

A simplified version of Ptolemy's planetary model (ignoring eccentricity and equants and not to scale) to show the relationship between the planets and the sun in Ptolemy's system. The centers of the epicycles of Mercury (Me) and Venus (Ve) are on the line joining the earth E to the sun S For the outer planets—Mars (Ma), Jupiter (Ju) and Saturn (Sa)—the line joining the planet to the center of its epicycle remains parallel to ES.

Ptolemy's researches on light are presented in his Optics Here he gives the correct form for the law of reflection, already known to Euclid, that the incident and reflected rays make the same angle with the surface normal, the perpendicular to the mirror at the point of incidence. His experiments led him to an empirical relation for the law of refraction, the bending of light when it passes from one medium to another. He found that when light passes into a denser medium, as from air to water or glass, the refracted ray makes a smaller angle with the surface normal than does the incident ray. He then used these laws to find the location, size, and form of images produced by reflection and refraction.

Ptolemy's Geography is the most comprehensive work on that subject to survive from the ancient world. One defect of his work is that his value for the circumference of the earth is too small by a factor of one-third. The most conspicuous error in his map of theecumenos, or the inhabited world, is the extension of the Eurasian landmass over 180 degrees of longitude instead of 120 degrees. Nevertheless, Ptolemy's treatise was by far the best geographical work produced in antiquity.

Galen of Pergamum (A.D. 130-after 204), the most renowned medical writer of antiquity, was a younger contemporary of Ptolemy's. Galen was born in Pergamum and studied there as well as in Smyrna, Corinth, and Alexandria. He served his medical apprenticeship at the healing shrine of Asclepios at Pergamum, where his work treating wounded gladiators gave him an unrivaled knowledge of human anatomy, physiology, and neurology. In 161 he moved to Rome, where he spent most of the rest of his life, serving as physician to the emperors Marcus Aurelius (r. A.D. 161-80), Lucius Verus (coemperor; r. A.D. 161-69), and Commodus (r. A.D. 180-92).

Galen's writings, translated successively into Arabic and Latin, served as the basis for explaining the anatomy and physiology of the human body until the seventeenth century, earning him the title of “Prince of Physicians.” The title of one of his treatises is That the Best Doctor Is Also a Philosopher His philosophical bent is evident in his medical writings, where he interprets the work of Plato, Aristotle, Epicurus, and others, and also in his treatises On Scientific Proof and Introduction to Logic He wrote on psychology as well, including an imaginative analysis of dreams, seventeen centuries before Freud. One of the psychic complaints recognized by Galen was lovesickness, which he believed to be the principal cause of insomnia, and he notes that “the quickening of the pulse at the name of the beloved gives the clue.”

Ptolemy's experimental investigation of refraction, where α is the angle of incidence and β the angle of refraction.

The last great mathematician of antiquity was Diophantus of Alexandria (fl. ca. A.D. 250), who did for algebra and number theory what Euclid had done for geometry. Little is known of his life beyond the bare facts given in an algebraic riddle about Diophantus in the Greek Anthology, dating from the fifth or sixth century: “God granted him to be a boy for the sixth part of his life, and adding a twelfth part to this He clothed his cheeks with down; He lit him the light of wedlock after a seventh part, and five years after his marriage He granted him a son. Alas! Late-born wretched child; after attaining the measure of half his father's life, chill Fate took him. After consoling his grief by this science of numbers for four years he ended his life.”

Assuming that the biographical facts in this riddle are correct, Diophantus lived to be eighty-four years old. His most important treatise is the Arithmetica, of which six of the original thirteen books have survived. This work was translated into Latin in 1621; six years later it inspired the French mathematician Pierre Fermat to create the modern theory of numbers. After reading Diophantus's solution of the problem of dividing a square into the sum of two squares, which is the Pythagorean theorem, Fermat made a notation in the margin of his copy of the Arithmetica, stating, “It is impossible to divide a cube into two other cubes, or a fourth power or, in general, any number which is a power greater than the second into two powers of the same denomination.” He then noted that he had discovered a remarkable proof “which this margin is too narrow to contain.” He never supplied the proof of what came to be called “Fermat's last theorem,” which was finally solved in 1995 by Andrew Wiles, a British mathematician working at Princeton University, the last link in a long chain of mathematical development that began in ancient Alexandria. The work of Diophantus is still a part of modern mathematics, studied under the heading of “Diophantine analysis.”

Pappus of Alexandria, who flourished in the first half of the fourth century A.D., is renowned for his work in mathematics, astronomy, music, and geography. His treatise entitled Synagogue (Collection) is the principal source of knowledge of the accomplishments of many of his predecessors in the Hellenistic era, most notably Euclid, Archimedes, Apollonius, and Ptolemy. His work in mathematics influenced both Descartes and Newton, and one of his discoveries, known today as the theorem of Pappus, is still taught in elementary calculus courses. In one of his works in mathematics he reflects on mathematics in nature, remarking that “Bees … by virtue of a certain geometrical forethought … know that the hexagon is greater than the square and the triangle and will hold more honey for the same expenditure of material.”

An oath signed by “Pappus, philosopher” in a collection of works on alchemy has been attributed to Pappus of Alexandria. If so, it may shed light on his religious views, which would seem to be a mixture of Christian, Pythagorean, alchemical, and astrological beliefs. It reads: “I therefore swear to thee, whoever thou art, the great oath, I declare God to be one in form, but not in number, the maker of heaven and earth, as well as the tetrad of the elements and things formed from them, who has furthermore harmonized our rational and intellectual souls with our bodies, who is borne upon the chariots of the cherubim and hymned by angelic throngs.”

During the Hellenistic period the pseudosciences of alchemy and astrology developed strong connections with magic, a mystical influence that would be passed on to Islam and medieval Europe. Much of the resulting lore is found in the so-called Corpus Hermeticum, a collection of writings on alchemy, astrology, and magic that takes its name from the legendary Hermes Trismegestus (Thrice Greatest), a syn-cretization of the Greek god Hermes and the Egyptian divinity Thoth. Most of these writings, once thought to be of ancient Egyptian origin, are now dated to about the second century A.D. The earliest full description of the Corpus Hermeticum is in the Stromata of Clement of Alexandria (ca. 150-ca. 220), who says that of the forty-two books four are astrological and the remainder religious, philosophical, and medical, all of them with a touch of the occult, as in one fragment that states that “philosophy and magic nourish the soul.”

The library in the Serapeum survived almost to the end of the fourth century, by which time the Museum seems to have vanished. The emperor Theodosius I issued a decree in 391 calling for the destruction of all pagan temples throughout the empire. Theophilus, bishop of Alexandria, took this opportunity to lead his fanatical followers in demolishing the Serapeum, including its library.

The last contemporary description of the Library is by Aelius Festus Aphthonius, who sometime after 391 paid tribute to its role in making Alexandria the center of Greek culture: “On the inner side of the colonnade were built rooms, some of which served as book stores and were open to those who devoted their life to the cause of learning. It was these study rooms that exalted the city to be the first in philosophy. Some other rooms were set up for the worship of the old gods.”

The last scholar known to have worked in the Museum and Library was Theon of Alexandria, who in the second half of the fourth century wrote commentaries on Euclid's Elements and Optica, as well as on Ptolemy's Almagest and Handy Tables. A passage in Theon's commentary on the Handy Tables led to an interesting development in Arabic astronomy. This is where Theon states that “certain ancient astrologers” believed that the points of the spring and fall equinox oscillate back and forth along the ecliptic, moving through an angle of 8 degrees over a period of 640 years. This erroneous notion was revived in the “trepidation theory” of Arab astronomers, and it survived in various forms into the sixteenth century, when it was discussed by Copernicus.

Theon was the father of Hypatia (ca. 370-415), the first woman to appear in the history of science. Hypatia was a professor of philosophy and mathematics, and circa 400 she became head of the Platonic school in Alexandria. She revised the third book of Theon's commentary on Ptolemy's Almagest, and she also wrote commentaries on the works of Apollonius and Diophantus, now lost. Her lectures on pagan philosophy aroused the anger of Saint Cyril, bishop of Alexandria, who in 415 instigated a riot by fanatical Christians in which Hypatia was killed.

The last pagan to head the Platonic school in Alexandria was Ammo-nius, who directed it from 485 until his death circa 517-26. A distinguished philosopher, astronomer, and mathematician, he was noted for his commentaries on Aristotle. He managed to remain on good terms with the Christian authorities in Alexandria, though he and several of his faculty were pagans. One of his two most famous students, the philosopher John Philoponus, who succeeded him as head of the Platonic school, was a Christian, probably from birth. His other famous student, Simplicius, renowned for his commentaries on Aristotle, seems to have remained a pagan.

Eutocius of Ascalon (born ca. 480) was also a student of Ammo-nius's, to whom he dedicated his commentary on the first book of Archimedes’ On the Sphere and the Cylinder Eutocius later wrote commentaries on two more works of Archimedes’—On the Measurement of a Circle and On the Equilibrium of Planes—as well as on the first four books of the Conics of Apollonius. His commentaries proved to be crucial in the survival of these works.

Ammonius was the last pagan philosopher of Alexandria, for by his time Christianity had triumphed over the old gods who had been worshipped in the Serapeum and the Greek philosophers whose works had been studied in the Museum and the Library. As Tertullian had written two centuries earlier in attacking pagan philosophy, rejecting research in favor of revelation:

What then has Athens to do with Jerusalem, the Academy with the Church, the heretic with the Christian? Our instruction comes from the Porch of Solomon who himself taught us that the Lord is to be sought in the simplicity of one's heart…. We have no need of curiosity after Jesus Christ, nor of research after the gospel. When we believe, we desire to believe no more. For we believe this first, that there is nothing else that we believe.

The ruins of the ancient Library of Alexandria were unearthed by Italian archaeologists in the 1990s. By that time a project had already begun to re-create the ancient Library near its original site, sponsored by UNESCO. This led to the creation of a new library known as the Bib-liotheca Alexandrina, which opened on 16 August 2002. The collection of the new institution includes all of the extant works of those who, like Euclid and Archimedes, were associated with the Ptolemaic Library, some of them in the medieval Arabic and Latin translations in which they were transmitted to western Europe, strands in the Ariadne's thread that links the scientific thought of the ancient and modern worlds.

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