Ideas

by Peter Watson

  The first librarian was Demetrius and by the time of the poet Callimachus, one of his better known successors, in the third century BC, the library comprised more than 400,000 mixed scrolls plus 90,000 single scrolls. Later, a daughter library, the Serapeion, housed in the temple of Serapis, a new Graeco-Egyptian cult, which may have been based on Hades, the Greek god of the dead, held another 40,000 scrolls. Callimachus installed the first subject catalogue in the world, the Pinakes, one effect of which was that by the fourth century AD, as many as one hundred scholars at a time came to the library to consult the books and discuss the texts with others. This distinguished community existed in all for some seven hundred years. The scholars wrote on papyrus, over which Alexandria had a monopoly for some time, and then on parchment when the king stopped exporting papyrus in an attempt to stifle rival libraries being built up elsewhere, notably at Pergamum.25 The papyrus and parchment books were written as scrolls (in length they were what we would mean by a chapter) and were stored in linen or leather jackets and kept in racks. By Roman times, not all the books were scrolls any more: the codex had been introduced, stored in wooden crates.26

  The library also boasted many charakitai, ‘scribblers’ as they were called, in effect translators. The kings of Alexandria–the Ptolemies–were very keen to acquire copies of all the books they did not possess, in their attempt to attain all the wisdom of Greece, Babylon, India and elsewhere. In particular, the agents for Ptolemy III Euergetes scoured the Mediterranean and he himself wrote to all the sovereigns of the known world, asking to borrow their books for copying. When he was lent works written by Euripides, Aeschylus and Sophocles from Athens, he held onto the originals, forfeiting his deposit, and returned the copies. In the same way all ships passing through the harbours of Alexandria were forced to deposit any books they were carrying at the library, where they were copied and catalogued as ‘from the ships’. For the most part the ships also had returned to them copies of the books that had been confiscated. This assiduous ‘collecting’ gave the Alexandrian library a pivotal role in the civilised world of antiquity.27

  Among the famous scholars who made their name at Alexandria were Euclid, who may have written his Elements during the reign of Ptolemy I (323–285 BC), Aristarchus, who proposed a heliocentric basis for the solar system, and Apollonius of Perga, ‘the great geometer’, who wrote his influential book on conic sections in the city. Apollonius of Rhodes was the author of the epic Argonautica (about 270 BC) and he introduced Archimedes of Syracuse, who spent time observing the rise and fall of the Nile, and inventing the screw for which he became famous. Archimedes also initiated hydrostatics and began his method of calculating area and volume that, 1,800 years later, would form the basis of the calculus.

  A later librarian, Eratosthenes (c. 276–196), was a geographer as well as a mathematician. A great friend of Archimedes, he believed that all the earth’s oceans were connected, that Africa might one day be circumnavigated and that India ‘could be reached by sailing westward from Spain’. It was Eratosthenes who calculated the correct duration of a year, who put forward the idea that the earth is round, and calculated its diameter to within an error of fifty miles. He did this by selecting two sites which were a known distance apart, Alexandria in the north and Syene (modern Aswan) in the south, which was assumed at that time to be exactly under the Tropic of Cancer, which meant that at the summer solstice the sun would be directly overhead and cast no shadow. At Alexandria on the same day, he used a skaph or bowl, a concave hemisphere, with a vertical rod or gnomon fixed at its centre. This cast a shadow which covered one-fiftieth of the surface of the bowl and so Eratosthenes calculated the circumference of the earth as 50 × 5,000 (= 250,000) stades (later amended to 252,000 stades, since it was more conveniently divisible by sixty). 250,000 stades was equal to 25,000 miles, not so far from the modern calculation of just under 26,000 miles.28 Eratosthenes also began the science of chronology, carefully establishing when the fall of Troy occurred (1184 BC), the first Olympiad (776 BC) and the outbreak of the Peloponnesian war (432 BC). He also initiated the calendar that Julius Caesar eventually installed and devised a method for identifying prime numbers. He was known among fellow scholars as ‘Beta’ (Plato was ‘Alpha’).29

  The Elements of Euclid is widely acknowledged as the most influential textbook of all time. Composed about 300 BC, some one thousand editions have been produced, making it perhaps the most republished book after the Bible (its contents are still taught in secondary schools today). Euclid (eu means ‘good’ and kleis–conveniently–means ‘key’) may well have studied at Plato’s Academy, if not with the great man in person (he was born in Athens around 330 BC), and although he produced no new ideas himself, Elements (Stoichia) is regarded as a history of Greek mathematics to that point.30 The book begins with a series of definitions: of a point (‘that which has no part’), a line (‘a length without breadth’), various angles and planes, followed by five postulates (‘a line can be drawn from any point to any other point’), and five axioms, such as ‘all things equal to the same thing are equal to one another’.31 The thirteen books, or chapters, that follow explore plane geometry, solid geometry, the theory of numbers, proportions, and his famous method of ‘exhaustion’.32 In this Euclid showed how to ‘exhaust’ the area of a circle by means of an inscribed polygon: ‘If we successfully double the number of sides in the polygon, we will eventually reduce the difference between the area of the polygon (known) and the area of the circle (unknown) to the point where it is smaller than any magnitude we choose’ (see Figure 8). One effect of Euclid’s work was that the Alexandrians, unlike the Athenians, treated mathematics as a subject wholly distinct from philosophy.33

  Figure 8: Euclid’s method of ‘exhaustion’ of a circle

  Apollonius of Perga was both a mathematician and an astronomer. Born at Perga in Pamphylia (southern Asia Minor), he studied at Pergamum, but flourished at Alexandria during the reign of Ptolemy Euergetes, dying in 200 BC. Several of his works have been lost but the Conics was the equal of Euclid’s Elements in that it survived throughout antiquity without being improved upon. A jealous man, he was known as ‘Epsilon’, because in the Mouseion he always used the room numbered 5 in the Greek alphabet. In the Conics Apollonius studied the ellipse, parabola and hyperbola–the plane figures generated when a circular cone is cut at acute, right and obtuse angles–and set out a new approach to their definition and description. Cones would become important in both optics and astronomy.34 In his astronomical works (which he sent to colleagues to critique before he released them generally), Apollonius built on the epicycles of Eudoxus of Cnidus to explain planetary motion. This system envisaged planets moving in small circles around a point, as the point moved in a larger circle around the earth. At this stage, before elliptical orbits were conceived, this was the only way mathematical theory could be made to fit observation.35

  The most interesting, as well as the most versatile of the Hellenistic mathematicians was Archimedes of Syracuse (c. 287–212 BC). He appears to have studied at Alexandria for quite a while, with the students of Euclid, and he was constantly in touch with the scholars there, though he lived mainly at Syracuse, where he died. During the second Punic war, Syracuse was caught up in the struggle between Rome and Carthage and, having sided with the latter power, the city was besieged by the Romans in 214–212 BC. During this war, we are told by Plutarch, in his life of the Roman general Marcellus, Archimedes invented a number of ingenious weapons to use against the enemy, including catapults and burning-mirrors to set fire to Roman ships. All to no avail, for the city eventually fell and, despite an order from Marcellus to spare Archimedes’ life, he was killed when a Roman soldier ran a sword through him while he was drawing a mathematical figure in the sand.

  He himself set little store by his innovations. He was more interested in ideas, and his range was remarkable. He wrote on levers, in On the Equilibrium of Planes, and on hydrostatics, in On Floating Bodies. This latter gave rise to
his famous lines: ‘Any solid lighter than a fluid will, if placed in a fluid, be so far immersed that the weight of the solid will be equal to the weight of the fluid displaced.’ And: ‘A solid heavier than a fluid will, if placed in it, descend to the bottom of the fluid, and the solid will, when weighed in the fluid, be lighter than its true weight by the weight of the fluid displaced.’36 He explored large numbers, a preoccupation that would lead centuries later to the invention of logarithms, and he achieved the most accurate rendering of pi.37

  The last of the great Hellenistic mathematicians at Alexandria was Claudius Ptolemy, who was active from AD 127 to 151. (The name Ptolemy here refers to the area of the city he came from; he was not related to the royal Ptolemies of Alexandria.) His great work was originally called Mathematical Syntaxis (System), thirteen books or chapters, but since this was often compared with other (lesser) collections by various authors, it became known as megiste, ‘the greatest’. Later, in the Muslim world, there was a custom of calling this book by the Arabic equivalent, Almagest, and it is by this name that Ptolemy’s work is usually known.38 The Almagest is primarily a work of trigonometry, that branch of mathematics associated with triangles, how the angles and lengths of the sides are related, and how they are all related to the circles which encompass them. In turn, these are related to the orbits of the heavenly bodies and the angles the planets present to the observer here on earth. Books 7 and 8 of the Almagest listed over one thousand stars, arranged according to forty-eight constellations.

  Towards the middle of the third century BC, Aristarchus of Samos had proposed putting the earth in motion about the sun. Most other astronomers, Ptolemy included, discounted this because they thought that if the earth moved by so much, the ‘fixed’ stars in the heavens should change their positions relative to one another. But they didn’t. Ptolemy, armed with his calculations of trigonometry–of chords and arcs (similar to sines)–went on to develop his system of planetary cycles and epicycles, known as the Ptolemaic system. This system envisaged a geocentric universe, with other bodies moving in a grand circle around a central point (the deferent), and in a smaller epicycle, as Eudoxus had imagined, all the while spinning on their axes.

  Ptolemy’s other great work was his Geography, in eight chapters. In Alexandria, geography had been put on the map, so to speak, by Strabo, who had written a history of the subject and of his travels, which showed for example that ‘Egypt’ had originally referred only to that strip of land or ‘bandage’ running along the Nile, but then extended further and further east and west, eventually taking in Cyprus. Strabo also noted the convexity of the sea.39 But Ptolemy was a more theoretical and innovative geographer. His Geography introduced the system of latitudes and longitudes as used today, and catalogued around eight thousand cities, rivers and other features of the earth. At the time there was in fact no satisfactory way to determine longitude and, as a result, Ptolemy seriously underestimated the size of the earth, opting for a circumference of 180,000 stadia given by Posidonius, a Stoic teacher of Pompey and Cicero, rather than the 252,000 stadia calculated by Eratosthenes and amended by Hipparchus. One of the major consequences of this error was that subsequent navigators and explorers assumed that a voyage westward to India would not be nearly so far as it was. Had Columbus not been misled in this way, he might never have risked the journey he did make. Ptolemy also developed the first projection of the earth–i.e., a representation of the globe on a flat surface.40

  Alexandria continued as the focus of Hellenistic mathematics: Menelaus of Alexandria, Heron of Alexandria, Diophantus of Alexandria, Pappus of Alexandria and Proclus of Alexandria all built on Euclid, Archimedes, Apollonius and Ptolemy. We should not forget that the great age of Greek maths and science lasted from the sixth century BC to the beginning of the sixth century AD, representing more than a millennium of great productivity. No other civilisation has produced so much over such a long period of time.41

  There was another–very important and very different–aspect to mathematics, or at least to numbers, in Alexandria. These were the so-called ‘Orphic mysteries’ with the emphasis on mysteries and mysticism. According to Marsilio Ficino, writing in the fifteenth century, there was a line of succession of the six great theologians in antiquity. Zoroaster was ‘the chief of the Magi’; the second was Hermes Trismegistus, the head of the Egyptian priesthood; Orpheus succeeded Trismegistus and was followed by Aglaophamus, who initiated Pythagoras into the secrets, who in turn confided in Plato. In Alexandria, Plato was built on by Clement and by Philo, to create what became known as Neoplatonism.

  Three ideas underlie the Orphic mysteries. One is the mystic power of number. The existence of numbers, their abstract quality and their behaviour, relating to so much in the universe, had an enduring fascination for the ancients, accounting as they did (so it was felt) for celestial harmony.42 The abstract nature of number also reinforced the idea of an abstract soul, which brought with it the further–all-important–idea of salvation, the belief that there was a future state of bliss, achieved by transmigration, or reincarnation. Finally, there was the principle of emanation–that there is an eternally-existent ‘good’, a unity or ‘monad’, from which all creation springs. Like number, this was felt to be an essentially abstract entity. The soul occupied a central position between the monad and the material world, between the totally abstract mind and the senses. According to the Orphics, the monad sent out (‘emanated’) projections of itself into the material world and it was the task of the soul, using the senses, to learn. In this way, via repeated reincarnations, the soul evolved to the point where further reincarnations were no longer needed. A series of ecstatic moments of deep insight resulted in a form of knowledge known as gnôsis, ‘in which the mind comes into a state of oneness with the thing perceived’. It can be seen that this idea, stemming originally from Zoroaster/Zarathustra, underlies many of the world’s major religions. It is another core belief, to add to the others considered in earlier chapters.

  Pythagoras believed in particular that the study of number and harmony could lead to gnosis. For Pythagoreans, one, 1, is not a true number but the ‘essence’ of number, out of which the number system emerges. Its division into two creates a triangle, a trinity, the most basic harmonic form, which would find echoes in so many religions. Plato, at his most mystical, believed that there was a ‘world soul’, also based on number and harmony, and out of which all creation arose. But he added the important refinement that the method to approach gnosis was by dialectic, the critical examination of opinions.43

  Traditionally, Christianity reached Alexandria in the middle of the first century AD when the evangelist St Mark arrived, to preach the new religion. The spiritual similarities between Platonism and Christianity had been most fully perceived by Clement of Alexandria (c. 150–c. 215) but it was Philo Judaeus who first worked out the new amalgamation. Pythagorean and Platonic schools of thought had existed in Alexandria for some time, with educated Jews well aware of the parallels between Jewish and Hellenistic ideas, so much so that many of them thought that Orphism was no more than ‘an unrecorded emanation of the Torah’. Philo was a typical Alexandrian who ‘never relied on the literal meaning of things, and looked for mystical and allegorical interpretation’. He thought that we can ‘connect’ with God through the divine ideas, that ideas were ‘the thoughts of God’ because they brought ‘unformed matter’ into order. Like Plato he had a dualistic notion of humanity: ‘Of the pure souls that inhabit ethereal space, those nearest earth are attracted by sensible beings and descend into their bodies.’ Souls are ‘the Godward side of man’. Salvation is achieved when the soul returns to God.44

  Philo’s ideas were built on by Ammonius Saccas (d. 242), who taught in Alexandria for more than fifty years. His pupils were both pagans and Christians and included some major thinkers, such as Plotinus, Longinus and Origen. For Ammonius, God was threefold: essence, intellect and power, the latter two being emanations of the essence (and in this way mirroring t
he behaviour of number). For Ammonius and other Neoplatonists the essence of God could not be known by intellect alone–this produced ‘only opinion and belief’. This was a major difference between the early Christians and the Greeks: for the Christians all that was needed, they said, was faith, belief. But this cut across the Greek tradition of reason. The Neoplatonists, like the Orphics before them, posited what was in effect a third form of knowledge, gnosis, which was experiential, and not wholly within the power of the intellect. Philosophy and theology helped one towards gnosis and the Christian idea–that only belief was needed–appeared to the Neoplatonists to be an undermining of spiritual evolution. Under Plotinus, who moved from Alexandria to Rome, gnosis–appreciation of the divine–could be achieved only by doing good, by experiencing good, and by use of the intellect in self-contemplation, self-awareness leading to the monad, or the One, or unity. This is not Christianity; but its mystical elements, its ideas about the Trinity, and the reasons for the Trinity (more difficult to grasp even than the Christian Trinity), and its use of the intellect and dialectic, did help to shape early Christian thought. The notion of biblical exegesis, the practice of asceticism, hermitism and monasticism are all founded in the Orphic mysteries, gnosis, and Neoplatonism.45 It is difficult for us to grasp (even to write about) and shows how different early Christianity was from the modern version.