Of Alexandria

Crucial to the further development of Greek mathematics and science was Alexander the Great's founding of Alexandria in 332 BC. Alexander stayed only for a few months in the city, after which he never saw it again. The city's development was undertaken by Alexander's viceroy who also arranged that Alexander's body be entombed there.

Situated on the Mediterranean, 12 miles west of the Canopic mouth of the Nile, the city was laid out in the shape of a T, the top of which was a former island made into a peninsular and the trunk of which formed a bridge between the peninsular and another stretch of land that separated the sea from a lake. The former island was known as Pharos, and the city's two large harbors made it the most important port on the Mediterranean. On Pharos stood the largest and most famous lighthouse of antiquity. (Whence the term for lighthouse or beam of light in certain languages.)74 However, Alexandria's most important feature was its library system, a center for Greek learning that would quickly eclipse Athens. Alexandria's intellectual hegemony lasted well into the Roman era, during which it was the most important Western city after Rome. Whereas Athens became utterly eclipsed by the Rome, Alexandria thrived as a fountainhead of Greek civilization. Its beacon would shine for over six hundred years. 45

Alexandria's first king, Ptolemy Soter, started the city's book collections. His successor, Ptolemy Philadelphia, an avid bibliophile, is thought by some to have acquired what remained of Aristotle's vast collection. Under Philadelphus, two libraries were established in separate buildings. The larger was located in the Bruchium quarter, which with the Museum formed an academy. The smaller was in another quarter called the Serapeum. Philadephus sought out valuable works in every part of the known world and copied them at great expense. The successor, Ptolemy Euergetes, was a ruthless sort who systematically seized all books brought into Egypt, returning only copies of originals to the unfortunate owners. According to the mathematician Erastothanes, who was the head librarian at one point, the Searpeum housed 42,800 papyrus rolls, and there were 490,000 rolls in the Bruchium.75 Of Alexandria's first five librarians, Zenodotus, Caliimachus, Erastophanes, Appolonius and Aristophanes; Erastophanes and Appolonius were preeminent mathematicians of the age.

The libraries were gradually despoiled as a result of a number of major fires. The first occurred in 46 BC, when Julius Caesar torched the Egyptian fleet in the harbor. The flames swept through the Bruchium, destroying the larger library and other facilities. In the mean time, a rival library had been founded at Pergamon, on the Turkish coast, which flourished despite the monopoly that the Egyptian kings held over the production and distribution of papyrus used in manuscript production. To placate Cleopatra, a few years later, Mark Anthony seized all 200,000 volumes from Pergamon and put them in a new building in the Bruchium. In 273, the main library was again destroyed by fire, on orders from the Emperor Aurelian, after which the Serapeum became the principal library. This, too, was set on fire on an edict of Theodosius in 398 or 390, after which, its remains were pillaged by the Christians. The Serapeum's final destruction is believed to have occurred about 640, when the city fell, following a 14 month siege to the Arab general Amu. On the capture of the great city, Amu reported to the Caliph Omar that the city contained 4000 palaces, 4000 baths and 400 theaters or places of amusement. Amu begged the Caliph to be awarded the Royal Library as a prize. The Caliph replied that if library's books contained Koranic teachings, they were superfluous, and if they did not, they violated the Koran and should be destroyed. This pronouncement resulted in the distribution of all the books to Alexandria's 4,000 public baths, where for the next six months they served to fuel the fires that warmed the waters.76

Of Euclid

Euclid's residence in Alexandria is reported by Pappus (320 AD), who observes that Apollonius spent many years studying with Euclid's pupils in that city. Writing more than a century later, Proclus, after a brief synopsis of the Platonic school, which included Hippocrates, Eudoxus, Menaechmus and Theaetetus; says of Euclid:

"Those who compiled histories carry the development of this science up to this point. Not much younger than these is Euclid, who put together the Elements, arranging in order many of Eudoxus' theorems, perfecting many of Theaetetus', and also bringing to irrefutable demonstration the things which had been only loosely proved by his predecessors. This man lived in the time of the first Ptolemy; for Archimedes, who came immediately after the first Ptolemy, makes mention of Euclid; and further they say that Ptolemy once asked him if there was in geometry a way shorter than that of the elements; he replied that there was no royal road to geometry. He is therefore younger that the pupils of Plato, but older than Eratosthemnes and Archimedes. For these men were contemporaries, as Eratosthenes somewhere says. In his aim he was a Platonist, being in sympathy with this philosophy, whence it comes that he made the end of the whole Elements the construction of the so-called Platonic figures. There are many other mathematical writings by this man, wonderful in their accuracy and replete with scientific investigations. Such are the Optics and Catoptrics, and the Elements of Music, and again the book on Divisions. He deserves admiration pre-eminently in the compilation of his Elements of Geometry on account of the order and of the selection both of the theorems and of the problems made with a view to the elements. For he included not everything which he could have said, but only such things a he could set down as elements. And he used all the various forms of syllogisms , some getting their plausibility from the fist principles, some setting out from demonstrative proofs, all being irrefutable and accurate and in harmony with science."

There is an aesthetic, minimalistic simplicity to Euclid. Starting with three simple statements - there is point, there is line, there is plane - the Elements advance step by step like a Jacob's ladder from the geometry of the triangle, to that of the circle, the parallelogram and the formation of areas. Propelled by the logic of ratios and the theory of irrational lines, Euclid arrives finally at the geometry of the sphere and the five Platonic solids that inhabit three-dimensional space. Sometimes plodding, at times repetitious, each wrung is anchored in an unshakable logic that rises from the elementary to the highly complex, at certain points taking astonishing turns and transformations that flip human perception inside-out and upside down.

One can argue that Euclid is a prime example of intellectual abstraction. In fact, many have contended that the Greeks were less concerned with resolving concrete problems than with the advancement of pure logic. This argument is often employed to explain Archimedes' works and attitude. Yet history attests that Archimedes addressed a great many practical problems involving warfare and weaponry.79 At the same time, while Euclidean mathematics and geometry may indeed epitomize pure abstraction, the last three books of the Elements lay the groundwork for modern physics. Even though the Greeks fashioned mathematics as an abstract construct, it would become a very potent tool for resolving pragmatic problems.

With Euclid, of course, there are no short-cuts. One must take the argument step by step, wrung by wrung, and climb the ladder slowly. Having reached the summit, one finally comprehends why people have scaled this structure for more than twenty centuries. One marvels at the way it affords such certainty in a uncertain universe where the gods are often fickle.

Permeating the structure, moreover, are two fundamental mathematical constructs, to which we have already been introduced. The combination of these would prove vital to the development of trigonometry - the point at which geometry becomes physics.

If the Pythagoreans had provided a general rule for the sides and hypotenuse of a triangle, it would be left to Euclid to present a final proof. The proof comes at the end of Book I, under the form of Preposition 47. Since Book I is comprised of 48 prepositions, the proof appears as a crowning moment in the gradual progression of the Elements of Book I. Later, in Book VI, the theorem is represented in a version of that has wider applications, through an argument that is entirely based on proportion. "If we listen," says Proclus, "to those who wish to recount ancient history, we may find some of them referring this theorem to Pythagoras...But for my part... I marvel more at the writer of the Elements, not only because he made it fast by a most lucid demonstration, but because he compelled assent to the still more general theorem by the irrefragable arguments of science in Book VI."

The general theorem in question is Proposition 31, Book VI, which states: "In right-angled triangles the figure on the side subtending the right angle is equal to the similar and similarly described figures on the sides of the right angle. As the figure shows, the construction for the proof is a fusion of the geometrical mean and the Pythagoras Theorem. (fig. 31)

There are over four hundred theorems in Euclid, and the Pythagoras Theorem is so pivotal that Euclid employs it explicitly thirty-eight times throughout The Elements. It is hard to estimate the number of times it is used implicitly.

The Golden Section reemerges, in a striking coda, in Book XIII, The Elements' final book. The book begins with a group of propositions concerning the Golden Section's properties. It then describes the inscription of the five Platonic solids in a sphere. It is in the last two of these figures, the icosahedron and the dodecahedron, that Golden Section comes into play. And it is in Proposition 8, Book XIII, that identifies its conspicuous and potent role in the pentagon: If in an equilateral and equiangular pentagon straight lines subtend two angles taken in order, they cut one another in extreme and mean ratio, and their greater segments are equal to the side of the pentagon.82 (fig. 32)

Here, with the construction of the Platonic solids within the perfect form of the sphere, Euclidean geometry culminates. The geometry is logical, pure, ethereal and abstract, and one may wonder what, if any, practical value it all has. However, four hundred and fifty years later, in 150 AD, Ptolemy, the last of the great Alexandrian mathematicians, supplied an answer when he mined Book XIII's construction of the pentagon and decagon to complete the first complete table of chords.

The very first theorem in Ptolemy's Almagest, antiquity's final word on astronomy, uses the Golden Section and the Theorem of Pythagoras repeatedly to determine the section of chord subtended by 36- and 72- degree arcs. Using these two values Ptolemy went on to assemble a table of chords from half a degree to 159 1/2 degrees. Not only was this the beginning of trigonometry, it marked the beginning of mathematical astronomy, since with this table of chords it was possible to fix precisely the longitude and latitude of celestial constellations. This development would make it possible to later navigate the oceans and eventually leave the planet.