For over a thousand years--from the fifth century B.C. to the fifth century A.D.--Greek mathematicians maintained a splendid tradition of work in the exact sciences: mathematics, astronomy, and related fields. Though the early synthesis of Euclid and some of the supremely brilliant works of Archimedes were known in the medieval west, this tradition really survived elsewhere. In Byzantium, the capital of the Greek-speaking Eastern empire, the original Greek texts were copied and preserved. In the Islamic world, in locales that ranged from Spain to Persia, the texts were studied in Arabic translations and fundamental new work was done. The Vatican Library has one of the richest collections in the world of the products of this tradition, in all its languages and forms. Both the manuscripts that the Vatican collected and the work done on them in Rome proved vital to the recovery of ancient science--which, in turn, laid the foundation for the Scientific Revolution of the 16th and 17th centuries. In the Roman Renaissance, science and humanistic scholarship were not only not enemies; they were natural allies.

Although Archimedes was a mathematician, he is known for his strategic role in ancient war and the development of military techniques. First the Carthaginians, then the Romans besieged Syracuse where Archimedes was born. While in the end Rome won and killed him, Archimedes put up a good defense. First he invented an engine that threw stones at the enemy, then he used glass to set the Roman ships on fire.

Archimedes probably studied mathematics in Alexandria with the successors of Euclid. The name Archimedes is connected to a pumping device now known as a Archimedes Screw, which he may have seen in operation in Egypt.

Euclid studied at Plato's Academy in Athens, then later moved to Alexandria where he taught mathematics.

He wrote Elements, a text book containing teachings on algebra, number theory, and especially geometry. Euclid proved his concepts logically, using definitions, axioms, and postulates. Proclus Diadochus wrote a commentary on his Elements that kept Euclid's works in circulation.

This is the oldest and best manuscript of a collection of early Greek astronomical works, mostly elementary, by Autolycus, Euclid, Aristarchus, Hypsicles, and Theodosius, as well as mathematical works. The most interesting, really curious, of these is Aristarchus's "On the Distances and Sizes of the Sun and Moon," in which he shows that the sun is between 18 and 20 times the distance of the moon. Shown here is Proposition 13, with many scholia, concerned with the ratio to the diameters of the moon and sun of the line subtending the arc dividing the light and dark portions of the moon in a lunar eclipse.

Apollonius's "Conics," written about 200 B.C., on conic sections, the ellipse, parabola, and hyperbola, is the most complex and difficult single work of all Greek mathematics and was all but unknown in the west until the fifteenth century. This magnificent copy, probably the most elegant of all Greek mathematical manuscripts, was made in 1536 for Pope Paul III. The pages on display show the particularly elaborate figures illustrating Propositions 2-4 of Book III on the equality of areas of triangles and quadrilaterals formed by tangents and diameters of conics, and by tangents and lines parallel to the tangents.