* These papyri are not older than Alexandria; but since they use the primitive digamma to represent 6, it is probable that the alphabetical notation antedated the Hellenistic age.

* Books I and II summarize the geometrical work of Pythagoras; Book III, Hippocrates of Chios; Book V, Eudoxus; Books IV, VI, XI, and XII, the later Pythagorean and Athenian geometricians. Books VII-X deal with higher mathematics.

* Cicero saw the apparatus two centuries later, and marveled at its complex synchronism. “When Gallus moved the globe,” he writes, “it was actually true that the moon was always as many revolutions behind the sun on the bronze contrivance as would agree with the number of days it was behind it in the sky. Thus the same eclipse of the sun happened on the globe as would happen in actuality.⁷

* Lucian is our earliest, and not quite reliable, authority for the story that Archimedes set Roman ships on fire by concentrating the sun’s rays upon them through the use of great concave mirrors.¹³

* Aristarchus estimated the volume of the sun as three hundred (it is over a million) times that of the earth—an estimate that seems low to us, but would have astonished Anaxagoras or Epicurus. He calculated the diameter of the moon as one third that of the earth—an error of eight per cent—and our distance from the sun as twenty (it is almost four hundred) times our distance from the moon. “When the sun is totally eclipsed,” reads one proposition, “the sun and the moon are then comprehended by one and the same cone, which has its vertex at our eye.”²⁹

* If it was not taken from his Babylonian predecessor Kidinnu.²²

† The equinoxes (lit., equal nights) are those two days of the year when the sun in its annual apparent motion crosses the equator northward (our vernal, Argentina’s autumnal, equinox), or southward (our autumnal equinox), making day and night equal for a day. The equinoctial points are those points in the sky where the equator of the celestial sphere intersects the ecliptic.

* A confluence of blood sinuses in the dura mater, or outer membrane of the brain.