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 work on the spire, a sort of anchor-ring surface described by Heron as being produced by the revolution of a circle around one of its chords as an axis. The sections of this surface yield peculiar curves called spiral sections, which, according to Geminus, were thought out by Perseus. These curves appear to be the same as the Hippopede of Eudoxus.

Probably somewhat later than Perseus lived Zenodorus. He wrote an interesting treatise on a new subject; namely, isoperimetrical figures. Fourteen propositions are preserved by Pappus and Theon. Here are a few of them: Of isoperimetrical, regular polygons, the one having the largest number of angles has the greatest area; the circle has a greater area than any regular polygon of equal periphery; of all isoperimetrical polygons of n sides, the regular is the greatest; of all solids having surfaces equal in area, the sphere has the greatest volume.

Hypsicles (between 200 and 100 B.C.) was supposed to be the author of both the fourteenth and fifteenth books of Euclid, but recent critics are of opinion that the fifteenth book was written by an author who lived several centuries after Christ. The fourteenth book contains seven elegant theorems on regular solids. A treatise of Hypsicles on Risings is of interest because it is the first Greek work giving the division of the circumference into 360 degrees after the fashion of the Babylonians.

Hipparchus of Nicæa in Bithynia was the greatest astronomer of antiquity. He established inductively the famous theory of epicycles and eccentrics. As might be expected, he was interested in mathematics, not per se, but only as an aid to astronomical inquiry. No mathematical writings of his are extant, but Theon of Alexandria informs us that Hipparchus originated the science of trigonometry, and that he calculated a "table of chords" in twelve books. Such calculations