preceding pair of triangles and therefore (¼)² of the original inscribed triangle. Proceeding thus, we have a series of areas exhausting the parabolic segment. Their sum, if we denote the first inscribed triangle by Δ, is
Δ {1 + ¼ + (¼)² + (¼)³ + . . . .}
Archimedes proves geometrically in Prop. 23 that the sum of this infinite series is 4⁄3Δ, and then confirms by reductio ad absurdum the equality of the area of the parabolic segment to this area.
CHAPTER V.
THE SANDRECKONER.
The Sandreckoner deserves a place by itself. It is not mathematically very important; but it is an arithmetical curiosity which illustrates the versatility and genius of Archimedes, and it contains some precious details of the history of Greek astronomy which, coming from such a source and at first hand, possess unique authority. We will begin with the astronomical data. They are contained in the preface addressed to King Gelon of Syracuse, which begins as follows:—
“There are some, King Gelon, who think that the