way in the four segments left over, he proves that their sum is ¼ of the sum of the 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.