The mechanical proof with the necessary preliminary propositions about the parabola (some of which are merely quoted, while two, evidently original, are proved, Props. 4, 5) extends down to Prop. 17; the geometrical proof with other auxiliary propositions completes the book (Props. 18-24). The mechanical proof recalls that of the _Method_ in some respects, but is more elaborate in that the elements of the area of the parabola to be measured are not straight lines but narrow strips. The figures inscribed and circumscribed to the segment are made up of such narrow strips and have a saw-like edge; all the elements are trapezia except two, which are triangles, one in each figure. Each trapezium (or triangle) is weighed where it is against another area hung at a fixed point of an assumed lever; thus the whole of the inscribed and circumscribed figures respectively are weighed against the sum of an indefinite number of areas all suspended from one point on the lever. The result is obtained by a real _integration_, confirmed as usual by a proof by the method of exhaustion. The geometrical proof proceeds thus. Drawing in the segment the inscribed triangle with the same base and height as the segment, Archimedes next inscribes triangles in precisely the same way in each of the segments left over, and proves that the sum of the two new triangles is 1/4 of the original inscribed triangle. Again, drawing triangles inscribed in the same way in the four segments left over, he proves that their sum is 1/4 of the sum of the preceding pair of triangles and therefore (1/4)^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 [Delta], is [Delta]{1 + 1/4 + (1/4)^2 + (1/4)^3 + . . . .} Archimedes proves geometrically in Prop. 23 that the sum of this infinite series is 4/3[Delta], 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:--