and (b) the surface of the cylinder, including its bases, is 1½ times the surface of the sphere.
Next come a few definitions, followed by certain Assumptions, two of which are well known, namely:—
- Of all lines which have the same extremities the straight line is the least (this has been made the basis of an alternative definition of a straight line).
- Of unequal lines, unequal surfaces and unequal solids the greater exceeds the less by such a magnitude as, when (continually) added to itself, can be made to exceed any assigned magnitude among those which are comparable [with it and] with one another (i.e. are of the same kind). This is the Postulate of Archimedes.
He also assumes that, of pairs of lines (including broken lines) and pairs of surfaces, concave in the same direction and bounded by the same extremities, the outer is greater than the inner. These assumptions are fundamental to his investigation, which proceeds throughout by means of figures inscribed and circumscribed