Archimedes was right. II (Q919115)

[For Part I see the preceding review.] The author proves, with a lot of detail, that the second method of Archimedes for calculating the area of a parabolic section suggests ideas and working methods in modern mathematics. The method we mentioned above contributes, also, to understanding the ``inner working'' of summability of sequences, and of integration theory. By this method, Archimedes proved, implicitly, that we can speak of the sum of sequences of positive terms, \(a_ 1+a_ 2+...+a_ n+...\) if and only if there exists a number A such that \(a_ 1+a_ 2+...+a_ n\leq A\) for each \(n\in {\mathbb{N}}\), and in some particular cases, Archimedes was able to find this sum. At the same time, the second method suggests the modern method of integration of positive functions starting with the class of step- functions. The author's conclusions are interesting and they offer a guide for important applications, even at the high school level. Indeed, it's a pity that many of Archimedes' ideas are not sufficiently explored.