Antiphon, Eudoxus, and Archimedes. In his Exercise 5 Allen raises the issue of whether Archimedes deserves to be called the inventor of calculus. That accolade is presumably based on his calculation (really, definition) of the area of circles by means of approximating polygons -- effectively a sort of Riemann sum. Although Archimedes was the first to make this topic quantitative -- thereby dragging Greek geometrical thinking a big step in the modern direction -- Allen points out elsewhere that Eudoxus had earlier made extensive use of this "method of exhaustion" and would even have understood "volumes of solids of revolution by slicing".
It seems worthwhile to note (from Heath, Vol. I, pp. 221-223) an even earlier reference to the method. Antiphon, a contemporary of Socrates, discussed the squaring of the circle this way: Inscribe a square in the circle; on each side of the square, as base, put an isosceles triangle with its vertex on the nearby arc of the circle; on each side of this octagon put an isosceles triangle; etc. According to the commentary of Simplicius, "Antiphon thought that in this way the area [of the circle] would be used up.... And, as we can make a square equal to any polygon ... we shall be in a position to make a square equal to a circle." Aristotle mentioned Antiphon's argument only in order to denounce it as a well-known fallacy, and of course it is a fallacy within the Euclidean ruler-and-compass framework. But Antiphon was clearly ahead of his time! Heath says, "But the objection [that the whole area will never be used up] is really no more than verbal; Euclid [presumably following Eudoxus] uses exactly the same construction in XII.2, only he expresses the conclusion in a different way, saying that, if the process be continued far enough, the small segments left over will be together less than any assigned area. Antiphon in effect said the same thing, which again we express by saying that the circle is the limit of such an inscribed polygon when the number of its sides is indefinitely increased." How close Antiphon, Eudoxus, and Euclid came, not only to Archimedes, but to Leibniz, Cauchy, Riemann, and Weierstrass!
Stay tuned for some feedback on the first week's homework and maybe some policy statements about future homework after I consult with the grader.