re-posted from reddit: news on Archimedes use of limits to find curved areas and volumes

In *The Method*, Archimedes was working out a way to compute the areas and volumes of objects with curved surfaces, which was also one of the problems that motivated Newton and Leibniz. Ancient mathematicians had long struggled to “square the circle” by calculating its exact area. That problem turned out to be impossible using only a straightedge and compass, the only tools the ancient Greeks allowed themselves. Nevertheless, Archimedes worked out ways of computing the areas of many other curved regions.

Such problems are tricky because solving them directly requires slicing up curved areas into infinitely many areas with straight boundaries. But the concept of infinity is a slippery and troublesome one that can quickly lead to paradox.

The Greek philosopher Aristotle built defenses against infinity’s vexing qualities by distinguishing between the “potential infinite” and the “actual infinite.” An infinitely long line would be actually infinite, whereas a line that could always be extended would be potentially infinite. Aristotle argued that the actual infinite didn’t exist.

Archimedes developed rigorous methods of dealing with infinity—still used today—in which he followed Aristotle’s injunction. For example, Archimedes proved that the area of a section of a parabola is four-thirds the area of the triangle inside it (shown in red in the diagram below). To do so, he built a straight-lined figure that’s an approximation of the curvy one. Then he showed that he could make the approximation as close as anyone could ever demand to both the section of the parabola and to four-thirds the area of the triangle.

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