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...mathematics is distinguished from all other sciences except only ethics, in standing in no need of ethics. Every other science, even logic, especially in its early stages, is in danger of evaporating into airy nothingness, degenerating, as the Germans say, into an arachnoid film, spun from the stuff that dreams are made of. There is no such danger for pure mathematics; for that is precisely what mathematics ought to be.
"The Essence of Mathematics" in J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
Loci: Convergence
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Archimedes' Method for Computing Areas and Volumes
Introduction
We owe many familiar area and volume formulas to Archimedes. For example, he proved that the area A of a circle equals half its circumference C multiplied by the radius r, i.e.
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This formula is often "explained" by the following picture, which shows that if you cut up the circle into an even number of equal sectors, the sectors can be reassembled into a figure that looks approximately like a parallelogram whose base is half the circumference of the circle and whose height is the radius.
Archimedes derived many formulas that are familiar to us today for computing relationships among volumes of spheres, cylinders, and paraboloids. How was he able to discover these formulas? About one hundred years ago, an old Greek manuscript by Archimedes was found which explained his Method, based on the Law of the Lever. The exciting story of its discovery and recent sale will appear on this site very soon.
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