As we mentioned in the Introduction to Chapter 9, the calculation of lengths of curved lines was one of the principal problems that led to the creation of the calculus. It was an old and intractable problem. Archimedes had used polygons inscribed in a circle to calculate π, but nothing further was discovered about curve lengths until the seventeenth century. In fact, even such a powerful mathematician as Descartes (1596–1650) had asserted that the length of no curve but the circle would ever be calculated. He was proven wrong, however, first by Torricelli in his work on the logarithmic spiral and then by the English architect, Christopher Wren, who established the length of the cycloid.
Unable to display preview. Download preview PDF.