For centuries, the chief battle fought on geometry's terrain was between the straight and the curved. The former marched under the banner of the certainties offered by formulas known and used since time immemorial, such as those giving the areas of polygons. The latter countered with a host of difficult problems that went so far as to call the very definition of length into question.
Yet as early as antiquity, Archimedes had laid the foundations of a gentlemen's agreement between the two, notably in calculating π, the ratio of a circle's circumference to its diameter.
A circle's curvature prevented its circumference from being calculated directly, but this limitation could be overcome using increasingly accurate approximations by inscribed and circumscribed regular polygons.
Admittedly, the price of reducing curves to straight lines in this way was an appeal to infinity, which in turn raised new questions that would take several centuries to settle once and for all. The final throes of this battle had nevertheless subsided in the 19th century, when infinity was finally tamed through the work of Augustin Louis Cauchy, Bernard Bolzano, Richard Dedekind, and above all Georg Cantor.
