5 - Quadrature

Summary

Three hundred fifty cities in the world

Just thirty teeth inside of our heads

These are the limits to our experience

It's scary, but it's all right

And everything is finite.

— from the album Feelings, David Byrne (1997)

Amid a flurry of discovery in the 1600s, European mathematicians began to recognize the underlying unity of their results. Quite often, a successful quadrature shed light on the mysteries inherent in series, or a clever use of geometry prompted advances in the theory of numbers. Each new connection fanned the intellectual fire. We see this effect in this chapter as we trace efforts to find the quadrature of the hyperbola.

Gregory studies hyperbolas

One way to define a hyperbola is as the collection of points whose horizontal and vertical components are inversely proportional, as in the modern notation xy = 1 or y = 1/x. Figure 5.1 depicts one of the two branches of a hyperbola. A successful quadrature of this curve would answer any question like, ‘What is the area labeled A in the figure?’

Gregory of Saint-Vincent (Belgium, born 1584) took the first step, finding an arithmetic sequence linked to a geometric sequence in Figure 5.1. A sequence of numbers is arithmetic if each member equals the previous member plus some common constant (described in more detail in exercise 2.1).