## Characteristic ratios of powers of x
Before calculus, areas of regions were thought of as fractions of a rectangle or parallelogram containing the region. For example, the area of a triangle is half the area of the parallelogram determined by two of the sides of the triangle.
Summation formulas derived by Al Hazen allowed the calculation of areas under curves defined by integer power of x. Rather than making the subdivisions smaller, as in modern calculus, the region under the curve grew (from 0 to k in this picture.) As k gets large, the ratio of the area under the curve y=x^n to the area of the rectangle approaches 1/n.
Susan Addington, Created with GeoGebra |