Proofs Using a Quadrature Method of Archimedes

dissection diagram

curve y =

-z

x

z

1

interval [n,n+1]

n

1

subintervals

q

2

layer

t

2

move polygon

v

1

Archimedes divided a segment of a parabola into increasingly many diminishing triangles. Having deduced the area of each triangle and observing that the areas formed a geometric sequence, he found the area by summing the geometric series [1].

This Demonstration divides the region under the curve

y=

-z

x

into nonoverlapping polygons and identifies the area of each polygon from the coordinates of its vertices touching the curve. The infinite sequence of polygonal areas is then written in terms of the coordinates and the summation is expressed as a series.

Whereas the triangles formed a known geometric series for an unknown area, we observe various Dirichlet series for known areas. The algebraic expression of the dissection identifies and proves many infinite series for numbers definable as areas; examples are logarithm values at integers, the Euler–Mascheroni constant

γ

,

ln

4

π

, zeta function values, and other generalizations of some of these.