Pythagoras and the Pythagoreans - Department of Mathematics

Pythagoras and the Pythagoreans 29
A
C
D
Lune
We wish to determine the area of the lune ABCD, wherethelarge
segment ABD is similar to the smaller segment (with base on one leg
of the right isosceles triangle 4ABC). Because segments are to each
other as the squares upon their bases, we have the
Proposition:TheareaofthelargeluneABCD is the area of the triangle
4ABC.
This proposition was among the first that determined the area of a curvilinear
figure in terms of a rectilinear figure. Quadratures were obtained
for other lunes, as well. There resulted great hope and encouragement
that the circle could be squared. This was not to be.
7 The Pythagorean Theory of Proportion
Besides discovering the five regular solids, Pythagoras also discovered
the theory of proportion. Pythagoras had probably learned in Babylon
the three basic means, the arithmetic, thegeometric, andthesubcontrary
(later to be called the harmonic).
Beginning with a>b>cand denoting b as the —mean of a
and c, they are:
1
a − b
b − c
= a
a
B
arithmetic a + c =2b