Pythagoras and the Pythagoreans
29
B
A
Lune
C
D
We wish to determine the area of the lune
ABCD
, where the large
segment
ABD
is similar to the smaller segment (with base on one leg
of the right isosceles triangle
4
ABC
). Because segments are to each
other as the squares upon their bases, we have the
Proposition:
The area of the large lune
ABCD
is the area of the triangle
4
ABC
.
This proposition was among the first that determined the area of a curvi-
linear 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
, the
geometric
, and the
subcon-
trary
(later to be called the
harmonic
).
Beginning with
a > b > c
and denoting
b
as the
—mean
of
a
and
c
, they are:
1
a
−
b
b
−
c
=
a
a
arithmetic
a
+
c
= 2
b