The Golden Thread, Part 1
MEAN:
In ancient Greece, a proportion
meant the equality of two ratios, as in
A:B = C:D. A proportion of three terms
meant one in which the inner two terms
were the same, as in A:B = B:C.
In this context A and C are extremes, and
B is a mean.
In case the extremes are known and the mean is
to be found, we must solve, A:X = X:C.
In our modern notation, this says that X
is the square root of the product of A and C,
or X = SQR ROOT (A*C).
This latter is also called the geometric mean
of A and C.
DEMR: Division in Extreme and Mean Ratio
means: divide a line segment of length A
into a longer part X and a shorter part A-X
so that A:X = X:A-X. This results in a quadratic
equation which the Babylonians knew how to solve.
- Case 1: let A = 1. The X = little phi, 0.618...,
which satisfies phi^2 + phi = 1
- Case 2: let A - X = 1. Then X = big PHI, 1.618...
which satisfies PHI^2 = PHI + 1
In either case, the construction has been known
since Renaissance times as the Golden Section.
Ralph H. Abraham, 03 May 1996.