# The Golden Section

## Projective Geometry Construction

*wrath**. noun. retributory punishment for an offense or a crime*

**:**divine chastisement. (Merriam-Webster)**mean**

**tone**(of voice) is a property of this temperament, but if for good purpose, then suppose that it is

*golden*. Hence, the

*golden meantone*is a temperament involving the divine proportion, also known as the

*golden mean*.

**"algebraic temperament"**Luke 2:19 in harmonic perspective. The red word is the longer and the blue word is the shorter, in the divine section, meaning that the proportion of the short to the long is the same as the long to the whole of the two together. In reality, the two words are in the same-sized boxes, but the nature of perspective reduces the appearance of things at length.

The formula for proportion, supposing the longer is a unit length of one pica, is given also as a quadratic:

X * X + X - 1 = 0,

where X is the shorter length.

The page in the photo is from *Biblia Sacra: Iuxta Vulgatam Versionem*, Deutsche Bibelgesellschaft.

The camera for the photo is a digital model with macro by Samsung.

Geometric calculations are performed using "Cinderella Version 2.9" software (http://cinderella.de) and are based on mathematical theory given in *Projective Geometry,* by T. Ewan Faulkner, Dover Publications.

(*) Lama Lodo Rimpoche. *Quintessence of the Animate and Inanimate: Discourse on the Holy Dharma. * page 114-115. KDK Publications: San Francisco.

# Neoplatonism

## Submodules of Z[x,y]

Let M = Z[1/2,1/5] (ring of polynomials evaluated at 1/2 and 1/5 with integer coefficients))

Then let K = Z[1/2] a submodule of M.

Does the following calculation hold for M/K?

Take 9/10 from M, then what is its congruence in M/K?

9/10 = 5/10 + 4/10 = 1/2 + 2/5 and since 1/2 is from K, then

9/10 is congruent to 2/5 + K in M/K?