# Geometry notes

See also Analytic Geometry.

Commensurability and Symmetry

Now published as “On Commensurability and Symmetry,”
*Journal of Humanistic Mathematics,* Volume **7,**
Issue 2 (July 2017), pages
90–148. DOI: 10.5642/jhummath.201702.06

In 2006, I looked up the Greek origins of the word symmetry,

using mainly the works of Aristotle. The resulting two pages of notes,
Συμμετρία in Aristotle,

are on my Aristotle page.

Ten years later, I greatly expanded that work, to 54 pages called now
Commensurability and Symmetry

(A5 paper, 61 pages, 12 point type,
dated July 28, 2016; updated and expanded from the edition of April 12,
which had been updated from the edition of April 7 by the new footnote 1
now on page 9, created thanks to a tweet
of Oxford Psychology
retweeted by Nevit
Dilmen):

From Euclid to Descartes

The following notes should be revised in the light of

- the April 2016
revision of my May 2013 talk,
Descartes as Model Theorist,

- the course Geometries (
*Geometriler*), fall, 2015–16, for which I kept a detailed record.

As it is now, the article called From Euclid to Descartes

is 8
pages of size A5, dated January 23, 2013.

These are some notes about the transition from Euclid's geometry to
Descartes's geometry. Descartes uses the theory of proportion, as in
Book V of Euclid's *Elements,* in order to define the product of
two line segments as another line segment (once a particular segment has
been chosen as unit.) In fact this product can be defined on the basis
of Book I of the *Elements* alone.