# Analitik Geometri // *Analytic Geometry*

“Analitik Geometri”

(20 Şubat 2020 taslağı, 33 sayfa, A5 boyu):

(13 Ocak 2020 taslağı, 15 sayfa, A5 boyu):

Bu notlarda Öklid’in *Öğeler*’inin birinci
kitabından Pappus, Desargues, ve Thales Teoremleri elde edilir.
Analitik geometri, bunlara dayanır.

## Analitik Geometri (MAT 104) dersi

## Ders ve konu hakkında /
*About the course and subject*

*Analitik Geometri* (Analytic Geometry, MAT 104) is taken in the second semester of the
first undergraduate year in our department. *Öklid Geometrisine
Giriş* (Introduction to Euclidean Geometry, MAT 113), will have been taken in the first
semester. The Euclidean geometry course is
literally that: students read Book I of Euclid’s *Elements* and
present the propositions. They are also asked to find their own proofs,
on the basis of that book, for results not found there.

While leading a section of the Euclid course for several years, I
researched the transition to analytic geometry. The Turkish notes above
use Book I of Euclid to prove Pappus’s Theorem, namely Lemma VIII
of the lemmas for Euclid’s *Porisms* in Book 7 of Pappus’s
*Collection.* I have been fortunate to read Pappus’s lemmas with
students in the course *Geometriler*
(Geometries); for the course, I translated 19 of the lemmas into Turkish.

As Hessenberg (1905) showed, Pappus’s Theorem yields Desargues’s Theorem.

As Hilbert (1899) and Artin (1957) showed, Pappus and Desargues yield a field over which the plane is an affine plane.

The notes above obtain vectors as equivalence classes of directed segments; the vectors compose an abelian group; ratios of vectors make Thales’s Theorem true; these ratios compose a field, over which the vectors compose a two-dimensional vector space.

Meanwhile, some preliminary ideas ended up in an 8-page note of
January 23, 2013 called From Euclid
to Descartes

; more extensive concerns are detailed in the 107
pages of Analytic
Geometry

(January 28, 2015). When I taught our analytic
geometry course for myself in the spring of 2014–15, I kept a
summary of what we did: Analitik Geometri Özeti,

in Turkish, on
the course webpage above, but with a 5-page
English foreward (and reformatted), here (64 pages, A5 paper, 12-point
type, dated April 8, 2016):