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):