Pure mathematical books on Analytic Geometry

Question

I am impregnated with the idea that pure mathematics is when you do mathematics in such a way that you begin with some set of axioms and definitions, and then you develop such things by means of logic.

So I am searching for a book or notes or a website where this is done in the case of analytic geometry. This "thing" must have at the beginning of each concept their formal definitions and the theorems, corollaries, lemmas, etc.

Can someone help me with something like this?

There are no specific axioms for Analytic Geometry; we have e.g. the Euclide axioms for plane geometry (see Hilbert's modern version) and then we define a mapping in $\mathbb R \times \mathbb R$ showing that it is a model of the euclidean plane. — Mauro ALLEGRANZA, Sep 30 '21 at 10:33
Sure, there are not axioms for analytic geometry, then, I suppose it is developed from certain definitions. Is there a treatment where that is done?
Mauro ALLEGRANZA — Rata mágica, Sep 30 '21 at 10:38
Maybe useful Felix Klein, Elementary Mathematics from a Higher Standpoint. Volume II: Geometry (1908) (2016, Springer), II. Foundations of Geometry: Relation to Analytic Geometry. — Mauro ALLEGRANZA, Sep 30 '21 at 10:38
Generally speaking calculus textbooks which have a lot of multivariable calculus also contain a lot of analytic geometry, so you just want one which starts with carefully formulated axioms and continues with carefully formulated definitions and proofs. For that purpose I like Fitzpatrick's "Advanced Calculus", which starts with the axioms for the real numbers. — Lee Mosher, Sep 30 '21 at 10:53
Regarding your first sentence, see Terence Tao's blog entry There’s more to mathematics than rigour and proofs. That said, maybe look at the following: Karol Borsuk, Multidimensional Analytic Geometry, Monografie Matematyczne #50, PWN - Polish Scientific Publishers, 1969, 443 pages (table of contents -- need to translate from Polish). — Dave L. Renfro, Sep 30 '21 at 15:17

score 2 · Answer 1 · 2021-09-30T15:53:53.663

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Analytical Geometry can be seen as a combination of calculus/analysis, algebra and geometry. There exist literature on Analytic Geometry where the focus on these three subjects vary.

The following books might be helpful (all full PDFs available by Google Scholar Search):

Calculus with Analytic Geometry
Analytic Geometry with Calculus
Crowell and Slesnick’s Calculus with Analytic Geometry
Analytic Geometry Notes, Quizzes & Exams (Izmir Economy University)
Exploring Analytic Geometry with Mathematica

The last one I like very much, since it is more technical in terms of algorithms and one can play with the commands directly and get visual output using Wolfram. Please check the Webportal WolframAlpha as well.

An extensive list of Analytic Geometry resources is provided by this web page of MAA (Mathematical Association of America)

edited Sep 30 '21 at 15:53

answered Sep 30 '21 at 12:38

You are totally right and I will fix it in my answer (for a moment I had the same thought, but put it anyway for the sake of completeness). Thanks for your hint! – Sep 30 '21 at 15:44
After thinking again I removed it and added a much more suitable resource collection that introductes Analytic Geometry much more gently (including quizzes and exams). – Sep 30 '21 at 15:55
more suitable resource collection --- In case anyone is interested in additional references, see this answer and the 3 books mentioned in my last comment to this question. However, these additional references do not fit the (in my opinion, artificial) constraints of the OP's question. – Dave L. Renfro Sep 30 '21 at 16:02

score 2 · Answer 2 · answered Sep 30 '21 at 16:18

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Try "Elementary Geometry From an Advanced Standpoint" by Edwin Moise.

It is not purely axiomatic but it is a rigorous development of the field and probably in the spirit of what you seek.

answered Sep 30 '21 at 16:18

John Douma

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Mike · Answer 3 · 2021-09-30T21:54:37.003

This is the exact purpose of the book Linear Algebra and Geometry by Dieudonné.

He starts with a set of assumptions about the real number system. Then he gives axioms for the Euclidean plane and Euclidean space, which amount to assuming given a two- or three-dimensional affine space whose associated vector space is endowed with an inner product.

The rest of the book is devoted to establishing analytic foundations for the usual geometric concepts.

The exercises have full solutions at the back of the book.

Pure mathematical books on Analytic Geometry

3 Answers3