I am currently writing an essay on the history of geometry. To educate myself on the subject, I sometimes read the following Wikipedia article on the history of Euclidean Geometry. It seems to me that, based on this article, many different branches in geometry have been given their own names. For instance, we have Analytic Geometry (developed in the 17th century by Descartes, Fermat and others), Projective Geometry (founded in the same century by Desargues), Affine Geometry as initiated by Euler in the 18th century and Non-Euclidean Geometry discovered by Bolyai, Lobachevsky and Gauss in the 19th century.

I was quite surprised however when I found out that the collection of algebraic and number-theoretic methods developed in the 18th century by Pierre Wantzel, Gauss and von Lindemann is not named. They proved that it is impossible to trisect the angle, showed why certain regular polygons can and others can't be constructed with compass and straightedge and proved the impossibility of squaring the squaring the circle, respectively. By doing so, they resolved long-standing problems by means of roughly similar methods. (See this article for more.)

So my question is perhaps fourfold: 1) Does this "branch" of geometry have a name? 2) If so: what is it, and if not: why not? 3) If not, what name would be apt? 4) Should it have a name at all?

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    $\begingroup$ The examples in the 2nd paragraph have to do with the field of so-called "constructible numbers": en.wikipedia.org/wiki/Constructible_number, which is the union of subfields of $\mathbb{C}$ that are iterated quadratic extensions of $\mathbb{Q}$. But as opposed to some of your other examples, this doesn't seem to fit within the Erlangen Program paradigm of considering geometries according to continuous groups of transformations that preserve geometric structure. The problems you mention are venerable but ancient, and now seem more number-theoretic to me than particularly "geometric". $\endgroup$ – Todd Trimble Jun 10 '16 at 14:09
  • $\begingroup$ 1) ... probably it is not considered a branch of geometry at all. But instead a branch of algebra. $\endgroup$ – Gerald Edgar Jun 10 '16 at 14:36
  • $\begingroup$ @ToddTrimble and Gerald Edgar: Thank you for the link, Todd. Perhaps we could call this branch of mathematics "Constructible (Algebraic) Number Theory" ? $\endgroup$ – Max Muller Jun 10 '16 at 14:37
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    $\begingroup$ You might enjoy (and find useful for your essay) the books by R. Hartshorne and M. Greenberg on Euclidean and non-Euclidean geometry. They discuss at great length the use of models based on various fields to prove the impossibility of various constructions or the independence of various continuity axioms from the usual incidence, betweenness, and congruence axioms. $\endgroup$ – Robert Bryant Jun 10 '16 at 15:26
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    $\begingroup$ Robert Bryant's comment gets an upvote. I'll add that one of the big 20th century developments which was a capstone to the classical axiomatic developments of Euclidean geometry, beginning in ancient Hellenic civilization and continuing through Hilbert, was Tarski's "elementary" (first-order) axioms, which are closely tied to his theory of real closed fields (essentially he showed that his axioms describe coordinate geometries over real closed fields). A main example is the field of real algebraic numbers; real constructible numbers form a subfield, one that is historically important. $\endgroup$ – Todd Trimble Jun 10 '16 at 17:11

It seems what you are talking about is a field called "counterexamples in Euclidean geometry", constructed via geometric algebra. If you adhere to the thesis that the Greeks possessed geometric algebra (a controversial one to be sure), then these late proofs may have been arguably accessible to the Greeks, though I personally am sceptical.

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    $\begingroup$ What do you call the content of Euclid's book II ? It is full of propositions that are not used anywhere in the geometric books. $\endgroup$ – Franz Lemmermeyer Sep 11 '16 at 14:27

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