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 straighedge 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?