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Recall the two paragraphs from Section 1.2 that we intended to spend time making sense of and working through:

Whereas Euclid's approach to geometry was additive (he started with basic definitions and axioms and proceeded to build a sequence of results depending on previous ones), Klein's approach was subtractive. He started with a space and a group of allowable transformations of that space. He then threw out all concepts that did not remain unchanged under these transformations. Geometry, to Klein, is the study of objects and functions that remain unchanged under allowable transformations.

Klein's approach to geometry, called the Erlangen Program after the university at which he worked at the time, has the benefit that all three geometries (Euclidean, hyperbolic and elliptic) emerge as special cases from a general space and a general set of transformations.

We now have both the space (\(\mathbb{C}^+\)) and the transformations (Möbius transformations), and are just about ready to embark on non-Euclidean adventures. Before doing so, however, one more phrase needs defining: group of transformations. This phrase has a precise meaning. Not every collection of transformations is lucky enough to form a group.