of the objects is not. The golden age and afterwards The word "geometry" (from Ancient Greek: geo- "earth", -metron "measurement") initially meant a practical way of processing lengths, regions and volumes in the space in which we live, but was then extended widely (as well as the notion of space in question here). According to Bourbaki, the period between 1795 (Géométrie descriptive of Monge) and 1872 (the "Erlangen programme" of Klein) can be called the golden age of geometry. The original space investigated by Euclid is now called three-dimensional Euclidean space. Its axiomatization, started by Euclid 23 centuries ago, was reformed with Hilbert's axioms, Tarski's axioms and Birkhoff's axioms. These axiom systems describe the space via primitive notions (such as "point", "between", "congruent") constrained by a number of axioms. Analytic geometry made great progress and succeeded in replacing theorems of classical geometry with computations via invariants of transformation groups. Since that time, new theorems of classical geometry have been of more interest to amateurs than to professional mathematicians. However, the heritage of classical geometry was not lost. According to Bourbaki, "passed over in its role as an autonomous and living science, classical geometry is thus transfigured into a universal language of contemporary mathematics". Simultaneously, numbers began to displace geometry as the foundation of mathematics. For instance, in Richard Dedekind's 1872 essay Stetigkeit und irrationale Zahlen (Continuity and irrational numbers), he asserts that points on a line ought to have the properties of Dedekind cuts, and that therefore a line was the same thing as the set of real numbers. Dedekind is careful to note that this is an assumption that is incapable of being proven. In modern treatments, Dedekind's assertion is often taken to be the definition of a line, thereby reducing geometry to arithmetic. Three-dimensional Euclidean space is defined to be an affine space whose associated vector space of differences of its elements is equipped with an inner product. A definition "from scratch", as in Euclid, is now not often used, since it does not reveal the relation of this space to other spaces. Also, a three-dimensional projective space is now defined as the space of all one-dimensional subspaces (that is, straight lines through the origin) of a four-dimensional vector space. This shift in foundations requires a new set of axioms, and if these axioms are adopted, the classical axioms of geometry become theorems. A space now consists of