[citation needed] The process of logical synthesis begins with some arbitrary but definite starting point.
Axioms of continuity and "betweenness" are also optional, for example, discrete geometries may be created by discarding or modifying them.
Following the Erlangen program of Klein, the nature of any given geometry can be seen as the connection between symmetry and the content of the propositions, rather than the style of development.
These structures introduced the field of non-Euclidean geometry where Euclid's parallel axiom is denied.
Another example concerns inversive geometry as advanced by Ludwig Immanuel Magnus, which can be considered synthetic in spirit.
Karl von Staudt showed that algebraic axioms, such as commutativity and associativity of addition and multiplication, were in fact consequences of incidence of lines in geometric configurations.
In 1955 Herbert Busemann and Paul J. Kelley sounded a nostalgic note for synthetic geometry: For example, college studies now include linear algebra, topology, and graph theory where the subject is developed from first principles, and propositions are deduced by elementary proofs.
Expecting to replace synthetic with analytic geometry leads to loss of geometric content.
Ernst Kötter published a (German) report in 1901 on "The development of synthetic geometry from Monge to Staudt (1847)";[9] Synthetic proofs of geometric theorems make use of auxiliary constructs (such as helping lines) and concepts such as equality of sides or angles and similarity and congruence of triangles.