The synthetic method is an approach to geometry in which coordinates are not used directly. Mainly relies on axioms and tools directly related to them. The term appeared with the advent of analytic geometry .
Geometry in the "Beginnings" of Euclid is a typical example of using the only synthetic method at that time.
Most nineteenth-century geometers preferred synthetic methods, in particular in projective geometry and non-Euclidean geometry , and analytical geometry methods were often seen as a sign of poor style.
Synthetic methods are also used in modern differential geometry , namely in global Riemannian geometry . Here, a certain set of theorems of Riemannian geometry takes the place of axioms. Most consistently, this approach was developed by the Aleksandrov school.