Synthetic geometry

Synthetic geometry (sometimes referred to as axiomatic or even pure geometry) is the study of geometry without the use of coordinates or formulas. It relies on the axiomatic method and the tools directly related to them, that is, compass and straightedge, to draw conclusions and solve problems.

Only after the introduction of coordinate methods was there a reason to introduce the term "synthetic geometry" to distinguish this approach to geometry from other approaches. Other approaches to geometry are embodied in analytic and algebraic geometries, where one would use analysis and algebraic techniques to obtain geometric results.

According to Felix Klein,

Synthetic geometry is that which studies figures as such, without recourse to formulas, whereas analytic geometry consistently makes use of such formulas as can be written down after the adoption of an appropriate system of coordinates.

Geometry, as presented by Euclid in the elements, is the quintessential example of the use of the synthetic method. It was the favoured method of Isaac Newton for the solution of geometric problems.

Synthetic methods were most prominent during the 19th century when geometers rejected coordinate methods in establishing the foundations of projective geometry and non-Euclidean geometries. For example the geometer Jakob Steiner (1796 – 1863) hated analytic geometry, and always gave preference to synthetic methods.

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