Projecting a sphere to a plane. 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 coordinate geometry proofs pdf 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.
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. The process of logical synthesis begins with some arbitrary but definite starting point. Primitives are the most basic ideas.
Typically they include both objects and relationships. The terms themselves are undefined. Hilbert once remarked that instead of points, lines and planes one might just as well talk of tables, chairs and beer mugs, the point being that the primitive terms are just empty placeholders and have no intrinsic properties.
Axioms are assumed true, and not proven. They are the building blocks of geometric concepts, since they specify the properties that the primitives have.