History of geometry
Classic geometry was focused in compass and straightedge constructions. Geometry was revolutionized by Euclid, who introduced mathematical rigor and the axiomatic method still in use today. His book, "The Elements" is widely considered the most influential textbook of all time, and was known to all educated people in the West until the middle of the 20th century.
Synthetic geometry
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.