Synthetic geometry is that which studies figures as such, without recourse to formulae, whereas analytic geometry consistently makes use of such formulae 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 geometerJakob Steiner hated analytic geometry, and always gave preference to synthetic methods.
Logical synthesis
The process of logical synthesis begins with some arbitrary but definite starting point. This starting point is the introduction of primitive notions or primitives and axioms about these primitives:
Primitives are the most basic ideas. Typically they include both objects and relationships. In geometry, the objects are things such as points, lines and planes, while a fundamental relationship is that of incidence – of one object meeting or joining with another. 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 statements about these primitives; for example, any two points are together incident with just one line. Axioms are assumed true, and not proven. They are the building blocks of geometric concepts, since they specify the properties that the primitives have.
From a given set of axioms, synthesis proceeds as a carefully constructed logical argument. When a significant result is proved rigorously, it becomes a theorem.
Properties of axiom sets
There is no fixed axiom set for geometry, as more than one consistent set can be chosen. Each such set may lead to a different geometry, while there are also examples of different sets giving the same geometry. With this plethora of possibilities, it is no longer appropriate to speak of "geometry" in the singular. Historically, Euclid's parallel postulate has turned out to be independent of the other axioms. Simply discarding it gives absolute geometry, while negating it yields hyperbolic geometry. Other consistent axiom sets can yield other geometries, such as projective, elliptic, spherical or affine geometry. Axioms of continuity and "betweeness" 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.
History
Euclid's original treatment remained unchallenged for over two thousand years, until the simultaneous discoveries of the non-Euclidean geometries by Gauss, Bolyai, Lobachevsky and Riemann in the 19th century led mathematicians to question Euclid's underlying assumptions. One of the early French analysts summarized synthetic geometry this way: The heyday of synthetic geometry can be considered to have been the 19th century, when analytic methods based on coordinates and calculus were ignored by some geometers such as Jakob Steiner, in favor of a purely synthetic development of projective geometry. For example, the treatment of the projective plane starting from axioms of incidence is actually a broader theory than is found by starting with a vector space of dimension three. Projective geometry has in fact the simplest and most elegant synthetic expression of any geometry. In his Erlangen program, Felix Klein played down the tension between synthetic and analytic methods: The close axiomatic study of Euclidean geometry led to the construction of the Lambert quadrilateral and the Saccheri quadrilateral. These structures introduced the field of non-Euclidean geometry where Euclid's parallel axiom is denied. Gauss, Bolyai and Lobachevski independently constructed hyperbolic geometry, where parallel lines have an angle of parallelism that depends on their separation. This study became widely accessible through the Poincaré disc model where motions are given by Möbius transformations. Similarly, Riemann, a student of Gauss's, constructed Riemannian geometry, of which elliptic geometry is a particular case. Another example concerns inversive geometry as advanced by Ludwig Immanuel Magnus, which can be considered synthetic in spirit. The closely related operation of reciprocation expresses analysis of the plane. 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. David Hilbert showed that the Desargues configuration played a special role. Further work was done by Ruth Moufang and her students. The concepts have been one of the motivators of incidence geometry. When parallel lines are taken as primary, synthesis produces affine geometry. Though Euclidean geometry is both an affine and metric geometry, in general affine spaces may be missing a metric. The extra flexibility thus afforded makes affine geometry appropriate for the study of spacetime, as discussed in the history of affine geometry. 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. Today's student of geometry has axioms other than Euclid's available: see Hilbert's axioms and Tarski's axioms. Ernst Kötter published a report in 1901 on "The development of synthetic geometry from Monge to Staudt ";
