Synthetic geometry (sometimes referred to asaxiomatic geometry or evenpure geometry) isgeometry without the use ofcoordinates. It relies on theaxiomatic method for proving all results from a few basic properties, initially calledpostulates and at present calledaxioms.
After the 17th-century introduction byRené Descartes of the coordinate method, which was calledanalytic geometry, the term "synthetic geometry" was coined to refer to the older methods that were, before Descartes, the only known ones.
According toFelix Klein,
Synthetic geometry is that which studiesfigures 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.
The first systematic approach to synthetic geometry isEuclid'sElements. However, it appeared at the end of the 19th century thatEuclid's postulates were not sufficient for characterizing geometry. The first completeaxiom system for geometry was given only at the end of the 19th century byDavid Hilbert. At the same time, it appeared that both synthetic methods and analytic methods can be used to build geometry. The fact that the two approaches are equivalent has been proved byEmil Artin in his bookGeometric Algebra.
Because of this equivalence, the distinction between synthetic and analytic geometry is no more in use, except at elementary level, or for geometries that are not related to any sort of numbers, such as somefinite geometries andnon-Desarguesian geometry.[citation needed]
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:
From a givenset of axioms, synthesis proceeds as a carefully constructed logical argument. When a significant result is proved rigorously, it becomes atheorem.
There is no fixed axiom set for geometry, as more than oneconsistent 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'sparallel postulate has turned out to beindependent of the other axioms. Simply discarding it givesabsolute geometry, while negating it yieldshyperbolic geometry. Otherconsistent axiom sets can yield other geometries, such asprojective,elliptic,spherical oraffine geometry.
Axioms of continuity and "betweenness" are also optional; for example,discrete geometries may be created by discarding or modifying them.
Following theErlangen program ofKlein, the nature of any given geometry can be seen as the connection betweensymmetry and the content of the propositions, rather than the style of development.
Euclid's original treatment remained unchallenged for over two thousand years, until the simultaneous discoveries of the non-Euclidean geometries byGauss,Bolyai,Lobachevsky andRiemann in the 19th century led mathematicians to question Euclid's underlying assumptions.[3]
The early French analystSylvestre François Lacroix (1816) summarized synthetic geometry this way:
The Elements of Euclid are treated by the synthetic method. This author, after having posed theaxioms, and formed the requisites, established the propositions which he proves successively being supported by that which preceded, proceeding always from thesimple to compound, which is the essential character of synthesis.[4]
The heyday of synthetic geometry is considered to have been the 19th century, when analytic methods based oncoordinates andcalculus were ignored by somegeometers such asJakob Steiner, in favor of a purely synthetic development ofprojective geometry. For example, the treatment of theprojective plane starting from axioms of incidence is actually a broader theory (with moremodels) than is found by starting with a third dimensionalvector space. Projective geometry has in fact the simplest and most elegant synthetic expression of any geometry.[5]
In hisErlangen program,Felix Klein played down the tension between synthetic and analytic methods:
The distinction between modern synthesis and modern analytic geometry must no longer be regarded as essential, inasmuch as both subject-matter and methods of reasoning have gradually taken a similar form in both. We choose therefore in the text as common designation of them both the term projective geometry. Although the synthetic method has more to do with space-perception and thereby imparts a rare charm to its first simple developments, the realm of space-perception is nevertheless not closed to the analytic method, and the formulae of analytic geometry can be looked upon as a precise and perspicuous statement of geometrical relations. On the other hand, the advantage to original research of a well formulated analysis should not be underestimated, - an advantage due to its moving, so to speak, in advance of the thought. But it should always be insisted that a mathematical subject is not to be considered exhausted until it has become intuitively evident, and the progress made by the aid of analysis is only a first, though a very important, step.[6]
The close axiomatic study ofEuclidean geometry led to the construction of theLambert quadrilateral and theSaccheri quadrilateral. These structures introduced the field ofnon-Euclidean geometry where Euclid's parallel axiom is denied.Gauss,Bolyai andLobachevski independently constructedhyperbolic geometry, where parallel lines have anangle of parallelism that depends on their separation. This study became widely accessible through thePoincaré disc model wheremotions are given byMöbius transformations. Similarly,Riemann, a student of Gauss's, constructedRiemannian geometry, of whichelliptic geometry is a particular case.
Another example concernsinversive geometry as advanced byLudwig Immanuel Magnus, which can be considered synthetic in spirit. The closely related operation ofreciprocation expresses analysis of the plane.
Karl von Staudt showed that algebraic axioms, such ascommutativity andassociativity of addition and multiplication, were in fact consequences ofincidence of lines ingeometric configurations.David Hilbert showed[7] that theDesargues configuration played a special role. Further work was done byRuth Moufang and her students. The concepts have been one of the motivators ofincidence geometry.
Whenparallel lines are taken as primary, synthesis producesaffine geometry. Though Euclidean geometry is both an affine andmetric geometry, in generalaffine spaces may be missing a metric. The extra flexibility thus afforded makes affine geometry appropriate for the study ofspacetime, as discussed in thehistory of affine geometry.
In 1955,Herbert Busemann and Paul J. Kelley sounded a nostalgic note for synthetic geometry:
Although reluctantly, geometers must admit that the beauty of synthetic geometry has lost its appeal for the new generation. The reasons are clear: not so long ago synthetic geometry was the only field in which the reasoning proceeded strictly from axioms, whereas this appeal — so fundamental to many mathematically interested people — is now made by many other fields.[5]
For example, college studies now includelinear algebra,topology, andgraph theory where the subject is developed from first principles, and propositions are deduced byelementary proofs. Expecting to replace synthetic withanalytic geometry leads to loss of geometric content.[8]
Today's student of geometry has axioms other than Euclid's available.
Ernst Kötter published a German report in 1901 onThe development of synthetic geometry fromMonge to Staudt (1847).[9]
Synthetic proofs of geometric theorems make use of auxiliary constructs (such ashelping lines) and concepts such as equality of sides or angles, andsimilarity andcongruence of triangles. Examples include theButterfly theorem,Angle bisector theorem,Apollonius' theorem,British flag theorem,Ceva's theorem,Equal incircles theorem,Geometric mean theorem,Heron's formula,Isosceles triangle theorem, andLaw of cosines.
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In conjunction withcomputational geometry, acomputational synthetic geometry has been founded, having close connection, for example, withmatroid theory.Synthetic differential geometry is an application oftopos theory to the foundations ofdifferentiable manifold theory.
