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((Short description|Geometry without using coordinates))
((General geometry |branches))

'''Synthetic geometry''' (sometimes referred to as '''axiomatic geometry''' or even '''pure geometry''') is  [[geometry]] without the use of [[coordinates]]. It relies on the [[axiomatic method]] for proving all results from a few basic properties initially called [[postulate]], and at present called [[axiom]]s. 

The term "synthetic geometry" was coined only after the 17th century, and the introduction by [[René Descartes]] of the coordinate method, which was called [[analytic geometry]]. So the term "synthetic geometry" was introduced to refer to the older methods that were, before Descartes, the only known ones.

According to [[Felix Klein]]
<blockquote>
Synthetic geometry is that which studies [[shape|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.<ref>((harvnb|Klein|1948|loc=p. 55))</ref>
</blockquote>

The first systematic approach for synthetic geometry is [[Euclid's Elements|Euclid's ''Elements'']].  However, it appeared at the end of the 19th century that [[Euclid]]'s postulates were not sufficient for characterizing geometry. The first complete [[axiom system]] for geometry was given only at the end of the 19th century by [[David 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 approches are equivalent has been proved by [[Emil Artin]] in his book ''[[Geometric Algebra (book)|Geometric 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 some [[finite geometry|finite geometries]] and [[non-Desarguesian geometry]].((cn|date=February 2023))

==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:
* '''[[Primitive notion|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. [[David Hilbert|Hilbert]] once remarked that instead of points, lines and planes one might just as well talk of tables, chairs and beer mugs,<ref>((harvnb|Greenberg|1974|loc=p. 59))</ref> the point being that the primitive terms are just empty [[Free variables and bound variables|placeholders]] and have no intrinsic properties.
* '''[[Axiom]]s''' are statements about these primitives; for example, ''any two points are together incident with just one line'' (i.e. that for any two points, there is just one line which passes through both of them). 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 [[consistency|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 [[Independence (mathematical logic)|independent]] of the other axioms. Simply discarding it gives [[absolute geometry]], while negating it yields [[hyperbolic geometry]].<!-- note in absolute geometry non intersecting lines and infinite long lines exists--> Other [[consistency|consistent axiom sets]] can yield other geometries, such as [[projective geometry|projective]], [[elliptic geometry|elliptic]], [[spherical geometry|spherical]] or [[affine geometry|affine]] geometry.

Axioms of continuity and "betweenness" are also optional, for example, [[discrete geometry|discrete geometries]] may be created by discarding or modifying them.

Following the [[Erlangen program]] of [[Felix Klein|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 [[Carl Friedrich Gauss|Gauss]], [[János Bolyai|Bolyai]], [[Nikolai Lobachevsky|Lobachevsky]] and [[Bernhard Riemann|Riemann]] in the 19th century led mathematicians to question Euclid's underlying assumptions.<ref>Mlodinow 2001, Part III The Story of Gauss</ref><!-- see also [[Bernhard Riemann#Riemannian geometry]] -->

One of the early French analysts summarized synthetic geometry this way:
:''The Elements'' of Euclid are treated by the synthetic method. This author, after having posed the ''axioms'', and formed the requisites, established the propositions which he proves successively being supported by that which preceded, proceeding always from the ''simple to compound'', which is the essential character of synthesis.<ref>[[S. F. Lacroix]] (1816) ''Essais sur L'Enseignement en Général, et sur celui des Mathématiques en Particulier'', page 207, Libraire pur les Mathématiques.</ref>

The heyday of synthetic geometry can be considered to have been the 19th century, when analytic methods based on [[Coordinate system|coordinates]] and [[calculus]] were ignored by some [[list of geometers|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 (with more [[model theory|models]]) 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.<ref name=HBPK/>

In his [[Erlangen program]], [[Felix Klein]] played down the tension between synthetic and analytic methods:
::On the Antithesis between the Synthetic and the Analytic Method in Modern Geometry:
: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.<ref>Felix Klein (1872) Ralf Stephan translator (2006) [https://arxiv.org/abs/0807.3161 "A comparative review of researches in geometry"]</ref>

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 [[motion (geometry)|motion]]s are given by [[Möbius transformation]]s. Similarly, [[Bernhard Riemann|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 [[multiplicative inverse|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 (geometry)|incidence]] of lines in [[geometric configuration]]s. [[David Hilbert]] showed<ref>[[David Hilbert]], 1980 (1899). ''[http://www.gutenberg.org/files/17384/17384-pdf.pdf The Foundations of Geometry]'', 2nd edition, §22 Desargues Theorem, Chicago: Open Court</ref> 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 space]]s may be missing a metric. The extra flexibility thus afforded makes affine geometry appropriate for the study of [[spacetime]], as discussed in the [[affine geometry#History|history 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.<ref name=HBPK>[[Herbert Busemann]] and [[Paul Kelly (mathematician)|Paul J. Kelly]] (1953) ''Projective Geometry and Projective Metrics'', Preface, page v, [[Academic Press]]</ref>

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 proof]]s. Expecting to replace synthetic with [[analytic geometry]] leads to loss of geometric content.<ref name=PS>((citation|first1=Victor|last1=Pambuccian|first2=Celia|last2=Schacht|title=The Case for the Irreducibility of Geometry to Algebra|journal=Philosophia Mathematica|volume=29|year=2021|issue=4|doi=10.1093/philmat/nkab022|url=https://academic.oup.com/philmat/advance-article-abstract/doi/10.1093/philmat/nkab022/6371269?redirectedFrom=fulltext))</ref>

Today's student of geometry has axioms other than Euclid's available: see [[Hilbert's axioms]] and [[Tarski's axioms]].

[[Ernst Kötter]] published a (German) report in 1901 on ''"The development of synthetic geometry from [[Monge]] to Staudt (1847)"'';<ref>((cite book| author=Ernst Kötter| title=Die Entwickelung der Synthetischen Geometrie von Monge bis auf Staudt (1847)| year=1901 |url=http://gdz-lucene.tc.sub.uni-goettingen.de/gcs/gcs?&action=pdf&metsFile=PPN37721857X_0005&divID=LOG_0035&pagesize=original&pdfTitlePage=http://gdz.sub.uni-goettingen.de/dms/load/pdftitle/?metsFile=PPN37721857X_0005%7C&targetFileName=PPN37721857X_0005_LOG_0035.pdf&)) (2012 Reprint as ((ISBN|1275932649)))</ref>

==Proofs using synthetic geometry==

Synthetic proofs of geometric theorems make use of auxiliary constructs (such as [[Auxiliary line|helping lines]]) and concepts such as equality of sides or angles and [[similarity (geometry)|similarity]] and [[congruence (geometry)|congruence]] of triangles. Examples of such proofs can be found in the articles [[Butterfly theorem]], [[Angle bisector theorem]], [[Apollonius' theorem]], [[British flag theorem]], [[Ceva's theorem]], [[Equal incircles theorem]], [[Geometric mean theorem]], [[Heron's formula]], [[Isosceles triangle theorem]], [[Law of cosines]], and others that are linked to [[:Category:Theorems in plane geometry|here]].

==Computational synthetic geometry==
((Unreferenced section|date=February 2019))
In conjunction with [[computational geometry]], a '''computational synthetic geometry''' has been founded, having close connection, for example, with [[matroid]] theory. [[Synthetic differential geometry]] is an application of [[topos]] theory to the foundations of [[differentiable manifold]] theory.

==See also==
* [[Foundations of geometry]]
* [[Incidence geometry]]
* [[Synthetic differential geometry]]

==Notes==
((Reflist))

==References==
* ((citation|first=Carl B.|last=Boyer|title=History of Analytic Geometry|year=2004|orig-year=1956|publisher=Dover|isbn=978-0-486-43832-0))
* ((citation|author-link=Marvin Greenberg|first=Marvin Jay|last=Greenberg|title=Euclidean and Non-Euclidean Geometries/Development and History|year=1974|publisher=W.H. Freeman|place=San Francisco|isbn=0-7167-0454-4))
* Halsted, G. B. (1896) [https://archive.org/details/elementarysynth00halsgoog Elementary Synthetic Geometry] via Internet Archive
* [[G. B. Halsted|Halsted, George Bruce]] (1906) [https://archive.org/details/syntheticproject00halsuoft Synthetic Projective Geometry], via [[Internet Archive]].
* Hilbert & Cohn-Vossen, ''Geometry and the imagination''.
* ((citation|first=Felix|last=Klein|title=Elementary Mathematics from an Advanced Standpoint/Geometry|publisher=Dover|place=New York|year=1948))
* ((citation|first=Leonard|last=Mlodinow|title=Euclid's Window/The Story of Geometry from Parallel Lines to Hyperspace|publisher=The Free Press|place=New York|year=2001|isbn=0-684-86523-8|url-access=registration|url=https://archive.org/details/euclidswindowsto00mlod))
*((Citation|first1=Victor|last1=Pambuccian|first2=Celia|last2=Schacht|title=The Case for the Irreducibility of Geometry to Algebra|journal=Philosophia Mathematica|volume=29|year=2021|issue=4|doi=10.1093/philmat/nkab022|url=https://academic.oup.com/philmat/advance-article-abstract/doi/10.1093/philmat/nkab022/6371269?redirectedFrom=fulltext))

((Authority control))

[[Category:Fields of geometry]]

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