Editing Synthetic geometry (section)

==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>{{cite arXiv |last=Klein |first=Felix C. |title=A comparative review of recent researches in geometry |date=2008-07-20 |class=math.HO |eprint=0807.3161}}</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|pages=1–31 |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>