Editing Poncelet–Steiner theorem (section)

== History ==
[[File:Frontispice(Steiner).png|thumb|right|350px|"''The geometrical constructions, carried out using the straight line and a fixed circle, as a subject of teaching at higher educational institutions and for practical use; by Jacob Steiner, Doctor of philosophy, Royal Prussian professor and distinguished teacher of mathematics at the commercial school in Berlin. With two copper plaques. Berlin, with [[Ferdinand_Dümmler_(publisher)|Ferdinand Dummler]]. 1833.''"]]

In the tenth century, the Persian mathematician [[Abu al-Wafa' Buzjani]] (940−998) considered geometric constructions using a straightedge and a compass with a fixed opening, a so-called ''rusty compass''. Constructions of this type appeared to have some practical significance as they were used by artists [[Leonardo da Vinci]] and [[Albrecht Dürer]] in Europe in the late fifteenth century. A new viewpoint developed in the mid sixteenth century when the size of the opening was considered fixed but arbitrary and the question of how many of Euclid's constructions could be obtained was paramount.<ref name=Eves205>{{harvnb|Eves|1963|loc=p.205}}</ref>

[[Renaissance]] mathematician [[Lodovico Ferrari]], a student of [[Gerolamo Cardano]] in a "mathematical challenge" against [[Niccolò Fontana Tartaglia]] was able to show that "all of Euclid" (that is, the straightedge and compass constructions in the first six books of [[Euclid's Elements]]) could be accomplished with a straightedge and rusty compass. Within ten years additional sets of solutions were obtained by Cardano, Tartaglia and Tartaglia's student [[Giambattista_Benedetti|Benedetti]].<ref>https://www.encyclopedia.com/people/science-and-technology/mathematics-biographies/giovanni-battista-benedetti</ref> During the next century these solutions were generally forgotten until, in 1673, [[Georg Mohr]] published (anonymously and in Dutch) ''Euclidis Curiosi'' containing his own solutions. Mohr had only heard about the existence of the earlier results and this led him to work on the problem.<ref>{{harvnb|Retz|Keihn|1989|loc=p.195}}</ref>

Showing that "all of Euclid" could be performed with straightedge and rusty compass is not the same as proving that ''all'' straightedge and compass constructions could be done with a straightedge and just a rusty compass. Such a proof would require the formalization of what a straightedge and compass could construct. This groundwork was provided by [[Jean Victor Poncelet]] in 1822, having been motivated by Mohr's work on the [[Mohr-Mascheroni theorem]]. He also conjectured and suggested a possible proof that a straightedge and rusty compass would be equivalent to a straightedge and compass, and moreover, the rusty compass need only be used once. The result of this theorem, that ''a straightedge and single circle with given centre is equivalent to a straightedge and compass'' was proved by [[Jakob Steiner]] in 1833.<ref name="Steiner1833">
{{cite book
|author=Jacob Steiner|authorlink=Jakob Steiner
|title=Die geometrischen Konstructionen, ausgeführt mittelst der geraden Linie und eines festen Kreises, als Lehrgegenstand auf höheren Unterrichts-Anstalten und zur praktischen Benutzung
|url=https://archive.org/details/bub_gb_xHc1T1W-ZXUC|accessdate=2 April 2013
|year=1833|publisher=Ferdinand Dümmler|location=Berlin
|language=german}}</ref><ref name=Eves205 />

Major contributions to the field were made by [[Lazare Carnot]], [[Karl von Staudt]], [[Giuseppe Peano]], [[Joseph Diez Gergonne]], and others in the mid-to-late 19th and early 20th centuries, including contributions by [[Francesco Severi]] as late as the mid 20th century, even as further research is ongoing.

=== Controversies ===
While the theorem itself was not directly implicated, Jean-Victor Poncelet was involved in a plagiarism controversy with Joseph-Diez Gergonne in 1827 over duality principles in projective geometry.<ref>{{cite journal |title=Polemics in Public: Poncelet, Gergonne, Plücker, and the Duality Controversy |journal=Science in Context |date=November 11, 2015 |volume=28 |issue=4 |pages=545-585 |doi=10.1017/S0269889715000289 |url=https://www.cambridge.org/core/journals/science-in-context/article/abs/polemics-in-public-poncelet-gergonne-plucker-and-the-duality-controversy/C305E6B9326F2AC4C8AC31D4194FEDC6 |ref=Cambridge University Press}}</ref> This unrelated controversy, the accusation of which was never proven, might occasionally surface in discussions about Poncelet's broader work.