Editing Equality (mathematics) (section)

=== Geometry ===
[[File:Congruent non-congruent triangles.svg|thumb|upright=1.35|The two triangles on the left are [[Congruence (geometry)|congruent]]. The third is [[Similarity (geometry)|similar]] to them. The last triangle is neither congruent nor similar to any of the others. ]]
In [[geometry]], formally, two figures are equal if they contain exactly the same [[Point (geometry)|points]]. However, historically, geometric-equality has always been taken to be much broader. [[Euclid]] and [[Archimedes]] used "equal" ({{lang|grc|ἴσος}} {{tlit|grc|isos}}) often referring to figures with the same area or those that could be cut and rearranged to form one another. For example, Euclid stated the [[Pythagorean theorem]] as "the square on the hypotenuse is equal to the squares on the sides, taken together"; and Archimedes said that "a circle is equal to the rectangle whose sides are the radius and half the circumference."<ref>{{Cite journal |last=Beeson |first=Michael |date=2023-09-01 |title=On the notion of equal figures in Euclid |journal=Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry |volume=64 |issue=3 |pages=581–625 |doi=10.1007/s13366-022-00649-9 |arxiv=2008.12643 |issn=2191-0383}}</ref>

This notion persisted until [[Adrien-Marie Legendre]], who introduced the term "equivalent" to describe figures of equal area and restricted "equal" to what we now call "[[Congruence (geometry)|congruent]]"—the same [[shape]] and [[size]], or if one has the same shape and size as the [[mirror image]] of the other.<ref>{{Cite book |last=Legendre |first=Adrien Marie |url=https://archive.org/details/cu31924001166341/page/n77/mode/2up |title=Elements of geometry |date=1867 |publisher=Baltimore, Kelly & Piet |others=Cornell University Library |page=68}}</ref><ref>{{cite dictionary |last1=Clapham |first1=C. |last2=Nicholson |first2=J. |year=2009 |dictionary=Oxford Concise Dictionary of Mathematics |entry=Congruent Figures |url=http://web.cortland.edu/matresearch/OxfordDictionaryMathematics.pdf |url-status=dead |archive-url=https://web.archive.org/web/20131029203826/http://web.cortland.edu/matresearch/OxfordDictionaryMathematics.pdf |archive-date=29 October 2013 |access-date=2 June 2017 |publisher=Addison-Wesley |page=167}}</ref> Euclid's terminology continued in the work of [[David Hilbert]] in his ''{{lang|De|[[Grundlagen der Geometrie]]}}'', who further refined Euclid's ideas by introducing the notions of polygons being "divisibly equal" ({{lang|de|zerlegungsgleich}}) if they can be cut into finitely many triangles which are congruent, and "equal in content" ({{lang|de|inhaltsgleichheit}}) if one can add finitely many divisibly equal polygons to each such that the resulting polygons are divisibly equal.<ref>{{Cite book |last=Hilbert |first=David |url=https://archive.org/details/grundlagendergeo00hilb/page/40/mode/2up |title=Grundlagen der Geometrie |date=1899 |publisher=B. G. Teubner |others=Wellesley College Library |page=40 |language=de}}</ref>

After the rise of set theory, around the 1960s, there was a push for a reform in [[mathematics education]] called [[New Math]], following [[Andrey Kolmogorov]], who, in an effort to restructure Russian geometry courses, proposed presenting geometry through the lens of [[Transformation geometry|transformations]] and set theory. Since a figure was seen as a set of points, it could only be equal to itself, as a result of Kolmogorov, the term "congruent" became standard in schools for figures that were previously called "equal", which popularized the term.<ref>[https://books.google.com/books?id=qwyBPybT4oMC Alexander Karp & Bruce R. Vogeli – Russian Mathematics Education: Programs and Practices, Volume 5], pp. 100–102</ref>

While Euclid addressed [[Proportionality (mathematics)|proportionality]] and figures of the same shape, it was not until the 17th century that the concept of [[Similarity (geometry)|similarity]] was formalized in the modern sense. Similar figures are those that have the same shape but can differ in size; they can be transformed into one another by [[Scaling (geometry)|scaling]] and congruence.<ref>{{Cite book |date=2020-02-10 |title=PreAlgebra |chapter=2.2.1: Similarity |chapter-url=https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_II_(Illustrative_Mathematics_-_Grade_8)/02:_Dilations_Similarity_and_Introducing_Slope/2.02:_New_Page/2.2.1:_Similarity |access-date=2025-03-04 |publisher=Mathematics LibreTexts}}</ref> Later a concept of equality of [[directed line segment]]s, [[Equipollence (geometry)|equipollence]], was advanced by [[Giusto Bellavitis]] in 1835.<ref>{{Cite web |title=Giusto Bellavitis – Biography |url=https://mathshistory.st-andrews.ac.uk/Biographies/Bellavitis/ |access-date=2025-03-04 |website=Maths History}}</ref>