Editing Euclidean distance (section)

== History ==
Euclidean distance is the distance in [[Euclidean space]]. Both concepts are named after ancient Greek mathematician [[Euclid]], whose [[Euclid's Elements|''Elements'']] became a standard textbook in geometry for many centuries.<ref>{{citation|title=Visualization for Information Retrieval|first=Jin|last=Zhang|publisher=Springer|year=2007|isbn=978-3-540-75148-9}}</ref> Concepts of [[length]] and [[distance]] are widespread across cultures, can be dated to the earliest surviving "protoliterate" bureaucratic documents from [[Sumer]] in the fourth millennium BC (far before Euclid),<ref>{{citation|last=Høyrup|first=Jens|author-link=Jens Høyrup|editor1-last=Jones|editor1-first=Alexander|editor2-last=Taub|editor2-first=Liba|editor2-link=Liba Taub|contribution=Mesopotamian mathematics|contribution-url=https://akira.ruc.dk/~jensh/Publications/2018%7Bj%7D_Mesopotamian%20Mathematics_S.pdf|pages=58–72|publisher=Cambridge University Press|title=The Cambridge History of Science, Volume 1: Ancient Science|year=2018|access-date=October 21, 2020|archive-date=May 17, 2021|archive-url=https://web.archive.org/web/20210517124414/http://akira.ruc.dk/~jensh/Publications/2018%7Bj%7D_Mesopotamian%20Mathematics_S.pdf|url-status=dead}}</ref> and have been hypothesized to develop in children earlier than the related concepts of speed and time.<ref>{{citation|last1=Acredolo|first1=Curt|last2=Schmid|first2=Jeannine|doi=10.1037/0012-1649.17.4.490|issue=4|journal=[[Developmental Psychology (journal)|Developmental Psychology]]|pages=490–493|title=The understanding of relative speeds, distances, and durations of movement|volume=17|year=1981}}</ref> But the notion of a distance, as a number defined from two points, does not actually appear in Euclid's ''Elements''. Instead, Euclid approaches this concept implicitly, through the [[Congruence (geometry)|congruence]] of line segments, through the comparison of lengths of line segments, and through the concept of [[Proportionality (mathematics)|proportionality]].<ref>{{citation|last=Henderson|first=David W.|author-link=David W. Henderson|journal=[[Bulletin of the American Mathematical Society]]|pages=563–571|title=Review of ''Geometry: Euclid and Beyond'' by Robin Hartshorne|url=https://www.ams.org/journals/bull/2002-39-04/S0273-0979-02-00949-7|volume=39|year=2002|issue=4 |doi=10.1090/S0273-0979-02-00949-7|doi-access=free}}</ref>

The [[Pythagorean theorem]] is also ancient, but it could only take its central role in the measurement of distances after the invention of [[Cartesian coordinates]] by [[René Descartes]] in 1637. The distance formula itself was first published in 1731 by [[Alexis Clairaut]].<ref>{{citation|last=Maor|first=Eli|author-link=Eli Maor|isbn=978-0-691-19688-6|pages=133–134|publisher=Princeton University Press|title=The Pythagorean Theorem: A 4,000-Year History|url=https://books.google.com/books?id=XuWZDwAAQBAJ&pg=PA133|year=2019}}</ref> Because of this formula, Euclidean distance is also sometimes called Pythagorean distance.<ref>{{citation|last1=Rankin|first1=William C.|last2=Markley|first2=Robert P.|last3=Evans|first3=Selby H.|date=March 1970|doi=10.3758/bf03210143|issue=2|journal=[[Perception & Psychophysics]]|pages=103–107|title=Pythagorean distance and the judged similarity of schematic stimuli|volume=7|doi-access=free }}</ref> Although accurate measurements of long distances on the Earth's surface, which are not Euclidean, had again been studied in many cultures since ancient times (see [[history of geodesy]]), the idea that Euclidean distance might not be the only way of measuring distances between points in mathematical spaces came even later, with the 19th-century formulation of [[non-Euclidean geometry]].<ref>{{citation|last=Milnor|first=John|author-link=John Milnor|doi=10.1090/S0273-0979-1982-14958-8|issue=1|journal=[[Bulletin of the American Mathematical Society]]|mr=634431|pages=9–24|title=Hyperbolic geometry: the first 150 years |url=https://www.ams.org/journals/bull/1982-06-01/S0273-0979-1982-14958-8/S0273-0979-1982-14958-8.pdf |volume=6|year=1982|doi-access=free}}</ref> The definition of the Euclidean norm and Euclidean distance for geometries of more than three dimensions also first appeared in the 19th century, in the work of [[Augustin-Louis Cauchy]].<ref>{{citation|title=Foundations of Hyperbolic Manifolds|volume=149|series=[[Graduate Texts in Mathematics]]|first=John G.|last=Ratcliffe|edition=3rd|publisher=Springer|year=2019|isbn=978-3-030-31597-9|page=32|url=https://books.google.com/books?id=yMO4DwAAQBAJ&pg=PA32}}</ref>