Editing Buckingham π theorem (section)

== History ==

Although named for [[Edgar Buckingham]], the {{pi}} theorem was first proved by the French mathematician [[Joseph Louis François Bertrand|Joseph Bertrand]] in 1878.<ref>{{cite journal|last=Bertrand|first=J.|year=1878|title=Sur l'homogénéité dans les formules de physique|url=https://archive.org/details/comptesrendusheb86acad|journal=Comptes Rendus|volume=86|number=15|pages=916–920}}</ref> Bertrand considered only special cases of problems from electrodynamics and heat conduction, but his article contains, in distinct terms, all the basic ideas of the modern proof of the theorem and clearly indicates the theorem's utility for modelling physical phenomena. The technique of using the theorem ("the method of dimensions") became widely known due to the works of [[John Strutt, 3rd Baron Rayleigh|Rayleigh]]. The first application of the {{pi}} theorem ''in the general case''<ref group=note>When in applying the {{pi}}–theorem there arises an ''arbitrary function'' of dimensionless numbers.</ref> to the dependence of pressure drop in a pipe upon governing parameters probably dates back to 1892,<ref>{{cite journal|last=Rayleigh|year=1892|title=On the question of the stability of the flow of liquids|url=http://gidropraktikum.narod.ru/Rayleigh-1892.djvu|journal=Philosophical Magazine|volume=34|issue=206|pages=59–70|doi=10.1080/14786449208620167}}</ref> a heuristic proof with the use of series expansions, to 1894.<ref>{{cite book|last=Strutt|first=John William|year=1896|title=The Theory of Sound|url=https://archive.org/details/theorysound05raylgoog|publisher=Macmillan|volume=II|edition=2nd}}</ref>

Formal generalization of the {{pi}} theorem for the case of arbitrarily many quantities was given first by {{ill|A. Vaschy|fr|Aimé Vaschy}} in 1892,<ref>Quotes from Vaschy's article with his statement of the pi–theorem can be found in: {{cite journal|last=Macagno|first=E. O.|year=1971|title=Historico-critical review of dimensional analysis| url=http://gidropraktikum.narod.ru/Macagno-1971.djvu|journal=Journal of the Franklin Institute|issue=6|volume=292|pages=391–402|doi=10.1016/0016-0032(71)90160-8}}</ref><ref>{{Cite journal |last=De A. Martins |first=Roberto |date=1981 |title=The origin of dimensional analysis |url=https://linkinghub.elsevier.com/retrieve/pii/0016003281904750 |journal=Journal of the Franklin Institute |language=en |volume=311 |issue=5 |pages=331–337 |doi=10.1016/0016-0032(81)90475-0}}</ref> then in 1911—apparently independently—by both A. Federman<ref>{{cite journal|last=Федерман|first=А.|year=1911|title=О некоторых общих методах интегрирования уравнений с частными производными первого порядка|url=http://gidropraktikum.narod.ru/Federman.djvu|journal=Известия Санкт-Петербургского политехнического института императора Петра Великого. Отдел техники, естествознания и математики|issue=1|volume=16|pages=97–155}} (Federman A., On some general methods of integration of first-order partial differential equations, Proceedings of the Saint-Petersburg polytechnic institute. Section of technics, natural science, and mathematics)</ref> and [[Dimitri Riabouchinsky|D. Riabouchinsky]],<ref>{{cite journal|last=Riabouchinsky|first=D.|year=1911|title=Мéthode des variables de dimension zéro et son application en aérodynamique|url=http://gidropraktikum.narod.ru/Riabouchinsky-Aerophile-1911.djvu|journal=L'Aérophile|pages=407–408}}</ref> and again in 1914 by Buckingham.{{sfn|Buckingham|1914}} It was Buckingham's article that introduced the use of the symbol "<math>\pi_i</math>" for the dimensionless variables (or parameters), and this is the source of the theorem's name.