The name of the method and its origin story have been debated by histroianshistorians.<ref name="Leissa">{{cite journal|last1=Leissa|first1=A.W.|title=The historical bases of the Rayleigh and Ritz methods|journal=Journal of Sound and Vibration|volume=287|issue=4–5|year=2005|pages=961–978| doi=10.1016/j.jsv.2004.12.021| bibcode=2005JSV...287..961L| url=https://www.sciencedirect.com/science/article/abs/pii/S0022460X05000362 |url-access=subscription}}</ref><ref name="Ilanko">{{cite journal|last1=Ilanko|first1=Sinniah|title=Comments on the historical bases of the Rayleigh and Ritz methods|journal=Journal of Sound and Vibration|volume=319|issue=1–2|year=2009|pages=731–733 | doi=10.1016/j.jsv.2008.06.001|bibcode=2009JSV...319..731I }}</ref> It has been called [[Ritz method]] after [[Walther Ritz]], since the numerical procedure has been published by [[Walther Ritz]] in 1908-1909. According to A. W. Leissa,<ref name="Leissa" /> [[Lord Rayleigh]] wrote a paper congratulating Ritz on his work in 1911, but stating that he himself had used Ritz's method in many places in his book and in another publication. This statement, although later disputed, and the fact that the method in the trivial case of a single vector results in the [[Rayleigh quotient]] make the case for the name ''Rayleigh–Ritz'' method. According to S. Ilanko,<ref name="Ilanko"/> citing [[Richard Courant]], both [[Lord Rayleigh]] and [[Walther Ritz]] independently conceived the idea of utilizing the equivalence between [[boundary value problem]]s of [[partial differential equations]] on the one hand and problems of the [[calculus of variations]] on the other hand for numerical calculation of the solutions, by substituting for the variational problems simpler approximating extremum problems in which a finite number of parameters need to be determined; see the article [[Ritz method]] for details. Ironically for the debate, the modern justification of the algorithm drops the [[calculus of variations]] in favor of the simpler and more general approach of [[orthogonal projection]] as in [[Galerkin method]] named after [[Boris Galerkin]], thus leading also to the [[Ritz-Galerkin method]] naming.{{cn|reason=need historian reference here|date=June 2024}}
