Levenberg's original article “A method for the solution of certain problems in least squares” Does there exist any digital copy of the original article (or a transcript) K. Levenberg, A method for the solution of certain problems in least-squares, Quart. Appl. Math. 2 (1944): 164-168?
It is (obviously!) cited in numerous works, but the paper itself appears to have completely vanished. There seems to be not even any publisher, library, etc. proposing to buy a hard copy or anything alike. Apart from the citations all over the places, the paper has ceased to exist?!
 A: This question has been asked and answered on MathOverflow. I have replicated the accepted answer by Christian Clason below.

As far as I can see, there's no official digitized copy (yet), but I have a hard time agreeing with your statement that no library has a hard copy. For example, looking at the German Zeitschriftendatenbank, you can see that many universities have a copy of that issue, and if you (or someone you know) are at a German university, will probably send you a scanned copy upon request (look for "Fernleihe: Ja").
EDIT: Since there's no full bibliographical detail for the articles in such an old journal, you have to search for the journal (i.e., "Quarterly of Applied Mathematics") and not the article. This will get you a search result (it's the second in the list; the first is to the online version), and clicking "Link zum Gateway" at the bottom of the tab "Details" will lead to the "Gateway Bayern", where you will find a link to "SFX", where you can click on "Dokumentlieferung: Fernleihe im Bibliotheksverbund Bayern". Since I no longer have an account at TUM, I cannot follow from there, but I expect you will be presented with a form where you can enter the precise reference (title, volume, issue, pages etc.) you want to have sent.
Of course, this is not nearly as convenient as getting a current online publication, but that's why librarians are still needed. Moral of the story: If you can't find it online (in reasonable time), take the full reference and ask your librarian. They're trained to help you with precisely such a problem!

A: This link might be better. The paper is now open access and the pdf is searchable.
