Noether's theorems/History/The precursors

Noether opens her paper by naming the people who had done parts of this before her. That list is the best evidence available about what she thought she was adding, and it is worth reading before anyone else's account of the same question.

Whom she named

Her second sentence says that the combination of methods she is using — the formal calculus of variations joined to Lie's theory of groups — is not new for special cases. She then names five parties, sorted by which kind of group they had handled.

For special finite groups, meaning groups with a fixed number of parameters: Hamel and Herglotz.

For special infinite groups, meaning groups depending on arbitrary functions: Lorentz and his pupils, with Fokker named as the example, and Weyl and Klein.

She adds in a footnote that a paper by Kneser had just appeared, dealing with the construction of invariants by a similar method. Elsewhere in the paper she cites Engel. And the whole apparatus of continuous groups is Lie's, cited by name in her third footnote to a specific work and later to a specific theorem inside it.

The loci she gives

Her footnotes carry the references, and they are short. Hamel is cited to Mathematische Annalen volume 59 and to the Zeitschrift für Mathematik und Physik volume 50. Herglotz is cited to Annalen der Physik, fourth series, volume 36, with section 9 at page 511 singled out. Fokker is cited to the Verslag of the Amsterdam Academy, 27 January 1917. Kneser is cited to Mathematische Zeitschrift volume 2. Engel is cited to the Göttingen Nachrichten for 1916 at page 270. For further literature she refers the reader to Klein's second note rather than compiling it herself.

Lie is cited to Grundlagen für die Theorie der unendlichen kontinuierlichen Transformationsgruppen, in the reports of the Saxon Academy for 1891, which she abbreviates as Grundlagen throughout. She uses it three times: for the definition of an infinite continuous group as one whose transformations are the general solutions of a system of partial differential equations; for the normalisation that puts the identity at zero parameter values; and, at the decisive point of her fourth section, for Theorem VII at page 391, which is what lets her pass from a group of infinitesimal transformations back to a group of finite ones.

A caution on Hamel. Secondary reference lists in this area sometimes give Hamel's relevant papers as Mathematische Annalen volume 66 and a 1917 Jahresbericht note. Those are not the loci Noether gives. Either she is pointing at different papers or one of the two lists is wrong. This has not been checked and the citations above are hers, read from her own footnotes.

What is not settled

Naming a precursor is not the same as measuring what they anticipated, and Noether does neither. Her sentence says the combination of methods is not new in special cases. It does not say how much of either theorem was already in Hamel, or in Herglotz, or in Lie.

This volume does not answer that question. Answering it means reading the eight cited papers against the two theorems, and deciding for each what it establishes and under what restriction — work that has not been done here.

What can be said is where the boundary of the claim sits. Both theorems are statements at a generality nobody in the list attempted: arbitrary numbers of independent variables, derivatives of arbitrary order, transformations mixing dependent and independent variables, and both converses. Anticipation of a special case is not anticipation of that. But this volume states that as the shape of the question, not as its answer.

Provenance: the names, the loci and the finite-against-infinite sorting are read directly from the German original, which is public domain. None of the cited precursor papers was read. No priority claim of any kind is made or implied by the ordering above. See Sources.