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The title of the question more or less says it all. The question asks for precise and scientific descriptions (submission-rules, editor-behavior, referee-recruitment, anonimity issues in an age where handwriting was still the norm, all the way to rejection or acceptance-and-concomitant-galley-proof-process), examples (maybe even scans of some handwritten referee reports from this journal and this time), or pointers to articles in historical journals on precisely this topic.

The motivation is mainly historical interest, together with my having to deliver a referee report posing some problems, and some hope on my part that even from seemingly-ancient examples one can learn new (or at least be remembered of known) useful ideas for the perennial problem of refereeing mathematical papers.

Remarks.

  • It would not surprise me if (a question equivalent to) the present question had been asked recently on the web, but I did not find it (though not searching around for long).
  • It would also not surprise me (see below) if there already was a book like "The Acta Mathematica---Then and Now", or something like that, in folio format and replete with high-resolution photographic reproductions. But I did not find one. There is a biography on Mittag-Leffler by A. Stubhaug. There is a book on letters between Poincaré and Mittag-Leffler by P. Nabonnand, but both seem not to contain information on the refereeing process of the Acta Mathematica in the early days, let alone a dedicated study.
  • I decided against phrasing the questions in the form "How were mathematical referee reports written in the pre-typewriter-age?" since this would involve the ill-defined concept pre-typewriter-age. Typewriters gradually grew into being during the 1800s, notable hot-beds having been Italy and the USA. One could have phrased the question as "What are concrete, well-documented examples of non-typewritten mathematical referee reports?" but it seemed to me that localizing at the Acta Mathematica could be a more fruitful and focused historical question. This journal's history is likely to be very well documented. The late 1800s is a period which sits squarely within what is conventionally called modernity, and many original documents are bound to be extant even today, especially in a country as---relatively---untouched by the turmoils of the twentieth century as Sweden. I chose the end-point 1918 of the time-interval asked about for no precise reason.

Most importantly, the journal is operational to this day, and there may be many Swedish mathematicians among MO users who are in the know, or even actively involved in the Acta Mathematica. I did decide upon asking it here---and not in some other, more historical forum---consciously.

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    $\begingroup$ This question may also be suitable for History of Science and Mathematics. $\endgroup$ – Danu Jul 12 '17 at 15:00
  • $\begingroup$ A good person to ask is June Barrow-Green, who wrote a book about Poincaré's work on the three-body problem. $\endgroup$ – Fernando Jul 18 '17 at 19:26
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Looking over the following references, I believe that they contain some of what is being asked after:

Nickerson, Sylvia (2012). Referees, publisher’s readers and the image of mathematics in nineteenth century England. Publishing History, 71, 27-67. Link.

(Sample quotation: "The article examines the processes of refereeing, and the role of the editor, at nineteenth century mathematical journals by looking at the Cambridge Mathematical Journal and Acta Mathematica," p. 27.)

One of the references in the above manuscript is Chapter 8 (pp. 139-164) in an earlier work:

Parshall, K. H. (2002). Mathematics Unbound: The evolution of an international mathematical research community, 1800-1945 (No. 23). American Mathematical Society. Chicago. Link.

(Chapter 8 is entitled: "Gosta Mittag-Leffler and the Foundation and Administration of Acta Mathematica.")

And among the many references in this latter chapter is the following article of potential interest:

Domar, Y. (1982). On the foundation of Acta Mathematica. Acta mathematica, 148(1), 3-8. Chicago. Link.

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    $\begingroup$ Thanks. Nickerson's article is certainly very relevant, and perhaps the closest thing there is to what I am asking for, even though the emphasis is clearly on British journals. One clear difference that seems very plausible in retrospect but which was erroneously expected by me, extrapolating from the present: authors apparently hardly ever received a report from a referee, all they got most of the time was a writ from the editor('s office). This renders the handwriting-issue more or less moot. $\endgroup$ – Peter Heinig Jul 14 '17 at 5:44
  • $\begingroup$ @PeterHeinig Yes; I would be surprised if there currently exists a more relevant article; I was a bit surprised to find it! Perhaps Nickerson would be worth contacting... $\endgroup$ – Benjamin Dickman Jul 14 '17 at 14:10
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(Not an answer, too long for a comment.)

Is it even clear that there was a refereeing system in place? As opposed to, say, the editors looking through each submission and giving a thumbs up/thumbs down.

There is an interesting story about a 1936 paper of Einstein and Rosen on gravitational waves, which suggests that even that recently, peer review worked very differently. Einstein and Rosen had discovered an erroneous argument proving mathematically that gravitational waves couldn't possibly exist, and they submitted it to Physical Review. The handling editor, John Tate Sr., had usually accepted Einstein's papers without any refereeing process, but felt skeptical about the result. So he sent it to Howard P. Robertson for review, who wrote a ten-page report explaining why the argument didn't hold water. Einstein was not happy about this, and wrote the following letter in response:

Dear Sir,

We (Mr. Rosen and I) had sent you our manuscript for publication and had not authorized you to show it to specialists before it is printed. I see no reason to address the—in any case erroneous—comments of your anonymous expert. On the basis of this incident I prefer to publish the paper elsewhere.

Respectfully,

P.S. Mr. Rosen, who has left for the Soviet Union, has authorized me to represent him in this matter.

Einstein never again published in Physical Review.

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    $\begingroup$ Great story. Did Einstein publish the erroneous paper as he threatened in his letter to Tate? By the way is this the father of Tate the mathematician? $\endgroup$ – Mikhail Katz Jul 13 '17 at 15:16
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    $\begingroup$ The paper did end up published but with completely different conclusions; Einstein did end up convinced that the argument was bogus. Yes, it's the father of John Tate, Jr., mathematician. $\endgroup$ – Dan Petersen Jul 13 '17 at 21:10
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    $\begingroup$ Thanks for this answer. While not being stricly relevant to the Acta Mathematics, it adds a very interesting unusual aspect that I was not consciously aware of: the expectation of an author not to be peer reviewed before being printed. Nowadays, more often than not, authors self-breach this expectation by self-publishing. $\endgroup$ – Peter Heinig Jul 14 '17 at 5:37
  • $\begingroup$ Really a great story! $\endgroup$ – Piotr Hajlasz Jan 15 at 4:16
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To get this wrapped up, I'll write a summarizing answer, acknowledging the input received from the answers given so far.

Briefly, the answers to the question in the title seem to be, more or less, the following:

  • Sadly, the hope for making available a trove of supercentenarian handwritten referee reports, locked away somewhere in Sweden, written by professional mathematicians from the 19th century, full of masterful work-techniques, useful also in the electronic age, will not be fulfilled.

  • Sadly, there appears little to be learnt from the early days of the Acta Mathematica for the very important question of contemporary mathematical refereeing. (Only speaking of this journal's prehistory, not criticizing it in its current form, needless to say.)

  • While one can of course not say that referee reports were never written or sent in those days, this was not the rule. (In both senses of that phrase.) Refereeing and editorial decisions were less formal than they are today. Most of the time, an author never received a referee report. This renders the anonimity/handwriting issue moot.

  • The answer to the question in the title, more or less, is: This is unknown, and probably forever will be. Flippantly put, the question in the title is equivalent to "How did Gösta Mittag-Leffler function?"

  • Dan Petersen kindly pointed out (see the answer) an unexpected aspect: there sometimes was the expectation on the part of the author for their manuscript to be printed before being shown to other scientists.

  • Benjamin Dickman kindly pointed out a very relevant professional historical article, namely

S. Nickerson: Referees, publisher’s readers and the image of mathematics in nineteenth century England. Publishing History, 71

The emphasis of this paper, as its title says, is on British publications, in particular books. But it contains a little which is directly relevant to this question. To keep this strictly relevant to this title's thread, I refrain from relating any of the content the article contains on British publications, and only give what is relevant for the Acta Mathematica in its early days, and of these, only what is relevant for its refereeing system (if one can call it "system", back then I mean), not for its foundation or funding.

Nickerson in loc. cit. quoting J. Barrow-Green: "Barrow-Green makes it clear in whose hands control of the journal rested:" [Now the quote of Barrow-Green's:

Once publication began, Mittag-Leffler became the driving force, and [Sophus] Lie effectively dropped out of the picture. And Lie was not the only member of the editorial board not to play an active part in the journal’s early years. Of the fourteen original members of the editorial board, only Malmsten and Zeuthen made any significant contribution. It was Mittag-Leffler who shouldered the responsibilities and Mittag-Leffler who ensured the journal’s survival.

Nickerson loc. cit. goes on to write

The journal was able to attract submissions from some of the mathematical community’s most talented members due to Mittag-Leffler’s many personal contacts. Moreover, the portrait of Norwegian mathematician Niels Henrik Abel on its frontispiece provided a standard to which the journal sought to have its submissions aspire. Mittag-Leffler had the ultimate say concerning what was published in Acta. As editor, he was the gatekeeper who ensured that the quality of articles in his journal remained high.

Nickerson then gives an example of the above: Mittag-Leffler's handling of Cantor's submissions (he asked Cantor to withdraw one, which hurt Cantor; see Nickerson's article for more).

Nickerson loc. cit. also says the following about the refereeing system:

As the editor of a mathematical journal, Mittag-Leffler held a great degree of control over what was published, although he likely consulted, on an informal basis, his network of mathematicians.

and calls this (again, this is a conclusion about the ancient modus operandi of the Acta Mathematica; it is not a summary of its current editorial system, which is more formal and less centralized):

a centralized and fluid approach when deciding what to publish

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    $\begingroup$ @BenjaminDickman: thanks for pointing out. I do not know why. I do not recall to have given this decision a thought. I now unaccepted and accepted yours. $\endgroup$ – Peter Heinig Aug 8 '17 at 15:39

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