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I've always been curious about how great forgotten ideas are rediscovered. This question: Are there written (19th century) sources expressing the belief that the intermediate value property is equivalent to continuity? led me to the following paper:

Schubring, Gert: "Bernard Bolzano--Not as unknown to His contemporaries as is commonly believed?" Historia Mathematica, 20(1):45-53, 1993. (Elsevier's paywall, sorry I couldn't find a freed version.)

which says that "Herman Hankel is credited with having been the first to bring Bolzano to the general attention of the mathematical community in 1871." (and Schubring goes on to argue that in fact, Bolzano's work had probably been known to Crelle's circle in Berlin around 1825 before being forgotten.)

How did Hankel come upon Bolzano's work and recognize that Bolzano had priority over Weierstrass?

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Fair warning: this answer doesn't completely answer the question, but I think it may answer the question as well as it is possible to do.

The article which Hankel wrote (published 1971) that is generally credited with "rediscovering" Bolzano's work was an article in section 1 Theil 90 (Gregorius - Grezin) of Allgemeine Encyclopädie der Wissenschaften und Künste, one of the largest encyclopedias ever written (spanning 167 volumes despite being uncompleted). You can read this section freely here on Google Books. Hankel's article is on "Grenze" ("limits").

The relevant paragraph concerning Bolzano is on pages 209-210, while discussing the history of analysis. I'll provide a rough translation in modern English of this paragraph. I should note that I know next-to no German, so the translation is a lot of guesswork, and I could be wrong in some places. Anyone who knows German should feel free to note errors.

Even worse was another contemporary who remained then and now almost entirely unknown among mathematicians: We have to reclaim the priority of the first rigorous development in the series of algebraic analysis in favor of the excellent Bernhard Bolzano. Bolzano's notions of the convergence of the series are quite clearly and correctly written, its operations with infinite series all strictly proved, and nothing is wrong with the development of those statements for real arguments, which he supposes everywhere. In the preface he gives an apt criticism of the previous derivations of the Binominal theorem and then of ordinary unrestricted use of infinite series. In short, this work was not only a French art, he should be placed in this respect on the same level with Cauchy, and stated his thoughts in a pleasant way. But Bolzano remained unknown and was soon forgotten; Cauchy was the lucky one, the one praised as a reformer of science and whose elegant writings in a short time found general dissemination.

In this paragraph, Hankel basically credits Bolzano with developing much of the foundations of analysis independently of (and years before) Cauchy. However, Bolzano's work remained unknown, while Cauchy, who was well-connected in French math circles, found it easy to communicate his work. Hankel makes no mention of where or how he found Bolzano's work.

Some historical commentary is in order here. 1871 is a significant year; specifically, it's the year in which the Franco-Prussian war, a time of strong national pride in Germany and general dislike of all things French. The encyclopedia Hankel was writing in was intended to be something of an encyclopedia "for and by the German people". Hankel surely would not have been happy having to give the credit of developing analysis to Cauchy, a Frenchman. It was far better to give it to Bolzano. Sure, Bolzano wasn't the ideal German mathematician, having spent most of his academic career in Austria, and being as much a philosopher and theologian as a mathematician (and a controversial one at that), but he did speak and write in German, and equally importantly, was not French. And Bolzano did really do (for the most part) the things Hankel attributed to him. To be clear, I'm not accusing Hankel of any wrongdoing by pointing this out, only saying that he had considerable interest in attributing as much as he could to Bolzano.

However, there is something of an issue attributing the development of limits in analysis to Bolzano over Cauchy, though it is more philosophical than mathematical. Bolzano probably had a very different interpretation of his theorems than later readers. Indeed, in "The Mathematical Works of Bernard Bolzano", Steve Russ argues that Bolzano would not have thought of his theorems in terms of limits at all, which he would have associated with the very infinities he was trying to do away with. From pages 146-147:

However, the modern recognition of Bolzano's work raises a historical problem. From Hankel's article in 1871 to the extracts in Bitkhoff (1973) commentators have been inclined to give particular credit to Bolzano for matters which at the time he saw in a very different light from these later critics. We are thinking here of the arithmetic concept of limit and the concept of the convergence of infinite series that are commonly adopted today. These concepts had been used in some form for a long time, and judging from other examples in his writings, Bolzano would not have been too modest to claim them as new and original if he had regarded them as so. He does not do so. Undoubtedly he had great confidence in these definitions; they satisfied his conceptual requirements, he knew they would be fruitful and effective in the development of analysis, but never does he claim them to be his own...

It is commonly assumed that following the introduction of quantities labelled as ω, or Ω, possibly with subscripts, there is outlined in BL §14 ff. a fairly standard theory of limits. The irony is that Bolzano, along with most of his contemporaries, would have associated limits with infinite processes (or infinitely small quantities). And so he would, at this time, have been horrified to be associated with such a theory. Similar remarks apply to his work on the convergence of series. He believed he was treating the binomial series for negative and rational exponents in a purely finite manner. The way in which he use his ω quantities—variable quantities that can become smaller than any given quantity, or that can become as small as we please, naturally appealed to an infinite range of values. We might call them 'arbitrarily small quantities' . Rusnock suggests that such a concept of a variable that can become as small as desired was common at the time. It is some sort of counterpart to a physical variable quantity. He suggests that Bolzano's ω's might be interpreted as ranges of values containing zero...

That is to say that Hankel's judgement of Bolzano as an independent discoverer of the rigorous theory of limits in analysis, while correct in terms of mathematical content, is surely false if we take into account the philosophical aspects of his work. But of course even if Hankel realized this, he had little to gain by pointing it out explicitly in his article. In any case, neither Bolzano's methods nor his theorems were any less rigorous than Cauchy's; just his interpretation of the definitions and content of the theorems was different.

In any case, you'll note that Hankel didn't mention Bolzano-Weierstrass specifically, nor the intermediate value theorem (which was Bolzano's ultimate goal, toward which Bolzano-Weierstrass was just a lemma). That isn't terribly surprising. While Hankel was likely aware of Weierstrass' result (they were well acquainted with each other, Hankel having worked with Weierstrass in Berlin in 1861 prior to his doctorate) it was probably too recent to appreciate its significance, especially in the context of this sort of publication. It isn't even clear that Hankel had read the parts of Bolzano's work related to the intermediate value theorem; the parts which he cites in the article are elsewhere. So it wasn't actually Hankel who established Bolzano's priority here.

After Hankel's original citation, some mathematicians went back and read Bolzano's various works, reinterpreting them in more modern language. Otto Stolz in particular is credited with rediscovering and republishing many of his mathematical works in 1881. This included the relevant article, Bedeutung in der Geschichte der Infinitesimalrechnung, which predates Weierstrass and even Cauchy, establishing Bolzano's priority for the intermediate value theorem and the Bolzano-Weierstrass theorem. A number of other influential German mathematicians and philosophers also read Bolzano's works, and a few other interesting mathematical results were found. His legacy was likely cemented in the various historical notes in the highly influential (at least at Göttingen) Enzyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen, which mention his work several times as being far ahead of his time.

That answers the question on Bolzano-Weierstrass, but there's still a lingering unanswered question, namely, how did Hankel even find Bolzano in the first place (which was a big part of your original question). I don't know the answer to that, and to the best of my knowledge, no one does. Perhaps he had some concept that there was an Eastern European school of analytic philosophy which in the early 19th century dealt with issues related to infinities and was unhappy with Leibniz's informal approach to calculus. Or maybe, in writing the article, he got into a conversation with someone (possibly someone who was familiar with Bolzano's work from the period in the 1820s mentioned in the article you quoted, or possibly not even a mathematician) who suggested he look in that direction. Hankel did a decent amount of study on the history of mathematics (though his historical works typically had notable errors), also pointing out the importance of the work of Hermann Grassmann in 1867 two decades after Grassman had essentially stopped doing mathematics, so he certainly had some broader understanding of the works of his predecessors than the average mathematician of his time. How exactly Hankel found Bolzano is anyone's guess, but once he did, it's pretty clear that he wasn't just going to ignore it in his article, regardless of what Bolzano did/didn't think about how to interpret his results. Hankel died in 1873, just 2 years after the publication of the article, and to the best of my knowledge he didn't ever comment on Bolzano's work again. While one might be able to track the movements of various mathematicians from 1817 to 1871 to try to figure out how the idea might have been transmitted to Hankel (a seemingly Herculean task, though not technically impossible), at best we'd end up with a guess, and the truth is quite possibly lost to history.

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