## Bützberger's Work on Steiner

The following account of Bützberger's work on Steiner is taken from the Thesis:

*C F Geiser and F Rudio: the men behind the First International Congress of Mathematicians*by Stefanie Eminger (2014) pp 95-101: http://www-history.mcs.st-and.ac.uk/Publications/Eminger.pdf.Bützberger was an avid Steiner scholar. His scientific estate in the ETH Library Archive: (Hs 194), contains a host of letters and notes connected to his research. The correspondence indicates that Bützberger's research lasted for several decades. Furthermore, it also shows that he was in touch with a number of his colleagues on the organising committee -- Carl Friedrich Geiser in particular -- as well as fellow mathematicians in Switzerland and abroad. Correspondence with Johann Heinrich Graf, Julius Gysel, and Georg Sidler deserves a special mention here, but the list also includes Moritz Cantor, Arnold Emch, Wilhelm Fiedler, and Theodor Reye.

As one would expect, there are sheets and slips of paper with notes on the topic of his Steiner research, some of them in French or English. There are also a number of manuscripts and drafts of papers relating to Steiner. Among these, a handwritten, 125-page biography of Steiner and a draft of a book on Steiner's mathematical manuscripts 1823-1826 stand out. It seems that this book,

*Jakob Steiners Nachlass aus den Jahren 1823-1826*, was never published. However, Bützberger did publish two papers on Steiner's life:

*Jakob Steiner bei Pestalozzi in Yverdon*(Jakob Steiner at Pestalozzi's in Yverdon) (1896), and

*Zum 100. Geburtstage Jakob Steiners*(On the Occasion of Jakob Steiner's 100th Birthday) (1896), in which he focuses on Steiner's notebooks as a student at Pestalozzi's school and at university. Letters from Geiser suggest that it may have been the same paper: In November 1895 he arranges for Bützberger's paper to be published in

*Schweizerische Pädagogische Zeitschrift*and puts him in touch with its editor, Friedrich Fritschi (1851-1921). Furthermore, he proposes a re-publication in

*Zeitschrift für mathematischen und naturwissenschaftlichen Unterricht*. Indeed, in February 1896 he recommends the paper to the journal's editor Immanuel Carl Volkmar Hoffmann, as Geiser himself 'could not provide [Hoffmann] with the desired article [on Steiner]'. These short letters nicely illustrate Geiser's remarkable networking skills.

There are several postcards in Bützberger's estate in which colleagues congratulate him on his paper -- "paper" in the singular, without any specification as to which one is being referred to. Nevertheless, a card by Cantor from 1896 represents the sentiment echoed in these cards:

Dear Colleague!

Thank you very much indeed for your exceptionally intriguing paper on Steiner's background. Like all our peers, I dare say, I eagerly anticipate the continuation of your publications.

Yours respectfully

M. Cantor

In fact, it was Geiser who suggested that Bützberger send his paper to Cantor: Thank you very much indeed for your exceptionally intriguing paper on Steiner's background. Like all our peers, I dare say, I eagerly anticipate the continuation of your publications.

Yours respectfully

M. Cantor

However, for some reason unknown to us the paper was not published in Cantor's journal. Perhaps Bützberger did not send it to Cantor after all, and Cantor commented on an already published version of it.Should you have reservations about entrusting your fine work to a non-mathem[atical]journal, do contact Prof. Cantor in Heidelberg, so that the "Zeitschrift für Mathematik & Physik" includes the article. Do not hesitate to say that I asked you to send the paper if you think that this would have an impact.

Bützberger must have spent much of his time trying to track down any remaining relatives and friends of Steiner, as well as pictures of and documents relating to him. Over the years he built up quite an extensive network of contacts that could provide him with information. Among them was his own father-in-law Johann Kohler, who lived about 20 km from Steiner's hometown. Letters in Bützberger's estate suggest that Kohler did a lot of research in the region. Friends of Bützberger, such as Johann Petri, and relatives of Steiner helped as well. Gysel tried to track down a certain Conrad Maurer for him, who seems to have been Steiner's mathematics teacher at Pestalozzi's school.

Bützberger showed a particular interest in Steiner's family tree, especially how Steiner and Geiser were related.

Another very profitable source of information was Sidler, who inherited some of Steiner's manuscripts and for whom Steiner had been a fatherly friend. Indeed, Bützberger reports in his Sidler biography that

In fact, in 1906 Bützberger asks Sidler's permission to include this anecdote in his 'soon to be completed work'. In his reply, Sidler suggests changes to the manuscript and gives background information on some points, such as Steiner's character traits and his heirs. Moreover, he recounts some anecdotes that explain Steiner's difficulties with writing and the end of his friendships with Carl Jacobi, Lejeune Dirichlet, and his doctor Johann Schneider. Sidler also comments on the rumour about Steiner's illegitimate daughter. Apparently Bützberger planned to ask Geiser for comments on the manuscript as well.Sidler visited[Steiner]almost daily during[Steiner's]bitter time of suffering. His dear widowed mother[...], who had lived with him since1861, ministered to the terminally ill geometer as well, for which he was very grateful.

Furthermore, Sidler responds to Bützberger's accusations against Graf and tries to calm him down: According to Bützberger, Graf included some of Bützberger's results regarding Steiner's years in Yverdon in his own papers but failed to reference them. In his Steiner biography, Graf remarks that the Yverdon section is based on Bützberger's paper. Comparing the two papers today, some passages are almost identical and would require better referencing. However, Bützberger seems most outraged about Graf's 1905 paper, in which he is not referred to at all. The matter of dispute is the year of Steiner's arrival in Yverdon: Bützberger, believing that he settled the issue, cannot understand why Graf revived the debate. This is explained in letters to one Dr Israel, in which Bützberger complains about Graf, and to Graf himself, in which he expresses his displeasure. In his reply, Graf insists that this must have been a misunderstanding and encloses a postcard with Steiner's birthplace, 'to prove [that he] is not cross with [Bützberger]' -- a rather curious reaction! Whilst Bützberger had a point, his reaction was rather dramatic given the scale of this academic dispute. In his letter, Sidler urges Bützberger to remain professional:

The other letters that Bützberger and Graf exchanged suggest that their relationship was generally professional.Your work should become a classic in memory of the great geometer & it should go without saying that the preface should be nobly written. As an example, look at the second part of Poncelet's "Propriétés des figures projectives". Surely everybody laments that Poncelet got too carried away with polemics there; this tarnishes Poncelet's memory.

In fact, both were members of the Steiner-Schläfli Committee, as was Geiser, along with Sidler and five more mathematicians. In a circular of October 1895 the committee explains that its main objective was to raise money for tombs for Steiner and Schläfli. Graf writes that Bützberger and a colleague, Christian Moser, found Steiner's lost grave, and Sidler donated a small tombstone. However, as the graveyard was closed down, the committee successfully applied for permission to exhume Steiner. The committee organised Steiner's re-interment and the erection of a grand tombstone on Schläfli's grave in 1896 to celebrate Steiner's centenary and the anniversary of Schläfli's death (who had died in 1895). It seems that the committee disbanded afterwards.

Letters from Reye and Emch to Bützberger suggest that he might have planned to publish Steiner's posthumous works. As Hollcroft writes, Bützberger 'died before he had completed the work of editing the Steiner manuscripts'. However, he did publish

*Über bizentrische Polygone, Steinersche Kreis- und Kugelreihen und die Erfindung der Inversion*(On Bicentric Polygons, Steiner Series of Circles and of Spheres, and the Invention of Inversion) in 1913, dedicating a separate section for each of the three topics. The 'carefully and clearly written' book was reviewed favourably, with particular praise for the historical background: 'The numerous historical and biographical facts add particular value to this book'. Furthermore, reviewers note that Bützberger used elementary geometric methods (instead of analytical methods), such as reciprocal radii. In the second section in particular, he treats 'Steiner series of circles and of spheres; here [he] follows Geiser's view: "Einleitung in die synthetische Geometrie", last chapter "Das Prinzip der reziproken Radien" '.

All the reviews available agree that the third section is the most intriguing one. Danzer for example writes:

Specifically, Bützberger cites a document that he found among Steiner's manuscripts, which proves that Steiner did invent the inversion, as had been suspected previously. Presumably in preparation for the book Bützberger copied down a dozen relevant papers in a scrapbook, from 1896 onwards. Among the papers are excerpts from J Plücker's papers on Steiner's solution of the Malfatti problem (by H Schröter),I think that Bützberger's book is very interesting; the first and second sections, in which hardly any new material is included, less so, but certainly the third section, which provides an insight into "Master Steiner's" workshop.

As an aside, a couple of letters that Bützberger exchanged with Sidler have mathematical content as well. Sidler lent his friend mathematical books from his extensive library and pointed out further reading on the subject of the geometry of triangles, Sidler's particular interest. In 1898 Sidler sent Bützberger a copy of a proof by Droz-Farny concerning a property of triangles, 'as I presume that you will take as much pleasure in it as I have'.

According to Sidler the theorem was first proved by 'Brocard junior' (presumably he meant Henri Brocard) in

*Mathésis*in 1896, but Droz-Farny's proof was much simpler.

The theorem is the following:

In the plane of any given triangle

*ABC*there are two (always real) points*P*and*Q*, which have the property that when one extends each of the rays*AP*,*BP*,*CP*or*AQ*,*BQ*,*CQ*up to the points of intersection with the opposite sides*A*',*B*',*C*' or*A*",*B*",*C*", the resulting 6 line segments are of equal length:*AA*' =*BB*' =*CC*' =*AA*" =*BB*" =*CC*". These points*P*and*Q*are the foci of the ellipse with least surface area circumscribed around the triangle*ABC*.