Remarks about J.G.Hermes

Johann Gustav Hermes was born on June, 20th 1846 in Königsberg (now Kaliningrad). After having served in the military during the Franco-Prussian war of 1870/71 he graduated with the examination for high school teachers in mathematics (in German called Staatsexamen ) in 1872. He achieved his Ph.D. in Königsberg 1878 with the thesis [Hermes 1879]. In 1873 he became teacher at the Progymnasium des königlichen Waisenhauses  (high school of the royal orphanage) in Königsberg. In 1883 he was promoted to the rank of a senior teacher at this school. In 1893 he became senior teacher at the Georgianum  (another high school} in Lingen (Ems)^[1]. There he was awarded the title Professor  in 1896^[2]. From 1899 to 1906 he was head teacher at the Ernst-Moritz-Arndt-High-School in Osnabrück. In 1906 Hermes retired because of poor health. He died in 1912 in Bad Oeynhausen (a spa town). Hermes' grave is located in Osnabrück. His work on cyclotomy was performed from 4.11.1879 till 15.4.1889 during his time in Königsberg.

In his paper on cyclotomy [Hermes 1894] Hermes writes about the Diarium  that has been archived at the University of Göttingen. This diary fills the so called Hermeskoffer  (case of Hermes), which is mentioned in numerous popular mathematical treatments. Most of the authors write about the contents of this suitcase by hearsay. The following description is based on my own personal inspection.

Physical Appearance

The case, which is about 50cm $\times$ 65cm $\times$ 10cm, is made of wood and coated with firm ocher fabric, some kind of twill. Five nails are driven into the front cover as armoring and ornament.

Within the case there is a loosely connected stack of paper of more than 200 sheets, each one about 47cm $\times$ 55cm. Some sheets, however, are folded, so that they are of double size.

Hermes apparantly adapted a big hardback register with checkered pages for his purpose. Those registers were frequently used in offices for bookkeeping in these times. Some of his writing is on the original pages, other writing and drawing was performed on separate sheets and these were glued carefully onto the pages. The binding suffered badly by this method and was repaired in a rough-and-ready-way at several spots.

The cover of the register is made of purple synthetic leather^[3]. Because of the binding one may take the whole volume out of the case at a single blow. If one does so, one can see the inner chemise of the case. It's made of linen showing blue and white stripes. In the middle of it, there is a clasp made of leather and below this someone has written the name of the owner J. Hermes  in large rounded letters and with dark ink right onto the linen.

A label has been glued to the middle of the cover. It has the form of two crossing ellipses, somewhat like the central symmetric variant of the logo of the Japonese car manufacturer Toyota. In Germany Toyota uses the claim Nichts ist unmöglich (nothing is impossible), which could well be the motto of Hermes' work. In English speaking countries Toyotas claim was for several years You asked for it! You got it!, maybe a nice motto for Hermes too. On this label you find in dark ink and pin sharp handwriting (Kurrentschrift):

Turning the cover you see the title page with the inscription:

Below these lines one finds a circle with a diameter of about 10cm. On its circumference several numbered points have been drawn. Along the inner side of the circumference Hermes wrote in tiny small caps a dedication:

Den Manen Richelot's gewidmet in dankbarer Verehrung. 4.11.1879 – 15.4.1889 – J. Hermes.

(Dedicated to the manes of Richelot with thankful reverence. 4.11.1879 – 15.4.1889 – J. Hermes.)

Leaving the circle on the right side, a radius is streched almost unto the margin af the page. The end point of the radius is marked, and a cryptic annotation is written there: Radius = der Einheit, die für 1km 168,32 wäre  ^[4]. I can hardly translate this, because I dont even understand the meaning of the German sentence. If someone has any idea what this is about, please let me know.

Contents

The pages, which are numbered with Pagina 1, Pagina 2  etc. are mostly filled with tables, some of them graphically designed and colored. For example, on Pagina 7  you find a matrix with 32 rows and 64 columns. Each entry of the matrix is drawn as a tiny circle. Each circle resembles a tiny clock face with numbers on the circumference, not with the twelf numbers of a regular clock but with only 3 or 5 different numbers at different positions. I did not try to make a meaning out of this.

Some pages contain stuff that is only marginally connected with the 65537-gon. So on Pagina 8a  the construction of the regular 17-gon is given (perhaps for the purpose of recreation). Hermes comments on it with the following remark:

…dieselbe ist mit der geringen Anzahl von 22 Gebilden (10 Geraden und 12 Kreise) ausgeführt, den gegebenen nicht mitgerechnet.

(…idem is accomplished with the small number of 22 figures, 10 straight lines and 12 circles, the given not included.}

Moreover there are endless columns of figures, tables of signs, some of them 256 rows and columns in size, looking like patterns for tailors, checkered pages with only a few fields marked with hatching or with color. On the other hand, mathematical prose, like theorems and proofs are very rare.

In [Hermes 1894] Hermes presents the table of contents of his Diarium (see fig.1). In the Diarium itself the same outline can be found, but as you notice, the arrangement is a little bit chaotic. Hermes' Diarium is surely not a very structured publication, but, as the name indicates, a diary. However you may extract from the outline that Hermes did proceed in a way comparable to the method described in this blog. For example he computed the numeric values of almost all Gaussian periods by trigonometry (Pag. 100-215), which make up more than half of the Diarium. We used these values exclusively for the determination of the sign of the square roots for the Gaussian periods (see Hermes Pag. 97). But it may be that Hermes used them for more advanced computations too (see paragraph below).

Verification

For a long time most mathematicians believed that nobody did ever check Hermes' work. But in the years around 2000 the Australian mathematician Joan Taylor found a method (without knowing the details of Gauss' work or Hermes' construction at that time), how to gain a single square root expression for $\cos{2\pi/n}$, which contains only square roots and integer fractions. The precision of the available computers was too low, however, to apply her program to the 65537-gon. So she started to go deeper into the theory and to write another program informed by Gauss' methods, which assembled step by step (similar to our program) the root expressions of each level with help of the Gaussian periods.

During this work Joan Taylor finally came across the 1894-paper of Hermes. She found out that Hermes used an almost identical approach and modified her second program to print analogous tables like those on Hermes' pages 178 to 180. The correspondence was excellent – only a few typos could be found.

In 2003 Joan Taylor summarized both her methods in the paper [Taylor 2003] and sent it to the Lawrence Berkeley National Laboratory in California, where the ARPREC package, an arbitrary precision  software, was in development. She could persuade the scientists, to run her first program adapted for the ARPREC package and applied to the 65537-gon. An extremely high number of valid decimal places was necessary, because the starting point for the computation had to be very precise values for the cosine-part of the roots of unity.

The result was an expression for $\cos{2\pi/65537},$ with square roots only nested up to a depth of 15 [Taylor 2004]. The numerators of the fractions within these roots, however, are very  big – up to 20000 digits. Such an expression, of course, cannot be used for an explicit construction of the polygon. But up to now it surely is the most compact root expression for $\cos{2\pi/65537}$ with only  12.5MB. This is much  less than the expression you could get, if you successively insert our root expressions into each other (see chap.8).

In summary one can say, that at least a part of Hermes' Diarium has been verified by the work of Joan Taylor.

Attempt of a Fair Judgment

As already said, Hermes' work is mentioned in almost every publication, when something is written about cyclotomy. However, Hermes is often seen as a crank, who wasted ten years of his life with a useless activity. For example, here is the judgment of George E. Martin in [Martin 1998] on page 47:

Connected with the Gauss-Wantzel Theorem, there is one story whose telling cannot be avoided. From the theory, it is evident that a regular 65537-gon can be constructed with the ruler and compass. One person supposedly did so. Oswald [sic] Hermes spent ten years of his life carrying out the necessary constructions; the manuscript was deposited at the University of Göttingen in Germany. Nothing more need be said about Hermes.

Probably, Martin caught the wrong line on the pages of the Mathematics Genealogy Project, where some Oswald Hermes shines up just below the entry for the right Johann Hermes.

The following story about Hermes is told frequently, we give it in the version of [Stewart 1988]:

Bell tells of an over-zealous research student being sent away to find a construction for the 65537-gon, and reapprearing with one twenty years later. This story, though apocryphal, ist not far from the truth; Professor Hermes of Lingen spent ten years on the problem, and his manuscripts are still (I believe) preserved at Göttingen.

I could not verify the source in the book of Bell [Bell 1965]. Searching for the number 65537 in this book, there are matches only on pages 65 and 67, when Bell writes about Fermat (provided that Google doesn't lie).

On the web you find numerous elaborations of this story. Felix Klein (1849-1925) as well as Edmund Landau (1877-1938) are put into the role of Hermes' supervisor. Indeed, Klein was marginally involved in Hermes' work: In [Klein 1895] he mentions Hermes' diary rather benevolently (see also fig.2). In 1879, when Hermes started his diary, Klein taught in Munich, in 1889, when Hermes completed his diary, Klein was already in Göttingen. So, with great probability, Klein was not the one who had sent away  Hermes. Landau, on the other hand, got his first appointment as lecturer in 1908, only four years before Hermes' dead, so he cannot be the supervisor either.

From the biografical data one can see that Hermes wrote his Ph.D. thesis when he already worked as a school teacher in Königsberg. In Germany this kind of conferral of a doctorate is known by the name externe Promotion. He started his teacher job in 1873 and completed his thesis in 1878. The Mathematics Genealogy Project  doesn't provide us with the name of Hermes' supervisor. But it is very suspicious that from 1843 till 1875 the mathematics chair in Königsberg was held by Friedrich Julius Richelot – the very same Richelot, who constructed the 257-gon (see [Richelot 1832]). Richelot died in 1875 at age 67, when he was still on duty. Hermes' thesis is about cyclotomy, so one might suppose that Hermes started his thesis under supervision of Richelot but had to complete it under someone else. This assumption is backed up by the somewhat strange dedication on the title page of the Hermeskoffer. Richelot's successor in Königsberg was Heinrich Weber (1842-1913), one of the teachers of David Hilbert (1862-1943).

In [Scharlau 1989] one finds more information about the mathematics in Königsberg at the time of Hermes. Richelot's predecessor on the mathematics chair was Carl Gustav Jacob Jacobi (1804-1851). Jacobi made the mathematics department of Königsberg one of the leading institutions in German speaking Europe. The list of his successors from 1827 till 1900 is impressive: Jacobi, Richelot, Weber, Lindemann, Hilbert, Minkowski, Hölder. A second mathematics chair was established not before 1899..

If Hermes wrote his thesis in Königsberg, which is almost sure, because Hermes added the line Königsberg, den 4.Februar 1878 to the end of it, the most probable supervisor is Weber. Maybe that Weber was not very interested in cyclotomy and did not encourage his scientific stepson. May also be that this is the truth behind the story of the over-zealous student. I leave the decision to the reader.

However, Hermes' thesis was published already in 1879 in (Crelles} Journal für die reine und angewandte Mathematik. So it seems that the editors of this renowned journal did not jugde Hermes' work as inferior. The same year Hermes started his Diarium. In [Hermes 1894], which is the tip of the iceberg of his Diarium, Hermes mentions his thesis explicitly. He writes:

Es schien nun die Frage von Interesse, ob sich bei beliebigem $\mu$ die Endformeln (d.s. die Perioden $\eta^{\nu}$ verschiedener Ordnung) als Formeln in $\mu$ darstellen lassen. Das ist von mir in Crelle's Journal 87 versucht und bis zur Ordnung $\nu \leq \mu+2$ durchgeführt worden. Die höheren Ordnungen boten Schwierigkeiten dar bei allgemeiner Behandlung und es wurde daher der noch übrige specielle Fall: $\mu = 4,$ welcher der Teilung des Kreises in 65537 Teile entspricht, in Angriff genommen.

The question arose, if for arbitrary $\mu$ the final formulae (the periods $\eta^{\nu}$ of different order) can be expressed by the $\mu$. I tried this in Crelle's Journal 87 and succeeded up to the order $\nu \leq \mu+2$. The higher orders became difficult, when treated by the general method, and so the missing special case $\mu = 4$, which corresponds to the partitioning of the circle in 65537 parts, was tackled [in the Diarium].

Hermes uses $\mu$ as the exponent of the Fermat prime $2^{2^{\mu}}+1$, the periods are the well known Gaussian periods, and the order  is called by Hermes, what is called by us level  of the period (Hermes starts his count of orders with $1$, where we start with $0$). Hermes obviously had to stop at order 6 (of 16) when treating the 65537-gon in his thesis. So he looks at his work in the Diarium as completion of his thesis. He had to change his approach in order to overcome the difficulties of the general method. He writes:

Zur wirklichen Durchführung reichen indessen die Zerlegungen nicht hin, vielmehr mußte auf die ältere Methode von Gauß zurückgegriffen werden, doch konnten dabei in der Rechnung Erleichterungen eintreten, so daß die Aufstellung und Auflösung der quadratischen Gleichungen verhältnismäßig einfach wurde. Zu diesen Vereinfachungen gehört:

a) die Einführung einer linearen Verbindung der Perioden statt des Quadrats einer Periode, […]

b) Die an sich zwar unwesentliche, hier aber […] zu empfehlende Darstellung einer ganzen Zahl als Summe aus Potenzen von 2.

For the effective execution the decompositions are not sufficient. One has to resort to the older method of Gauss. Nevertheless, the calculations could be made somewhat easier, so the composition and solution of the quadratic equations became relatively simple. The following points were essential for the simplifications:

a) introduction of a linear combination of the periods instead of the square of a period.

b) the representation of an integer as a sum of powers of 2, which is not essential but highly recommended in this case.

So, there is some evidence that Hermes started the work on his Diarium on his own accord, because he wanted to fill the gaps in his thesis, and not because some ill tempered professor sent him away  with a futile job.

Neither his thesis nor his work on the 65537-gon are a simple matter of diligence. While we could succeed in finding the linear combinations for the products of Gaussian periods by brute force with aid of the computer, this was no option for Hermes. He had to spend more igenuity, to have a chance of finding the desired expressions. Without this ingenuity his first publication would not have been accepted as a thesis or for publication in Crelles Journal. However, one must admit that the continuation of his thesis within the Diarium went far beyond the scope of ordinary mathematical publications.

If one wants to speculate about the reason, why someone spent ten years of his life to solve a problem, nobody was really interested in, the following theory seems to be much more plausible than the story of the over-zealous student:

Hermes worked on his Diarium from 1879 till 1889, from his 33rd to his 43rd year, while he taughed at the royal orphanage in Königsberg. The lessons, he had to give there, surely were of a very elementary level. As a consequence, Hermes was not challenged by his job. How could a teacher in the 19th century, notoriously underpaid, find an occupation for his restless mind? Theater, musical performances or traveling he could not afford. Radio, TV and the internet were not available. What else could he do than try his wits with some demanding work. On the other hand, the work should not be too  demanding, so that he could spend some hours on it, even when he had a hard day at school – think of the many pages he filled with trigonometric calculations to find the numeric values of the periods.

By no means are the papers about cyclotomy the only publications of Johann Gustav Hermes. In [Lemmermeyer 2000] another paper of Hermes [Hermes 1889] is listed as the 59th proof of the law of quadratic reciprocity ^[5]. Searching for J. Hermes in the Jahrbuch über die Fortschritte der Mathematik, which was a predecessor of the Zentralblatt für Mathematik  in the years from 1868 till 1932, one finds a total of 16 papers, beginning in 1882 and ending in 1906. Some of them were published in prestiguous journals like Mathematische Annalen. Starting in 1897, Hermes interests and the topics of his publications shifted more and more towards questions of teaching mathematics in high school.

Hermes certainly was not a crank. Cranks try to prove Fermats Last Theorem, using only elementary mathematics. Hermes, on the other hand, fulfilled all the requirements, to cope with the problems, he tried to solve. And, very probably, he did succeed, even if the final proof of the correctness of his calculations will never be given, and the 200 pages buried in the Hermeskoffer will never be published.

Postscriptum

By coincidence ^[6] it happened, that on August, 16th 2012, when the manuscript of the German edition of this blog was almost finished, an essay about the Hermeskoffer was published in the German newspaper Die Zeit. The essay was written by Frank Fischer (see [Fischer 2012]) with some photographs of the Hermeskoffer by Christian Malsch.

The very readable essay contains some variant of the story of the over-zealous student (with Felix Klein as bad guy), but Fischer shares our opinion that the story is rather a myth. The pictures show that the name of the owner J. Hermes  is also written on the outside of the suitcase, a fact, strange to say, not remembered by me, when I wrote this blog. Moreover, Frank Fischer tells his readers that the Mathematical Institute in Göttingen nowadays is very restrictive, showing the case to interested persons, because of the case's alarming state of conservation. In the seventies of the 20th century, when I was a student in Göttingen, the practice was much more liberal. In those days every student simply could ask the custodian at the entry of the library in the Mathematical Institute at the Bunsenstraße in Göttingen for the Hermeskoffer. The suitcase was then taken out of a separate cabinet and delivered to the student without any further question. The order Bring it back hither! were the only words spoken by the custodian.

After some remarks about the aesthetic qualities of the diagrams and tables in the Hermeskoffer, Frank Fischer writes at the end of his essay:

Vollständig nachgerechnet hat Hermes' Kofferkonvolut bisher niemand. Wozu auch? Wenn man wollte, könnte man heute die Konstruktion recht problemlos mit einem Computer programmieren. Doch wer braucht schon ein regelmäßiges 65537-Eck?

A complete check of Hermes' convolute is still missing. Wherefore should it be done? Today one could with relatively little effort write a computer program to achieve the construction, if one really wants to. But who does need a regular 65537-gon?

Without any doubt, this statement tells the truth, and at the same time it is a good close for this blog. Why should anybody write a program, to print the construction of a 65537-gon? The standard answer for that kind of questions was, is and will always be: Because one can!. And this, definitely, is a very good close for this blog.