Literature and Links

Because this bibliography has been originally written for the German version of this site, several of the reviewed books and articles are in German. If I could find one, I have added a link to an English translation. However, for some of the items, like the original papers [Hermes 1879] or [Hermes 1889], no translations are available. On the other hand, several books and most of the websites reviewed on this page were already in English at the outset.

• [Bachmann 1872]
• Bachmann, P. :
• Die Lehre von der Kreistheilung und ihre Beziehungen zur Zahlentheorie.
• Teubner, Leipzig (1872).
• In spite of its age, the book presents in its first part a comprehensable introduction into Gauss' cyclotomic theory. The second part is more advanced and deals with the connection of cyclotomy to other parts of mathematics, in particular to the theory of quadratic residues. (⇒ Google Books)

• [Bell 1965]
• Bell, E.T. :
• Men of Mathematics.
• Penguin, Harmondsworth, Middlesex. (1965)
• A classic, that had its first edition in 1937. However you have to tolerate the rather pathetic style of Bell. (⇒ Google Books)

• [Bewersdorff 2006]
• Bewersdorff, J. :
• Galois Theory for Beginners
• American Mathematical Society (2006)
• I have read only the German original Algebra für Einsteiger. The latter is a readable introduction into abstract algebra with focus on Galois theory. Moreover the book contains a detailled classical presentation of cyclotomy, as a kind of motivation for Galois theory. (⇒ AMS)

• [Biermann 1990]
• Biermann, K.-R. :
• Der ›Fürst der Mathematiker‹ in Briefen und Gesprächen,.
• Urania-Verlag, Leipzig / Jena / Berlin (1990)
• Everybody, who wants to learn more about Gauss as a human being, should read his letters. Unfortunately, there seems to be no English translation of Biermann's book, but several letters of Gauss (translated into English) can be found at different places in the internet. (⇒ Google Books)

• [Brakke 2011]
• Brakke, K. :
• Constructing 17, 257, and 65537 sided polygons.
• Webseite
• Brakke gives a very dense description of an algorithm, which is similar to the one we presented in chap.6, and even publishes square root expressions of the 17-gon, the 257-gon and the 65537-gon. Comparing the expressions with those of the raw output of our program, you will find many correlations. I did so for several dozen samples with the numerical values given by Brakke, and found no discrepancies (within the achievable precision) to the values produced by the program Hermes
  However, Brakke obviously spent more effort into the simplification of his root expressions. Some of them contain negative coefficients, thereby being shorter, and at the same time reducing the number of square roots needed. Hence, the program (which is not published) can be expected to be more complicated than our Hermes. (⇒ Algorithm and ⇒ Roots of the 65537-gon)

• [Conway 1996]
• Conway, J. H. :
• The Book of Numbers.
• Copernicus (An Imprint of Springer-Verlag), New York (1996)
• Several self-contained and entertaining articles about various kinds of numbers. The topic constructability of regular polygons is discussed in the chapter about Fermat numbers. (⇒ Google Books)

• [Cooke 2011]
• Cooke R.L. :
• The History of Mathematics: A Brief Course.
• John Wiley & Sons. Hoboken, New Jersey (2011)
• As far as I know this book is the only one that contains a reference to the work of Joan Taylor (see [Taylor 2003] and [Taylor 2004], namely in a footnote on page 189. (⇒ Google Books).

• [Duden 2010]
• Anonymus:
• Lernhelfer Restklassen.
• Ein Angebot von Duden, Mannheim (2010)
• Probably this is only of interest for German readers. It's a site of the Duden team, containing a list of topics that a German high-schools student should know at his or her final exams. The link has been added, because one of the topics is about congruence classes, which may complement the presentation in chap.4. (⇒ Duden)

• [Fischer 2012]
• Fischer, F. :
• Ein Koffer voller Zahlen.
• Die Zeit, Ausgabe 34/2012 vom 16. August 2012,
• An article in the German weekly newspaper Die Zeit about Hermes and his Hermeskoffer (with several pictures of the latter). (⇒ to the article or ⇒ Umblätterer).

• [Gauß 1801]
• Gauß, C.F. :
• Disquisitiones Arithmeticae. Paperback.
• Translated by Arthur A. Clarke. Yale University Press 1965
• I know only the German translation (from the Latin original) by H. Maser. To quote Maser, the book contains die herrlichen Geisteserzeugnisse unseres unsterblichen Gauß (the splendid intellectual productions of our immortal Gauss ). (⇒ Amazon)

• [Gottlieb 1999]
• Gottlieb, Ch. :
• The Simple and Straightforward Construction of the Regular 257-gon.
• Mathematical Intelligencer Vol 21, Number 1, Springer, New York (1999)
• A very condensed presentation of the algorithms used by cyclotomy in case of the 257-gon. (⇒ Researchgate.net)

• [Hermes 1879]
• Hermes, J. :
• Zurückführung des Problems der Kreistheilung auf lineare Gleichungen für Primzahlen von der Form 2 hoch m plus 1.
• Crelles Journal für die reine und angewandte Mathematik, Band 87, Zeitschriftenband (1879), 84-115
• Hermes' doctoral thesis of 1878. Note that the paper had been published only within a year after completeion in one of the most decent journals of that time. (⇒ Göttinger Digitalisierungszentrum)

• [Hermes 1889]
• Hermes, J. :
• Beweis des quadratischen Reciprocitätsgesetzes durch Umkehrung.
• Arch. Math. Phys. (2), 5 (1889), 190-198; FdM
• One of Hermes' papers that were written at the same time as his diary. As already mentioned, it contains the 59th proof of the law of quadratic reciprocity (due to the count of [Lemmermeyer 2000]) and is a by-product of his work about cyclotomy.

• [Hermes 1894]
• Hermes, J. :
• Über die Teilung des Kreises in 65537 gleiche Teile.
• Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-physikalische Klasse (1894), 170-186
• Hermes' summary of his construction of the regular 65537-gon. It's a very condensed extract of the material filed in the Hermeskoffer. (⇒ Göttinger Digitalisierungszentrum)

• [Hudson 1969]
• Hudson, H.P. :
• in: Squaring the Circle and Other Monographs.
• Chelsea Publishing Comany (1969)
• The article Ruler and Compasses in this monograph is particularly interesting, because it contains a classical proof of Wantzel's theorem, that the polygon constructions discovered by Gauss are indeed the only possible ones. (⇒ Google Books)

• [Klein 1895]
• Klein, F. :
• Famous Problems of Elementary Geometry.
• Ginn & Company (1897)
• This is the English translation of Klein's book Vorträge über ausgewählte Fragen der Elementargeometrie. Its contains not only the benevolent review of Hermes' work, we mentioned in chap.10, but also a rather detailled discussion of the construction of regular polygons, in particular of the 17-gon. Other topics are the quadrature of the circle or the duplication of the cube. (⇒ Google Books)

• [Kratz, Wörle 1968]
• Kratz, J. Wörle, K. :
• Geometrie. Ein Lehr- und Arbeitsbuch II. Teil.
• Bayerischer Schulbuch-Verlag, München (1968)
• The reason, why this reference has been added, is given in chap.0. No link can be given, because the book appears in irregular intervals on the pages of the well-known second-hand bookshops.

• [Lemmermeyer 2000]
• Lemmermeyer, F. :
• Reciprocity Laws: From Euler to Eisenstein.
• Springer, Heidelberg (2000)
• Cyclotomy is treated in this book only as a marginal note to the several proofs of the law of quadratic reciprocity. The link has been added, to demonstrate that some papers of Hermes (like [Hermes 1889]) are indeed accepted by the mathematical community. (⇒ Google Books)

• [Martin 1998]
• Martin, G.E. :
• Geometric Constructions.
• Springer, New York (1998)
• In addition to classical geometric constructions with ruler and compass, you will find in this book constructions with unusual tools, like a ruler with marks or a fixed compass. (⇒ Google Books)

• [Mitzscherling 1913]
• Mitzscherling, A. :
• Das Problem der Kreisteilung.
• Teubner, Leipzig (1913)
• Mitzscherling refers to [Bachmann 1872] as theoretical basis, hence offers no new approach. However, he gives constructions of a lot of polygons in many different ways together with detailled descriptions. If you want to learn, how much ingenuity has been spent by mathematicians, to simplify the cumbersome constructions, that are deduced directly from the Gaussian theory, you will find a wealth of material. There seems to be no English translation of this book. (⇒ Google Books)

• [Nahin 1998]
• Nahin, P. J. :
• An Imaginary Tale: The Story of $\sqrt{-1}$
• Princeton University Press, Princeton N.J. (1998)
• A very entertainig but also detailled introduction into the field of complex numbers, their history and their relevance for mathematics and physics. The first four chapters should not be difficult to read for someone having read this blog. Starting with chapter 5, however, hardcore mathematics is presented. (⇒ Google Books)

• [Nahin 2006]
• Nahin, P. J. :
• Dr. Eulers Fabulous Formula.
• Princeton University Press, Princeton N.J. (2006)
• One of my favourite books, taking Eulers famous formula $e^{i\pi}+1 = 0$ as central theme to connect several mathematical topics, like the geometry of complex numbers, the irrationality of $\pi$ or even Fourier series. Concerning cyclotomy you will find a readable and rather detailled presentation in chapter 1.6 Regular n-gons and primes. Some chapters require a background far beyond high-school math. (⇒ Google Books)

• [Richelot 1832]
• Richelot, F.J. :
• De resolutione algebraica aequationis $x^{257} = 1,$ sive de divisione circuli per bisectionam anguli septies repetitam in partes 257 inter se aequales commentatio coronata.
• Crelles Journal IX (1832) 1-26, 145-161, 209-230, 337-356
• Richelot's paper about his construction of the 257-gon, whose title will fill the heart of every friend of the Latin language with delight. (⇒ Göttinger Digitalisierungszentrum)

• [Scharlau 1989]
• Scharlau, W. (Hrsg.):
• Mathematische Institute in Deutschland 1800-1945.
• Vieweg, Braunschweig/Wiesbaden 1989
• A kind of Who's Who of German university mathematics of the years 1800-1945. (⇒ Google Books)

• [Stewart 1988]
• Stewart, I. :
• Galois Theory, Fourth Edition.
• CRC Press, London, New York (2015).
• A readable introduction to Galois theory. The setup is a bit more formal than that of [Bewersdorff 2006], it's more like a conventional lecture note, but the style is relaxed and not too technical. (⇒ Google Books)

• [Taylor 2003]
• Taylor, J.M. :
• Constructible Polygons.
• Private Correspondence (not published up to now)
• The description of Joan Taylors method to combine the square roots representing $\cos{2\pi/n}$ into a highly compressed expression. By this paper she could convince the Lawrence Berkeley National Laboratory in California, to perform the computations of the 65537-gon with their arbitrary precision software. (⇒ see PDF)

• [Taylor 2004]
• Taylor, J.M. :
• High Precision Solves Ancient Problem.
• Private Correspondence (not published up to now)
• A three pages summary of Joan Taylor's representation of $\cos{2\pi/65537}$ as expression of no more than 15 nested square roots and fractions, whose numerators are integers with up to 20000 digits. (⇒ The Summary). The original output of the ARPREC-program as (⇒ ASCII text) of 211245 lines.

• [Tijsma 2016]
• Tijsma, D.J.:
• Gaussian periods.
• Faculty of Science Utrecht University (Bachelor Thesis 2016)
• A bachelor thesis, which places many of the properties of Gaussian periods into a wider context, using more advanced methods than ours. For questions of cyclotomy the chapters 5.2 to 5.4 are the most interesting. (⇒ Utrecht Universiy)

• [Trott 2000]
• Trott, M. :
• $\cos{2\pi/257}$ a la Gauss.
• Mathematica in education and research, Vol.4 Nr.2 (2000)
• The peculiar influence of Trott's article concerning the genesis of this blog has already been recognized in chap.0. It's sad to say that the journal Mathematica in education and research does not exist any more. However, the said article seems to have found its way into the book The Mathematica GuideBook for Symbolics (Springer 2005) ISBN 978-0387950204. (⇒ Google Books)

• [Wantzel 1837]
• Wantzel, P.L. :
• Recherches sur les moyens de reconnaître si un Problème de Géomètrie se résoudre avec la règle et le compas.
• J. Math. 2 (1837) 366-372
• The original paper of Wantzel with the proof of the reverse of Gauss' theorem, namely that only regular polygons having $n=2^k\cdot p_1 \cdot p_2 \cdot \dots \cdot p_s$ vertices, with distinct Fermat primes $p_i$, can be constructed with ruler and compass. (⇒ visualiseur)

• [Winter 2011]
• Winter, B. :
• Zur Konstruktion regulärer Polygone, insbesondere des regulären 17-Ecks, 257-Ecks und 65537-Ecks.
• Website
• It's a short summary about the basics of cyclotomy. You may regard it as a very condensed presentation of the topics discussed in this blog. You may also find several interesting links, e.g. to a video, showing the construction of a 257-gon with Geogebra. Furthermore I recommend this site because of some rare pictures of the Hermeskoffer and of its content. (⇒ Website)

• [Wußing 2011]
• Wußing, H. :
• Carl Friedrich Gauß, Biographie und Dokumente,6.Auflage
• Edition am Gutenbergplatz, Leipzig (2011)
• I restrict myself to refer to this German biography of Gauss, because I did not read any English one. However, you will find nearly a dozen English books about Gauss' life on the pages of Amazon alone. (⇒ Google Books)

• [Ziegler 2013]
• Ziegler, G.M. :
• Do I Count?: Stories from Mathematics
• CRC Press, Boca Raton (2013)
• I have read only the German original Darf ich Zahlen?. It's a very entertaining and anecdotal compilation of short articles about a great number of mathematical topics from past to present. (⇒ Google Books)