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Gauss: English and German translations

Read the English collection, 1100 pages or the German collection, 1133 pages.

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Both languages contain Volume I in full: the historical title matter, dedication, preface, all seven sections of the Disquisitiones Arithmeticae (Articles 1–366), five historical additions, Tables I–III, the historical contents, the manuscript appendix, the historical editor's closing note and both imprints. Historical page references retain the source pagination.

Volume II contains the historical title matter, its first five complete papers:

The complete self-reviews of May 12 and September 19, 1808, March 10, 1817, April 11, 1825, and April 23, 1831, follow these five papers.

Seven complete reviews of other writings follow: March 11, 1809; March 23, 1812; November 3, 1814; November 7, 1816; August 9, 1817; December 19, 1825; and July 9, 1831. They concern a work on irrational radicals, the factor tables of Chernac and Burckhardt, Erchinger’s construction of the regular seventeen-sided polygon, and Seeber’s positive ternary quadratic forms.

The complete manuscript work Solution of the Congruence X^m−1 ≡ 0, the first part of Chapter VI of Analysis of Residues (Articles 237–252), follows the reviews.

General Investigations of Congruences, Chapter VIII of Analysis of Residues, contains Articles 330–375 as printed, including the unfinished ending of Article 375. R. Dedekind's complete historical remarks follow. This portion treats congruences between functions, division and common divisors, prime functions, factorization and counting, power transformations and periods, two proofs of quadratic reciprocity, equal roots and divisors, and separation of factors by degree.

Further Development of the Investigations of Pure Equations follows in full as printed, with its title, Articles 1–19 and R. Dedekind's historical remarks. It treats proper and improper roots, products and rational expressions formed from their powers, consistent radical choices, periods, division of ratios and angles, indices and least residues, power expansions, trigonometric period expressions, worked numerical examples and the sign ambiguity in quadratic sums.

The fragment Proof of Some Theorems Concerning the Periods of Classes of Binary Quadratic Forms contains both theorems, the unfinished ending of the second proof and Stern's historical remark.

The texts marked [I.] and [II.] of On the Connection between the Number of Classes into Which Binary Quadratic Forms Are Distributed and Their Determinant, First Paper are each included in full as printed, with their titles and Articles 1–8. The text marked [II.] was presented in 1837. These texts treat counts of integer points in regions, limiting ratios of counts to areas, and mean numbers of representations by binary quadratic forms. Text [I.] includes prime-factor rules, series and infinite products. Text [II.] gives the boundary-square argument, distinguishes approximation of ratios from approximation of differences, and treats counts with congruence and parity conditions.

Fragments III–X follow in full as printed, with their historical editorial notes. They treat representation counts, average and actual numbers of classes, and the distribution of quadratic residues in eighths and twelfths, including the historical tables. The complete historical remarks on [I.]–[X.], signed by R. Dedekind, discuss circle-counting formulas and quoted rules, divisor and representation counts, the series for π/4, prime elimination and mean values, infinite products and the qualification concerning their rigor, trigonometric summations and class numbers, and hyperbolic sectors for positive determinants. They also give the biquadratic-character argument for 2 and the full class-number relations for quadratic residues in eighths and twelfths, with the correspondence tables and both congruence cases.

Geometrical Aspect of Ternary Forms follows with its complete geometric definitions and relations, the ellipsoid and tetrahedron notes, the crystallization law, transformation tables and complex projections, and the hexakisoctahedron formulas, value table and two editable diagrams. Schering’s complete historical remarks include the crystal measurements, the quoted theorem on transformations of an indefinite ternary form, its parameter conditions and the account of the transformation table.

The posthumous texts [I.] and [II.] of On the Theory of Biquadratic Residues follow. Text [I.] contains its title and complete Articles 1–19, including all five editable diagrams. Text [II.], Preparations for the General Theory of Biquadratic Residues, contains both preparatory sections (1) and (2) and complete Articles 1–12 as printed. They treat oriented lines, winding and boundary conventions, residue classes, decidents and their composition, the residue 1+i, and the point, square and ligature constructions for biquadratic characters. Their originally German prose is rendered in modern German in the German reader; the original German bodies and diagrams are supplied separately.

Text [III.], Theory of Biquadratic Residues, follows in full as printed, with its title and complete Articles 1–14, including the joint heading 7–8. It treats least-residue classes, decidents and the stated reciprocity theorem, coordinate bounds, parity sums and all boundary exceptions. Text [IV.] follows in full as printed, Articles 1–5, with its coordinate relations, two bound tables, intensor sums and sign table; its second method ends after the fourth step. The German reader renders the original German in modern German; the historical originals are supplied separately.

Text [V.], Theory of Biquadratic Residues, follows in full as printed, Articles 1–17, with its editable lattice diagram, numerical examples, parity tables and 128-case table. Text [VI.] follows in full as printed, with its title and complete Articles 1–6: complete residue systems, associated quarter systems, the character and its independence, boundary conventions and the four-case construction. Their original German prose is rendered in modern German in the German reader; the historical originals are supplied separately.

Schering’s complete historical remarks on the six biquadratic manuscript texts [I.]–[VI.] follow, with his dating qualifications, geometric interpretations, summation derivations and signature. The originally German remarks are rendered in modern German in the German reader; their historical originals are supplied separately.

On the Theory of Complex Numbers follows in full as printed: six fragments [I.]–[VI.] on the decomposition of cubes, the nearest integer, measures of numbers, units and congruences, followed by Schering’s historical remarks. The originally German passages are rendered in modern German in the German reader; their historical originals are supplied separately.

The complete historical table of the quadratic character follows: primes from 2 to 997 as residues with respect to the primes from 3 to 503 as divisors. All historical entries and blanks are retained in editable tables.

The complete historical table for converting common fractions with denominators from the first thousand into decimal fractions follows, including every printed expansion and the primitive-root bands. The complete historical table of the frequency of prime numbers follows, with its matrices, integral values and closing observations.

The complete letter from Gauss to Enke of 24 December 1849 follows, with its tables and discussion of prime-number frequency. The German reader renders the originally German letter in modern German; its historical original is supplied separately.

The complete historical table of the number of classes of binary quadratic forms follows, including negative and positive determinant panels, hundred and thousand groups, and summary tables. Its headings and labels are translated; historical values and notation are retained. The German reader renders the originally German title in modern German; its source text is supplied separately.

The complete historical Table for Circle Division (Cyklotechnie) follows, with all nine factor-table bands and prime-index lists. A New Quadrature of the Circle includes the four printed calculations and the table of arcs; On Calculating Common Logarithms includes its fraction tables and historical editorial account. The First Ten Thousand Squares, Indices of Primes in the Higher Number Domain and the auxiliary table for solving the indeterminate equation by the method of exclusion follow in full as printed. Historical values, blanks and notation are retained.

The printed Eighth Section, Articles 367–373, follows, including its interruption in Article 371 and the complete ending of Article 373. Both complete letters from Gauss to Dirichlet of 13 September 1826 and 30 May 1828 follow, with their tables, dates and signatures. The originally German prose is rendered in modern German in the German reader; the historical originals are supplied separately.

Schering’s complete historical closing remarks follow, with his account of the published papers, manuscript sources, tables and historical corrections, followed by the complete historical contents and Göttingen imprint. Every historical table entry, expression and reference is retained; historical page references use the source pagination. The German reader gives a modern German rendering of the originally German passages; their historical originals are supplied separately. The reported historical corrections remain part of Schering’s account.

Volume III contains the historical title matter and the complete 1799 dissertation, Articles 1–24, all ten historical footnotes, the closing October 1797 statement and all four figures in editable LaTeX. Gauss’s discussion of earlier proofs, geometric proof and concluding alternative argument retain their formulations and qualifications. Its originally German title matter is rendered in modern German in the German reader, with the historical German and Latin source bodies and editable figures supplied separately.

The complete second proof follows with its title matter, Articles 1–20 and both historical footnotes. Presented in 1815 and published in 1816, it develops Gauss’s determinant, symmetric-function and degree-reduction argument, including his criticism of circular proofs, the polynomial divisibility identities and the concluding real linear or quadratic factor argument.

Later Volume-III works and other volumes remain outside these readers. This is a partial collected edition.

The translations follow the retained original-language LaTeX editions and preserve Gauss's mathematical formulations and historical values. Originally German passages in the Volume-I appendix and editorial closing, and the German Volume-II title matter, all twelve reviews, the manuscript division title and the historical remarks of R. Dedekind and Stern are identified as modern German renderings. Their historical originals are supplied separately in the source package.

The source tree contains editable text, formulas, tables and diagrams, together with the genuine portraits used in the readers. Build main.tex in each language and volume directory with XeLaTeX, rerunning until references and table widths stabilize. The collected readers present the volumes in numerical order; the individual volume readers are also available.

Historical sources: Carl Friedrich Gauss, Werke, Volume I, Göttingen, 1863; Volume II, second printing, Göttingen, 1876. The Disquisitiones Arithmeticae was first published in Leipzig in 1801. The original-language Volume-I edition is available separately.

AI-translated into English and German using OpenAI GPT-6.1 Sol; AI-produced typesetting.

The originally German Fragments III–IV and VIII–X, the geometric notes and the historical remarks are rendered in modern German in the German reader. Their historical originals are supplied separately in the source package.

The corresponding editable original-language editions are available in the Gauss original-language collection: Volume I, Volume II, Volume III. Their text, formula and diagram TeX files correspond byte for byte to the retained editable sources used for these translations. Exact translation-source archive versions and SHA-256 identities are recorded in CITATIONS.bib; the published original-language archives have their own packaging and archive identities.

Translation sources and terminology

These English and German texts are AI-translated from Gauss's original-language editions. Earlier translations are consulted to standardize mathematical terminology and usage; they are not reproduced as this edition's running text. The original mathematical formulations remain those of Gauss's source.

H. Maser, German editor and translator, Carl Friedrich Gauss' Untersuchungen über höhere Arithmetik, Berlin: Julius Springer, 1889, is a reference for German arithmetical terminology and corresponding passages. https://archive.org/details/carlfriedrichga00gausgoog

James Caddall Morehead and Adam Miller Hiltebeitel, translators, General Investigations of Curved Surfaces of 1827 and 1825, Princeton University Library, 1902 (Project Gutenberg 36856), and Paul Taylor's English translation of Gauss's second proof of the fundamental theorem of algebra, with corrections by Bernard Leak, 1983, are references for English mathematical terminology and usage. Taylor–Leak is a corresponding translation of the 1815 second proof, and a related terminology reference for the arithmetic papers and 1799 dissertation. Morehead–Hiltebeitel supplies related geometrical terminology. Neither replaces Gauss’s original mathematics or exposition. The consulted Taylor–Leak online edition identifies its translation as December 1983, its LaTeX version as August 2003 and its underlying source as last modified on 8 June 2007. Its acknowledgements credit Mark Wainwright for returning Taylor’s notes and Nikita Danilov for typo corrections. https://www.gutenberg.org/ebooks/36856 https://www.paultaylor.eu/misc/gauss.html

Contemporary German terminology is also informed by Ulrich Görtz, Algebra, Wintersemester 2021/22, Section 3.1, and Lineare Algebra II, Sommersemester 2021, Section 18.4; and Nikolay Moshchevitin, Eine Bemerkung über positiv definite quadratische Formen und rationale Punkte, arXiv:1510.06214v3. These are terminology references, not translations of the present works. https://math.ug/a-ws2122/sect0006.html https://math.ug/la2-ss21/sec-quotient-ring.html https://arxiv.org/abs/1510.06214v3

Quellen und Terminologie der Übersetzungen

Die englischen und deutschen Texte sind aus Gauss' originalsprachlichen Ausgaben KI-übersetzt. Frühere Übersetzungen dienen dem Abgleich der mathematischen Terminologie und des Sprachgebrauchs; ihr Wortlaut wird nicht als fortlaufender Text dieser Ausgabe übernommen. Die mathematischen Formulierungen bleiben die der Originalvorlage. Die oben genannten Übersetzer und Autoren werden für diese terminologische Nutzung gewürdigt.

Crystallographic terminology references: Ian O. Angell and Moreton Moore, Projections of cubic crystals, IUCr, Table 2, https://www.iucr.org/what-we-do/education/pamphlets/projections-of-cubic-crystals ; Th. Hahn, H. Klapper, U. Müller and M. I. Aroyo, Point groups and crystal classes, International Tables for Crystallography A (2016), chapter 3.2, https://doi.org/10.1107/97809553602060000930 . These references guide terminology and usage; their running prose is not reproduced.

AI authorship and sources

These English and German editions are AI-translated using OpenAI GPT-6.1 Sol, with AI-produced typesetting. Gauss is the author of the historical mathematical text; signed historical editorial contributions retain their authorship. The named human translations and writings cited above are credited for their stated terminology and usage roles.

These editions are freely available for study and teaching.

Scholarly citation metadata identifies the original editions and the named terminology references.

The citation metadata records the exact editable transcription archives used for the translations, with their SHA-256 identities, alongside the historical editions and named terminology references.