Systematic Bibliography

Primary Editions, Classical Treatises & Foundational Commentaries

Antonio De Zolt — Principii della eguaglianza di poligoni

Milano, Briola, 1881.

Formulates the polygonal area postulate (Euclid C.N. 5 for dissection) proved deductively by Hilbert as Theorem 30 in § 20.

Aristotle — Analytica Posteriora (Posterior Analytics)

Book I, Chs. 1–10 (71a1–77a4), Classical Corpus Aristotelicum, c. 335 BC.

Classical theory of demonstrative science (episteme) contrasted by Hoenen (Caput V § 1.2) with Hilbert’s formal axiomatics.

Bertrand Russell — The Principles of Mathematics

Part VI: Space, Ch. XLVI, p. 405, Cambridge University Press, 1903.

Critique of Euclidean superposition quoted by Heath on Euclid I.4 alongside Hilbert’s Axiom IV 6.

Camille Jordan — Cours d’analyse de l’École Polytechnique (2e édition)

Tome I, pp. 90–99, Gauthier-Villars, Paris, 1893.

Locus classicus of the Jordan curve theorem cited in § 4 (polygon separation) and the 1902 Appendix.

David Hilbert — Grundlagen der Geometrie (Festschrift zur Feier der Enthüllung des Gauss-Weber-Denkmals in Göttingen, Erster Theil)

pp. 1–92, B. G. Teubner, Leipzig, 1899.

First German edition digitized in verbatim archival full text for the Deutsch track of this Book Codex.

David Hilbert — Les principes fondamentaux de la géométrie (traduit par L. Laugel)

Annales scientifiques de l’École Normale Supérieure (3), Tome 17, pp. 103–209 (and separate monograph), Gauthier-Villars, Paris, 1900.

Authorized French edition introducing the Axiom of Completeness (Vollständigkeit) in § 8 and Dehn’s polyhedra note in the Conclusion.

David Hilbert — Über den Zahlbegriff

Jahresbericht der Deutschen Mathematiker-Vereinigung, Bd. 8, pp. 180–184, B. G. Teubner, Leipzig, 1900.

Companion axiomatization of the real number continuum pairing Archimedes’s Axiom with the Axiom of Completeness.

David Hilbert — Über die Grundlagen der Geometrie

Mathematische Annalen, Bd. 56, pp. 381–422, B. G. Teubner, Leipzig, 1902.

Topological characterization of the Euclidean and hyperbolic planes via Jordan curves and transformation groups without differentiability; summarized in Townsend’s 1902 Appendix.

David Hilbert — Grundlagen der Geometrie (Siebente umgearbeitete und vermehrte Auflage)

pp. vii + 326, B. G. Teubner, Leipzig & Berlin, 1930.

Definitive lifetime revision by Hilbert, cited by Peter Hoenen in De Noetica Geometriae (1954).

David Hilbert (trans. E. J. Townsend) — The Foundations of Geometry

pp. vii + 1–132, The Open Court Publishing Company, Chicago, 1902.

Authorized English translation forming the verbatim English track of this Book Codex.

Eliakim Hastings Moore — On the Projective Axioms of Geometry

Transactions of the American Mathematical Society, Vol. 3, No. 1, pp. 142–158, American Mathematical Society, 1902.

Proved that Hilbert’s 1899 linear order Axiom II 4 is redundant (deducible from planar incidence I 1–2 and Pasch’s Axiom II 5), cited by Townsend (§ 3 fn.) and Hoenen (Caput V § 3.2.A).

Euclid of Alexandria — Elements (Stoicheia)

Books I, V, VI, X, XI, Critical ed. J. L. Heiberg (Teubner, 1883–1888); English ed. T. L. Heath (Cambridge, 1908), c. 300 BC.

Foundational Greek treatise whose axioms, congruence proofs, proportion theory, polygonal area theorems, and straightedge-and-compass constructions Hilbert reconstructs throughout §§ 1–39.

Friedrich Schur — Über den Fundamentalsatz der projectiven Geometrie

Mathematische Annalen, Bd. 51, pp. 401–409, B. G. Teubner, Leipzig, 1898.

Spatial metric proof of Pascal’s/Pappus’s Theorem without Archimedes’s Axiom cited by Hilbert in § 14.

Giuseppe Peano — Die Grundzüge des geometrischen Calculs (trans. A. Schepp) & I principii di geometria logicamente esposti

Turin (1889) / Leipzig (1891), Fratelli Bocca / B. G. Teubner, 1889–1891.

Symbolic axiomatization of geometric segments and vector calculus cited in Hilbert’s Introduction and § 3.

Giuseppe Veronese — Grundzüge der Geometrie von mehreren Dimensionen und mehreren Arten gradliniger Einheiten in elementarer Form entwickelt (trans. A. Schepp)

Leipzig (Italian original: Fondamenti di geometria, Padova, 1891), B. G. Teubner, Leipzig, 1891 / 1894.

Synthetic investigation of non-Archimedean geometry cited by Hilbert in § 12.

Godfrey Harold Hardy — Mathematical Proof

Mind, New Series, Vol. 38, No. 149, pp. 1–25, Oxford University Press, 1929.

Quoted by Hoenen (Caput V § 3.2.B, fn. 9) on the indispensable intuitive role of spatial symbol order and geometric figures in Hilbert’s axiomatics.

Gottlob Frege — Über die Grundlagen der Geometrie

Jahresbericht der Deutschen Mathematiker-Vereinigung, Bd. 12, pp. 319–324, 368–375, B. G. Teubner, Leipzig, 1903.

Public continuation of the 1899–1900 Frege–Hilbert correspondence debating implicit definitions, consistency, and geometric truth.

Hans Freudenthal — Zur Geschichte der Grundlagen der Geometrie (Zugleich eine Besprechung der 8. Aufl. von Hilberts ‘Grundlagen der Geometrie’)

Nieuw Archief voor Wiskunde (3), Vol. 5, pp. 105–142, Noordhoff, Groningen, 1957.

Definitive historical and mathematical retrospective on Hilbert’s 1899 Grundlagen and its evolution through subsequent editions.

Henri Poincaré — Les fondements de la géométrie

Bulletin des sciences mathématiques (2), Tome 26, pp. 249–272, Gauthier-Villars, Paris, 1902.

Classic philosophical and mathematical review of Hilbert’s Grundlagen.

Hermann von Helmholtz — Über die Thatsachen, die der Geometrie zum Grunde liegen

Nachrichten von der Königl. Gesellschaft der Wissenschaften zu Göttingen, pp. 193–221, Göttingen, 1868.

Initiated the group-theoretic free-mobility characterization of space revisited by Sophus Lie and by Hilbert in the 1902 Appendix.

Hermann Wiener — Über Grundlagen und Aufbau der Geometrie / Weiteres über Grundlagen und Aufbau der Geometrie

Jahresbericht der Deutschen Mathematiker-Vereinigung, Bd. 1 (1891), pp. 45–48; Bd. 3 (1893), pp. 70–80, Georg Reimer, Berlin, 1891–1893.

Proposed Desargues’s and Pascal’s (Pappus’s) theorems as incidence-based foundations of projective geometry, directly motivating Hilbert’s Chapters III, V, and VI.

Immanuel Kant — Kritik der reinen Vernunft (Critique of Pure Reason)

Transcendental Dialectic, A 702 / B 730, Johann Friedrich Hartknoch, Riga, 1781 / 1787.

Source of Hilbert’s epigraph: ‘So fängt denn alle menschliche Erkenntniss mit Anschauungen an, geht von da zu Begriffen und endigt mit Ideen.’

Jakob Steiner — Die geometrischen Constructionen, ausgeführt mittelst der geraden Linie und eines festen Kreises

Berlin, Ferdinand Dümmler, 1833.

Poncelet–Steiner fixed-circle theorem generalized and refined by Hilbert in Chapter VII (§§ 36–39).

Julius Sommer — Hilbert’s Foundations of Geometry

Bulletin of the American Mathematical Society, Vol. 6, pp. 287–299, American Mathematical Society, 1900.

Early Göttingen review of Hilbert’s Festschrift; Sommer’s 3D independence model for IV 6 is cited by Hilbert in § 11.

Max Dehn — Über raumgleiche Polyeder

Nachrichten von der Königl. Gesellschaft der Wissenschaften zu Göttingen, Math.-phys. Klasse, pp. 345–354 (and Math. Annalen, Bd. 55, 1901, pp. 465–478), Vandenhoeck & Ruprecht / B. G. Teubner, 1900.

Solves Hilbert’s Third Problem, proving that tetrahedra of equal base and altitude need not be equidecomposable or equicomplementable—cited by Hilbert in the 1900/1902 Conclusion.

Max Dehn — Die Legendre’schen Sätze über die Winkelsumme im Dreieck

Mathematische Annalen, Bd. 53, pp. 404–439, B. G. Teubner, Leipzig, 1900.

Göttingen doctoral dissertation under Hilbert constructing non-Legendrian and non-Archimedean geometries.

Moritz Pasch — Vorlesungen über neuere Geometrie

§§ 1–4, pp. 1–35, B. G. Teubner, Leipzig, 1882.

First rigorous axiomatization of linear and planar order; source of Hilbert’s Axiom II 5 (‘Pasch’s Axiom’) in § 3.

Otto Stolz — Die ebenen Vielecke und die Winkel mit Einschluss der Berührungswinkel als Systeme von absoluten Grössen

Monatshefte für Mathematik und Physik, Jahrg. 5, pp. 233–240, Wien, 1894.

Cited by Hilbert in § 12 and § 18 on non-Archimedean magnitudes and polygon area equivalence.

Pappus of Alexandria — Synagoge (Mathematical Collection)

Book VII, Propositions 138–139 (ed. F. Hultsch, Vol. II, 1877, pp. 882–886), Weidmann, Berlin (GEOMETOR pappus), c. 340 AD.

Original Greek locus of Pappus’s Hexagon Theorem—the degenerate conic theorem Hilbert calls ‘Pascal’s Theorem’ (Pascalscher Satz) in Chapters III and VI.

Petrus Hubertus Jacobus Hoenen, S.J. — De Noetica Geometriae: Origine Theoriae Cognitionis

Analecta Gregoriana, Vol. LXIII (Caput I § 6, Caput IV Prelim., Caput V §§ 1–5), Apud Aedes Universitatis Gregorianae, Romae, 1954.

Epistemological treatise in GEOMETOR hoenen critically examining Hilbert’s primitive terms, Erklärungen, linear order axioms, and completeness postulate.

Sir Thomas Little Heath — The Thirteen Books of Euclid’s Elements, Translated from the Text of Heiberg with Introduction and Commentary

Vol. I: pp. 200, 228–230, 242, 249–250, 328, Cambridge University Press (2nd ed. 1926), 1908.

Primary commentary in GEOMETOR euclid analyzing Hilbert’s Congruence Axioms, SAS proof, right-angle congruence (Post. 4), continuity, and area dissection.

Sophus Lie (bearbeitet von Friedrich Engel) — Theorie der Transformationsgruppen, Dritter und letzter Abschnitt

Abtheilung V: Untersuchungen über die Grundlagen der Geometrie, pp. 393–543, B. G. Teubner, Leipzig, 1893.

Lie’s analytic solution of the Riemann–Helmholtz problem cited by Hilbert in the Introduction and Appendix.

Wilhelm Killing — Einführung in die Grundlagen der Geometrie

Bd. I (1893), Bd. II (1898), Ferdinand Schöningh, Paderborn, 1893–1898.

Comprehensive treatise on space forms and non-Archimedean/group-theoretic foundations cited by Hilbert in § 12.