1See also Schur, Math. Annalen, Vol. 55, p. 265.—Tr.
2The following is a summary of a paper by Professor Hilbert which is soon to appear in full in the Math. Annalen.—Tr.
3Compare the comprehensive and explanatory report of G. Veronese, Grundzüge der Geometrie, German translation by A. Schepp, Leipzig, 1894 (Appendix). See also F. Klein, “Zur ersten Verteilung des Lobatschefskiy-Preises,” Math. Ann., Vol. 50.
4These axioms were first studied in detail by M. Pasch in his Vorlesungen über neuere Geometrie, Leipsic, 1882. Axiom II, 5 is in particular due to him.
5Added by Professor Hilbert in the French translation.—Tr.
6See Hilbert, “Ueber den Zahlenbegriff,” Berichte der deutschen Mathematiker-Vereinigung, 1900.
7The mutual independence of Hilbert’s system of axioms has also been discussed recently by Schur and Moore. Schur’s paper, entitled “Ueber die Grundlagen der Geometrie” appeared in Math. Annalem, Vol. 55, p. 265, and that of Moore, “On the Projective Axioms of Geometry,” is to be found in the Jan. (1902) number of the Transactions of the Amer. Math. Society.—Tr.
8See my lectures upon Euclidean Geometry, winter semester of 1898–1899, which were reported by Dr. Von Schaper and manifolded for the members of the class.
9In his very scholarly book,—Grundzüge der Geometrie, German translation by A. Schepp, Leipzig, 1894,—G. Veronese has also attempted the construction of a geometry independent of the axiom of Archimedes.
10See also Schur, Math. Annalen, Vol. 55, p. 265.—Tr.
11See also Schur, Math. Annalen, Vol. 55, p. 265.—Tr.
12F. Schur has published in the Math. Ann., Vol. 51, a very interesting proof of the theorem of Pascal, based upon the axioms I–II, IV.
13In connection with the theory of areas, we desire to call attention to the
following works of M. Gérard: Thèse de Doctorat sur la géométrie non
euclidienne (1892) and Géométrie plane (Paris, 1898). M. Gérard has
developed a theory concerning the measurement of polygons analogous to
that presented in section 4 of the present work. The difference is that M.
Gérard makes use of parallel transversals, while I use transversals emanating
from the vertex. The reader should also compare the following works of
F. Schur, where he will find a similar development: Sitzungsberichte der
Dorpater Naturf. Ges., 1892, and Lehrbuch der analytischen Geometrie,
Leipzig, 1898 (introduction). Finally, let me refer to an article by O. Stolz in
Monatshefte für Math, und Phys., 1894. (Note by Professor Hilbert in French
ed.)
M. Gérard has also treated the subject of areas in various ways in the following
journals: Bulletin de Math, spciales (May, 1895), Bulletin de la Société
mathématique de France (Dec., 1895), Bulletin Math, élémentaires (January,
1896, June, 1897, June, 1898). (Note in French ed.)
14Sitzungsberichte der Dorpater Naturf. Ges. 1892.
15Grundlagen der Geometrie, Vol. 2, Chapter 5, section 1, 1898.
16Monatshefte für Math, und Phys. 1894.
17See also a recent paper by F. R. Moulton on “Simple Non-desarguesian Geometry,” Transactions of the Amer. Math. Soc., April, 1902.—Tr.
18Discussed also by Moore in a paper before the Am. Math. Soc., Jan, 1902. See Trans. Am. Math. Soc.—Tr.
19Figures 46, 47, and 48 were designed by Dr. Von Schaper, as have also the details of the demonstrations relating to these figures.
20“Ueber die Theorie der relativquadratischen Zahlkörper,” Jahresbericht der Deutschen Math. Vereinigung, Vol. 6, 1899, and Math. Annalen, Vol. 51. See, also, “Ueber die Theorie der relativ-Abelschen Zahlkörper” Nachr. der K. Ges. der Wiss. zu Göttingen, 1898.
21See “Ueber ternäre definite Formen,” Acta mathematica, Vol. 17.
22Math. Annalen, Vol. 53 (1900).
23The following is a summary of a paper by Professor Hilbert which is soon to appear in full in the Math. Annalen.—Tr.
24The following is a summary of a paper by Professor Hilbert which is soon to appear in full in the Math. Annalen.—Tr.
25See Lie-Engel, Theorie der Transformationsgruppen, Vol. 3, Chapter 5.
26By the following investigation is answered also, as I believe, a general question concerning the theory of groups, which I proposed in my address on “MathematischeProbleme,” Göttinger Nachrichten, 1900, p. 17.
27Concerning the broader statement of the conception of the plane see my note, “Ueber die Grundlagen der Geometrie,” Göttinger Nachrichten, 1901.
29Lie makes this assumption to contain the condition that the group of displacements be generated by infinitesimal transformations. The opposite assumption would assist essentially the demonstration in so far as the “true straight line” could then be defined as the locus of those points which remain unchanged by a displacement changing the sense in which the curve is traversed (Umklappung).
30The term “rotation” is used here in the sense of a rotatory displacement; that is to say, only the initial and final stages and not the aggregate of the intermediate stages of the transition enter into consideration.—Tr.
31It is sufficient to assume that axiom III holds for sufficiently small domains as Lie has done. My method of proof may be so changed as to make use of only this narrower assumption.
32The complete proof will appear later in the Math. Annalen.
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