Book I

., with a Commentary_ (Cambridge, 1905)--the commentary contains copious extracts from the ancient commentators. The next period of really substantive importance is that of the 18th century. The leading authors are: G. Saccheri, S.J., _Euclides ab omni naevo vindicatus_ (Milan, 1733). Saccheri was an Italian Jesuit who unconsciously discovered non-Euclidean geometry in the course of his efforts to prove its impossibility. J.H. Lambert, _Theorie der Parallellinien_ (1766); A.M. Legendre, _Elements de geometrie_ (1794). An adequate account of the above authors is given by P. Stackel and F. Engel, _Die Theorie der Parallellinien von Euklid bis auf Gauss_ (Leipzig, 1895). The next period of time (roughly from 1800 to 1870) contains two streams of thought, both of which are essential to the modern analysis of the subject. The first stream is that which produced the discovery and investigation of non-Euclidean geometries, the second stream is that which has produced the geometry of position, comprising both projective and descriptive geometry not very accurately discriminated. The leading authors on non-Euclidean geometry are K.F. Gauss, in private letters to Schumacher, cf. Stackel and Engel, _loc. cit._; N. Lobatchewsky, rector of the university of Kazan, to whom the honour of the effective discovery of non-Euclidean geometry must be assigned. His first publication was at Kazan in 1826. His various memoirs have been re-edited by Engel; cf. _Urkunden zur Geschichte der nichteuklidischen Geometrie_ by Stackel and Engel, vol. i. "Lobatchewsky." J. Bolyai discovered non-Euclidean geometry apparently in independence of Lobatchewsky. His memoir was published in 1831 as an appendix to a work by his father W. Bolyai, _Tentamen juventutem...._ This memoir has been separately edited by J. Frischauf, _Absolute Geometrie nach J. Bolyai_ (Leipzig, 1872); B. Riemann, _Uber die Hypothesen, welche der Geometrie zu Grunde liegen_ (1854); cf. _Gesamte Werke_, a translation in The Collected Papers of W.K. Clifford. This is a fundamental memoir on the subject and must rank with the work of Lobatchewsky. Riemann discovered elliptic metrical geometry, and Lobatchewsky hyperbolic geometry. A full account of Riemann's ideas, with the subsequent developments due to Clifford, F. Klein and W. Killing, will be found in _The Boston Colloquium for 1903_ (New York, 1905), article "Forms of Non-Euclidean Space," by F.S. Woods. A. Cayley, _loc. cit._ (1859), and F. Klein, "Uber die sogenannte nichteuklidische Geometrie," _Math. Annal._ vols. iv. and vi. (1871 and 1872), between them elaborated the projective theory of distance; H. Helmholtz, "Uber die thatsachlichen Grundlagen der Geometrie" (1866), and "Uber die Thatsachen, die der Geometrie zu Grunde liegen" (1868), both in his _Wissenschaftliche Abhandlungen_, vol. ii., and S. Lie, _loc. cit._ (1890 and 1893), between them elaborated the group theory of congruence.

The numberless works which have been written to suggest equivalent alternatives to Euclid's parallel axioms may be neglected as being of trivial importance, though many of them are marvels of geometric ingenuity.

The second stream of thought confined itself within the circle of ideas of Euclidean geometry. Its origin was mainly due to a succession of great French mathematicians, for example, G. Monge, _Geometrie descriptive_ (1800); J.V. Poncelet, _Traite des proprietes projectives des figures_ (1822); M. Chasles, _Apercu historique sur l'origine et le developpement des methodes en geometrie_ (Bruxelles, 1837), and _Traite de geometrie superieure_ (Paris, 1852); and many others. But the works which have been, and are still, of decisive influence on thought as a store-house of ideas relevant to the foundations of geometry are K.G.C. von Staudt's two works, _Geometrie der Lage_ (Nurnberg, 1847); and _Beitrage zur Geometrie der Lage_ (Nurnberg, 1856, 3rd ed. 1860).

The final period is characterized by the successful production of exact systems of axioms, and by the final solution of problems which have occupied mathematicians for two thousand years. The successful analysis of the ideas involved in serial continuity is due to R. Dedekind, _Stetigkeit und irrationale Zahlen_ (1872), and to G. Cantor, _Grundlagen einer allgemeinen Mannigfaltigkeitslehre_ (Leipzig, 1883), and _Acta math._ vol. 2.

Complete systems of axioms have been stated by M. Pasch, _loc. cit._; G. Peano, _loc. cit._; M. Pieri, _loc. cit._; B. Russell, _Principles of Mathematics_; O. Veblen, _loc. cit._; and by G. Veronese in his treatise, _Fondamenti di geometria_ (Padua, 1891; German transl. by A. Schepp, _Grundzuge der Geometrie_, Leipzig, 1894). Most of the leading memoirs on special questions involved have been cited in the text; in addition there may be mentioned M. Pieri, "Nuovi principii di geometria projettiva complessa," _Trans. Accad. R. d. Sci._ (Turin, 1905); E.H. Moore, "On the Projective Axioms of Geometry," _Trans. Amer. Math. Soc._, 1902; O. Veblen and W.H. Bussey, "Finite Projective Geometries," _Trans. Amer. Math. Soc._, 1905; A.B. Kempe, "On the Relation between the Logical Theory of Classes and the Geometrical Theory of Points," _Proc. Lond. Math. Soc._, 1890; J. Royce, "The Relation of the Principles of Logic to the Foundations of Geometry," _Trans. of Amer. Math. Soc._, 1905; A. Schoenflies, "Uber die Moglichkeit einer projectiven Geometrie bei transfiniter (nichtarchimedischer) Massbestimmung," _Deutsch. M.-V. Jahresb._, 1906.

For general expositions of the bearings of the above investigations, cf. Hon. Bertrand Russell, _loc. cit._; L. Couturat, _Les Principes des mathematiques_ (Paris, 1905); H. Poincare, _loc. cit._; Russell and Whitehead, _Principia mathematica_ (Cambridge, Univ. Press). The philosophers whose views on space and geometric truth deserve especial study are Descartes, Leibnitz, Hume, Kant and J.S. Mill. (A. N. W.)

FOOTNOTES:

[1] For Egyptian geometry see EGYPT, S _Science and Mathematics_.

[2] Cf. A.N. Whitehead, _Universal Algebra_, Bk. vi. (Cambridge, 1898).

[3] Cf. A.N. Whitehead, _loc. cit._

[4] Cf. A.N. Whitehead, "The Geodesic Geometry of Surfaces in non-Euclidean Space," _Proc. Lond. Math. Soc._ vol. xxix.

[5] Cf. Klein, "Zur nicht-Euklidischen Geometrie," _Math. Annal._ vol. xxxvii.

[6] On the theory of parallels before Lobatchewsky, see Stackel und Engel, _Theorie der Parallellinien von Euklid bis auf Gauss_ (Leipzig, 1895). The foregoing remarks are based upon the materials collected in this work.

[7] See Stackel und Engel, _op. cit._, and "Gauss, die beiden Bolyai, und die nicht-Euklidische Geometrie," _Math. Annalen_, Bd. xlix.; also Engel's translation of Lobatchewsky (Leipzig, 1898), pp. 378 ff.

[8] Lobatchewsky's works on the subject are the following:--"On the Foundations of Geometry," _Kazan Messenger_, 1829-1830; "New Foundations of Geometry, with a complete Theory of Parallels," _Proceedings of the University of Kazan_, 1835 (both in Russian, but translated into German by Engel, Leipzig, 1898); "Geometrie imaginaire," Crelle's Journal, 1837; _Theorie der Parallellinien_ (Berlin, 1840; 2nd ed., 1887; translated by Halsted, Austin, Texas, 1891). His results appear to have been set forth in a paper (now lost) which he read at Kazan in 1826.

[9] Translated by Halsted (Austin, Texas, 4th ed., 1896.)

[10] _Abhandlungen d. Konigl. Ges. d. Wiss. zu Gottingen_, Bd. xiii.; _Ges. math. Werke_, pp. 254-269; translated by Clifford, _Collected Mathematical Papers_.

[11] Cf. _Gesamm. math. und phys. Werke_, vol. i. (Leipzig, 1894).

[12] _Wiss. Abh._ vol. ii. pp. 610, 618 (1866, 1868).

[13] _Mind_, O.S., vols. i. and iii.; _Vortrage und Reden_, vol. ii. pp. 1, 256.

[14] His papers are "Saggio di interpretazione della geometria non-Euclidea," _Giornale di matematiche_, vol. vi. (1868); "Teoria fondamentale degli spazii di curvatura costante," _Annali di matematica_, vol. ii. (1868-1869). Both were translated into French by J. Houel, _Annales scientifiques de l'Ecole Normale superieure_, vol. vi. (1869).

[15] Beltrami shows also that this definition agrees with that of Gauss.

[16] "Sur la theorie des foyers," _Nouv. Ann._ vol. xii.

[17] _Math. Annalen_, iv. vi., 1871-1872.

[18] For an investigation of these and similar properties, see Whitehead, _Universal Algebra_ (Cambridge, 1898), bk. vi. ch. ii. The polar form was independently discovered by Simon Newcomb in 1877.

[19] For an analysis of Leibnitz's ideas on space, cf. B. Russell, _The Philosophy of Leibnitz_, chs. viii.-x.

[20] Cf. Hon. Bertrand Russell, "Is Position in Time and Space Absolute or Relative?" _Mind_, n.s. vol. 10 (1901), and A.N. Whitehead, "Mathematical Concepts of the Material World," _Phil. Trans._ (1906), p. 205.

[21] Cf. _Critique of Pure Reason_, 1st section: "Of Space," conclusion A, Max Muller's translation.

[22] Cf. Ernst Mach, _Erkenntniss und Irrtum_ (Leipzig); the relevant chapters are translated by T.J. McCormack, _Space and Geometry_ (London, 1906); also A. Meinong, _Uber die Stellung der Gegenstandstheorie im System der Wissenschaften_ (Leipzig, 1907).

[23] Cf. Russell, _Principles of Mathematics_, S 352 (Cambridge, 1903).

[24] Cf. A.N. Whitehead, _The Axioms of Projective Geometry_, S 3 (Cambridge, 1906).

[25] Cf. Russell, _Princ. of Math._, ch. i.

[26] Cf. Russell, _loc. cit._, and G. Frege, "Uber die Grundlagen der Geometrie," _Jahresber. der Deutsch. Math. Ver._ (1906).

[27] This formulation--though not in respect to number--is in all essentials that of M. Pieri, cf. "I principii della Geometria di Posizione," _Accad. R. di Torino_ (1898); also cf. Whitehead, _loc. cit._

[28] Cf. G. Peano, "Sui fondamenti della Geometria," p. 73, _Rivista di matematica_, vol. iv. (1894), and D. Hilbert, _Grundlagen der Geometrie_ (Leipzig, 1899); and R.F. Moulton, "A Simple non-Desarguesian Plane Geometry," _Trans. Amer. Math. Soc._, vol. iii. (1902).

[29] Cf. "Sui postulati fondamentali della geometria projettiva," _Giorn. di matematica_, vol. xxx. (1891); also of Pieri, _loc. cit._, and Whitehead, _loc. cit._

[30] Cf. Hilbert, _loc. cit._; for a fuller exposition of Hilbert's proof cf. K.T. Vahlen, _Abstrakte Geometrie_ (Leipzig, 1905), also Whitehead, _loc. cit._

[31] Cf. H. Wiener, _Jahresber. der Deutsch. Math. Ver._ vol. i. (1890); and F. Schur, "Uber den Fundamentalsatz der projectiven Geometrie," _Math. Ann._ vol. li. (1899).

[32] Cf. Hilbert, _loc. cit._, and Whitehead, _loc. cit._

[33] Cf. Dedekind, _Stetigkeit und irrationale Zahlen_ (1872).

[34] Cf. v. Staudt, _Geometrie der Lage_ (1847).

[35] Cf. Pasch, _Vorlesungen uber neuere Geometrie_ (Leipzig, 1882), a classic work; also Fiedler, _Die darstellende Geometrie_ (1st ed., 1871, 3rd ed., 1888); Clebsch, _Vorlesungen uber Geometrie_, vol. iii.; Hilbert, _loc. cit._; F. Schur, _Math. Ann. Bd._ lv. (1902); Vahlen, _loc. cit._; Whitehead, _loc. cit._

[36] Cf. _loc. cit._

[37] Cf. _I Principii di geometria_ (Turin, 1889) and "Sui fondamenti della geometria," _Rivista di mat._ vol. iv. (1894).

[38] Cf. _loc. cit._

[39] Cf. Vailati, _Rivista di mat._ vol. iv. and Russell, _loc. cit._ S 376.

[40] Cf. O. Veblen, "On the Projective Axioms of Geometry," _Trans. Amer. Math. Soc._ vol. iii. (1902).

[41] Cf. P. Stackel and F. Engel, _Die Theorie der Parallellinien von Euklid bis auf Gauss_ (Leipzig, 1895).

[42] Cf. Pasch, _loc. cit._, and R. Bonola, "Sulla introduzione degli enti improprii in geometria projettive," _Giorn. di mat._ vol. xxxviii. (1900); and Whitehead, _Axioms of Descriptive Geometry_ (Cambridge, 1907).

[43] The original idea (confined to this particular case) of ideal points is due to von Staudt (_loc. cit._).

[44] Cf. _Critique_, "Trans. Aesth." Sect. I.

[45] Cf. _loc. cit._

[46] Cf. _Uber die Grundlagen der Geometrie_ (Leipzig, Ber., 1890); and _Theorie der Transformationsgruppen_ (Leipzig, 1893), vol. iii.

[47] Cf. A. Cayley, "A Sixth Memoir on Quantics," _Trans. Roy. Soc._, 1859, and _Coll. Papers_, vol. ii.; and F. Klein, _Math. Ann._ vol. iv., 1871.

[48] Cf. _loc. cit._

[49] For similar deductions from a third set of axioms, suggested in essence by Peano, Riv. mat. vol. iv. _loc. cit._ cf. Whitehead, _Desc. Geom. loc. cit._

[50] Cf. H. Poincare, _La Science et l'hypothese_, ch. iii.