FOOTNOTES:

FOOTNOTES:[5]V.Mémoires de l'Académie royale des Sciences de l'lnstitut de France, T.XII.1833, for a full statement of his results, with references to former writings.[6]This bolder method, it appears, had been suggested, nearly a century earlier, by an Italian, Saccheri. His work, which seems to have remained completely unknown until Beltrami rediscovered it in 1889, is called"Euclides ab omni naevo vindicatus, etc."Mediolani, 1733. (See Veronese, Grundzüge der Geometrie, German translation, Leipzig, 1894, p. 636.) His results included spherical as well as hyperbolic space; but they alarmed him to such an extent that he devoted the last half of his book to disproving them.[7]Klein's first account of elliptic Geometry, as a result of Cayley's projective theory of distance, appeared in two articles entitled"Ueber die sogenannte Nicht-Euklidische Geometrie, I, II,"Math. Annalen 4, 6 (1871–2). It was afterwards independently discovered by Newcomb, in an article entitled "Elementary Theorems relating to the geometry of a space of three dimensions, and of uniform positive curvature in the fourth dimension," Crelle'sJournal für die reine und angewandte Mathematik, Vol. 83 (1877). For an account of the mathematical controversies concerning elliptic Geometry, see Klein's"Vorlesungen über Nicht-Euklidische Geometrie,"Göttingen 1893,I.p. 284 ff. A bibliography of the relevant literature up to the year 1878 was given by Halsted in the American Journal of Mathematics, Vols. 1, 2.[8]Veronese (op. cit. p. 638) denies the priority of Gauss in the invention of a non-Euclidean system, though he admits him to have been the first to regard the axiom of parallels as indemonstrable. His grounds for the former assertion seem scarcely adequate: on the evidence against it, see Klein,Nicht-Euklid,I.pp. 171–174.[9]V.Briefwechsel mit Schumacher, Bd.II.p. 268.[10]f. Helmholtz, Wiss. Abh.II.p. 611.[11]Crelle's Journal, 1837.[12]Theorie der Parallellinien, Berlin, 1840. Republished, Berlin, 1887. Translated by Halsted, Austin, Texas, U.S.A. 4th edition, 1892.[13]Frischauf,Absolute Geometrie, nach Johann Bolyai, Leipzig, 1872. Halsted, The Science Absolute of Space, translated from the Latin, 4th edition, Austin, Texas, U.S.A. 1896.[14]Both Lobatchewsky and Bolyai, as Veronese remarks, start rather from the point-pair than from distance. See Frischauf, Absolute Geometrie, Anhang.[15]Compare Stallo,Concepts of Modern Physics, p. 248.[16]Gesammelte Werke, pp. 255–268.[17]On the history of this word, see Stallo,Concepts of Modern Physics, p. 258. It was used by Kant, and adapted by Herbart to almost the same meaning as it bears in Riemann. Herbart, however, also uses the wordReihenformto express a similar idea. SeePsychologie als Wissenschaft,I.§ 100 andII.§ 139, where Riemann's analogy with colours is also suggested.[18]Compare Erdmann's"Grössenbegriff vom Raum."[19]Compare Veronese, op. cit. p. 642:"Riemann ist in seiner Definition des Begriffs Grösse dunkel."See also Veronese's whole following criticism.[20]Vorträge und Reden, Vol.II.p. 18.[21]Cf. Klein,Nicht-Euklid,I.p. 160.[22]Since we are considering the curvature at a point, we are only concerned with the first infinitesimal elements of the geodesics that start from such a point.[23]Disquisitiones generales circa superficies curvas, Werke, Bd.IV.SS. 219–258, 1827.[24]Nevertheless, the Geometries of different surfaces of equal curvature are liable to important differences. For example, the cylinder is a surface of zero curvature, but since its lines of curvature in one direction are finite, its Geometry coincides with that of the plane only for lengths smaller than the circumference of its generating circle (see Veronese, op. cit. p. 644). Two geodesics on a cylinder may meet in many points. For surfaces of zero curvature on which this is not possible, the identity with the plane may be allowed to stand. Otherwise, the identity extends only to the properties of figures not exceeding a certain size.[25]For we may consider two different parts of the same surface as corresponding parts of different surfaces; the above proposition then shows that a figure can be reproduced in one part when it has been drawn in another, if the measures of curvature correspond in the two parts.[26]Crelle, Vols,XIX.,XX., 1839–40.[27]In this formula,u,vmay be the lengths of lines, or the angles between lines, drawn on the surface, and having thus no necessary reference to a third dimension.[28]In what follows, I have given rather Klein's exposition of Riemann, than Riemann's own account. The former is much clearer and fuller, and not substantially different in any way. V. Klein,Nicht-Euklid,I.pp. 206 ff.[29]See§§ 69–73.[30]Grundlagen der Geometrie,I.andII., Leipziger Berichte, 1890; v. end of present chapter,§ 45.[31]Nicht-Euklid,I.pp. 258–9.[32]Giornale di Matematiche, Vol.VI., 1868. Translated into French by J. Hoüel in the"Annales Scientifiques de l'École Normale Supérieure,"Vol.VI.1869.[33]Crelle's Journal, Vols.XIX.XX., 1839–40.[34]Nicht-Euklid,I.p. 190.[35]This article is more trigonometrical and analytical than the German book, and therefore makes the above interpretation peculiarly evident.[36]Such surfaces are by no means particularly remote. One of them, for example, is formed by the revolution of the common Tractrixx=asinφ,y=a(log tanφ2+ cosφ).[37]"Teoria fondamentale degli spazii di curvatura costanta," Annali di Matematica,II.Vol. 2, 1868–9. Also translated by J. Hoüel,loc. cit.[38]See Klein,Nicht-Euklid,I.p. 47 ff., and the references there given.[39]See quotation below, from his British Association Address.[40]Compare the opening sentence, due to Cayley, of Salmon's Higher Plane Curves.[41]V.Nicht-Euklid,I.Chaps.I.andII.[42]See p. 9 of Cayley's address to the Brit. Ass. 1883. Also a quotation from Klein in Erdmann'sAxiome der Geometrie, p. 124 note.[43]Nature, Vol.XLV.p. 407.[44]Nicht-Euklid,I.p. 200.[45]I.e. the equationAB+BC=AC, for three points in one straight line.[46]The formula substituted by Klein for Cayley's inverse sine or cosine. The two are equivalent, but Klein's is mathematically much the more convenient.[47]Elements of Projective Geometry, Second Edition, Oxford, 1893, Chap.IX.[48]Chap.III.Section B.[49]SeeNicht-Euklid,I.p. 338 ff.[50]See hisGeometrie der Lage, § 8, Harmonische Gebilde.[51]The anharmonic ratio of four numbers,p,q,r,s, is defined as(p-q).(r-s)/(p-r).(q-s).[52]I.e.as transformable into each other by a collineation. SeeChap.III.Sec. A, § 110.[53]SeeChap.III.Sec. A.[54]It follows from this, that the reduction of metrical to projective properties, even when, as in hyperbolic Geometry, the Absolute is real, is only apparent, and has a merely technical validity.[55]Sir R. Ball does not regard his non-Euclidean content as a possible space (v. op. cit.p. 151). In this important point I disagree with his interpretation, holding such a content to be a space as possible,à priori, as Euclid's, and perhaps actually true within the margin due to errors of observation.[56]SeeNicht-Euklid,I.p. 97 ff. and p. 292 ff.[57]Newcomb says (loc. cit.p. 293): "The system here set forth is founded on the following three postulates."1. I assume that space is triply extended, unbounded, without properties dependent either on position or direction, and possessing such planeness in its smallest parts that both the postulates of the Euclidean Geometry, and our common conceptions of the relations of the parts of space are true for every indefinitely small region in space."2. I assume that this space is affected with such curvature that a right line shall always return into itself at the end of a finite and real distance 2Dwithout losing, in any part of its course, that symmetry with respect to space on all sides of it which constitutes the fundamental property of our conception of it."3. I assume that if two right lines emanate from the same point, making the indefinitely small angleawith each other, their distance apart at the distancerfrom the point of intersection will be given by the equations=2aDπsinrπ2D.The right line thus has this property in common with the Euclidean right line that two such lines intersect only in a single point. It may be that the number of points in which two such lines can intersect admit of being determined from the laws of curvature, but not being able so to determine it, I assume as a postulate the fundamental property of the Euclidean right line."It is plain that in the absence of the determination spoken of, the possibility of elliptic space is not established. It may be possible, for example, to prove that, in a space where there is a maximum to distance, there must be an infinite number of straight lines joining two points of maximum distance. In this event, elliptic space would become impossible.[58]For an elucidation of this term, see Klein,Nicht-Euklid,I.p. 99 ff.[59]Cf. p. 9 of Report: "My own view is that Euclid's twelfth axiom, in Playfair's form of it, does not need demonstration, but is part of our notion of space, of the physical space of our experience, but which is the representation lying at the bottom of all external experience."[60]The exception to this axiom, in spherical space, presupposes metrical Geometry, and does not destroy the validity of the axiom for projective Geometry. SeeChap.III.Sec. B, § 171.[61]Mathematicians of Lie's school have a habit, at first somewhat confusing, of speaking of motions of space instead of motions of bodies, as though space as a whole could move. All that is meant is, of course, the equivalent motion of the coordinate axes,i.e.a change of axes in the usual elementary sense.[62]"Ueber die Grundlagen der Geometrie,"Leipziger Berichte, 1890. The problem of these two papers is really metrical, since it is concerned, not with collineations in general, but with motions. The problem, however, is dealt with by the projective method, motions being regarded as collineations which leave the Absolute unchanged. It seemed impossible, therefore, to discuss Lie's work, until some account had been given of the projective method.[63]Lie's premisses, to be accurate, are the following:Letx1=f(x,y,z,a1,a2...)x2=φ(x,y,z,a1,a2...)x3=ψ(x,y,z,a1,a2...)give an infinite family of real transformations of space, as to which we make the following hypotheses:A. The functionsf,φ,ψ, areanalyticalfunctions ofx,y,z,a1,a2....B. Two pointsx1y1z1,x2y2z2possess an invariant,i.e.Ω(x1,y1,z1,x2,y2,z2) = Ω(x1′,y1′,z1′,x2′,y2′,z2′)wherex1′...,x2′..., are the transformed coordinates of the two points.C. Free Mobility:i.e., any point can be moved into any other position; when one point is fixed, any other point of general position can take up ∞2positions; when two points are fixed, any other of general position can take up ∞1positions; when three, no motion is possible—these limitations being results of the equations given by the invariant Ω.[64]On this point, cf. Klein,Höhere Geometrie, Göttingen, 1893,II.pp. 225–244, especially pp. 230–1.[65]AxiomII.of the metrical triad corresponds to AxiomIII.of the projective, andvice versâ.[66]Cf. Helmholtz, Wiss. Abh. Vol.II.p. 640, note:"Die Bearbeiter der Nicht-Euklidischen Geometrie (haben) deren objective Wahrheit nie behauptet."

[5]V.Mémoires de l'Académie royale des Sciences de l'lnstitut de France, T.XII.1833, for a full statement of his results, with references to former writings.

[6]This bolder method, it appears, had been suggested, nearly a century earlier, by an Italian, Saccheri. His work, which seems to have remained completely unknown until Beltrami rediscovered it in 1889, is called"Euclides ab omni naevo vindicatus, etc."Mediolani, 1733. (See Veronese, Grundzüge der Geometrie, German translation, Leipzig, 1894, p. 636.) His results included spherical as well as hyperbolic space; but they alarmed him to such an extent that he devoted the last half of his book to disproving them.

[7]Klein's first account of elliptic Geometry, as a result of Cayley's projective theory of distance, appeared in two articles entitled"Ueber die sogenannte Nicht-Euklidische Geometrie, I, II,"Math. Annalen 4, 6 (1871–2). It was afterwards independently discovered by Newcomb, in an article entitled "Elementary Theorems relating to the geometry of a space of three dimensions, and of uniform positive curvature in the fourth dimension," Crelle'sJournal für die reine und angewandte Mathematik, Vol. 83 (1877). For an account of the mathematical controversies concerning elliptic Geometry, see Klein's"Vorlesungen über Nicht-Euklidische Geometrie,"Göttingen 1893,I.p. 284 ff. A bibliography of the relevant literature up to the year 1878 was given by Halsted in the American Journal of Mathematics, Vols. 1, 2.

[8]Veronese (op. cit. p. 638) denies the priority of Gauss in the invention of a non-Euclidean system, though he admits him to have been the first to regard the axiom of parallels as indemonstrable. His grounds for the former assertion seem scarcely adequate: on the evidence against it, see Klein,Nicht-Euklid,I.pp. 171–174.

[9]V.Briefwechsel mit Schumacher, Bd.II.p. 268.

[10]f. Helmholtz, Wiss. Abh.II.p. 611.

[11]Crelle's Journal, 1837.

[12]Theorie der Parallellinien, Berlin, 1840. Republished, Berlin, 1887. Translated by Halsted, Austin, Texas, U.S.A. 4th edition, 1892.

[13]Frischauf,Absolute Geometrie, nach Johann Bolyai, Leipzig, 1872. Halsted, The Science Absolute of Space, translated from the Latin, 4th edition, Austin, Texas, U.S.A. 1896.

[14]Both Lobatchewsky and Bolyai, as Veronese remarks, start rather from the point-pair than from distance. See Frischauf, Absolute Geometrie, Anhang.

[15]Compare Stallo,Concepts of Modern Physics, p. 248.

[16]Gesammelte Werke, pp. 255–268.

[17]On the history of this word, see Stallo,Concepts of Modern Physics, p. 258. It was used by Kant, and adapted by Herbart to almost the same meaning as it bears in Riemann. Herbart, however, also uses the wordReihenformto express a similar idea. SeePsychologie als Wissenschaft,I.§ 100 andII.§ 139, where Riemann's analogy with colours is also suggested.

[18]Compare Erdmann's"Grössenbegriff vom Raum."

[19]Compare Veronese, op. cit. p. 642:"Riemann ist in seiner Definition des Begriffs Grösse dunkel."See also Veronese's whole following criticism.

[20]Vorträge und Reden, Vol.II.p. 18.

[21]Cf. Klein,Nicht-Euklid,I.p. 160.

[22]Since we are considering the curvature at a point, we are only concerned with the first infinitesimal elements of the geodesics that start from such a point.

[23]Disquisitiones generales circa superficies curvas, Werke, Bd.IV.SS. 219–258, 1827.

[24]Nevertheless, the Geometries of different surfaces of equal curvature are liable to important differences. For example, the cylinder is a surface of zero curvature, but since its lines of curvature in one direction are finite, its Geometry coincides with that of the plane only for lengths smaller than the circumference of its generating circle (see Veronese, op. cit. p. 644). Two geodesics on a cylinder may meet in many points. For surfaces of zero curvature on which this is not possible, the identity with the plane may be allowed to stand. Otherwise, the identity extends only to the properties of figures not exceeding a certain size.

[25]For we may consider two different parts of the same surface as corresponding parts of different surfaces; the above proposition then shows that a figure can be reproduced in one part when it has been drawn in another, if the measures of curvature correspond in the two parts.

[26]Crelle, Vols,XIX.,XX., 1839–40.

[27]In this formula,u,vmay be the lengths of lines, or the angles between lines, drawn on the surface, and having thus no necessary reference to a third dimension.

[28]In what follows, I have given rather Klein's exposition of Riemann, than Riemann's own account. The former is much clearer and fuller, and not substantially different in any way. V. Klein,Nicht-Euklid,I.pp. 206 ff.

[29]See§§ 69–73.

[30]Grundlagen der Geometrie,I.andII., Leipziger Berichte, 1890; v. end of present chapter,§ 45.

[31]Nicht-Euklid,I.pp. 258–9.

[32]Giornale di Matematiche, Vol.VI., 1868. Translated into French by J. Hoüel in the"Annales Scientifiques de l'École Normale Supérieure,"Vol.VI.1869.

[33]Crelle's Journal, Vols.XIX.XX., 1839–40.

[34]Nicht-Euklid,I.p. 190.

[35]This article is more trigonometrical and analytical than the German book, and therefore makes the above interpretation peculiarly evident.

[36]Such surfaces are by no means particularly remote. One of them, for example, is formed by the revolution of the common Tractrixx=asinφ,y=a(log tanφ2+ cosφ).

[36]Such surfaces are by no means particularly remote. One of them, for example, is formed by the revolution of the common Tractrix

x=asinφ,y=a(log tanφ2+ cosφ).

[37]"Teoria fondamentale degli spazii di curvatura costanta," Annali di Matematica,II.Vol. 2, 1868–9. Also translated by J. Hoüel,loc. cit.

[38]See Klein,Nicht-Euklid,I.p. 47 ff., and the references there given.

[39]See quotation below, from his British Association Address.

[40]Compare the opening sentence, due to Cayley, of Salmon's Higher Plane Curves.

[41]V.Nicht-Euklid,I.Chaps.I.andII.

[42]See p. 9 of Cayley's address to the Brit. Ass. 1883. Also a quotation from Klein in Erdmann'sAxiome der Geometrie, p. 124 note.

[43]Nature, Vol.XLV.p. 407.

[44]Nicht-Euklid,I.p. 200.

[45]I.e. the equationAB+BC=AC, for three points in one straight line.

[46]The formula substituted by Klein for Cayley's inverse sine or cosine. The two are equivalent, but Klein's is mathematically much the more convenient.

[47]Elements of Projective Geometry, Second Edition, Oxford, 1893, Chap.IX.

[48]Chap.III.Section B.

[49]SeeNicht-Euklid,I.p. 338 ff.

[50]See hisGeometrie der Lage, § 8, Harmonische Gebilde.

[51]The anharmonic ratio of four numbers,p,q,r,s, is defined as(p-q).(r-s)/(p-r).(q-s).

[51]The anharmonic ratio of four numbers,p,q,r,s, is defined as

(p-q).(r-s)/(p-r).(q-s).

[52]I.e.as transformable into each other by a collineation. SeeChap.III.Sec. A, § 110.

[53]SeeChap.III.Sec. A.

[54]It follows from this, that the reduction of metrical to projective properties, even when, as in hyperbolic Geometry, the Absolute is real, is only apparent, and has a merely technical validity.

[55]Sir R. Ball does not regard his non-Euclidean content as a possible space (v. op. cit.p. 151). In this important point I disagree with his interpretation, holding such a content to be a space as possible,à priori, as Euclid's, and perhaps actually true within the margin due to errors of observation.

[56]SeeNicht-Euklid,I.p. 97 ff. and p. 292 ff.

[57]Newcomb says (loc. cit.p. 293): "The system here set forth is founded on the following three postulates."1. I assume that space is triply extended, unbounded, without properties dependent either on position or direction, and possessing such planeness in its smallest parts that both the postulates of the Euclidean Geometry, and our common conceptions of the relations of the parts of space are true for every indefinitely small region in space."2. I assume that this space is affected with such curvature that a right line shall always return into itself at the end of a finite and real distance 2Dwithout losing, in any part of its course, that symmetry with respect to space on all sides of it which constitutes the fundamental property of our conception of it."3. I assume that if two right lines emanate from the same point, making the indefinitely small angleawith each other, their distance apart at the distancerfrom the point of intersection will be given by the equations=2aDπsinrπ2D.The right line thus has this property in common with the Euclidean right line that two such lines intersect only in a single point. It may be that the number of points in which two such lines can intersect admit of being determined from the laws of curvature, but not being able so to determine it, I assume as a postulate the fundamental property of the Euclidean right line."It is plain that in the absence of the determination spoken of, the possibility of elliptic space is not established. It may be possible, for example, to prove that, in a space where there is a maximum to distance, there must be an infinite number of straight lines joining two points of maximum distance. In this event, elliptic space would become impossible.

[57]Newcomb says (loc. cit.p. 293): "The system here set forth is founded on the following three postulates.

"1. I assume that space is triply extended, unbounded, without properties dependent either on position or direction, and possessing such planeness in its smallest parts that both the postulates of the Euclidean Geometry, and our common conceptions of the relations of the parts of space are true for every indefinitely small region in space.

"2. I assume that this space is affected with such curvature that a right line shall always return into itself at the end of a finite and real distance 2Dwithout losing, in any part of its course, that symmetry with respect to space on all sides of it which constitutes the fundamental property of our conception of it.

"3. I assume that if two right lines emanate from the same point, making the indefinitely small angleawith each other, their distance apart at the distancerfrom the point of intersection will be given by the equation

s=2aDπsinrπ2D.

The right line thus has this property in common with the Euclidean right line that two such lines intersect only in a single point. It may be that the number of points in which two such lines can intersect admit of being determined from the laws of curvature, but not being able so to determine it, I assume as a postulate the fundamental property of the Euclidean right line."

It is plain that in the absence of the determination spoken of, the possibility of elliptic space is not established. It may be possible, for example, to prove that, in a space where there is a maximum to distance, there must be an infinite number of straight lines joining two points of maximum distance. In this event, elliptic space would become impossible.

[58]For an elucidation of this term, see Klein,Nicht-Euklid,I.p. 99 ff.

[59]Cf. p. 9 of Report: "My own view is that Euclid's twelfth axiom, in Playfair's form of it, does not need demonstration, but is part of our notion of space, of the physical space of our experience, but which is the representation lying at the bottom of all external experience."

[60]The exception to this axiom, in spherical space, presupposes metrical Geometry, and does not destroy the validity of the axiom for projective Geometry. SeeChap.III.Sec. B, § 171.

[61]Mathematicians of Lie's school have a habit, at first somewhat confusing, of speaking of motions of space instead of motions of bodies, as though space as a whole could move. All that is meant is, of course, the equivalent motion of the coordinate axes,i.e.a change of axes in the usual elementary sense.

[62]"Ueber die Grundlagen der Geometrie,"Leipziger Berichte, 1890. The problem of these two papers is really metrical, since it is concerned, not with collineations in general, but with motions. The problem, however, is dealt with by the projective method, motions being regarded as collineations which leave the Absolute unchanged. It seemed impossible, therefore, to discuss Lie's work, until some account had been given of the projective method.

[63]Lie's premisses, to be accurate, are the following:Letx1=f(x,y,z,a1,a2...)x2=φ(x,y,z,a1,a2...)x3=ψ(x,y,z,a1,a2...)give an infinite family of real transformations of space, as to which we make the following hypotheses:A. The functionsf,φ,ψ, areanalyticalfunctions ofx,y,z,a1,a2....B. Two pointsx1y1z1,x2y2z2possess an invariant,i.e.Ω(x1,y1,z1,x2,y2,z2) = Ω(x1′,y1′,z1′,x2′,y2′,z2′)wherex1′...,x2′..., are the transformed coordinates of the two points.C. Free Mobility:i.e., any point can be moved into any other position; when one point is fixed, any other point of general position can take up ∞2positions; when two points are fixed, any other of general position can take up ∞1positions; when three, no motion is possible—these limitations being results of the equations given by the invariant Ω.

[63]Lie's premisses, to be accurate, are the following:

Let

x1=f(x,y,z,a1,a2...)x2=φ(x,y,z,a1,a2...)x3=ψ(x,y,z,a1,a2...)

give an infinite family of real transformations of space, as to which we make the following hypotheses:

A. The functionsf,φ,ψ, areanalyticalfunctions of

x,y,z,a1,a2....

B. Two pointsx1y1z1,x2y2z2possess an invariant,i.e.

Ω(x1,y1,z1,x2,y2,z2) = Ω(x1′,y1′,z1′,x2′,y2′,z2′)

wherex1′...,x2′..., are the transformed coordinates of the two points.

C. Free Mobility:i.e., any point can be moved into any other position; when one point is fixed, any other point of general position can take up ∞2positions; when two points are fixed, any other of general position can take up ∞1positions; when three, no motion is possible—these limitations being results of the equations given by the invariant Ω.

[64]On this point, cf. Klein,Höhere Geometrie, Göttingen, 1893,II.pp. 225–244, especially pp. 230–1.

[65]AxiomII.of the metrical triad corresponds to AxiomIII.of the projective, andvice versâ.

[66]Cf. Helmholtz, Wiss. Abh. Vol.II.p. 640, note:"Die Bearbeiter der Nicht-Euklidischen Geometrie (haben) deren objective Wahrheit nie behauptet."

51.We have now traced the mathematical development of the theory of geometrical axioms, from the first revolt against Euclid to the present day. We may hope, therefore, to have at our command the technical knowledge required for the philosophy of the subject. The importance of Geometry, in the theories of knowledge which have arisen in the past, can scarcely be exaggerated. In Descartes, we find the whole theory of method dominated by analytical Geometry, of whose fruitfulness he was justly proud. In Spinoza, the paramount influence of Geometry is too obvious to require comment. Among mathematicians, Newton's belief in absolute space was long supreme, and is still responsible for the current formulation of the laws of motion. Against this belief on the one hand, and against Leibnitz's theory of space on the other, and not, as Caird has pointed out[67], against Hume's empiricism, was directed that keystone of the Critical Philosophy, the Kantian doctrine of space. Thus Geometry has been, throughout, of supreme importance in the theory of knowledge.

But in a criticism of representative modern theories of Geometry, which is designed to be, not a history of the subject, but an introduction to, and defence of, the views of the author, it will not be necessary to discuss any more ancient theory than that of Kant. Kant's views on this subject, true or false, have so dominated subsequent thought, that whether they wereaccepted or rejected, they seemed equally potent in forming the opinions, and the manner of exposition, of almost all later writers.

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