Chapter 26

Riemann’s work contains two fundamental conceptions, that of a manifold and that of themeasure of curvatureof a continuous manifold possessed of what he calls flatness in the smallest parts. By means of these conceptions space is made to appearDefinition of a manifold.at the end of a gradual series of more and more specialized conceptions. Conceptions of magnitude, he explains, are only possible where we have a general conception capable of determination in various ways. The manifold consists of all these various determinations, each of which is an element of the manifold. The passage from one element to another may be discrete or continuous; the manifold is called discrete or continuous accordingly. Where it is discrete two portions of it can be compared, as to magnitude, by counting; where continuous, by measurement. But measurement demands superposition, and consequently some magnitude independent of its place in the manifold. In passing, in a continuous manifold, from one element to another in a determinate way, we pass through a series of intermediate terms, which form a one-dimensional manifold. If this whole manifold be similarly caused to pass over into another, each of its elements passes through a one-dimensional manifold, and thus on the whole a two-dimensional manifold is generated. In this way we can proceed to n dimensions. Conversely, a manifold of n dimensions can be analysed into one of one dimension and one of (n − 1) dimensions. By repetitions of this process the position of an element may be at last determined by n magnitudes. We may here stop to observe that the above conception of a manifold is akin to that due to Hermann Grassmann in the first edition (1847) of hisAusdehnungslehre.11

Both concepts have been elaborated and superseded by the modern procedure in respect to the axioms of geometry, and by the conception of abstract geometry involved therein. Riemann proceeds to specialize the manifold by considerationsMeasure of curvature.as to measurement. If measurement is to be possible, some magnitude, we saw, must be independent of position; let us consider manifolds in which lengths of lines are such magnitudes, so that every line is measurable by every other. The coordinates of a point being x1, x2, ... xn, let us confine ourselves to lines along which the ratios dx1: dx2: ... : dxnalter continuously. Let us also assume that the element of length, ds, is unchanged (to the first order) when all its points undergo the same infinitesimal motion. Then if all the increments dx be altered in the same ratio, ds is also altered in this ratio. Hence ds is a homogeneous function of the first degree of the increments dx. Moreover, ds must be unchanged when all the dx change sign. The simplest possible case is, therefore, that in which ds is the square root of a quadratic function of the dx. This case includes space, and is alone considered in what follows. It is called the case of flatness in the smallest parts. Its further discussion depends upon the measure of curvature, the second of Riemann’s fundamental conceptions. This conception, derived from the theory of surfaces, is applied as follows. Any one of the shortest lines which issue from a given point (say the origin) is completely determined by the initial ratios of the dx. Two such lines, defined by dx and δx say, determine a pencil, or one-dimensional series, of shortest lines, any one of which is definedby λdx + μδx, where the parameter λ : μ may have any value. This pencil generates a two-dimensional series of points, which may be regarded as a surface, and for which we may apply Gauss’s formula for the measure of curvature at any point. Thus at every point of our manifold there is a measure of curvature corresponding to every such pencil; but all these can be found when n·n − 1/2 of them are known. If figures are to be freely movable, it is necessary and sufficient that the measure of curvature should be the same for all points and all directions at each point. Where this is the case,ifα be the measure of curvature, the linear element can be put into the form

ds = √(Σdx²) / (1 + ¼αΣx²).

If α be positive, space is finite, though still unbounded, and every straight line is closed—a possibility first recognized by Riemann. It is pointed out that, since the possible values of a form a continuous series, observations cannot prove that our space is strictly Euclidean. It is also regarded as possible that, in the infinitesimal, the measure of curvature of our space should be variable.

There are four points in which this profound and epoch-making work is open to criticism or development—(1) the idea of a manifold requires more precise determination; (2) the introduction of coordinates is entirely unexplained and the requisite presuppositions are unanalysed; (3) the assumption that ds is the square root of a quadratic function of dx1, dx2, ... is arbitrary; (4) the idea of superposition, or congruence, is not adequately analysed. The modern solution of these difficulties is properly considered in connexion with the general subject of the axioms of geometry.

The publication of Riemann’s dissertation was closely followed by two works of Hermann von Helmholtz,12again undertaken in ignorance of the work of predecessors. In these aHelmholtz.proof is attempted that ds must be a rational integral quadratic function of the increments of the coordinates. This proof has since been shown by Lie to stand in need of correction (see VII.Axioms of Geometry). Helmholtz’s remaining works on the subject13are of almost exclusively philosophical interest. We shall return to them later.

The only other writer of importance in the second period is Eugenio Beltrami, by whom Riemann’s work was brought into connexion with that of Lobatchewsky and Bolyai. As he gave, by an elegant method, a convenientBeltrami.Euclidean interpretation of hyperbolic plane geometry, his results will be stated at some length14. TheSaggioshows that Lobatchewsky’s plane geometry holds in Euclidean geometry on surfaces of constant negative curvature, straight lines being replaced by geodesics. Such surfaces are capable of a conformal representation on a plane, by which geodesics are represented by straight lines. Hence if we take, as coordinates on the surface, the Cartesian coordinates of corresponding points on the plane, the geodesics must have linear equations.

Hence it follows thatds² = R²w−4{(α² − v²) du² + 2uvdudv + (α² − u²)dv²}where w² = α² − u² − v², and −1/R² is the measure of curvature of our surface (note that k = γ as used above). The angle between two geodesics u = const., v = const. is θ, wherecos θ = uv / √ {(α² − u²) (α² − v²)}, sin θ = aw / √ {(a² − u²) (a² − v²)}.Thus u = 0 is orthogonal to all geodesies v = const., and vice versa. In order that sin θ may be real, w² must be positive; thus geodesics have no real intersection when the corresponding straight lines intersect outside the circle u² + v² = α². When they intersect on this circle, θ = 0. Thus Lobatchewsky’s parallels are represented by straight lines intersecting on the circle. Again, transforming to polar coordinates u = r cos μ, v = r sin μ, and calling ρ the geodesic distance of u, v from the origin, we have, for a geodesic through the origin,dρ = Radr / (a² − r²), ρ = ½R loga + r, r = a tan h (ρ / R).a − rThus points on the surface corresponding to points in the plane on the limiting circle r = a, are all at an infinite distance from the origin. Again, considering r constant, the arc of a geodesic circle subtending an angle μ at the origin isσ = Rrμ / √ (a² − r²) = μR sin h (ρ/R),whence the circumference of a circle of radius ρ is 2πR sin h (ρ/R). Again, if α be the angle between any two geodesicsV − v = m (U − u), V − v = n (U − u),thentan α = a (n − m)w / {(1 + mn)a² − (v − mu) (v − nu)}.Thus α is imaginary when u, v is outside the limiting circle, and is zero when, and only when, u, v is on the limiting circle. All these results agree with those of Lobatchewsky and Bolyai. The maximum triangle, whose angles are all zero, is represented in the auxiliary plane by a triangle inscribed in the limiting circle. The angle of parallelism is also easily obtained. The perpendicular to v = 0 at a distance δ from the origin is u = a tan h (δ/R), and the parallel to this through the origin is u = v sin h (δ/R). Hence Π (δ), the angle which this parallel makes with v = 0, is given bytan Π(δ) . sin h (δ/R) = 1, or tan ½Π(δ) = e−δ/Rwhich is Lobatchewsky’s formula. We also obtain easily for the area of a triangle the formula R²(π − A − B − C).Beltrami’s treatment connects two curves which, in the earlier treatment, had no connexion. These are limit-lines and curves of constant distance from a straight line. Both may be regarded as circles, the first having an infinite, the second an imaginary radius. The equation to a circle of radius ρ and centre u0v0is(a² − uu0− vv0)² = cos h² (ρ/R) w0²w² = C²w²(say).This equation remains real when ρ is a pure imaginary, and remains finite when w0= 0, provided ρ becomes infinite in such a way that w0cos h (ρ/R) remains finite. In the latter case the equation represents a limit-line. In the former case, by giving different values to C, we obtain concentric circles with the imaginary centre u0v0. One of these, obtained by putting C = 0, is the straight line a² − uu0− vv0= 0. Hence the others are each throughout at a constant distance from this line. (It may be shown that all motions in a hyperbolic plane consist, in a general sense, of rotations; but three types must be distinguished according as the centre is real, imaginary or at infinity. All points describe, accordingly, one of the three types of circles.)The above Euclidean interpretation fails for three or more dimensions. In theTeoria fondamentale, accordingly, where n dimensions are considered, Beltrami treats hyperbolic space in a purely analytical spirit. The paper shows that Lobatchewsky’s space of any number of dimensions has, in Riemann’s sense, a constant negative measure of curvature. Beltrami starts with the formula (analogous to that of theSaggio)ds² = R²x−2(dx² + dx1² + dx2² + ... + dxn²)wherex² + x1² + x2² + ... + xn² = a².He shows that geodesics are represented by linear equations between x1, x2, ..., xn, and that the geodesic distance ρ between two points x and x′ is given bycos hρ=a² − x1x′1− x2x′2− ... − xnx′nR{(a² − x1² − x2² − ... − xn²) (a² − x′1² − x′2² − ... − x′n²)}1/2(a formula practically identical with Cayley’s, though obtained by a very different method). In order to show that the measure of curvature is constant, we make the substitutionsx1= rλ1, x2= rλ2... xn= rλn, where Σλ² = 1.Henceds² = (Radr /a² − r²)² + R²r²dΔ² / (a² − r²).wheredΔ² = Σdλ².Also calling ρ the geodesic distance from the origin, we havecos h (ρ/R) =a, sin h (ρ/R) =r.√(a² − r²)√(a² − r²)Henceds² = dρ² + (R sin h (ρ/R))² dΔ².Puttingz1= ρλ1, z2= ρλ2, ... zn= ρλn,we obtainds² = Σdz² +1{ (Rsinhρ)²− 1}Σ (zidzk− zkdzi)².ρ²ρRHence when ρ is small, we have approximatelyds² = Σdz² +1Σ (zidzk− zkdzi)²3R²(1).Considering a surface element through the origin, we may choose our axes so that, for this element,z3= z4= ... = zn= 0.Thusdz1² + dz2² +1(z1dz2− z2dz1)²3R²(2).Now the area of the triangle whose vertices are (0, 0), (z1, z2), (dz1, dz2) is ½(z1, dz2− z2dz1). Hence the quotient when the terms of the fourth order in (2) are divided by the square of this triangle is4/3R²; hence, returning to general axes, the same is the quotient when the terms of the fourth order in (1) are divided by the square of the triangle whose vertices are (0, 0, ... 0), (z1, z2, z3, ... zn), (dz1, dz2, dz3... dzn). But −¾ of this quotient is defined by Riemann as the measure of curvature.15Hence the measure of curvature is −1/R²,i.e.is constant and negative. The properties of parallels, triangles, &c., are as in theSaggio. It is also shown that the analogues of limit surfaces have zero curvature; and that spheres of radius ρ have constant positive curvature 1/R² sinh² (ρ/R), so that spherical geometry may be regarded as contained in the pseudo-spherical (as Beltrami calls Lobatchewsky’s system).

Hence it follows that

ds² = R²w−4{(α² − v²) du² + 2uvdudv + (α² − u²)dv²}

where w² = α² − u² − v², and −1/R² is the measure of curvature of our surface (note that k = γ as used above). The angle between two geodesics u = const., v = const. is θ, where

cos θ = uv / √ {(α² − u²) (α² − v²)}, sin θ = aw / √ {(a² − u²) (a² − v²)}.

Thus u = 0 is orthogonal to all geodesies v = const., and vice versa. In order that sin θ may be real, w² must be positive; thus geodesics have no real intersection when the corresponding straight lines intersect outside the circle u² + v² = α². When they intersect on this circle, θ = 0. Thus Lobatchewsky’s parallels are represented by straight lines intersecting on the circle. Again, transforming to polar coordinates u = r cos μ, v = r sin μ, and calling ρ the geodesic distance of u, v from the origin, we have, for a geodesic through the origin,

Thus points on the surface corresponding to points in the plane on the limiting circle r = a, are all at an infinite distance from the origin. Again, considering r constant, the arc of a geodesic circle subtending an angle μ at the origin is

σ = Rrμ / √ (a² − r²) = μR sin h (ρ/R),

whence the circumference of a circle of radius ρ is 2πR sin h (ρ/R). Again, if α be the angle between any two geodesics

V − v = m (U − u), V − v = n (U − u),

then

tan α = a (n − m)w / {(1 + mn)a² − (v − mu) (v − nu)}.

Thus α is imaginary when u, v is outside the limiting circle, and is zero when, and only when, u, v is on the limiting circle. All these results agree with those of Lobatchewsky and Bolyai. The maximum triangle, whose angles are all zero, is represented in the auxiliary plane by a triangle inscribed in the limiting circle. The angle of parallelism is also easily obtained. The perpendicular to v = 0 at a distance δ from the origin is u = a tan h (δ/R), and the parallel to this through the origin is u = v sin h (δ/R). Hence Π (δ), the angle which this parallel makes with v = 0, is given by

tan Π(δ) . sin h (δ/R) = 1, or tan ½Π(δ) = e−δ/R

which is Lobatchewsky’s formula. We also obtain easily for the area of a triangle the formula R²(π − A − B − C).

Beltrami’s treatment connects two curves which, in the earlier treatment, had no connexion. These are limit-lines and curves of constant distance from a straight line. Both may be regarded as circles, the first having an infinite, the second an imaginary radius. The equation to a circle of radius ρ and centre u0v0is

(a² − uu0− vv0)² = cos h² (ρ/R) w0²w² = C²w²

(say).

This equation remains real when ρ is a pure imaginary, and remains finite when w0= 0, provided ρ becomes infinite in such a way that w0cos h (ρ/R) remains finite. In the latter case the equation represents a limit-line. In the former case, by giving different values to C, we obtain concentric circles with the imaginary centre u0v0. One of these, obtained by putting C = 0, is the straight line a² − uu0− vv0= 0. Hence the others are each throughout at a constant distance from this line. (It may be shown that all motions in a hyperbolic plane consist, in a general sense, of rotations; but three types must be distinguished according as the centre is real, imaginary or at infinity. All points describe, accordingly, one of the three types of circles.)

The above Euclidean interpretation fails for three or more dimensions. In theTeoria fondamentale, accordingly, where n dimensions are considered, Beltrami treats hyperbolic space in a purely analytical spirit. The paper shows that Lobatchewsky’s space of any number of dimensions has, in Riemann’s sense, a constant negative measure of curvature. Beltrami starts with the formula (analogous to that of theSaggio)

ds² = R²x−2(dx² + dx1² + dx2² + ... + dxn²)

where

x² + x1² + x2² + ... + xn² = a².

He shows that geodesics are represented by linear equations between x1, x2, ..., xn, and that the geodesic distance ρ between two points x and x′ is given by

(a formula practically identical with Cayley’s, though obtained by a very different method). In order to show that the measure of curvature is constant, we make the substitutions

x1= rλ1, x2= rλ2... xn= rλn, where Σλ² = 1.

Hence

ds² = (Radr /a² − r²)² + R²r²dΔ² / (a² − r²).

where

dΔ² = Σdλ².

Also calling ρ the geodesic distance from the origin, we have

Hence

ds² = dρ² + (R sin h (ρ/R))² dΔ².

Putting

z1= ρλ1, z2= ρλ2, ... zn= ρλn,

we obtain

Hence when ρ is small, we have approximately

(1).

Considering a surface element through the origin, we may choose our axes so that, for this element,

z3= z4= ... = zn= 0.

Thus

(2).

Now the area of the triangle whose vertices are (0, 0), (z1, z2), (dz1, dz2) is ½(z1, dz2− z2dz1). Hence the quotient when the terms of the fourth order in (2) are divided by the square of this triangle is4/3R²; hence, returning to general axes, the same is the quotient when the terms of the fourth order in (1) are divided by the square of the triangle whose vertices are (0, 0, ... 0), (z1, z2, z3, ... zn), (dz1, dz2, dz3... dzn). But −¾ of this quotient is defined by Riemann as the measure of curvature.15Hence the measure of curvature is −1/R²,i.e.is constant and negative. The properties of parallels, triangles, &c., are as in theSaggio. It is also shown that the analogues of limit surfaces have zero curvature; and that spheres of radius ρ have constant positive curvature 1/R² sinh² (ρ/R), so that spherical geometry may be regarded as contained in the pseudo-spherical (as Beltrami calls Lobatchewsky’s system).

TheSaggio, as we saw, gives a Euclidean interpretation confined to two dimensions. But a consideration of the auxiliary plane suggests a different interpretation, which may be extended to any number of dimensions. If, insteadTransition to the projective method.of referring to the pseudosphere, we merelydefinedistance and angle, in the Euclidean plane, as those functions of the coordinates which gave us distance and angle on the pseudosphere, we find that the geometry of our plane has become Lobatchewsky’s. All the points of the limiting circle are now at infinity, and points beyond it are imaginary. If we give our circle an imaginary radius the geometry on the plane becomes elliptic. Replacing the circle by a sphere, we obtain an analogous representation for three dimensions. Instead of a circle or sphere we may take any conic or quadric. With this definition, if the fundamental quadric be Σxx= 0, and if Σxx′ be the polar form of Σxx, the distance ρ between x and x′ is given by the projective formula

cos(ρ/k) = Σxx′ / {Σxx·Σx′x′}1/2.

That this formula is projective is rendered evident by observing that e−2iρ/kis the anharmonic ratio of the range consisting of the two points and the intersections of the line joining them with the fundamental quadric. With this we are brought to the third or projective period. The method of this period is due to Cayley; its application to previous non-Euclidean geometry is due to Klein. The projective method contains a generalization of discoveries already made by Laguerre16in 1853 as regards Euclidean geometry. The arbitrariness of this procedure of deriving metrical geometry from the properties of conics is removed by Lie’s theory of congruence. We then arrive at the stage of thought which finds its expression in the modern treatment of the axioms of geometry.

The projective method leads to a discrimination, first made by Klein,17of two varieties of Riemann’s space; Klein calls these elliptic and spherical. They are also called the polar and antipodal forms of elliptic space. The latterThe two kinds of elliptic space.names will here be used. The difference is strictly analogous to that between the diameters and the points of a sphere. In the polar form two straight lines in a plane always intersect in one and only one point; in the antipodal form they intersect always in two points, which are antipodes. According to the definition of geometry adopted in section VII. (Axioms of Geometry), the antipodal form is not to be termed “geometry,” since any pair of coplanar straight lines intersect each other in two points. It may be called a “quasi-geometry.” Similarly in the antipodal form two diameters always determine a plane, but two points on a sphere do not determine a great circle when they are antipodes, and two great circles always intersect in two points. Again, a plane does not form a boundary among lines through a point: we can pass from any one such line to any other without passing through the plane. But a great circle does divide the surface of a sphere. So, in the polar form, a complete straight line does not divide a plane, and a plane does not divide space, and does not, like a Euclidean plane, have two sides.18But, in the antipodal form, a plane is, in these respects, like a Euclidean plane.

It is explained in section VII. in what sense the metrical geometry of the material world can be considered to be determinate and not a matter of arbitrary choice. The scientific question as to the best available evidence concerning the nature of this geometry is one beset with difficulties of a peculiar kind. We are obstructed by the fact that all existing physical science assumes the Euclidean hypothesis. This hypothesis has been involved in all actual measurements of large distances, and in all the laws of astronomy and physics. The principle of simplicity would therefore lead us, in general, where an observation conflicted with one or more of those laws, to ascribe this anomaly, not to the falsity of Euclidean geometry, but to the falsity of the laws in question. This applies especially to astronomy. On the earth our means of measurement are many and direct, and so long as no great accuracy is sought they involve few scientific laws. Thus we acquire, from such direct measurements, a very high degree of probability that the space-constant, if not infinite, is yet large as compared with terrestrial distances. But astronomical distances and triangles can only be measured by means of the received laws of astronomy and optics, all of which have been established by assuming the truth of the Euclidean hypothesis. It therefore remains possible (until a detailed proof of the contrary is forthcoming) that a large but finite space-constant, with different laws of astronomy and optics, would have equally explained the phenomena. We cannot, therefore, accept the measurements of stellar parallaxes, &c., as conclusive evidence that the space-constant is large as compared with stellar distances. For the present, on grounds of simplicity, we may rightly adopt this view; but it must remain possible that, in view of some hitherto undiscovered discrepancy, a slight correction of the sort suggested might prove the simplest alternative. But conversely, a finite parallax for very distant stars, or a negative parallax for any star, could not be accepted as conclusive evidence that our geometry is non-Euclidean, unless it were shown—and this seems scarcely possible—that no modification of astronomy or optics could account for the phenomenon. Thus although we may admit a probability that the space-constant is large in comparison with stellar distances, a conclusive proof or disproof seems scarcely possible.

Finally, it is of interest to note that, though it is theoretically possible to prove, by scientific methods, that our geometry is non-Euclidean, it is wholly impossible to prove by such methods that it is accurately Euclidean. For the unavoidable errors of observation must always leave a slight margin in our measurements. A triangle might be found whose angles were certainly greater, or certainly less, than two right angles; but to prove themexactlyequal to two right angles must always be beyond our powers. If, therefore, any man cherishes a hope of proving the exact truth of Euclid, such a hope must be based, not upon scientific, but upon philosophical considerations.

Bibliography.—The bibliography appended to section VII. should be consulted in this connexion. Also, in addition to the citations already made, the following works may be mentioned.For Lobatchewsky’s writings, cf.Urkunden zur Geschichte der nichteuklidischen Geometrie, i.,Nikolaj Iwanowitsch Lobatschefsky, by F. Engel and P. Stäckel (Leipzig, 1898). For John Bolyai’sAppendix, cf.Absolute Geometrie nach Johann Bolyai, by J. Frischauf (Leipzig, 1872), and also the new edition of his father’s large work,Tentamen..., published by the Mathematical Society of Budapest; the second volume contains the appendix. Cf. also J. Frischauf,Elemente der absoluten Geometrie(Leipzig, 1876); M.L. Gérard,Sur la géométrie non-Euclidienne(thesis for doctorate) (Paris, 1892); de Tilly,Essai sur les principes fondamentales de la géométrie et de la mécanique(Bordeaux, 1879); Sir R.S. Ball, “On the Theory of Content,”Trans. Roy. Irish Acad.vol. xxix. (1889); F. Lindemann, “Mechanik bei projectiver Maasbestimmung,”Math. Annal.vol. vii.; W.K. Clifford, “Preliminary Sketch of Biquaternions,”Proc. of Lond. Math. Soc.(1873), andColl. Works; A. Buchheim, “On the Theory of Screws in Elliptic Space,”Proc. Lond. Math. Soc.vols. xv., xvi., xvii.; H. Cox, “On the Application of Quaternions and Grassmann’s Algebra to different Kinds of Uniform Space,”Trans. Camb. Phil. Soc.(1882); M. Dehn, “Die Legendarischen Sätze über die Winkelsumme im Dreieck,” Math. Ann. vol. 53 (1900), and “Über den Rauminhalt,”Math. Annal.vol. 55 (1902).For expositions of the whole subject, cf. F. Klein,Nicht-Euklidische Geometrie(Göttingen, 1893); R. Bonola,La Geometria non-Euclidea(Bologna, 1906); P. Barbarin,La Géométrie non-Euclidienne(Paris, 1902); W. Killing,Die nicht-Euklidischen Raumformen in analytischer Behandlung(Leipzig, 1885). The last-named work also deals with geometry of more than three dimensions; in this connexion cf. also G. Veronese,Fondamenti di geometria a più dimensioni ed a più speciedi unità rettilinee... (Padua, 1891, German translation, Leipzig, 1894); G. Fontené,L’Hyperespace à (n-1) dimensions(Paris, 1892); and A.N. Whitehead,loc. cit.Cf. also E. Study, “Über nicht-Euklidische und Liniengeometrie,”Jahr. d. Deutsch. Math. Ver.vol. xv. (1906); W. Burnside, “On the Kinematics of non-Euclidean Space,”Proc. Lond. Math. Soc.vol. xxvi. (1894). A bibliography on the subject up to 1878 has been published by G.B. Halsted,Amer. Journ. of Math.vols. i. and ii.; and one up to 1900 by R. Bonola,Index operum ad geometriam absolutam spectantium... (1902, and Leipzig, 1903).

Bibliography.—The bibliography appended to section VII. should be consulted in this connexion. Also, in addition to the citations already made, the following works may be mentioned.

For Lobatchewsky’s writings, cf.Urkunden zur Geschichte der nichteuklidischen Geometrie, i.,Nikolaj Iwanowitsch Lobatschefsky, by F. Engel and P. Stäckel (Leipzig, 1898). For John Bolyai’sAppendix, cf.Absolute Geometrie nach Johann Bolyai, by J. Frischauf (Leipzig, 1872), and also the new edition of his father’s large work,Tentamen..., published by the Mathematical Society of Budapest; the second volume contains the appendix. Cf. also J. Frischauf,Elemente der absoluten Geometrie(Leipzig, 1876); M.L. Gérard,Sur la géométrie non-Euclidienne(thesis for doctorate) (Paris, 1892); de Tilly,Essai sur les principes fondamentales de la géométrie et de la mécanique(Bordeaux, 1879); Sir R.S. Ball, “On the Theory of Content,”Trans. Roy. Irish Acad.vol. xxix. (1889); F. Lindemann, “Mechanik bei projectiver Maasbestimmung,”Math. Annal.vol. vii.; W.K. Clifford, “Preliminary Sketch of Biquaternions,”Proc. of Lond. Math. Soc.(1873), andColl. Works; A. Buchheim, “On the Theory of Screws in Elliptic Space,”Proc. Lond. Math. Soc.vols. xv., xvi., xvii.; H. Cox, “On the Application of Quaternions and Grassmann’s Algebra to different Kinds of Uniform Space,”Trans. Camb. Phil. Soc.(1882); M. Dehn, “Die Legendarischen Sätze über die Winkelsumme im Dreieck,” Math. Ann. vol. 53 (1900), and “Über den Rauminhalt,”Math. Annal.vol. 55 (1902).

For expositions of the whole subject, cf. F. Klein,Nicht-Euklidische Geometrie(Göttingen, 1893); R. Bonola,La Geometria non-Euclidea(Bologna, 1906); P. Barbarin,La Géométrie non-Euclidienne(Paris, 1902); W. Killing,Die nicht-Euklidischen Raumformen in analytischer Behandlung(Leipzig, 1885). The last-named work also deals with geometry of more than three dimensions; in this connexion cf. also G. Veronese,Fondamenti di geometria a più dimensioni ed a più speciedi unità rettilinee... (Padua, 1891, German translation, Leipzig, 1894); G. Fontené,L’Hyperespace à (n-1) dimensions(Paris, 1892); and A.N. Whitehead,loc. cit.Cf. also E. Study, “Über nicht-Euklidische und Liniengeometrie,”Jahr. d. Deutsch. Math. Ver.vol. xv. (1906); W. Burnside, “On the Kinematics of non-Euclidean Space,”Proc. Lond. Math. Soc.vol. xxvi. (1894). A bibliography on the subject up to 1878 has been published by G.B. Halsted,Amer. Journ. of Math.vols. i. and ii.; and one up to 1900 by R. Bonola,Index operum ad geometriam absolutam spectantium... (1902, and Leipzig, 1903).

(B. A. W. R.; A. N. W.)

VII. Axioms of Geometry

Until the discovery of the non-Euclidean geometries (Lobatchewsky, 1826 and 1829; J. Bolyai, 1832; B. Riemann, 1854), geometry was universally considered as being exclusively the science of existent space. (See sectionTheories of space.VI.Non-Euclidean Geometry.) In respect to the science, as thus conceived, two controversies may be noticed. First, there is the controversy respecting the absolute and relational theories of space. According to the absolute theory, which is the traditional view (held explicitly by Newton), space has an existence, in some sense whatever it may be, independent of the bodies which it contains. The bodies occupy space, and it is not intrinsically unmeaning to say that any definite body occupiesthispart of space, and notthatpart of space, without reference to other bodies occupying space. According to the relational theory of space, of which the chief exponent was Leibnitz,19space is nothing but a certain assemblage of the relations between the various particular bodies in space. The idea of space with no bodies in it is absurd. Accordingly there can be no meaning in saying that a body ishereand notthere, apart from a reference to the other bodies in the universe. Thus, on this theory, absolute motion is intrinsically unmeaning. It is admitted on all hands that in practice only relative motion is directly measurable. Newton, however, maintains in thePrincipia(scholium to the 8th definition) that it is indirectly measurable by means of the effects of “centrifugal force” as it occurs in the phenomena of rotation. This irrelevance of absolute motion (if there be such a thing) to science has led to the general adoption of the relational theory by modern men of science. But no decisive argument for either view has at present been elaborated.20Kant’s view of space as being a form of perception at first sight appears to cut across this controversy. But he, saturated as he was with the spirit of the Newtonian physics, must (at least in both editions of theCritique) be classed with the upholders of the absolute theory. The form of perception has a type of existence proper to itself independently of the particular bodies which it contains. For example he writes:21“Space does not represent any quality of objects by themselves, or objects in their relation to one another,i.e.space does not represent any determination which is inherent in the objects themselves, and would remain, even if all subjective conditions of intuition were removed.”

The second controversy is that between the view that the axioms applicable to space are known only from experience, and the view that in some sense these axioms are givena priori. Both these views, thus broadly stated,Axioms.are capable of various subtle modifications, and a discussion of them would merge into a general treatise on epistemology. The cruder forms of thea prioriview have been made quite untenable by the modern mathematical discoveries. Geometers now profess ignorance in many respects of the exact axioms which apply to existent space, and it seems unlikely that a profound study of the question should thus obliteratea prioriintuitions.

Another question irrelevant to this article, but with some relevance to the above controversy, is that of the derivation of our perception of existent space from our various types of sensation. This is a question for psychology.22

Definition of Abstract Geometry.—Existent space is the subject matter of only one of the applications of the modern science of abstract geometry, viewed as a branch of pure mathematics. Geometry has been defined23as “the study of series of two or more dimensions.” It has also been defined24as “the science of cross classification.” These definitions are founded upon the actual practice of mathematicians in respect to their use of the term “Geometry.” Either of them brings out the fact that geometry is not a science with a determinate subject matter. It is concerned with any subject matter to which the formal axioms may apply. Geometry is not peculiar in this respect. All branches of pure mathematics deal merely with types of relations. Thus the fundamental ideas of geometry (e.g.those ofpointsand ofstraight lines) are not ideas of determinate entities, but of any entities for which the axioms are true. And a set of formal geometrical axioms cannot in themselves be true or false, since they are not determinate propositions, in that they do not refer to a determinate subject matter. The axioms are propositional functions.25When a set of axioms is given, we can ask (1) whether they are consistent, (2) whether their “existence theorem” is proved, (3) whether they are independent. Axioms are consistent when the contradictory of any axiom cannot be deduced from the remaining axioms. Their existence theorem is the proof that they are true when the fundamental ideas are considered as denoting some determinate subject matter, so that the axioms are developed into determinate propositions. It follows from the logical law of contradiction that the proof of the existence theorem proves also the consistency of the axioms. This is the only method of proof of consistency. The axioms of a set are independent of each other when no axiom can be deduced from the remaining axioms of the set. The independence of a given axiom is proved by establishing the consistency of the remaining axioms of the set, together with the contradictory of the given axiom. The enumeration of the axioms is simply the enumeration of the hypotheses26(with respect to the undetermined subject matter) of which some at least occur in each of the subsequent propositions.

Any science is called a “geometry” if it investigates the theory of the classification of a set of entities (the points) into classes (the straight lines), such that (1) there is one and only one class which contains any given pair of the entities, and (2) every such class contains more than two members. In the two geometries, important from their relevance to existent space, axioms which secure an order of the points on any line also occur. These geometries will be called “Projective Geometry” and “Descriptive Geometry.” In projective geometry any two straight lines in a plane intersect, and the straight lines are closed series which return into themselves, like the circumference of a circle. In descriptive geometry two straight lines in a plane do not necessarily intersect, and a straight line is an open series without beginning or end. Ordinary Euclidean geometry is a descriptive geometry; it becomes a projective geometry when the so-called “points at infinity” are added.

Projective Geometry.

Projective geometry may be developed from two undefined fundamental ideas, namely, that of a “point” and that of a “straight line.” These undetermined ideas take different specific meanings for the various specific subject matters to which projective geometry can be applied. The number of the axioms is always to some extent arbitrary, being dependent upon the verbal forms of statement which are adopted. They willbe presented27here as twelve in number, eight being “axioms of classification,” and four being “axioms of order.”

Axioms of Classification.—The eight axioms of classification are as follows:

1. Points form a class of entities with at least two members.

2. Any straight line is a class of points containing at least three members.

3. Any two distinct points lie in one and only one straight line.

4. There is at least one straight line which does not contain all the points.

5. If A, B, C are non-collinear points, and A′ is on the straight line BC, and B′ is on the straight line CA, then the straight lines AA′ and BB′ possess a point in common.

Definition.—If A, B, C are any three non-collinear points, theplaneABC is the class of points lying on the straight lines joining A with the various points on the straight line BC.

6. There is at least one plane which does not contain all the points.

7. There exists a plane α, and a point A not incident in α, such that any point lies in some straight line which contains both A and a point in α.

Definition.—Harm. (ABCD) symbolizes the following conjoint statements: (1) that the points A, B, C, D are collinear, and (2) that a quadrilateral can be found with one pair of opposite sides intersecting at A, with the other pair intersecting at C, and with its diagonals passing through B and D respectively. Then B and D are said to be “harmonic conjugates” with respect to A and C.

8. Harm. (ABCD) implies that B and D are distinct points.

In the above axioms 4 secures at least two dimensions, axiom 5 is the fundamental axiom of the plane, axiom 6 secures at least three dimensions, and axiom 7 secures at most three dimensions. From axioms 1-5 it can be proved that any two distinct points in a straight line determine that line, that any three non-collinear points in a plane determine that plane, that the straight line containing any two points in a plane lies wholly in that plane, and that any two straight lines in a plane intersect. From axioms 1-6 Desargue’s well-known theorem on triangles in perspective can be proved.

The enunciation of this theorem is as follows: If ABC and A′B′C′ are two coplanar triangles such that the lines AA′, BB′, CC′ are concurrent, then the three points of intersection of BC and B′C′ of CA and C′A′, and of AB and A′B′ are collinear; and conversely if the three points of intersection are collinear, the three lines are concurrent. The proof which can be applied is the usual projective proof by which a third triangle A″B″C″ is constructed not coplanar with the other two, but in perspective with each of them.It has been proved28that Desargues’s theorem cannot be deduced from axioms 1-5, that is, if the geometry be confined to two dimensions. All the proofs proceed by the method of producing a specification of “points” and “straight lines” which satisfies axioms 1-5, and such that Desargues’s theorem does not hold.It follows from axioms 1-5 that Harm. (ABCD) implies Harm. (ADCB) and Harm. (CBAD), and that, if A, B, C be any three distinct collinear points, there exists at least one point D such that Harm. (ABCD). But it requires Desargues’s theorem, and hence axiom 6, to prove that Harm. (ABCD) and Harm. (ABCD′) imply the identity of D and D′.

It has been proved28that Desargues’s theorem cannot be deduced from axioms 1-5, that is, if the geometry be confined to two dimensions. All the proofs proceed by the method of producing a specification of “points” and “straight lines” which satisfies axioms 1-5, and such that Desargues’s theorem does not hold.

It follows from axioms 1-5 that Harm. (ABCD) implies Harm. (ADCB) and Harm. (CBAD), and that, if A, B, C be any three distinct collinear points, there exists at least one point D such that Harm. (ABCD). But it requires Desargues’s theorem, and hence axiom 6, to prove that Harm. (ABCD) and Harm. (ABCD′) imply the identity of D and D′.

The necessity for axiom 8 has been proved by G. Fano,29who has produced a three dimensional geometry of fifteen points,i.e.a method of cross classification of fifteen entities, in which each straight line contains three points, and each plane contains seven straight lines. In this geometry axiom 8 does not hold. Also from axioms 1-6 and 8 it follows that Harm. (ABCD) implies Harm. (BCDA).

Definitions.—When two plane figures can be derived from one another by a single projection, they are said to be inperspective. When two plane figures can be derived one from the other by a finite series of perspective relations between intermediate figures, they are said to beprojectivelyrelated. Any property of a plane figure which necessarily also belongs to any projectively related figure, is called aprojectiveproperty.The following theorem, known from its importance as “the fundamental theorem of projective geometry,” cannot be proved30from axioms 1-8. The enunciation is: “A projective correspondence between the points on two straight lines is completely determined when the correspondents of three distinct points on one line are determined on the other.” This theorem is equivalent31(assuming axioms 1-8) to another theorem, known as Pappus’s Theorem, namely: “If l and l′ are two distinct coplanar lines, and A, B, C are three distinct points on l, and A′, B′, C′ are three distinct points on l′, then the three points of intersection of AA′ and B′C, of A′B and CC′, of BB′ and C′A, are collinear.” This theorem is obviously Pascal’s well-known theorem respecting a hexagon inscribed in a conic, for the special case when the conic has degenerated into the two lines l and l′. Another theorem also equivalent (assuming axioms 1-8) to the fundamental theorem is the following:32If the three collinear pairs of points, A and A′, B and B′, C and C′, are such that the three pairs of opposite sides of a complete quadrangle pass respectively through them,i.e.one pair through A and A′ respectively, and so on, and if also the three sides of the quadrangle which pass through A, B, and C, are concurrent in one of the corners of the quadrangle, then another quadrangle can be found with the same relation to the three pairs of points, except that its three sides which pass through A, B, and C, are not concurrent.Thus, if we choose to take any one of these three theorems as an axiom, all the theorems of projective geometry which do not require ordinal or metrical ideas for their enunciation can be proved. Also a conic can be defined as the locus of the points found by the usual construction, based upon Pascal’s theorem, for points on the conic through five given points. But it is unnecessary to assume here any one of the suggested axioms; for the fundamental theorem can be deduced from the axioms of order together with axioms 1-8.

The following theorem, known from its importance as “the fundamental theorem of projective geometry,” cannot be proved30from axioms 1-8. The enunciation is: “A projective correspondence between the points on two straight lines is completely determined when the correspondents of three distinct points on one line are determined on the other.” This theorem is equivalent31(assuming axioms 1-8) to another theorem, known as Pappus’s Theorem, namely: “If l and l′ are two distinct coplanar lines, and A, B, C are three distinct points on l, and A′, B′, C′ are three distinct points on l′, then the three points of intersection of AA′ and B′C, of A′B and CC′, of BB′ and C′A, are collinear.” This theorem is obviously Pascal’s well-known theorem respecting a hexagon inscribed in a conic, for the special case when the conic has degenerated into the two lines l and l′. Another theorem also equivalent (assuming axioms 1-8) to the fundamental theorem is the following:32If the three collinear pairs of points, A and A′, B and B′, C and C′, are such that the three pairs of opposite sides of a complete quadrangle pass respectively through them,i.e.one pair through A and A′ respectively, and so on, and if also the three sides of the quadrangle which pass through A, B, and C, are concurrent in one of the corners of the quadrangle, then another quadrangle can be found with the same relation to the three pairs of points, except that its three sides which pass through A, B, and C, are not concurrent.

Thus, if we choose to take any one of these three theorems as an axiom, all the theorems of projective geometry which do not require ordinal or metrical ideas for their enunciation can be proved. Also a conic can be defined as the locus of the points found by the usual construction, based upon Pascal’s theorem, for points on the conic through five given points. But it is unnecessary to assume here any one of the suggested axioms; for the fundamental theorem can be deduced from the axioms of order together with axioms 1-8.

Axioms of Order.—It is possible to define (cf. Pieri,loc. cit.) the property upon which the order of points on a straight line depends. But to secure that this property does in fact range the points in a serial order, some axioms are required. A straight line is to be a closed series; thus, when the points are in order, it requires two points on the line to divide it into two distinct complementary segments, which do not overlap, and together form the whole line. Accordingly the problem of the definition of order reduces itself to the definition of these two segments formed by any two points on the line; and the axioms are stated relatively to these segments.

Definition.—If A, B, C are three collinear points, the points on thesegmentABC are defined to be those points such as X, for which there exist two points Y and Y′ with the property that Harm. (AYCY′) and Harm. (BYXY′) both hold. Thesupplementary segmentABC is defined to be the rest of the points on the line. This definition is elucidated by noticing that with our ordinary geometrical ideas, if B and X are any two points between A and C, then the two pairs of points, A and C, B and X, define an involution with real double points, namely, the Y and Y′ of the above definition. The property of belonging to a segment ABC is projective, since the harmonic relation is projective.

The first three axioms of order (cf. Pieri,loc. cit.) are:

9. If A, B, C are three distinct collinear points, the supplementary segment ABC is contained within the segment BCA.

10. If A, B, C are three distinct collinear points, the common part of the segments BCA and CAB is contained in the supplementary segment ABC.

11. If A, B, C are three distinct collinear points, and D lies In the segment ABC, then the segment ADC is contained within the segment ABC.

From these axioms all the usual properties of a closed order follow. It will be noticed that, if A, B, C are any three collinear points, C is necessarily traversed in passing from A to B by one route along the line, and is not traversed in passing from A to B along the other route. Thus there is no meaning, as referred to closed straight lines, in the simple statement that C lies between A and B. But there may be a relation of separation between two pairs of collinear points, such as A and C, and B and D. The couple B and D is said to separate A and C, ifthe four points are collinear and D lies in the segment complementary to the segment ABC. The property of the separation of pairs of points by pairs of points is projective. Also it can be proved that Harm. (ABCD) implies that B and D separate A and C.

Definitions.—A series of entities arranged in a serial order, open or closed, is said to becompact, if the series contains no immediately consecutive entities, so that in traversing the series from any one entity to any other entity it is necessary to pass through entities distinct from either. It was the merit of R. Dedekind and of G. Cantor explicitly to formulate another fundamental property of series. The Dedekind property33as applied to an open series can be defined thus: An open series possesses the Dedekind property, if, however, it be divided into two mutually exclusive classes u and v, which (1) contain between them the whole series, and (2) are such that every member of u precedes in the serial order every member of v, there is always a member of the series, belonging to one of the two, u or v, which precedes every member of v (other than itself if it belong to v), and also succeeds every member of u (other than itself if it belong to u). Accordingly in an open series with the Dedekind property there is always a member of the series marking the junction of two classes such as u and v. An open series iscontinuousif it is compact and possesses the Dedekind property. A closed series can always be transformed into an open series by taking any arbitrary member as the first term and by taking one of the two ways round as the ascending order of the series. Thus the definitions of compactness and of the Dedekind property can be at once transferred to a closed series.

12. The last axiom of order is that there exists at least one straight line for which the point order possesses the Dedekind property.

It follows from axioms 1-12 by projection that the Dedekind property is true for all lines. Again theharmonic systemABC, where A, B, C are collinear points, is defined34thus: take the harmonic conjugates A′, B′, C′ of each point with respect to the other two, again take the harmonic conjugates of each of the six points A, B, C, A′, B′, C′ with respect to each pair of the remaining five, and proceed in this way by an unending series of steps. The set of points thus obtained is called the harmonic system ABC. It can be proved that a harmonic system is compact, and that every segment of the line containing it possesses members of it. Furthermore, it is easy to prove that the fundamental theorem holds for harmonic systems, in the sense that, if A, B, C are three points on a line l, and A′, B′, C′ are three points on a line l′, and if by any two distinct series of projections A, B, C are projected into A′, B′, C′, then any point of the harmonic system ABC corresponds to the same point of the harmonic system A′B′C′ according to both the projective relations which are thus established between l and l′. It now follows immediately that the fundamental theorem must hold for all the points on the lines l and l′, since (as has been pointed out) harmonic systems are “everywhere dense” on their containing lines. Thus the fundamental theorem follows from the axioms of order.

A system of numerical coordinates can now be introduced, possessing the property that linear equations represent planes and straight lines. The outline of the argument by which this remarkable problem (in that “distance” is as yet undefined) is solved, will now be given. It is first proved that the points on any line can in a certain way be definitely associated with all the positive and negative real numbers, so as to form with them a one-one correspondence. The arbitrary elements in the establishment of this relation are the points on the line associated with 0, 1 and ∞.

This association35is most easily effected by considering a class of projective relations of the line with itself, called by F. Schur (loc. cit.)prospectivities.

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