Let l (fig. 69) be the given line, m and n any two lines intersecting at U on l, S and S′ two points on n. Then a projective relation between l and itself is formed by projecting l from S on to m, and then by projecting m from S′ back on to l. All such projective relations, however m, n, S and S′ be varied, are called “prospectivities,” and U is the double point of the prospectivity. If a point O on l is related to A by a prospectivity, then all prospectivities, which (1) have the same double point U, and (2) relate O to A, give the same correspondent (Q, in figure) to any point P on the line l; in fact they are all the same prospectivity, however m, n, S, and S′ may have been varied subject to these conditions. Such a prospectivity will be denoted by (OAU²).The sum of two prospectivities, written (OAU²) + (OBU²), is defined to be that transformation of the line l into itself which is obtained by first applying the prospectivity (OAU²) and then applying the prospectivity (OBU²). Such a transformation, when the two summands have the same double point, is itself a prospectivity with that double point.With this definition of addition it can be proved that prospectivities with the same double point satisfy all the axioms of magnitude. Accordingly they can be associated in a one-one correspondence with the positive and negative real numbers. Let E (fig. 70) be any point on l, distinct from O and U. Then the prospectivity (OEU²) is associated with unity, the prospectivity (OOU²) is associated with zero, and (OUU²) with ∞. The prospectivities of the type (OPU²), where P is any point on the segment OEU, correspond to the positive numbers; also if P′ is the harmonic conjugate of P with respect to O and U, the prospectivity (OP′U²) is associated with the corresponding negative number. (The subjoined figure explains this relation of the positive and negative prospectivities.) Then any point P on l is associated with the same number as is the prospectivity (OPU²).It can be proved that the order of the numbers in algebraic order of magnitude agrees with the order on the line of the associated points. Let the numbers, assigned according to the preceding specification, be said to be associated with the points according to the “numeration-system (OEU).” The introduction of a coordinate system for a plane is now managed as follows: Take any triangle OUV in the plane, and on the lines OU and OV establish the numeration systems (OE1U) and (OE2V), where E1and E2are arbitrarily chosen. Then (cf. fig. 71) if M and N are associated with the numbers x and y according to these systems, the coordinates of P are x and y. It then follows that the equation of a straight line is of the form ax + by + c = 0. Both coordinates of any point on the line UV are infinite. This can be avoided by introducing homogeneous coordinates X, Y, Z, where x = X/Z, and y = Y/Z, and Z = 0 is the equation of UV.The procedure for three dimensions is similar. Let OUVW (fig. 72) be any tetrahedron, and associate points on OU, OV, OW with numbers according to the numeration systems (OE1U), (OE2V), and (OE3W). Let the planes VWP, WUP, UVP cut OU, OV, OW in L, M, N respectively; and let x, y, z be the numbers associated with L, M, N respectively. Then P is the point (x, y, z). Also homogeneous coordinates can be introduced as before, thus avoiding the infinities on the plane UVW.The cross ratio of a range of four collinear points can now be defined as a number characteristic of that range. Let the coordinates of any point Prof the range P1P2P3P4beλra + μr+ a′,λrb + μrb′,λrc + μrc′, (r = 1, 2, 3, 4)λr+ μrλr+ μrλr+ μrand let (λrμs) be written for λrμs-λsμr. Then the cross ratio {P1P2P3P4} is defined to be the number (λ1μ2)(λ3μ4) / (λ1μ4)(λ3μ2). The equality of the cross ratios of the ranges (P1P2P3P4) and (Q1Q2Q3Q4) is proved to be the necessary and sufficient condition for their mutual projectivity. The cross ratios of all harmonic ranges are then easily seen to be all equal to -1, by comparing with the range (OE1UE′1) on the axis of x.Thus all the ordinary propositions of geometry in which distance and angular measure do not enter otherwise than in cross ratios can now be enunciated and proved. Accordingly the greater part of the analytical theory of conics and quadrics belongs to geometryat this stage The theory of distance will be considered after the principles of descriptive geometry have been developed.
Let l (fig. 69) be the given line, m and n any two lines intersecting at U on l, S and S′ two points on n. Then a projective relation between l and itself is formed by projecting l from S on to m, and then by projecting m from S′ back on to l. All such projective relations, however m, n, S and S′ be varied, are called “prospectivities,” and U is the double point of the prospectivity. If a point O on l is related to A by a prospectivity, then all prospectivities, which (1) have the same double point U, and (2) relate O to A, give the same correspondent (Q, in figure) to any point P on the line l; in fact they are all the same prospectivity, however m, n, S, and S′ may have been varied subject to these conditions. Such a prospectivity will be denoted by (OAU²).
The sum of two prospectivities, written (OAU²) + (OBU²), is defined to be that transformation of the line l into itself which is obtained by first applying the prospectivity (OAU²) and then applying the prospectivity (OBU²). Such a transformation, when the two summands have the same double point, is itself a prospectivity with that double point.
With this definition of addition it can be proved that prospectivities with the same double point satisfy all the axioms of magnitude. Accordingly they can be associated in a one-one correspondence with the positive and negative real numbers. Let E (fig. 70) be any point on l, distinct from O and U. Then the prospectivity (OEU²) is associated with unity, the prospectivity (OOU²) is associated with zero, and (OUU²) with ∞. The prospectivities of the type (OPU²), where P is any point on the segment OEU, correspond to the positive numbers; also if P′ is the harmonic conjugate of P with respect to O and U, the prospectivity (OP′U²) is associated with the corresponding negative number. (The subjoined figure explains this relation of the positive and negative prospectivities.) Then any point P on l is associated with the same number as is the prospectivity (OPU²).
It can be proved that the order of the numbers in algebraic order of magnitude agrees with the order on the line of the associated points. Let the numbers, assigned according to the preceding specification, be said to be associated with the points according to the “numeration-system (OEU).” The introduction of a coordinate system for a plane is now managed as follows: Take any triangle OUV in the plane, and on the lines OU and OV establish the numeration systems (OE1U) and (OE2V), where E1and E2are arbitrarily chosen. Then (cf. fig. 71) if M and N are associated with the numbers x and y according to these systems, the coordinates of P are x and y. It then follows that the equation of a straight line is of the form ax + by + c = 0. Both coordinates of any point on the line UV are infinite. This can be avoided by introducing homogeneous coordinates X, Y, Z, where x = X/Z, and y = Y/Z, and Z = 0 is the equation of UV.
The procedure for three dimensions is similar. Let OUVW (fig. 72) be any tetrahedron, and associate points on OU, OV, OW with numbers according to the numeration systems (OE1U), (OE2V), and (OE3W). Let the planes VWP, WUP, UVP cut OU, OV, OW in L, M, N respectively; and let x, y, z be the numbers associated with L, M, N respectively. Then P is the point (x, y, z). Also homogeneous coordinates can be introduced as before, thus avoiding the infinities on the plane UVW.
The cross ratio of a range of four collinear points can now be defined as a number characteristic of that range. Let the coordinates of any point Prof the range P1P2P3P4be
and let (λrμs) be written for λrμs-λsμr. Then the cross ratio {P1P2P3P4} is defined to be the number (λ1μ2)(λ3μ4) / (λ1μ4)(λ3μ2). The equality of the cross ratios of the ranges (P1P2P3P4) and (Q1Q2Q3Q4) is proved to be the necessary and sufficient condition for their mutual projectivity. The cross ratios of all harmonic ranges are then easily seen to be all equal to -1, by comparing with the range (OE1UE′1) on the axis of x.
Thus all the ordinary propositions of geometry in which distance and angular measure do not enter otherwise than in cross ratios can now be enunciated and proved. Accordingly the greater part of the analytical theory of conics and quadrics belongs to geometryat this stage The theory of distance will be considered after the principles of descriptive geometry have been developed.
Descriptive Geometry.
Descriptive geometry is essentially the science of multiple order for open series. The first satisfactory system of axioms was given by M. Pasch.36An improved version is due to G. Peano.37Both these authors treat the idea of the class of points constituting the segment lyingbetweentwo points as an undefined fundamental idea. Thus in fact there are in this system two fundamental ideas, namely, of points and of segments. It is then easy enough to define the prolongations of the segments, so as to form the complete straight lines. D. Hilbert’s38formulation of the axioms is in this respect practically based on the same fundamental ideas. His work is justly famous for some of the mathematical investigations contained in it, but his exposition of the axioms is distinctly inferior to that of Peano. Descriptive geometry can also be considered39as the science of a class of relations, each relation being a two-termed serial relation, as considered in the logic of relations, ranging the points between which it holds into a linear open order. Thus the relations are the straight lines, and the terms between which they hold are the points. But a combination of these two points of view yields40the simplest statement of all. Descriptive geometry is then conceived as the investigation of an undefined fundamental relation between three terms (points); and when the relation holds between three points A, B, C, the points are said to be “in the [linear] order ABC.”
O. Veblen’s axioms and definitions, slightly modified, are as follows:—
1. If the points A, B, C are in the order ABC, they are in the order CBA.
2. If the points A, B, C are in the order ABC, they are not in the order BCA.
3. If the points A, B, C are in the order ABC, A is distinct from C.
4. If A and B are any two distinct points, there exists a point C such that A, B, C are in the order ABC.
Definition.—ThelineAB (A ≠ B) consists of A and B, and of all points X in one of the possible orders, ABX, AXB, XAB. The points X in the order AXB constitute thesegmentAB.
Definition.—ThelineAB (A ≠ B) consists of A and B, and of all points X in one of the possible orders, ABX, AXB, XAB. The points X in the order AXB constitute thesegmentAB.
5. If points C and D (C ≠ D) lie on the line AB, then A lies on the line CD.
6. There exist three distinct points A, B, C not in any of the orders ABC, BCA, CAB.
7. If three distinct points A, B, C (fig. 73) do not lie on the same line, and D and E are two distinct points in the orders BCD and CEA, then a point F exists in the order AFB, and such that D, E, F are collinear.
Definition.—If A, B, C are three non-collinear points, theplaneABC is the class of points which lie on any one of the lines joining any two of the points belonging to theboundaryof the triangle ABC, the boundary being formed by the segments BC, CA and AB. Theinteriorof the triangle ABC is formed by the points in segments such as PQ, where P and Q are points respectively on two of the segments BC, CA, AB.
Definition.—If A, B, C are three non-collinear points, theplaneABC is the class of points which lie on any one of the lines joining any two of the points belonging to theboundaryof the triangle ABC, the boundary being formed by the segments BC, CA and AB. Theinteriorof the triangle ABC is formed by the points in segments such as PQ, where P and Q are points respectively on two of the segments BC, CA, AB.
8. There exists a plane ABC, which does not contain all the points.
Definition.—If A, B, C, D are four non-coplanar points, the space ABCD is the class of points which lie on any of the lines containing two points on the surface of the tetrahedron ABCD, thesurfacebeing formed by the interiors of the triangles ABC, BCD, DCA, DAB.
Definition.—If A, B, C, D are four non-coplanar points, the space ABCD is the class of points which lie on any of the lines containing two points on the surface of the tetrahedron ABCD, thesurfacebeing formed by the interiors of the triangles ABC, BCD, DCA, DAB.
9. There exists a space ABCD which contains all the points.
10. The Dedekind property holds for the order of the points on any straight line.
It follows from axioms 1-9 that the points on any straight line are arranged in an open serial order. Also all the ordinary theorems respecting a point dividing a straight line into two parts, a straight line dividing a plane into two parts, and a plane dividing space into two parts, follow.
Again, in any plane α consider a line l and a point A (fig. 74).Fig. 74.Let any point B divide l into two half-lines l1and l2. Then it can be proved that the set of half-lines, emanating from A and intersecting l1(such as m), are bounded by two half-lines, of which ABC is one. Let r be the other. Then it can be proved that r does not intersect l1. Similarly for the half-line, such as n, intersecting l2. Let s be its bounding half-line. Then two cases are possible. (1) The half-lines r and s are collinear, and together form one complete line. In this case, there is one and only one line (viz. r + s) through A and lying in α which does not intersect l. This is the Euclidean case, and the assumption that this case holds is theEuclidean parallel axiom. But (2) the half-lines r and s may not be collinear. In this case there will be an infinite number of lines, such as k for instance, containing A and lying in α, which do not intersect l. Then the lines through A in α are divided into two classes by reference to l, namely, thesecantlines which intersect l, and thenon-secantlines which do not intersect l. The two boundary non-secant lines, of which r and s are respectively halves, may be called the two parallels to l through A.The perception of the possibility of case 2 constituted the starting-point from which Lobatchewsky constructed the first explicit coherent theory of non-Euclidean geometry, and thus created a revolution in the philosophy of the subject. For many centuries the speculations of mathematicians on the foundations of geometry were almost confined to hopeless attempts to prove the “parallel axiom” without the introduction of some equivalent axiom.41
Again, in any plane α consider a line l and a point A (fig. 74).
Let any point B divide l into two half-lines l1and l2. Then it can be proved that the set of half-lines, emanating from A and intersecting l1(such as m), are bounded by two half-lines, of which ABC is one. Let r be the other. Then it can be proved that r does not intersect l1. Similarly for the half-line, such as n, intersecting l2. Let s be its bounding half-line. Then two cases are possible. (1) The half-lines r and s are collinear, and together form one complete line. In this case, there is one and only one line (viz. r + s) through A and lying in α which does not intersect l. This is the Euclidean case, and the assumption that this case holds is theEuclidean parallel axiom. But (2) the half-lines r and s may not be collinear. In this case there will be an infinite number of lines, such as k for instance, containing A and lying in α, which do not intersect l. Then the lines through A in α are divided into two classes by reference to l, namely, thesecantlines which intersect l, and thenon-secantlines which do not intersect l. The two boundary non-secant lines, of which r and s are respectively halves, may be called the two parallels to l through A.
The perception of the possibility of case 2 constituted the starting-point from which Lobatchewsky constructed the first explicit coherent theory of non-Euclidean geometry, and thus created a revolution in the philosophy of the subject. For many centuries the speculations of mathematicians on the foundations of geometry were almost confined to hopeless attempts to prove the “parallel axiom” without the introduction of some equivalent axiom.41
Associated Projective and Descriptive Spaces.—A region of a projective space, such that one, and only one, of the two supplementary segments between any pair of points within it lies entirely within it, satisfies the above axioms (1-10) of descriptive geometry, where the points of the region are the descriptive points, and the portions of straight lines within the region are the descriptive lines. If the excluded part of the original projective space is a single plane, the Euclidean parallel axiom also holds, otherwise it does not hold for the descriptive space of the limited region. Again, conversely, starting from an original descriptive space an associated projective space can be constructed by means of the concept ofideal points.42These are also calledprojective points, where it is understood that the simple points are the points of the original descriptive space. Anideal pointis the class of straight lines which is composed of two coplanar lines a and b, together with the lines of intersection of all pairs of intersecting planes which respectively contain a and b, together with the lines of intersection with the plane ab of all planes containing any one of the lines (other than a or b) already specified as belonging to the ideal point. It is evident that, if the two original lines a and b intersect, the corresponding ideal point is nothing else than the whole class of lines which are concurrent at the point ab. But the essence of the definition is that an ideal point has an existence when the lines a and b do not intersect, so long as they are coplanar. An ideal point is termedproper, if the lines composing it intersect; otherwise it isimproper.
A theorem essential to the whole theory is the following: if any two of the three lines a, b, c are coplanar, but the three lines are not all coplanar, and similarly for the lines a, b, d, then c and d are coplanar. It follows that any two lines belonging to an ideal point can be used as the pair of guiding lines in the definition. An ideal point is said to becoherentwith a plane, if any of the lines composing it lie in the plane. Anideal lineis the class of ideal points each of which is coherent with two given planes.If the planes intersect, the ideal line is termedproper, otherwise it isimproper. It can be proved that any two planes, with which any two of the ideal points are both coherent, will serve as the guiding planes used in the definition. The ideal planes are defined as in projective geometry, and all the other definitions (for segments, order, &c.) of projective geometry are applied to the ideal elements. If an ideal plane contains some proper ideal points, it is calledproper, otherwise it isimproper. Every ideal plane contains some improper ideal points.
It can now be proved that all the axioms of projective geometry hold of the ideal elements as thus obtained; and also that the order of the ideal points as obtained by the projective method agrees with the order of the proper ideal points as obtained from that of the associated points of the descriptive geometry. Thus a projective space has been constructed out of the ideal elements, and the proper ideal elements correspond element by element with the associated descriptive elements. Thus the proper ideal elements form a region in the projective space within which the descriptive axioms hold. Accordingly, by substituting ideal elements, a descriptive space can always be considered as a region within a projective space. This is the justification for the ordinary use of the “points at infinity” in the ordinary Euclidean geometry; the reasoning has been transferred from the original descriptive space to the associated projective space of ideal elements; and with the Euclidean parallel axiom the improper ideal elements reduce to the ideal points on a single improper ideal plane, namely, the plane at infinity.43
Congruence and Measurement.—The property of physical space which is expressed by the term “measurability” has now to be considered. This property has often been considered as essential to the very idea of space. For example, Kant writes,44“Space is represented as an infinite givenquantity.” This quantitative aspect of space arises from the measurability of distances, of angles, of surfaces and of volumes. These four types of quantity depend upon the two first among them as fundamental. The measurability of space is essentially connected with the idea ofcongruence, of which the simplest examples are to be found in the proofs of equality by the method of superposition, as used in elementary plane geometry. The mere concepts of “part” and of “whole” must of necessity be inadequate as the foundation of measurement, since we require the comparison as to quantity of regions of space which have no portions in common. The idea of congruence, as exemplified by the method of superposition in geometrical reasoning, appears to be founded upon that of the “rigid body,” which moves from one position to another with its internal spatial relations unchanged. But unless there is a previous concept of the metrical relations between the parts of the body, there can be no basis from which to deduce that they are unchanged.
It would therefore appear as if the idea of the congruence, or metrical equality, of two portions of space (as empirically suggested by the motion of rigid bodies) must be considered as a fundamental idea incapable of definition in terms of those geometrical concepts which have already been enumerated. This was in effect the point of view of Pasch.45It has, however, been proved by Sophus Lie46that congruence is capable of definition without recourse to a new fundamental idea. This he does by means of his theory of finite continuous groups (seeGroups, Theory of), of which the definition is possible in terms of our established geometrical ideas, remembering that coordinates have already been introduced. The displacement of a rigid body is simply a mode of defining to the senses a one-one transformation of all space into itself. For at any point of space a particle may be conceived to be placed, and to be rigidly connected with the rigid body; and thus there is a definite correspondence of any point of space with the new point occupied by the associated particle after displacement. Again two successive displacements of a rigid body from position A to position B, and from position B to position C, are the same in effect as one displacement from A to C. But this is the characteristic “group” property. Thus the transformations of space into itself defined by displacements of rigid bodies form a group.
Call this group of transformations a congruence-group. Now according to Lie a congruence-group is defined by the following characteristics:—
1. A congruence-group is a finite continuous group of one-one transformations, containing the identical transformation.
2. It is a sub-group of the general projective group,i.e.of the group of which any transformation converts planes into planes, and straight lines into straight lines.
3. An infinitesimal transformation can always be found satisfying the condition that, at least throughout a certain enclosed region, any definite line and any definite point on the line are latent,i.e.correspond to themselves.
4. No infinitesimal transformation of the group exists, such that, at least in the region for which (3) holds, a straight line, a point on it, and a plane through it, shall all be latent.
The property enunciated by conditions (3) and (4), taken together, is named by Lie “Free mobility in the infinitesimal.” Lie proves the following theorems for a projective space:—
1. If the above four conditions are only satisfied by a group throughout part of projective space, this part either (α) must be the region enclosed by a real closed quadric, or (β) must be the whole of the projective space with the exception of a single plane. In case (α) the corresponding congruence group is the continuous group for which the enclosing quadric is latent; and in case (β) an imaginary conic (with a real equation) lying in the latent plane is also latent, and the congruence group is the continuous group for which the plane and conic are latent.2. If the above four conditions are satisfied by a group throughout the whole of projective space, the congruence group is the continuous group for which some imaginary quadric (with a real equation) is latent.By a proper choice of non-homogeneous co-ordinates the equation of any quadrics of the types considered, either in theorem 1 (α), or in theorem 2, can be written in the form 1 + c(x² + y² + z²) = 0, where c is negative for a real closed quadric, and positive for an imaginary quadric. Then the general infinitesimal transformation is defined by the three equations:dx/dt = u − ω3y + ω2z + cx (ux + vy + wz),(A)dy/dt = v − ω1z + ω3x + cy (ux + vy + wz),dz/dt = w − ω2x + ω1y + cz (ux + vy + wz).In the ease considered in theorem 1 (β), with the proper choice of co-ordinates the three equations defining the general infinitesimal transformation are:dx/dt = u − ω3y + ω2z,(B)dy/dt = v − ω1z + ω3x,dz/dt = w − ω2x + ω1y.In this case the latent plane is the plane for which at least one of x, y, z are infinite, that is, the plane 0.x + 0.y + 0.z + a = 0; and the latent conic is the conic in which the cone x² + y² + z² = 0 intersects the latent plane.
1. If the above four conditions are only satisfied by a group throughout part of projective space, this part either (α) must be the region enclosed by a real closed quadric, or (β) must be the whole of the projective space with the exception of a single plane. In case (α) the corresponding congruence group is the continuous group for which the enclosing quadric is latent; and in case (β) an imaginary conic (with a real equation) lying in the latent plane is also latent, and the congruence group is the continuous group for which the plane and conic are latent.
2. If the above four conditions are satisfied by a group throughout the whole of projective space, the congruence group is the continuous group for which some imaginary quadric (with a real equation) is latent.
By a proper choice of non-homogeneous co-ordinates the equation of any quadrics of the types considered, either in theorem 1 (α), or in theorem 2, can be written in the form 1 + c(x² + y² + z²) = 0, where c is negative for a real closed quadric, and positive for an imaginary quadric. Then the general infinitesimal transformation is defined by the three equations:
In the ease considered in theorem 1 (β), with the proper choice of co-ordinates the three equations defining the general infinitesimal transformation are:
In this case the latent plane is the plane for which at least one of x, y, z are infinite, that is, the plane 0.x + 0.y + 0.z + a = 0; and the latent conic is the conic in which the cone x² + y² + z² = 0 intersects the latent plane.
It follows from theorems 1 and 2 that there is not one unique congruence-group, but an indefinite number of them. There is one congruence-group corresponding to each closed real quadric, one to each imaginary quadric with a real equation, and one to each imaginary conic in a real plane and with a real equation. The quadric thus associated with each congruence-group is called theabsolutefor that group, and in the degenerate case of 1 (β) the absolute is the latent plane together with the latent imaginary conic. If the absolute is real, the congruence-group ishyperbolic; if imaginary, it iselliptic; if the absolute is a plane and imaginary conic, the group is parabolic. Metrical geometry is simply the theory of the properties of some particular congruence-group selected for study.
The definition of distance is connected with the corresponding congruence-group by two considerations in respect to a range of five points (A1, A2, P1, P2, P3), of which A1and A2are on the absolute.Let {A1P1A2P2} stand for the cross ratio (as defined above) of the range (A1P1A2P2), with a similar notation for the other ranges. Then(1)log {A1P1A2P2} + log {A1P2A2P3} = log {A1P1A2P3},and(2), if the points A1, A2, P1, P2are transformed into A′1, A′2, P′1, P′2by any transformation of the congruence-group, (α) {A1P1A2P2= {A′1P′1A′2P′2}, since the transformation is projective, and (β) A′1, A′2are on the absolute since A1and A2are on it. Thus if we definethe distance P1P2to be ½k log {A1P1A2P2}, where A1and A2are the points in which the line P1P2cuts the absolute, and k is some constant, the two characteristic properties of distance, namely, (1) the addition of consecutive lengths on a straight line, and (2) the invariability of distances during a transformation of the congruence-group, are satisfied. This is the well-known Cayley-Klein projective definition47of distance, which was elaborated in view of the addition property alone, previously to Lie’s discovery of the theory of congruence-groups. For a hyperbolic group when P1and P2are in the region enclosed by the absolute, log {A1P1A2P2} is real, and therefore k must be real. For an elliptic group A1and A2are conjugate imaginaries, and log {A1P1A2P2} is a pure imaginary, and k is chosen to be κ/ι, where κ is real and ι = √ −.Similarly the angle between two planes, p1and p2, is defined to be (1/2ι) log (t1p1t2p2), where t1and t2are tangent planes to the absolute through the line p1p2. The planes t1and t2are imaginary for an elliptic group, and also for an hyperbolic group when the planes p1and p2intersect at points within the region enclosed by the absolute. The development of the consequences of these metrical definitions is thesubjectof non-Euclidean geometry.The definitions for the parabolic case can be arrived at as limits of those obtained in either of the other two cases by making k ultimately to vanish. It is also obvious that, if P1and P2be the points (x1, y1, z1) and (x2, y2, z2), it follows from equations (B) above that {(x1− x2)² + (y1− y2)² + (z1− z2)²}1/2is unaltered by a congruence transformation and also satisfies the addition property for collinear distances. Also the previous definition of an angle can be adapted to this case, by making t1and t2to be the tangent planes through the line p1p2to the imaginary conic. Similarly if p1and p2are intersecting lines, the same definition of an angle holds, where t1and t2are now the lines from the point p1p2to the two points where the plane p1p2cuts the imaginary conic. These points are in fact the “circular points at infinity” on the plane. The development of the consequences of these definitions for the parabolic case gives the ordinary Euclidean metrical geometry.
The definition of distance is connected with the corresponding congruence-group by two considerations in respect to a range of five points (A1, A2, P1, P2, P3), of which A1and A2are on the absolute.
Let {A1P1A2P2} stand for the cross ratio (as defined above) of the range (A1P1A2P2), with a similar notation for the other ranges. Then
(1)
log {A1P1A2P2} + log {A1P2A2P3} = log {A1P1A2P3},
and
(2), if the points A1, A2, P1, P2are transformed into A′1, A′2, P′1, P′2by any transformation of the congruence-group, (α) {A1P1A2P2= {A′1P′1A′2P′2}, since the transformation is projective, and (β) A′1, A′2are on the absolute since A1and A2are on it. Thus if we definethe distance P1P2to be ½k log {A1P1A2P2}, where A1and A2are the points in which the line P1P2cuts the absolute, and k is some constant, the two characteristic properties of distance, namely, (1) the addition of consecutive lengths on a straight line, and (2) the invariability of distances during a transformation of the congruence-group, are satisfied. This is the well-known Cayley-Klein projective definition47of distance, which was elaborated in view of the addition property alone, previously to Lie’s discovery of the theory of congruence-groups. For a hyperbolic group when P1and P2are in the region enclosed by the absolute, log {A1P1A2P2} is real, and therefore k must be real. For an elliptic group A1and A2are conjugate imaginaries, and log {A1P1A2P2} is a pure imaginary, and k is chosen to be κ/ι, where κ is real and ι = √ −.
Similarly the angle between two planes, p1and p2, is defined to be (1/2ι) log (t1p1t2p2), where t1and t2are tangent planes to the absolute through the line p1p2. The planes t1and t2are imaginary for an elliptic group, and also for an hyperbolic group when the planes p1and p2intersect at points within the region enclosed by the absolute. The development of the consequences of these metrical definitions is thesubjectof non-Euclidean geometry.
The definitions for the parabolic case can be arrived at as limits of those obtained in either of the other two cases by making k ultimately to vanish. It is also obvious that, if P1and P2be the points (x1, y1, z1) and (x2, y2, z2), it follows from equations (B) above that {(x1− x2)² + (y1− y2)² + (z1− z2)²}1/2is unaltered by a congruence transformation and also satisfies the addition property for collinear distances. Also the previous definition of an angle can be adapted to this case, by making t1and t2to be the tangent planes through the line p1p2to the imaginary conic. Similarly if p1and p2are intersecting lines, the same definition of an angle holds, where t1and t2are now the lines from the point p1p2to the two points where the plane p1p2cuts the imaginary conic. These points are in fact the “circular points at infinity” on the plane. The development of the consequences of these definitions for the parabolic case gives the ordinary Euclidean metrical geometry.
Thus the only metrical geometry for the whole of projective space is of the elliptic type. But the actual measure-relations (though not their general properties) differ according to the elliptic congruence-group selected for study. In a descriptive space a congruence-group should possess the four characteristics of such a group throughout the whole of the space. Then form the associated ideal projective space. The associated congruence-group for this ideal space must satisfy the four conditions throughout the region of the proper ideal points. Thus the boundary of this region is the absolute. Accordingly there can be no metrical geometry for the whole of a descriptive space unless its boundary (in the associated ideal space) is a closed quadric or a plane. If the boundary is a closed quadric, there is one possible congruence-group of the hyperbolic type. If the boundary is a plane (the plane at infinity), the possible congruence-groups are parabolic; and there is a congruence-group corresponding to each imaginary conic in this plane, together with a Euclidean metrical geometry corresponding to each such group. Owing to these alternative possibilities, it would appear to be more accurate to say that systems of quantities can be found in a space, rather than that space is a quantity.
Lie has also deduced48the same results with respect to congruence-groups from another set of defining properties, which explicitly assume the existence of a quantitative relation (the distance) between any two points, which is invariant for any transformation of the congruence-group.49
The above results, in respect to congruence and metrical geometry, considered in relation to existent space, have led to the doctrine50that it is intrinsically unmeaning to ask which system of metrical geometry is true of the physical world. Any one of these systems can be applied, and in an indefinite number of ways. The only question before us is one of convenience in respect to simplicity of statement of the physical laws. This point of view seems to neglect the consideration that science is to be relevant to the definite perceiving minds of men; and that (neglecting the ambiguity introduced by the invariable slight inexactness of observation which is not relevant to this special doctrine) we have, in fact, presented to our senses a definite set of transformations forming a congruence-group, resulting in a set of measure relations which are in no respect arbitrary. Accordingly our scientific laws are to be stated relevantly to that particular congruence-group. Thus the investigation of the type (elliptic, hyperbolic or parabolic) of this special congruence-group is a perfectly definite problem, to be decided by experiment. The consideration of experiments adapted to this object requires some development of non-Euclidean geometry (see section VI.,Non-Euclidean Geometry). But if the doctrine means that, assuming some sort of objective reality for the material universe, beings can be imagined, to whomeitherall congruence-groups are equally important,orsome other congruence-group is specially important, the doctrine appears to be an immediate deduction from the mathematical facts. Assuming a definite congruence-group, the investigation of surfaces (or three-dimensional loci in space of four dimensions) with geodesic geometries of the form of metrical geometries of other types of congruence-groups forms an important chapter of non-Euclidean geometry. Arising from this investigation there is a widely-spread fallacy, which has found its way into many philosophic writings, namely, that the possibility of the geometry of existent three-dimensional space being other than Euclidean depends on the physical existence of Euclidean space of four or more dimensions. The foregoing exposition shows the baselessness of this idea.
Bibliography.—For an account of the investigations on the axioms of geometry during the Greek period, see M. Cantor,Vorlesungen über die Geschichte der Mathematik, Bd. i. and iii.; T.L. Heath,The Thirteen Books of Euclid’s Elements, a New Translation from the Greek, with Introductory Essays and Commentary, Historical, Critical, and Explanatory(Cambridge, 1908)—this work is the standard source of information; W.B. Frankland,Euclid, 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,Éléments de géométrie(1794). An adequate account of the above authors is given by P. Stäckel 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. Stäckel 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 Geometrieby Stäckel 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,Über 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 inThe 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, “Über die sogenannte nichteuklidische Geometrie,”Math. Annal.vols. iv. and vi. (1871 and 1872), between them elaborated the projective theory of distance; H. Helmholtz, “Über die thatsächlichen Grundlagen der Geometrie” (1866), and “Über die Thatsachen, die der Geometrie zu Grunde liegen” (1868), both in hisWissenschaftliche 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 asuccession of great French mathematicians, for example, G. Monge,Géométrie descriptive(1800); J.V. Poncelet,Traité des proprietés projectives des figures(1822); M. Chasles,Aperçu historique sur l’origine et le développement des méthodes en géométrie(Bruxelles, 1837), andTraité de géométrie supérieure(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(Nürnberg, 1847); andBeiträge zur Geometrie der Lage(Nürnberg, 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), andActa 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,Grundzüge 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, “Über die Möglichkeit 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 mathématiques(Paris, 1905); H. Poincaré,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.
Bibliography.—For an account of the investigations on the axioms of geometry during the Greek period, see M. Cantor,Vorlesungen über die Geschichte der Mathematik, Bd. i. and iii.; T.L. Heath,The Thirteen Books of Euclid’s Elements, a New Translation from the Greek, with Introductory Essays and Commentary, Historical, Critical, and Explanatory(Cambridge, 1908)—this work is the standard source of information; W.B. Frankland,Euclid, 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,Éléments de géométrie(1794). An adequate account of the above authors is given by P. Stäckel 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. Stäckel 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 Geometrieby Stäckel 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,Über 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 inThe 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, “Über die sogenannte nichteuklidische Geometrie,”Math. Annal.vols. iv. and vi. (1871 and 1872), between them elaborated the projective theory of distance; H. Helmholtz, “Über die thatsächlichen Grundlagen der Geometrie” (1866), and “Über die Thatsachen, die der Geometrie zu Grunde liegen” (1868), both in hisWissenschaftliche 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 asuccession of great French mathematicians, for example, G. Monge,Géométrie descriptive(1800); J.V. Poncelet,Traité des proprietés projectives des figures(1822); M. Chasles,Aperçu historique sur l’origine et le développement des méthodes en géométrie(Bruxelles, 1837), andTraité de géométrie supérieure(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(Nürnberg, 1847); andBeiträge zur Geometrie der Lage(Nürnberg, 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), andActa 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,Grundzüge 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, “Über die Möglichkeit 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 mathématiques(Paris, 1905); H. Poincaré,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.)
1For Egyptian geometry seeEgypt, §Science and Mathematics.2Cf. A.N. Whitehead,Universal Algebra, Bk. vi. (Cambridge, 1898).3Cf. A.N. Whitehead,loc. cit.4Cf. A.N. Whitehead, “The Geodesic Geometry of Surfaces in non-Euclidean Space,”Proc. Lond. Math. Soc.vol. xxix.5Cf. Klein, “Zur nicht-Euklidischen Geometrie,”Math. Annal.vol. xxxvii.6On the theory of parallels before Lobatchewsky, see Stäckel und Engel,Theorie der Parallellinien von Euklid bis auf Gauss(Leipzig, 1895). The foregoing remarks are based upon the materials collected in this work.7See Stäckel 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.8Lobatchewsky’s works on the subject are the following:—“On the Foundations of Geometry,”Kazañ Messenger, 1829-1830; “New Foundations of Geometry, with a complete Theory of Parallels,”Proceedings of the University of Kazañ, 1835 (both in Russian, but translated into German by Engel, Leipzig, 1898); “Géométrie 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 Kazañ in 1826.9Translated by Halsted (Austin, Texas, 4th ed., 1896.)10Abhandlungen d. Königl. Ges. d. Wiss. zu Göttingen, Bd. xiii.;Ges. math. Werke, pp. 254-269; translated by Clifford,Collected Mathematical Papers.11Cf.Gesamm. math. und phys. Werke, vol. i. (Leipzig, 1894).12Wiss. Abh.vol. ii. pp. 610, 618 (1866, 1868).13Mind, O.S., vols. i. and iii.;Vorträge und Reden, vol. ii. pp. 1, 256.14His 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. Hoüel,Annales scientifiques de l’École Normale supérieure, vol. vi. (1869).15Beltrami shows also that this definition agrees with that of Gauss.16“Sur la théorie des foyers,”Nouv. Ann.vol. xii.17Math. Annalen, iv. vi., 1871-1872.18For 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 1877.19For an analysis of Leibnitz’s ideas on space, cf. B. Russell,The Philosophy of Leibnitz, chs. viii.-x.20Cf. 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. 205.21Cf.Critique of Pure Reason, 1st section: “Of Space,” conclusion A, Max Müller’s translation.22Cf. Ernst Mach,Erkenntniss und Irrtum(Leipzig); the relevant chapters are translated by T.J. McCormack,Space and Geometry(London, 1906); also A. Meinong,Über die Stellung der Gegenstandstheorie im System der Wissenschaften(Leipzig, 1907).23Cf. Russell,Principles of Mathematics, § 352 (Cambridge, 1903).24Cf. A.N. Whitehead,The Axioms of Projective Geometry, § 3 (Cambridge, 1906).25Cf. Russell,Princ. of Math., ch. i.26Cf. Russell,loc. cit., and G. Frege, “Über die Grundlagen der Géométrie,”Jahresber. der Deutsch. Math. Ver.(1906).27This 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.28Cf. 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).29Cf. “Sui postulati fondamentali della geometria projettiva,”Giorn. di matematica, vol. xxx. (1891); also of Pieri, loc. cit., and Whitehead,loc. cit.30Cf. Hilbert,loc. cit.; for a fuller exposition of Hilbert’s proof cf. K.T. Vahlen,Abstrakte Geometrie(Leipzig, 1905), also Whitehead,loc. cit.31Cf. H. Wiener,Jahresber. der Deutsch. Math. Ver.vol. i. (1890); and F. Schur, “Über den Fundamentalsatz der projectiven Geometrie,”Math. Ann.vol. li. (1899).32Cf. Hilbert,loc. cit., and Whitehead,loc. cit.33Cf. Dedekind,Stetigkeit und irrationale Zahlen(1872).34Cf. v. Staudt,Geometrie der Lage(1847).35Cf. Pasch,Vorlesungen über neuere Geometrie(Leipzig, 1882), a classic work; also Fiedler,Die darstellende Geometrie(1st ed., 1871, 3rd ed., 1888); Clebsch,Vorlesungen über Geometrie, vol. iii.; Hilbert,loc. cit.; F. Schur,Math. Ann. Bd.lv. (1902); Vahlen,loc. cit.; Whitehead,loc. cit.36Cf.loc. cit.37Cf.I Principii di geometria(Turin, 1889) and “Sui fondamenti della geometria,”Rivista di mat.vol. iv. (1894).38Cf.loc. cit.39Cf. Vailati,Rivista di mat.vol. iv. and Russell,loc. cit.§ 376.40Cf. O. Veblen, “On the Projective Axioms of Geometry,”Trans. Amer. Math. Soc.vol. iii. (1902).41Cf. P. Stäckel and F. Engel,Die Theorie der Parallellinien von Euklid bis auf Gauss(Leipzig, 1895).42Cf. 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).43The original idea (confined to this particular case) of ideal points is due to von Staudt (loc. cit.).44Cf.Critique, “Trans. Aesth.” Sect. I.45Cf.loc. cit.46Cf.Über die Grundlagen der Geometrie(Leipzig, Ber., 1890); andTheorie der Transformationsgruppen(Leipzig, 1893), vol. iii.47Cf. A. Cayley, “A Sixth Memoir on Quantics,”Trans. Roy. Soc., 1859, andColl. Papers, vol. ii.; and F. Klein,Math. Ann.vol. iv., 1871.48Cf.loc. cit.49For 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.50Cf. H. Poincaré,La Science et l’hypothèse, ch. iii.
1For Egyptian geometry seeEgypt, §Science and Mathematics.
2Cf. A.N. Whitehead,Universal Algebra, Bk. vi. (Cambridge, 1898).
3Cf. A.N. Whitehead,loc. cit.
4Cf. A.N. Whitehead, “The Geodesic Geometry of Surfaces in non-Euclidean Space,”Proc. Lond. Math. Soc.vol. xxix.
5Cf. Klein, “Zur nicht-Euklidischen Geometrie,”Math. Annal.vol. xxxvii.
6On the theory of parallels before Lobatchewsky, see Stäckel und Engel,Theorie der Parallellinien von Euklid bis auf Gauss(Leipzig, 1895). The foregoing remarks are based upon the materials collected in this work.
7See Stäckel 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.
8Lobatchewsky’s works on the subject are the following:—“On the Foundations of Geometry,”Kazañ Messenger, 1829-1830; “New Foundations of Geometry, with a complete Theory of Parallels,”Proceedings of the University of Kazañ, 1835 (both in Russian, but translated into German by Engel, Leipzig, 1898); “Géométrie 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 Kazañ in 1826.
9Translated by Halsted (Austin, Texas, 4th ed., 1896.)
10Abhandlungen d. Königl. Ges. d. Wiss. zu Göttingen, Bd. xiii.;Ges. math. Werke, pp. 254-269; translated by Clifford,Collected Mathematical Papers.
11Cf.Gesamm. math. und phys. Werke, vol. i. (Leipzig, 1894).
12Wiss. Abh.vol. ii. pp. 610, 618 (1866, 1868).
13Mind, O.S., vols. i. and iii.;Vorträge und Reden, vol. ii. pp. 1, 256.
14His 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. Hoüel,Annales scientifiques de l’École Normale supérieure, vol. vi. (1869).
15Beltrami shows also that this definition agrees with that of Gauss.
16“Sur la théorie des foyers,”Nouv. Ann.vol. xii.
17Math. Annalen, iv. vi., 1871-1872.
18For 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.
19For an analysis of Leibnitz’s ideas on space, cf. B. Russell,The Philosophy of Leibnitz, chs. viii.-x.
20Cf. 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.
21Cf.Critique of Pure Reason, 1st section: “Of Space,” conclusion A, Max Müller’s translation.
22Cf. Ernst Mach,Erkenntniss und Irrtum(Leipzig); the relevant chapters are translated by T.J. McCormack,Space and Geometry(London, 1906); also A. Meinong,Über die Stellung der Gegenstandstheorie im System der Wissenschaften(Leipzig, 1907).
23Cf. Russell,Principles of Mathematics, § 352 (Cambridge, 1903).
24Cf. A.N. Whitehead,The Axioms of Projective Geometry, § 3 (Cambridge, 1906).
25Cf. Russell,Princ. of Math., ch. i.
26Cf. Russell,loc. cit., and G. Frege, “Über die Grundlagen der Géométrie,”Jahresber. der Deutsch. Math. Ver.(1906).
27This 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.
28Cf. 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).
29Cf. “Sui postulati fondamentali della geometria projettiva,”Giorn. di matematica, vol. xxx. (1891); also of Pieri, loc. cit., and Whitehead,loc. cit.
30Cf. Hilbert,loc. cit.; for a fuller exposition of Hilbert’s proof cf. K.T. Vahlen,Abstrakte Geometrie(Leipzig, 1905), also Whitehead,loc. cit.
31Cf. H. Wiener,Jahresber. der Deutsch. Math. Ver.vol. i. (1890); and F. Schur, “Über den Fundamentalsatz der projectiven Geometrie,”Math. Ann.vol. li. (1899).
32Cf. Hilbert,loc. cit., and Whitehead,loc. cit.
33Cf. Dedekind,Stetigkeit und irrationale Zahlen(1872).
34Cf. v. Staudt,Geometrie der Lage(1847).
35Cf. Pasch,Vorlesungen über neuere Geometrie(Leipzig, 1882), a classic work; also Fiedler,Die darstellende Geometrie(1st ed., 1871, 3rd ed., 1888); Clebsch,Vorlesungen über Geometrie, vol. iii.; Hilbert,loc. cit.; F. Schur,Math. Ann. Bd.lv. (1902); Vahlen,loc. cit.; Whitehead,loc. cit.
36Cf.loc. cit.
37Cf.I Principii di geometria(Turin, 1889) and “Sui fondamenti della geometria,”Rivista di mat.vol. iv. (1894).
38Cf.loc. cit.
39Cf. Vailati,Rivista di mat.vol. iv. and Russell,loc. cit.§ 376.
40Cf. O. Veblen, “On the Projective Axioms of Geometry,”Trans. Amer. Math. Soc.vol. iii. (1902).
41Cf. P. Stäckel and F. Engel,Die Theorie der Parallellinien von Euklid bis auf Gauss(Leipzig, 1895).
42Cf. 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).
43The original idea (confined to this particular case) of ideal points is due to von Staudt (loc. cit.).
44Cf.Critique, “Trans. Aesth.” Sect. I.
45Cf.loc. cit.
46Cf.Über die Grundlagen der Geometrie(Leipzig, Ber., 1890); andTheorie der Transformationsgruppen(Leipzig, 1893), vol. iii.
47Cf. A. Cayley, “A Sixth Memoir on Quantics,”Trans. Roy. Soc., 1859, andColl. Papers, vol. ii.; and F. Klein,Math. Ann.vol. iv., 1871.
48Cf.loc. cit.
49For 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.
50Cf. H. Poincaré,La Science et l’hypothèse, ch. iii.