determinant, s2 + s3 ... times in every first minor, s3 + ... times in
every second minor, and so on; the meaning of each exponent is then
perfectly definite. Every factor of the type ([lambda] - [alpha])^s is
called an _elementartheil_ (elementary divisor) of the determinant,
and the condition of equivalence of two pairs of quadratics is simply
that their determinants have the same elementary divisors. We write
the pair of forms symbolically thus [(s1s2 ...), (t1t2 ...), ...],
letters in the inner brackets referring to the same factor. Returning
now to the two quadratics representing the complex, the sum of the
exponents will be six, and two complexes are put in the same class if
they have the same symbolical expression; i.e. the actual values of
the roots of the determinantal equation need not be the same for both,
but their manner of occurrence, as far as here indicated, must be
identical in the two. The enumeration of all possible cases is thus
reduced to a simple question in combinatorial analysis, and the actual
study of any particular case is much facilitated by a useful rule of
Klein's for writing down in a simple form two quadratics belonging to
a given class--one of which, of course, represents the equation
connecting line coordinates, and the other the equation of the
complex. The general complex is naturally [111111]; the complex of
tangents to a quadric is [(111), (111)] and that of lines meeting a
conic is [(222)]. Full information will be found in Weiler's memoir,
_Math. Ann._ vol. vii.
The detailed study of each variety of complex opens up a vast subject;
we only mention two special cases, the harmonic complex and the
tetrahedral complex.
The harmonic complex, first studied by Battaglini, is generated in an
infinite number of ways by the lines cutting two quadrics
harmonically. Taking the most general case, and referring the quadrics
to their common self-conjugate tetrahedron, we can find its equation
in a simple form, and verify that this complex really depends only on
seventeen constants, so that it is not the most general quadratic
complex. It belongs to the general type in so far as it is discussed
above, but the roots of the determinant are in involution. The
singular surface is the "tetrahedroid" discussed by Cayley. As a
particular case, from a metrical point of view, we have L.F. Painvin's
complex generated by the lines of intersection of perpendicular
tangent planes of a quadric, the singular surface now being Fresnel's
wave surface. The tetrahedral or Reye complex is the simplest and best
known of proper quadratic complexes. It is generated by the lines
which cut the faces of a tetrahedron in a constant cross ratio, and
therefore by those subtending the same cross ratio at the four
vertices. The singular surface is made up of the faces or the vertices
of the fundamental tetrahedron, and each edge of this tetrahedron is a
double line of the complex. The complex was first discussed by K.T.
Reye as the assemblage of lines joining corresponding points in a
homographic transformation of space, and this point of view leads to
many important and elegant properties. A (metrically) particular case
of great interest is the complex generated by the normals to a family
of confocal quadrics, and for many investigations it is convenient to
deal with this complex referred to the principal axes. For example,
Lie has developed the theory of curves in a Reye complex (i.e. curves
whose tangents belong to the complex) as solutions of a differential
equation of the form (b - c)xdydz + (c - a)ydzdx + (a - b)zdxdy = 0,
and we can simplify this equation by a logarithmic transformation.
Many theorems connecting complexes with differential equations have
been given by Lie and his school. A line complex, in fact, corresponds
to a Mongian equation having [oo]^3 line integrals.
Congruences.
As the coordinates of a line belonging to a congruence are functions
of two independent parameters, the theory of congruences is analogous
to that of surfaces, and we may regard it as a fundamental inquiry to
find the simplest form of surface into which a given congruence can be
transformed. Most of those whose properties have been extensively
discussed can be represented on a plane by a birational
transformation. But in addition to the difficulties of the theory of
algebraic surfaces, a subject still in its infancy, the theory of
congruences has other difficulties in that a congruence is seldom
completely represented, even by two equations.
A fundamental theorem is that the lines of a congruence are in general
bitangents of a surface; in fact, since the condition of intersection
of two consecutive straight lines is ld[lambda] + dmd[mu] + dnd[nu] =
0, a line l of the congruence meets two adjacent lines, say l1 and l2.
Suppose l, l1 lie in the plane pencil (A1a1) and l, l2 in the plane
pencil (A2a2), then the locus of the A's is the same as the envelope
of the a's, but a2 is the tangent plane at A1 and a1 at A2. This
surface is called the focal surface of the congruence, and to it all
the lines l are bitangent. The distinctive property of the points A is
that two of the congruence lines through them coincide, and in like
manner the planes a each contain two coincident lines. The focal
surface consists of two sheets, but one or both may degenerate into
curves; thus, for example, the normals to a surface are bitangents of
the surface of centres, and in the case of Dupin's cyclide this
surface degenerates into two conics.
In the discussion of congruences it soon becomes necessary to
introduce another number r, called the rank, which expresses the
number of plane pencils each of which contains an arbitrary line and
two lines of the congruence. The order of the focal surface is 2m(n -
1) - 2r, and its class is m(m - 1) - 2r. Our knowledge of congruences
is almost exclusively confined to those in which either m or n does
not exceed two. We give a brief account of those of the second order
without singular lines, those of order unity not being especially
interesting. A congruence generally has singular points through which
an infinite number of lines pass; a singular point is said to be of
order r when the lines through it lie on a cone of the rth degree. By
means of formulae connecting the number of singular points and their
orders with the class m of quadratic congruence Kummer proved that the
class cannot exceed seven. The focal surface is of degree four and
class 2m; this kind of quartic surface has been extensively studied by
Kummer, Cayley, Rohn and others. The varieties (2, 2), (2, 3), (2, 4),
(2, 5) all belong to at least one Reye complex; and so also does the
most important class of (2, 6) congruences which includes all the
above as special cases. The congruence (2, 2) belongs to a linear
complex and forty different Reye complexes; as above remarked, the
singular surface is Kummer's sixteen-nodal quartic, and the same
surface is focal for six different congruences of this variety. The
theory of (2, 2) congruences is completely analogous to that of the
surfaces called cyclides in three dimensions. Further particulars
regarding quadratic congruences will be found in Kummer's memoir of
1866, and the second volume of Sturm's treatise. The properties of
quadratic congruences having singular lines, i.e. degenerate focal
surfaces, are not so interesting as those of the above class; they
have been discussed by Kummer, Sturm and others.
Ruled surfaces.
Since a ruled surface contains only [infinity]^1 elements, this theory
is practically the same as that of curves. If a linear complex
contains more than n generators of a ruled surface of the nth degree,
it contains all the generators, hence for n = 2 there are three
linearly independent complexes, containing all the generators, and
this is a well-known property of quadric surfaces. In ruled cubics the
generators all meet two lines which may or may not coincide; these two
cases correspond to the two main classes of cubics discussed by Cayley
and Cremona. As regards ruled quartics, the generators must lie in one
and may lie in two linear complexes. The first class is equivalent to
a quartic in four dimensions and is always rational, but the latter
class has to be subdivided into the elliptic and the rational, just
like twisted quartic curves. A quintic skew may not lie in a linear
complex, and then it is unicursal, while of sextics we have two
classes not in a linear complex, viz. the elliptic variety, having
thirty-six places where a linear complex contains six consecutive
generators, and the rational, having six such places.
The general theory of skews in two linear complexes is identical with
that of curves on a quadric in three dimensions and is known. But for
skews lying in only one linear complex there are difficulties; the
curve now lies in four dimensions, and we represent it in three by
stereographic projection as a curve meeting a given plane in n points
on a conic. To find the maximum deficiency for a given degree would
probably be difficult, but as far as degree eight the space-curve
theory of Halphen and Nother can be translated into line geometry at
once. When the skew does not lie in a linear complex at all the theory
is more difficult still, and the general theory clearly cannot advance
until further progress is made in the study of twisted curves.
REFERENCES.--The earliest works of a general nature are Plucker, _Neue
Geometrie des Raumes_ (Leipzig, 1868); and Kummer, "Uber die
algebraischen Strahlensysteme," _Berlin Academy_ (1866). Systematic
development on purely synthetic lines will be found in the three
volumes of Sturm, _Liniengeometrie_ (Leipzig, 1892, 1893, 1896); vol.
i. deals with the linear and Reye complexes, vols. ii. and iii. with
quadratic congruences and complexes respectively. For a highly
suggestive review by Gino Loria see _Bulletin des sciences
mathematiques_ (1893, 1897). A shorter treatise, giving a very
interesting account of Klein's coordinates, is the work of Koenigs,
_La Geometrie reglee et ses applications_ (Paris, 1898). English
treatises are C.M. Jessop, _Treatise on the Line Complex_ (1903);
R.W.H.T. Hudson, _Kummer's Quartic_ (1905). Many references to memoirs
on line geometry will be found in Hagen, _Synopsis der hoheren
Mathematik_, ii. (Berlin, 1894); Loria, _Il passato ed il presente
delle principali teorie geometriche_ (Milan, 1897); a clear resume of
the principal results is contained in the very elegant volume of
Pascal, _Repertorio di mathematiche superiori_, ii. (Milan, 1900).
Another treatise dealing extensively with line geometry is Lie,
_Geometrie der Beruhrungstransformationen_ (Leipzig, 1896). Many
memoirs on the subject have appeared in the _Mathematische Annalen_; a
full list of these will be found in the index to the first fifty
volumes, p. 115. Perhaps the two memoirs which have left most
impression on the subsequent development of the subject are Klein,
"Zur Theorie der Liniencomplexe des ersten und zweiten Grades," _Math.
Ann._ ii.; and Lie, "Uber Complexe, insbesondere Linien- und
Kugelcomplexe," _Math. Ann._ v. (J. H. Gr.)
VI. NON-EUCLIDEAN GEOMETRY
The various metrical geometries are concerned with the properties of the
various types of congruence-groups, which are defined in the study of
the _axioms_ of _geometry_ and of their immediate consequences. But this
point of view of the subject is the outcome of recent research, and
historically the subject has a different origin. Non-Euclidean geometry
arose from the discussion, extending from the Greek period to the
present day, of the various assumptions which are implicit in the
traditional Euclidean system of geometry. In the course of these
investigations it became evident that metrical geometries, each
internally consistent but inconsistent in many respects with each other
and with the Euclidean system, could be developed. A short historical
sketch will explain this origin of the subject, and describe the famous
and interesting progress of thought on the subject. But previously a
description of the chief characteristic properties of elliptic and of
hyperbolic geometries will be given, assuming the standpoint arrived at
below under VII. _Axioms of Geometry_.
First assume the equation to the absolute (cf. _loc. cit._) to be w^2 -
x^2 - y^2 - z^2 = 0. The absolute is then real, and the geometry is
hyberbolic.
The distance (d12) between the two points (x1, y1, z1, w1) and (x2,
y2, z2, w2) is given by
cosh (d12/[gamma]) = (w1w2 - x1x2 - y1y2 - z1z2)/[(w1^2 - x1^2 - y1^2 - Z1^2)
(w2^2 - x2^2 - y2^2 - z2^2)]1/2 (1)
The only points to which the metrical geometry applies are those
within the region enclosed by the quadric; the other points are
"improper ideal points." The angle ([theta]12) between two planes, l1x
+ m1y + n1z + r1w = 0 and l2x + m2y + n2z + r2w = 0, is given by
cos [theta]12 = (l1l2 + m1m2 + n1n2 - r1r2)/{(l1^2 + m1^2 + n1^2 - r1^2)
(l2^2 + m2^2 + n2^2 - r2^2)}^1/2 (2)
These planes only have a real angle of inclination if they possess a
line of intersection within the actual space, i.e. if they intersect.
Planes which do not intersect possess a shortest distance along a line
which is perpendicular to both of them. If this shortest distance is
[delta]12, we have
cosh ([delta]12/[gamma]) = (l1l2 + m1m2 + n1n2 - r1r2)/(l1^2 + m1^2 + n1^2 - r1^2)
(l2^2 + m2^2 + n2^2 - r2^2)^1/2 (3)
[Illustration: FIG. 67.]
Thus in the case of the two planes one and only one of the two,
[theta]12 and [delta]12, is real. The same considerations hold for
coplanar straight lines (see VII. _Axioms of Geometry_). Let O (fig.
67) be the point (0, 0, 0, 1), OX the line y = 0, z = 0, OY the line z
= 0, x = 0, and OZ the line x = 0, y = 0. These are the coordinate
axes and are at right angles to each other. Let P be any point, and
let [rho] be the distance OP, [theta] the angle POZ, and [phi] the
angle between the planes ZOX and ZOP. Then the coordinates of P can be
taken to be
sinh ([rho]/[gamma]) sin [theta] cos [phi], sinh ([rho]/[gamma]) sin [theta]
sin [phi], sinh ([rho]/[gamma]) cos[theta], cosh ([rho]/[gamma]).
[Illustration: FIG. 68.]
If ABC is a triangle, and the sides and angles are named according to
the usual convention, we have
sinh (a/[gamma])/sin A = sinh (b/[gamma])/sin B = sinh (c/[gamma])/sin C, (4)
and also
cosh (a/[gamma]) = cosh (b/[gamma]) cosh (c/[gamma]) -
sinh (b/[gamma]) sinh (c/[gamma]) cos A, (5)
with two similar equations. The sum of the three angles of a triangle
is always less than two right angles. The area of the triangle ABC is
[lambda]^2([pi] - A - B - C). If the base BC of a triangle is kept
fixed and the vertex A moves in the fixed plane ABC so that the area
ABC is constant, then the locus of A is a line of equal distance from
BC. This locus is not a straight line. The whole theory of similarity
is inapplicable; two triangles are either congruent, or their angles
are not equal two by two. Thus the elements of a triangle are
determined when its three angles are given. By keeping A and B and the
line BC fixed, but by making C move off to infinity along BC, the
lines BC and AC become parallel, and the sides a and b become
infinite. Hence from equation (5) above, it follows that two parallel
lines (cf. Section VII. _Axioms of Geometry_) must be considered as
making a zero angle with each other. Also if B be a right angle, from
the equation (5), remembering that, in the limit,
cosh (a/[gamma])/cosh (b/[gamma]) = cosh (a/[gamma])/sinh (b/[gamma]) = 1,
we have cos A = tanh (c/2[gamma]) .... (6).
The angle A is called by N.I. Lobatchewsky the "angle of parallelism."
The whole theory of lines and planes at right angles to each other is
simply the theory of conjugate elements with respect to the absolute,
where ideal lines and planes are introduced.
Thus if l and l' be any two conjugate lines with respect to the
absolute (of which one of the two must be improper, say l'), then any
plane through l' and containing proper points is perpendicular to l.
Also if p is any plane containing proper points, and P is its pole,
which is necessarily improper, then the lines through P are the
normals to P. The equation of the sphere, centre (x1, y1, z1, w1) and
radius [rho], is
(w1^2- x1^2- y1^2- z1^2)(w^2 - x^2 - y^2 - z^2) cosh^2([rho]/[gamma]) = (w1w -
x1x - y1y -z1z)^2 (7).
The equation of the surface of equal distance ([sigma]) from the plane
lx + my + nz + rw = 0 is
(l^2 + m^2 + n^2 - r^2)(w^2 - x^2 - y^2 - z^2) sinh^2([sigma]/[gamma]) = (rw +
lx + my + nz)^2 (8).
A surface of equal distance is a sphere whose centre is improper; and
both types of surface are included in the family
k^2(w^2 - x^2 - y^2 - z^2) = (ax + by + cz + dw)^2 (9).
But this family also includes a third type of surfaces, which can be
looked on either as the limits of spheres whose centres have
approached the absolute, or as the limits of surfaces of equal
distance whose central planes have approached a position tangential to
the absolute. These surfaces are called limit-surfaces. Thus (9)
denotes a limit-surface, if d^2 - a^2 - b^2 - c^2 = 0. Two
limit-surfaces only differ in position. Thus the two limit-surfaces
which touch the plane YOZ at O, but have their concavities turned in
opposite directions, have as their equations
w^2 - x^2 - y^2 - z^2 = (w [+-] x)^2.
The geodesic geometry of a sphere is elliptic, that of a surface of
equal distance is hyperbolic, and that of a limit-surface is parabolic
(i.e. _Euclidean_). The equation of the surface (cylinder) of equal
distance ([delta]) from the line OX is
(w^2 - x^2) tanh^2([delta]/[gamma]) - y^2 - z^2 = 0.
This is not a ruled surface. Hence in this geometry it is not possible
for two straight lines to be at a constant distance from each other.
Secondly, let the equation of the absolute be x^2 + y^2 + z^2 + w^2 =
0. The absolute is now imaginary and the geometry is elliptic.
The distance (d12) between the two points (x1, y1, z1, w1) and (x2,
y2, z2, w2) is given by
cos (d12/[gamma]) = [+-](x1x2 + y1y2 + z1z2 + w1w2)
/ {(x1^2 + y1^2 + z1^2 + w1^2) {(x2^2 + y2^2 + z2^2 + w2^2)}^1/2 (10).
Thus there are two distances between the points, and if one is d12,
the other is [pi][gamma]-d12. Every straight line returns into itself,
forming a closed series. Thus there are two segments between any two
points, together forming the whole line which contains them; one
distance is associated with one segment, and the other distance with
the other segment. The complete length of every straight line is
[pi][gamma].
The angle between the two planes l1x + m1y + n1z + r + 1w = 0 and l2x
+ m2y + n2z + r2w = 0 is
cos [theta]12 = (l1l2 + m1m2 + n1n2 + r1r2)/ {(l1^2 + m1^2 + n1^2 +r1^2)
(l2^2 + m2^2 + n2^2 + r2^2)}^1/2 (11).
The polar plane with respect to the absolute of the point (x1, y1, z1,
w1) is the real plane x1x + y1y + z1z + w1w = 0, and the pole of the
plane l1x + m1y + n1z + r1w = 0 is the point (l1, m1, n1, r1). Thus
(from equations 10 and 11) it follows that the angle between the polar
planes of the points (x1, ...) and (x2, ...) is d12/[gamma], and that
the distance between the poles of the planes (l1, ...) and (l2, ...)
is [gamma][theta]12. Thus there is complete reciprocity between points
and planes in respect to all properties. This complete reign of the
principle of duality is one of the great beauties of this geometry.
The theory of lines and planes at right angles is simply the theory of
conjugate elements with respect to the absolute. A tetrahedron
self-conjugate with respect to the absolute has all its intersecting
elements (edges and planes) at right angles. If l and l' are two
conjugate lines, the planes through one are the planes perpendicular
to the other. If P is the pole of the plane p, the lines through P are
the normals to the plane p. The distance from P to p is 1/2[pi][gamma].
Thus every sphere is also a surface of equal distance from the polar
of its centre, and conversely. A plane does not divide space; for the
line joining any two points P and Q only cuts the plane once, in L
say, then it is always possible to go from P to Q by the segment of
the line PQ which does not contain L. But P and Q may be said to be
separated by a plane p, if the point in which PQ cuts p lies on the
shortest segment between P and Q. With this sense of "separation," it
is possible[2] to find three points P, Q, R such that P and Q are
separated by the plane p, but P and R are not separated by p, nor are
Q and R.
Let A, B, C be any three non-collinear points, then four triangles are
defined by these points. Thus if a, b, c and A, B, C are the elements
of any one triangle, then the four triangles have as their elements:
(1) a, b, c, A, B, C.
(2) a, [pi][gamma] - b, [pi][gamma] - c, A, [pi] - B, [pi] - C.
(3) [pi][gamma] - a, b, [pi][gamma] - c, [pi] - A, B, [pi] - C.
(4) [pi][gamma] - a, [pi][gamma] - b, c, [pi] - A, [pi] - B, C.
The formulae connecting the elements are
sin A/sin (a/[gamma]) = sin B/sin (b/[gamma]) = sin C/sin (c/[gamma]),
(12)
and
cos (a/[gamma]) = cos (b/[gamma]) cos (c/[gamma]) + sin (b/[gamma])
sin (c/[gamma]) cos A, (13)
with two similar equations.
Two cases arise, namely (I.) according as one of the four triangles
has as its sides the shortest segments between the angular points, or
(II.) according as this is not the case. When case I. holds there is
said to be a "principal triangle."[3] If all the figures considered
lie within a sphere of radius 1/4[pi][gamma] only case I. can hold, and
the principal triangle is the triangle wholly within this sphere, also
the peculiarities in respect to the separation of points by a plane
cannot then arise. The sum of the three angles of a triangle ABC is
always greater than two right angles, and the area of the triangle is
[gamma]^2(A + B + C--[pi]). Thus as in hyperbolic geometry the theory
of similarity does not hold, and the elements of a triangle are
determined when its three angles are given. The coordinates of a point
can be written in the form
sin ([rho]/[gamma]) sin [Phi] cos [phi], sin ([rho]/[gamma]) sin [Phi]
sin [phi], sin ([rho]/[gamma]) cos [Phi], cos ([rho]/[gamma]),
where [rho], [Phi] and [phi] have the same meanings as in the
corresponding formulae in hyperbolic geometry. Again, suppose a watch
is laid on the plane OXY, face upwards with its centre at O, and the
line 12 to 6 (as marked on dial) along the line YOY. Let the watch be
continually pushed along the plane along the line OX, that is, in the
direction 9 to 3. Then the line XOX being of finite length, the watch
will return to O, but at its first return it will be found to be face
downwards on the other side of the plane, with the line 12 to 6
reversed in direction along the line YOY. This peculiarity was first
pointed out by Felix Klein. The theory of parallels as it exists in
hyperbolic space has no application in elliptic geometry. But another
property of Euclidean parallel lines holds in elliptic geometry, and
by the use of it parallel lines are defined. For the equation of the
surface (cylinder) of equal distance ([delta]) from the line XOX is
(x^2 + w^2) tan^2([delta]/[gamma]) - (y^2 + z^2) = 0.
This is also the surface of equal distance, 1/2[pi][gamma]-[delta],
from the line conjugate to XOX. Now from the form of the above
equation this is a ruled surface, and through every point of it two
generators pass. But these generators are lines of equal distance from
XOX. Thus throughout every point of space two lines can be drawn which
are lines of equal distance from a given line l. This property was
discovered by W.K. Clifford. The two lines are called Clifford's right
and left parallels to l through the point. This property of
parallelism is reciprocal, so that if m is a left parallel to l, then
l is a left parallel to m. Note also that two parallel lines l and m
are not coplanar. Many of those properties of Euclidean parallels,
which do not hold for Lobatchewsky's parallels in hyperbolic geometry,
do hold for Clifford's parallels in elliptic geometry. The geodesic
geometry of spheres is elliptic, the geodesic geometry of surfaces of
equal distance from lines (cylinders) is Euclidean, and surfaces of
revolution can be found[4] of which the geodesic geometry is
hyperbolic. But it is to be noticed that the connectivity of these
surfaces is different to that of a Euclidean plane. For instance there
are only [&infin]^2 congruence transformations of the cylindrical
surfaces of equal distance into themselves, instead of the [&infin]^3
for the ordinary plane. It would obviously be possible to state
"axioms" which these geodesics satisfy, and thus to define
independently, and not as loci, quasi-spaces of these peculiar types.
The existence of such Euclidean quasi-geometries was first pointed out
by Clifford.[5]
In both elliptic and hyperbolic geometry the spherical geometry, i.e.
the relations between the angles formed by lines and planes passing
through the same point, is the same as the "spherical trigonometry" in
Euclidean geometry. The constant [gamma], which appears in the formulae
both of hyperbolic and elliptic geometry, does not by its variation
produce different types of geometry. There is only one type of elliptic
geometry and one type of hyperbolic geometry; and the magnitude of the
constant [gamma] in each case simply depends upon the magnitude of the
arbitrary unit of length in comparison with the natural unit of length
which each particular instance of either geometry presents. The
existence of a natural unit of length is a peculiarity common both to
hyperbolic and elliptic geometries, and differentiates them from
Euclidean geometry. It is the reason for the failure of the theory of
similarity in them. If [gamma] is very large, that is, if the natural
unit is very large compared to the arbitrary unit, and if the lengths
involved in the figures considered are not large compared to the
arbitrary unit, then both the elliptic and hyperbolic geometries
approximate to the Euclidean. For from formulae (4) and (5) and also
from (12) and (13) we find, after retaining only the lowest powers of
small quantities, as the formulae for any triangle ABC,
a/ sin A = b/ sin B = c/ sin C,
and
a^2 = b^2 + c^2 - 2bc cos A,
with two similar equations. Thus the geometries of small figures are in
both types Euclidean.
Theory of parallels before Gauss.
_History._--"In pulcherrimo Geometriae corpore," wrote Sir Henry Savile
in 1621, "duo sunt naevi, duae labes nec quod sciam plures, in quibus
eluendis et emaculendis cum veterum tum recentiorum ... vigilavit
industria." These two blemishes are the theory of parallels and the
theory of proportion. The "industry of the moderns," in both respects,
has given rise to important branches of mathematics, while at the same
time showing that Euclid is in these respects more free from blemish
than had been previously credible. It was from endeavours to improve the
theory of parallels that non-Euclidean geometry arose; and though it has
now acquired a far wider scope, its historical origin remains
instructive and interesting. Euclid's "axiom of parallels" appears as
Postulate V. to the first book of his _Elements_, and is stated thus,
"And that, if a straight line falling on two straight lines make the
angles, internal and on the same side, less than two right angles, the
two straight lines, being produced indefinitely, meet on the side on
which are the angles less than two right angles." The original Greek is
[Greek: kai ean eis duo eutheias eutheia empiptousa tas entos kai epi ta
auta mere gonias duo orthon elassonas poie, ekballomenas tas duo
eutheias ep' apeiron sympiptein, eph' ha mere eisin hai ton duo orthon
elassones].
To Euclid's successors this axiom had signally failed to appear
self-evident, and had failed equally to appear indemonstrable. Without
the use of the postulate its converse is proved in Euclid's 28th
proposition, and it was hoped that by further efforts the postulate
itself could be also proved. The first step consisted in the discovery
of equivalent axioms. Christoph Clavius in 1574 deduced the axiom from
the assumption that a line whose points are all equidistant from a
straight line is itself straight. John Wallis in 1663 showed that the
postulate follows from the possibility of similar triangles on different
scales. Girolamo Saccheri (1733) showed that it is sufficient to have a
single triangle, the sum of whose angles is two right angles. Other
equivalent forms may be obtained, but none shows any essential
superiority to Euclid's. Indeed plausibility, which is chiefly aimed at,
becomes a positive demerit where it conceals a real assumption.
Saccheri.
A new method, which, though it failed to lead to the desired goal,
proved in the end immensely fruitful, was invented by Saccheri, in a
work entitled _Euclides ab omni naevo vindicatus_ (Milan, 1733). If the
postulate of parallels is involved in Euclid's other assumptions,
contradictions must emerge when it is denied while the others are
maintained. This led Saccheri to attempt a _reductio ad absurdum_, in
which he mistakenly believed himself to have succeeded. What is
interesting, however, is not his fallacious conclusion, but the
non-Euclidean results which he obtains in the process. Saccheri
distinguishes three hypotheses (corresponding to what are now known as
Euclidean or parabolic, elliptic and hyperbolic geometry), and proves
that some one of the three must be universally true. His three
hypotheses are thus obtained: equal perpendiculars AC, BD are drawn from
a straight line AB, and CD are joined. It is shown that the angles ACD,
BDC are equal. The first hypothesis is that these are both right angles;
the second, that they are both obtuse; and the third, that they are both
acute. Many of the results afterwards obtained by Lobatchewsky and
Bolyai are here developed. Saccheri fails to be the founder of
non-Euclidean geometry only because he does not perceive the possible
truth of his non-Euclidean hypotheses.
Lambert.
Some advance is made by Johann Heinrich Lambert in his _Theorie der
Parallellinien_ (written 1766; posthumously published 1786). Though he
still believed in the necessary truth of Euclidean geometry, he
confessed that, in all his attempted proofs, something remained
undemonstrated. He deals with the same three hypotheses as Saccheri,
showing that the second holds on a sphere, while the third would hold on
a sphere of purely imaginary radius. The second hypothesis he succeeds
in condemning, since, like all who preceded Bernhard Riemann, he is
unable to conceive of the straight line as finite and closed. But the
third hypothesis, which is the same as Lobatchewsky's, is not even
professedly refuted.[6]
Three periods of non-Euclidean geometry.
Non-Euclidean geometry proper begins with Karl Friedrich Gauss. The
advance which he made was rather philosophical than mathematical: it was
he (probably) who first recognized that the postulate of parallels is
possibly false, and should be empirically tested by measuring the angles
of large triangles. The history of non-Euclidean geometry has been aptly
divided by Felix Klein into three very distinct periods. The
first--which contains only Gauss, Lobatchewsky and Bolyai--is
characterized by its synthetic method and by its close relation to
Euclid. The attempt at indirect proof of the disputed postulate would
seem to have been the source of these three men's discoveries; but when
the postulate had been denied, they found that the results, so far from
showing contradictions, were just as self-consistent as Euclid. They
inferred that the postulate, if true at all, can only be proved by
observations and measurements. Only one kind of non-Euclidean space is
known to them, namely, that which is now called hyperbolic. The second
period is analytical, and is characterized by a close relation to the
theory of surfaces. It begins with Riemann's inaugural dissertation,
which regards space as a particular case of a _manifold_; but the
characteristic standpoint of the period is chiefly emphasized by Eugenio
Beltrami. The conception of measure of curvature is extended by Riemann
from surfaces to spaces, and a new kind of space, finite but unbounded
(corresponding to the second hypothesis of Saccheri and Lambert), is
shown to be possible. As opposed to the second period, which is purely
metrical, the third period is essentially projective in its method. It
begins with Arthur Cayley, who showed that metrical properties are
projective properties relative to a certain fundamental quadric, and
that different geometries arise according as this quadric is real,
imaginary or degenerate. Klein, to whom the development of Cayley's work
is due, showed further that there are two forms of Riemann's space,
called by him the elliptic and the spherical. Finally, it has been shown
by Sophus Lie, that if figures are to be freely movable throughout all
space in [oo]^6 ways, no other three-dimensional spaces than the above
four are possible.
Gauss.
Gauss published nothing on the theory of parallels, and it was not
generally known until after his death that he had interested himself in
that theory from a very early date. In 1799 he announces that Euclidean
geometry would follow from the assumption that a triangle can be drawn
greater than any given triangle. Though unwilling to assume this, we
find him in 1804 still hoping to prove the postulate of parallels. In
1830 he announces his conviction that geometry is not an a priori
science; in the following year he explains that non-Euclidean geometry
is free from contradictions, and that, in this system, the angles of a
triangle diminish without limit when all the sides are increased. He
also gives for the circumference of a circle of radius r the formula
[pi]k(e^(r/k) - e^(r-/k)), where k is a constant depending upon the
nature of the space. In 1832, in reply to the receipt of Bolyai's
_Appendix_, he gives an elegant proof that the amount by which the sum
of the angles of a triangle falls short of two right angles is
proportional to the area of the triangle. From these and a few other
remarks it appears that Gauss possessed the foundations of hyperbolic
geometry, which he was probably the first to regard as perhaps true. It
is not known with certainty whether he influenced Lobatchewsky and
Bolyai, but the evidence we possess is against such a view.[7]
Lobatchewsky.
The first to publish a non-Euclidean geometry was Nicholas Lobatchewsky,
professor of mathematics in the new university of Kazan.[8] In the place
of the disputed postulate he puts the following: "All straight lines
which, in a plane, radiate from a given point, can, with respect to any
other straight line in the same plane, be divided into two classes, the
_intersecting_ and the _non-intersecting_. The _boundary line_ of the
one and the other class is called _parallel to the given line_." It
follows that there are two parallels to the given line through any
point, each meeting the line at infinity, like a Euclidean parallel.
(Hence a line has two distinct points at infinity, and not one only as
in ordinary geometry.) The two parallels to a line through a point make
equal acute angles with the perpendicular to the line through the point.
If p be the length of the perpendicular, either of these angles is
denoted by [Pi](p). The determination of [Pi](p) is the chief problem
(cf. equation (6) above); it appears finally that, with a suitable
choice of the unit of length,
tan 1/2 [Pi](p) = e^(-p).
Before obtaining this result it is shown that spherical trigonometry is
unchanged, and that the normals to a circle or a sphere still pass
through its centre. When the radius of the circle or sphere becomes
infinite all these normals become parallel, but the circle or sphere
does not become a straight line or plane. It becomes what Lobatchewsky
calls a limit-line or limit-surface. The geometry on such a surface is
shown to be Euclidean, limit-lines replacing Euclidean straight lines.
(It is, in fact, a surface of zero measure of curvature.) By the help of
these propositions Lobatchewsky obtains the above value of [Pi](p), and
thence the solution of triangles. He points out that his formulae result
from those of spherical trigonometry by substituting ia, ib, ic, for the
sides a, b, c.
Bolyai.
John Bolyai, a Hungarian, obtained results closely corresponding to
those of Lobatchewsky. These he published in an appendix to a work by
his father, entitled _Appendix Scientiam spatii absolute veram exhibens:
a veritate aut falsitate Axiomatis XI. Euclidei (a priori haud unquam
decidenda) independentem: adjecta ad casum falsitatis, quadratura
circuli geometrica_.[9] This work was published in 1831, but its
conception dates from 1823. It reveals a profounder appreciation of the
importance of the new ideas, but otherwise differs little from
Lobatchewsky's. Both men point out that Euclidean geometry as a limiting
case of their own more general system, that the geometry of very small
spaces is always approximately Euclidean, that no a priori grounds exist
for a decision, and that observation can only give an approximate
answer. Bolyai gives also, as his title indicates, a geometrical
construction, in hyperbolic space, for the quadrature of the circle, and
shows that the area of the greatest possible triangle, which has all its
sides parallel and all its angles zero, is [pi][iota]^2, where i is what
we should now call the space-constant.
Riemann.
The works of Lobatchewsky and Bolyai, though known and valued by Gauss,
remained obscure and ineffective until, in 1866, they were translated
into French by J. Houel. But at this time Riemann's dissertation, _Uber
die Hypothesen, welche der Geometrie zu Grunde liegen_,[10] was already
about to be published. In this work Riemann, without any knowledge of
his predecessors in the same field, inaugurated a far more profound
discussion, based on a far more general standpoint; and by its
publication in 1867 the attention of mathematicians and philosophers was
at last secured. (The dissertation dates from 1854, but owing to changes
which Riemann wished to make in it, it remained unpublished until after
his death.)
Definition of a manifold.
Riemann's work contains two fundamental conceptions, that of a manifold
and that of the _measure of curvature_ of a continuous manifold
possessed of what he calls flatness in the smallest parts. By means of
these conceptions space is made to appear 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 his _Ausdehnungslehre_.[11]
Measure of curvature.
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 considerations 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, ... x_n, let us confine ourselves to lines along which the
ratios dx1 : dx2 : ... : dx_n alter 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 [delta]x say, determine a pencil,
or one-dimensional series, of shortest lines, any one of which is
defined by [lambda]dx + [mu][delta]x, where the parameter [lambda] :
[mu] 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, if [alpha] be the measure of curvature, the linear element can be
put into the form
ds = [root]([Sigma]dx^2)/(1 + 1/4[alpha][Sigma]x^2).
If [alpha] 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.
Helmholtz.
The publication of Riemann's dissertation was closely followed by two
works of Hermann von Helmholtz,[12] again undertaken in ignorance of the
work of predecessors. In these a 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 subject[13] are of almost exclusively philosophical interest. We
shall return to them later.
Beltrami.
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 convenient
Euclidean interpretation of hyperbolic plane geometry, his results will
be stated at some length[14]. The _Saggio_ shows 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 that
ds^2 = R^2w^(-4){([alpha]^2 - v^2)du^2 + 2uvdudv + ([alpha]^2 - u^2)dv^2}
where w^2 = [alpha]^2 - u^2 - v^2, and (-1)/R^2 is the measure of
curvature of our surface (note that k = [gamma] as used above). The
angle between two geodesics u = const., v = const. is [theta], where
cos [theta] = uv/[root]{([alpha]^2 - u^2)([alpha]^2 - v^2)}, sin [theta] =
aw/[root]{(a^2 - u^2)(a^2 - v^2)}.
Thus u = 0 is orthogonal to all geodesies v = const., and vice versa.
In order that sin [theta] may be real, w^2 must be positive; thus
geodesics have no real intersection when the corresponding straight
lines intersect outside the circle u^2 + v^2 = [alpha]^2. When they
intersect on this circle, [theta] = 0. Thus Lobatchewsky's parallels
are represented by straight lines intersecting on the circle. Again,
transforming to polar coordinates u = r cos [mu], v = r sin [mu], and
calling [rho] the geodesic distance of u, v from the origin, we have,
for a geodesic through the origin,
d[rho] = Radr/(a^2 - r^2), [rho] = 1/2R log(a + r)/(a - r), r = a tan h
([rho]/R).
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 [mu] at the origin is
[sigma] = Rr[mu]/[root](a^2 - r^2) = [mu]R sin h ([rho]/R),
whence the circumference of a circle of radius [rho] is 2[pi]R sin h
([rho]/R). Again, if [alpha] be the angle between any two geodesics
V - v = m(U - u), V - v = n(U - u),
then tan [alpha] = a(n - m)w/{(1 + mn)a^2 - (v - mu) (v - nu)}.
Thus [alpha] 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 [delta] from the origin is u = a tan h ([delta]/R), and the
parallel to this through the origin is u = v sin h ([delta]/R). Hence
[Pi] ([delta]), the angle which this parallel makes with v = 0, is
given by
tan [Pi]([delta]) . sin h ([delta]/R) = 1, or tan 1/2[Pi]([delta]) =
e^(-[delta]/R)
which is Lobatchewsky's formula. We also obtain easily for the area of
a triangle the formula R^2([pi] - 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 [rho] and centre u0v0 is
(a^2 - uu0 - vv0)^2 = cos h^2 ([rho]/R)w0^2w^2 = C^2w^2 (say).
This equation remains real when [rho] is a pure imaginary, and remains
finite when w0 = 0, provided [rho] becomes infinite in such a way that
w0 cos h ([rho]/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
a^2 - 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 the _Teoria 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 the
_Saggio_)
ds^2 = R^2x^(-2)(dx^2 + dx1^2 + dx2^2 + ... + dx_n^2)
where x^2 + x1^2 + x2^2 + ... + x_n^2 = a^2.
He shows that geodesics are represented by linear equations between
x1, x2, ..., x_n, and that the geodesic distance [rho] between two
points x and x' is given by
[rho] a^2 - x1x'1 - x2x'2 - ... - x_n x'_n
cosh ----- = ---------------------------------------------------------------------------
R {(a^2 - x1^2 - x2^2 - ... - x_n^2)(a^2 - x'1^2 - x'2^2 - ... - x'_n^2)}^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 substitutions
x1 = r[lambda]1, x2 = r[lambda]2 ... x_n = r[lambda]_n, where
[Sigma][lambda]^2 = 1.
Hence
_________
ds^2 = (Radr/(a^2 - r^2)])^2 + R^2r^2d[Delta]^2/(a^2 - r^2).
where
d[Delta]^2 = [Sigma]d[lambda]^2.
Also calling [rho] the geodesic distance from the origin, we have
[rho] a [rho] r
cosh ----- = -----------------, sinh ----- = -----------------.
R [root](a^2 - r^2) R [root](a^2 - r^2)
Hence
ds^2 = d[rho]^2 + (R sin h ([rho]/R))^2d[Delta]^2.
Putting
z1 = [rho][lambda]1, z2 = [rho][lambda]2, ... z_n = [rho][lambda]_n,
we obtain
_ _
1 | / R [rho]\^2 |
ds^2 = [Sigma]dz^2 + ------ | ( ----- sinh ----- ) - 1| [Sigma](z_i dz_k - z_k dz_i)^2.
[rho]^2 |_ \[rho] R / _|
Hence when [rho] is small, we have approximately
1
ds^2 = [Sigma]dz^2 + ----[Sigma](z_i dz_k - z_k dz_i)^2 (1).
3R^2
Considering a surface element through the origin, we may choose our
axes so that, for this element,
z3 = Z4 = ... = z_n = 0.
Thus
1
ds^2 = dz1^2 + dz2^2 + ----(z1dz2 - z2dz1)^2 (2).
3R^2
Now the area of the triangle whose vertices are (0, 0), (z1, z2),
(dz1, dz2) is 1/2(z1, dz2 - z2dz1). Hence the quotient when the terms
of the fourth order in (2) are divided by the square of this triangle
is 4/3R^2; 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, ... z_n),
(dz1, dz2, dz3 ... dz_n). But -3/4 of this quotient is defined by
Riemann as the measure of curvature.[15] Hence the measure of
curvature is -1/R^2, i.e. is constant and negative. The properties of
parallels, triangles, &c., are as in the _Saggio_. It is also shown
that the analogues of limit surfaces have zero curvature; and that
spheres of radius [rho] have constant positive curvature 1/R^2 sinh^2
([rho]/R), so that spherical geometry may be regarded as contained in
the pseudo-spherical (as Beltrami calls Lobatchewsky's system).
Transition to the projective method.
The _Saggio_, 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, instead of referring to the pseudosphere, we merely
_define_ distance 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 [Sigma]_(xx) = 0, and if [Sigma]_(xx)' be
the polar form of [Sigma]_(xx), the distance [rho] between x and x' is
given by the projective formula
cos([rho]/k) = [Sigma]_xx'/{[Sigma]_(xx).[Sigma]_x'x'}^1/2.
That this formula is projective is rendered evident by observing that
e^(-2i[rho]/k) is 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 Laguerre[16] in
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 two kinds of elliptic space.
The projective method leads to a discrimination, first made by
Klein,[17] of 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 latter 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.[18] But, 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 them _exactly_ equal 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. Stackel (Leipzig, 1898). For John Bolyai's
_Appendix_, 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.
Gerard, _Sur la geometrie non-Euclidienne_ (thesis for doctorate)
(Paris, 1892); de Tilly, _Essai sur les principes fondamentales de la
geometrie et de la mecanique_ (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), and _Coll. 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 Satze uber die
Winkelsumme im Dreieck," Math. Ann. vol. 53 (1900), and "Uber den
Rauminhalt," _Math. Annal._ vol. 55 (1902).
For expositions of the whole subject, cf. F. Klein, _Nicht-Euklidische
Geometrie_ (Gottingen, 1893); R. Bonola, _La Geometria non-Euclidea_
(Bologna, 1906); P. Barbarin, _La Geometrie 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 piu dimensioni ed a piu specie_
_di unita rettilinee_ ... (Padua, 1891, German translation, Leipzig,
1894); G. Fontene, _L'Hyperespace a (n-1) dimensions_ (Paris, 1892);
and A.N. Whitehead, _loc. cit._ Cf. also E. Study, "Uber
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
Theories of space.
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
section 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 occupies _this_ part of space, and not _that_ part of space,
without reference to other bodies occupying space. According to the
relational theory of space, of which the chief exponent was
Leibnitz,[19] space 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 is _here_ and not _there_, 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 the _Principia_ (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.[20]
Kant'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
the _Critique_) 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."
Axioms.
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 given _a priori_. Both these views, thus
broadly stated, are capable of various subtle modifications, and a
discussion of them would merge into a general treatise on epistemology.
The cruder forms of the _a priori_ view 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
obliterate _a priori_ intuitions.
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
defined[23] as "the study of series of two or more dimensions." It has
also been defined[24] as "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 of _points_ and of _straight
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.[25] When 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 hypotheses[26] (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
will be presented[27] here 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, the
_plane_ ABC 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 [alpha], and a point A not incident in [alpha],
such that any point lies in some straight line which contains both A and
a point in [alpha].
_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 proved[28] that 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,[29] who 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 in _perspective_. 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
be _projectively_ related. Any property of a plane figure which
necessarily also belongs to any projectively related figure, is called
a _projective_ property.
The following theorem, known from its importance as "the fundamental
theorem of projective geometry," cannot be proved[30] from 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 equivalent[31] (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:[32] If 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
the _segment_ ABC 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. The _supplementary segment_ ABC 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, if
the 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 be _compact_, 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 property[33] as 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 is _continuous_ if 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 the _harmonic system_ ABC, where A, B, C are
collinear points, is defined[34] thus: 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 [oo].
This association[35] is most easily effected by considering a class of
projective relations of the line with itself, called by F. Schur (_loc.
cit._) _prospectivities_.
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 (OAU^2).
[Illustration: FIG. 69.]
The sum of two prospectivities, written (OAU^2) + (OBU^2), is defined
to be that transformation of the line l into itself which is obtained
by first applying the prospectivity (OAU^2) and then applying the
prospectivity (OBU^2). Such a transformation, when the two summands
have the same double point, is itself a prospectivity with that double
point.
[Illustration: FIG. 70]
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 (OEU^2) is
associated with unity, the prospectivity (OOU^2) is associated with
zero, and (OUU^2) with [infinity]. The prospectivities of the type
(OPU^2), 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'U^2) 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
(OPU^2).
[Illustration: FIG. 71.]
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 E1 and E2 are 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.
[Illustration: FIG. 72.]
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 P_r of the range P1 P2 P3 P4 be
[lambda]_r a + [mu]_r + a' [lambda]_r b + [mu]_r b'
-------------------------, ------------------------,
[lambda]_r + [mu]_r [lambda]_r + [mu]_r
[lambda]_r c + [mu]_r c'
------------------------, (r = 1, 2, 3, 4)
[lambda]_r + [mu]_r
and let ([lambda]_r [mu]_s) be written for [lambda]_r [mu]_s
-[lambda]_s [mu]_r. Then the cross ratio {P1 P2 P3 P4} is defined to
be the number ([lambda]1[mu]2) ([lambda]3[mu]4) / ([lambda]1[mu]4)
([lambda]3[mu]2). The equality of the cross ratios of the ranges (P1
P2 P3 P4) and (Q1 Q2 Q3 Q4) 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 geometry at 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.[36] An improved version is due to G. Peano.[37] Both these
authors treat the idea of the class of points constituting the segment
lying _between_ two 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's[38] formulation 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 considered[39] as 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 yields[40] the 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._--The _line_ AB (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 the _segment_ AB.
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.
[Illustration: FIG. 73.]
_Definition._--If A, B, C are three non-collinear points, the _plane_
ABC is the class of points which lie on any one of the lines joining
any two of the points belonging to the _boundary_ of the triangle ABC,
the boundary being formed by the segments BC, CA and AB. The
_interior_ of 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, the _surface_ being
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 [alpha] consider a line l and a point A (fig. 74).
[Illustration: FIG. 74.]
Let any point B divide l into two half-lines l1 and 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 [alpha] which does not intersect l. This is the Euclidean
case, and the assumption that this case holds is the _Euclidean
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 [alpha], which do not intersect l.
Then the lines through A in [alpha] are divided into two classes by
reference to l, namely, the _secant_ lines which intersect l, and the
_non-secant_ lines 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 of _ideal points_.[42] These are
also called _projective points_, where it is understood that the simple
points are the points of the original descriptive space. An _ideal
point_ is 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
termed _proper_, if the lines composing it intersect; otherwise it is
_improper_.
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 be _coherent_ with a plane, if any of the lines composing it
lie in the plane. An _ideal line_ is the class of ideal points each of
which is coherent with two given planes. If the planes intersect, the
ideal line is termed _proper_, otherwise it is _improper_. 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 called _proper_, otherwise it is _improper_. 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 given _quantity_." 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 of _congruence_, 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.[45] It has, however, been proved by Sophus Lie[46] that
congruence is capable of definition without recourse to a new
fundamental idea. This he does by means of his theory of finite
continuous groups (see GROUPS, 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 ([alpha]) must
be the region enclosed by a real closed quadric, or ([beta]) must be
the whole of the projective space with the exception of a single
plane. In case ([alpha]) the corresponding congruence group is the
continuous group for which the enclosing quadric is latent; and in
case ([beta]) 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 ([alpha]), or in
theorem 2, can be written in the form 1 +c(x^2 + y^2 + z^2) = 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 - [omega]3y + [omega]2z + cx(ux + vy + wz), \
dy/dt = v - [omega]1z + [omega]3x + cy(ux + vy + wz), > (A)
dz/dt = w - [omega]2x + [omega]1y + cz(ux + vy + wz). /
In the ease considered in theorem 1 ([beta]), with the proper choice
of co-ordinates the three equations defining the general infinitesimal
transformation are:
dx/dt = u - [omega]3y + [omega]2z, \
dy/dt = v - [omega]1z + [omega]3x, > (B)
dz/dt = w - [omega]2x + [omega]1y. /
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 x^2 + y^2 + z^2 = 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 the _absolute_ for that group, and
in the degenerate case of 1 ([beta]) the absolute is the latent plane
together with the latent imaginary conic. If the absolute is real, the
congruence-group is _hyperbolic_; if imaginary, it is _elliptic_; 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 A1 and A2 are 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, P2 are transformed into A'1, A'2, P'1,
P'2 by any transformation of the congruence-group, ([alpha])
{A1P(1}A2P2 = {A'1P'1A'2P'2}, since the transformation is projective,
and ([beta]) A'1, A'2 are on the absolute since A1 and A2 are on it.
Thus if we define the distance P1P2 to be 1/2k log {A1P1A2P2}, where
A1 and A2 are the points in which the line P1P2 cuts 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 definition[47] of 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 P1 and P2 are
in the region enclosed by the absolute, log {A1P1A2P2} is real, and
therefore k must be real. For an elliptic group A1 and A2 are
conjugate imaginaries, and log {A1P1A2P2} is a pure imaginary, and k
is chosen to be [kappa]/[iota], where [kappa] is real and [iota] =
[root]-.
Similarly the angle between two planes, p1 and p2, is defined to be
(1/2[iota]) log (t1p1t2p2), where t1 and t2 are tangent planes to the
absolute through the line p1p2. The planes t1 and t2 are imaginary for
an elliptic group, and also for an hyperbolic group when the planes p1
and p2 intersect at points within the region enclosed by the absolute.
The development of the consequences of these metrical definitions is
the subject of 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 P1 and P2 be 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}^1/2 is 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 t1 and t2 to be the tangent planes
through the line p1p2 to the imaginary conic. Similarly if p1 and p2
are intersecting lines, the same definition of an angle holds, where
t1 and t2 are now the lines from the point p1p2 to the two points
where the plane p1p2 cuts 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 deduced[48] the 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 doctrine[50]
that 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 whom _either_ all
congruence-groups are equally important, _or_ some 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 uber 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, _Elements de
geometrie_ (1794). An adequate account of the above authors is given
by P. Stackel and F. Engel, _Die Theorie der Parallellinien von Euklid
bis auf Gauss_ (Leipzig, 1895). The next period of time (roughly from
1800 to 1870) contains two streams of thought, both of which are
essential to the modern analysis of the subject. The first stream is
that which produced the discovery and investigation of non-Euclidean
geometries, the second stream is that which has produced the geometry
of position, comprising both projective and descriptive geometry not
very accurately discriminated. The leading authors on non-Euclidean
geometry are K.F. Gauss, in private letters to Schumacher, cf. Stackel
and Engel, _loc. cit._; N. Lobatchewsky, rector of the university of
Kazan, to whom the honour of the effective discovery of non-Euclidean
geometry must be assigned. His first publication was at Kazan in 1826.
His various memoirs have been re-edited by Engel; cf. _Urkunden zur
Geschichte der nichteuklidischen Geometrie_ by Stackel and Engel, vol.
i. "Lobatchewsky." J. Bolyai discovered non-Euclidean geometry
apparently in independence of Lobatchewsky. His memoir was published
in 1831 as an appendix to a work by his father W. Bolyai, _Tentamen
juventutem...._ This memoir has been separately edited by J.
Frischauf, _Absolute Geometrie nach J. Bolyai_ (Leipzig, 1872); B.
Riemann, _Uber die Hypothesen, welche der Geometrie zu Grunde liegen_
(1854); cf. _Gesamte Werke_, a translation in The Collected Papers of
W.K. Clifford. This is a fundamental memoir on the subject and must
rank with the work of Lobatchewsky. Riemann discovered elliptic
metrical geometry, and Lobatchewsky hyperbolic geometry. A full
account of Riemann's ideas, with the subsequent developments due to
Clifford, F. Klein and W. Killing, will be found in _The Boston
Colloquium for 1903_ (New York, 1905), article "Forms of Non-Euclidean
Space," by F.S. Woods. A. Cayley, _loc. cit._ (1859), and F. Klein,
"Uber die sogenannte nichteuklidische Geometrie," _Math. Annal._ vols.
iv. and vi. (1871 and 1872), between them elaborated the projective
theory of distance; H. Helmholtz, "Uber die thatsachlichen Grundlagen
der Geometrie" (1866), and "Uber die Thatsachen, die der Geometrie zu
Grunde liegen" (1868), both in his _Wissenschaftliche Abhandlungen_,
vol. ii., and S. Lie, _loc. cit._ (1890 and 1893), between them
elaborated the group theory of congruence.
The numberless works which have been written to suggest equivalent
alternatives to Euclid's parallel axioms may be neglected as being of
trivial importance, though many of them are marvels of geometric
ingenuity.
The second stream of thought confined itself within the circle of
ideas of Euclidean geometry. Its origin was mainly due to a succession
of great French mathematicians, for example, G. Monge, _Geometrie
descriptive_ (1800); J.V. Poncelet, _Traite des proprietes projectives
des figures_ (1822); M. Chasles, _Apercu historique sur l'origine et
le developpement des methodes en geometrie_ (Bruxelles, 1837), and
_Traite de geometrie superieure_ (Paris, 1852); and many others. But
the works which have been, and are still, of decisive influence on
thought as a store-house of ideas relevant to the foundations of
geometry are K.G.C. von Staudt's two works, _Geometrie der Lage_
(Nurnberg, 1847); and _Beitrage zur Geometrie der Lage_ (Nurnberg,
1856, 3rd ed. 1860).
The final period is characterized by the successful production of
exact systems of axioms, and by the final solution of problems which
have occupied mathematicians for two thousand years. The successful
analysis of the ideas involved in serial continuity is due to R.
Dedekind, _Stetigkeit und irrationale Zahlen_ (1872), and to G.
Cantor, _Grundlagen einer allgemeinen Mannigfaltigkeitslehre_
(Leipzig, 1883), and _Acta math._ vol. 2.
Complete systems of axioms have been stated by M. Pasch, _loc. cit._;
G. Peano, _loc. cit._; M. Pieri, _loc. cit._; B. Russell, _Principles
of Mathematics_; O. Veblen, _loc. cit._; and by G. Veronese in his
treatise, _Fondamenti di geometria_ (Padua, 1891; German transl. by A.
Schepp, _Grundzuge der Geometrie_, Leipzig, 1894). Most of the leading
memoirs on special questions involved have been cited in the text; in
addition there may be mentioned M. Pieri, "Nuovi principii di
geometria projettiva complessa," _Trans. Accad. R. d. Sci._ (Turin,
1905); E.H. Moore, "On the Projective Axioms of Geometry," _Trans.
Amer. Math. Soc._, 1902; O. Veblen and W.H. Bussey, "Finite Projective
Geometries," _Trans. Amer. Math. Soc._, 1905; A.B. Kempe, "On the
Relation between the Logical Theory of Classes and the Geometrical
Theory of Points," _Proc. Lond. Math. Soc._, 1890; J. Royce, "The
Relation of the Principles of Logic to the Foundations of Geometry,"
_Trans. of Amer. Math. Soc._, 1905; A. Schoenflies, "Uber die
Moglichkeit einer projectiven Geometrie bei transfiniter
(nichtarchimedischer) Massbestimmung," _Deutsch. M.-V. Jahresb._,
1906.
For general expositions of the bearings of the above investigations,
cf. Hon. Bertrand Russell, _loc. cit._; L. Couturat, _Les Principes
des mathematiques_ (Paris, 1905); H. Poincare, _loc. cit._; Russell
and Whitehead, _Principia mathematica_ (Cambridge, Univ. Press). The
philosophers whose views on space and geometric truth deserve especial
study are Descartes, Leibnitz, Hume, Kant and J.S. Mill. (A. N. W.)
FOOTNOTES:
[1] For Egyptian geometry see EGYPT, S _Science and Mathematics_.
[2] Cf. A.N. Whitehead, _Universal Algebra_, Bk. vi. (Cambridge,
1898).
[3] Cf. A.N. Whitehead, _loc. cit._
[4] Cf. A.N. Whitehead, "The Geodesic Geometry of Surfaces in
non-Euclidean Space," _Proc. Lond. Math. Soc._ vol. xxix.
[5] Cf. Klein, "Zur nicht-Euklidischen Geometrie," _Math. Annal._
vol. xxxvii.
[6] On the theory of parallels before Lobatchewsky, see Stackel und
Engel, _Theorie der Parallellinien von Euklid bis auf Gauss_
(Leipzig, 1895). The foregoing remarks are based upon the materials
collected in this work.
[7] See Stackel und Engel, _op. cit._, and "Gauss, die beiden Bolyai,
und die nicht-Euklidische Geometrie," _Math. Annalen_, Bd. xlix.;
also Engel's translation of Lobatchewsky (Leipzig, 1898), pp. 378 ff.
[8] Lobatchewsky's works on the subject are the following:--"On the
Foundations of Geometry," _Kazan Messenger_, 1829-1830; "New
Foundations of Geometry, with a complete Theory of Parallels,"
_Proceedings of the University of Kazan_, 1835 (both in Russian, but
translated into German by Engel, Leipzig, 1898); "Geometrie
imaginaire," Crelle's Journal, 1837; _Theorie der Parallellinien_
(Berlin, 1840; 2nd ed., 1887; translated by Halsted, Austin, Texas,
1891). His results appear to have been set forth in a paper (now
lost) which he read at Kazan in 1826.
[9] Translated by Halsted (Austin, Texas, 4th ed., 1896.)
[10] _Abhandlungen d. Konigl. Ges. d. Wiss. zu Gottingen_, Bd. xiii.;
_Ges. math. Werke_, pp. 254-269; translated by Clifford, _Collected
Mathematical Papers_.
[11] Cf. _Gesamm. math. und phys. Werke_, vol. i. (Leipzig, 1894).
[12] _Wiss. Abh._ vol. ii. pp. 610, 618 (1866, 1868).
[13] _Mind_, O.S., vols. i. and iii.; _Vortrage und Reden_, vol. ii.
pp. 1, 256.
[14] His papers are "Saggio di interpretazione della geometria
non-Euclidea," _Giornale di matematiche_, vol. vi. (1868); "Teoria
fondamentale degli spazii di curvatura costante," _Annali di
matematica_, vol. ii. (1868-1869). Both were translated into French
by J. Houel, _Annales scientifiques de l'Ecole Normale superieure_,
vol. vi. (1869).
[15] Beltrami shows also that this definition agrees with that of
Gauss.
[16] "Sur la theorie des foyers," _Nouv. Ann._ vol. xii.
[17] _Math. Annalen_, iv. vi., 1871-1872.
[18] For an investigation of these and similar properties, see
Whitehead, _Universal Algebra_ (Cambridge, 1898), bk. vi. ch. ii. The
polar form was independently discovered by Simon Newcomb in 1877.
[19] For an analysis of Leibnitz's ideas on space, cf. B. Russell,
_The Philosophy of Leibnitz_, chs. viii.-x.
[20] Cf. Hon. Bertrand Russell, "Is Position in Time and Space
Absolute or Relative?" _Mind_, n.s. vol. 10 (1901), and A.N.
Whitehead, "Mathematical Concepts of the Material World," _Phil.
Trans._ (1906), p. 205.
[21] Cf. _Critique of Pure Reason_, 1st section: "Of Space,"
conclusion A, Max Muller's translation.
[22] Cf. Ernst Mach, _Erkenntniss und Irrtum_ (Leipzig); the relevant
chapters are translated by T.J. McCormack, _Space and Geometry_
(London, 1906); also A. Meinong, _Uber die Stellung der
Gegenstandstheorie im System der Wissenschaften_ (Leipzig, 1907).
[23] Cf. Russell, _Principles of Mathematics_, S 352 (Cambridge,
1903).
[24] Cf. A.N. Whitehead, _The Axioms of Projective Geometry_, S 3
(Cambridge, 1906).
[25] Cf. Russell, _Princ. of Math._, ch. i.
[26] Cf. Russell, _loc. cit._, and G. Frege, "Uber die Grundlagen der
Geometrie," _Jahresber. der Deutsch. Math. Ver._ (1906).
[27] This formulation--though not in respect to number--is in all
essentials that of M. Pieri, cf. "I principii della Geometria di
Posizione," _Accad. R. di Torino_ (1898); also cf. Whitehead, _loc.
cit._
[28] Cf. G. Peano, "Sui fondamenti della Geometria," p. 73, _Rivista
di matematica_, vol. iv. (1894), and D. Hilbert, _Grundlagen der
Geometrie_ (Leipzig, 1899); and R.F. Moulton, "A Simple
non-Desarguesian Plane Geometry," _Trans. Amer. Math. Soc._, vol.
iii. (1902).
[29] Cf. "Sui postulati fondamentali della geometria projettiva,"
_Giorn. di matematica_, vol. xxx. (1891); also of Pieri, _loc. cit._,
and Whitehead, _loc. cit._
[30] Cf. Hilbert, _loc. cit._; for a fuller exposition of Hilbert's
proof cf. K.T. Vahlen, _Abstrakte Geometrie_ (Leipzig, 1905), also
Whitehead, _loc. cit._
[31] Cf. H. Wiener, _Jahresber. der Deutsch. Math. Ver._ vol. i.
(1890); and F. Schur, "Uber den Fundamentalsatz der projectiven
Geometrie," _Math. Ann._ vol. li. (1899).
[32] Cf. Hilbert, _loc. cit._, and Whitehead, _loc. cit._
[33] Cf. Dedekind, _Stetigkeit und irrationale Zahlen_ (1872).
[34] Cf. v. Staudt, _Geometrie der Lage_ (1847).
[35] Cf. Pasch, _Vorlesungen uber neuere Geometrie_ (Leipzig, 1882),
a classic work; also Fiedler, _Die darstellende Geometrie_ (1st ed.,
1871, 3rd ed., 1888); Clebsch, _Vorlesungen uber Geometrie_, vol.
iii.; Hilbert, _loc. cit._; F. Schur, _Math. Ann. Bd._ lv. (1902);
Vahlen, _loc. cit._; Whitehead, _loc. cit._
[36] Cf. _loc. cit._
[37] Cf. _I Principii di geometria_ (Turin, 1889) and "Sui fondamenti
della geometria," _Rivista di mat._ vol. iv. (1894).
[38] Cf. _loc. cit._
[39] Cf. Vailati, _Rivista di mat._ vol. iv. and Russell, _loc. cit._
S 376.
[40] Cf. O. Veblen, "On the Projective Axioms of Geometry," _Trans.
Amer. Math. Soc._ vol. iii. (1902).
[41] Cf. P. Stackel and F. Engel, _Die Theorie der Parallellinien von
Euklid bis auf Gauss_ (Leipzig, 1895).
[42] Cf. Pasch, _loc. cit._, and R. Bonola, "Sulla introduzione degli
enti improprii in geometria projettive," _Giorn. di mat._ vol.
xxxviii. (1900); and Whitehead, _Axioms of Descriptive Geometry_
(Cambridge, 1907).
[43] The original idea (confined to this particular case) of ideal
points is due to von Staudt (_loc. cit._).
[44] Cf. _Critique_, "Trans. Aesth." Sect. I.
[45] Cf. _loc. cit._
[46] Cf. _Uber die Grundlagen der Geometrie_ (Leipzig, Ber., 1890);
and _Theorie der Transformationsgruppen_ (Leipzig, 1893), vol. iii.
[47] Cf. A. Cayley, "A Sixth Memoir on Quantics," _Trans. Roy. Soc._,
1859, and _Coll. Papers_, vol. ii.; and F. Klein, _Math. Ann._ vol.
iv., 1871.
[48] Cf. _loc. cit._
[49] For similar deductions from a third set of axioms, suggested in
essence by Peano, Riv. mat. vol. iv. _loc. cit._ cf. Whitehead, _Desc.
Geom. loc. cit._
[50] Cf. H. Poincare, _La Science et l'hypothese_, ch. iii.