and others. Of cosingular complexes of higher degree nothing is known.

Following J. Plücker, we give an account of the lines of a quadratic complex that meet a given line.

The cones whose vertices are on the given line all pass through eight fixed points and envelop a surface of the fourth degree; the conics whose planes contain the given line all lie on a surface of the fourth class and touch eight fixed planes. It is easy to see by elementary geometry that these two surfaces are identical. Further, the given line contains four singular points A1, A2, A3, A4, and the planes into which their cones degenerate are the eight common tangent planes mentioned above; similarly, there are four singular planes, 41, 42, 43, 44, through the line, and the eight points into which their conics degenerate are the eight common points above. The locus of the pole of the line with respect to all the conics in planes through it is a straight line called the polar line of the given one; and through this line passes the polar plane of the given line with respect to each of the cones. The name polar is applied in the ordinary analytical sense; any line has an infinite number of polar complexes with respect to the given complex, for the equation of the latter can be written in an infinite number of ways; one of these polars is a straight line, and is the polar line already introduced. The surface on which lie all the conics through a line l is called the Plücker surface of that line: from the known properties of (2, 2) correspondences it can be shown that the Plücker surface of / cuts in a range of the same cross ratio as that of the range in which the Plücker surface of cuts 1. Applying this to the case in which is the polar of 1, we find that the cross ratios of (A1, A2, A3, A4) and (a, a, a, a) are equal. The identity of the locus of the A's with the envelope of the a's follows at once; moreover, a line meets the singular surface ip four points having the same cross ratio as that of the four tangent planes drawn through the line to touch the surface. The Plucker surface has eight nodes, eight singular tangent planes, and is a double line. The relation between a line and its polar line is not a reciprocal one with respect to the complex; but W. Stahl has pointed out that the relation is reciprocal as far as the singular surface is concerned.

To facilitate the discussion of the general quadratic complex we introduce Klein's canonical form. We have, in fact, to Quadratic deal with two quadratic equations in six variables; and by suitable linear transformations these can be reduced to the

complexes.

form

We have practically to study the intersection of two quadrics F and F in six variables, and to classify the different cases arising we make use of the results of Karl Weierstrass on the equivalence conditions of two pairs of quadratics. As far as at present required, they are as follows: Suppose that the factorized form of the deter minantal equation Disct (F+AF')=0 is

(^~a)'s +°2+*8 · · · (λ −8) 's +'?+'s+•••..... where the root a occurs si+s2+5.... times in the determinant, $+sa... times in every first minor, st... times in every second minor, and so on; the meaning of each exponent is then perfectly definite. Every factor of the type (-a) 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 [(sis... (...),...), 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 the simplest and best known of proper quadratic complexes. It is being Fresnel's wave surface. The tetrahedral or Reye complex is 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 com plex 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=o, 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 line integrals.

@1x12+a; x2+0x32+a«x«2+axs2+ax2=0 x2+ x;2+ x82+ x62+ xs2+ x2=0 subject to certain exceptions, which will be mentioned later. Taking the first equation to be that of the complex, we remark that both equations are unaltered by changing the sign of any coordinate; the geometrical meaning of this is, that the quadratic complex is its own reciprocal with respect to each of the six funda-generated by the lines which cut the faces of a tetrahedron in a mental complexes, for changing the sign of a coordinate is equivalent to taking the polar of a line with respect to the corresponding fundamental complex. It is easy to establish the existence of six systems of bitangent linear complexes, for the complex 4x+hx2+2x+4x+x+x=o is a bitangent when 42 1,2 142 4=0, and + + + and its lines of contact are conjugate lines with respect to the first fundamental complex. We therefore infer the existence of six systems of bitangent lines of the complex, of which the first is given by X,2 x42 хо2 = 0.

x=0,

+

x2

+

=0,

+ + + Each of these lines is a bitangent of the singular surface, which is therefore completely determined as being the focal surface of the (2, 2) congruence above. It is thence easy to verify that the two complexes Zaxo and Ebxo are cosingular if b1 = a,λ+u/a,v+p. The singular surface of the general quadratic complex is the famous quartic, with sixteen nodes and sixteen singular tangent planes, first discovered by E. E. Kümmer.

Congru ences..

As the coordinates of a line belonging to a congruence are functions We cannot give a full account of its properties here, but we deduce of two independent parameters, the theory of congruences is analogous at once from the above that its bitangents break up into six (2, 2) to that of surfaces, and we may regard it as a fundamental congruences, and the six linear complexes containing these are inquiry to find the simplest form of surface into which mutually in involution. The nodes of the singular surface are points a given congruence can be transformed. Most of those whose complex cones are coincident planes, and the complex conic whose properties have been extensively discussed can be represented in a singular tangent plane consists of two coincident points. This on a plane by a birational transformation. But in addition to the configuration of sixteen points and planes has many interesting difficulties of the theory of algebraic surfaces, a subject still in its properties; thus each plane contains six points which lie on a conic, infancy, the theory of congruences has other difficulties in that a while through each point there pass six planes which touch a quadric congruence is seldom completely represented, even by two equations. cone. In many respects the Kummer quartic plays a part in three A fundamental theorem is that the lines of a congruence are in dimensions analogous to the general quartic curve in two; it further general bitangents of a surface; in fact, since the condition of intergives a natural representation of certain relations between hyper-section of two consecutive straight lines is dλ+dmdu+dndv=0, 3 elliptic functions (cf. R. W. H. T. Hudson, Kümmer's Quartic, 1905). line of the congruence meets two adjacent lines, say and h. As might be expected from the magnitude of a form in six variables, Suppose 1, 4 lie in the plane pencil (A,a,) and I, in the plane pencil the number of projectivally distinct varieties of quadratic complexes (A), then the locus of the A's is the same as the envelope of the is very great; and in fact Adolf Weiler, by whom the a's, but a, is the tangent plane at A, and a, at A2. This surface is Classifica question was first systematically studied on lines indicated called the focal surface of the congruence, and to it all the lines tion of by Klein, enumerated no fewer than forty-nine different quadratic are bitangent. The distinctive property of the points A is that two types. But the principle of the classification is so im- of the congruence lines through them coincide, and in like manner complexes. portant, and withal so simple, that we give a brief sketch the planes a each contain two coincident lines. The focal surface which indicates its essential features. 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.

VI. NON-EUCLIDEAN GEOMETRY

The various metrical geometries are concerned with the In the discussion of congruences it soon becomes necessary to properties of the various types of congruence-groups, which are introduce another number, called the rank, which expresses the defined in the study of the axioms of geometry and of their number of plane pencils each of which contains an arbitrary line and two lines of the congruence. The order of the focal surface is immediate consequences. But this point of view of the subject 2m(n-1)-27, and its class is m(m-1)-27. Our knowledge of is the outcome of recent research, and historically the subject congruences is almost exclusively confined to those in which either has a different origin. Non-Euclidean geometry arose from the m or n does not exceed two. We give a brief account of those of discussion, extending from the Greek period to the present day, the second order without singular lines, those of order unity not being especially interesting. A congruence generally has singular of the various assumptions which are implicit in the traditional points through which an infinite number of lines pass; a singular Euclidean system of geometry. In the course of these investigapoint is said to be of order when the lines through it lie on a cone tions it became evident that metrical geometries, each internally of the rth degree. By means of formulae connecting the number of consistent but inconsistent in many respects with each other singular points and their orders with the class m of quadratic congruence Kümmer proved that the class cannot exceed seven. The and with the Euclidean system, could be developed. A short focal surface is of degree four and class 2m; this kind of quartic historical sketch will explain this origin of the subject, and surface has been extensively studied by Kümmer, Cayley, Rohn and describe the famous and interesting progress of thought on the others. The varieties (2, 2), (2, 3), (2, 4), (2, 5) all belong to at subject. But previously a description of the chief characteristic least one Reye complex; and so also does the most important class of (2, 6) congruences which includes all the above as special cases. properties of elliptic and of hyperbolic geometries will be given, The congruence (2, 2) belongs to a linear complex and forty different assuming the standpoint arrived at below under VII. Axioms Reye complexes; as above remarked, the singular surface is of Geometry. Kümmer'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 Kümmer'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 Kümmer, Sturm and others.

Ruled

Since a ruled surface contains only 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, surfaces. 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 inay 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.

For a

First assume the equation to the absolute (cf. loc. cit.) to be w2-x2-y2-22-0. The absolute is then real, and the geometry is hyberbolic.

The distance (d2) between the two points (x, y, z, w1) and (x2, Y's, 22, wa) is given by

cosh (dly)=(w1W2 − x1X2 - Y1Y2 – Z12)/ { (w)2 — x)2 —•yıa — 2,2)
(W22 - x23 — y1⁄23 - 222)}}

(1)
The only points to which the metrical geometry applies are those
within the region enclosed by the quadric; the other points are
1x+my+ni+r1w=0 and 2x+my+n,+r2w=0, is given by
improper ideal points." The angle (012) between two planes,
cos 012 = (lil2+m1m2+n1n2− r1#2}{{{h2+mi2+nj3 −12)

(2)

(3)

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
812, we have
cosh (812/7)=(lika+m ̧m2+n ̧n2 − 7172)/ {(l,2+m,2+n,2 − r‚2)
(122+mz2+ng2-r22)!
Thus in the case of the two planes one and only one of the two,
and 812, 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,
2=0, OY the line zo, x=o, and
OZ the line x=o, y=o. These are
the coordinate axes and are at
right angles to each other. Let
P be any point, and let o be the
distance OP, 0 the angle POZ, and
the angle between the planes
ZOX and ZOP. Then the co-
ordinates of P can be taken to be

and also

FIG. 67.

(4)

sinh (p/v) sine cos, sinh (p/y) sin ℗ sino, sinh (p/v) coso, cosh (p/Y). If ABC is a triangle, and the REFERENCES.-The earliest works of a general nature are Plücker, sides and angles are named accordNeue Geometrie des Raumes (Leipzig, 1868); and Kummer," Übering to the usual convention, we have die algebraischen Strahlensysteme," Berlin Academy (1866). Systemsinh (a/Y)/sin A=sinh (b/v)/sin B = sinh (c/v)/sin C, atic 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. cosh (a/y)=cosh (b/v) cosh (c/v)—sinh (b/y) sinh (c/v) cos A, (5) with quadratic congruences and complexes respectively. highly suggestive review by Gino Loria see Bulletin des sciences with two similar equations. The sum of the three angles of a triangle mathématiques (1893, 1897). A shorter treatise, giving a very is (-A-B-C). If the base BC of a triangle is kept fixed is always less than two right angles. The area of the triangle ABC interesting account of Klein's coordinates, is the work of Koenigs, and the vertex A moves in the fixed plane ABC so that the area La Géométrie réglée et ses applications (Paris, 1898). English treatises are C. M. Jessop, Treatise on the Line Complex (1903); R. W. H. T. ABC is constant, then the locus of A is a line of equal distance from Hudson, Kümmer's Quartic (1905). Many references to memoirs on BC. This locus is not a straight line. The whole theory of similarity line geometry will be found in Hagen, Synopsis der, höheren Mathe- is inapplicable; two triangles are either congruent, or their angles matik, ii. (Berlin, 1894); Loria, Il passato ed il presente delle principali determined when its three angles are are not equal two by two. Thus the elements of a triangle are teorie geometriche (Milan, 1897); a clear résumé of the principal di mathematiche superiori, ii. (Milan, 1900). Another treatise dealing off to infinity along BC, the lines BC results is contained in the very elegant volume of Pascal, Repertorio given. By keeping A and B and the line BC fixed, but by making C move extensively with line geometry is Lie, Geometrie der Berührungstransformationen (Leipzig, 1896). Many memoirs on the subject have and AC become parallel, and the sides a and b become infinite. Hence from 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 equation (5) above, it follows that two two memoirs which have left most impression on the subsequent parallel lines (cf. Section VII. Axioms of development of the subject are Klein," Zur Theorie der Linien-Geometry) must be considered as making a zero angle with each complexe des ersten und zweiten Grades," Math. Ann. ii.; and Lie, other. Also if B be a right angle, from the equation (5), remem "Uber Complexe, insbesondere Linien- und Kugelcomplexe," bering that, in the limit, Math. Ann. v. J. H. GR.)

FIG. 68.

cosh (a/r)/cosh (b/y) = cosh (a/v)/sinh (b/7) = 1,

to C

(6). The angle A is called by N. I. Lobatchewsky the "angle of parallel

ism.

"

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 and be any two conjugate lines with respect to the absolute (of which one of the two must be improper, say '), then any plane through and containing proper points is perpendicular to. 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 (x, y1, Si, Wi) and radius p. is

(w ̧2—x¡2—‚μ‚2—2 ̧3) (w2—x2 — y2—22) cosh2(p/v) =
(w,w-xx-y1y-212) (7).
The equation of the surface of equal distance () from the plane
Ix+my+ne+rw=o is

(P+m2+n2—r2) (w2—x2 — y2 —z2) sinh2(σ/v) =

A surface of equal distance is a sphere whose centre is improper; and both types of surface are included in the family

by the plane p, but P and R are not separated by p, nor are Q
Let A, B, C be any three non-collinear points, then four triangles
and R.
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
B.

elements:

a,

A,

C.

A.

-B,

-C.

(3) πγ-α,
(4) πγ-0, πγ -6,

b,

A,

B,

T-C.

-A,

-B,

C.

The formulae connecting the elements are

and

sin A/sin (4/7)=sin B/sin (b/y) = sin C/sin (c/v),

cos (a/r)= =cos (by) cos (c/y) +sin (b/y) sin (c/y) 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, When case I. holds there or (II.) according as this is not the case. (rw+lx+my+nz)2 (8). If all the figures considered lie is said to be a "principal triangle." within a sphere of radius ry 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 (A+B+C-T). 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

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-a-b2-co. 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

sin (ply) sin cos ø, sin (p/y) sin @ sin ø, sin (p/Y) cose, cos (ply). where p, 0 and 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 the direction 9 to 3. Then the line XOX being of finite length, the continually pushed along the plane along the line OX, that is, in 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 This is not a ruled surface. Hence in this geometry it is not possible another property of Euclidean parallel lines holds in elliptic geo metry, and by the use of it parallel lines are defined. For the equafor two straight lines to be at a constant distance from each other. Secondly, let the equation of the absolute be 2+12+22+tion of the surface (cylinder) of equal distance (8) from the line w2o. The absolute is now imaginary and the geometry is XOX is elliptic.

Thus there are two distances between the points, and if one is di2, the other is y-di. 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 Y. The angle between the two planes x+my+ni+wo and b2x+my+n2+r2w=0 is

cos 012= (lila+m1m2+n1na+r1ra)}/{{k,2+m,2+n,2+r,2)'

The polar plane with respect to the absolute of the point (x, y, z, w1)
is the real plane xx+y+is+wwo, and the pole of the plane
41x+my+nis+wo is the point (l, m, n, r). Thus (from
equations 10 and 11) it follows that the angle between the polar
planes of the points (x,...) and (x,...) is di/y, and that the
distance between the poles of the planes (...) and () is
YO. 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 I 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 ry. 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 PO cuts plies on the shortest
With this sense of "separation," it is
segment between P and Q.
possible to find three points P. Q, R such that P and Q are separated
1 Cf. A. N. Whitehead, Universal Algebra, Bk. vi. (Cambridge,

(x2+w2) tan 2(8/v) − (y2+22) = 0.

This is also the surface of equal distance, ay-8, 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
covered by W. K. Clifford. The two lines are called Clifford's right
lines of equal distance from a given line. This property was dis-
and left parallels to through the point. This property of parallel-
parallel to m. Note also that two parallel lines and m are not
ism is reciprocal, so that if m is a left parallel to 1, then is a left
not hold for Lobatchewsky's parallels in hyperbolic geometry, do
coplanar. Many of those properties of Euclidean parallels, which 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 of which the geodesic geometry is hyper-
bolic. 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
congruence transformations of the cylindrical surfaces of equal
It would obviously be possible to state "axioms" which these
distance into themselves, instead of the ∞ for the ordinary plane.
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.

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 Y, 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 in each case simply depends upon the magnitude of the arbitrary unit of length in comparison with the natural unit of length Cf. A. N. Whitehead, loc. cit.

Cf. A. N. Whitehead, "The Geodesic Geometry of Surfaces in non-Euclidean Space," Proc. Lond. Math. Soc. vol. xxix. Cf. Klein, "Zur nicht-Euklidischen Geometrie," Math. Annal. vol. xxxvii.

which each particular instance of either geometry presents. | equal. The first hypothesis is that these are both right angles; 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 y 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 formulac (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,