Foundation of Euclidean and Non-Euclidean Geometries according to F. Klein

401 Pages·1968·15.03 MB·English

by  Klein

FOUNDATION OF EUCLIDEAN
AND NON-EUCLIDEAN
GEOMETRIES
ACCORDING TO F. KLEIN
BY
L. RÉDEI
MATHEMATICAL INSTITUTE, JOZSEF ATTILA UNIVERSITY
SZEGED, HUNGARY
PERGAMON PRESS
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Copyright © 1968
AKADÉMIAI KIADO, BUDAPEST
First English edition 1968
Library of Congress Catalog Card No. 67-18486
3928/68
PREFACE
WHAT is our space like? How is it constructed? These are the questions of
crucial importance that geometers have been striving to answer throughout
the whole period of our civilization, and efforts made in this direction have
been almost continuous. It was a great achievement and to the benefit of
mankind when in the last century this question was settled, although there
were alternative answers, i.e. we have the choice of three possible geometri-
cal systems. The first of these was set up by the Greek Euclid, while of
the other two, one was constructed by the Hungarian J. Bolyai and
the Russian Ν. I. Lobachevski, and the other by the German B. Riemann:
the names of all these heroes of mathematics are written in flaming charac-
ters in the sky for ever. Many other outstanding mathematicians have rend-
ered great services by contributing to the logical clarification of these three
geometries, and nowadays we use the terminology of F. Klein and
call them the parabolic, hyperbolic and elliptic geometries, respectively.
We shall mention only a few of these men, such as D. Hilbert, M. Dehn,
M. Pasch, F. Schur, F. Klein, while disregarding many other great names.
There are various methods of laying the foundation of the three geome-
tries (new ones have even been appearing recently) but we do not want to
sum up all of them; rather, we confine ourselves in this respect to the few
references at the end of the book. Among these we especially draw attention
to the recently published book of F. Bachmann in which the development of
(plane) geometry is based on reflections.
The method followed by F. Klein in his lectures, which leads to the goal
through a projective extension of space, has not yet found a satisfactory
treatment in the literature. The first step in this direction was made in the
two volumes of Fr. Schilling's book — written with great enthusiasm —
in which, however, only plane geometry is dealt with, and even that in a
somewhat sketchy manner. It is the aim of our present book to remedy this
deficiency so that the ideas of F. Klein obtain the place they merit in the
literature of mathematics. The filling-in of gaps and the extension of the
considerations to space has required an unexpectedly great amount of work.
So in order not to make our book too long, we have confined ourselves
mainly to the foundation of geometry by developing the group of mo-
tions and the proof of consistency; this we look upon as complete, ac-
cording to the ideas laid down in the "Erlangen Programme" of F. Klein.
Nevertheless, at the end we have added three sections dealing with an
introduction to measurements of segments and angles (according to the
ix
χ PREFACE
principles of Cayley-Klein) as well as some notions of trigonometry.
For further study the reader may avail himself of the references at the end of
the book.
The development of projective geometry is covered in Chapters I to V,
in the course of which we have aimed at a restriction in the number of
ideas used; thus some simple concepts of projective geometry have
not been introduced, since we did not in fact need them.
Throughout the book, however, we do use set-theoretical ideas, as well
as the current notation of this theory. Acquaintance with some well-known
concepts of algebra, and analysis and complex numbers is presumed. Fami-
liarity with the methods of analytical geometry is also assumed.
To some extent we have deviated from the common terminology. For
example, we deal with axioms of "incidence" (containedness) instead of those
of "connection", since we define lines and planes as sets of points; further,
we speak of axioms of "betweenness" instead of those of "order", because
they are based on the statement "lies between", while the notion "ordering"
appears only later as a derived concept.
It has always been our aim to use the simplest possible methods for the
proofs of statements; we are not, however, convinced that we have succeeded
in all cases. Among other methods, we mention the notion of the "associated"
Desargues configuration, which was new to us and the use of which has
enabled us to obtain a significant shortening of many proofs. Another
innovation worth mentioning is that when dealing with space we have
introduced plane-coordinates first, and point-coordinates only in the further
development of the subject. The fundamental theorem of projective geo-
metry has first been proved for the plane, and then — very easily, of course —
for space, while the proof for lines (and more generally, for all basic projec-
tive configurations of the first degree) comes last.
L. Rédei
CHAPTER I
AXIOMS
§ 1. Axioms of Incidence
Let us agree that if we mention a number of elements, one, two, etc. (of a
set), these elements shall be looked upon as distinct, except when we explicitly
indicate the contrary or it is clear from the context. The statement "JC is
contained by y", where χ is an element or a subset of a set y (i.e. x £ y οτ
χ £j y), may also be expressed by saying: "x belongs to y9 or (x( lies in y"
etc,, or even — using a typically geometrical mode of expression — that
"y passes through x".
The empty set is denoted by 0. Unit-sets (i.e. sets consisting of one ele-
ment only) are identified by their element: this can be done without running
the risk of a misunderstanding.
The following axioms will be called "axioms of incidence":
AXIOM I Space—denoted by ΐϋ—is the set composed of all points.
V
AXIOM 1. A straight line is a subset of 9? consisting of at least two points.
2
AXIOM I . For any two points there exists exactly one straight line passing
3
through them.
AXIOM I . Any plane is a subset of ίϋ and contains three points which do
4
not all lie in any one straight line.
AXIOM I . Just one plane passes through three points which do not all lie
5
in any one straight line.
FIG. 1
AXIOM I . A plane contains any straight line two points of which belong to it.
e
1
2 AXIOMS
AXIOM I . Any two planes with a common point contain at least two common
7
points.
AXIOM I . There exist four points which do not all lie in any one plane.
8
Let us refer to these axioms as thç System of Axioms I. We shall often
represent the points, lines and planes (as well as some notions appearing
later) by simple figures which, however, will not form an organic part of the
treatment. Thus e.g. Axiom I may be represented by Fig. 1. For a "straight
e
line" we use the synonyms "line" or "axis", the latter only in certain con-
nections.
(Instead of "points on a line" we use "points in aline" since lines are sets
(of points).)
§ 2. Axioms of Betweenness
The following axioms will be called "axioms of betweenness" \
AXIOM The basic domain—denoted by W—is a subset of dt not con-
tained in any one plane.
Β
FIG. 2 FIG. 3
AXIOM II . If one of three points lies between the other two, then they are
2
all contained in 9ΐ' and they lie on a line.
AXIOM II . If a point A lies between points Β and C, then it lies also between
3
C and B.
AXIOM II . Just one of three points of W—contained by a line—lies be-
4
tween the other two.
AXIOM II . For any two points A and Β of 9ΐ', the point Β lies between A and
5
some further point.
AXIOM II . Let A, B, C be three points offfi not in a line and g a line not
6
passing through any of these points but lying together with it in a common
plane and passing through a point lying between A and B, then g passes
through a point lying between A and C, or between Β and C.
AXIOMS OF ΜΟΉΟΝ 3
Let us refer to these axioms as the System of Axioms IL We may illus-
trate these: e.g. Axioms II and II are represented by Figs. 2 and 3, respec-
2 e
tively. (On Fig. 2 we have also indicated W but on the subsequent figures we
9
will omit it.)
§ 3. Axiom of Continuity
The axiom of continuity reads as follows:
AXIOM III. Inside 9Î' there exist two points A and Β with the property that
if the (denumerably) infinite series of points P P,... lies between A and B
l9 2 9
where P lies between A and P, and every P(i ^ 2) lies between P and
± 2 t imml
P then there exists a point Η {lying between A and Β or identical with B)
i+l9
such that for every point X lying between A and Η at least one P lies between
t
X and Η (Figs. 4, 4a).
A P P P Χ Ρ Η Β Α Ρ, P X P B~H
1 2 3 { 3 {
FIG. 4 FIG. 4a
Instead of "Axiom III" we sometimes say "System of Axioms III": e.g.
when we mention "Systems of Axioms I, II, III" we mean the Systems of
Axioms I, II and Axiom III.
§ 4. Axioms of Motion
The axioms of motion are as follows:
AXIOM IV^ Any motion is a one-to-one mapping of?R onto itself such that
every three points of a line will be transformed into (three) points of a line.
AXIOM IV . The identical mapping of SR is a motion.
2
AXIOM IV. The product of two motions is a motion.
3
AXIOM IV. The inverse mapping of a motion is also a motion.
4
AXIOM IV. If we have two planes a, a', two lines g, g' and two points P
5 9
P' such that Ρ Ç g da and P' Ç g' CI a', then there exists a motion mapping
a onto a', g onto g' and Ρ onto P'.
AXIOM IV. There is a plane a, a line g and a point P, where P f^ Ca
6
such that there exist just four motions mapping cc g and Ρ onto themselves,
9
respectively, and not more than two of these motions may have every point of
g as a fixed point, while there is only ope of them (i.e. the identity) for which
every point of a is fixed.
4 AXIOMS
AXIOM IV . There exist three points Α, Β, Ρ offfi lying in a line g such that
7
Ρ is between A and Β and for every point C(^P) between A and Β there is
a point D lying between C and Ρ for which no motion with Ρ as fixed point
can be found that will map C onto a point lying between D and P.
AXIOM IV. There exists a line g such that g Π 9î'#0 having the following
8
property: If A, B, C are three points ofgCilfl' and Β lies between A and C, and
M is a motion such that AM = B> (AMZ =) BM = C, mapping the set γ of points
between A and Β onto the set of points lying between Β and C: then every point
of g is either one of the AM i or a point of one of the γΜ' (/=» 0, ± 1 ,...)·
(Here and also later, when we wish to denote the images of elements
or subsets of a set prodoced by mappings, the mappings will be written
as exponents.) We shall refer to these axioms as System of Axioms IV.
We can see that Axioms Γν can be linked to form the (combined)
2 34
statement that the motions form a group, called the group of motions of 9Î.
The following discussion has been arranged so that first of all we take only
the System of Axioms I as established, and then we shall add System II,
Axiom III and System IV in succession.
CHAPTER II
CONSEQUENCES OF THE SYSTEM OF AXIOMS I
HENCEFORTH the System of Axioms I will be presumed. In this chapter
we shall obtain some conclusions from it; at the same time we shall intro-
duce some ideas which will be used later on, all these being related to
space % even if this is not explicitly stated.
§ 5. Simple Properties of Straight Lines and Planes
A (geometric) configuration (figure) or, more exactly, a point-configura-
tion is a subset of SR. Accordingly, a set of lines or planes will be called a
configuration of lines or planes, respectively.
A set of configurations lying in a line or in a plane is called collinear or
coplanar, respectively. A set of lines or planes passing through a point may
be called concurrent, while planes having a line in common are called co-
axial.
We say that two point-configurations 21 and 93 intersect (in 31 Π S3) if
neither of them is contained by the other and 21 Π 93^0, which means that
21 and 93 must each contain at least two points and that neither 21 nor S3
is equal to 9Î. If 21 and 93 intersect and their intersection 21 Π 93 happens
to be a point or a Une, we call this the point of intersection or line of inter-
section of 2Ï and 93.
The line passing through A and B, which according to Axiom I is unique-
3
ly determined, will be denoted by AB (= ΒΑ). Similarly the plane deter-
mined uniquely by three non-collinear points (according to Axiom I ) has
5
the following notation: ABC (= ACB = BAC =...):
FIG. 5
THEOREM 1. Two intersecting lines intersect in one point.
This theorem states that two Unes a and b having a common point Ρ
(Fig. 5) cannot have any other common points. To prove this let us suppose
that we have another point Q (#P), where QÇ.a,b. Since P Q £ a and
9
5
6 CONSEQUENCES OF THE SYSTEM OF AXIOMS I
Λ Q £ b we have (according to Axiom I ): a = PQ as well as b = Ρβ, i.e.
3
a = b. This is a contradiction, and Theorem 1 is proved.
FIG. 6 FIG. 7
THEOREM 2. ^4 plane and a line intersect, if at all, in one point.
Let α be a plane and g a line not lying in it with a common point Ρ (Fig. 6).
We have to prove that α Π g = P. Unless this is true, we have a point
Q (ΦΡ) such that β Ç a, g. Then since P, β Ç we have ^ = Ρβ. Hence
since Ρ, β £ α it follows from Axiom I that # lies in a. This contradiction
e
proves the theorem.
THEOREM 3. Two intersecting planes intersect in one line.
To prove this let us consider two planes α, β with a common point P.
By Axiom I they have another common point Q (φΡ) (Fig. 7) which fur-
7
ther implies, by Axiom I , that PQCKx and PQ (Ζ β, i.e. α Π β Ξ> Ρβ. We
e
must now show that in this formula the equality holds. Unless this is so,
it follows that there exists a point R outside the Une Ρβ and lying in α Π β,
consequently also in α and β. Owing to Axiom I this would mean that both
5
α and β were equal to PQR. This contradiction proves Theorem 3.
THEOREM 4. A point and a line are coplanar and so are two intersecting
9
lines.
We can restrict ourselves to the proof of the second part of the statement.
FIG. 8
Let a and b be two intersecting lines with a common point Ρ (Fig. 8). We
take two further points A and Β in a and b, respectively, and consider the