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by David Hilbert 

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Title: The Foundations of Geometry 

Author: David Hilbert 

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Language: English 

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The 

Foundations of Geometry 

BY 
DAVID HILBERT, PH. D. 

PROFESSOR OF MATHEMATICS, UNIVERSITY OF GOTTINGEN 



AUTHORIZED TRANSLATION 

BY 
E. J. TOWNSEND, PH. D. 

UNIVERSITY OF ILLINOIS 



REPRINT EDITION 

THE OPEN COURT PUBLISHING COMPANY 

LA SALLE ILLINOIS 

1950 



TRANSLATION COPYRIGHTED 



The Open Court Publishing Co. 



PREFACE. 

The material contained in the following translation was given in substance by Professor Hilbert 
as a course of lectures on euclidean geometry at the University of Gottingen during the winter 
semester of 1898-1899. The results of his investigation were re-arranged and put into the form 
in which they appear here as a memorial address published in connection with the celebration at 
the unveiling of the Gauss- Weber monument at Gottingen, in June, 1899. In the French edition, 
which appeared soon after. Professor Hilbert made some additions, particularly in the concluding 
remarks, where he gave an account of the results of a recent investigation made by Dr. Dehn. 
These additions have been incorporated in the following translation. 

As a basis for the analysis of our intuition of space. Professor Hilbert commences his discussion 
by considering three systems of things which he calls points, straight lines, and planes, and sets 
up a system of axioms connecting these elements in their mutual relations. The purpose of his 
investigations is to discuss systematically the relations of these axioms to one another and also the 
bearing of each upon the logical development of euclidean geometry. Among the important results 
obtained, the following are worthy of special mention: 

1. The mutual independence and also the compatibility of the given system of axioms is fully 
discussed by the aid of various new systems of geometry which are introduced. 

2. The most important propositions of euclidean geometry are demonstrated in such a manner 
as to show precisely what axioms underlie and make possible the demonstration. 

3. The axioms of congruence are introduced and made the basis of the definition of geometric 
displacement. 

4. The significance of several of the most important axioms and theorems in the development 
of the euclidean geometry is clearly shown; for example, it is shown that the whole of the eu- 
clidean geometry may be developed without the use of the axiom of continuity; the significance of 
Desargues's theorem, as a condition that a given plane geometr}' may be regarded as a part of a 
geometry of space, is made apparent, etc. 

5. A variety of algebras of segments are introduced in accordance with the laws of arithmetic. 

This development and discussion of the foundation principles of geometry is not only of math- 
ematical but of pedagogical importance. Hoping that through an English edition these important 
results of Professor Hilbert's investigation may be made more accessible to English speaking stu- 
dents and teachers of geometry, 1 have undertaken, with his permission, this translation. In its 
preparation, 1 have had the assistance of many valuable suggestions from Professor Osgood of 
Harvard, Professor Moore of Chicago, and Professor Halsted of Texas. 1 am also under obligations 
to Mr. Henry Coar and Mr. Arthur Bell for reading the proof. 

e. j. townsend 
University of Illinois. 



CONTENTS 



PAGE 

Introduction 1 

CHAPTER I. 
THE FIVE GROUPS OF AXIOMS. 

§ 1. The elements of geometry and the five groups of axioms 2 

§ 2. Group I: Axioms of connection 2 

§ 3. Group II: Axioms of Order 3 

§ 4. Consequences of the axioms of connection and order 5 

§ 5. Group III: Axiom of Parallels {Euclid's axiom) 7 

§ 6. Group IV: Axioms of congruence 8 

§ 7. Consequences of the axioms of congruence 10 

§ 8. Group V: Axiom of Continuity (Archimedes's axiom) 15 

CHAPTER II. 
THE COMPATIBILITY AND MUTUAL INDEPENDENCE OF THE AXIOMS. 

§ 9. Compatibility of the axioms 17 

§10. Independence of the axioms of parallels. Non-euclidean geometry ... 19 

§11. Independence of the axioms of congruence 20 

§12. Independence of the axiom of continuity. Non-archimedean geometry 21 

CHAPTER HI. 
THE THEORY OF PROPORTION. 

§13. Complex number-systems 23 

§14. Demonstration of Pascal's theorem 24 

§15. An algebra of segments, based upon Pascal's theorem 29 

§16. Proportion and the theorems of similitude 32 

§ 17. Equations of straight lines and of planes 34 

CHAPTER IV. 

THE THEORY OF PLANE AREAS. 

§18. Equal area and equal content of polygons 37 

§19. Pai'allelo grams and triangles having equal bases and equal altitudes . 39 

§20. The measure of area of triangles and polygons 40 

§21. Equality of content and the measure of area 43 



CHAPTER V. 

DESARGUES'S THEOREM. 

§22. Desargues's theorem and its demonstration for plane geometry 

by aid of the axioms of congruence 46 

§23. The impossibihty of demonstrating Desargues's theorem for the 

plane without the help of the axioms of congruence 47 

§24. Introduction of an algebra of segments based upon Desargues's theorem 

and independent of the axioms of congruence 51 

§25. The commutative and the associative law of addition for our new 

algebra of segments 53 

§26. The associative law of multiplication and the two distributive laws 

for the new algebra of segments 54 

§27. Equation of the straight line, based upon the new algebra of segments ... 58 

§28. The totality of segments, regarded as a complex number system 61 

§29. Construction of a geometry of space by aid of a 

desarguesian number system 62 

§30. Significance of Desargues's theorem 64 

CHAPTER VI. 
PASCAL'S THEOREM. 

§31. Two theorems concerning the possibility of proving Pascal's theorem .... 65 
§32. The commutative law of multiplication for an 

archimedean number system 65 

§33. The commutative law of multiplication for a 

non- archimedean number system 67 

§34. Proof of the two propositions concerning Pascal's theorem. 

Non-pascalian geometry 69 

§35. The demonstration, by means of the theorems of Pascal and Desar'gues, 

of any theorem relating to points of intersection 69 

CHAPTER VII. 
GEOMETRICAL CONSTRUCTIONS BASED UPON THE AXIOMS I-V. 

§36. Geometrical constructions by means of a straight-edge and a 

transferer of segments 71 

§37. Analytical representation of the co-ordinates of points 

which can be so constructed 73 

§38. The representation of algebraic numbers and of integral rational functions 

as sums of squares 74 

§39. Criterion for the possibility of a geometrical construction by means of 

a straight-edge and a transferer of segments 77 

Conclusion 80 



"All human knowledge begins with intu- 
itions, thence passes to concepts and ends 
with ideas." 

Kant, Kritik der reinen Vernunft. 
Elementariehre, Part 2, Sec. 2. 



INTRODUCTION. 

Geometry, like arithmetic, requires for its logical development only a small number of 
simple, fundamental principles. These fundamental principles are called the axioms of 
geometry. The choice of the axioms and the investigation of their relations to one another 
is a problem which, since the time of Euclid, has been discussed in numerous excellent 
memoirs to be found in the mathematical literature.-^ This problem is tantamount to the 
logical analysis of our intuition of space. 

The following investigation is a new attempt to choose for geometry a simple and 
complete set of independent axioms and to deduce from these the most important geo- 
metrical theorems in such a manner as to bring out as clearly as possible the significance 
of the different groups of axioms and the scope of the conclusions to be derived from the 
individual axioms. 



Compare the comprehensive and explanatory report of G. Veronese, Grundziige der Geometric, Ger- 
man translation by A. Schepp, Leipzig, 1894 (Appendix). See also F. Klein, "Zur ersten Verteilung des 
Lobatschefskiy-Preises," Math. Ann., Vol. 50. 



THE FIVE GROUPS OF AXIOMS. 



§ 1. THE ELEMENTS OF GEOMETRY AND THE FIVE GROUPS OF AXIOMS. 

Let us consider three distinct systems of things. The things composing the first system, 
we wiU call points and designate them by the letters A, B, C,. . . : those of the second, we 
will call straight lines and designate them by the letters a, b, c,. . . ; and those of the third 
system, we will call planes and designate them by the Greek letters a, /?, 7,. . . The points 
are called the elements of linear geometry: the points and straight lines, the elements of 
plane geometry; and the points, lines, and planes, the elements of the geometry of space 
or the elements of space. 

We think of these points, straight lines, and planes as having certain mutual relations, 
which we indicate by means of such words as "are situated," "between," "parallel," "con- 
gruent," "continuous," etc. The complete and exact description of these relations follows 
as a consequence of the axioms of geometry. These axioms may be arranged in five gi'oups. 
Each of these groups expresses, by itself, certain related fundamental facts of our intuition. 
We will name these groups as follows: 

Axioms of connection. 

Axioms of order. 

Axiom oi parallels (Euclid's axiom). 

Axioms of congruence. 

Axiom of continuity (Archimedes's axiom). 



§ 2. GROUP I: AXIOMS OF CONNECTION. 

The axioms of this gi'oup establish a connection between the concepts indicated above; 
namely, points, straight lines, and planes. These axioms are as follows: 

I, 1. Two distinct points A and B always completely determine a straight line a. We 
write AB = a or BA = a. 

Instead of "determine," we may also employ other forms of expression; for example, 
we may say A "lies upon" a, A "is a point of" a, a "goes through" A "and through" B, 
a "joins" A "and" or "with" B, etc. If A lies upon a and at the same time upon another 
straight line 6, we make use also of the expression: "The straight lines" a "and" b "have 
the point A in common," etc. 

I, 2. Any two distinct points of a straight line completely determine that line; that is, if 
AB = a and AC = a, where B ^ C, then is also BC = a. 

I, 3. Three points A, B, C not situated in the same straight line always completely deter- 
mine a plane a. We write ABC = a. 



I, 


1-7. 


11, 


1-5. 


III. 




IV, 


1-6. 


V. 





We employ also the expressions: A, B, C, "lie in" a: A, B, C "are points of" a, etc. 

I, 4. Any three points A, B, C of a plane a, which do not lie in the same straight line, 
completely determine that plane. 

I, 5. If two points A, B of a straight line a lie in a plane a, then every point of a lies in 
a. 

In this case we say: "The straight line a lies in the plane a," etc. 

I, 6. If two planes a, (3 have a point A in common, then they have at least a second point 
B in common. 

I, 7. Upon every straight line there exist at least two points, in every plane at least three 
points not lying in the same straight line, and in space there exist at least four points 
not lying in a plane. 

Axioms I, 1-2 contain statements concerning points and straight lines only; that is, 
concerning the elements of plane geometry. We will call them, therefore, the plane axioms 
of group /, in order to distinguish them from the axioms I, 3-7, which we will designate 
briefly as the space axioms of this group. 

Of the theorems which follow from the axioms I, 3—7, we shall mention only the fol- 
lowing: 

Theorem 1. Two straight lines of a plane have either one point or no point in 
common; two planes have no point in common or a straight line in common; a plane 
and a straight line not lying in it have no point or one point in common. 

Theorem 2. Through a straight line and a point not lying in it, or through two 
distinct straight lines having a common point, one and only one plane may be made 
to pass. 



§ 3. GROUP II: AXIOMS OF ORDER.^ 

The axioms of this group define the idea expressed by the word "between," and make 
possible, upon the basis of this idea, an order of sequence of the points upon a straight 
line, in a plane, and in space. The points of a straight line have a certain relation to one 
another which the word "between" serves to describe. The axioms of this group are as 
follows: 

II, 1. If A, B, C are points of a straight line and B lies between A and C, then B lies 
also between C and A. 



"These axioms were first studied in detail by M. Pasch in his Vorlesungen iiber neuere Geometrie, 
Leipsic, 1882. Axiom II, 5 is in particulEir due to him. 



Fig. 1. 



II, 2. If A and C are two points of a straight line, then there exists at least one point B 
lying between A and C and at least one point D so situated that C lies between A 
and D. 



Fig. 2. 



II, 3. Of any three points situated on a straight line, there is always one and only one 
which lies between the other two. 

II, 4. Any four points A, B, C, D of a straight line can always be so arranged that B 
shall lie between A and C and also between A and D, and. furthermore, that C shall 
lie between A and D and also between B and D. 

Definition. We wall call the system of two points A and S, lying upon a straight 
line, a segment and denote it by AB or BA. The points lying between A and B are called 
the points of the segment AB or the points lying within the segment AB. All other points 
of the straight line are referred to as the points lying outside the segment AB. The points 
A and B are called the extremities of the segment AB. 




Fig. .3. 



II, 5. Let A, B. C he three points not lying in the same straight line and let a he a 
straight line lying in the plane ABC and not passing through any of the points A, 
B, C. Then, if the straight line a passes through a point of the segment AB, it will 
also pass through either a point of the segment BC or a point of the segment AC. 
Axioms II, 1-4 contain statements concerning the points of a straight line only, and, 
hence, we will call them the linear axioms of group II. Axiom II, 5 relates to the 
elements of plane geometry and, consequently, shall be called the plane axiom of 
group II. 



§ 4. CONSEQUENCES OF THE AXIOMS OF CONNECTION AND ORDER. 

By the aid of the four linear axioms II, 1-4, we can easily deduce the following theorems: 

Theorem 3. Between any two points of a straight line, there always exists an 
unlimited number of points. 

Theorem 4. If we have given any finite number of points situated upon a straight 
Hne, we can always arrange them in a sequence A., B, C, D, E,. . ., K so that B shall 
lie between A and C, D, E,. . ., K; C between A, B and D, E,. . . , K: D between 
A, B, C and E,. . . K, etc. Aside from this order of sequence, there exists but one 
other possessing this property namely, the reverse order K,. . . , E, D, C, B, A. 

A B C D E K 

1 1 i 1 1 1 — 



Fig. 4. 



Theorem 5. Every straight line a, which lies in a plane a, divides the remaining 
points of this plane into two regions having the following properties: Every point A 
of the one region determines with each point B of the other region a segment AB 
containing a point of the straight line a. On the other hand, any two points A, A' 
of the same region determine a segment AA' containing no point of a. 




6 

If A, A', O, B are four points of a straight line a, where O hes between A and B but 
not between A and A' , then we may say: The points A, A' are situated on the line a upon 
one and the same side of the point O, and the points A, B are situated on the straight 
line a upon different sides of the point O. 



Fig. 



All of the points of a which lie upon the same side of O, when taken together, are 
called the half-ray emanating from O. Hence, each point of a straight line divides it into 
two half-rays. 

Making use of the notation of theorem 5, we say: The points A, A' lie in the plane a 
upon one and the same side of the straight line a, and the points A, B lie in the plane a 
upon different sides of the straight line a. 

Definitions. A system of segments AB, BC, CD, . . . , KL is called a broken line 
joining A with L and is designated, briefly, as the broken line ABCDE ...KL. The 
points lying within the segments AB, BC, CD, . . . , KL, as also the points A, B, C, D, 
. . . , K, L, are called the points of the broken line. In particular, if the point A coincides 
with L, the broken line is called a polygon and is designated as the polygon ABCD . . . K. 
The segments AB, BC, CD, . . . , KA are called the sides of the polygon and the points 
A, B, C , D, . . . , K the vertices. Polygons having 3, 4, 5, . . . , n vertices are called, 
respectively, triangles, guadrangles, pentagons, ..., n-gons. If the vertices of a polygon 
are all distinct and none of them lie within the segments composing the sides of the 
polygon, and, furthermore, if no two sides have a point in common, then the polygon is 
called a simple polygon. 

With the aid of theorein 5, we may now obtain, without serious difficulty, the following 
theoreins: 

Theorem 6. Every simple polygon, whose vertices all lie in a plane a, divides 
the points of this plane, not belonging to the broken line constituting the sides 
of the polygon, into two regions, an interior and an exterior, having the following 
properties: If _A is a point of the interior region (interior point) and B a point of 
the exterior region (exterior point), then any broken line joining A and B must have 
at least one point in common with the polygon. If, on the other hand. A, A' are 
two points of the interior and B, B' two points of the exterior region, then there are 
always broken lines to be found joining A with A' and B with B' without having a 
point in common with the polygon. There exist straight lines in the plane a which 
lie entirely outside of the given polygon, but there are none which lie entirely within 
it. 

Theorem 7. Every plane a divides the remaining points of space into two regions 
having the following properties: Every point A of the one region determines with 
each point B of the other region a segment AH, within which lies a point of q. 
On the other hand, any two points A, A' lying within the same region determine a 
segment A A' containing no point of a. 




Fis. 7. 



Making use of the notation of theorem 7, we may now say: The points A, A' are situated 
in space upon one and the same side of the plane a, and the points A, B are situated in 
space upon different sides of the plane a. 

Theorem 7 gives us the most important facts relating to the order of sequence of the 
elements of space. These facts are the results, exclusively, of the axioms already considered, 
and, hence, no new space axioms are required in group II. 



5. GROUP III: AXIOM OF PARALLELS. (EUCLID'S AXIOM.) 



The introduction of this axiom simplifies greatly the fundamental principles of geometry 
and facilitates in no small degree its development. This axiom may be expressed as follows: 

III. In a plane a there can he drawn through any point A, lying outside of a straight 
line a, one and only one straight line which does not intersect the line a. This 
straight line is called the parallel to a through the given point A. 

This statement of the axiom of parallels contains two assertions. The first of these 
is that, in the plane a, there is always a straight line passing through A which does not 
intersect the given line a. The second states that only one such line is possible. The latter 
of these statements is the essential one, and it may also be expressed as follows: 

Theorem 8. If two straight lines a, 6 of a plane do not meet a third straight line c 
of the same plane, then they do not meet each other. 

For, if a, h had a point A in common, there would then exist in the same plane with c 
two straight lines a and b each passing through the point A and not meeting the straight 
line c. This condition of affairs is, however, contradictory to the second assertion contained 



in the axiom of parallels as originally stated. Conversely, the second part of the axiom of 
pai'allels, in its original form, follows as a consequence of theorem 8. 
The axiom of parallels is a plane axiom. 



§ 6. GROUP IV. AXIOMS OF CONGRUENCE. 

The axioms of this group define the idea of congruence or displacement. 

Segments stand in a certain relation to one another which is described by the word 
^''congruent." 

IV, I. // A, B are two points on a straight line a, and if A' is a point upon the 
same or another straight line a' , then, upon a given side of A' on the straight line 
a' , we can always find one and only one point B' so that the segment AB (or BA) 
is congruent to the segment A'B' . We indicate this relation by writing 

AB = A'B'. 

Every segment is congruent to itself; that is, we always have 

AB = AB. 

We can state the above axiom briefly by saying that every segment can be laid off 
upon a given side of a given point of a given straight line in one and and only one way. 

IV, 2. // a segment AB is congruent to the segment A'B' and also to the segment 
A" B" , then the segment A'B' is congruent to the segment A"B" : that is. if AB = 
A'B and AB = A"B", then A'B' = A"B". 

IV, 3. Let AB and BC he two segments of a straight line a which have no points 
in common aside from the point B, and, furthermore, let A'B' and B'C he two 
segments of the same or of another straight line a' having, likewise, no point other 
than B' in common. Then, if AB = A'B' and BC = B'C, we have AC = A'C. 



Fig. 8. 



Definitions. Let a be any arbitrary plane and h, k any two distinct half-rays lying 
in a and emanating from the point O so as to form a part of two different straight lines. 
We call the system formed by these two half-rays h, k an angle and represent it by the 
symbol Z(/i, k) or /^{k, h). From axioms II, 1—5, it follows readily that the half-rays h and 



9 

k, taken together with the point O, divide the remaining points of the plane a into two 
regions having the following property: If yl is a point of one region and B a point of the 
other, then every broken line joining A and B either passes through O or has a point in 
common with one of the half-rays h, k. If, however, A, A' both lie within the same region, 
then it is always possible to join these two points by a broken line which neither passes 
through O nor has a point in common with either of the half-rays h, k. One of these two 
regions is distinguished from the other in that the segment joining any two points of this 
region lies entirely within the region. The region so chai'acterised is called the interior of 
the angle (/i, k). To distingTiish the other region from this, we call it the exterior of the 
angle [h, k). The half rays h and k are called the sides of the angle, and the point O is 
called the vertex of the angle. 

IV, 4. Let an angle (h, k) be given in the plane a and let a straight line a' be given in 
a plane a' . Suppose also that, in the plane a. a definite side of the straight line a' 
be assigned. Denote by h' a half-ray of the straight line a' emanating from a point 
O' of this line. Then in the plane a' there is one and only one half-ray k' such that 
the angle {h,k). or {k,h), is congruent to the angle {h',k') and at the same time 
all interior points of the angle (h', k') lie upon the given side of a' . We express this 
relation by means of the notation 

Z{h,k) = Z{h',k') 

Every angle is congruent to itself: that is, 

Z{h,k) = Z{h,k) 

or 

Z{h,k) = Z{k,h) 

We say, briefly, that every angle in a given plane can be laid off upon a given side of 
a given half-ray in one and only one way. 

IV, 5. // the angle (h, k) is congruent to the angle (h' , k') and to the angle (/i", k"), then 
the angle (h',k') is congruent to the angle {h",k"); that is to say, if /l[h,k) = 
Z(h', k') and Z{h, k) = Z(h", k"), then Z(h', k') = Z{h", k"). 

Suppose we have given a triangle ABC . Denote by h, k the two half-rays emanating 
from A and passing respectively through B and C . The angle {/i, fc) is then said to be the 
angle included by the sides AB and AC , or the one opposite to the side BC in the triangle 
ABC . It contains all of the interior points of the triangle ABC and is represented by the 
symbol ABAC, or by AA. 

IV, 6. //. in the two triangles ABC and A'B'C the congruences 

AB = A'B', AC = A'C, ABAC = ZB'A'C 

hold, then the congruences 

ZABC = AA'B'C and ZACB = ZA'C'B' 

also hold. 



10 

Axioms IV, 1-3 contain statements concerning the congruence of segments of a straight 
line only. They may, therefore, be called the linear axioms of group IV. Axioms IV, 4, 5 
contain statements relating to the congruence of angles. Axiom IV, 6 gives the connection 
between the congruence of segments and the congruence of angles. Axioms IV, 4-6 contain 
statements regarding the elements of plane geometry and may be called the plane axioms 
of group IV. 



§ 7. CONSEQUENCES OF THE AXIOMS OF CONGRUENCE. 

Suppose the segment AB is congruent to the segment A'B' . Since, according to axiom 
IV, 1, the segment AB is congruent to itself, it follows from axiom IV, 2 that A'B' is 
congruent to AB\ that is to say, if AB = A'B', then A'B' = AB. We say, then, that the 
two segments are congruent to one another. 

Let A, B, C, D, . . . , K, L and A' , B' , C , D', . . . , K', V be two series of points on the 
straight lines a and a', respectively, so that all the corresponding segments AB and A'B', 
AC and A'C , BC and B'C' , . . . , KL and K' L' are respectively congruent, then the two 
series of points are said to he congruent to one another. A and A' , B and B' , . . . , L and 
L' are called corresponding points of the two congruent series of points. 

From the linear axioms IV, 1-3, we can easily deduce the follo"wing theorems: 

Theorem 9. If the first of two congTuent series of points A, B, C, D, . . . , K, L and 
A', B', C , D', . . . , K', L' is so arranged that B lies between A and C, D, . . . , K, L, 
and C between A, B and D, . . . , K, L, etc., then the points A' , B' , C , D' , . . . , K' , 
V of the second series are arranged in a similar way; that is to say, B' lies between 
A! and C, -D', . . . , K' , L', and C lies between A', B' and D',..., K', L', etc. 

Let the angle (h, k) be congruent to the angle [h' , k'). Since, according to axiom IV, 4, 
the angle [h, k) is congruent to itself, it follows from axiom IV, 5 that the angle (/i', k') is 
congruent to the angle (/?,, k). We say, then, that the angles [h, k) and {/?/, k') are congruent 
to one another. 

Definitions. Two angles having the same vertex and one side in common, while 
the sides not common form a straight line, are called supplementary angles. Two angles 
having a common vertex and whose sides form straight lines are called vertical angles. An 
angle which is congruent to its supplementary angle is called a right angle. 

Two triangles ABC and A'B'C are said to be congruent to one another when all of 
the following congruences are fulfilled: 

AB = A'B', AC = A'C, BC = B'C, 
LA = LA!, LB = LB', LC = LC . 

Theorem 10. {First theorem of congruence for triangles). If, for the two triangles 
ABC and A'B'C, the congruences 

AB = A'B', AC = A'C', LA = LA' 

hold, then the two triangles are congruent to each other. 



11 



Proof. From axiom IV, 6, it follows that the two congruences 

ZB = ZB' and ZC= ZC 

are fulfilled, and it is, therefore, sufficient to show that the two sides EC and B'C' are 
congruent. We will assume the contrary to be true, namely, that BC and B'C are not 
congruent, and show that this leads to a contradiction. We take upon B'C a point D' 
such that BC = B'D'. The two triangles ABC and A'B'D' have, then, two sides and 
the included angle of the one agreeing, respectively, to two sides and the included angle 
of the other. It follows from axiom IV, 6 that the two angles BAC and B'A'D' are also 
congruent to each other. Consequently, by aid of axiom IV, 5, the two angles B'A'C and 
B'A'D' must be congruent. 




B A'- 

Fig. 9. 




This, however, is impossible, since, by axiom IV, 4, an angle can be laid off in one and 
only one way on a given side of a given half-ray of a plane. From this contradiction the 
theorem follows. 

We can also easily demonstrate the following theorem: 

Theorem 1 1 . (Second theorem of congruence for triangles) . If in any two triangles 
one side and the two adjacent angles are respectively congruent, the triangles are 
congruent. 

We are now in a position to demonstrate the following important proposition. 

Theorem 12. If two angles ABC and A'B'C are congruent to each other, their 
supplementary angles CBD and C'B'D' are also congruent. 




Fig. 10. 



Proof. Take the points A', C, D' upon the sides passing through B' in such a way 



that 



A'B' = AB, C'B' = CB, D'B' = DB. 



Then, in the two triangles ABC and A'B'C, the sides AB and BC are respectively 
congruent to A'B' and C'B'. Moreover, since the angles included by these sides are 



12 

congruent to each other by hypothesis, it follows from theorem 10 that these triangles are 
congruent: that is to say, we have the congruences 

AC = A'C, ABAC = ZB'A'C 

On the other hand, since by axiom IV, 3 the segments AD and A'D' are congruent to each 
other, it follows again from theorem 10 that the triangles CAD and C'A'D' are congruent, 
and, consequently, we have the congruences: 

CD = CD', ZADC = ZA'D'C. 

From these congruences and the consideration of the triangles BCD and B'C'D', it follows 
by virtue of axiom IV, 6 that the angles CBD and C'B'D' are congruent. 

As an immediate consequence of theorem 12, we have a similar theorem concerning 
the congruence of vertical angles. 

Theorem 13. Let the angle (/i, k) of the plane a be congTuent to the angle (/?/, A;') 
of the plane a', and, furthermore, let / be a half-ray in the plane a emanating from 
the vertex of the angle [h, k) and lying within this angle. Then, there always exists 
in the plane a' a half-ray I' emanating from the vertex of the angle [h' , k') and lying 
within this angle so that we have 

Z{hJ) = Z{h',l'), Z{k,l) = Z{k',l'). 




Proof. We will represent the vertices of the angles (/i, k) and [h', k') by O and O', 
respectively, and so select upon the sides h, k, h' , k' the points A, B, A', B' that the 
congruences 

OA = O'A', OB = O'B' 

are fulfilled. Because of the congruence of the triangles OAB and O'A'B', we have at once 

AB = A'B', Z.OAB = O'A'B', ZOBA = ZO'B'A'. 

Let the straight line AB intersect / in C. Take the point C upon the segment A'B' so 
that A'C = AC. Then, O'C is the required half-ray. In fact, it follows directly from 
these congruences, by aid of axiom IV, 3, that BC = B'C . Furthermore, the triangles 
OAC and O' A'C are congruent to each other, and the same is true also of the triangles 
OCB and O'B'C . With this our proposition is demonstrated. 
In a similar manner, we obtain the following proposition. 



13 



Theorem 14. Let h, k, I and h', k' , I' be two sets of three half-rays, where those 
of each set emanate from the same point and he in the same plane. Then, if the 
congruences 

ZihJ)^Z{h'j'), Z{k,l)^Zik'j') 

are fulfilled, the following congruence is also valid; viz.: 

Z{h,k) = Z{h',k'). 

By aid of theorems 12 and 13, it is possible to deduce the following simple theorem, 
which Euclid held-although it seems to me wrongly-to be an axiom. 

Theorem 15. All right angles are congruent to one another. 

Proof. Let the angle BAD be congruent to its supplementary angle CAD, and, 
likewise, let the angle B'A'D' be congruent to its supplementary angle C'A'D'. Hence the 
angles BAD, CAD, B'A'D', and C'A'D' are all right angles. We will assume that the 
contrary of our proposition is true, namely, that the right angle B'A'D' is not congruent 
to the right angle BAD, and will show that this assumption leads to a contradiction. We 
lay off the angle B'A'D' upon the half-ray AB in such a manner that the side AD" arising 
from this operation falls either within the angle BAD or within the angle CAD. Suppose, 
for example, the first of these possibilities to be true. Because of the congruence of the 
angles B'A'D' and BAD" , it follows from theorem 12 that angle C'A'D' is congruent to 
angle CAD" , and, as the angles B'A'D' and C'A'D' are congTuent to each other, then, by 
IV, 5, the angle BAD" must he congruent to CAD". 



,-i D 




A' 



Fig. 12. 



Furthermore, since the angle BAD is congruent to the angle CAD, it is possible, by 
theorem 13, to find within the angle CAD a half-ray AD'" emanating from A, so that 
the angle BAD" will be congruent to the angle CAD'" , and also the angle DAD" will be 
congruent to the angle DAD'" . The angle BAD" was shown to be congruent to the angle 
CAD" and, hence, by axiom IV, 5, the angle CAD", is congruent to the angle CAD"'. 
This, however, is not possible; for, according to axiom IV, 4, an angle can be laid off in 
a plane upon a given side of a given half-ray in only one way. With this our proposition 
is demonstrated. We can now introduce, in accordance with common usage, the terms 
^''acute angle''' and ^''obtuse angle.'''' 



14 

The theorem relating to the congruence of the base angles A and B of an equilateral 
triangle ABC follows immediately by the application of axiom IV, 6 to the triangles ABC 
and BAC. By aid of this theorem, in addition to theorem 14, we can easily demonstrate 
the following proposition. 

Theorem 16. (Third theorem of congruence for triangles.) If two triangles have 
the three sides of one congruent respectively to the corresponding three sides of the 
other, the triangles are congruent. 

Any finite number of points is called a figure. If all of the points lie in a plane, the 
figure is called a plane figure. 

Two figures are said to be congruent if their points can be arranged in a one-to-one 
correspondence so that the corresponding segments and the corresponding angles of the 
two figures are in every case congruent to each other. 

Congruent figures have, as may be seen from theorems 9 and 12, the following proper- 
ties: Three points of a figure lying in a straight line are likewise in a straight line in every 
figure congruent to it. In congruent figTires, the arrangement of the points in correspond- 
ing planes with respect to corresponding lines is always the same. The same is true of the 
sequence of corresponding points situated on corresponding lines. 

The most general theorems relating to congruences in a plane and in space may be 
expressed as follows: 

Theorem 17. If {A, B,C,. . .) and (A', B', C, . . .) are congruent plane figures and 
P is a point in the plane of the first, then it is always possible to find a point P in 
the plane of the second figure so that {A, B,C, . . . , P) and {A', B', C, . . . , P') shall 
likewise be congruent figures. If the two figures have at least three points not lying 
in a straight line, then the selection of P' can be made in only one way. 

Theorem 18. If (A,B,C,...) and (A',B',C', . . . = are congruent figures and P 
represents any arbitrary point, then there can always be found a point P' so that 
the two figures (A, B,C, . . . , P) and [A', B' , C , . . . , P') shall likewise be congruent. 
If the figure [A, B,C, . . . , P) contains at least four points not lying in the same plane, 
then the determination of P' can be made in but one way. 

This theorem contains an important result; namely, that all the facts concerning space 
which have reference to congruence, that is to say, to displacements in space, are (by the 
addition of the axioms of groups I and II) exclusively the consequences of the six linear 
and plane axioms mentioned above. Hence, it is not necessary to assume the axiom of 
parallels in order to establish these facts. 

If we take, in, addition to the axioms of congruence, the axiom of parallels, we can 
then easily establish the following propositions: 

Theorem 19. If two parallel lines are cut by a third straight line, the alternate- 
interior angles and also the exterior-interior angles are congruent Conversely, if the 
alternate- interior or the exterior- interior angles are congruent, the given lines are 
parallel. 



15 

Theorem 20. The sum of the angles of a triangle is two right angles. 

Definitions. If M is an arbitrary point in the plane a, the totality of all points A, 
for which the segments MA are congruent to one another, is called a circle. M is called 
the centre of the circle. 

From this definition can he easily deduced, with the help of the axioms of groups III 
and IV, the known properties of the circle; in particular, the possibility of constructing a 
circle through any three points not lying in a straight line, as also the congruence of all 
angles inscribed in the same segment of a circle, and the theorem relating to the angles of 
an inscribed quadrilateral. 



§ 8. GROUP V. AXIOM OF CONTINUITY. (ARCHIMEDEAN AXIOM.) 

This axiom makes possible the introduction into geometry of the idea of continuity. In 
order to state this axiom, we must first establish a convention concerning the equality 
of two segments. For this purpose, we can either base our idea of equality upon the 
axioms relating to the congruence of segments and define as '''' equaV the correspondingly 
congruent segments, or upon the basis of groups I and II, we may determine how, by 
suitable constructions (see Chap. V, § 24), a segment is to be laid off from a point of a 
given straight line so that a new, definite segment is obtained "eg-ua/" to it. In conformity 
with such a convention, the axiom of Archimedes may be stated as follows: 

V. Let Ai he any point upon a straight line between the arbitrarily chosen points 
A and B. Take the points A2, A^, A4, . . . so that Ai lies between A and A2, 
A2 between Ai and As, As between A2 and A4 etc. Moreover, let the segments 

AAi, A1A2, A2A3, A3A4, ... 

be equal to one another. Then, among this series of points, there always exists 
a certain point A^ such that B lies between A and A^- 

The axiom of Archimedes is a linear axiom. 

Remark.^ To the preceeding five groups of axioms, we may add the following one, 
which, although not of a purely geometrical nature, merits particular attention from a 
theoretical point of view. It may he expressed in the following form: 

Axiom of Completeness.^ (Vollstdndigkeit): To a system of points, straight 
lines, and planes, it is impossible to add other elements in such a manner that 
the system thus generalized shall form a new geometry obeying all of the five 
groups of axioms. In other words, the elements of geometry form a system 
which is not susceptible of extension, if we regard the five groups of axioms as 
valid. 



Added by Professor Hilbert in the French translation. — Tr. 

See Hilbert, "Ueber den Zahlenbegriff," Berichte der deutschen Mathematiker-Vereinigung, 1900. 



16 

This axiom gives us nothing directly concerning the existence of limiting points, or of 
the idea of convergence. Nevertheless, it enables us to demonstrate Bolzano's theorem by 
virtue of which, for all sets of points situated upon a straight line between two definite 
points of the same line, there exists necessarily a point of condensation, that is to say, 
a limiting point. From a theoretical point of view, the value of this axiom is that it 
leads indirectly to the introduction of limiting points, and, hence, renders it possible to 
establish a one-to-one correspondence between the points of a segment and the system 
of real numbers. However, in what is to follow, no use will be made of the "axiom of 
completeness." 



17 



COMPATIBILITY AND MUTUAL INDEPENDENCE OF THE 

AXIOMS. 



§ 9. COMPATIBILITY OF THE AXIOMS. 

The axioms, which we have discussed in the previous chapter and have divided into five 
groups, are not contradictory to one another; that is to say, it is not possible to deduce 
from tliese axioms, by any logical process of reasoning, a proposition which is contradictory 
to any of them. To demonstrate this, it is sufficient to construct a geometry where all of 
the five groups are fulfilled. 

To this end, let us consider a domain Q consisting of all of those algebraic numbers 
which may be obtained by beginning with the number one and applying to it a finite 
number of times the four arithmetical operations {addition, subtraction, multiplication, 
and division) and the operation vl + o;^, where to represents a number arising from the 
five operations already given. 

Let us regard a pair of numbers (x, y) of the domain Q as defining a point and the 
ratio of three such numbers [u : v : w) of S7, where u, v are not both equal to zero, as 
defining a straight line. Furthermore, let the existence of the equation 

ux -\- vy -\- w = 

express the condition that the point (x, y) lies on the straight line {u : v : w). Then, as one 
readily sees, axioms I, 1-2 and III are fulfilled. The numbers of the domain O are all real 
numbers. If now we take into consideration the fact that these numbers may be arranged 
according to magnitude, we can easily make such necessary conventions concerning our 
points and straight lines as will also make the axioms of order (group II) hold. In fact, 
if [xi, yi), [x2, ^2)) (^3) Vs), ■ ■ ■ cire any points whatever of a straight line, then this may 
be taken as their sequence on this straight line, providing the numbers xi, X2, X3, . . . , or 
the numbers yi, ?/2, 2/3, ■ ■ ■ , either all increase or decrease in the order of sequence given 
here. In order that axiom II, 5 shall be fulfilled, we have merely to assume that all points 
corresponding to values of x and y which make ux -\- vy -\- w less than zero or greater 
than zero shall fall respectively upon the one side or upon the other side of the straight 
line [u : V : w). We can easily convince ourselves that this convention is in accordance 
with those which precede, and by which the sequence of the points on a straight line has 
already been determined. 

The laying off of segments and of angles follows by the known methods of analytical 
geometry. A transformation of the form 

X = X -\- a 
y = y^b 

produces a translation of segments and of angles. 



18 



n 



A2 



An-i B A, 



Fig. 13. 



Furthermore, if, in the accompanying figure, we represent the point (0,0) by O and 
the point (1,0) by £, then, corresponding to a rotation of the angle COE about O as a 
center, any point [x^y] is transformed into another point {x',y') so related that 



X = 



y = 



=x y, 

b a 

y- 



Va2 + 62 ^^2 ^ ^2 



Since the number 



62 =ai 1 



belongs to the domain fi, it follows that, under the conventions which we have made, 
the axioms of congruence (group IV) are all fulfilled. The same is true of the axiom of 
Archimedes. 



0(0,0 J 



(x.'y) 




From these considerations, it follows that every contradiction resulting from our system 
of axioms must also appear in the arithmetic related to the domain O. 

The corresponding considerations for the geometry of space present no difficulties. 

If, in the preceding development, we had selected the domain of all real numbers 
instead of the domain O, we should have obtained likewise a geometry in which all of the 
axioms of groups 1 — V are valid. For the purposes of our demonstration, however, it was 
sufficient to take the domain O, containing on an enumerable set of elements. 



19 

§ 10. INDEPENDENCE OF THE AXIOMS OF PARALLELS. 
(NON-EUCLIDEAN GEOMETRY.)^ 

Having shown that the axioms of the above system are not contradictory to one another, 
it is of interest to investigate the question of their mutual independence. In fact, it may be 
shown that none of them can be deduced from the remaining ones by any logical process 
of reasoning. 

First of all, so far as the particular axioms of groups I, II, and IV are concerned, it 
is easy to show that the axioms of these groups are each independent of the other of the 
same group. ^ 

According to our presentation, the axioms of groups I and II form the basis of the 
remaining axioms. It is sufficient, therefore, to show that each of the groups II, IV, and 
V is independent of the others. 

The first statement of the axiom of parallels can be demonstrated by aid of the axioms 
of groups I, II, and IV. In order to do this, join the given point A with any arbitrary point 
B of the straight line a. Let C be any other point of the given straight line. At the point 
A on AB, construct the angle ABC so that it shall lie in the same plane a as the point 
(7, but upon the opposite side of AB from it. The straight line thus obtained through A 
does not meet the given straight line a: for, if it should cut it, say in the point D, and 
if we suppose B to be situated between C and D, we could then find on a a point D' so 
situated that B would lie between D and D', and, moreover, so that we should have 

AD = BD' 

Because of the congTuence of the two triangles ABD and BAD', we have also 

ZABD = ZBAD', 

and since the angles ABD' and ABD are supplementary, it follows from theorem 12 that 
the angles BAD and BAD' are also supplementary. This, however, cannot be true, as, 
by theorem 1, two straight lines cannot intersect in more than one point, which would be 
the case if BAD and BAD' were supplementary. 

The second statement of the axiom of parallels is independent of all the other axioms. 
This may be most easily shown in the following well known manner. As the individual 
elements of a geometry of space, select the points, straight lines, and planes of the ordinary 
geometry as constructed in § 9, and regard these elements as restricted in extent to the 
interior of a fixed sphere. Then, define the congruences of this geometry by aid of such 
linear transformations of the ordinary geometry as transform the fixed sphere into itself. 
By suitable conventions, we can make this ^''non-euclidean geometry" obey all of the axioms 
of our system except the axiom of Euclid (group III). Since the possibility of the ordinary 
geometry has already been established, that of the non-euclidean geometry is now an 
immediate consequence of the above considerations. 



The mutual independence of Hilbert's system of axioms has also been discussed recently by Schur and 
Moore. Schur's paper, entitled "Ueber die Grundlagen der Geometrie" appeared in Math. Annalem, Vol. 
55, p. 265, and that of Moore, "On the Projective Axioms of Geometry," is to be found in the Jan. (1902) 
number of the Transactions of the Amer. Math. Society. — Tr. 

See my lectures upon Euclidean Geometry, winter semester of 1898—1899, which were reported by Dr. 
Von Schaper and manifolded for the members of the class. 



20 



§ 11. INDEPENDENCE OF THE AXIOMS OF CONGRUENCE. 

We shall show the independence of the axioms of congruence by demonstrating that axiom 
IV, 6, or what amounts to the same thing, that the first theorem of congruence for triangles 
(theorem 10) cannot he deduced from the remaining axioms I, II, III, IV 1-5, V by any 
logical process of reasoning. 

Select, as the points, straight lines, and planes of our new geometry of space, the 
points, straight lines, and planes of ordinary geometry, and define the laying off of an 
angle as in ordinary geometry, for example, as explained in § 9. We will, however, define 
the laying off of segments in another manner. Let Ai, A2 be two points which, in ordinary 
geometry, have the co-ordinates xi, ?/i, zi and X2, y2, Z2, respectively. We will now define 
the length of the segment A1A2 as the positive value of the expression 



\/{xi - x2 + yi- y2f + (yi - 2/2)^ + {-1 - ^2)^ 

and call the two segments A1A2 and A'-^A'2 congruent when they have equal lengths in the 
sense just defined. 

It is at once evident that, in the geometry of space thus defined, the axioms I, II, III, 
IV 1-2, 4-5, V are all fulfilled. 

In order to show that axiom IV, 3 also holds, we select an arbitrary straight line a and 
upon it three points Ai, ^2, ^3 so that A2 shall lie between Ai and A-^. Let the points a:, 
y, z of the straight line a be given by means of the equations 

X = \t-^ A', 
y = fit + fi', 
z = ut -\- u , 

where A, A', /i, ^', u, v' represent certain constants and T is a parameter. If ti, ^2 (< ^i), 
^3 (< ^2) are the values of the parameter corresponding to the points Ai, A2, A3 we have 
as the lengths of the three segments A1A2 A2A3 and A1A3 respectively, the following 
values : 

(tl-t2) \/{A + ^)2 + ^2_^i.2 



(t2-t-^ \/{A + ^.)2 + ^2+i.2 



(h-t.^ x/{A + ^.)' + ^' + ^' 

Consequently, the length of Ai A3 is equal to the sum of the lengi;hs of the segments Ai A2 
and A2A3. But this result is equivalent to the existence of axiom IV, 3. 

Axiom IV, 6, or rather the first theorem of congruence for triangles, is not always 
fulfilled in this geometry. Consider, for example, in the plane = 0, the four points 

O, having the co-ordinates a: = 0, ?/ = 
A, " " " :c = 1, y = 

S, " " " :c = 0, y = 1 

o, X — 2, y - 2 



21 



.B(0,I) 




cfih 



0(0,0) 



AU.O) 



Fig. 15. 



Then, in the right triangles OAC and OBC, the angles at C as also the adjacent sides AC 
and BC are respectively congruent; for, the side OC is common to the two triangles and 
the sides AC and BC have the same length, namely, ^. However, the third sides OA and 
OB have the lengths 1 and \/2, respectively, and are not, therefore, congruent. It is not 
difficult to find in this geometry two triangles for which axiom IV, 6, itself is not valid. 



12. INDEPENDENCE OF THE AXIOM OF CONTINUITY. 
(NON-ARCHIMEDEAN GEOMETRY.) 



In order to demonstrate the independence of the axiom of Archimedes, we must produce a 
geometry in which all of the axioms are fulfilled with the exception of the one in question. ' 

For this purpose, we construct a domain Q[t) of all those algebraic functions of t 
which may be obtained from t by means of the four arithmetical operations of addition, 
subtraction, multiplication, division, and the fifth operation vl + o;^, where to represents 
any function arising from the application of these five operations. The elements of Q.{t) — 
just as was previously the case for fi — constitute an enumerable set. These five operations 
may all be performed without introducing imaginaries, and that in only one way. The 
domain Q[t) contains, therefore, only real, single- valued functions of t. 

Let c be any function of the domain Q.(t). Since this function c is an algebraic function 
of t, it can in no case vanish for more than a finite number of values of t, and, hence, for 
sufficiently large positive values of t, it must remain always positive or always negative. 

Let us now regard the functions of the domain Q,{t) as a kind of complex numbers. 
In the system of complex numbers thus defined, all of the ordinary rules of operation 
evidently hold. Moreover, if a, h are any two distinct numbers of this system, then a is 
said to be greater than, or less than, b (written a > b or a < b) according as the difference 
c = a — b IS always positive or always negative for sufficiently large values of t. By the 
adoption of this convention for the numbers of our system, it is possible to arrange them 



In his very scholarly book, — Grundziige der Geometrie, German translation by A. Schepp, Leipzig, 
1894, — G. Veronese has also attempted the construction of a geometry independent of the axiom of 
Archimedes. 



22 

according to their magnitude in a manner analogous to that employed for real numbers. 
We readily see also that, for this system of complex numbers, the validity of an inequality 
is not destroyed by adding the same or equal numbers to both members, or by multiplying 
both members by the same number, providing it is greater than zero. 

If n is any arbitrary positive integral rational number, then, for the two numbers n 
and t of the domain Q.{t), the inequality n < t certainly holds; for, the difference n — t, 
considered as a function of t, is always negative for sufficiently large values of t. We express 
this fact in the following manner: The two numbers I and t of the domain 0(i), each of 
which is greater than zero, possess the property that any multiple whatever of the first 
always remains smaller than the second. 

From the complex numbers of the domain 0(i), we now proceed to construct a geome- 
try in exactly the same manner as in § 9, where we took as the basis of our consideration the 
algebraic numbers of the domain O. We will regard a system of three numbers (x, y, z) of 
the domain 0(f) as defining a point, and the ratio of any four such numbers [u : v : 7d : r) , 
where u, v, w are not all zero, as defining a plane. Finally, the existence of the equation 

xu -\- yv -\- zw + r = 

shall express the condition that the point (x, y, z) lies in the plane {u :v : w : r). Let the 
straight line be defined in our geometry as the totality of all the points lying simultaneously 
in the same two planes. If now we adopt conventions corresponding to those of § 9 
concerning the arrangement of elements and the laying off of angles and of segments, 
we shall obtain a "non-archimedean" geometry where, as the properties of the complex 
number system already investigated show, all of the axioms, with the exception of that 
of Archimedes, are fulfilled. In fact, we can lay off successively the segment 1 upon the 
segment t an arbitrary number of times without reaching the end point of the segment t, 
which is a contradiction to the axiom of Archimedes. 



23 

THE THEORY OF PROPORTION.^ 

§ 13. COMPLEX NUMBER-SYSTEMS. 

At the beginning of this chapter, we shah present briefly certain prehminary ideas con- 
cerning complex number systems which will later be of service to us in our discussion. 

The real numbers form, in their totality, a system of things having the following prop- 
erties: 

THEOREMS OF CONNECTION (1-12). 

1. From the number a and the number 6, there is obtained by "addition" a definite number 

c, which we express by writing 

a-\-h = c or c = a-\-b. 

2. There exists a definite number, which we call 0, such that, for every number a, we have 

a + = a and + a = a . 

3. If a and h are two given numbers, there exists one and only one number x^ and also 

one and only one number y, such that we have respectively 

a -\- X = h, y -\- a = h. 

4. From the number a and the number 6, there may be obtained in another way, namely, 

by "multiplication," a definite number c, which we express by wi'iting 

ah = c or c = ah. 

5. There exists a definite number, called 1, such that, for every number a, we have 

a ■ 1 = a and 1 ■ a = a. 

6. If a and b are any arbitrarily given numbers, where a is different from 0, then there 

exists one and only one number x and also one and only one number y such that we 
have respectively 

ax = h, ya = h. 

If a, h, c are arbitrary numbers, the following laws of operation always hold: 

7. a^{b^c) = {a-\-h)^c 

8. a + & = h-\-a 

9. a(hc) = {ah)c 

10. a{h + c) = ah -\- ac 

11. (a + h)c = ac -\- he 

12. ah = ha. 



See also Scliur, Math. Annalen, Vol. 55, p. 265. — Tr. 



24 
THEOREMS OF ORDER (13-16). 

13. If a, b are any two distinct numbers, one of these, say a, is always greater {>) than 

the other. The other number is said to be the smaller of the two. We express this 
relation by writing 

a > b and b < a. 

14. If a > 6 and b > c, then is also a > c. 

15. If a > 6, then is also a -\- c > b -\- c and c-\- a > c -\- b. 

16. li a > b and c > 0, then is also ac > be and ca > cb. 

THEOREM OF ARCHIMEDES (17). 



17. If a, b are any two arbitrary numbers, such that a > and 6 > 0, it is always possible 
to add a to itself a sufficient number of times so that the resulting sum shall have 
the property that 

a-\-a-\-a-\-----\-a> b. 

A system of things possessing only a portion of the above properties (1-17) is called a 
complex number system, or simply a number system. A number system is called archimedean, 
or non-archimedean, according as it does, or does not, satisfy condition (17). 

Not every one of the properties (1—17) given above is independent of the others. The 
problem arises to investigate the logical dependence of these properties. Because of their 
great importance in geometry, we shall, in §§ 32, 33, pp. 65-68, answer two definite 
questions of this character. We will here merely call attention to the fact that, in any 
case, the last of these conditions (17) is not a consequence of the remaining properties, 
since, for example, the complex number system 0.{t), considered in § 12, possesses all of 
the properties (1—16), but does not fulfil the law stated in (17). 



§ 14. DEMONSTRATION OF PASCAL'S THEOREM. 

In this and the following chapter, we shall take as the basis of our discussion all of the plane 
axioms with the exception of the axiom of Archimedes; that is to say, the axioms I, 1—2 
and II-IV. In the present chapter, we propose, by aid of these axioms, to establish Euclid's 
theory of proportion; that is, we shall establish it for the plane and that independently of 
the axiom of Archimedes. 

For this purpose, we shall first demonstrate a proposition which is a special case of 
the well known theorem of Pascal usually considered in the theory of conic sections, and 
which we shall hereafter, for the sake of brevity, refer to simply as Pascal's theorem. This 
theorem may he stated as follows: 



25 




Fig. 16. 



Theorem 21. (Pascal's theorem.) Given the two sets of points A, B, C and A', B', 
C' so situated respectively upon two intersecting straight hues that none of them fall 
at the intersection of these lines. If CB' is parallel to BC and CA' is also parallel 
to AC, then BA' is parallel to AB' .^ 

In order to demonstrate this theorem, we shall first introduce the following notation. 
In a right triangle, the base a is uniquely determined by the hypotenuse c and the base 
angle a included by a and c. We will express this fact briefly by writing 

a = ac. 




Fig. 17. 



Hence, the symbol ac always represents a definite segment, providing c is any given 
segment whatever and a is any given acute angle. 

Furthermore, if c is any arbitrary segment and a, (3 are any two acute angles whatever, 
then the two segments a/3c and ^ac are always congruent: that is, we have 

a. (3c = /?ac, 

and, consequently, the symbols a and (3 are interchangeable. 



F. Schur has published in the Math. Ann., Vol. 51, a very interesting proof of the theorem of Pascal, 
based upon the axioms I— II, IV. 



26 




Fig. 18. 



In order to prove this statement, we take the segment c = AB, and with A as a vertex 
lay off upon the one side of this segment the angle a and upon the other the angle /?. 
Then, from the point B, let fall upon the opposite sides of the a and 3 the perpendiculars 
BC and BD, respectively. Finally, join C with D and let fall from A the perpendicular 
AE upon CD. 

Since the two angles ACB and ADB are right angles, the four points A, B, C, D are 
situated upon a circle. Consequently, the angles ACD and ABD, being inscribed in the 
same segment of the circle, are congruent. But the angles ACD and CAE, taken together, 
make a right angle, and the same is true of the two angles ABD and BAD. Hence, the 
two angles CAE and BAD are also congTuent; that is to say, 

ZCAE = (3 

and, therefore, 

IDAE = a. 

From these considerations, we have immediately the follo\'idng congruences of segments: 
I3c = AD, ac = AC, 



a 



(3c = a{AD) = AE, (3ac = (3(AC) = AE. 



From these, the \'alidity of the congruence in question follows. 

Returning now to the figure in connection with Pascal's theorem, denote the intersec- 
tion of the two given straight lines by O and the segments OA, OB, DC, OA', OB', OC , 
CB', BC, CA', AC, BA', AB' by a, h, c, a', b' , c' , I, /*, m, m\ n, n\ respectively. 

Let fall from the point O a perpendicular upon each of the segments /, m, n. The 
perpendicular to / will form with the straight lines OA and OA' acute angles, which we 
shall denote by A' and A, respectively. Likewise, the perpendiculars to m and n form with 
these same lines OA and OA' acute angles, which we shall denote by jj,' , /i and i/, u, 
respectively. If we now express, as indicated above, each of these perpendiculars in terms 
of the hypotenuse and base angle, we have the three following congruences of segments: 



27 




Fig. 19. 



Xb' = X'c 
^■a' = ji'c 



(1) 
(2) 

(3) va' = v'h. 

But since, according to our hypothesis, / is paraUel to /* and m is parallel to m*, the 
perpendiculars from O falling upon I* and nt* must coincide with the perpendiculars from 
the same point falling upon / and r/i, and consequently, we have 

(4) Xc' = A'6, 



(5) 



jic' = ji'a. 



Multiplying both members of congTuence (3) by the symbol A'//, and remembering that, 
as we have already seen, the symbols in question are commutative, we have 

vX' fia' = i^' fiX'h. 

In this congruence, we may replace fia' in the first member by its value given in (2) and 
X'b in the second member by its value given in (4), thus obtaining as a result 

i/X' fi'c = i/fiXc'j 



or 



i/fi' X'c = i/ Xfic' . 

Here again in this congruence we can, by aid of {!), replace X'c by Xb', and, by aid of (5), 
we may replace in the second member jic' by ji'a. We then have 

i^ji Xb = ly Xjj, a, 



or 



Xfi. vb = Xfi i> a. 



Because of the significance of our symbols, we can conclude at once from this congruence 
that 

^'vb' = fi'iy'a, 



28 



and, consequently, that 

(6) i^b' = i/a. 

If now we consider the perpendicular let fall from O upon n and draw perpendiculars to 
this same line from the points A and B' , then congruence (6) shows that the feet of the 
last two perpendiculars must coincide; that is to say, the straight line n* = AB' makes a 
right angle with the perpendicular to n and, consequently, is parallel to n. This establishes 
the truth of Pascal's theorem. 

Having given any straight line whatever, together with an arbitrary angle and a point 
lying outside of the given line, we can, by constructing the given angle and drawing a 
parallel line, find a straight line passing through the given point and cutting the given 
straight line at the given angle. By means of this construction, we can demonstrate 
Pascal's theorem in the following very simple manner, for which, however, I am indebted 
to another source. 




Through the point B, draw a straight line cutting OA' in the point D' and making 
with it the angle OCA' , so that the congruence 
(1*) ZOCA' = ZOD'B 

is fulfilled. Now, according to a well known property of circles, CBD'A' is an inscribed 
quadrilateral, and, consequently, by aid of the theorem concerning the congruence of angles 
inscribed in the same segment of a circle, we have the congruence 
(2*) ZOBA' = ZOD'C. 

Since, by hypothesis, CA' and AC are parallel, we have 
(3*) ZOCA' = ZOAC, 



29 

and from (1*) and (3*) we obtain the congruence 

ZOD'B= ZOAC. 

However, BAD'C' is also an inscribed quadrilateral, and, consequently, by virtue of the 
theorem relating to the angles of such a quadrilateral, we have the congruence 
(4*) Z.OAD' = ZOC'B. 

But as CB' is, by hypothesis, parallel to BC, we have also 
(5*) ZOB'C = ZOC'B. 

From (4*) and (5*), we obtain the congruence 

ZOAD' = ZOB'C, 

which shows that CAD'B' is also an inscribed quadrilateral, and, hence, the congruence 
(6*) ZOAB' = ZOD'C, 

is valid. From (2*) and (6*), it follows that 

ZOBA' = ZOAB', 

and this congruence shows that BA' and AB' are parallel as Pascal's theorem demands. In 
case D' coincides with one of the points A' , B' , C , it is necessary to make a modification 
of this method, which evidently is not difficult to do. 



§ 15. AN ALGEBRA OF SEGMENTS, BASED UPON PASCAL'S THEOREM. 

Pascal's theorem, which was demonstrated in the last section, puts us in a position to 
introduce into geometry a method of calculating with segments, in which all of the rules 
for calculating with real numbers remain valid without any modification. 

Instead of the word "congruent" and the sign =, we make use, in the algebra of 
segments, of the word "equal" and the sign =. 

^ ^ —^ b ^ 



c-at b 



Fig. 21. 



If A, B, C are three points of a straight line and if B lies between A and C, then we 
say that c = AC is the sum of the two segments a = AB and h = BC . We indicate this 
by writing 

c = a ^h. 



30 



The segments a and b are said to be smaller than c, which fact we indicate by writing 

a < c, b < c. 
On the other hand, c is said to be larger than a and b, and we indicate this by WTiting 

c> a, c > b. 

From the linear axioms of congruence (axioms IV, 1-3), we easily see that, for the 
above definition of addition of segments, the associative law 



+ (6 + c) = (a + 6) + c, 



as well as the commutative law 



a + h = b 



is valid. 

In order to define geometrically the product of 
two segments a and 6, we shall make use of the fol- 
lowing construction. Select any convenient segment, 
which, having been selected, shall remain constant 
throughout the discussion, and denote the same by 
1. Upon the one side of a right angle, lay off from 
the vertex O the segment 1 and also the segment 
b. Then, from O lay off upon the other side of the 
right angle the segment a. Join the extremities of 
the segments 1 and a by a straight line, and from the 
extremity of b draw a line parallel to this straight 
line. This parallel will cut off from the other side of 

the right angle a segment c. We call this segment c the product of the segments a and 6, 
and indicate this relation by writing 

c = ab. 




Fig. 22. 



31 



We shall now demonstrate that, for this defi- 
nition of the multiplication of segments, the com- 
mutative law 

ah = ha 

holds. For this purpose, we construct in the 
above manner the product ab. Furthermore, lay 
off from upon the first side (1) of the right an- 
gle the segment a and upon the other side (11) 
the segment b. Connect by a straight line the ex- 
tremity of the segment 1 with the extremity of 6, 
situated on 11, and draw through the endpoint of 
a, on I, a line parallel to this straight line. This 
parallel will determine, by its intersection with 
the side II, the segment ha. But, because the 
two dotted lines are, by Pascal's theorem, paral- 
lel, the segment ba just found coincides with the 

segment ab previously constructed, and our proposition is established. In order to show 

that the associative law 

a{hc) = {ah)c 

holds for the multiplication of segments, we construct first of all the segment d = be, then 
da, after that the segment e = ba, and finally ec. By virtue of Pascal's theorem, the 
extremities of the segments da and ec coincide, as may be clearly seen from figure 24. 




da=(bcla 




Fig. 24. 



If now we apply the commutative law which we have just demonstrated, we obtain the 
above formula, which expresses the associative law for the multiplication of two segments. 
Finally, the distributive law 

a{h + c) = ah -\- ac 

also holds for our algebra of segments. In order to demonstrate this, we construct the 
segments, ah, ac, and a{h + c), and draw through the extremity of the segment c (Fig. 25) 
a straight line parallel to the other side of the right angle. From the congruence of the two 



32 



8t.b+C) 




b 1 
a{b ^ c)^= ab -^ ac 

Fig. 25. 



right-angled triangles which are shaded in the figure and the application of the theorem 
relating to the equality of the opposite sides of a par'allelogram, the desired result follows. 
If h and c are any two arbitrary segments, there is always a segment a to he found such 
that c = ah. This segment a is denoted by | and is called the quotient of c by h. 



16. PROPORTION AND THE THEOREMS OF SIMILITUDE. 



By aid of the preceding algebra of segTnents, we can establish Euclid's theory of proportion 
in a manner free from objections and without making use of the axiom of Archimedes. 
If a, h, a', h' are any four segments whatever, the proportion 

a : b = a : b 

expresses nothing else than the validity of equation 

ah = ha . 

Definition. Two triangles are called similar when the corresponding angles are 
congruent. 

Theorem 22. If a,b and a',b' are homologous sides of two similar triangles, we 
have the proportion 

a : h = a : b 

Proof. We shall first consider the special case where the angle included between a 
and b and the one included between a' and b' are right angles. Moreover, we shall assume 
that the two triangles are laid off in one and the same right angle. Upon one of the sides 
of this right angle, we lay off from the vertex the segment 1, and through the extremity 
of this segment, we draw a straight line parallel to the hypotenuses of the two triangles. 



33 



This parallel determines upon the other side 
of the right angle a segment e. Then, according 
to our definition of the product of two segments, 
we have 

h = ea, b' = ea', 

from which we obtain 

ab' = ba' , 

that is to say, 

a:b = a' :b'. 

Let us now return to the general case. In 
each of the two similar triangles, find the point of 
intersection of the bisectors of the three angles. 
Denote these points by S and S' . From these 
points let fall upon the sides of the triangles the 
perpendiculars r and r', respectively. Denote the 
segments thus determined upon the sides of the 
triangles by 




Fig. 26. 



Q^, (2^, C*^, C*Q, Cq_, C^ 



and 




Fig. 27. 



respectively. The special case of our proposition, 
demonstrated above, gives us then the following proportions: 

ai,:r = af^:r, be : r = b^ : r , 

ac : r = a^ : r , ba : r = b^ : r . 

By aid of the distributive law, we obtain from these proportions the following: 

a : r = a' : r, b : r = b' : r' . 

Consequently, by virtue of the commutative law of multiplication, we have 

a : b = a : b . 

From the theorem just demonstrated, we can easily deduce the fundamental theorem 
in the theory of proportion. This theorem may he stated as follows: 

Theorem 23. If two parallel lines cut from the sides of an arbitrary angle the 
segments a,b and a',b' respectively, then we have always the proportion 

a : b = a : b . 

Conversely, if the four segments a,b,a',b fulfill this proportion and if a, a' and b,b' 
are laid off upon the two sides respectively of an arbitrary angle, then the straight 
lines joining the extremities of a and b and of a' and b' are parallel to each other. 



34 



17. EQUATIONS OF STRAIGHT LINES AND OF PLANES. 



To the system of segments already discussed, let us now add a second system. We will 
distinguish the segments of the new system from those of the former one by means of a 
special sign, and will call them ''^negative'''' segments in contradistinction to the ^'' •positive''' 
segments already considered. If we introduce also the segment O, which is determined 
by a single point, and make other appropriate conventions, then all of the rules deduced 
in § 13 for calculating with real numbers will hold equally well here for calculating with 
segments. We call special attention to the following pai'ticular propositions: 

We have always a- 1 = 1 ■ a = a. 

If a ■ 6 = 0, then either a = 0, or 6 = 0. 

If a > & and c > 0, then ac > be. 

In a plane a, we now take two straight lines cutting each other in O at right angles as 
the fixed axes of rectangular co-ordinates, and lay off from O upon these two straight lines 
the ai'bitrary segments x and y. We lay off these segTnents upon the one side or upon the 
other side of O, according as they are positive or negative. At the extremities of x and y, 
erect perpendiculars and determine the point P of their intersection. The segments x and 
y are called the co-ordinates of P. Every point of the plane a is uniquely determined by 
its co-ordinates x, y, which may be positive, negative, or zero. 

Let / be a straight line in the plane a, such that it shall pass through O and also 
through a point C having the co-ordinates a, b. If x, y are the co-ordinates 




Fig. 28. 



35 

of any point on I, it follows at once from theorem 22 that 

a : b = X : y, 

or 

bx — ay = 0, 

is the equation of the straight line I. If /' is a straight line parallel to I and cutting off 
from the a;-axis the segment c, then we may obtain the equation of the straight line /' by 
replacing, in the equation for /, the segTnent x by the segment x — c. The desired equation 
will then be of the form 

bx — ay — be = 0. 

From these considerations, we may easily conclude, independently of the axiom of 
Archimedes, that every straight line of a plane is represented by an equation which is 
linear in the co-ordinates x, y, and, conversely, every such linear equation represents a 
straight line when the co-ordinates are segments appertaining to the geometry in question. 
The corresponding results for the geometry of space may be easily deduced. 

The remaining parts of geometry may now be developed by the usual methods of 
analytic geometry. 

So far in this chapter, we have made absolutely no use of the axiom of Archimedes. 
If now we assume the validity of this axiom, we can arrange a definite correspondence 
between the points on any straight line in space and the real numbers. This may be 
accomplished in the following manner. 

We first select on a straight line any two points, and assign to these points the numbers 
and 1. Then, bisect the segment (0,1) thus determined and denote the middle point 
by the number ^. In the same way, we denote the middle of (0, 2) by ^, etc. After 
applying this process n times, we obtain a point which corresponds to ^w- Now, lay off m 
times in both directions from the point O the segment (O, 2^). We obtain in this manner 
a point corresponding to the numbers ^) and — ^). From the axiom of Archimedes, 
we now easily see that, upon the basis of this association, to each arbitrary point of a 
straight line there corresponds a single, definite, real number, and, indeed, such that this 
correspondence possesses the following property: If A, B, C are any three points on a 
straight line and ct, /?, 7 are the corresponding real numbers, and, if B lies between A and 
(7, then one of the inequalities, 

a < /? < 7 or a > /3 > 7, 

is always fulfilled. 

From the development given in § 9, p. 15, it is evident, that to every number belonging 
to the field of algebraic numbers H, there must exist a corresponding point upon the 
straight line. Whether to every real number there corresponds a point cannot in general 
be established, but depends upon the geometry to which we have reference. 

However, it is always possible to generalize the original system of points, straight lines, 
and planes by the addition of "ideal" or "irrational" elements, so that, upon any straight 
line of the corresponding geometry, a point corresponds without exception to every system 
of three real numbers. By the adoption of suitable conventions, it may also be seen that. 



36 

in this generalized geometry, all of the axioms I-V are valid. This geometry, generalized 
by the addition of irrational elements, is nothing else than the ordinary analytic geometry 
of space. 



37 

THE THEORY OF PLANE AREAS. 

§ 18. EQUAL AREA AND EQUAL CONTENT OF POLYGONS. ^° 

We shall base the investigations of the present chapter upon the same axioms as were made 
use of in the last chapter, §§ 13-17, namely, upon the plane axioms of all the groups, with 
the single exception of the axiom of Archimedes. This involves then the axioms I, 1-2 and 
II-IV. 

The theory of proportion as developed in §§ 13-17 together with the algebra of segments 
introduced in the same chapter, puts us now in a position to establish Euclid's theory of 
areas by means of the axioms already mentioned; that is to say, for the plane geometry, 
and that independently of the axiom of Archimedes. 

Since, by the development given in the last chapter, pp. 23-35, the theory of proportion 
was made to depend essentially upon Pascal's theorem (theorem 21), the same may then 
be said here of the theory of areas. This manner of establishing the theory of areas seems 
to me a very remarkable application of Pascal's theorem to elementary geometry. 

If we join two points of a polygon P by any arbitrary broken line lying entirely within 
the polygon, we shall obtain two new polygons Pi and P2 whose interior points all lie 
within P. We say that P is decomposed into Pi and P2, or that the polygon P is composed 
of Pi and P2. 

Definition. Two polygons are said to be of equal area when they can be decomposed 
into a finite number of triangles which are respectively congruent to one another in pairs. 

Definition. Two polygons are said to be of equal content when it is possible, by 
the addition of other polygons having equal area, to obtain two resulting polygons having 
equal area. 

From these definitions, it follows at once that by combining polygons having equal 
area, we obtain as a result polygons having equal area. However, if from polygons having 
equal area we take polygons having equal area, we obtain polygons which are of equal 
content. 

Furthermore, we have the following propositions: 

Theorem 24. If each of two polygons Pi and P2 is of equal area to a third polygon 
P3 then Pi and P2 are themselves of equal area. If each of two polygons is of equal 
content to a third, then they are themselves of equal content. 



In connection with the theory of areas, we desire to call attention to the following works of M. Gerard: 
These de Doctorat sur la geometrie non euclidienne (1892) and Geometrie plane (Paris, 1898). M. Gerard 
has developed a theory concerning the measurement of polygons analogous to that presented in § 20 of the 
present work. The difference is that M. Gerard makes use of parallel transversals, while I use transversals 
emanating from the vertex. The reader should also compare the following works of F. Schur, where he 
will find a similar development: Sitzungsberichte der Dorpater Naturf. Ges., 1892, and Lehrbuch der 
analytischen Geometrie, Leipzig, 1898 (introduction). Finally, let me refer to an article by O. Stolz in 
Monatshefte fiir Math, und Phys., 1894. (Note by Professor Hilbert in French ed.) 

M. Gerard has also treated the subject of areas in various ways in the following journals: Bulletin de 
Math, spciales (May, 1895), Bulletin de la Societe mathematique de France (Dec, 1895), Bulletin Math, 
elenientaires (January, 1896, June, 1897, June, 1898). (Note in French ed.) 



38 



Proof. By hypothesis, we can so decompose each of the polygons Pi and P2 into 
such a system of triangles that any triangle of either of these systems will be congruent to 
the corresponding triangle of a system into which P3 may be decomposed. If we consider 
simultaneously the two decompositions of P3 we see that, in general, each triangle of the 
one decomposition is broken up into polygons by the segments which belong to the other 
decomposition. Let ms add to these segments a sufficient number of others to reduce each 
of these polygons to triangles, and apply the two resulting methods of decompositions 
to Pi and P2, thus breaking them up into corresponding triangles. Then, evidently the 
two polygons Pi and P2 are each decomposed into the same number of triangles, which 
are respectively congruent by pairs. Consequently, the two polygons are, by definition, of 
equal area. 




Fig. 29. 



The proof of the second part of the theorem follows without difficulty. 
We define, in the usual manner, the terms: rectangle, base and height of a parallelogram, 
base and height of a triangle. 



39 



§ 19. PARALLELOGRAMS AND TRIANGLES HAVING EQUAL BASES AND 

EQUAL ALTITUDES. 

The well known reasoning of Euclid, illustrated by the accompanying figure, furnishes a 
proof for the following theorem: 

Theorem 25. Two parallelograms having equal bases and equal altitudes are also 
of equal content. 




Fig. 30. 



We have also the following well kno^'vii proposition: 

Theorem 26. Any triangle ABC is always of equal area to a certain parallelogram 
having an equal base and an altitude half as great as that of the triangle. 




Fig. 31. 



Proof. Bisect AC in D and BC in E, and extend the line DE to F, making EF 
equal to DE. Then, the triangles DEC and FBE are congruent to each other, and, 
consequently, the triangle ABC and the parallelogram ABFD are of equal area. 

From theorems 25 and 26, we have at once, by aid of theorem 24, the following propo- 
sition. 



Theorem 27. Two triangles having equal bases and equal altitudes have also equal 
content. 



40 

It is usual to show that two triangles having equal bases and equal altitudes are always 
of equal area. It is to be remarked, however, that this demonstration cannot be made 
without the aid of the axiom of Archimedes. In fact, we may easily construct in our non- 
archimedean geometry (see § 12, p. 21) two triangles so that they shall have equal bases 
and equal altitudes and, consequently, by theorem 27, must be of equal content, but which 
are not, however, of equal area. As such an example, we may take the two triangles ABC 
and ABD having each the base AB = 1 and the altitude 1, where the vertex of the first 
triangle is situated perpendicularly above A, and in the second triangle the foot F of the 
perpendicular let fall from the vertex D upon the base is so situated that AF = t. The 
remaining propositions of elementary geometry concerning the equal content of polygons, 
and in particular the pythagorean theorem, are all simple consequences of the theorems 
which we have already given. In the further development of the theory of area, we meet, 
however, with an essential difficulty. In fact, the discussion so far leaves it still in doubt 
whether all polygons are not, perhaps, of equal content. In this case, all of the propositions 
which we have given would be devoid of meaning and hence of no value. Furthermore, 
the more general question also arises as to whether two rectangles of equal content and 
having one side in common, do not also have their other sides congruent; that is to say, 
whether a rectangle is not definitely determined by means of a side and its area. As a 
closer investigation shows, in order to answer this question, we need to make use of the 
converse of theorem 27. This may be stated as follows: 

Theorem 28. If two triangles have equal content and equal bases, they have also 
equal altitudes. 

This fundamental theorem is to be found in the ffi'st book of Euclid's Elements as 
proposition 39. In the demonstration of this theorem, however, Euclid appeals to the 
general proposition relating to magnitudes: ^''Kal to dXov toj) jiepovi; jiciCoi/ eaniy" — a 
method of procedure which amounts to the same thing as introducing a new geometrical 
axiom concerning areas. 

It is now possible to establish the above theorem and hence the theory of areas in the 
manner we have proposed, that is to say, with the help of the plane axioms and without 
making use of the axiom of Archimedes. In order to show this, it is necessary to introduce 
the idea of the measure of area. 



20. THE MEASURE OF AREA OF TRIANGLES AND POLYGONS. 



Definition. If in a triangle ABC, having the sides a, b, c, we construct the two altitudes 
ha = AD, hfj = BE, then, according to theorem 22, it follows from the similarity of the 
triangles BCE and ACD, that we have the proportion 

a : hh = b : ha\ 



Fig. 32. 



41 




that is, we have 



a- hr. = b ■ hf 



This shows that the product of the base and the corresponding altitude of a triangle is 
the same whichever side is selected as the base. The half of this product of the base and 
the altitude of a triangle A is called the measure of area of the triangle A and we denote 
it by F{A). A segment joining a vertex of a triangle with a point of the opposite side is 
called a transversal. A transversal divides the given triangle into two others having the 
same altitude and having bases which lie in the same straight line. Such a decomposition 
is called a transversal decomposition of the triangle. 

Theorem 29. If a triangle A is decomposed by means of arbitrary straight lines 
into a finite number of triangles A^, then the measure of area of A is equal to the 
sum of the measures of area of the separate triangles A^. 

Proof. From the distributive law of our algebra of segments, it follows immediately 
that the measure of area of an arbitrary triangle is equal to the sum of the measures of area 
of two such triangles as arise from any transversal decomposition of the given triangle. 
The repeated application of this proposition shows that the measure of area of any triangle 
is equal to the sum of the measures of area of all the triangles arising by applying the 
transversal decomposition an arbitrary number of times in succession. 

In order to establish the corresponding proof for an 
arbitrary decomposition of the triangle A into the tri- 
angles A;;, draw from the vertex A of the given triangle 
A a transversal through each of the points of division 
of the required decomposition; that is to say, draw a 
transversal through each vertex of the triangles A/^. By 
means of these transversals, the given triangle A is de- 
composed into certain triangles A^. Each of these trian- 
gles A^ is broken up by the segments which determined 
this decomposition into certain triangles and quadrilat- 
erals. If, now, in each of the quadrilaterals, we draw 
a diagonal, then each triangle A^ is decomposed into 
certain other triangles A^^. We shall now show that the decomposition into the triangles 
Ats is for the triangles A^, as well as for the triangles A^, nothing else than a series of 




Fig. 33. 



42 




transversal decompositions. In fact, it is at once evident that any decomposition of a 
triangle into partial triangles may always be affected by a series of transversal decomposi- 
tions, providing, in this decomposition, points of division do not exist within the triangle, 
and further, that at least one side of the triangle remains free from points of division. 

We easily see that these conditions hold for the tri- 
angles Af. In fact, the interior of each of these triangles, 
as also one side, namely, the side opposite the point A, 
contains no points of division. 

Likewise, for each of the triangles A/,., the decom- 
position into Ats is reducible to transversal decompo- 
sitions. Let us consider a triangle A/^. Among the 
transversals of the triangle A emanating from the point 
A, there is always a definite one to be found which either 
coincides with a side of A/^, or which itself divides A^^ 
into two triangles. In the first case, the side in question 
always remains free from further points of division by 
the decomposition into the triangles A^^. In the second 
case, the segment of the transversal contained within 
the triangle A/^ is a side of the two triangles arising 
from the division, and this side certainly remains free 
from further points of division. 
According to the considerations set forth at the beginning of this demonstration, the 
measure of area F{A) of the triangle A is equal to the sum of the measures of area F[At) 
of all the triangles A^ and this sum is in turn equal to the sum of all the measures of area 
F[Ats)- However, the sum of the measures of area F[A/;) of all the triangles A/^ is also 
equal to the sum of the measures of area F{Ats)- Consequently, we have finally that the 
measure of area F{A) is also equal to the sum of all the measures of area F(A^), and with 
this conclusion our demonstration is completed. 

Definition. If we defhie the measure of area F[P) of a polygon as the sum of the 
measures of area of all the triangles into which the polygon is, by a definite decomposition, 
divided, then upon the basis of theorem 29 and by a process of reasoning similar to that 
which we have employed in 18 to prove theorem 24, we know that the measure of area of 
a polygon is independent of the manner of decomposition into triangles and, consequently, 
is definitely determined by the polygon itself. From this we obtain, by aid of theorem 29, 
the result that polygons of equal area have also equal measures of area. 

Furthermore, if P and Q are two polygons of equal content, then there must exist, 
according to the above definition, two other polygons P' and Q' of equal area, such that 
the polygon composed of P and P' shall be of equal area with the polygon formed by 
combining the polygons Q and Q' . From the two equations 



Fig. 34. 



F{P + P') = FiQ^Q'] 
F{P') = F{Q'\ 



we easily deduce the equation 



F{P) = F{Q); 



43 

that is to say, polygons of equal content have also equal measure of area. 

From this last statement, we obtain immediately the proof of theorem 28. If we 
denote the equal bases of the two triangles by g and the corresponding altitudes by h and 
h', respectively, then we may conclude from the assumed equality of content of the two 
triangles that they must also have equal measures of area; that is to say, it follows that 

— oh = -ah 
2 2 

and, consequently, dividing by 2P, we get 

h = h ^ 
which is the statement made in theorem 28. 



§ 21. EQUALITY OF CONTENT AND THE MEASURE OF AREA. 

In § 20 we have found that polygons having equal content have also equal measures of 
area. The converse of this is also true. 

In order to prove the converse, let us consider two triangles ABC and AB'C having 
a common right angle at A. The measures of area of these two triangles are expressed by 
the formulee 

F{ABC) = \aB ■ AC, 

F{AB'C') = ^AB'-AC'. 

We now assume that these measures of area are equal to each other, and consequently we 
have 

ABAC = AB' ■ AC, 

or 

AB : AB' = AC : AC. 

From this proposition, it follows, according to theorem 23, that the two straight lines 
BC and B'C are parallel, and hence, by theorem 27, the two triangles BC'B' and BC'C 
are of equal content. By the addition of the triangle ABC , it follows that the two triangles 
ABC and AB'C are of equal content. We have then shown that two right triangles having 
the same measure of area are always of equal content. 

Take now any arbitrary triangle having the base g and the altitude h. Then, according 
to theorem 27, it has equal content with a right triangle having the two sides g and h. 
Since the original triangle had evidently the same measure of area as the right triangle, it 
follows that, in the above consideration, the restriction to right triangles was not necessary. 
Hence, two arbitrary triangles with equal measures of area are also of equal content. 



44 




Moreover, let us suppose P to be any polygon having the measure of area g and let 
P be decomposed into n triangles having respectively the measures of area gi, g^-, gs, ■ . ., 
Qn- Then, we have 

5 = 5i +P2 +53 H \-9n- 

Construct now a triangle ABC having the base AB = g and the altitude h = 1. Take, 
upon the base of this triangle, the points Ai, A2, . . ., An-i so that gi = AAi, gi = A1A2, 




Fig. 36. 



Since the triangles composing the polygon P have respectively the same measures of 
area as the triangles AAiC, A1A2C, . . ., yl„_2A„_i(7, A^^iBC, it follows from what 
has already been demonstrated that they have also the same content as these triangles. 
Consequently, the polygon P and a triangle, having the base g and the altitude h = 1 
are of equal content. From this, it follows, by the application of theorem 24, that two 
polygons having equal measures of area are always of equal content. 

We can now combine the proposition of this section with that demonstrated in the 
last, and thus obtain the following theorem: 

Theorem 30. Two polygons of equal content have always equal measures of area. 
Conversely, two polygons having equal measures of area are always of equal content. 



45 

In particular, if two rectangles ai'e of equal content and have one side in common, then 
their remaining sides are respectively congruent. Hence, we have the following proposition: 

Theorem 31 . If we decompose a rectangle into several triangles by means of straight 
lines and then omit one of these triangles, we can no longer make up completely the 
rectangle from the triangles which remain. 

This theorem has been demonstrated by F. Schur-'--'- and by W. Killing, ■'■^ but by making 
use of the axiom of Archimedes. By O. Stolz,-^'^ it has been regarded as an axiom. In the 
foregoing discussion, it has been shown that it is completely independent of the axiom of 
Archimedes. However, when we disregard the axiom of Archimedes, this theorem (31) is 
not sufficient of itself to enable us to demonstrate Euclid's theorem concerning the equality 
of altitudes of triangles having equal content and equal bases. (Theorem 28.) 

In the demonstration of theorems 28, 29, and 30, we have employed essentially the 
algebra of segments introduced in § 15, p. 29, and as this depends substantially upon 
Pascal's theorem (theorem 21), we see that this theorem is really the corner-stone in the 
theory of areas. We may, by the aid of theorems 27 and 28, easily establish the converse 
of Pascal's theorem. 

Of two polygons P and Q, we call P the smaller or larger in content according as 
the measure of area F{P) is less or gTeater than the measure of area F{Q). From what 
has already been said, it is clear that the notions, equal content, smaller content, larger 
content, are mutually exclusive. Moreover, we easily see that a polygon, which lies wholly 
within another polygon, must always be of smaller content than the exterior one. 

With this we have established the important theorems in the theory of areas. 



Sitzungsberichte der Dorpater Naturf. Ges. 1892. 
~ Grundlagen der Geometric, Vol. 2, Chapter 5, § 5, 1! 
^^ Monatshefte fiir Math, und Phys. 1894. 



46 

DESARGUES'S THEOREM. 



§ 22. DESARGUES'S THEOREM AND ITS DEMONSTRATION FOR PLANE 
GEOMETRY BY AID OF THE AXIOMS OF CONGRUENCE. 

Of the axioms given in §§ 1-8, pp. 1-14, those of groups II-V are in part linear and in part 
plane axioms. Axioms 3-7 of group I are the only space axioms. In order to show clearly 
the significance of these axioms of space, let us assume a plane geometry and investigate, 
in general, the conditions for which this plane geometry may be regarded as a part of a 
geometry of space in which at least the axioms of groups I-III are all fulfilled. 

Upon the basis of the axioms of groups I— III, it is well known that the so-called theorem 
of Desargues may be easily demonstrated. This theorem relates to points of intersection 
in a plane. Let us assume in particular that the straight line, upon which are situated 
the points of intersection of the homologous sides of the two triangles, is the straight line 
which we call the straight line at infinity. We will designate the theorem which arises in 
this case, together with its converse, as the theorem of Desargues. This theorem is as 
follows : 

Theorem 32. (Desargues's theorem.) When two triangles are so situated in a 
plane that their homologous sides are respectively parallel, then the lines joining 
the homologous vertices pass through one and the same point, or are parallel to one 
another . 

Conversely, if two triangles are so situated in a plane that the straight lines joining 
the homologous vertices intersect in a common point, or are parallel to one another, and, 
furthermore, if two pairs of homologous sides are parallel to each other, then the third 
sides of the two triangles are also parallel to each other. 




Fig. 37. 



As we have already mentioned, theorem 32 is a consequence of the axioms I— III. Be- 
cause of this fact, the validity of Desargues's theorem in the plane is, in any case, a nec- 
essary condition that the geometry of this plane may be regarded as a part of a geometry 
of space in which the axioms of groups I-III are all fulfilled. 



47 

Let us assume, as in §§ 13-21, pp. 23-45, that we have a plane geometry in which the 
axioms I, 1—2 and II— IV all hold and, also, that we have introduced in this geometry an 
algebra of segments conforming to § 15. 

Now, as has already been established in § 17, there may be made to correspond to 
each point in the plane a pair of segments {x,y) and to each straight line a ratio of three 
segments [u : v : w), so that the linear equation 

ux -\- vy -\- w = 

expresses the condition that the point is situated upon the straight line. The system 
composed of all the segments in our geometry forms, according to § 17, a domain of 
numbers for which the properties (1-16), enumerated in § 13, are valid. We can, therefore, 
by means of this domain of numbers, construct a geometry of space in a manner similar 
to that already employed in § 9 or in § 12, where we made use of the systems of numbers 
Q and ri(i), respectively. For this purpose, we assume that a system of three segments 
{x,y,z) shall represent a point, and that the ratio of four segments [u : v : 7d : r) shall 
represent a plane, while a straight line is defined as the intersection of two planes. Hence, 
the linear equation 

ux -\- vy -\- wz -|- r = 

expresses the fact that the point [x^y^z) lies in the plane [u : v : w : r). Finally, we 
determine the arrangement of the points upon a straight line, or the points of a plane 
with respect to a straight line situated in this plane, or the arrangement of the points in 
space with respect to a plane, by means of inequalities in a manner similar to the method 
employed for the plane in § 9. 

Since we obtain again the original plane geometry by putting i; = 0, we know that 
our plane geometry can be regarded as a part of geometry of space. Now, the validity of 
Desargues's theorem is, according to the above considerations, a necessary condition for 
this result. Hence, in the assumed plane geometry, it follows that Desargues's theorem 
must also hold. 

It will he seen that the result just stated may also be deduced without difficulty from 
theorem 23 in the theory of proportion. 



§ 23. THE IMPOSSIBILITY OF DEMONSTRATING DESARGUES'S THEOREM 
FOR THE PLANE WITHOUT THE HELP OF THE AXIOMS OF 

CONGRUENCE. ^^ 

We shall now investigate the question whether or no in plane geometry Desargues's the- 
orem may he deduced without the assistance of the axioms of congruence. This leads us 
to the following result: 



See also a recent paper by F. R. Moulton on "Siinple Non-desarguesian Geometry," Transactions of 
the Amer. Math. Soc, April, 1902.— Tr. 



48 

Theorem 33. A plane geometry exists in which the axioms I 1—2, II— III, IV 1-5, 
V, that is to say, all linear and all plane axioms with the exception of axiom IV, 6 
of congruence, are fulfilled, but in which the theorem of Desargues {theorem 32) is 
not valid. Desargues's theorem is not, therefore, a consequence solely of the axioms 
mentioned; for, its demonstration necessitates either the space axioms or all of the 
axioms of congruence. 

Proof. Select in the ordinary plane geometry (the possibility of which has already 
been demonstrated in § 9, pp. 15-17) any two straight lines perpendicular to each other 
as the axes of x and y. Construct about the origin O of this system of co-ordinates an 
ellipse having the major and minor axes equal to 1 and ^^ respectively. Finally, let F 
denote the point situated upon the positive a;-axis at the distance ^ from O. Consider all 
of the circles which cut the ellipse in four real points. These points may be either distinct 
or in any way coincident. Of all the points situated upon these circles, we shall attempt to 
determine the one which lies upon the x-axis farthest from the origin. For this purpose, let 
us begin with an arbitrary circle cutting the ellipse in four distinct points and intersecting 
the positive a;-axis in the point C. Suppose this circle then turned about the point C in 
such a manner that two or more of the four points of intersection with the ellipse finally 
coincide in a single point A, while the rest of them remain real. Increase now the resulting 
tangent circle in such a way that A always remains a point of tangency with the ellipse. 
In this way we obtain a circle which is either tangent to the ellipse in also a second point 
B, or which has with the ellipse a four-point contact in A. Moreover, this circle cuts the 
positive X-axis in a point more remote than C. The desired farthest point will accordingly 
be found among those points of intersection of the positive x-axis by circles lying exterior 
to the ellipse and being doubly tangent to it. All such circles must lie, as we can easily 
see, symmetrically with respect to the y-axis. Let a, b be the co-ordinates of any point 
on the ellipse. Then an easy calculation shows that the circle, which is symmetrical with 
respect to y-axis and tangent to the ellipse at this point, cuts off from the positive a;-axis 
the segment 

X =1 \/l + 3&2 I . 

The greatest possible value of this expression occurs for b = ^ and, hence, is equal to 
2 I v7 I . Since the point on the a:-axis which we have denoted by F has for its abscissa 
the value ^ > 2 I ^7 |, it follows that among the circles cutting the ellipse four times there 
is certainly none which passes through the point F. 

We will now construct a new plane geometry in the following manner. As points in 
this new geometry, let us take the points of the (a;y)-plane. We will define a straight line 
of our new geometry in the following manner. Every straight line of the {xy)-plane which 
is either tangent to the fixed ellipse, or does not cut it at all, is taken unchanged as a 
straight line of the new geometry. However, when any straight line g of the {xy)-pla,ne 
cuts the ellipse, say in the points P and Q, we will then define the corresponding 

straight line of the new geometry as follows. Construct a circle passing through the 
points P and Q and the fixed point F. From what has just been said, this circle will have 
no other point in common with the ellipse. We will now take the broken line, consisting 
of the arc PQ just mentioned and the two parts of the straight line g extending outward 



49 




Fig. 38. 



indefinitely from the points P and Q, as the required straight hne in our new geometry. 
Let us suppose all of the broken lines constructed which correspond to straight lines of 
the (a;?/)-plane. We have then a system of broken lines which, considered as straight lines 
of our new geometry, evidently satisfy the axioms I, 1-2 and III. By a convention as to 
the actual arrangement of the points and the straight lines in our new geometry, we have 
also the axioms II fulfilled. 

Moreover, we will call two segments AB and A'B' congruent in this new geometry, if 
the broken line extending between A and B has equal length, in the ordinary sense of the 
word, with the broken line extending from A' to B' . 

Finally, we need a convention concerning the congruence of angles. So long as neither 
of the vertices of the angles to be compared lies upon the ellipse, we call the two angles 
congruent to each other, if they are equal in the ordinary sense. In all other cases we make 
the following convention. Let A, B, C be points which follow one another in this order 
upon a straight line of our new geometry, and let A', B', C be also points which lie in this 
order upon another straight line of our new geometry. Let D be a point lying outside of 
the straight line ABC and D' be a point outside of the straight A'B'C . We will now say 
that, in our new geometry, the angles between these straight lines fulfill the congruences 

ZABD = ZA'B'D' and ZCBD = ZC'B'D' 



whenever the natural angles between the corresponding broken lines of the ordinary ge- 
ometry fulfill the proportion 

ZABD : ZCBD = ZA'B'D' : ZC'B'D'. 

These conventions render the axioms IV, 1-5 and V valid. 



50 




Fig. 39. 



In order to see that Desargues's theorem does not hold for our new geometry, let us 
consider the following three ordinary straight lines of the (a:y)-plane; viz., the axis of a:, 
the axis of y, and the straight line joining the two points of the ellipse (5,5) and (~ 5; ~ 5)- 
Since these three ordinary straight lines pass through the origin, we can easily construct 
two triangles so that their vertices shall lie respectively upon these three straight lines and 
their homologous sides shall be parallel and all three sides shall lie exterior to the ellipse. 
As we may see from figure 40, or show by an easy calculation, the broken lines arising from 
the three straight lines in question do not intersect in a common point. Hence, it follows 
that Desargues's theorem certainly does not hold for this particular plane geometry in 
which we have constructed the two triangles just considered. 

This new geometry serves at the same time as an example of a plane geometry in which 
the axioms 1, 1-2, II-IIl, IV, 1-5, V all hold, but which cannot be considered as a part of 
a geometry of space. 



51 




Fig. 40. 



} 24. INTRODUCTION OF AN ALGEBRA OF SEGMENTS BASED UPON 
DESARGUES'S THEOREM AND INDEPENDENT OF THE AXIOMS OF 

CONGRUENCE. ^^ 



In order to see fully the significance of Desargues's theorem (theorem 32), let us take as 
the basis of our consideration a plane geometry where all of the axioms 1 1-2, II— III are 
valid, that is to say, where all of the plane axioms of the first three groups hold, and 
then introduce into this geometry, in the following manner, a new algebra of segments 
independent of the axioms of congruence. 

Take in the plane two fixed straight lines intersecting in O, and consider only such 
segments as have O for their origin and their other extremity in one of the fixed lines. We 
will regard the point O itself as a segment and call it the segment O. We will indicate 
this fact by WTiting 

00 = 0, or = OO. 

Let E and E' be two definite points situated respectively upon the two fixed straight 
lines through O. Then, define the two segments OE and OE' as the segment 1 and write 
accordingly 

OE = OE' = loi l = OE = OE'. (1) 

We will call the straight line EE' , for brevity, the unit-line. If, furthermore, A and A' are 
points upon 

the straight lines OE and OE', respectively, and, if the straight line AA' joining them 
is parallel to EE', then we will say that the segments OA and OA' are equal to one 



Discussed also by Moore in a paper before the Am. Math. Soc, Jan, 1902. See Trans. Am. Math. 
Soc. — Tr. 



52 



atb.C' 




another, and write 



Fig. 41. 



OA = OA' or OA' = OA. 



In order now to define the sum of the segments a = OA and h = OB, we construct 
AA' paraUel to the unit-line EE' and draw through A' a parallel to OE and through B 
a parallel to OE'. Let these two parallels intersect in A". Finally, draw through A" a 
straight line parallel to the unit-line EE' . Let this parallel cut the two fixed lines OE and 
OE' in C and C respectively. Then c = OC = OC is called the sum of the segments 
a = OA and b = OB. We indicate this by writing 



c = a 



or a 



b = c. 



In order to define the product of a segment a = OA by a segment b = OB, we make use 
of exactly the same construction as employed in § 15, except that, in place of the sides 
of a right angle, we make use here of the straight lines OE and OE' . The construction 



ab.c; 




Fig. 42. 



is consequently as follows. Determine upon OE' a point A' so that AA' is parallel to the 
unit-line EE', and join E with A' . Then draw through B a straight line parallel to EA' . 
This parallel will intersect the fixed straight line OE' in the point C , and we call c = OC 



53 



the product of the segment a = OA by the segment b = OB. We indicate this relation by 
writing 



c = 



or ab = c. 



25. THE COMMUTATIVE AND THE ASSOCIATIVE LAW OF ADDITION 
FOR OUR NEW ALGEBRA OF SEGMENTS. 



In this section, we shall investigate the laws of operation, as enumerated in § 13, in order to 
see which of these hold for our new algebra of segments, when we base our considerations 
upon a plane geometry in which axioms I 1-2, II-III are all fulfilled, and, moreover, in 
which Desargues's theorem also holds. 

First of all, we shall show that, for the addition of segments as defined in § 24, the 
commutative law 

a -\- b = b -\- a 

holds. Let 

a = OA = OA' 
b = OB = OB' 

Hence, AA' and BB' are, according to our conven- 
tion, parallel to the unit-line. Construct the points A" 
and B" by drawing A' A" and B'B" parallel to OA and 
also AB" and BA' parallel to OA. We see at once that 
the line A" B" is parallel to AA' as the commutative 
law requires. We shall show the validity of this state- 
ment by the aid of Desargues's theorem in the following 
manner. Denote the point of intersection of AB" and 
A' A" by F and that of BA" and B'B" by D. Then, 
in the triangles AA' F and BB' D., the homologous sides 
are parallel to each other. By Desargues's theorem, it 
follows that the three points O, F, D lie in a straight 
line. In consequence of this condition, the two triangles OAA' and DB" A" lie in such a 
way that the lines joining the corresponding vertices pass through the same point F, and 
since the homologous sides OA and DB", as also OA' and DA" , are parallel to each other, 
then, according to the second part of Desargues's theorem (theorem 32), the third sides 
AA' and B" A" are parallel to each other. 
To prove the associative law of addition 

a + (6 + c) = (a + 6) + c, 

we shall make use of figure 44. In consequence of the commutative law of addition just 
demonstrated, the above formula states that the straight line A" B" must he parallel to 
the unit-line. The validity of this statement is evident, since the shaded part of figure 44 
corresponds exactly with figure 43. 




Fig. 43. 



54 



« + b+c 




a*b b*-c 



Fig. 44. 



26. THE ASSOCIATIVE LAW OF MULTIPLICATION AND THE TWO 
DISTRIBUTIVE LAWS FOR THE NEW ALGEBRA OF SEGMENTS. 



The associative law of multiplication 



a{hc) = {ab)c 



has also a place in our new algebra of segments. 

Let there be given upon the first of the two fixed straight lines through O the segments 

l=OA. b = OC, c = OA' 



abc, D 




a {dc) ^^ {ad) c 
Fig. 45. 



55 

and upon the second of these straight lines, the segments 

a = OG, b = OB. 

In order to construct the segments 

be = OB', and be = OC, 
ah = OD, 
{ab)c = OD', 

in accordance with §24, draw A'B' parallel to AB, B'C' parallel to BC, CD parallel to 
AG, and A'D' parallel to AD. We see at once that the given law amounts to the same 
as saying that CD must also be parallel to CD' . Denote the point of intersection of the 
straight lines A'D' and B'C by F' and that of the straight Hues AD and BC by F. Then 
the triangles ABF and A'B'F' have their homologous sides parallel to each other, and, 
according to Desargues's theorem, the three points O, F, F' must lie in a straight line. 
Because of these conditions, we can apply the second part of Desargues's theorem to the 
two triangle CDF and C'D'F', and hence show that, in fact, CD is parallel to CD' . 

Finally, upon the basis of Desargues's theorem, we shall show that the two distributive 
laws 

a(6 + c) = ab -\- ac 

and 

[a -\- b)c = ac^hc 

hold for our algebra of segments. 

In the proof of the first one of these laws, we shall make use of figure 46. ■'■^ In this 
figure, we have 

h = OA',c = OC, 
ab = OB', ab = OA", ac = OC" , etc. 

In the same figure, B" D2 is parallel to C"Di which is parallel to the fixed straight line OA' , 
and B'Di is parallel to CD2, which is parallel to the fixed straight line OA". Moreover, 
we have A' A" parallel to C'C", and A'B" parallel to B'A", parallel to F'D2, parallel to 
F"Di. 

Our proposition amounts to asserting that we must necessarily have also 

F'F" parallel to A' A" and to C'C". 

We construct the following auxiliary lines: 

F".J parallel to the fixed straight line OA', 

rpf J u u u a u u jQ /I" 

Let us denote the points of intersection of the straight lines C'Di and C'D2, C"Di 
and F' J , CD2 and F" J by G, Hi, H2, respectively. Finally, we obtain the other auxiliary 
lines indicated in the figure by joining the points already constructed. 



Figures 46, 47, and 48 were designed by Dr. Von Schaper, as have also the details of the demonstrations 
relating to these figures. 



56 




a{b '\- c)^=ab -^ ac 

Fig. 46. 



In the two triangles A'B"C" and F'D2G, the straight lines joining homologous vertices 
are parallel to each other. According to the second part of Desargues's theorem, it follows, 
therefore, that 

A'C" is parallel to F'G. 

In the two triangles A'C" F" and F'GH2, the straight lines joining the homologous vertices 
are also parallel to each other. From the properties already demonstrated, it follows by 
virtue of the second part of Desargues's theorem that we must have 

A'F" parallel to F'H2. 

Since in the two horizontally shaded triangles OA'F' and JH2F' the homologous sides 
are parallel, Desargues's theorem shows that the three straight lines joining the homologous 
vertices, viz.: 

OJ, A'H2, F"F' (2) 

all intersect in one and the same point, say in P. 
In the same way, we have necessarily 

A"F' parallel to F"H^ 

and since, in the two obliquely shaded triangles OA"F' and JHiF", the homologous sides 
are parallel, then, in consequence of Desargues's theorem, the three straight lines joining 
the homologous vertices, viz.: 

OJ,A"Hi,F'F" 

all intersect likewise in the same point, namely, in point P. 

Moreover, in the triangles OA'A" and JH2H1, the straight lines joining the homologous 
vertices all pass through this same point P, and, consequently, it follows that we have 

H1H2 parallel to A'yl", 



57 



and, therefore, 

H1H2 pai-allelto C'C", 

Finally, let us consider the figure F" H2C' C" HiF' F" . Since, in this figure, we have 

F"H2 parallel to C'F', parallel to C"Hi, 
GH2C' " " F"C", " " HiF', 
C'C" " " H1H2, 

we recognize here again figure 43, which we have already made use of in § 25 to prove 
the commutative law of addition. The conclusions, analogous to those which we reached 
there, show that we must have 

F'F" par-allel to H1H2 

and, consequently, we must have also 

F'F" parallel toA' A" , 

which result concludes our demonstration. 

To prove the second formula of the distributive law, we make use of an entirely different 
figure, — -figure 47. In this figure, we have 




Fig. 47. 



1 = OD, a = OA, a = OB, h = OG, c = OD', 
ac = OA', ac = OB', BC = OG', etc., 
and, furthermore, we have 

GH parallel to G'H', parallel to the fixed line OA, 
GH " " A'H', " " " " " OB, 



58 

We have also 

AB parallel to A'B' 

ED " " B'D' 

DG " " D'G' 

HJ " " H'J'. 

That which we are to prove amounts, then, to demonstrating that 

DJ must he parallel to D J . 

Denote the points in which BD and GD intersect the straight line AH by C and F, 
respectively, and the points in which B'D' and G'D' intersect the straight line A'H' by 
C and F' , respectively. Finally, draw the auxiliary lines FJ and F'J' , indicated in the 
figure by dotted lines. 

In the triangles ABC and A'B'C', the homologous sides are parallel and, consequently, 
by Desargues's theorem the three points O, C, C lie on a straight line. Then, by con- 
sidering in the same way the triangles CDF and CD'F', it follows that the points O, 
F, F' lie upon the same straight line and likewise, from a consideration of the triangles 
FGH and F'G'H', we find the points O, H, H' to be situated on a straight line. Now, in 
the triangles FHJ and F'H'J', the straight lines joining the homologous vertices all pass 
through the same point O, and, hence, as a consequence of the second part of Desargues's 
theorem, the straight lines FJ and F'J' must also be parallel to each other. Finally, a 
consideration of the triangles DFJ and D'F'J' shows that the straight lines DJ and D'J' 
are parallel to each other and with this our proof is completed. 



§ 27. EQUATION OF THE STRAIGHT LINE, BASED UPON THE NEW 

ALGEBRA OF SEGMENTS. 

In §§ 24-26, we have introduced into the plane geometry an algebra of segments in which 
the commutative law of addition and that of multiplication, as well as the two distributive 
laws, hold. This was done upon the assumption that the axioms cited in § 24, as also 
the theorem of Desargues, were valid. In this section, we shall show how an analytical 
representation of the point and straight line in the plane is possible upon the basis of this 
algebra of segTnents. 

Definition. Take the two given fixed straight lines lying in the plane and intersecting 
in O as the axis of x and of y, respectively. Let us suppose any point P of the plane deter- 
mined by the two segments x, y which we obtain upon the x-axis and y-axis, respectively, 
by drawing through P parallels to these axes. These segments are called the co-ordinates 
of the point P. Upon the basis of this new algebra of segments and by aid of Desargues's 
theorem, we shall deduce the following proposition. 



59 



Theorem 34. The co-ordinates x, y oi a, point on an arbitrary straight hne always 
satisfy an equation in these segments of the form 



ax 



by + c = 0. 



In this equation, the segments a and b stand necessarily to the left of the co-ordinates 
X and y. The segments a and b are never both zero and c is an arbitrary segment. 

Conversely, every equation in these segments and of this form represents always a 
straight line in the plane geometry under consideration. 

Proof. Suppose that the straight line / passes through the origin O. Furthermore, 
let C be a definite point upon I different from O, and P any arbitrary point of l. Let 
OA and OB be the co-ordinates of C and x, y be the co-ordinates of P. We ^^dll denote 
the straight line joining the extremities of the segments x, y by g. Finally, through the 
extremity of the segment 1, laid off on the a:-axis, draw a straight line h parallel to AB. 
This parallel cuts off upon the y-axis the segment e. From the second part of 




Fig. 48. 



Desargues's theorem, it follows that the straight line g is also always parallel to AB. 
Since g is always parallel to h, it follows that the co-ordinates x, y of the point P must 
satisfy the equation 

ex = g. 

Moreover, in figure 49 let /' he any arbitrary straight line in our plane. This straight 
line will cut off on the a:-axis the segment c = OO' . Now, in the same figure, draw through 
O the straight line I parallel to /'. Let P' be an arbitrary point on the line I' . The straight 
line through P' , parallel to the a;-axis, intersects the straight line / in P and cuts off upon 
the y-axis the segTnent y = OB. Finally, through P and P' let parallels to the y-axis cut 
off on the a;-axis the segments x = OA and x' = OA' . 

We shall now undertake to show that the equation 



X = X 



60 



is fulfilled by the segments in question. For this purpose, draw O'C parallel to the unit-line 
and likewise CD parallel to the x-sxis and AD parallel to the y-axis. 




Fig. 49. 



Then, to prove our proposition amounts to showing that we must have necessarily 

A'D parallel to O'C. 

Let D' be the point of intersection of the straight lines CD and A' P' and draw O'C' 
parallel to the y-axis. 

Since, in the triangles OCP and O'C'P', the straight lines joining the homologous 
vertices are parallel, it follows, by virtue of the second part of Desargues's theorem, that 
we must have 

CP parallel to C'P'. 

In a similar way, a consideration of the triangles ACP and A'C'P' shows that we must 
have 

AC parallel to A'C 

Since, in the triangles ACD and C'A'O', the homologous sides ai'e parallel to each other, 
it follows that the straight lines AC, CA' and DO' intersect in a common point. A 
consideration of the triangles C'A'D and ACO' then shows that A'D and CO' are parallel 
to each other. From the two equations already obtained, viz.: 



ex = y and x = a; + c. 



(3) 



follows at once the equation 



ex = y + ec. 



(4) 



If we denote, finally, by n the segment which added to the segment 1 gives the segment 
0, then, from this last equation, we may easily deduce the following 



ex -\- ny -\- nee = 0, 
and this equation is of the form required by theorem 34. 



(5) 



61 

We can now show that the second part of the theorem is equally true; for, every linear 
equation 

ax -\- by -\- c = (6) 

may evidently be brought into the required form 

ex -\- ny + 7iec = (7) 

by a left-sided multiplication by a properly chosen segment. 

It must be expressly stated, however, that, by our hypothesis, an equation of segments 
of the form 

xa^yb^c = 0, (8) 

where the segments a, b stand to the right of the co-ordinates x, y does not, in general, 
represent a straight line. 

In Section 30, we shall make an important application of theorem 34. 



§ 28. THE TOTALITY OF SEGMENTS, REGARDED AS A COMPLEX 

NUMBER SYSTEM. 

We see immediately that, for the new algebra of segments established in Section 24, 
theorems 1-6 of Section 13 are fulfilled. Moreover, by aid of Desargues's theorem, we have 
already shown in Sections 25 and 26 that the laws 7—11 of operation, as given in Section 
13, are all valid in this algebra of segments. With the single exception of the commutative 
law of multiplication, therefore, all of the theorems of connection hold. 

Finally, in order to make possible an order of magnitude of these segments, we make 
the following convention. Let A and B he any two distinct points of the straight line OE. 
Suppose then that the four points O, E, A, B stand, in conformity with axiom II, 4, in a 
certain sequence. If this sequence is one of the following six possible ones, viz.: 

ABOE, AOBE, AOEB, OABE, OAEB, OEAB, 

then we will call the segment a = OA smaller than the segment b = OB and indicate 
the same by writing 

a < b. 
On the other hand, if the sequence is one of the six following ones, viz.: 

BAOE, BOAE, BOEA, OBAE, OBEA, OEBA, 

then we will call the segment a = OA greater than the segment b = OB, and we write 
accordingly 

a> b. 



62 

This convention remains in force wlienever A or B coincides with O or E, only then 
the coinciding points are to be regarded as a single point, and, consequently, we have only 
to consider the order of three points. 

Upon the basis of the axioms of group II, we can easily show also that, in our algebra 
of segments, the laws 13-16 of operation given in Section 13 are fulfilled. Consequently, 
the totality of all the different segments forms a complex number system for which the 
laws 1-11, 13-16 of Section 13 hold; that is to say, all of the usual laws of operation except 
the commutative law of multiplication and the theorem of Archimedes. We will call such 
a system, briefly, a desarguesian number system. 



§ 29. CONSTRUCTION OF A GEOMETRY OF SPACE BY AID OF A 
DESARGUESIAN NUMBER SYSTEM. 

Suppose we have given a desarguesian number system D. Such a system makes possible 
the construction of a geometry of space in which axioms I, II, III are all fulfilled. 

In order to show this, let us consider any system of three numbers {x,y,z) of the 
desarguesian number system D as a point, and the ratio of four such numbers {u : v : 7jU : r) , 
of which the first three are not 0, as a plane. However, the systems (u : v : w : r) and 
[ati : av : aw ; ar), where a is any number of D different from 0, represent the same plane. 
The existence of the equation 

ux -\- vy -\- wz + r = 

expresses the condition that the point (x, y, z) shall lie in the plane {u : v : w : r\ Finally, 
we define a straight line by the aid of a system of two planes {u' : v' : w' : r') and 
{u" : v" : w" : r"), where we impose the condition that it is impossible to find in D two 
numbers a', a" different from zero, such that we have simultaneously the relations 

/ / unit null ti II 
a u = a u ,a V = a v ,aw = a w , 

A point {x,y,z) is said to be situated upon this straight line [[u' : v' : vj' : r'), {u" : v" : 
w" : r")], if it is common to the two planes {v! : v' : w' : r') and {u" : v" : w" : r"). Two 
straight lines which contain the same points are not regarded as being distinct. 

By application of the laws 1-11 of § 13, which by hypothesis hold for the numbers of 
D, we obtain without difficulty the result that the geometry of space which we have just 
constructed satisfies all of the axioms of groups I and III. 

In order that the axioms (II) of order may also be valid, we adopt the following 
conventions. Let 

be any three points of a straight line 

\i I I I i\ I " II 'I '/\i 
[[u : V : w : r ),(u : v : w '■ r )\. 



63 

Then, the point [x2, 2/2,-^2) is said to He between the other two, if we have fulfilled at least 
one of the six following double inequalities: 



3:1 < ^2 < xs, a;i > a;2 > x^, (9) 

yi <y2< y-i, yi> y2> ys, (lo) 

Zi < Z2 < Z3, Zi> Z2> Z3, (11) 

If one of the two double inequalities (1) exists, then we can easily conclude that either 
yi = y2 = ys or one of the two double inequalities (2) exists, and, consequently, either 
zi = Z2 = 23 or one of the double inequalities (3) must exist. In fact, from the equations 



I'xi -\- v' 



u'Xi + v'ui -\- w' Zi + r' = 0, 



vl'xi + v"yi + w"zi + r" = 0, 
(i = l,2,3) 

we may obtain, by a left-sided multiplication of these equations by numbers suitably 
chosen from D and then adding the resulting equations, a system of equations of the form 

v!"xi + v"'yi + r'" = 0, (i = 1, 2, 3). (12) 

In this system, the coefficient v'" is certainly different from zero, since otherwise the three 
numbers xi^ X2-, a^s would be mutually equal. 
From 

it follows that 

/// < /// < /// 

U X\ = X U2 = X Us, 

and, hence, as a consequence of (4), we have 

v'"yi+r"'^v"'y2^r"'fv"'ys^r"' 

and, therefore, 

fi/ < in < !if 

V yi = v y2 = v ys. 

Since v'" is different from zero, we have 

yi ^ 2/2 ^ y-i- 

In each of these double inequalities, we must take either the upper sign throughout, or 
the middle sign throughout, or the lower sign throughout. 

The preceding considerations show, that, in our geometry, the linear axioms 11, 1—4 
of order are all valid. However, it remains yet to show that, in this geometry, the plane 
axiom II, 5 is also valid. 

For this purpose let a plane [u : v : 7ju : r) and a straight line [{u : v : w : r), {v! : v' : 
w' : r')] in this plane be given. Let us assume that all the points {x,y,z) of the plane 
{u : V : w : r), for which we have the expression u'x + v'y + w'z + r' greater than or less 



64 

than zero, lie respectively upon the one side or upon the other side of the given straight 
line. We have then only to show that this convention is in accordance with the preceding 
statements. This, however, is easily done. We have thus shown that all of the axioms of 
groups I, II, III are fulfilled in the geometry of space which we have obtained in the above 
indicated manner from the desarguesian number system D. Remembering now that the 
theorem of Desargues is a consequence of the axioms I, II, III, we see that the proposition 
just stated is exactly the converse of the result reached in Section 28. 



§ 30. SIGNIFICANCE OF DESARGUES'S THEOREM. 

If, in a plane geometry, axioms I, 1—2, II, III are all fulfilled and, moreover, if the theorem 
of Desargues holds, then, according to §§ 24—28, it is always possible to introduce into 
this geometry an algebra of segments to which the laws 1—11, 13-16 of § 13 are applicable. 
We will now consider the totality of these segments as a complex number system and 
construct, upon the basis of this system, a geometry of space, in accordance with § 29, in 
which all of the axioms I, II, III hold. 

In this geometry of space, we shall consider only the points [x, y, 0) and those straight 
lines upon which only such points lie. We have then a plane geometry which must, if 
we take into account the proposition established in § 27, coincide exactly with the plane 
geometry proposed at the beginning. Hence, we are led to the following proposition, which 
may be regarded as the objective point of the entire discussion of the present chapter. 

Theorem 35. If, in a plane geometry, axioms I, 1-2, II, III are all fulfilled, then 
the existence of Desargues's theorem is the necessary and sufficient condition that 
this plane geometry may be regarded as a part of a geometry of space in which all 
of the axioms I, II, III ai'e fulfilled. 

The theorem of Desargues may be characterized for plane geometry as being, so to 
speak, the result of the elimination of the space axioms. 

The results obtained so far put us now in the position to show that every geometry 
of space in which axioms I, II, III are all fulfilled may be always regarded as a part of a 
"geometry of any number of dimensions whatever." By a geometry of an arbitrary number 
of dimensions is to be understood the totality of all points, straight lines, planes, and other 
linear elements, for which the corresponding axioms of connection and of order, as well as 
the axiom of parallels, are all valid. 



65 

PASCAL'S THEOREM. 



§ 31. TWO THEOREMS CONCERNING THE POSSIBILITY OF PROVING 

PASCAL'S THEOREM. 

As is well known, Desargues's theorem (theorem 32) may be demonstrated by the aid of 
axioms I, II, III; that is to say, by the use, essentially, of the axioms of space. In § 23, we 
have shown that the demonstration of this theorem without the aid of the space axioms of 
group I and without the axioms of congruence (gi'oup IV) is impossible, even if we make 
use of the axiom of Archimedes. 

Upon the basis of axioms I, 1-2, II, III, IV and, hence, by the exclusion of the axioms 
of space but with the assistance, essentially, of the axioms of congi'uence, we have, in § 14, 
deduced Pascal's theorem and, consequently, according to § 22, also Desargues's theorem. 
The question arises as to whether Pascal's theorem can be demonstrated without the 
assistance of the axioms of congruence. Our investigation will show that in this respect 
Pascal's theorem is very different from Desargues's theorem; for, in the demonstration of 
Pascal's theorem, the admission or exclusion of the axiom of Archimedes is of decided 
influence. We may combine the essential results of our investigation in the two following 
theorems. 

Theorem 36. Pascal's theorem (theorem 21) may be demonstrated by means of the 
axioms I, II, III, V; that is to say, without the assistance of the axioms of congruence 
and with the aid of the axiom of Archimedes. 

Theorem 37. Pascal's theorem (theorem 21) cannot be demonstrated by means of 
the axioms I, II, III alone; that is to say, by exclusion of the axioms of congruence 
and also the axiom of Archimedes. 

In the statement of these two theorems, we may, by virtue of the general theorem 35, 
replace the space axioms I, 3-7 by the plane condition that Desargues's theorem (theorem 
32) shall be valid. 



§ 32. THE COMMUTATIVE LAW OF MULTIPLICATION FOR AN 
ARCHIMEDEAN NUMBER SYSTEM. 

The demonstration of theorems 36 and 37 rests essentially upon certain mutual relations 
concerning the laws of operation and the fundamental propositions of arithmetic, a knowl- 
edge of which is of itself of interest. We will state the two following theorems. 

Theorem 38. For an archimedean number system, the commutative law of multi- 
plication is a necessary consequence of the remaining laws of operation: that is to 
say, if a number system possesses the properties 1—11, 13-17 given in § 13, it follows 
necessarily that this system satisfies also formula 12. 



66 

Proof. Let us observe first of all that, if a is an arbitrary number of the system, and, 
if 

n= 1 + 1 + --- + 1 

is a positive integral rational number, then for n and a the commutative law of multipli- 
cation always holds. In fact, we have 

an = a(l + IH hi) 

= a-l-\-a-l-\-----\-a- 1 
= a -\- a -\- ■ ■ ■ -\- a 

and likewise 

na = (1 + H \-l)a 

= l-a-\-l-a-\-----\-l-a 
= a -\- a -\- ■ ■ ■ -\- a. 

Suppose now, in contradiction to our hypothesis, a,b to be numbers of this system, for 
which the commutative law of multiplication does not hold. It is then at once evident 
that we may make the assumption that we have 

a > 0, h > 0, ab - ba > 0. 

By virtue of condition 6 of § 13, there exists a number c(> 0), such that 

(a + 6 + l)c = ab — ba. 

Finally, if we select a number rf, satisfying simultaneously the inequalities 

d> 0, d<l, d < c, 

and denote by m and n two such integral rational numbers > that we have respectively 

md < a < (773, + l)d 

and 

nd < b < (n+ l)d 

then the existence of the numbers m and n is an immediate consequence of the theorem of 
Archimedes {theorem 17, § 13). Recalling now the remark made at the beginning of this 
proof, we have by the multiplication of the last inequalities 

ab ^ mnd + [m -\- n -\- l)d 
ba > mnd' , 

and, hence, by subtraction 

ab — ba ^ {rn + n + \)d . 



67 



We have, however, 
and, consequently, 



I.e.. 



or, since d < c, we have 



md < a, nd < b, , d < 1 

(m + n + l)rf < a + 6 + 1; 
ah — ha < (a -[- 6 + l)d, 
ah — ha < {a -\-h -\- l)c. 



This inequality stands in contradiction to the definition of the nuinber c, and, hence, the 
validity of the theorem 38 follows . 



§ 33. THE COMMUTATIVE LAW OF MULTIPLICATION FOR A 
NON- ARCHIMEDEAN NUMBER SYSTEM. 

Theorem 39. For a non-ar chime dean nuinber system, the commutative law of 
multiplication is not a necessary consequence of the remaining laws of operation: 
that is to say, there exists a system of numbers possessing the properties 1-11, 13- 
16 mentioned in § 13, but for which the commutative law (12) of multiplication is 
not valid. A desarguesian nuinber system, in the sense of § 28, is such a system. 

Proof. Let f be a parameter and T any expression containing a finite or infinite 
number of terms, say of the form 

T = rot" + T^e+-' + r2t"+' + r3r+3 + . . . , 

where ro(^ 0),ri,r2... are arbitrary rational numbers and n is an arbitrary integral 
rational number = 0. Moreover, let s be another parameter and S any expression having 
a finite or infinite nuinber of terms, say of the form 



= S Jo + S ^ ii + S ^ i; 



where To{^ 0),Ti,T2, . . . denote arbitrary expressions of the form T and m is again an 
arbitrary integral rational number = 0. We will regard the totality of all the expressions 
of the form S' as a complex number system Q{s,t), for which we will assume the following 
laws of operation; namely, we will operate with s and t according to the laws 7-11 of § 13, 
as with parameters, while in place of rule 12 we will apply the formula 

ts = 2st. (13) 

If, now, S", S" are any two expressions of the form S, say 

s' = s'^''^ + s^^'+^r; + s'^'+'^T!^ + . . . , 

s" = s'^"t;; + .s-"+iT^' + s'""+2t^' + . . . , 



68 

then, by combination, we can evidently form a new expression S' + S" which is of the 
form 5', and is, moreover, miiquely determined. This expression S' + S" is called the sum 
of the numbers represented by S' and S" . 

By the multiplication of the two expressions S' and S" term by term, we obtain another 
expression of the form 



S'S" = s^''%s^"t;; + {s'^'T;ys'--'"+^T'l - s'^'+^T[s'--'"Ti;) 

I / m' rp! m"+2rjil/ I m'+ln-il m"+lrplf . m'+2rpl m" rpll\ , 

-T \S IqS ±2 -r s 1-^s 1-^ ^ s I2S 1q ) -\- . . . 

This expression, by the aid of formula (1), is evidently a definite single-valued expression 
of the form S and we will call it the product of the numbers represented by S' and S" . 

This method of calculation shows at once the validity of the laws 1—5 given in Section 
13 for calculating with numbers. The validity of law 6 of that section is also not difficult 
to establish. To this end, let us assume that 

S' = s'^'t;, + s'^'+^Ti + s'^'+^T^ + ■ ■ ■ 

and 

S'" = s^"'T^" + s^'"'+^T{" + s^"'+^Ti" ■ ■ ■ 

are two expressions of the form 5', and let us suppose, further, that the coefficient Tq of Tq 
is different from zero. By equating the like powers of s in the two members of the equation 

S'S" = s'", 

we find, first of all, in a definite manner an integral number m" as exponent, and then 
such a succession of expressions 

r^', Ti", T^ . . 

that, by aid of formula (1), the expression 

S" = s^^'Ti; + s^'"+^Ti' + .s"^"+2r^' . . . 

satisfies equation (2). With this our theorem is established. 

In order, finally, to render possible an order of sequence of the numbers of our system 
Q[s, t), we make the following conventions. Let a number of this system be called greater 
or less than according as in the expression S, which represents it, the first coefficient 
To of To is greater or less than zero. Given any two numbers a, b of the complex number 
system under consideration, we say that a < b or a > b according as we have a — 6 < or 
> 0. It is seen immediately that, with these conventions, the laws 13-16 of § 13 are valid: 
that is to say, Q.{s, t) is a desarguesian number system (see § 28). 

As equation (1) shows, law 12 of § 13 is not fulfilled by our complex number system 
and, consequently, the validity of theorem 39 is fully established. 

In conformity with theorem 38, Archimedes's theorem (theorem 17, § 13) does not hold 
for the number system fl[s,t) which we have just constructed. 

We wish also to call attention to the fact that the number system n(s, t), as well as the 
systems O and fl{t) made use of in § 9 and § 12, respectively, contains only an enumerable 
set of numbers. 



69 



§ 34. PROOF OF THE TWO PROPOSITIONS CONCERNING PASCAL'S 
THEOREM. (NON-PASCALIAN GEOMETRY.) 

If, in a geometry of space, all of the axioms I, II, III are fulfilled, then Desargues's theorem 
(theorem 32) is also valid, and, consequently, according to §§ 24-26, pp. 50—58, it is 
possible to introduce into this geometry an algebra of segments for which the rules 1-11, 
13-16 of § 13 are all valid. If we assume now that the axiom (V) of Archimedes is valid 
for our geometry, then evidently Archimedes's theorem (theorem 17 of § 13) also holds for 
our algebra of segments, and, consequently, by virtue of theorem 38, the commutative law 
of multiplication is valid. Since, however, the definition of the product of two segments, 
as introduced in § 24 (figure 42) and which is the definition here also under discussion, 
agrees with the definition in § 15 (figure 22), it follows from the construction made in § 
15 that the commutative law of multiplication is here nothing else than Pascal's theorem. 
Consequently, the validity of theorem 36 is established. 

In order to demonstrate theorem 37, let us consider again the desarguesian number 
system Q{s,t) introduced in § 33, and construct, in the manner described in § 29, a 
geometry of space for which all of the axioms I, II, III are fulfilled. However, Pascal's 
theorem will not hold for this geometry; for, the commutative law of multiplication is 
not valid in the desarguesian number system Q.{s,t). According to theorem 36, the non- 
pascalian geometry is then necessarily also a non-ar chime dean geometry. 

By adopting the hypothesis we have, it is evident that we cannot demonstrate Pascal's 
theorem, providing we regard our geometry of space as a part of a geometry of an arbitrary 
number of dimensions in which, besides the points, straight lines, and planes, still other 
linear elements are present, and providing there exists for these elements a corresponding 
system of axioms of connection and of order, as well as the axiom of parallels. 



§ 35. THE DEMONSTRATION, BY MEANS OF THE THEOREMS OF PASCAL 
AND DESARGUES, OF ANY THEOREM RELATING TO POINTS OF 

INTERSECTION. 

Every proposition relating to points of intersection in a plane has necessarily The form: 
Select, first of all, an arbitrary system of points and straight lines satisfying respectively 
the condition that certain ones of these points are situated on certain ones of the straight 
lines. If, in some known manner, we construct the straight lines joining the given points 
and determine the points of intersection of the given lines, we shall obtain finally a definite 
system of three straight lines, of which our proposition asserts that they all pass through 
the same point. 

Suppose we now have a plane geometry in which all of the axioms I 1-2, II . . . , V are 
valid. According to § 17, pp. 33-35, we may now find, by making use of a rectangular 
pair of axes, for each point a corresponding pair of numbers {x,y) and for each straight 
line a ratio of tliree definite numbers (u : v : w). Here, the numbers x, y, u, v, w are all 



70 

real numbers, of which u, v cannot both be zero. The condition showing that the given 
point is situated upon the given straight Hne, viz.: 

ux -\- vy -\- w = (14) 

is an equation in the ordinary sense of the word. Conversely, in case x, y, u, v, w are 
numbers of the algebraic domain fi of § 9, and zi, v are not both zero, we may certainly 
assume that each pair of numbers (x, y) gives a point and that each ratio of three numbers 
[u : V : w) gives a straight line in the geometry in question. 

If, for all the points and straight lines which occur in connection with any theorem 
relating to intersections in a plane, we introduce the corresponding pairs and triples of 
numbers, then such a theorem asserts that a definite expression yl(pi,p2, ■ ■ ■ ,Pr) with 
real coefficients and depending rationally upon certain parameters pi,p2, ■ ■ ■ ,Pr always 
vanishes as soon as we put for each of these parameters a number of the main Q considered 
in § 9. We conclude from this that the expression A{pi,p2, ■ ■ ■ .iPr) must also vanish 
identically in accordance with the laws 7-12 of § 13. 

Since, according to § 32, Desargues's theorem holds for the geometry in question, 
it follows that we certainly can make use of the algebra of segments introduced in § 
24, and because Pascal's theorem is equally valid in this case, the commutative law of 
multiplication is also. Hence, for this algebra of segments, all of the laws 7—12 of § 13 are 
valid. 

If we take as our axes in this new algebra of seginents the co-ordinate axes already used 
and consider the unit points E, E' as suitably established, we see that the new algebra of 
segments is nothing else than the system of co-ordinates previously employed. 

In order to show that, for the new algebra of segments, the expression A{pi, p2, ■ ■ . , Pr) 
vanishes identically, it is sufficient to apply the theorems of Pascal and Desargues. Con- 
sequently we see that: 

Every proposition relative to points of intersection in the geometry in question must 
always, by the aid of suitably constructed auxiliary points and straight lines, turn out to be 
a combination of the theorems of Pascal and Desargues. Hence for the proof of the validity 
of a theorem relating to points of intersection, we need not have resource to the theorems 
of congruence. 



71 



GEOMETRICAL CONSTRUCTIONS BASED UPON THE 

AXIOMS I-V. 



§ 36. GEOMETRICAL CONSTRUCTIONS BY MEANS OF A STRAIGHT-EDGE 
AND A TRANSFERER OF SEGMENTS. 

Suppose we have given a geometry of space, in which all of the axioms I-V are valid. For 
the sake of simplicity, we shall consider in this chapter a a plane geometry which is con- 
tained in this geometry of space and shall investigate the question as to what elementary 
geometrical constructions may be carried out in such a geometry. 

Upon the basis of the axioms of group I, the following constructions are always possible. 

Problem 1. To join two points with a straight line and to find the intersection of two 
straight lines, the lines not being parallel. 

Axiom in renders possible the following construction; 

Problem 2. Through a given point to draw a parallel to a given straight line. 

By the assistance of the axioms (IV) of congruence, it is possible to lay off segments 
and angles; that is to say, in the given geometry we may solve the following problems: 

Problem 3. To lay off from a given point upon a given straight line a given segment. 

Problem 4. To lay off on a given straight line a given angle; or what is the same 
thing, to construct a straight line which shall cut a given straight line at a given angle. 

It is impossible to make any new constructions by the addition of the axioms of groups 
II and V. Consequently, when we take into consideration merely the axioms of groups 
I-V, all of those constructions and only those are possible, which may be reduced to the 
problems 1-4 given above. 

We will add to the fundamental problems 1—4 also the following: 

Problem 5. To draw a perpendicular to a given straight line. 

We see at once that this construction can be made in different ways by means of the 
problems 1-4. 

In order to carry out the construction in problem 1, we need to make use of only a 
straight edge. An instrument which enables us to make the construction in problem 3, we 
will call a transferer of segments. We shall now show that problems 2, 4, and 5 can be 
reduced to the constructions in problems 1 and 3 and, consequently, all of the problems 1- 
5 can be completely constructed by means of a straight-edge and a transferer of segments. 
We arrive, then, at the following result: 

Theorem 40. Those problems in geometrical construction, which may he solved by 
the assistance of only the axioms I-V, can always be carried out by the use of the 
straight-edge and the transferer of segments. 

Proof. In order to reduce problem 2 to the solution of problems 1 and 3, we join the 
given point P with any point A of the given straight line and produce PA to C, making 
AC = PA. Then, join C with any other point B of the given straight line and produce 
CB to Q, making, BQ = CB. The straight line PQ is the desired parallel. 



72 




Fig. 50. 



We can solve problem 5 in the following manner. Let A he an arhitrary point of the 
given straight line. Then upon this straight line, lay off in both directions from A the two 
equal segments AB and AC. Determine, upon any two straight lines passing through the 
point A, the points E and D so that the segments AD and AE will equal AB and AC. 




Suppose the straight lines BD and CE intersect in F and the straight lines BE and 
CD intersect in H . FH is then the desired perpendicular. In fact, the angles BDC and 
BEC, being inscribed in a semicircle having the diameter BC, are both right angles, and, 
hence, according to the theorem relating to the point of intersection of the altitudes of a 
triangle, the straight lines FH and BC are perpendicular to each other. 

Moreover, we can easily solve problem 4 simply by the drawing of straight lines and 
the laying off of segments. We will employ the following method which requires only the 
drawing of parallel lines and the erection of perpendiculars. Let (3 be the angle to be laid 
off and A its vertex. Draw through A a straight line / parallel to the given straight line, 
upon which we are to lay off the given angle /?. From an arbitrary point B of one side of 
the angle /?, let fall a perpendicular upon the other side of this angle and also one upon I. 



73 




Fig. 52. 



Denote the feet of these perpendiculars by D and C respectively. The construction of these 
perpendiculars is acconipUshed by means of problems 2 and 5. Then, let fall from A a 
perpendicular upon CD, and let its foot be denoted by E. According to the demonstration 
given in Section § 14, the angle CAE equals /3. Consequently, the construction in 4 is 
made to depend upon that of 1 and 3 and with this our proposition is demonstrated. 



37. ANALYTICAL REPRESENTATION OF THE CO-ORDINATES OF 
POINTS WHICH CAN BE SO CONSTRUCTED. 



Besides the elementai'y geometrical problems considered in § 36, there exists a long series 
of other problems whose solution is possible by the drawing of straight lines and the laying 
off of segments. In order to get a general survey of the scope of the problems which may 
be solved in this manner, let us take as the basis of our consideration a system of axes 
in rectangTilar co-ordinates and suppose that the co-ordinates of the points are, as usual, 
represented by real numbers or by functions of certain arbitrary parameters. In order to 
answer the question in respect to all the points capable of such a construction, we employ 
the following considerations. 

Let a system of definite points be given. Combine the co-ordinates of these points 
into a domain R. This domain contains, then, certain real numbers and certain arbitrary 
parameters p. Consider, now, the totality of points capable of construction by the drawing 
of straight lines and the laying off of definite segments, making use of the system of points 
in question. We will call the domain formed from the co-ordinates of these points n(i?), 
which will then contain real numbers and functions of the arbitrary parameters p. 

The discussion in § 17 shows that the drawing of straight lines and of parallels amounts, 
analytically, to the addition, subtraction, multiplication, and division of segments. Fur- 
thermore, the well known formula given in § 9 for a rotation shows that the laying off of 
segments upon a straight line does not necessitate any other analytical operation than the 



74 

extraction of the square root of the sum of the squares of two segments whose bases have 
been previously constructed. Conversely, in consequence of the pythagorean theorem, we 
can always construct, by the aid of a right triangle, the square root of the sum of the 
squai'es of two segments by the mere laying off of segments. 

From these considerations, it follows that the domain n(i?) contains all of those and 
only those real numbers and functions of the parameters p, which arise from the numbers 
and parameters in R by means of a finite number of applications of the five operations; 
viz., the four elementary operations of arithmetic and, in addition, the fifth operation 
of extracting the square root of the sum of two squares. We may express this result as 
follows: 

Theorem 41. A problem in geometrical construction is, then, possible of solution 
by the drawing of straight lines and the laying off of segments, that is to say, by 
the use of the straight-edge and a transferer of segments, when and only when, by 
the analytical solution of the problem, the co-ordinates of the desired points are 
such functions of the co-ordinates of the given points as may be determined by the 
rational operations and, in addition, the extraction of the square root of the sum of 
two squares. 

From this proposition, we can at once show that not every problem which can be solved 
by the use of a compass can also be solved by the aid of a transferer of segments and a 
straight-edge. For the purpose of showing this, let us consider again that geometry which 
was constructed in § 9 by the help of the domain Q. of algebraic numbers. In this geometry, 
there exist only such segments as can be constructed by means of a straight-edge and a 
transferer of segments, namely, the segments determined by the numbers of the domain 

n. 

Now, if a; is a number of the domain Q, we easily see from the definition of fi that every 
algebraic number conjugate to lo must also lie in Q. Since the numbers of the domain Q 
are evidently all real, it follows that it can contain only such real algebraic numbers as 
have their conjugates also real. 

Let us now consider the following problem; viz., to construct a right triangle having the 



hypotenuse 1 and one side |\/2| — 1- The algebraic number v/2|v2| — 2, which expresses 
the numerical value of the other side, does not occur in the domain H, since the conjugate 



number \/ — 2|\/2| — 2 is imaginary. This problem is, therefore, not capable of solution in 
the geometry in question and, hence, cannot be constructed by means of a straight-edge 
and a transferer of segments, although the solution by means of a compass is possible. 



§ 38. THE REPRESENTATION OF ALGEBRAIC NUMBERS AND OF 
INTEGRAL RATIONAL FUNCTIONS AS SUMS OF SQUARES. 

The question of the possibility of geometrical constructions by the aid of a straight-edge 
and a transferer of segments necessitates, for its complete treatment, particular theorems 



75 

of an arithmetical and algebraic character, which, it appears to me, are themselves of 
interest. Since the time of Fermat, it has been known that every positive integral rational 
number can be represented as the sum of four squares. This theorem of Fermat permits 
the following remarkable generalization: 

Definition. Let k be an arbitrary number field and let m be its degree. We will 
denote by k', k", . . . , fe''"~-^' the m — 1 number fields conjugate to k. If, among the m 
fields k, fc', fc", . . . , fc''"~-^' there is one or more formed entirely of real numbers, then we 
call these fields real. Suppose that the fields k, k', . . . , k^^~^' are such. A number a of 
the field k is called in this case totally positive in k, whenever the s numbers conjugate to 
a, contained respectively in fc, fc', k", . . . , k^^~^' are all positive. However, if in each of 
the m fields k, k', k", . . . , fc''"~-^' there are also imaginary numbers present, we call every 
number a in k totally positive. 

We have, then, the following proposition: 

Theorem 42. Every totally positive number in k may be represented as the sum 
of four squares, whose bases are integral or fractional numbers of the field k. 

The demonstration of this theorem presents serious difficulty. It depends essentially upon 
the theory of relatively quadratic number fields, which I have recently developed in several 
papers. ■'■^ We will here call attention only to that proposition in this theory which gives 
the condition that a ternary diophantine equation of the form 

aC' + Pv' + 7C' = 

can he solved when the coefficients a, /3, 7 are given numbers in k and ^, 7], (^ are the 
required numbers in k. The demonstration of theorem 42 is accomplished by the repeated 
application of the proposition just mentioned. 

From theorem 42 follow a series of propositions concerning the representation of such 
rational functions of a variable, with rational coefficients, as never have negative values. I 
will mention only the following theorem, which will be of service in the following sections. 

Theorem 43. Let, f{x} be an integral rational function of x whose coefficients are 
rational numbers and which never becomes negative for any real value of x. Then 
f{x) can always be represented as the quotient of two sums of squares of which the 
bases are all integral rational functions of x with rational coefficients. 

Proof. We will denote the degree of the function f{x) by m, which, in any case, must 
evidently be even. When m = 0, that is to say, when f{x) is a rational number, the validity 
of theorem 43 follows immediately from Fermat's theorem concerning the representation 
of a positive number as the sum of four squares. We will assume that the proposition is 
already established for functions of degree 2, 4, 6, . . . , m — 2, and show, in the following 
manner, its validity for the case of a function of the m*^ degree. 

Let us, first of all, consider briefiy the case where f{x) breaks up into the product 
of two or more integral functions of x with rational coefficients. Suppose p{x) to be one 



17(( 



Ueber die Theorie der relativquadratischen Zahlkorper," Jahresbericht der Deutschen Math. Vere- 
inigung, Vol. 6, 1899, and Math. Annalen, Vol. 51. See, also, "Ueber die Theorie der relativ-Abelschen 
Zahlkorper" Nachr. der K. Ges. der Wiss. zu G ottingen, 1898. 



76 

of those functions contained in f{x), which itself cannot he furtlier decomposed into a 
product of integral functions having rational coefficients. It then follows at once from the 
"definite" character which we have given to the function f{x), that the factor p{x) must 
either appear in f{x) to an even degree or p{x) must be itself "definite"; that is to say, 
must be a function which never has negative values for any real values of x. In the first 
case, the quotient r , U2 and, in the second case, both p{x) and (j , are "definite," and 
these functions have an even degree < m. Hence, according to our hypothesis, in the first 
case, I- , --I2 and, in the last case, p{x) and A^ may be represented as the quotient of the 
sum of squares of the character mentioned in theorem 43. Consequently, in both of these 
cases, the function f{x} admits of the required representation. 

Let us now consider the case where f{x) cannot be broken up into the product of 
two integral functions having rational coefficients. The equation f{9) = defines, then, a 
field of algebraic numbers k{0) of the m*^ degi'ee, which, together with all their conjugate 
fields, are imaginary. Since, according to the definition given just before the statement of 
theorem 42, each number given in k{6}, and hence also —1 is totally positive in k{$}, it 
follows from theorem 42 that the number —1 can be represented as a sum of the squares 
of four definite numbers in k[6). Let, for example 

-l = a2 + /32 + 7'+.5^ (1) 

where a, /?, 7, 6 are integral or fractional numbers in k{0). Let us put 

a = 016*"^-! + a2<9'""^ + ■ ■ ■ + a„^ = 4>(e), 
/3 = 6i^--i + 62^"^-' + ■ ■ ■ + &m = HO), 

e = diO^'-^ + d2e^-'' + ■ ■ ■ ^ dm = p(e); 

where oi, 02, . . . , Om? ■ ■ ■ ? ^i-, ^2? ■ ■ ■ j f^m are the rational numerical coefficients and 
(j>{0\ 'ip{0), x{0), p{0) the integral rational functions in question, having the degTee (m— 1) 
in^. 

From (1), we have 

1 + imr + mo)r + {xwr + {pm' = 

Because of the irreducibility of the equation f{x} = 0, the expression 

F{x) = 1 + imr + {tmr + {x{e)r + w^)}^ 

represents, necessarily, an integral rational function of x which is divisible by f{x). F{x) 
is, then, a "defoiite" function of the degree [2m — 2) or lower. Hence, the quotient -jj^ 
is a "definite" function of the degree [m — 2) or lower in x, having rational coefficients. 
Consequently, by the hypothesis we have made, -jj^ can be represented as the quotient 
of two sums of squares of the kind mentioned in theorem 43 and, since F{x) is itself such 
a sum of squares, it follows that, f{x) must also be a quotient of two sums of squares of 
the required kind. The validity of theorem 43 is accordingly established. 

It would be perhaps difficult to formulate and to demonstrate the corresponding propo- 
sition for integral functions of two or more variables. However, I will here merely remark 



77 

that I have demonstrated in an entirely different manner the possibihty of representing any 
"definite" integral rational function of two variables as the quotient of sums of squares 
of integral functions, upon the hypothesis that the functions represented may have as 
coefficients not only rational but any real numbers. ■'■^ 



§ 39. CRITERION FOR THE POSSIBILITY OF A GEOMETRICAL 
CONSTRUCTION BY MEANS OF A STRAIGHT-EDGE AND A TRANSFERER 

OF SEGMENTS. 

Suppose we have given a problem in geometrical construction which can be affected by 
means of a compass. We shall attempt to find a criterion which will enable us to decide, 
from the analytical nature of the problem and its solutions, whether or not the construction 
can be carried out by means of only a straight-edge and a transferer of segments. Our 
investigation will lead us to the following proposition. 

Theorem 44. Suppose we have given a problem in geometrical construction, which 
is of such a character that the analytical treatment of it enables us to determine 
uniquely the co-ordinates of the desired points from the co-ordinates of the given 
points by means of the rational operations and the extraction of the square root. Let 
n be the smallest number of square roots which suffice to calculate the co-ordinates 
of the points. Then, in order that the required construction shall be possible by the 
drawing of straight lines and the laying off of segments, it is necessary and sufficient 
that the given geometrical problem shall have exactly 2^ real solutions for every 
position of the given points; that is to say, for all values of the arbitrary parameter 
expressed in terms of the co-ordinates of the given points. 

Proof. We shall demonstrate this proposition merely for the case where the co- 
ordinates of the given points are rational functions, having rational coefficients, of a single 
parameter p. 

It is at once evident that the proposition gives a necessary condition. In order to 
show that it is also sufficient, let us assume that it is fulfilled and then, among the n 
squai'e roots, consider that one which, in the calculation of the co-ordinates of the desired 
points, is first to be extracted. The expression under this radical is a rational function 
fi{p), having rational coefficients, of the parameter p. This rational function cannot have 
a negative value for any real value of the parameter p; for, otherwise the problem must 
have imaginary solutions for certain values of p, which is contrary to the given hypothesis. 
Hence, from theorem 43, we conclude that fi{p) can be represented as a quotient of the 
sums of squares of integral rational functions. 

Moreover, the formulee 



i2+62 + c2 = W{\/a2+62) 



a2 + 62 + c2 + rf2 = V/ { Va2 + 62 + c2)2 + rf2 



See "Ueber ternare definite Formen," Acta mathematica, Vol. 17. 



78 



show that, in general, the extraction of the square root of a sum of any number of squares 
may always be reduced to the repeated extraction of the square root of the sum of two 
squai'es. 

If now we combine this conclusion with the preceding results, it follows that the ex- 
pression V /i(p) can certainly be constructed by means of a straight-edge and a transferer 
of segments. Among the n square roots, consider now the second one to be extracted in 
the process of calculating the co-ordinates of the required points. The expression under 
this radical is a rational function /2(p, Vfi) of the parameter p and the square root first 
considered. This function /2 can never be negative for any real arbitrary value of the pa- 
rameter p and for either sign of \//i; for, otherwise among the 2" solutions of our problem, 
there would exist for certain values of p also imaginary solutions, which is contrary to our 
hypothesis. It follows, therefore, that /2 must satisfy a quadratic equation of the form 

/|-92(rt/2+^l(p)=0, 

where 0i [p) and tl'i [p) are, necessarily, such rational functions of p as have rational coef- 
ficients and for real values of ^ never become negative. From this equation, we have 

<?i{p) 

Now, according to theorem 43, the functions 0i(p) and ipi{p) must again be the quo- 
tient of the sums of squares of rational functions, and, on the other hand, the expression 
/2 may be, from the above considerations, constructed by means of a straight-edge and a 
transferer of segments. The expression found for /2 shows, therefore, that /2 is a quotient 
of the sum of squares of functions which may be constructed in the same way. Hence, the 
expression ^/J2 can also be constructed by means of a straight-edge and a transferer of 
segments. 

Just as with the expression /2, any other rational function (f'2{p-,Vfi) of P and i//i 
may be shown to be the quotient of two sums of squares of functions which may be 
constructed, provided this rational function (1)2 possesses the property that for real values 
of the parameter p and for either sign of \/Ji, it never becomes negative. 

This remark permits us to extend the above method of reasoning in the follo"wing 
manner. 

Let /3(p, \//i, \//2) be such an expression as depends in a rational manner upon the 
three arguments p, \//i; Vf2 and of which, in the analytical calculation of the co-ordinates 
of the desired points, the square root is the third to be extracted. As before, it follows 
that /s can never have negative values for real values of p and for either sign of i//i and 
\//2- This condition of affairs shows again that /s must satisfy a quadratic equation of 
the form 

fi - Mp, \/Ti)f3 - Mp, \/a) = 0, 



where (f>2 and 1^2 are such rational functions of p and vA as never become negative for 
any real value of p and either sign of \//i. But, according to the preceding remark, the 



79 

functions d)2 and ?/?2 are the quotients of two sums of squares of functions which may be 
constructed and, hence, it follows that the expression 

<?2(p,Vfl) 

is likewise possible of construction by aid of a straight-edge and a transferer of segments. 

The continuation of this method of reasoning leads to the demonstration of theorem 
44 for the case of a single parameter p. 

The truth of theorem 44 for the general case depends upon whether or not theorem 
43 can be generalized in a similar manner to cover the case of two or more variables. 

As an example of the application of theorem 44, we may consider the regular polygons 
which may he constructed by means of a compass. In this case, the arbitrary parameter, 
p does not occur, and the expressions to be constructed all represent algebraic numbers. 
We easily see that the criterion of theorem 44 is fulfilled, and, consequently, it follows that 
the above-mentioned regular polygons can be constructed by the drawing of straight lines 
and the laying off of segments. We might deduce this result also directly from the theory 
of the division of the circle [Kreisteilung] . 

Concerning the other known problems of construction in the elementary geometry, we 
will here only mention that the problem of Malfatti may he constructed by means of a 
straight-edge and a transferer of segments. This is, however, not the case with the contact 
problems of Appolonius. 



80 



CONCLUSION. 

The preceding work treats essentially of the problems of the euclidean geometry only; that 
is to say, it is a discussion of the questions which present themselves when we admit the 
validity of the axiom of parallels. It is none the less important to discuss the principles 
and the fundamental theorems when we disregard the axiom of parallels. We have thus 
excluded from our study the important question as to whether it is possible to construct 
a geometry in a logical manner, without introducing the notion of the plane and the 
straight line, by means of only points as elements, making use of the idea of groups of 
transformations, or employing the idea of distance. This last question has recently been 
the subject of considerable study, due to the fundamental and prolific works of Sophus 
Lie. However, for the complete elucidation of this question, it would be well to divide 
into several parts the axiom of Lie, that space is a numerical multiplicity. First of all, it 
would seem to me desirable to discuss thoroughly the hypothesis of Lie, that functions 
which produce transformations are not only continuous, but may also be differentiated. 
As to myself, it does not seem to me probable that the geometrical axioms included in the 
condition for the possibility of differentiation are all necessary. 

In the treatment of all questions of this character, I believe the methods and the 
principles employed in the preceding work will be of value. As an example, let me call 
attention to an investigation undertaken at my suggestion by Mr. Dehn, and which has 
already appeared. ■'■^ In this ai'ticle, he has discussed the known theorems of Legendre 
concerning the sum of the angles of a triangle, in the demonstration of which that geometer 
made use of the idea of continuity. 

The investigation of Mr. Dehn rests upon the axioms of connection, of order, and of 
congruence: that is to say, upon the axioms of groups I, II, IV. However, the axiom of 
parallels and the axiom of Archimedes are excluded. Moreover, the axioms of order are 
stated in a more general maimer than in the present work, and in substance as follows: 
Among four points A,B,C, D of a straight line, there are always two, for example A,C, 
which are separated from the other two and conversely. Five points A, B,C,D, E upon 
a straight line may always be so arranged that A,C shall be separated from B,E and 
from B, D. Consequently, A, D are always separated from B, E and from C, E, etc. The 
(elliptic) geometry of Riemann, which we have not considered in the present work, is in 
this way not necessarily excluded. 

Upon the basis of the axioms of connection, order, and congruence, that is to say, 
the axioms I, II, IV, we may introduce, in the well known manner, the elements called 
ideal, — -ideal points, ideal straight lines, and ideal planes. Having done this, Mr. Dehn 
demonstrates the following theorem. 

If, with the exception of the straight line t and the points lying upon it, we regard 
all of the straight lines and all of the points (ideal or real) of a plane as the elements 
of a new geometry, we may then define a new kind of congruence so that all of the 



^^Math. Annalen, Vol. 53 (1900). 



81 

axioms of connection, order, and congruence, as well as the axiom of Euclid, shall 
be fulfilled. In this new geometry, the straight line t takes the place of the straight 
line at infinity. 

This euclidean geometry, superimposed upon the non-euclidean plane, may be called 
a pseudo-geometry and the new kind of congruence a pseudo- congruence. 

By aid of the preceding theorem, we may now introduce an algebra of segments relating 
to the plane and depending upon the developments made in §15, pp. 29-31. This algebra 
of segments permits the demonstration of the following important theorem: 

If, in any triangle whatever, the sum of the angles is greater than, equal to, or less 
than, two right angles, then the same is true for all triangles. 

The case where the sum of the angles is equal to two right angles gives the well known 
theorem of Legendre. However, in his demonstration, Legendre makes use of continuity. 

Mr. Dehn then discusses the connection between the three different hypotheses relative 
to the sum of the angles and the three hypotheses relative to parallels. 

He arrives in this manner at the following remarkable propositions. 

Upon the hypothesis that through a given point we may draw an infinity of lines 
parallel to a given straight line, it does not follow, when we exclude the axiom of 
Archimedes, that the sum of the angles of a triangle is less than two right angles, 
but on the contrai'y, this sum may be 

(a) greater than two right angles, or 

(6) equal to two right angles. 

In order to demonstrate part (a) of this theorem, Mr. Dehn constructs a geometry 
where we may draw through a point an infinity of lines parallel to a given straight line 
and where, moreover, all of the theorems of Riemann's (elliptic) geometry are valid. This 
geometry may be called non-legendrian, for it is in contradiction with that theorem of 
Legendre by virtue of which the sum of the angles a triangle is never greater than two 
right angles. From the existence of this non-legendrian geometry, it follows at once that it is 
impossible to demonstrate the theorem of Legendre just mentioned without employing the 
axiom of Archimedes, and in fact, Legendre made use of continuity in his demonstration 
of this theorem. 

For the demonstration of case (6), Mr. Dehn constructs a geometry where the axiom 
of parallels does not hold, but where, nevertheless, all of the theorems of the euclidean 
geometry are valid. Then, we have the sum of the angles of a triangle equal to two right 
angles. There exist also similar triangles, and the extremities of the perpendiculars having 
the same length and their bases upon a straight line all lie upon the same straight line, 
etc. The existence of this geometry shows that, if we disregard the axiom of Archimedes, 
the axiom of parallels cannot he replaced by any of the propositions which we usually 
regard as equivalent to it. 

This new geometry may be called a semi-euclidean geometry. As in the case of the 
non-legendrian geometry, it is clear that the semi-euclidean geometry is at the same time 
a non-ar chime dean geometry. 



82 



Mr. Dehn finally arrives at the following surprising theorem: 

Upon the hypothesis that there exists no parallel, it follows that the sum of the 
angles of a triangle is greater than two right angles. 

This theorem shows that, with respect to the axiom of Archimedes, the two non- 
euclidean hypotheses concerning parallels act very differently. 
We may combine the preceding results in the follo"wing table. 





THOUGH A GIVEN POINT 


WE MAY DRAW 


THE SUM OF 


NO PARALLELS 


ONE PARALLEL 


AN INFINITY OF PARALLELS 


THE ANGLES 


TO A 


TO A 


TO A STRAIGHT LINE 


OF A TRIANGLE IS 


STRAIGHT LINE 


STRAIGHT LINE 




> 2 right 


Riemann's 


This case is 


Non-legendrian geometry 


angles 


(elliptic) geometry 


impossible 




< 2 right 


This case is 


Euclidean 


Semi-euclidean geometry 


angles 


impossible 


(parabolic) geometry 




= 2 right 


This case is 


This case is 


Geometrv of Lobatschewski 


angles 


impossible 


impossible 


(hyperbolic) 



However, as I have already remarked, the present work is rather a critical investigation 
of the principles of the euclidean geometry. In this investigation, we have taken as a 
guide the following fundamental principle: viz., to make the discussion of each question 
of such a character as to examine at the same time whether or not it is possible to 
answer this question by following out a previously determined method and by employing 
certain limited means. This fundamental rule seems to me to contain a general law and 
to conform to the nature of things. In fact, whenever in our mathematical investigations 
we encounter a problem or suspect the existence of a theorem, our reason is satisfied only 
when we possess a complete solution of the problem or a rigorous demonstration of the 
theorem, or, indeed, when we see clearly the reason of the impossibility of the success and, 
consequently, the necessity of failure. 

Thus, in the modern mathematics, the question of the impossibility of solution of 
certain problems plays an important role, and the attempts made to answer such questions 
have often been the occasion of discovering new and fruitful fields for research. We recall 
in this connection the demonstration by Abel of the impossibility of solving an equation 
of the fifth degree by means of radicals, as also the discovery of the impossibility of 
demonstrating the axiom of parallels, and, finally, the theorems of Hermite and Lindeman 
concerning the impossibility of constructing by algebraic means the numbers e and tt. 

This fundamental principle, which we ought to bear in mind when we come to discuss 
the principles underlying the impossibility of demonstrations, is intimately connected with 
the condition for the "purity" of methods in demonstration, which in recent times has been 
considered of the highest importance by many mathematicians. The foundation of this 
condition is nothing else than a subjective conception of the fundamental principle given 
above. In fact, the preceding geometrical study attempts, in general, to explain what are 
the axioms, hypotheses, or means, necessary to the demonstration of a truth of elementary 
geometry, and it only remains now for us to judge from the point of view in which we 
place ourselves as to what are the methods of demonstration which we should prefer. 



83 



APPEND IX.2° 

The investigations by Riemann and Helmholtz of the foundations of geometry led Lie 
to take up the problem of the axiomatic treatment of geometry as introductory to the 
study of groups. This profound mathematician introduced a system of axioms which he 
showed by means of his theory of transformation groups to be sufficient for the complete 
development of geometry.^-'- 

As the basis of his transformation groups, Lie made the assumption that the functions 
defining the group can be differentiated. Hence in Lie's development, the question remains 
uninvestigated as to whether this assumption as to the differentiability of the functions in 
question is really unavoidable in developing the subject according to the axioms of geome- 
try, or whether, on the other hand, it is not a consequence of the group-conception and of 
the remaining axioms of geometry. In consequence of his method of development. Lie has 
also necessitated the express statement of the axiom that the group of displacements is 
produced by infinitesimal transformations. These requirements, as well as essential parts 
of Lie's fundamental axioms concerning the nature of the equation deffiiing points of equal 
distance, can be expressed geometrically in only a very unnatural and complicated man- 
ner. Moreover, they appear only through the analytical method used by Lie and not as a 
necessity of the problem itself. 

In what follows, I have therefore attempted to set up for plane geometry a system of 
axioms, depending likewise upon the conception of a group, ^^ which contains only those 
requirements which are simple and easily seen geometrically. In particular they do not 
require the differentiability of the functions defining displacement. The axioms of the 
system which I set up are a special di\'ision of Lie's, or, as I believe, are at once deducible 
from his. 

My method of proof is entirely different from Lie's method. I make use particularly of 
Cantor's theory of assemblages of points and of the theorem of C. Jordan, according to 
which every closed continuous plane curve free from double points divides the plane into 
an inner and an outer region. 

To be sure, in the system set up by me, particular parts are unnecessary. However, I 
have turned aside from the further investigation of these conditions to the simple statement 
of the axioms, and above all because I wish to avoid a comparatively too complicated proof, 
and one which is not at once geometrically evident. 

In what follows I shall consider only the axioms relating to the plane, although I 
suppose that an analogous system of axioms for space can be set up which will make 
possible the construction of the geometry of space in a similar manner. 

We establish the following convention, namely: We will understand by number-plane 
the ordinary plane having a rectangular system of co-ordinates x,y. 



The following is a summary of a paper by Professor Hilbert which is soon to appear in full in the 
Math. Annalen. — Tr. 

~ See Lie-Engel, Theorie der Transformations gruppen, Vol. .3, Chapter 5. 

By the following investigation is answered also, as I believe, a general question concerning the theory 
of groups, which I proposed in my address on "MathematischeProbleme," Gottinger Nachrichten, 1900, p. 
17. 



84 

A continuous curve lying in this number-plane and being free from double points and 
including its end points is called a Jordan curve. If the Jordan curve is closed, the interior 
of the region of the number-plane bounded by it is called a Jordan region. 

For the sake of easier representation and comprehension, 1 shall in the following inves- 
tigation formulate the definition of the plane in a more restricted sense than my method 
of proof requires, ^^ namely: I shall assume that it is possible to map^^ in a reversible, 
single- valued manner all of the points of our geometry at the same time upon the points 
lying in the finite region of the number-plane, or upon a definite partial system of the 
same. Hence, each point of our geometry is characterized by a definite pair of numbers 
a:, y. We formulate this statement of the idea of the plane as follows: 

Definitions of the Plane. The plane is a system of points which can be mapped in 
a reversible, single- valued manner upon the points lying in the finite region of the number- 
plane, or upon a certain partial system of the same. To each point A of our geometry, 
there exists a Jordan curve in whose interior the map of A lies and all of whose points 
likewise represent points of our geometry. This Jordan region is called the domain of the 
point A. Each Jordan region contained in a Jordan region which includes the point A is 
likewise called a domain of A. If B is any point in a domain of A, then this domain is at 
the same time called also a domain of B. 

If A and B are any two points of our geometry, then there always exists a domain 
which contains at the same time both of the points A and B. 

We will define a displacement as a reversible, single- valued transformation of a plane 
into itself. Evidently we may distinguish two kinds of reversible, single- valued, continuous 
transformations of the number-plane into itself. If we take any closed Jordan curve in 
the number-plane and think of its being traversed in a definite sense, then by such a 
transformation this curve goes over into another closed Jordan curve which is also traversed 
in a certain sense. We shall assume in the present investigation that it is traversed in 
the same sense as the original Jordan curve, when we apply a transformation of the 
number-plane into itself, which defines a displacement. This assumption^'^ necessitates 
the following statement of the conception of a displacement. 

Definition of Displacement. A displacement is a reversible, single- valued, contin- 
uous transformation of the maps of the given points upon the number-plane into them- 
selves in such a manner that a closed Jordan curve is traversed in the same sense after 
the transformation as before. A displacement by which the point M remains unchanged 
is called a rotation^^ about the point Af . 

In accordance with the conventions setting forth the notions "plane" and "displace- 
ment," we set up the three following axioms: 



" Concerning the broader statement of the conception of the plane see my note, "Ueber die Grundlagen 
der Geometrie," Gbttinger Nachrichten, 1901. 
Abbilden. 

Lie makes this assumption to contain the condition that the group of displacements be generated by 
infinitesimal transformations. The opposite assumption would assist essentially the demonstration in so 
far as the "true straight line" could then be defined as the locus of those points which remain unchanged 
by a displacement changing the sense in which the curve is traversed (Umklappung). 

~ The term "rotation" is used here in the sense of a rotatory displacement; that is to say, only the initial 
and final stages and not the aggregate of the intermediate stages of the transition enter into consideration. — 
Tr. 



85 

Axiom I. // two displacements are followed out one after the other, then the resulting 
map of the plane upon itself is again a displacement. 

We say briefly: 

Axiom I. The Displacements Form a Group. 

Axiom II. // A and M are two arbitrary points distinct from each other, then by a 
rotation about M we can always bring A into an infinite number of different positions. 

If in our geometry we define a true circle as the totality of those points which arise by 
rotating about M a point different from Af, then we can express the statement made in 
axiom II as follows: 

Axiom II. Every True Circle Consists of an Infinite Number of Points. 

As preliminary to axiom III, we make the following explanations: 

Let A be a definite point in our geometry and Ai, A2, A3, ...any infinite system 
of points. With the same letters we will also denote the maps of these points upon the 
number-plane. About the point A in the number-plane take an arbitrarily small domain 
a. If then any of the map-points Ai fall within the domain a, we say that there are points 
Ai arbitrarily near the point A. 

Let A, S be a definite pair of points in our geometry, and let AiSi, A2B2, A3S3, . . . 
be any infinite system of pairs of points. With the same letters we will denote the maps 
of these pairs of points upon the number-plane. Select about each of the points A and 
B in the number-plane an arbitrarily small domain a and /3, respectively. If then there 
are pairs of points AiBi such that Ai falls within the domain a and at the same time Bi 
falls within the domain /?, we say that there are segments AiBi lying arbitrarily near' the 
segment AB. 

Let ABC a definite triad of points in our geometry, and let AiBiCi, A2B2C2, A3B3C3, 
... be any infinite system of triads of points. With the same letters we will also denote 
the maps of these triads of points upon the number-plane. About each of the points A, 
B, C in the number-plane take an arbitrarily small domain a, /?, 7, respectively. If then 
there are triads of points AiBiCi such that Ai falls in the domain a, and likewise Bi in 
the domain (3 and Ci in the domain 7, then we say that there are triangles AiBiCi lying 
arbitrarily near to the triangle ABC . 

Axiom III. If there are displacements of such a kind that triangles arbitrarily near 
the triangle ABC can he brought arbitrarily near to the triangle A'B'C , then there al- 
ways exists a displacement by which the triangle ABC goes over exactly into the triangle 
A'B'C'.'''' 

The content of this axiom can be briefly expressed as follows: 

Axiom III. The Displacements Form a Closed System. 

We call special attention to the following particular cases of axiom III. 

If there ar'e rotations about a point M of the kind that segments lying arbitrarily near 
the segment AB can be brought arbitrarily near the segment A'B', then there is always 
such a rotation about M possible by which the segment AB goes over exactly into the 
segment A'B' . 



~ It is sufficient to assume that axiom III holds for sufficiently small domains as Lie has done. My 
method of proof may be so changed as to make use of only this narrower assumption. 



86 

If there are displacements of the kind that segments arbitrarily near the segment AB 
can be brought arbitrarily near to the segment A'B' , then there is always a displacement 
possible by which the segment AB goes over exactly into the segment A'B' . 

If there are rotations about the point M of the kind that points arbitrarily near the 
point A can be brought arbitrarily near the point A', then there is always such a rotation 
about M possible by which A goes over exactly into the point A' . 

1 now prove the following proposition: 

A geometry in which axioms I-llI are fulfilled is either the euclidean or the 
bolyai-lobatchefskian geometry. 

If we wish to obtain only the euclidean geometry, it is necessary merely to make in 
connection with axiom I the additional statement that the groups of displacements shall 
possess an invariant sub-group. This additional statement takes the place of the axioms 
of parallels. 

In what follows, I will briefiy outline the general idea of my method of proof.^^ 
Within the domain of a certain point M construct in a particular manner a certain point- 
configuration kk, and upon this configuration construct a certain point K. We then base 
our investigation upon the true circle k about M and passing through K. It may be easily 
shown that the true circle k is an assemblage of points which is closed and in itself dense. 
It constitutes, therefore, a perfect assemblage of points. 

The next objective point in our demonstration is to show that the true circle k is 
a closed Jordan curve. We do this in that we first show the possibility of a cyclical 
arrangement of the points of the true circle k, from which it follows that we may map in 
a reversible, single- valued manner the points of k upon the points of an ordinary circle. 
Finally, we show that this map must necessarily be a continuous one. Furthermore, it 
follows also that the originally constructed point-configuration kk is identical with the 
true circle k. Moreover, the law holds that each true circle inside of k is likewise a closed 
Jordan curve. 

We turn now to the investigation of the group of all the displacements which by the 
rotation of the plane about M transforms a definite true circle k into itself. This group 
possesses the following properties: (1) Every displacement which leaves one point of k 
undisturbed, leaves all points of k undisturbed. (2) There always exists a displacement 
which changes any given point of k into any other given point of k. (3) The group 
of displacements is a continuous one. These three properties determine completely the 
construction of the group of transformations of all the displacements of the true circle into 
itself. We set up the following proposition: The gi'oup of all the displacements of the true 
circle into itself, which are rotations about M, is holoedric, isomorphic with the group of 
ordinary rotations of the ordinary circle into itself. 

Moreover, we investigate the group of displacements of all the points of our plane by 
a rotation about M . The law holds that, aside from the identity, there is no rotation of 
the plane about M which leaves every point of the true circle undisturbed. We now see 
that every true circle is a Jordan curve and deduce formulae for the transformation of that 



The complete proof will appear later in the Math. Annalen. 



87 

group of all the rotations. Finally, the proposition easily follows that: If any two points 
remain fixed by a displacement of the plane, then all points remain fixed; that is to say, 
the displacement is the identity. Each point of the plane may be indeed made to go over 
into any other point of the plane by means of a displacement. 

Our further important objective point is to define the idea of the true straight line 
in our geometry and deduce those properties of it which are necessary in the further 
development of geometry. First of all, the notions "semi-rotation" and "middle of a 
segment" are defined. A segment has at most one middle, and, when we know the middle 
of one segment, then every smaller segment possesses a middle. 

In order to pass judgment as to the position of the middle of a segment, we need 
particular propositions concerning true circles which are mutually tangent, and indeed 
the question depends upon the construction of two congruent circles tangent to each other 
externally in one and only one point. We derive also a more general proposition concerning 
circles which are tangent to each other internally and consequently a theorem covering 
the special case where the circle which is tangent internally to a second passes through 
the centre of that circle. 

Moreover, a sufficiently small definite segment is taken as a unit segment, and from 
this by continued bisection and semi-rotation a system of points is constructed of the kind 
that to each point of this system a definite number a corresponds, which is rational and 
has as denominator some power of 2. By setting up a law concerning this correspondence, 
the points of the above system are so arranged that the above laws concerning mutually 
tangent circles are valid. It is now shown that the points corresponding to the numbers 
2, 4, g, . . . converge toward the point 0. This result is generalized step by step until 
it is finally shown that every series of points of our system converges, so soon as the 
corresponding series of numbers converges. 

From what has been said, the definition of the true straight line follows as a system of 
points which arise from two fundamental points, if we apply repeatedly a semi- rot at ion, 
take the middle point, and add to the assemblage the points of condensation of the system 
of points which arises. We can then prove that the true straight line is a continuous curve, 
possessing no double points and having with any other true straight line at most one 
point in common. Furthermore, it can be shown that the true straight line cuts each circle 
drawn about one of its points, and from this it follows that any two arbitrary points of 
the plane can always be joined by a true straight line. We see also that in our geometry 
the laws of congruence hold, by which however two triangles are proven to be congruent 
if they are traversed in the same sense. 

With regard to the position of the systems of all the true straight lines with respect 
to one another, there are two cases to distinguish, according as the axiom of parallels 
holds, or through each point there exists two straight lines which separate the straight 
lines which cut the given straight line from those which do not cut it. In the first case we 
have the euclidean and in the second the holyai- lob at s chefs kian geometry. 



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