希尔伯特及其《几何学基础》电子版（英文PDF），
    希尔伯特的巨著《几何学基础》是现代公理化数学的源头与楷模。
   在此书发表一百周年之际，国外有人将其翻译成英文版发布在互联网上，任其传播，很有意义。
   在数学发展史上，希尔伯特《几何学基础》发表具有里程碑的意义。从此。数学进入公理化时代。
   本文附件是希尔伯特《几何学基础》电子版（英文PDF），值得阅读。值得注意的是，在希尔伯特公理系统中，点、线、面是不加定义的基本概念，只要满足该公理系统即可。
    国内数学教育当局制定数学教学大纲背离了希尔伯特的公理化思想，反其道而行之。可悲也。
袁萌  陈启清   3月7日
附件: 希尔伯特及其《几何学基础》电子版（英文PDF），
The Foundations of Geometry
BY DAVID HILBERT, PH. D.
PROFESSOR OF MATHEMATICS, UNIVERSITY OF GÖTTINGEN
AUTHORIZED TRANSLATION BY E. J. TOWNSEND, PH. D.
UNIVERSITY OF ILLINOIS
REPRINT EDITION
THE OPEN COURT PUBLISHING COMPANY
LA SALLE ILLINOIS
1950
TRANSLATION COPYRIGHTED
BY
The Open Court Publishing Co.
1902.
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 Göttingen 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 Göttingen, 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. Theaxiomsofcongruenceareintroducedandmadethebasisofthedefinitionofgeometric 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 euclidean 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 geometry may be regarded as a part of a geometry of space, is made apparent, etc. 5. Avarietyofalgebrasofsegmentsareintroducedinaccordancewiththelawsofarithmetic. This development and discussion of the foundation principles of geometry is not only of mathematical 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 students and teachers of geometry, I have undertaken, with his permission, this translation. In its preparation, I have had the assistance of many valuable suggestions from Professor Osgood of Harvard, Professor Moore of Chicago, and Professor Halsted of Texas. I 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 III. THE THEORY OF PROPORTION.
§13. Complex number-systems ............................................ 23
§14. Demonstration of Pascal’s theorem ............................ 25
§15. An algebra of segments, based upon Pascal’s theorem ................ 30 §16. Proportion and the theorems of similitude ............................ 33
§17. Equations of straight lines and of planes ............................. 35
CHAPTER IV. THE THEORY OF PLANE AREAS.
§18. Equal area and equal content of polygons ............................ 38
§19. Parallelograms and triangles having equal bases and equal altitudes . 40 §20. The measure of area of triangles and polygons ................ 41
§21. Equality of content and the measure of area ...................... 44
CHAPTER V. DESARGUES’S THEOREM.
§22. Desargues’s theorem and its demonstration for plane geometry by aid of the axioms of congruence ...................................... 48
§23. The impossibility of demonstrating Desargues’s theorem for the plane without the help of the axioms of congruence ............. 50§24. Introduction of an algebra of segments based upon Desargues’s theorem and independent of the axioms of congruence ........................... 53
§25. The commutative and the associative law of addition for our new algebra of segments ...................................................... 55
§26. The associative law of multiplication and the two distributive laws for the new algebra of segments ......................................... 56
§27. Equation of the straight line, based upon the new algebra of segments ... 61 §28. The totality of segments, regarded as a complex number system ......... 64 §29. Construction of a geometry of space by aid of a desarguesian number system ............................................ 65
§30. Significance of Desargues’s theorem ............................... 67
CHAPTER VI. PASCAL’S THEOREM.
§31. Two theorems concerning the possibility of proving Pascal’s theorem .... 68 §32. The commutative law of multiplication for an archimedean number system ............................................. 68
§33. The commutative law of multiplication for a non-archimedean number system ........................................ 70 §34. Proof of the two propositions concerning Pascal’s theorem. Non-pascalian geometry. ................................................. 72 §35. The demonstration, by means of the theorems of Pascal and Desargues, of any theorem relating to points of intersection ......................... 73
CHAPTER VII. GEOMETRICAL CONSTRUCTIONS BASED UPON THE AXIOMS I–V.
§36. Geometrical constructions by means of a straight-edge and a transferer of segments ................................................... 74
§37. Analytical representation of the co-ordinates of points which can be so constructed ............................................. 76 §38. The representation of algebraic numbers and of integral rational functions as sums of squares ....................................................... 78 §39. Criterion for the possibility of a geometrical construction by means of a straight-edge and a transferer of segments .. 80 Conclusion ............................ 83
“All human knowledge begins with intuitions, 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.1 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 geometrical 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.
1Compare the comprehensive and explanatory report of G. Veronese, Grundzüge der Geometrie, German translation by A. Schepp, Leipzig, 1894 (Appendix). See also F. Klein, “Zur ersten Verteilung des Lobatschefskiy-Preises,” Math. Ann., Vol. 50.
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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 will 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 α, β, γ,...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,” “congruent,” “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 groups. Each of these groups expresses, by itself, certain related fundamental facts of our intuition. We will name these groups as follows: I, 1–7. Axioms of connection. II, 1–5. Axioms of order. III. Axiom of parallels (Euclid’s axiom). IV, 1–6. Axioms of congruence. V. Axiom of continuity (Archimedes’s axiom).
§2. GROUP I: AXIOMS OF CONNECTION.
The axioms of this group 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.
Insteadof“determine,” wemayalsoemployotherformsofexpression; forexample, 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 b, 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 B6= C, then is also BC = a.
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I, 3. Three points A, B, C not situated in the same straight line always completely determine a plane α. We write ABC = a.
We employ also the expressions: A, B, C, “lie in” α; A, B, C “are points of” α, etc.
I, 4. Anythreepoints A, B, C ofaplane α,whichdonotlieinthesamestraightline,completely determine that plane. I, 5. If two points A, B of a straight line a lie in a plane α, then every point of a lies in α.
In this case we say: “The straight line a lies in the plane α,” etc.
I, 6. If two planes α, β 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, concerningtheelementsofplanegeometry. Wewillcallthem,therefore,theplaneaxioms of group I, 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 following:
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.2
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:
4
Fig. 1.
II, 1. If A, B,Carepointsofastraightlineand Bliesbetween AandC,then Bliesalsobetween C and A.
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 will call the system of two points A and B, 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.
II, 5. Let A, B, C be three points not lying in the same straight line and let a be 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. 2These axioms were first studied in detail by M. Pasch in his Vorlesungen über neuere Geometrie, Leipsic, 1882. Axiom II, 5 is in particular due to him.
5
Fig. 3.
§4. CONSEQUENCES OF THE AXIOMS OF CONNECTION AND ORDER.
BytheaidofthefourlinearaxiomsII,1–4,wecaneasilydeducethefollowingtheorems:
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 line, 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.
Fig. 4.
Theorem 5. Every straight line a, which lies in a plane α, 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, A0 of the same region determine a segment AA0 containing no point of a. If A, A0, O, B are four points of a straight line a, where O lies between A and B but not between A and A0, then we may say: The points A, A0 are situated on the line a upon
6
Fig. 5.
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. 6.
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, A0 lie in the plane α upon one and the same side of the straight line a, and the points A, B lie in the plane α upon different sides of the straight line a. Definitions. Asystemofsegments AB, BC,CD,...,KLiscalledabrokenlinejoining 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 calledthepointsofthebrokenline. Inparticular,ifthepoint Acoincideswith L,thebroken 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, quadrangles, pentagons, ..., n-gons. Iftheverticesofapolygonarealldistinctandnone 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 theorem 5, we may now obtain, without serious difficulty, the following theorems:
Theorem 6. Every simple polygon, whose vertices all lie in a plane α, 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, A0 are two points of the interior and B, B0 two points of the exterior region, then there are
7
always broken lines to be found joining A with A0 and B with B0 without having a point in common with the polygon. There exist straight lines in the plane α which lie entirely outside of the given polygon, but there are none which lie entirely within it.
Fig. 7.
Theorem 7. Every plane α 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 α. On the other hand, any two points A, A0 lying within the same region determine a segment AA0 containing no point of α. Makinguseofthenotationoftheorem7, wemaynowsay: Thepoints A, A0 aresituated in space upononeandthesamesideoftheplane α, and the points A, B are situated in space upon different sides of the plane α. 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.)
Theintroductionofthisaxiomsimplifiesgreatlythefundamentalprinciplesofgeometry and facilitates in no small degree its development. This axiom may be expressed as follows: III. In a plane α there can be 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.
8
This statement of the axiom of parallels contains two assertions. The first of these is that, in the plane α, 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, b 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, b 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 parallels, 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. If A, B are two points on a straight line a, and if A0 is a point upon the same or anotherstraightline a0,then,uponagivensideof A0 onthestraightline a0,wecanalways findoneandonlyonepoint B0 sothatthesegment AB(or BA)iscongruenttothesegment A0B0. We indicate this relation by writing AB≡ A0B0. 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. If a segment AB is congruent to the segment A0B0 and also to the segment A00B00, then the segment A0B0 is congruent to the segment A00B00; that is, if AB ≡ A0B and AB≡ A00B00, then A0B0 ≡ A00B00. IV,3. Let ABand BCbetwosegmentsofastraightlineawhichhavenopointsincommon aside from the point B, and, furthermore, let A0B0 and B0C0 be two segments of the same or of another straight line a0 having, likewise, no point other than B0 in common. Then, if AB≡ A0B0 and BC ≡ B0C0, we have AC ≡ A0C0.
9
Fig. 8.
Definitions. Let α be any arbitrary plane and h, k any two distinct half-rays lying in α 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 ∠(h,k) or ∠(k,h). From axioms II, 1–5, it follows readily that the half-rays h and k, taken together with the point O, divide the remaining points of the plane a into two regions having the following property: If A 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, A0 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 characterised is called the interior of the angle (h,k). To distinguish 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 α and let a straight line a0 be given in a plane α0. Suppose also that, in the plane α, a definite side of the straight line a0 be assigned. Denote by h0 a half-ray of the straight line a0 emanating from a point O0 of this line. Then in the plane α0 there is one and only one half-ray k0 such that the angle (h,k), or (k,h), is congruent to the angle (h0,k0) and at the same time all interior points of the angle (h0,k0) lie upon the given side of a0. We express this relation by means of the notation ∠(h,k)≡∠(h0,k0) Every angle is congruent to itself; that is, ∠(h,k)≡∠(h,k) or ∠(h,k)≡∠(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. If the angle (h,k) is congruent to the angle (h0,k0) and to the angle (h00,k00), then the angle (h0,k0) is congruent to the angle (h00,k00); that is to say, if ∠(h,k) ≡∠(h0,k0) and ∠(h,k)≡∠(h00,k00), then∠(h0,k0)≡∠(h00,k00).
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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 (h,k) 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. Itcontainsalloftheinteriorpointsofthetriangle ABC andisrepresented by the symbol ∠BAC, or by ∠A. IV, 6. If, in the two triangles ABC and A0B0C0 the congruences AB≡ A0B0, AC ≡ A0C0, ∠BAC ≡∠B0A0C0 hold, then the congruences ∠ABC ≡∠A0B0C0and∠ACB≡∠A0C0B0 also hold.
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 A0B0. Since, according to axiom IV, 1, the segment AB is congruent to itself, it follows from axiom IV, 2 that A0B0 is congruent to AB; that is to say, if AB ≡ A0B0, then A0B0 ≡ AB. We say, then, that the two segments are congruent to one another. Let A, B, C, D,..., K, L and A0, B0, C0, D0,..., K0, L0 be two series of points on the straight lines a and a0, respectively, so that all the corresponding segments AB and A0B0, AC and A0C0, BC and B0C0,..., KL and K0L0 are respectively congruent, then the two series of points are said to be congruent to one another. A and A0, B and B0,..., L and L0 are called corresponding points of the two congruent series of points. From the linear axioms IV, 1–3, we can easily deduce the following theorems:
Theorem 9. If the first of two congruent series of points A, B, C, D,..., K, L and A0, B0, C0, D0,..., K0, L0 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 A0, B0, C0, D0,..., K0, L0 of the second series are arranged in a similar way; that is to say, B0 lies between A0 and C0, D0,..., K0, L0, and C0 lies between A0, B0 and D0,..., K0, L0, etc.
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Let the angle (h,k) be congruent to the angle (h0,k0). Since, according to axiom IV, 4, the angle (h,k) is congruent to itself, it follows from axiom IV, 5 that the angle (h0,k0) is congruenttotheangle (h,k). Wesay,then,thattheangles (h,k)and (h0,k0)arecongruent 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 A0B0C0 are said to be congruent to one another when all of the following congruences are fulfilled: AB≡ A0B0, AC ≡ A0C0, BC ≡ B0C0, ∠A≡∠A0, ∠B≡∠B0, ∠C ≡∠C0. Theorem 10. (First theorem of congruence for triangles). If, for the two triangles ABC and A0B0C0, the congruences AB≡ A0B0, AC ≡ A0C0, ∠A≡∠A0 hold, then the two triangles are congruent to each other.
Proof. From axiom IV, 6, it follows that the two congruences ∠B≡∠B0and∠C ≡∠C0 are fulfilled, and it is, therefore, sufficient to show that the two sides BC and B0C0 are congruent. We will assume the contrary to be true, namely, that BC and B0C0 are not congruent, and show that this leads to a contradiction. We take upon B0C0 a point D0 such that BC ≡ B0D0. The two triangles ABC and A0B0D0 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 B0A0D0 are also congruent to each other. Consequently, by aid of axiom IV, 5, the two angles B0A0C0 and B0A0D0 must be congruent.
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:
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Theorem 11. (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 A0B0C0 are congruent to each other, their supplementary angles CBD and C0B0D0 are also congruent.
Fig. 10.
Proof. Take the points A0, C0, D0 upon the sides passing through B0 in such a way that A0B0 ≡ AB, C0B0 ≡CB, D0B0 ≡ DB. Then, in the two triangles ABC and A0B0C0, the sides AB and BC are respectively congruent to A0B0 and C0B0. Moreover, since the angles included by these sides are 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 ≡ A0C, ∠BAC ≡∠B0A0C0. On the other hand, since by axiom IV, 3 the segments AD and A0D0 are congruent to each other, it follows again from theorem 10 that the triangles CAD and C0A0D0 are congruent, and, consequently, we have the congruences: CD ≡C0D0, ∠ADC ≡∠A0D0C0. From these congruences and the consideration of the triangles BCD and B0C0D0, it follows by virtue of axiom IV, 6 that the angles CBD and C0B0D0 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 (h, k) of the plane α be congruent to the angle (h0, k0) of theplane α0, and, furthermore, let l beahalf-rayintheplane α emanatingfromthe vertex of the angle (h, k) and lying within this angle. Then, there always exists in the plane α0 a half-ray l0 emanating from the vertex of the angle (h0, k0) and lying within this angle so that we have ∠(h,l)≡∠(h0,l0), ∠(k,l)≡∠(k0,l0).
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Fig. 11.
Proof. We will represent the vertices of the angles (h, k) and (h0, k0) by O and O0, respectively, and so select upon the sides h, k, h0, k0 the points A, B, A0, B0 that the congruences OA≡O0A0, OB≡O0B0 are fulfilled. Because of the congruence of the triangles OAB and O0A0B0, we have at once AB≡ A0B0, ∠OAB≡O0A0B0, ∠OBA≡∠O0B0A0. Let the straight line AB intersect l in C. Take the point C0 upon the segment A0B0 so that A0C0 ≡ AC. Then, O0C0 is the required half-ray. In fact, it follows directly from these congruences, by aid of axiom IV, 3, that BC ≡ B0C0. Furthermore, the triangles OAC and O0A0C0 are congruent to each other, and the same is true also of the triangles OCB and O0B0C0. With this our proposition is demonstrated. In a similar manner, we obtain the following proposition.
Theorem 14. Let h, k, l and h0, k0, l0 be two sets of three half-rays, where those of each set emanate from the same point and lie in the same plane. Then, if the congruences ∠(h,l)≡∠(h0,l0), ∠(k,l)≡∠(k0,l0) are fulfilled, the following congruence is also valid; viz.: ∠(h,k)≡∠(h0,k0). 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 B0A0D0 be congruent to its supplementary angle C0A0D0. Hence the angles BAD, CAD, B0A0D0, and C0A0D0 are all right angles. We will assume that the contrary of our proposition is true, namely, that the right angle B0A0D0 is not congruent to the right angle BAD, and will show that this assumption leads to a contradiction. We layofftheangle B0A0D0 uponthehalf-ray ABinsuchamannerthattheside AD00 arising fromthisoperationfallseitherwithintheangle BADorwithintheangleCAD. Suppose, for example, the first of these possibilities to be true. Because of the congruence of the
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angles B0A0D0 and BAD00, it follows from theorem 12 that angle C0A0D0 is congruent to angle CAD00, and, as the angles B0A0D0 and C0A0D0 are congruent to each other, then, by IV, 5, the angle BAD00 must be congruent to CAD00.
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 AD000 emanating from A, so that the angle BAD00 will be congruent to the angle CAD000, and also the angle DAD00 will be congruent to the angle DAD000. The angle BAD00 was shown to be congruent to the angle CAD00 and, hence, by axiom IV, 5, the angle CAD00, is congruent to the angle CAD000. 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.” The theorem relating to the congruence of the base angles A and B of an equilateral triangle ABC followsimmediatelybytheapplicationofaxiomIV, 6 tothetriangles 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 properties: Three points of a figure lying in a straight line are likewise in a straight line in every figure congruent to it. In congruent figures, the arrangement of the points in corresponding 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:
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Theorem 17. If (A,B,C,...) and (A0,B0,C0,...) 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 (A0,B0,C0,...,P0) shall likewise be congruent figures. If the two figures have at least three points not lying in a straight line, then the selection of P0 can be made in only one way. Theorem 18. If (A,B,C,...) and (A0,B0,C0,... = are congruent figures and P represents any arbitrary point, then there can always be found a point P0 so that the two figures (A,B,C,...,P) and (A0,B0,C0,...,P0) 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 P0 can be made in but one way. Thistheoremcontainsanimportantresult;namely,thatallthefactsconcerningspace 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 alternateinterior 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. Theorem 20. The sum of the angles of a triangle is two right angles.
Definitions. If M is an arbitrary point in the plane α, 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 be 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 axiomsrelatingtothecongruenceofsegmentsanddefineas“equal”thecorrespondingly
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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 givenstraightlinesothatanew,definitesegmentisobtained“equal”toit. Inconformity with such a convention, the axiom of Archimedes may be stated as follows:
V. Let A1 be any point upon a straight line between the arbitrarily chosen points A and B. Take the points A2, A3, A4,... so that A1 lies between A and A2, A2 between A1 and A3, A3 between A2 and A4 etc. Moreover, let the segments AA1, A1A2, A2A3, A3A4, ...
be equal to one another. Then, among this series of points, there always exists a certain point An such that B lies between A and An.
The axiom of Archimedes is a linear axiom. Remark.3 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 be expressed in the following form:
Axiom of Completeness.4 (Vollständigkeit): Toasystemofpoints,straightlines, 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.
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.”
3Added by Professor Hilbert in the French translation.—Tr. 4See Hilbert, “Ueber den Zahlenbegriff,” Berichte der deutschen Mathematiker-Vereinigung, 1900.
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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 these axioms, by any logical process of reasoning, a proposition which is contradictorytoanyofthem. Todemonstratethis,itissufficienttoconstructageometry where all of the five groups are fulfilled. To this end, let us consider a domain Ω 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 √1+ω2, where ω represents a number arising from the five operations already given. Let us regard a pair of numbers (x,y) of the domain Ω as defining a point and the ratio of three such numbers (u : v : w) of Ω, where u, v are not both equal to zero, as defining a straight line. Furthermore, let the existence of the equation ux+vy+w = 0 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 Ω 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 (x1,y1), (x2,y2), (x3,y3), ... are any points whatever of a straight line, then this may be taken as their sequence on this straight line, providing the numbers x1, x2, x3,..., or the numbers y1, y2, y3,..., 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 sideofthestraightline (u : v : w). Wecaneasilyconvinceourselvesthatthisconvention is in accordance with those which precede, and by which the sequence of the points on a straight line has already been determined. Thelayingoffofsegmentsandofanglesfollowsbytheknownmethodsofanalytical geometry. A transformation of the form x0 = x+a y0 = y+b produces a translation of segments and of angles.
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Fig. 13.
Furthermore, if, in the accompanying figure, we represent the point (0,0) by O and the point (1,0) by E, then, corresponding to a rotation of the angle COE about O as a center, any point (x,y) is transformed into another point (x0,y0) so related that
x0 =
a √a2 +b2x−
b √a2 +b2y,
y0 =
b √a2 +b2x+
a √a2 +b2y.
Since the number
pa2 +b2 = as1+b a2 belongs to the domain Ω, 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.
Fig. 14.
From these considerations, it follows that every contradiction resulting from our system of axioms must also appear in the arithmetic related to the domain Ω. 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 Ω, we should have obtained likewise a geometry in which all of the axioms of groups I—V are valid. For the purposes of our demonstration, however, it was sufficient to take the domain Ω, containing on an enumerable set of elements.
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§10. INDEPENDENCE OF THE AXIOMS OF PARALLELS. (NON-EUCLIDEAN GEOMETRY.)5
Havingshownthattheaxiomsoftheabovesystemarenotcontradictorytooneanother, 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.6 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. Thefirststatementoftheaxiomofparallelscanbedemonstratedbyaidoftheaxioms 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 α as the point C, 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 D0 so situated that B would lie between D and D0, and, moreover, so that we should have AD ≡ BD0 Because of the congruence of the two triangles ABD and BAD0, we have also ∠ABD ≡∠BAD0, and since the angles ABD0 and ABD are supplementary, it follows from theorem 12 that the angles BAD and BAD0 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 BAD0 were supplementary. Thesecondstatementoftheaxiomofparallelsisindependentofalltheotheraxioms. 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 ordinarygeometryasconstructedin§9,andregardtheseelementsasrestrictedinextent 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 5The 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. 6See 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.
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itself. By suitable conventions, we can make this “non-euclideangeometry” 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.
§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 be 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 A1, A2 be two points which, in ordinary geometry, have the co-ordinates x1, y1, z1 and x2, y2, z2, respectively. We will now define the length of the segment A1A2 as the positive value of the expression q(x1−x2 +y1−y2)2 +(y1−y2)2 +(z1−z2)2 and call the two segments A1A2 and A0 1A0 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 A1, A2, A3 so that A2 shall lie between A1 and A3. Let the points x, y, z of the straight line a be given by means of the equations
x = λt+λ0, y = µt+µ0, z = νt+ν0,
where λ, λ0, µ, µ0, ν, ν0 represent certain constants and T is a parameter. If t1, t2 (< t1), t3 (< t2) arethevaluesoftheparametercorrespondingtothepoints A1, A2, A3 wehave as the lengths of the three segments A1A2 A2A3 and A1A3 respectively, the following values: (t1−t2