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    The Foundations of Geometry
    The Foundations of Geometry
    The Foundations of Geometry
    Ebook187 pages2 hours

    The Foundations of Geometry

    By David Hilbert

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    This early work by David Hilbert was originally published in the early 20th century and we are now republishing it with a brand new introductory biography. David Hilbert was born on the 23rd January 1862, in a Province of Prussia. Hilbert is recognised as one of the most influential and universal mathematicians of the 19th and early 20th centuries. He discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry. He also formulated the theory of Hilbert spaces, one of the foundations of functional analysis.
    LanguageEnglish
    PublisherRead Books Ltd.
    Release dateMay 6, 2015
    ISBN9781473395947
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      Book preview

      The Foundations of Geometry - David Hilbert

      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.

      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:

      § 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.

      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 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 B ≠ C, then is also BC = a.

      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. Any three points A, B, C of a plane α, 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 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, concerning the elements of plane geometry. We will call them, therefore, the plane axioms 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:

      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.

      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 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.

      Fig. 3.

      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.

      § 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 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, A′ of the same region determine a segment AA′ containing no point of a.

      Fig. 5.

      If A, A′, O, B are four points of a straight line a, where O lies 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. 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, A′ 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. 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

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