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<!doctype html>
<html lang="en">
<head>
  <meta charset="utf-8">
  <title>Learning the box model</title>
</head>
<body>
  <header>
    <nav>
      <ul>
        <li><a href="#chapter1">Chapter 1</a></li>
        <li><a href="#chapter2">Chapter 2</a></li>
        <li><a href="#chapter3">Chapter 2</a></li>
        <li><a href="#chapter4">Chapter 2</a></li>
        <li><a href="#chapter5">Chapter 2</a></li>
      </ul>
    </nav>
    <img src="images/image1.jpg" alt="image 1">
    <h1>THE FOUNDATIONS OF GEOMETRY.</h1>
    <h2>London: <span>C. J. CLAY AND SONS<span></h2>
      <address>CAMBRIDGE UNIVERSITY PRESS WAREHOUSE,<br>
  AVE MARIA LANE.<br>
  Glasgow: 263, ARGYLE STREET.
</address>
  </header>
  <main>
    <div>
    <p>

      <img src="images/image2.jpg" alt="image 2">When a long established system is attacked, it usually happens
that the attack begins only at a single point, where the weakness of
the established doctrine is peculiarly evident. But criticism, when
once invited, is apt to extend much further than the most daring, at
first, would have wished.</p>
<blockquote>First cut the liquefaction, what comes last,
      But Fichte's clever cut at God himself?</blockquote>
    <p>So it has been with Geometry. The liquefaction of Euclidean orthodoxy
is the axiom of parallels, and it was by the refusal to admit this
axiom without proof that Metageometry began. The first effort in
this direction, that of Legendre[5], was inspired by the hope of
deducing this axiom from the others--a hope which, as we now know,
was doomed to inevitable failure. Parallels are defined by Legendre
as lines in the same plane, such that, if a third line cut them, it
makes the sum of the interior and opposite angles equal to two right
angles. He proves without difficulty that such lines would not meet,
but is unable to prove that non-parallel lines in a plane must meet.
Similarly he can prove that the sum of the angles of a triangle
cannot exceed two right angles, and that if any one triangle has a
sum equal to two right angles, all triangles have the same sum; but
he is unable to prove the existence of this one triangle.</p>
<p>Thus Legendre's attempt broke down; but mere failure could
prove nothing. A bolder method, suggested by Gauss, was carried out
by Lobatchewsky and Bolyai[6]. If the axiom of parallels is logically
deducible from the others, we shall, by denying it and maintaining
the rest, be led to contradictions. These three mathematicians,
accordingly, attacked the problem indirectly: they denied the axiom
of parallels, and yet obtained a logically consistent Geometry. They
inferred that the axiom was logically independent of the others, and
essential to the Euclidean system. Their works, being all inspired by
this motive, may be distinguished as forming the first period in the
development of Metageometry.
</p>
</div>
<div>
  <img src="images/image4.jpg" alt="image 4">
  <p>The originator of the whole system, _Gauss_, does not appear,
as regards strictly non-Euclidean Geometry, in any of his hitherto
published papers, to have given more than results; his proofs remain
unknown to us. Nevertheless he was the first to investigate the
consequences of denying the axiom of parallels[8], and in his letters
he communicated these consequences to some of his friends, among whom
was Wolfgang Bolyai. The first mention of the subject in his letters
occurs when he was only 18; four years later, in 1799, writing to
W. Bolyai, he enunciates the important theorem that, in hyperbolic
Geometry, there is a maximum to the area of a triangle. From later
writings it appears that he had worked out a system nearly, if not
quite, as complete as those of Lobatchewsky and Bolyai[9].</p>
<p>
It is important to remember, however, that Gauss's work on curvature,
which _was_ published, laid the foundation for the whole method of
the second period, and was undertaken, according to Riemann and
Helmholtz[10], with a view to an (unpublished) investigation of the
foundations of Geometry. His work in this direction will, owing to
its method, be better treated of under the second period, but it is
interesting to observe that he stood, like many pioneers, at the head
of two tendencies which afterwards diverged.</p>
</div>
    <aside>
      <img src="images/image6.jpg" alt="image 6">
      <h1>TABLE OF CONTENTS</h1>
        <h2>INTRODUCTION</h2>
        <p>OUR PROBLEM DEFINED BY ITS RELATIONS TO LOGIC,
        PSYCHOLOGY AND MATHEMATICS.</p>
        <ul>
          <li>The problem first received a modern form through Kant, who
        connected the priori_ with the subjective</li>
        <li>A mental state is subjective, for Psychology, when its immediate
        cause does not lie in the outer world</li>
        <li>A piece of knowledge is priori_, for Epistemology, when
        without it knowledge would be impossible</li>
        <li>The subjective and the priori_ belong respectively to
        Psychology and to Epistemology. The latter alone will be
        investigated in this essay</li>
        <li>My test of the _à priori_ will be purely logical: what knowledge
        is necessary for experience? </li>
        <li>But since the necessary is hypothetical, we must include, in
        the priori_, the ground of necessity
      </li>
    </ul>
  </aside>
  </main>
  <footer>
    <dl>
    <dt>Title</dt>
    <dd>An essay on the foundations of geometry</dd>
    <dt>Author</dt>
    <dd>Bertrand Russell</dd>
    <dt>Release Date</dt>
    <dd>May 17, 2016 [EBook #52091]</dd>
    <dt>Language:</dt>
    <dd>English</dd>
  </dl>
  <p>Copyright &copy; Johan Martin <a href="mailto:webdevfun@johan-martin.com">My Email</a></p>
  <img src="images/image5.jpg" alt="image 5">
  <footer>
</body>
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