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INTRODUCTORY MODERN GEOMETRY

MODERN GEOMETRY

OF

POINT, RAY, AND CIRCLE

PARTI

BY

WILLIAM BENJAMIN SMITH, A.M., Ph.D. (Goett.)

PROFESSOR OF MATHEMATICS AND ASTRONOMY

UNIVERSITY OF THE STATE OF MISSOURI

Aeltestes bewahrt mit Treue,
Freundlich aufgefaßtes Neue.

Goethe.

New York
MACMILLAN & CO.

AND LONDON

1892

All rights reserved

LOPYRIGHT, 1892,
BY MACMILLAN AND CO.

TYPOGRAPHY BY J. S. CUSHING & Co., Boston, U.S.A.

PRESSWORK BY BERWICK & SMITH, BOSTON, U.S.A.

12.31-1932

GEOMETRY.

☺ / - - TIMEN

INTRODUCTION.

every other

1. Geometry is the Doctrine of Space.

What is Space? On opening our eyes we see objects around us in endless number and variety : the book here, the table there, the tree yonder. This vision of a world outside of us is quite involuntary — we cannot prevent it, nor modify it in any way; it is called the Intuition (or Perception or Envisagement) of Space. Two objects precisely alike, as two copies of this book, so as to be indistinguishable in

respect, yet are not the same, because they differ in place, in their positions in Space: the one is here, the other is not here, but there. In between and all about these objects that thus differ in place, there lies before us an apparently unoccupied region, where it seems that nothing is, but where anything might be. We may imagine or suppose all these objects to vanish or to fade away, but we cannot imagine this region, either where they were or where they were not, to vanish or to change in any way. This region, whether occupied or unoccupied, where all these objects are and where countless others might be, is called Space.

2. There are certain elementary facts, that is, facts that cannot be resolved into any simpler facts, about this Space, and these deserve special notice.

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