MATH/STAT. T. SUNDARA ROW S Geometric Exercises in Paper Folding EditedandRevisedby WOOSTER WOODRUFF BEMAN PROFESSOROFMATHEMATICSIN THEUNIVERSITYOFMICHIGAK and DAVID EUGENE SMITH PROFESSOROFMATHEMATICSINTEACHERS1COLLEGE OFCOLUMBIAUNIVERSITY WITH ILLUSTRATIONS 87 THIRD EDITION CHICAGO LONDON ::: THE OPEN COURT PUBLISHING COMPANY 1917 & XC? 4255 COPYRIGHTBY THEOPENCOURTPUBLISHINGCo, 1901 PRINTED IN THE UNITED STATES OF AMERICA hn HATH EDITORS PREFACE. OUR attention was first attracted to Sundara Rows Geomet ricalExercises in Paper Foldingby a reference in Kleins Vor- lesungen iiber ausgezucihlte Fragen der Elementargeometrie. An examination of the book, obtained after manyvexatiousdelays, convinced us of its undoubted merits and of its probable value to American teachers and students of geometry. Accordingly we sought permission of the author to bring out an edition in this country, wnich permission was most generously granted. The purpose of the book is so fully set forth in the authors introduction that we need only to say that it is sure to prove of interest toevery wide-awake teacher of geometry from thegraded school to the college. The methods are so novel and the results so easily reached that they cannot fail to awaken enthusiasm. Ourwork as editors in this revision has been confined tosome slight modifications of the proofs, some additions in the way of references, and the insertion of a considerable numberof half-tone reproductions of actual photographsinstead of the line-drawings of the original. W. W. BEMAN. D. E. SMITH. OK02 CONTENTS. PAGE Introduction vn m I. The Square i II. The Equilateral Triangle 9 III. Squaresand Rectangles 14 IV. The Pentagon 3 V. The Hexagon , 35 VI. The Octagon 39 VII. The Nonagon 45 VIII. The Decagon and the Dodecagon 47 IX. The Pentedecagon 5 X. Series 52 XI. Polygons 67 XII. General Principles 82 XIII. The Conic Sections. Section i. TheCircle 102 Section n. The Parabola 115 Section in. The Ellipse 121 Section iv. The Hyperbola 126 XIV. MiscellaneousCurves 131