TABLE OF CONTENTS Xlll 



PAGE 



A Dozen Typical String Tricks 332 



The Lizard Twist or Caroline Catch 332 



The Chippewa Eelease 333 



Cheating the Halter 333 



The Threading of a Closed Loop 333 



The Fly on the Nose 334 



The Interlaced Handcuffs 335 



The Button-Hole Trick 335 



The Waistcoat Trick 336 



Chapter XVI. Three Classical Geometrical Problems. 



Statement of the Problems 337 



The Duplication of the Cube. Legendary Origin of the Problem . . 338 



Hippocrates's Lemma 340 



Solutions by Archytas, Plato, Menaechmus, Apollonius, and Diocles . 340 



Solutions by Vieta, Descartes, Gregory of St Vincent, and Newton . 343 



The Trisection of an Angle 344 



Ancient Solutions quoted by Pappus 344 



Solutions by Pappus, Descartes, Newton, Clairaut, and Chasles . . 345 



The Quadrature of the Circle 346 



Incommensurability of w 347 



Definitions of n- 348 



Origin of symbol it 349 



Methods of approximating to the numerical value of ir . . . 349 



Geometrical methods of Approximation 349 



Besults of Egyptians, Babylonians, Jews 350 



Besults of Archimedes and other Greek Writers .... 350 



Besults of Roman Surveyors and Gerbert 351 



Besults of Indian and Eastern Writers 351 



Besults of European Writers, 1200-1630 353 



Theorems of Wallis and Brouncker 355 



Analytical methods of Approximation. Gregory's series . . . 356 



Besults of European Writers, 1699-1873 356 



Geometrical Approximation 358 



Approximations by the Theory of Probability 358 



Index 360 



