xiv CONTENTS 7.The Greek Rationalization of Nature,145 1.The Inspiration for Greek Mathematics.145 2 The Beginnings of a Rational View of Nature,146 3 The Development of the Belief in Mathematical Design.147 4.Greek Mathematical Astronomy,154 5 Geography,160 6.Mechanics,162 7 Optics,166 8 Astrology.168 8.The Demise of the Greek World,171 1.A Review of the Greek Achievements,171 2.The Limitationsof Greek Mathematics,173 3.The Problems Bequeathed by the Greeks,176 4 The Demise of the Greek Civilization,177 9.The Mathematics of the Hindus and Arabs,183 1 Early Hindu Mathematics,183 2.Hindu Arithmetic and Algebra of the Period A.D.200-1200,184 3.Hindu Geometry and 'Trigonometry of the Period A.D.200-1200,188 4.The Arabs,190 5.Arabic Arithmetic and Algebra,191 6.Arabic Geometry and Trigonometry.195 7 Mathematics circa 1300.197 10.The Medieval Period in Europe,200 1.The Beginnings of a European Civilization,200 2.The Materials Available for Learning.201 3.The Role of Mathematics in Early Medieval Europe,202 4 The Stagnation in Mathematics,203 5.The First Revival of the Greek Works,205 6 The Revival of Rationalism and Interest in Nature,206 7.Progress in Mathematics Proper,209 8.Progress in Physical Science,211 9.Summary,213 11.The Renaissance,216 1 Revolutionary Influences in Europe,216 2.The New Intellectual Outlook,218 3.The Spread of Learning,220 4 Humanistic Activity in Mathematics,221 5.The Clamor for the Reform of Science,223 6 The Rise of Empiricism,227 12.Mathematical Contributions in the Renaissance,231 I Perspective,231 2.Geometry Proper,234 3 Algebra,236 4.Trigonometry, 237 5.The Major Scientific Progress in the Renaissance,240 6 Remarks on the Renaissance.247 13.Arithmetic and Algebra in the Sixteenth and Seventeenth Centuries,250 1.Introduction,250 2.The Status of the Number System and Arithmetic,251 Symbolism,259 4.The Solution of Third and Fourth Degree Equations.263 5.The 'Theory of Equations,270 6 The Binomial Theorem and Allied Topics,272 7 The Theory of Numbers,274 8.The Relationship of Algebra to Geometry,278 14.The Beginnings of Projective Geometry,285 netry,285 hekene work of Desg principles. 288 The Emergence of New