Contents 1.Mathematics in Mesopotamia,3 1 Where Did Mathematics Begin?3 2.Political History in Mesopotamia.4 3 The Number Symbols,5 4.Arithmetic Operations,7 5.Babylonian Algebra,8 6.Babylonian Geometry,10 7.The Uses of Mathematics in Babylonia,11 8 Evaluation of Babylonian Mathematics,13 2.Egyptian Mathematics,15 3.Algebra and Geometry,18 4.Egyptian ematics,2 3.The Creation of Classical Greek Mathematics,24 1.Background,24 2.The General Sources,25 3.The Major Schools of the Classical Period,27 4 The lonian School,28 5 The Pythagoreans,28 6.The Eleatic School,34 7 The Sophist School,37 8.The Platonic School,42 9.The School of Eudoxus,48 10.Aristotle and His School,51 4.Euclid and Apollonius,56 1.Introduction,56 2 The Background of Euclid's Elements,57 3.The Definitions and Axioms of the Elements,58 4.Books I to IV of the Elements,60 5 Book V. The Theory of Proportion,68 6.Book VI.Similar Figures,73 7.Books VII, VIIl,and IX:The Theory of Numbers.77 8.Book X:The Classification of nsurables.80 9.Books XI,XII,and XIII.Solid Geometry and the Method of Exhaustion.81 10.The Merits and Defects of the Elements.86 11.Other Mathematical Works by Euclid,88 12.The Mather matical Work of Apollonius,89 5.The Alexandrian Greek Period:Geometry and Trigonometry,101 1 The Foundingof Alexandria,Cha Mathe acter of Alexandrian Greck Area and Volumes in the Work of Archimedes,10 umes in the Work of Heron,116 5.Some Exceptional Curves,117 6.The and Volumes Creation of Trigonometry,119 7.Late Alexandrian Activity in Geometry,126 6.The Alexandrian Period:The Reemergence of Arithmetic and Algebra,131 1.The Symbols and Operations of Greek Arithmetic,131 2.Arithmetic and Algebra as an Independent Development,135