AMSCO'S Geometry. New York - Rye High School

CHAPTERPROVINGSTATEMENTS INGEOMETRYAfter proposing 23 definitions, Euclid listed fivepostulates and five “common notions.” These definitions,postulates, and common notions provided thefoundation for the propositions or theorems for whichEuclid presented proof. Modern mathematicians haverecognized the need for additional postulates to establisha more rigorous foundation for these proofs.David Hilbert (1862–1943) believed that mathematicsshould have a logical foundation based on twoprinciples:1. All mathematics follows from a correctly chosenfinite set of assumptions or axioms.2. This set of axioms is not contradictory.Although mathematicians later discovered that it isnot possible to formalize all of mathematics, Hilbertdid succeed in putting geometry on a firm logical foundation.In 1899, Hilbert published a text, Foundations ofGeometry, in which he presented a set of axioms thatavoided the limitations of Euclid.3CHAPTERTABLE OF CONTENTS3-1 Inductive Reasoning3-2 Definitions as Biconditionals3-3 Deductive Reasoning3-4 Direct and Indirect Proofs3-5 Postulates,Theorems, andProof3-6 The Substitution Postulate3-7 The Addition and SubtractionPostulates3-8 The Multiplication andDivision PostulatesChapter SummaryVocabularyReview ExercisesCumulative Review93