1 Cultural Connection The Oikoumene The Inhabited World – ca. 550 B.C. – 476 A.D. Student led discussion.

2 5 – Pythagorean Mathematics The student will learn about Greek mathematics after the time of Alexander the Great.

3 §5-1 Alexandria Student Discussion.

4 §5-2 Euclid Student Discussion.

5 §5-3 Euclid’s “Elements” Student Discussion.

6 §5-3 Euclid’s “Elements” First written copy dates to about 500 A.D. First printing Since then over 1,000 editions. A compilation and systematic arrangement of works of earlier writers.

7 §5-4 Contents of the “Elements”. Student Discussion.

8 §5-4 Contents of the “Elements”. Geometry, Number Theory, & Elementary Algebra in thirteen books of 465 propositions. Based on 5 postulates and several axioms which are true in all sciences. High School Geometry is contained in books I, III, IV, VI, XI, XII. A great deal more geometry was know than that contained in the elements.

9 §5-4 Contents of the “Elements”. Book I – Basic constructions, triangles, and parallels Book II – Algebraic identities Book III – Circles, chords, secants & tangents. Book IV – Construction of polygons. Book V - Eudoxus’ theory of proportion. Book VI – Proportion applied to plane geometry. Book VII, VIII, IX – Number theory. Book X – Irrationals & Fundamental Thm Arithmetic.. Book XI, XII, XIII – Solid Geometry.

10 §5-5 Theory of Proportion Student Discussion.

11 §5-5 Theory of Proportion Proposition VI, 1 – The area of triangles with the same altitude are to one another as their bases.

12 §5-6 Regular Polygons Student Discussion.

13 §5-6 Regular Polygons The “Elements” showed construction of regular polygons of 3, 4, 5, 6, and 15 sides. This implies that 2 n, 3 · 2 n, 5 · 2 n, 15 · 2 n sided polygons could be conducted. Gauss (1796) proved that a regular polygon with prime number of sides of the form Could be constructed. [ 3, 5, 17, 257, 65637, …]

14 §5-7 Formal Aspects of the “Elements” Student Discussion.

15 §5-7 Formal Aspects of the “Elements” Although the content of the “Elements” was very important the form was equally important. It was the first recorded use of deductive reasoning. The use of axioms or self evident statements – see p The basis of five postulates. – see p. 153.

16 §5 – 8 Euclid’s Other Works Student Discussion.

17 Assignment Prepare presentation of papers for next class.