Chapter #1 Presentation

1. Chapter #1 Presentation Hippocrates’ Quadrature of the Lune Bennett Laxton

2. Presentation Overview: • History of Demonstrative Mathematics • Quadrature Overview • What exactly is a Lune • Quadrature of the Lune • Trying to solve the impossible: Quadrature of the Circle • Why quadrature of the circle is impossible

3. Appearance of Demonstrative Mathematics: • The Invention of Agriculture • 15,000 to 10,000 B.C. • Addressed two fundamental concepts of Mathematics • Multiplicity • Space • Formed the two great branches of Mathematics • Arithmetic • Geometric

4. Ancient Egyptians • Used mathematics as a practical facilitator • Building, Trade, Agriculture • 2000 B.C. number system and triangles • Right Angles

5. Pythagorean or Not • Not quite Pythagorean Theorem, which came later • More of an example of the converse of the statement Theorem: If triangle BAC is a right triangle, then A^2=B^2 + C^2. Converse: If A^2=B^2 + C^2, then triangle BAC is a right triangle.

6. Egyptian Truncated Square Pyramid

7. Babylonians • Understood the Pythagorean Theorem • 5-12-13 and the 65-72-97 right triangles • Base 60 numerical system • Still seen today in time and angles • Like the Egyptians they focused on the “How” and not the why

8. The “Great” Greeks • Thriving civilization for more that 2000 years that we still admire today • Thales • One of “Seven Wise Men” of Antiquity • Father of Demonstrative mathematics • Earliest known mathematician • Supplied the “why” with the “How” • Not the kindest of men

9. Thales proved the following: • Vertical angles are equal • The angle sum of a triangle equals two right angles • The base angles of an isosceles triangle are equal • An angle inscribed in a semicircle is a right angle • PROOF:

10. Pythagoras • Next great Greek thinker after Thales • Gave us two great mathematical discoveries: • Pythagorean Theorem (Of Course) • Idea of Commensurable • AB and CD are Commensurable if there is a smaller segment EF that goes evenly into AB and CD • Also discovered that the side of a square and its diagonal are not commensurable. • This discovery shattered many proofs

11. Hippocrates of Chios • Earliest mathematical proof that has survived in authentic form • Born in 5th century B.C. • Aristotle stated, “While a talented geometer he seems in other respects to have been stupid and lacking in sense.”

12. Quadrature Quadrature: (or squaring) of a plane figure is the construction using only a straightedge and a compass of a square having area equal to that of the original plane. • Planes are reduced to area of a square, which is easy to compute

13. Quadrature of the Rectangle Proof:

14. Quadrature of the Triangle Proof:

15. Quadrature of the Polygon Proof: We can subdivide any given polygon into triangles with areas (B,C,D). So polygon has the area B+C+D. By the last proof we can construct squares that have equal area to B,C,D with sides b,c,d respectively. Construct a right triangle as shown… Thus y^2=B+C+D.

16. Great Quadrature of the Lune • Lune: is a plane figure bounded by two circular arcs, that is a crescent. • Proof is based on 3 previously proven axioms: • Pythagorean Theorem • Angle inscribed in a semicircle is right • Areas of two circles are to each other as the squares on their diameters.

17. The Proof:

18. Quadrature of the Circle??? • Possibilities! • Start with a circle and create a larger circle with Radius (Larger) = Diameter (Smaller) • Alexander pointed out that the proof is based upon a square inscribed in a circle not a hexagon. • Proof: Quadrature of the circle is impossible Assume that circles can be squared… Contradiction…

19. Work Cited Dunham, William. Journey Through Genius: The Great Theorems of Mathematics. New York: Penguin, 1990. Print.