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
and the 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, Print.