Transcript File - Bennett Laxton

Hippocrates’ Quadrature of the
Lune
Bennett Laxton
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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
 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
 Used mathematics as a practical facilitator
 Building, Trade, Agriculture
 2000 B.C. number system and triangles
 Right Angles
 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.
 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
 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
 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:
 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
 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.”
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
Proof:
Proof:
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.
 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.
 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…
Dunham, William. Journey Through Genius: The Great
Theorems of Mathematics. New York: Penguin,
1990. Print.