Archimedes

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Archimedes Determination of Circular Area225
B.C.

• by
• James McGraw
• Geoff Kenny
• Kelsey Currie

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Contents

• What else is happening?
• Biography of Archimedes
• Area of a circle
• Archimedes Masterpiece On the Sphere and
  Cylinder
• Other contributions from Archimedes
• Questions/comments

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What else is happening in 300-200 BC?

• China
• In 247 Ying Zheng took the thrown as King of the
  state of Qin
• 230 he set out in a battle for supremacy over the
  other Chinese states
• Largest battle between Qin and Chu states with
  over 1000000 troops combined
• 221 declared himself the first Chinese Emperor

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• Rome
• 225 BC Battle of Telamon
• Invasion of an alliance of Gauls
• Well organized alliances and defences
• Contained approximately 150,000 troops combined
• 264-146 BC Punic wars
• Largest war of ancient times up to that point

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Archimedes

• Born 287 BC in Syracuse, Sicily
• His father Phidias was an astronomer
• Studied at the Library of Alexandria
• Known for contributions to math, physics,
  engineering
• Details of his life lost

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Known for

• Absent-mindedness
• The Golden Crown
• Defense mechanisms
• Archimedes Claw
• Steam Cannon
• Catapults
• Heat Ray?

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Da Vinci drawing of steam cannon
Archimedes Claw
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Some other discoveries

• Archimedes Screw
• Law of the Lever

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Great Theorem Area of the Circle

• This has been a well know fact and geometers of
  that time would have known this.
• Modern mathematicians such as you and I would
  denote this ratio as

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• This was the ratio of circumference to diameter,
  but what about the ratio of area to diameter?
• Euclid knew there was a value "k" that was the
  ratio of area to diameter, but did not make the
  connection between that and the value Pi

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Theorem The area of a regular polygon is 1/2hQ
where Q is the perimeter

• Assume a polygon with n sides with sides of lenth
  b, then the area would be n times the area of the
  triangle created by side b and hight h.
• This gives

• where (b b ..... b) is the perimeter of the
  polygon
• QED

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Proposition 1

• The area of any circle is equal to the area of a
  right angled triangle in which one of the sides
  of the triangle is equal to circumference and the
  other side equal to its radius. (Proved by
  reductio ad absurdum)?

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Case 1 AgtT

• This is a contradiction.

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Case 2 AltT
This is also a contradiction Q.E.D.
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By proving AT1/2rC, He was able to provide a
link between the two dimensional concept of area
with the concept of circumference.

• Thus

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Proposition 3 The ratio of the circumference of
any circle to its diameter is less than 3 and
1/7 but greater than 3 and 10/71.
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Archimedes MasterpieceOn the Sphere and
Cylinder

• Proposition 13
• The surface of any right circular cylinder
  excluding the bases is equal to a circle whose
  radius is a mean proportional between the side of
  the cylinder and the diameter of the base.
• Or
• Lateral surface (cylinder of radius r and
  height h)
• Area (circle of radius x)

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• Proposition 13 continued
• Where h/x x/2r x2 2rh , therefore
• Lateral surface (cylinder) Area (circle)
• px2 2prh

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• Proposition 33
• The surface of any sphere is equal to four
  times the greatest circle in it.
• Used double reductio ad absurdum
• Surface area (sphere) 4pr2

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• Proposition 34
• Any sphere is equal to four times the cone
  which has its base equal to the greatest circle
  in the sphere and its height equal to the radius
  of the sphere
• Let r be the radius of the sphere
• Volume (cone) 1/3pr2h 1/3pr2r 1/3pr3
• Volume (sphere) 4 volume (cone) 4/3pr3
• Note volume constant from Euclids proposition
  XII.18
• 4/3pr3 volume (sphere) mD3 m(2r)3 8mr3
• Mp/6

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• The sphere and Cylinder
• Climax of work
• Used both other great propositions 33 34
• Cylinder 1.5 the volume and surface area of its
  sphere

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• The sphere and Cylinder
• Total cylindrical surface 2prh pr2 pr2
• 2pr(2r) 2pr2 6pr2
• 3/2(4pr2)
• 3/2(spherical surface)

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• The sphere and Cylinder
• Cylindrical volume 2pr3
• 3/2(4/3pr3) 3/2 (spherical volume)

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Other Contributions to Mathematics

• Quadrature of the Parabola

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• On Spirals
• Squaring the circle
• Archimedean Spiral
• r a b? a,b?R

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Numbers

• The Sandreckoner
• Approximation of v3

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Archimedean Solids

• Credit given to Archimedes by Pappus of
  Alexandria
• Truncated Platonic solids

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Strange but true

• Half the length of the sides and truncate

OR
OR
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Conclusion

• Archimedes died in 212 BC
• Died from a soldier when he refused to cooperate
  until he finished his math problem
• Cylinder and sphere placed on his tomb