Transcript Math ca. 300

Archimedes’ Determination of Circular Area
225 B.C.
by
James McGraw
Geoff Kenny
Kelsey Currie
Contents
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What else is happening?
Biography of Archimedes
Area of a circle
Archimedes’ Masterpiece: On the Sphere and
Cylinder
• Other contributions from Archimedes
• Questions/comments
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
• 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
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
Known for…
• Absent-mindedness
• The Golden Crown
• Defense mechanisms
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Archimedes Claw
Steam Cannon
Catapults
Heat Ray?
Da Vinci drawing of steam
cannon
Archimedes Claw

Some other discoveries…
• Archimedes’ Screw
• Law of the Lever
d1W1  d2W 2
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:
• 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
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
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)

Case 1: A>T
This is a contradiction.
• Case 2:
A<T
This is also a
contradiction
Q.E.D.
By proving A=T=1/2rC, He was able to provide a link
between the two dimensional concept of area with the
concept of circumference.

Thus
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.
Archimedes’ Masterpiece:
On 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)
• Proposition 13 continued…
• Where h/x = x/2r
x2 = 2rh , therefore:
Lateral surface (cylinder) = Area
(circle)
=πx2 = 2πrh
• 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) = 4πr2
• 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
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Let r be the radius of the sphere
Volume (cone) = 1/3πr2h = 1/3πr2r = 1/3πr3
Volume (sphere) = 4 volume (cone) = 4/3πr3
Note: volume constant from Euclid’s proposition XII.18
4/3πr3 = volume (sphere) = mD3 = m(2r)3 = 8mr3
M=π/6
• 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
• The sphere and Cylinder
Total cylindrical surface = 2πrh + πr2 + πr2
= 2πr(2r) + 2πr2 = 6πr2
=3/2(4πr2)
=3/2(spherical surface)
• The sphere and Cylinder
Cylindrical volume = 2πr3
= 3/2(4/3πr3) = 3/2 (spherical volume)
Other Contributions to Mathematics
• Quadrature of the Parabola
• On Spirals
• Squaring the circle
• Archimedean Spiral
• r = a + b
a,bR
Numbers
• The Sandreckoner
• Approximation of √3
Archimedean Solids
• Credit given to Archimedes by Pappus of Alexandria
• Truncated Platonic solids
Strange but true…
• Half the length of the sides and truncate
OR
OR
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