archimedes the archimedes portrait xvii century domenico fetti
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Archimedes
The Archimedes Portrait
XVII century
Domenico Fetti
Ancient Greece
At around 600 BC, the nature of Mathematics began to change in Greece…
- Polis, city-states;
- Exposure to other cultures;
The Greek were eager to learn.
- Mathematics valued as the basis for the study of the physical world;
- Idea of mathematical proof;
Life of Archimedes (287-212 BC)
- son of an Phidias, an astronomer
- related to King Hierro II of Syracuse?- spent time in Alexandria?
- Archimedes Screw
- Second Punic War; Plutarch’s biography of General Marcellus
Dont disturb my circles!
Lever
- Archimedes created a mathematical model of the lever; he was the first mathematician to derive quantitative results from the creation of mathematical models of physical problems.
- equal weights at equal distances from the fulcrum of a lever balance
- lever is rigid, but weightless
- fulcrum and weights are points
Law of the lever:
Magnitudes are in equilibrium at distances reciprocally proportional to their weights:
A B
a b
A x a = B x b
Golden Crown
Law of buoyancy (Archimedes’ Principle):
The buoyant force is equal to the weight of the displaced fluid.
Estimation of π
Proposition 3:
The ratio of the circumference of any circle to its diameter is less
than but greater than .7
13
71
103
Inscribed polygon of perimeter p
Circumscribed polygon of perimeter P
Circle of diameter 1 – circumference π
p ≤ π ≤ P
sn = sin( )
Estimation of π (adapted)
Inscribed 2n-gon
- 2n sides each of length sn
- perimeter pn=2n sn
Circumscribed 2n-gon
- 2n sides each of length Tn
- perimeter Pn=2n Tn
Circle of diameter 1
- circumference π
pn ≤ π ≤ Pn
θ
Tn = tan( )θ
θ 2θ
θ
21
Estimation of π (adapted)
nn
nn
Pp
Pp
2P 1n
11 nnn Ppp
222 p
4P2
n pn Pn
2 2.82842 4
3 3.06146 3.31370
4 3.12145 3.18259
5 3.13654 3.15172
pn ≤ π ≤ Pn
Archimedes’ tombstone
Known to Democritus:
3
1
3 : 2 : 1
rA
D
C
B
F
E
2r
2r
rA
D
C
B
F
E
2r
2r
H
G
P
Q
X
T
R
x
y
rA
D
C
B
F
EH
G
P
X
T
R
x
2r
2r
Q
A B
R
x
y
X
2r
222 ARyx
222)2( RByxr
222 )2( rRBAR
rxyx 222
rA
D
C
B
F
EH
G
P
X
T
R
x
2r
2r
Q
rxyx 222
2r
Sphere
- circle C2 of radius y
Cone
- circle C1 of radius x
Cylinder
- circle C of radius 2r
rxyx 222
Sphere
- circle C2 of radius y
Cone
- circle C1 of radius x
Cylinder
- circle C of radius 2r
x
A
C2
C1
C
r
x
r
yx
2)2( 2
22
rA
D
C
B
F
EH
G
P
X
T
R
x
Q
2r
r
x
C
CC
221
Sphere
- circle C2 of radius y
Cone
- circle C1 of radius x
Cylinder
- circle C of radius 2r
2
1
221
r
r
C
CC
)(2 21 CCC
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