Archimedes The Archimedes Portrait XVII century Domenico Fetti.
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Transcript of Archimedes The Archimedes Portrait XVII century Domenico Fetti.
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