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