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 ( 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: AB ab 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. Inscribed polygon of perimeter p Circumscribed polygon of perimeter P Circle of diameter 1 – circumference π p ≤ π ≤ P

s n = sin( ) Estimation of π (adapted) Inscribed 2 n -gon - 2 n sides each of length s n - perimeter p n =2 n s n Circumscribed 2 n -gon - 2 n sides each of length T n - perimeter P n =2 n T n Circle of diameter 1 - circumference π pn ≤ π ≤ Pnpn ≤ π ≤ Pn T n = tan( )

Estimation of π (adapted) npnpn PnPn pn ≤ π ≤ Pnpn ≤ π ≤ Pn

Archimedes’ tombstone Known to Democritus: 3 : 2 : 1

r A D C B F E 2r

r A D C B F E H G P Q X T R x y

r A D C B F E H G P X T R x Q A B R x y X

r A D C B F E H G P X T R x Q Sphere - circle C 2 of radius y Cone - circle C 1 of radius x Cylinder - circle C of radius 2r

Sphere - circle C 2 of radius y Cone - circle C 1 of radius x Cylinder - circle C of radius 2r x A C2C2 C1C1 C r A D C B F E H G P X T R x Q 2r

Sphere - circle C 2 of radius y Cone - circle C 1 of radius x Cylinder - circle C of radius 2r