SciRev. X102 wk. 3 maths exam with 100% accurate solutions + diagrams

incommensurability

-major discovery of the PYTHAGOREAN SCHOOL

-the side of a square doesn't match the diagonal

-there is no common submultiple

-challenged the role of whole numbers as a measure of reality

who was good at math at the time in the ancient Greek world?

-Egyptians and Babylonians

-focused on arithmetic and simple geometry

EUCLID

-of alexandria

-300 BCE

-enters a new era with the NOTION OF "FORMAL PROOF"

-main works is the "Elements"

Euclid's Elements presents geometry how?

-in axiomatic form

-provided the first axiomatic deductive system, based on theorems that descend necessarily from definitions and postulates

axiomatic system

A logical system based on undefined terms, definitions, axioms or postulates, and theorems.

elements

-euclid

-5 postulates

1. a line can be drawn from any point to another

2. a straight line can extend continuously from either end

3. a circle can be made around any point

4. all right angles are equal

5. straight lines intersect

archimedes

-of syracuse

-287 - 212 BCE

-most creative and flamboyant mathematician

-worked in pure mathematics (geometry) and also in mechanics

-axiomatic theory of balance

-had a treatise on FLOATING BODIES

-developed Euclid's idea of exhaustion (making a polygon into a circle) for curves and parabolas

-killed by a roman soldier

eudoxus

-of cnidos

-390 - 337 BCE

-early astronomer (his contemporary was Aristotle)

-formulated systems of CONCENTRIC SPHERES

-searched for mathematical order, not physical structure

Eudoxus' concentric circles showed what

concentric circles rotating around the earth to explain qualitatively planetary motion

Eudoxus vs Aristotle on the spheres

-Eudoxus: purely mathematical construction (considered planets on a one-by-one basis)

-Aristotle: attributed a physical reality to the spheres; put them together and considered the system as a whole, ending up w/ 55 of them acting and counteracting each other

Hellenistic world

the area of the world that was influenced by the reign of Alexander and continued after he died

hipparchus

-of nicea

-140 BCE

-merged the Babylonian numerical astronomical tradition w/ the Greek geometrical one

-focused on geometrical models, numerical, instrumental, and observational dimensions of astronomy

what was Greek astronomy originally focused on

-real or imaginary geometrical models (rather than accurate numerical predictions)

retrograde motion

the apparent reversal of motion of a planet as seen from the earth against the background of the fixed stars

what is hard to explain w/ concentric circles?

retrograde motion

claudius Ptolemy

-geography and Almagest (2 of his works)

-Almagest relied on circles rather than spheres

-was a tension in Ptolemy

what was the most elaborate astronomical work from antiquity?

-Ptolemy's Almagest

-where he relied on circles rather than spheres to explain geometrical motions

what was the tension in Ptolemy?

-was never fully solved

-tension between producing an accurate mathematical astronomy and constructing a credible physical structure for the heavens which he never truly solved

deferent-with-epicycle

-accounts for retrograde motion

-ptolemy

-non-uniform motions in order to "save the phenomena"

-explained changes in apparent distances of planets from the earth

was the deferent-with-epicycle successful mathematically or physically?

mathematically; lost physical meaning