Lecture 1 Flashcards
Greek antiquity
Geometry, philosophy and the world
Before Greek antiquity
Temple wisdom in mesapotamia and Egypt
With the Greek philosophers - part of philosophy, intended to understand the world
Plato
427-347: projections of the real world, the word of ideas incommensurable numbers
- focus on geometry = measuring the world
Can spot mystic origin:
Quadrature of the circle
Doubling the cube
Hippocrates of Chios
470-410
Squaring moon-shaped shapes in an attempt to square the circle
Euclid
300 BC
Thirteen books, theorems and proofs.
Starting with definitions and axioms
Euclid book 1
A point is that which has no parts
A line is breadth less length
A straight line is a line which lies evenly with the points on itself
A surface is that which has length and breadth only
A rectangle, a circle, a gnomon
Notions:
Things which equal the same thing also equal one another
If equals are added to equals, the wholes are equal
The whole is greater than the part
Book 1 postulates
Book 1 theorems
Construction 1 - regular triangle
Theorem 20 - triangle inequality
Theorem 47 - pythagoraen theorem
Theorem 48 - reverse of pythagoraen theorem
Book 2
Theorem 4: If a straight line is cut at random, the square on
the whole equals the squares on the segments plus twice the
rectangle contained by the segments.
Theorem 14: To construct a square equal to a given rectilinear
figure.
Corollary XIII.18
Proves the existence of exactly five platonic solids
Book VII
Theorem 1&2 - finding the greatest common divisor
Unity is not a number
Existence is proof
Number is worldly. Ratios based on music
Book XII
Theorem 2: circles are to one another like the squares on their diameters
Archimedes (287-212)
Quadrature of the parabola
On method: establishing the volume of a sphere, a cone and a cylinder
What exactly was appreciated ?
End of tenth century, Gerbert of Aurillac
18th century, Gerard de Lairesse
Early nineteenth century, Oliver Byrnes
End of nineteenth century, Felix Klein
Around 2000 AD, classical geometry in education
