1. A Short History of Geometry, http://geometryalgorithms.com/history.htm
Ch. 1: Some History
References
Modern Geometries: Non-Euclidean, Projective, and Discrete 2nd ed by Michael Henle
Euclidean and Non-Euclidean Geometries: Development and History 4th ed by Marvin Jay Greenberg
A Short History of Geometry,
Gilbert Lecture Notes,
Modern Geometries: The History of Geometry, Part 3 by Monica Garcia Pinillos

2. What is Geometry? “geometry” comes from the Greek geometrein
geo – “earth”
metrein – “to measure”
Several thousand years old – used in agriculture and building.
Size, shape and position of two dimensional plane figures and three dimensional objects.

3. Some Characteristics of Euclidean Geometry
Deductive science
Visualization
Intuition
Practical

4. Ancient Egyptians
Herodotus (Greek, 5th cent. B.C.) : Egyptian surveyors originated geometry
Construction of the pyramids
Egyptian geometry: collection of rules for calculation (some correct, some incorrect) without justification

5. Babylonians More interested in arithmetic (used base 60)
Had calculations for areas and volumes
Pythagorean theorem
Corresponding sides of similar triangles are proportional
360o in a circle

6. Ancient India Shapes and sizes of altars and temples
Sulbasutra (2000 B.C.) contains Pythagorean theorem
The number zero!

7. Ancient Chinese
Jiuzhang suanshu (Nine Chapters on the Mathematical Arts)
Surveying, agriculture, engineering, taxation
Pythagorean theorem with a diagram to help explain why it is correct

8. Knowledge of these ancient civilizations
Calculate the area of simple rectilinear shapes
The ratio of circumference to diameter in circles is constant & rough approximations to that constant
The area of a circle is half the circumference times half the diameter
Developed to solve practical problems
Evolved out of experiments

9. Ancient Greeks Thales of Miletus (6th cent. B.C.)
Development of theorems with proofs about abstract entities
Dialectics: the art of arguing well

10. Greek Mathematicians Pythagoras of Samos (569-475 BC)
Hippocrates of Chios ( BC)
Plato ( BC)
Theaetetus of Athens ( BC)
Eudoxus of Cnidus ( BC)
Euclid of Alexandria ( BC)
Archimedes of Syracuse ( BC)
Hypatia of Alexandria ( AD)

11. Euclid of Alexandria (325-265 BC)
The Elements
I-IV, VI : plane geometry
XI-XIII : solid geometry
V: Eudoxus’ theory of proportions
VII-IX : the theory of whole numbers
X : Theatetus’ classification of certain types of irrationals

12. Euclid’s Postulates (Henle, pp. 7-8)
A straight line may be drawn from a point to any other point.
A finite straight line may be produced to any length.
A circle may be described with any center and any radius.
All right angles are equal.
If a straight line meet two other straight lines so that as to make the interior angles on one side less than two right angles, the other straight lines meet on that side of the first line.

13. Euclid’s Parallel Postulate
“Given a line and a point not on that line, there exists one and only one line through the given point parallel to the given line.”
Contains two ideas: existence and uniqueness.
No proof of a contradiction from the denial of either the existence or the uniqueness part of the of the parallel postulate is possible.

14. Two alternatives to the parallel postulate
Elliptic Geometry: There is no line parallel to some line through some point.
(Chapter 11 in Henle)
Hyperbolic Geometry: There exist at least two lines parallel to some line through some point.
(Chapters 7-10 in Henle)

15. Further Developments Rene Descartes (1596-1650)
Pierre de Fermat ( )
Girard Desargues ( )
Blaise Pascal ( )
Leonhard Euler ( )
Carl Friedrich Gauss ( )
Hermann Grassmann ( )
Arthur Cayley ( )
Bernhard Riemann ( )
Felix Klein ( )
David Hilbert ( )
Donald Coxeter ( )