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, http://geometryalgorithms.com/history.htm Gilbert Lecture Notes, http://www.math.sc.edu/~sharpley/math532/Book_Gilbert.pdf Modern Geometries: The History of Geometry, Part 3 by Monica Garcia Pinillos http://cerezo.pntic.mec.es/mgarc144/marcohistoria/Modern%20Geometries.pdf

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.

Some Characteristics of Euclidean Geometry Deductive science Visualization Intuition Practical

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

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

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

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

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

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

Greek Mathematicians Pythagoras of Samos (569-475 BC) Hippocrates of Chios (470-410 BC) Plato (427-347 BC) Theaetetus of Athens (417-369 BC) Eudoxus of Cnidus (408-355 BC) Euclid of Alexandria (325-265 BC) Archimedes of Syracuse (287-212 BC) Hypatia of Alexandria (370-415 AD)

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

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.

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.

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)

Further Developments Rene Descartes (1596-1650) Pierre de Fermat (1601-1665) Girard Desargues (1591-1661) Blaise Pascal (1623-1662) Leonhard Euler (1707-1783) Carl Friedrich Gauss (1777-1855) Hermann Grassmann (1809-1877) Arthur Cayley (1821-1895) Bernhard Riemann (1826-1866) Felix Klein (1849-1925) David Hilbert (1862-1943) Donald Coxeter (1907-2003)