Ch. 1: Some History References Modern Geometries: Non-Euclidean, Projective, and Discrete 2 nd ed by Michael Henle Euclidean and Non-Euclidean Geometries: Development and History 4 th ed by Marvin Jay Greenberg A Short History of Geometry, References Modern Geometries: Non-Euclidean, Projective, and Discrete 2 nd ed by Michael Henle Euclidean and Non-Euclidean Geometries: Development and History 4 th ed by Marvin Jay Greenberg A Short History of Geometry,

Shocking Possibilities “The effect of the discovery of hyperbolic geometry on our ideas of truth and reality have been so profound,” wrote the great Canadian geometer H.S.M. Coxeter, “that we can hardly imagine how shocking the possibility of a geometry different from Euclid’s must have seemed in 1820.” (Greenberg, xxv) “The effect of the discovery of hyperbolic geometry on our ideas of truth and reality have been so profound,” wrote the great Canadian geometer H.S.M. Coxeter, “that we can hardly imagine how shocking the possibility of a geometry different from Euclid’s must have seemed in 1820.” (Greenberg, xxv)

Ancient Egyptians “geometry” comes from the Greek geometrein geo – “earth” metrein – “to measure” Herodotus (Greek, 5 th cent. B.C.) : Egyptian surveyors originated geometry Construction of the pyramids Egyptian geometry: collection of rules for calculation (some correct, some incorrect) without justification “geometry” comes from the Greek geometrein geo – “earth” metrein – “to measure” Herodotus (Greek, 5 th 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 360 o in a circle More interested in arithmetic (used base 60) Had calculations for areas and volumes Pythagorean theorem Corresponding sides of similar triangles are proportional 360 o in a circle

Ancient India Shapes and sizes of altars and temples Sulbasutra (2000 B.C.) contains Pythagorean theorem The number zero! 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 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 0f simple rectilinear shapes The ratio of circumference to diameter in circles is constant & rough approximations of that constant The area of a circle is half the circumference times half the diameter Developed to solve practical problems Evolved out of experiments Calculate the area 0f simple rectilinear shapes The ratio of circumference to diameter in circles is constant & rough approximations of 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 (6 th cent. B.C.) Development of theorems with proofs about abstract entities Dialectics: the art of arguing well Thales of Miletus (6 th cent. B.C.) Development of theorems with proofs about abstract entities Dialectics: the art of arguing well

Greek Mathematicians Pythagoras of Samos ( 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) Pythagoras of Samos ( 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)

Euclid of Alexandria ( 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 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) 1.A straight line may be drawn from a point to any other point. 2.A finite straight line may be produced to any length. 3.A circle may be described with any center and any radius. 4.All right angles are equal. 5.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. 1.A straight line may be drawn from a point to any other point. 2.A finite straight line may be produced to any length. 3.A circle may be described with any center and any radius. 4.All right angles are equal. 5.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.

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