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Ch. 1: Some History
•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,
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)
Ancient Egyptians
“geometry” comes from the Greek geometrein
geo – “earth”
metrein – “to measure”
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
• More interested in arithmetic (used base
• 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,
• Pythagorean theorem with a diagram to
help explain why it is correct
Knowledge of these ancient
• Calculate the area 0f simple rectilinear
• 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 (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)
1. A straight line may be drawn from a point to
any other point.
2. A finite straight line may be produced to any
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 (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)