- When and where did Thales live?
- Thales was the first person to do what?
- What mathematical discoveries is Thales famous for?
- What mathematical field was his specialty?
- What discoveries (theorems/postulates) of his do we use in mathematics on a day-to-day basis?

- When and where did Pythagoras live? (Greece, 500 B.C.)
- Did he do his work alone? (no, he had a whole school of followers who attributed their work to him.)
- What great mathematical theory bears his name?
- Did he invent it? How is it useful?
- What did the Pythagoreans discover regarding sqrt(2)?

- What was Desargues's theorem? (Perspective Theorem)
- What polygon did Desargues's Theorem relate to? (Triangle)
- What type of geometry did Desargues invented? (Projective or Modern?)
- Name two contemporary mathematicians to Desargues who recognized his work. (Mersenne, Descartes, Pascal).
- How many new terms did Desargues introduce during the times of his writing? and what was the only one preserved? (70. involution.)
- What was Desargues's occupation and nationality (French architect and military engineer.)

- What is John Playfair known for? (John Playfair is known for his parallel postulate that was based on Euclidean's fifth postulate)
- How does Playfair's postulate based on non-Euclidean geometries relate to Euclid's fifth postulate?
- Can Playfairs Parallel Postulate be proved from other Euclids postulates? Why or why not?
- How is Playfair's Parallel Postulate different than Euclid's 5th Postulate?
- Is Playfair's postulate still used today? If so. when and how? If not, why not?

- What were his contributions towards Geometry?
- When and where did Legendre live.
- What important discovery did he publish in 1806 (which Gauss wrote about in 1802)?
- What other great mathemeticians influenced him? How?
- How did the French Revolution affect his life?

- What were his major contributions to the mathematical community?
- Which of his works or studies is he best known for?
- Did Gauss devise or discover any axioms, postulates, theorems, etc.? If so, which one(s) is/are he credited for?
- Did Gauss work with any other people? Who? Why?
- How do Gauss' discoveries impact math today, and do they tie in at all with what we're learning in Geometry class?

- When and where did Dedekind live?
- What was his important contribution to defining the real numbers? (Dedekind Cut)
- What is a Dedekind Cut?
- Joint paper? Zeta functions??

- What other famous mathematician lectured Riemann? (Gauss)
- What General Theory of Riemann's formed the basis of some of his most important work? (General Theory Of Complex Variables)
- Euler's Zeta Function was examined by Riemann, who changed it to form Riemann's Hypothesis helps to describe what? (The Distribution of Prime Numbers)
- Riemann's work later influenced what person famous for his Theory of Relativity? (Einstein)
- When and where did Riemann live?
- Riemannian Geometry now means something different than it did. Explain.

- How did Pasch `axiomate' Geometry?
- What was Pasch's Theorem, and why was it necessary?
- How did Peano and Hilbert refine parts of his work?
- How did his work affect other branches of mathematics (i.e. calculus)?
- What mathematical discoveries did he realize had been based on nothing from an axiomatic point of view?

- When and where did Lobachevsky live?
- What was Lobachevsky best known for? His work in the field of Non-Euclidean Geometry, sometimes called Lobachevkian Geometry of H?/E? Geometry)
- What 2 natural disasters struck the University of Kazan while Lobachevsky was the "rector of Kazan"? (A cholera epidemic in 1830 and a big fire in 1842.)
- Which Euclidean postulate did Lobachevsky show false in his own geometry? Euclid's fifth postulate
- What was his parallel postulate? (There exist more than one lines parallel to a given line through a given point not on the line.)

- What geometry was he (co) founder of? (Hyperbolic)
- Who else discovered this geometry/when relative to him/independently or no?
- What assumption did he make?
- What famous mathimatician was he friends with? (Gauss)
- What axiom was he interested in? (Parallel Axiom)

- When ard where did Poincare live?
- What mathematical area is Poincare the founder of?
- What mathematical contest did Poincare win even though he did not provide a complete solution?
- What did this contest question deal with? (Answer: Dynamic Stability of the Solar System.)

- When and where did Klein live?
- What important discoveries/contributions did Klein make to non-euclidean Geometry?
- Sketch/describe a Klein Bottle.

- What happened in 1900 to make Hilbert famous? (23 Problems speech in Paris)
- What area of mathematics did Hilbert make special contributions. (Geometry, Grundlagen, 1899?)
- How many axioms did Hilbert have? Why so many more than Euclid?
- What contribution he made to mathematics?
- Where he came from and how he grew up?

- Personal information - born in Michigan, died in Cambridge Massachusetts, lived 1884 to 1944, etc
- Where and how did he get his education? - He attended havard and the University of Chicago. In 1907 he got a PhD from Chicago.
- What were his contributions to geometry?
- How many axioms did he have and why/how does this compare to Hilbert.
- How was he honored? - The Grorge David Birkhoff Prize est 1967, awarded by SIAM and AMS for "outstanding contribution to applied mathematics in the highest and broadest sense."

- e-mail: calkins@andrews.edu
- voice/mail: 269 471-6629/ BCM&S Smith Hall 106; Andrews University;
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- URL: http://www.andrews.edu/~calkins/math/webtexts/geom00ae.htm
- Copyright ©2002, Keith G. Calkins. Revised on or after November 19, 2002.