[Today’s and Tuesday’s classes: Big quiz on Tuesday, worksheet on symmetry,

worksheet on algebraic symbolism (syncopated algebra) of Diophantus,

Robert Recorde’s use of = (equals) sign and other arithmetic and algebraic notation, 

Euclid’s Elements, idea of an axiomatic system, review of Euclidian geometry, especially as presented in Book I of Euclid’s Elements, straightedge and compass constructions, two-column proofs]



Mathematics 115

Homework Assignment #9

Due Wednesday, January 23, 2002 – really!


[Y]ou can never make a lawyer if you do not understand what 'demonstrate' means;

and I left my situation in Springfield, went home to my father's house and stayed there

till I could give any proposition in the six books of Euclid at sight.  I then found out

what 'demonstrate' means, and went back to my law studies. - Abraham Lincoln



Prof. Beery's office hours this week:         Monday 1/21   10:30 a.m.-12:30 p.m., 4-5 p.m.                                                  

                                                            Tuesday 1/22 10:30 a.m.-12:30 p.m., 4-5 p.m.                                          

                                                            Wednesday 1/23          4 – 5 p.m. (Sorry!)

                                                              Thursday 1/24   11 a.m.-12:30 p.m., 4-5 p.m.

                                                                  Friday 1/25   2:30 - 4:30 p.m.

                                                      and by appointment   Hentschke 203D, x3118

Tutorial sessions:  Tuesday and Wednesday mornings, 11 a.m., Hentschke 202 (Sasha)



Read from the Mathematics of Ancient Greece Section:

            "The Greeks" (pages 401-404: "Euclid of Alexandria")

            "The Paradoxes of Zeno"

            "Euclid's Proof of the Pythagorean Theorem"


Do:    A.  Exercise 13 from page 409 of "The Greeks"  


         B.  Complete problems 1, 2, and 3 from the "Euclidian Geometry" worksheet.

         You'll find definitions 1, 2, 4, 10, 15, and 23, along with the common notions,

         the postulates, and Proposition 1.1 on pages 32-37 of the "Euclid's Proof of the

         Pythagorean Theorem" reading, as well as on the handout from class.  Definition 3

         is as follows. 

         Definition 3:  "The extremities of a line are points."


         C.  Answer the following questions about "Euclid's Proof of the Pythagorean


         1.   In addition to SSS, SAS, and ASA, which triangle congruence does Euclid

               prove in Book I of the Elements?

         2.   What are vertical angles and what is the relationship between them?  Draw a

               picture, and cite the relevant proposition from Book I of Euclid's Elements.

         3.   In which proposition does Euclid prove that the sum of the angles in any triangle

               is 180 degrees?  Explain.

         4.   Which proposition guarantees that if a triangle has sides of lengths  5, 12, and 13,

               then the triangle must be a right triangle, Proposition I.47 or Proposition I.48? 

               Explain.  In the 5-12-13 triangle, which angle is the right angle?  Draw a picture.


Tuesday's quiz       will be a 50-point quiz covering all of your classwork, reading, and 

                               homework since the last 50-point quiz, plus abacus calculations


For Tuesday's class, please bring your compass and straightedge (ruler).


Wednesday's class will begin at 1:30 p.m. and will end at the usual time.





Classical Greece Timeline:  800 - 200 BCE


Approximately 700s BCE:  Homer -     wrote Iliad and Odyssey, about the Trojan War

                                                            of the 1200s BCE (maybe)

624-547 BCE:  Thales -   first Greek mathematician and philosopher, first to give

                                        logical (deductive) proofs of mathematical statements

572-497 BCE:  Pythagoras - leader of "brotherhood" with motto "All is number"

Approximately 494 BCE:  King Darius -     of then world power Persia (remember the 

                                                                  Darius vase depicting his royal treasurer using

                                                                  a counting board?) repelled by Athenian army

                                                                  at Marathon

400s BCE:  Zeno - pointed out several physical / logical / mathematical paradoxes

431-404 BCE - Peloponnesian War:  Athens intellectuals versus warriors of Sparta

469-399 BCE:  Socrates - Plato's teacher, taught by asking questions of his students

429-347 BCE:  Plato - founded Academy in Athens specializing in math and philosophy

                                    with motto, “Let no one ignorant of geometry enter here.”

408-355 BCE:  Eudoxus - developed a theory of proportions (ratios)

384-322 BCE:  Aristotle - student of Plato, first codified principles of logical argument

356-323 BCE:  Alexander the Great -    tutored by Aristotle; his father, Philip II of 

                                                             Macedonia (382-336 BCE), had "unified"

                                                             (conquered) the rest of Greece; Alexander

                                                             conquered Egypt, founding Alexandria, Egypt,

                                                             and its Library in 332 BCE, then conquered Persia

Approximately 300 BCE:  Euclid -     probably first mathematician at Library of Alexandria,

                                                         organized then known mathematics into The Elements

287-212 BCE:  Archimedes -   estimated π and computed volumes of many solids; is

                                                considered one of the three greatest mathematicians ever



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