[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]
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.
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
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