[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
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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)
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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
Theorem."
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.
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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