[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