- I haven't finished grading the first exam. By next time.
- Please hand in your Modular Arithmetic homework.
- Your next assignment:
- Read (what you can get through easily of) Fit the Fifth -- "Send Me the Next Book....". Read for enjoyment. For next time.
- Problems, due 10/22:
- p. 160-162: solve/explain the "Geometrical Puzzles"
- p. 166: the coloring maps problem. Try to construct a map that requires more than 4 colors. You can do it, but you'll have to "cheat" somehow. Find a way to cheat!;)
- p. 169: solve the isosceles triangle problem
- p. 28: complete the proof that Charles began. Since we don't have his next page, you're on your own! But I'm sure that Charles managed to complete it. Notice that he makes an error (that he corrects) in his write-up....
- Changes in Dodgson's life
- How about educational policy?
- The mathematics... especially Euclid.
- Puns so bad they hurt!
- Euclid
- Historical and Geographical backdrop
- Greece (of about 500 BCE)
- Plato
(428/427 BC - 348/347 BC)
- The Grove of Academia: Plato's Academy (intellectual center of Greece for 900 years!)
- Plato gave mathematics a favored place in the Academy, but only pure mathematics -- not applied (sallying the elevating effect of mathematics).
- About 300 BC: Rival "Museum" of Ptolemy in Alexandria takes over the mathematical focus: the heart of mathematics moves to Alexandria from Athens.
- The end of came in 529 AD by the Christian Emperor Justinian "as a place of pagan and perverse learning"
- Philip II of Macedonia finally unified Greece by defeating its combined forces at Chaeronea in 338 BCE.
- Then Philip's son Alexander the Great (356-323 BCE) finished the job, and carried Greek civilization to the ends of the known world.
- The time from "...Alexander the Great into the first
century B.C. ... formed a brilliant period of history known
... as the Hellenistic Age." The period from 300 to 100 B.C. is
considered one of the richest in mathematical history, on a par
with the period from Kepler to Gauss (1600-1850).
- At Alexander's death, Ptolemy took over Alexandria, and founded the University of Alexandria. "Demetrius Pahlereus was invited from Athems to take charge of the Library, and it is probable... that Euclid was invited with him to open the mathematical school." (Cajori, A History of Elementary Mathematics")
- Euclid
(circa 300 BC) produced the epic work of the Museum, the Elements
of Geometry (this version from the Greek text
- i.e., that edited by J.L. Heiberg (1883-1885) - accompanied
by a modern English translation,).
Perhaps a more useful version is by D. E. Joyce, of Clark University, including Java applets permitting a little interaction with the text (and with some helpful guidance, and concordance).
- Oldest Greek mathematical treatise to reach us in its entirety.
- Euclid's genius was in organizing the mass of disparate results of the ages. "Elements of Geometry" had appeared as early as the fifth century BC.
- For awhile, Theon of Alexandria (A.D. 365) had the earliest version (his own revision), until a more ancient version was found in a Vatican manuscript swiped by Napoleon.
- Over a thousand editions have appeared.
- Alexandria was sacked in A.D. 641, and along with it went the Museum.
- Mathematics
- Plato is credited with "laying down the law" that only a
straightedge and compass were to be used in the
constructions, and, in particular, neither device was
allowed to "transfer distances" (the straightedge is
unmarked, and the compass collapses upon use), and each
could only be used a finite number of times ("horror of
the infinite").
- "It is also reported that Ptolemy once asked Euclid if
there was not a shorter road to geometry that through
the Elements, and Euclid replied that there was no
royal road to geometry." (Joyce)
- "But what shall I get by learning these things?"... Euclid
called his servant and said, "Give this man a coin,
since he must make a profit from what he learns."
- Four Impossible Problems (reminds me of a White
Queen we know!)
- "squaring the circle" -- constructing a square with the same area as a given circle.
- "doubling the cube" -- constructing a cube with twice the volume of a given cube.
- trisecting an angle -- cutting a given angle into three equal angles.
- Deriving the parallel postulate from the other four postulates (and common notions, etc.)
- Skills:
- Bisecting a given segment
- Dropping a perpendicular to a given line from a given point
- Bisecting a given angle
- Drawing a parallel to a given line through a given point.
- Constructing a regular triangle (equilateral triangle)
- Together:
- Trisecting a 90o angle (see "homework", p. 28)
- Constructing a regular hexagon (see p. 28)
- Proof of the asses' bridge (p. 84)
- Proof of the Pythagorean Theorem (p. 82)
- Plato is credited with "laying down the law" that only a
straightedge and compass were to be used in the
constructions, and, in particular, neither device was
allowed to "transfer distances" (the straightedge is
unmarked, and the compass collapses upon use), and each
could only be used a finite number of times ("horror of
the infinite").
- Historical and Geographical backdrop
-
- Lewis Carroll liked to have fun in all things:
- "If AB were to be divided into two parts at C..."
"It would be drownded."
- The Euclid Debate: "De Morgan raised the question whether
Euclid be, as many suppose, the best elementary
treatise on geometry, or whether it be a mockery,
delusion, snare, hindrance, pitfall, shoal, shallow,
and snake in the grass." -- An Anti-Euclid Association
was formed....
- What's Dodgson's point in the Minos sketch on page 93?
- What do you think of Dodgson's alternative postulate to
the parallel postulate (p. 98)?
- How about his approach to attacking circle squarers?
