- Homework through section 3.2 (change from 3.3!) will be due on (Wednesday).
- New homework for 4.1-2:
- Read Wilson's "Fit the Fourth", from Lewis Carroll in Numberland.
- Problems pp. 170--, #1, 3, 6, 8, 10, 11, 12
- Wrapping up section 3.4, and section 3.5
- Commercial (Jason); Adam for Wednesday. The commercials I've received electronically are now on-line! Please send others as you get the chance.
- Lessons from section 4.1
- Background
- Hippocrates of Chios (460-380 BC -- not the doctor) "dominated the second half of the fifth century B.C."
- Aristotle: "It is well known that persons stupid in one respect are by no means so in all others; thus Hippocrates, though a competent geometer, seems in other regards to be stupid and lacking sense."
- Hippocrates wrote an "Elements of Geometry" which predated Euclid's.
- Hippocrates arrived in Athens to find the three construction problems in play.
- 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").
- Each of the following "constructions" turns out to be impossible, but we awaited proofs (1837, 1882, 1837) of this. In the meantime (2200 years or so), mathematicians struggled to find these impossible solutions -- in fact, some "mathematicians" still are....
- Hippocrates did solve an interesting associated problem -- the quadrature of the lune. His work has been transmitted fairly faithfully by Simplicius (circa 550 AD).
- Squaring the Circle
- "Is it possible to construct a square whose area shall be equal to the area of a given circle?"
- Hippocrates also thought that he'd squared the circle, using lunes. No such luck!
- Squaring the circle was proven impossible when
was shown to be transcendental (what's that mean?), in 1882.
- Doubling the Cube
- "...if upon the diagonal of a given square a new square is constructed, then the new square has exactly twice the area of the original square." Does that suggest a new problem?
- Hippocrates made "progress" on this impossible problem,
too. He reduced the problem to finding "mean proportionals":
(notice that this results in numbers in sequence
- Trisecting the Angle
- Hippocrates had no luck here.
- There are some angles that are easy to trisect (e.g. a right angle).
- Archimedes managed to trisect an arbitrary angle, but he had to "transfer distances" (p. 128). When I did this in class, I introduced a parallel line to DA through B (but was in too much of a hurry to actually construct it). How can I do this?
- The Quadratrix, invented by Hippias of Elis (circa 420 BC)
- Hippias was a sophist, trained in the art of disputation. Sophists were sought after teachers of the elites.
- Interesting that, although negatively portayed in Plato's writing, the caricatures might have contained some truth ("enough of Hippias's eccentricities that his contemporaries would have recognized him." -- p. 133)
- The Quadratrix was designed by Hippias to trisect an angle. This was the "...first example of a curve that could not be drawn by the traditionally required straightedge and compass but had to be plotted point by point." (p. 133).
- Furthermore, it involved the concept of a point that could only be determined in the limit.
- How did it work?
- The quadratix could also be used to square the circle (Pappus, circa 300 AD) -- "square-forming".
- Plato (428 BC - 348)
- The "Grove of Academia" (named after the hero Academos): Plato's Academy became the intellectual center of Greece for 900 years!
- Plato gave mathematics a favored place in the Academy, but
only pure mathematics -- not applied (dirtying the
elevating effect of mathematics):
"Let no man ignorant of geometry enter here."
- 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 the Academy came in 529 A.D., closed by the Christian Emperor Justinian "as a place of pagan and perverse learning" (p. 137).
Or, as Robin Wilson puts it in "Lewis Carroll in Numberland", "Here's lookin' at Euclid".
- History/Geography
- Philip II of Macedonia finally unified Greece by defeating its combined forces at Chaeronea in 338BCE.
- 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." (p. 88)
- 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")
- Our author credits the period from 300 to 100 B.C. with being the richest in mathematical history, on a par with the period from Kepler to Gauss (1600-1850). (p. 144)
- 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.
- His only other extant work is Data, a collection of 95 exercises for students of the elements (p. 145), although he was the other of other manuscripts.
- 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.
- The "Museum" (seat of the Muses) produced many scholars:
- Archimedes
- Eratosthenes
- Apollonius
- Pappus
- Diophantus
- Alexandria was sacked in A.D. 641, and along with it went the Museum.
- Mathematics -- we'll get into this in section 4.2
- "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."
- Definitions, Postulates, and Common Notions
- Book I -- Elementary Plane Geometry
- Proposition 1 -- we've already done!
- Proposition 5 -- the asses' bridge
- Proposition 47 -- "the windmill"