- The return of Section 2.4 (1a, 3, 6, 9)
- #1a: make sure that you have units of a volume when you're done ("cubic cubits", which is fun to say anyway!).
- #6: compute relative error to compare:
- #9b: use bounds:
for positive angles
- Worksheet for selections from sections 4.1/4.2
- Homework through section 3.2 (change from 3.3!) will be due on our next "pink day" (next Wednesday)
- Wrapping up section 3.3
- Commercial (Chris); who's up for Monday?
The commercials I've received electronically are now on-line!
- Lessons from section 3.4/3.5
- The Pythagorean theorem
- "This is a theorem that may have more known proofs than any other (the law of quadratic reciprocity being also a contender for that distinction); the book Pythagorean Proposition, by Elisha Scott Loomis, contains 367 proofs." (source)
- Earliest proof perhaps from China;
- Suggestion of the general solution from Babylonia, as we've seen previously, from Plimpton 322 (1900-1600 BCE): a tablet containing "Pythagorean triples" (many years before Pythagoras, 6th century BCE)
- Geometric proof (p. 107)
- Algebraic solutions for special cases
- Pythagoras: using successive squares
- Plato: using consecutive odd squares
- We already derived the general form of a solution:
- The "Crisis of Incommensurable Quantities"
- That
is irrational. (Proof by contradiction -- famous and easy!) Careful: this one can get you killed....
- Theon of Smyrna concocts an approximation scheme
- Let's look at it from another perspective....
- Newton improved on it, with his "method".
- Eudoxus of Cnidus concocts a theoretical paradigm for
dealing with "the unutterable" at his school in Cyzicus (p. 117). Among his other contributions:
- The study of the golden section
- Method of Exhaustion
- That
- 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".
- 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).
- How did it work?
- The quadratix could also be used to square the circle (Pappus, circa 300 AD) -- "square-forming".
- Plato
- 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" (p. 137).