Pre-comments today:  Please read these answers sometime (they are on-line, you see) - many of your personal questions are answered here.  

please remember that for future teachers, your topic for your research topic needs to be a topic that is discussed for no more than a week at the level you intend to teach.  I recommend pulling it from the NYS standards since they have precise details in them and provide a good list of topics.

For non-future teachers, your situation is different.  Do the following - think about your college math. classes (here or elsewhere) and a topic that you discussed for no more than a week in one of those classes that you found particularly interesting.   Your research project is to find the history of that topic.  Please try to make it a topic that you understand already - you probably don't want to be learning mathematics in this project as well as learning the history.  

Remember, your topic is due by - which isn't terribly far away.  Good to be thinking of such things now.  

Thank you to those of you (most of you) who submit your reading reactions at least 12 hours before class.  It is making my preparations much easier.  

Overall Suzuki comment - he includes important history, and enough history to make the connections with the mathematicians meaningful.  If we say that this mathematician worked for this political figure (king &c) it is only meaningful if you know something about the kingdom and setting.  Ultimately all of the history is to place the mathematics in context.  It is the point of the book and our course.  This is our (my, Suzuki, and the courses) general premise.  If you do not accept this premise, the course will be pointless.

Quick answers -2.1.2

Again - if the book mentioned everything it could, it would never end.  The comment "why doesn't he discuss this" still continues.  Remember also one of the reasons that more is not mentioned about some of these things is that they are ancient and we don't know much more.  

I will talk extensively about music.  Please note there is no discussion of chords in the Pythagoreans, only intervals.  

Incommensurability is *very* important, especially to the Pythagoreans who believed that everything is number, by which they mean a whole number.  Incommensurables cannot be represented by rational numbers, therefore cannot be the ratio of two whole numbers.  You should think of irrational and incommensurable as basically being the same thing.   Hippasus is credited for noticing that some lengths were incommensurable - and probably he suffered somehow for noticing this.  

What's the status of the Pythagorean theorem?  Definitely he wasn't the first to know it to be true.  Did he personally prove it?  We don't know.  Did he know a proof?  We don't know, but it seems likely.  Did his group prove it?  We don't know, but seems quite likely.  Euclid has a proof, and Euclid mainly gathered others results.  He didn't give much credit for where I he got things, though.  Euclid's proof is the oldest known.

Euclid definitely does *not* assume side-angle-side congruence for triangles.  His 5 postulates are on p. 31.  They are famous.  Especially future teachers (and anyone interested in geometry) should know them by number.  

Inversion of chords is usually done with three (or more) pitches.  You can merely invert an interval by moving the top note down an octave.  This results in the (mathematical) inversion of the ration.  

Dissection theory can be perfectly formal - there are proofs of dissection equivalence (which are stronger than area equivalence, but weaker than congruence).  This is one use of "equal" for the greeks.  The other two are what you call congruent, and a weaker idea of only same length, volume, area, or something.  

Who proved what Thales said?  Unclear who proved it first.  Euclid had recorded proofs, that's for sure, but now clear who devised them.  

Why were the Pythagoreans secret?  It's more fun that way?  Makes them feel more special by excluding others?  Unclear.  In this case "school" probably merely means a gathering of people discussing things; closer to a school of thought than a formal institution.  

Logistics is methods for computation and measurement, whereas geometry is about proving general properties of spatial figures.  I don't think it's that logistics isn't as important as geometry, but not as notable.  Probably in fact logistics is more "important" in a practical way, but geometry is partly responsible for leading mathematics to become what mathematics is.  

There are accounts that say Pythagoras was captured by the Persians and taken to Babylon.  These accounts have been repeated.  This makes it a tradition.  Like "oral tradition".

Draw two intersecting lines:  Thales #3 says the opposite angles are congruent.  The are called "vertical" because they are opposite of the vertex.  You might think of them as "vertexal".  Thales #4 as given in our book is what you know from HS as ASA triangle congruence.

Thales surely had some definitions - clearly isosceles, probably circle, but probably not angle (by the way, I encourage each of you to check what your definition of angle is in 335 - it continues to be a problem).  Thales doesn't claim to not have proven his results, but it appears that he didn't.  Other sources claim he was the first to prove anything.  Clearly this history is somewhat murky and speculative on either side.  Especially in a time in which people were not proving things, merely noticing something to be true seems valuable.  I believe it is still a valuable step today.  He probably noticed them by looking at many different examples and seeing the results always hold true.  It seems possible that people didn't notice that these things always happened before Thales noticed it.  Remember again, this is still a long time ago - and also remember that you can't really formulate such things until you have language and notation for discussing them.  People wrote about Thales and his discoveries (such as Herotodus) and it is from them that we know what Thales did.

There is definitely a belief that the Greeks were able to prove things in mathematics because they didn't need to worry about their own survival as much.  This allowed them to focus on the more philosophical and less practical aspects.  

Why do people fight wars?  The answer is probably the same today as it was then - because someone else has something they want.  Or because they don't like how others do things.  I guess.  

Suzuki does raise important concerns about the pythagorean hammers.  Somehow he came across the basic idea - either in bells, strings, or something.  Not clear how.  

Much from ancient times doesn't survive - it's the way.  We're lucky to have as much as we do.  

In historical notation when someone's dates are given as "fl." in indicates that they "flourished" at that time - i.e. they were around and an active member of society, in the prime of their life.  Thales "fl. 6th c. BCE" according to Suzuki.

I'm not sure that the greeks did measure angles.  I'd have to see some evidence of this - and I don't know any.  You don't need to measure angles to talk of a right angle (which happens when two lines intersect so that all four angles are equal).  The greeks seemed - by reading Euclid - to work in units of right angles (notice that the Pythagoreans #1 is in terms of "two right angles").  

There is not an error in the story of Ionia at the top of p. 16.  Please reread carefully.  

One student reports "I heard the term "rich as Croesus" the other day and had no idea what it meant."  Since I've never heard this before, I'm surprised someone did, and apparently it is still in use.  

Pythagoreans #4 says that we can tesselate with regular triangles, squares or hexagons, i.e. that six equilateral triangles fit around a point, four squares, or three regular hexagons.  This lead to platonic solids by considering what happens when  there are fewer than these numbers around a point (three squares give a cube, three triangles give a tetrahedron &c).

One doesn't use the Pythagorean theorem, or a^2+b^2 = c^2 to construct triples of numbers satisfying this that are all whole numbers.  The Babylonians had list of such numbers and the Pythagoreans had ways of constructing arbitrarily many of them (although not all of them).

There is a map of the region on p. 23.

If you want to see examples of Greek proofs - go read a copy of Euclid.  We won't do terribly much of it, but we'll do a tiny bit, but not really in a contemporary context.  It's pretty easy to go find Euclid, and I strongly encourage it for anyone who is interested.  

Theorem 2.1 is proven by changing either into the one (and only) rectangle with the same base and height.  Loosely this is done by cutting a triangle off the end and putting it on the other side.  It's a little more complicated because it could be too slanted for this to work, but it's the basic idea.  

There is not an error "On page 19 in the book, Suzuki has a minor typo after the second paragraph where B to c."  Lowercase c is used to indicate the c an octave above middle C.  Please see the footnote on the page.  

The historical tension between Babylonian Jews and Palestinian Jews is not the same as the current tension between Jews and Muslims and Christians in the middle east (the latter two groups did not exist in 500 BCE).

Many other (definitely not most) mathematicians after Thales were military engineers.  

As Suzuki says, rope stretchers had a practical way of making a right angle, by creating a cord with knots at 3, 4 and 5 lengths and then stretching this into a triangle.  

There's a typo in the last sentence on page 18 ("inerted" instead of "inverted").