Pre-comments:  I say again, I hope for the last time:  Take notes on the reading.   Our conversations will supplement them, but cannot possibly replace the breadth of the reading.  For each section you read, have some idea what mathematics they were doing and why.  Once we get to Greece probably take 1-2 people as particular examples that you find interesting and keep those.  You are definitely not responsible for all the details in Suzuki or everything that we discuss in class.  But, take enough of it so that you have a good collection of examples and motivations for reference.   Naturally the people and topics that Suzuki and/or I discuss at more length are more important.  Now, please, can we just focus on learning and let go of grading for a while?

Here's another view of this same point - you have research topics due a week from today.  Aside from that, we don't have an exam for over a month.  Can we let it go for a month and just focus on learning at least that long?  Please.  And for that matter, when we do have an exam it will be for only 50 minutes.  How much can I expect you to tell me in 50 minutes?  I've also already told you you will have several large-scale questions to pick from and you will be encouraged to pick your own examples for them.  Please, do not worry about exams.    

As a consequence of this - I will not credit any reading reactions that ask "do we need to know this?"  The same goes for "why is there so much history here (in this book)?"

Ultimately, we are studying liberal arts.  I believe in Plato's response to those who question the value of knowledge.  I have some coins for those who want value.  

Quick answers 2.1.3-

Anaxagoras determined the sun was farther away because the moon passed in front of it (eclipses) and thus that the sun was larger because it appeared about the same size when the moon passed in front of it.  I'm not sure if he's the first to notice these facts, but I expect he is the first to reason this way.  

I don't think finding a way to measure angles is significant, and I don't know if there is any records of how that is done.  A related question is what units are used, and I don't know the history of when degrees began appearing.  I emphasise again that one can easily work in either fractions of a circle or of right angles without needing another measurement system.  

Eartosthenes knew the world was round by making observations like those he used to compute the circumference.  The "formula" for circumference merely says that circumference is proportional to diameter, which seems to be long known.  The formula for circular area is coming soon.  There are several greek approximations to pi, most famously is Archimedes, which you will read for Monday.  I'll say a few more things about the significance of pi then.  

I'm not sure how to elaborate on area is proportional to square of the diameter (Theorem 2.2).  A = kd^2?  It is *not* proven using lunes, it is used to prove results on lunes.  For lunes, please reconsider our in-class discussion from Wednesday.  Lunes are called lunes because they look like moon shapes.  

I am intentionally avoiding showing "images" of people before we have ones produced contemporaneously.  There are images here:  http://www-groups.dcs.st-andrews.ac.uk/~history/ of ancient greeks, but mostly they were produced in the Renaissance or later by people who surely had no ideas what they looked like.  When we get to the point of having contemporary images (roughly late medieval times) I will definitely be sharing them with you.  Until then, the artists' guesses seem a little silly to me.  

The quadratrix cannot be constructed with compass and straightedge - that is one of the notable features about it.  

Euclid's geometry wasn't much of his own proving (of his numerous theorems, many were simple things we would ask a HS student to prove for HW, and many of those that were sophisticated were merely his presentation of others' work), but it was much of his organisation and presentation.  He gave an appealing way of looking at mathematics that hadn't been presented before.  He did, also, present some original algebra and number theory.  Unfortunately this regularly gets overshadowed by his not-so-original geometry.  I will try to highlight his main contributions in algebra and number theory.  That all being said, I don't think Euclid was taking credit - there wasn't so much of a concern over credit at the time.  He was merely trying to present it.  Ultimately, he wrote a very good textbook - and deserves credit for that, but not for creating geometry.  I do not know anything about how long this took him.  

In addition to trisecting the angle (which we'll talk about in class) Hippias' Quadratrix could be used to square the circle (to construct [with compass and straighedge] a square of area equal to a circle).

Sometimes conic sections are taken from two cones attached at the vertex (this is why the hyperbola has two pieces).  When not, the cone is called "single-napped".

Plato was a philosopher, but remember philosophy and mathematics are never far apart.  They are close now, and probably closer then.  If one were to map disciplines, mathematics sits between physics (with chemistry on the other side) and philosophy (and probably with art or literature on the other side).  

I don't know who the others (aside from Appolonius) who bettered Eratosthenes were.  

The restriction to compass and straightedge is to the simplest and surest of figures as starting points (although the straightedge might not be so sure).  They are reflected in Euclid's 1-3 Postulates.  

Perfect numbers are probably more interesting than important.  Please remember that mathematics, on its border with philosophy, is frequently more concerned with what is interesting than what is useful.   

We will see a detailed example of Archimedes using Eudoxus's method of exhaustion on Monday.  

The story about Menaechmus and "royal road" is obviously very similar to the story about Euclid and "royal road".  Suzuki, at least, appears to doubt the second story.  

On p. 18 Suzuki refers to Pythagoras and the 6:4:3 harmonic ratio and the 4:3:2 arithmetic progression.  These properties weren't noticed by Pythagoras, they were noticed by Archytas.  This is not so important to us.  

The difference between manual arts and liberal arts were first discussed on p. 20.  As a consequence of this being a liberal arts school (and this class surely falling in that) we are much more attending to what is interesting than what is useful.  

Plato's academy was more like our school than Pythagoras's was.  Surely still not the same, but it was a place for training.  I do not know how people were selected for this academy.  And, in some sense, yes, Plato's academy only taught mathematics.  I do not know of any other schools at this time.  But, I am surely not expert enough to say there weren't any.   

I believe most of our sources for ancient Greek mathematics are second hand, but reliable and duplicated from several sources.  I'm not an expert on this, so I don't know much more.  

The Greeks are definitely *not* doing algebra the way we do now.  They were doing a kind of geometrical algebra - everything was represented by lengths and measurements of some kind.  They didn't use variables the way we do.  

Suzuki says that Hippocrates was the first professional teacher of mathematics.  What makes him a professional?  What makes anyone a professional?  If they are paid to do it.  Hippocrates is the first person we know that was paid to teach mathematics.  

Archytas's rattle is sometimes called a clapper and does appear to mostly have been what Aristotle said and not to actually appear on vases.  A transliteration of this word reads "platagn" in case that helps anyone.   

We know little of Democritus, therefore don't know if he had any negative consequence to his study of incommensurables, but at some point that must have waned.  I'm guessing it's only the first instance (likely Hippasus) that was really rebuked.  

Plato's mathematics included everything I said on Wednesday was part of mathematics.  

I have read no serious scholarship indicating Euclid to be a woman.  

Peloponnese is a large region in southern Greece.  I suggest searching for a map.  

The stadia - miles conversion is known by distances that the greeks knew between ciites &c, not circularly based on Eratosthenes's circumference computation.  

The fate of the library of Alexandria will be revealed in later reading.  

Suzuki does seem to be bold in his statements - I agree.  I happen to find this engaging and make the reading more lively, but I also think it's important to recognise that he may not be entirely accurate in his sweeping statements.  I think it's a matter of style.  

The Pythagoreans preceded Plato's academy.  And it's my impression that the library at Alexandria was more a gathering place than a research place (for finding new ideas).  I don't really see any signs of competition in Greek mathematics, but I may be missing or forgetting something.  

The Greeks appeared to primarily study geometry because it was accessible to logical organisation and reasoning.  It seems a consequence of the Ionian logic and seeing the world around them.  Ultimately, it's probably the most tangible mathematics which is the most accessible to logic.  Perhaps this only means it's what we see first, and since humans are so dependent on their sense of sight, it comes first.  

I think before mathematics had a name it probably wasn't referred to as an entity.  It isn't like needing a name for an ox that you see running in front of you - concepts sometimes don't really get formed until they have names.  

The procession toward real numbers is long and complicated.  The Greeks surely aren't there, and we won't get there, really, until the 19th or 20th century CE.  

What happened to Greek knowledge that we lived on a sphere?  Good question - much knowledge was lost, as we will see in the passage through the time before the renaissance.