Patrick and Milan:

  Your presentation and paper on Saccheri turned out pretty well, although in the paper
  I would have preferred to see a bit more of your own explanations of the proofs of
  Saccheri's propositions and a bit less of the direct quotation from McCleary's book. 
  In particular, the paper could have used some more details about exactly what the 
  contradiction was that Saccheri claimed to have found.  You explained that pretty 
  well in the talk after we discussed it on Sunday, but that did not really make it into 
  the paper.

  The presentation improved a lot from Sunday evening to Monday, but it was still a 
  bit uncertain at times.  

  Specific comments:

  Page 1: "One such great Mathematician, Giovanni Girolamo Saccheri, attempted to validate 
  Postulate V through contradiction" -- First, "mathematician" is not capitalized (it's not 
  a title).  Second, "validate through contradiction" sounds like a contradiction in terms(!)
  A better way to say this would be "... attempted to prove Postulate V using a proof by 
  contradiction."  I shouldn't have to tell math majors this, but in case what I'm saying
  is not clear, a proof by contradiction is one where you start by assuming the negation 
  of the statement to be proved and you show that leads to a contradiction.  The statement
  to be proved must then be true.  

  Page 1:  "We will then cover in greater detail Saccheri’s key arguments such as the 
  Hypothesis of Acute Angle and the Saccheri Quadrilateral" -- Neither of these things
  is an *argument* so this sentence does not really make sense.  The HAA is a hypothesis,
  that is, a starting assumption.  Saccheri does *arguments* (i.e. reasoning in proofs) based 
  on that hypothesis.  The Saccheri quadrilateral is a particular type of geometric figure.
  You can make arguments about its properties, but it's not itself an argument either. 

  Page 2:  "For example, the midpoint of a line segment can be defined as the point that 
  divides a line segment into equal segments; this is ‘definitiones quid rei’ since it is 
  more than a definition, for the midpoint of a line can actually be physically
  constructed."  I'm reasonably sure what you are saying here correctly reproduces what 
  your source said, but did you think carefully about this?  Does this distinction seem 
  like a meaningful or useful one to make?  After all *any* definition in mathematics is 
  defining a *name* for something ("quid nominis").  Isn't the existence of a construction 
  of the actual "thing" ("quid rei") a separate issue and not a property of the definition 
  itself?
  
  Page 3:  "... which discredited their proofs that were based on definitions that
  shouldn’t have been assumed to be true" -- continuing this line of thought, can a
  definition be "true" at all?  I think of a definition as just saying how a particular
  word is going to be used (what it will denote or mean in a following discussion).  
  Truth or falsity doesn't come into it (at least not the way most present-day 
  mathematicians would approach the question of definitions).  
  
  Page 6:  "Saccheri’s work is novel in the sense that he does exactly this when no one
  before him could" -- The discussion of Omar Khayyam's work that follows effectively
  disproves that claim.  Did you notice this?  I think the explanation is that the work
  of Islamic mathematicians in the Middle Ages was not known very well in Europe for a 
  long time.  So the author of the note at the start of the translation of Saccheri's
  book you were looking at may simply not have been aware of Omar Khayyam's works on
  these questions(!)

  Page 6:  "Like Saccheri he considered the three possible cases of the quadrilateral 
  and through a number of theorems he also correctly refuted the obtuse and acute cases 
  from his postulate" -- This statement cannot be correct: He can't have "correctly 
  refuted the acute case" (the HAA) because that is not possible.  There's *no contradiction* 
  inherent in Postulates I - IV plus HAA.  

  Page 8:  "so Saccheri began his lengthy struggle trying to deduce a contradiction under 
  the HAA assumption" -- the "lengthy struggle" here is a paraphrase of a statement in
  Saccheri's own book.

  Page 13:  It would have been good to say a bit more about how you can say the first
  four postulates of Euclid hold in the elliptic geometry case, because it's not 
  quite obvious.  One usual model of elliptic geometry is to take the points on 
  one hemisphere of a unit sphere as the points (or equivalently to identify antipodal
  points on the sphere) and the lines to be the arcs of great circles.  If you go
  more than distance pi along one of these circles, you come back to points you have
  encountered before.  So the maximum distance between any two points in this geometry
  is pi.  When you do this, it becomes slightly non-obvious how to interpret Postulates
  II and III, and indeed the usual way to do this is to alter II slightly to give a form
  that holds in all three geometries (elliptic, Euclidean, and hyperbolic).  The idea is to 
  avoid building in an assumption which forces distances along lines to have to become 
  infinite if you extend as far as possible along a line because that is not possible 
  here.  Instead you can say something like:

  II.' Given any line segment AB and any other line segment CD, there exists a segment
  along the line AB starting at A, in the direction of B, having the same length as CD.

  III.' Given a point A and a line segment CD, there exists a circle with center A and
  radius equal to CD.  (This one is pretty much exactly Euclid's Postulate III, but note
  that it doesn't force you to say the measure of the radius of the circle can be any
  positive real number.  The radius of a circle must be less than pi.  
  
  Final Project: 

  Annotated Bibliography -- 9/10
   
  Presentation -- 32/35  

  Paper -- 50/55

  Total -- 91/100   (letter: A-)