Charles and Kyle, 

  Your presentation and paper on cartography, Ptolemy's maps, and some other map
  projections have some good features.   But they are pretty superficial and 
  they don't go into the level of detail that I was hoping for in this project.  There 
  is also a major error in what you are saying about one of the map projections.

  To give just one big example of the superficiality to start, on page 7, you state that 
  "Mapping a sphere to a plane is not only difficult, but McCleary actually states it’s 
  impossible without losing the integrity of the size and the angles in the geometry."  The 
  proof is given on page 139, and it's not that complicated.  It would have been a very good 
  addition to your topic to explain more about this, based on the idea of an isometry -- 
  a mapping between surfaces preserving the first fundamental forms.  We did all 
  the background in class you need to work through and understand that proof and 
  other features of these map projections as mappings from regions on the sphere
  to regions on the plane.  You just haven't made use of any of that background.
  
  For another example, you work out the formula for the usual form of stereographic
  projection, but then you don't use it for anything.  You might have shown, for
  instance, that this mapping preserves angles (it's "conformal"), but it does
  not preserve areas or distances.  The conformality property is also derived in
  McCleary's book.

  Finally, the description of the Mercator projection that you gave is just
  NOT CORRECT.  (What you described would be a central projection onto the cylinder.)  
  Look at pages 150-151 in McCleary carefully.  The Mercator projection is NOT obtained 
  just by projecting linearly from the center of the sphere onto the circumscribed 
  cylinder.  The formula for the y-coordinate on the map in terms of the spherical 
  coordinates theta, phi is given by a more complicated formula:  

                 (x,y) = (theta, ln tan(pi/4 + phi/2))

  The main feature of this projection (the property that leads to this particular
  choice) is that it makes the so-called "rhumb lines" (curves making constant angles
  with the lines of longitude on the sphere) into straight lines in the map.  This makes 
  Mercator maps useful for navigation even though they distort distances and areas a lot near
  the poles.  I was trying to ask you about this in the run-though because I knew the 
  Mercator projection should have something like this property, but I hadn't looked
  at it in a while and I didn't remember the formula above or the derivation 
  on pages 150-151 of McCleary.  

  I'm sorry I didn't catch the error in the description of how the Mercator projection
  is constructed before.  But ultimately you are responsible for the correctness of what
  you are trying to say in a presentation like this too.        
  
  Specific comments:

  Page 1:  Your first sentence calls for some explanation.  This is certainly
  just an estimate, so "in 2300 B.C." is too definite/specific.  The first paragraph
  as a whole is also kind of a jumble with no coherent development of ideas.  Why did you 
  put all the stuff in between the first sentence and the later discussion of where
  the earliest traces of maps come from, how they created, etc.?  Also the point that
  it is difficult to create maps of a 3D world on flat surfaces is true, but the
  degree to which it's true depends on how accurately you are trying to reproduce
  the geometry (relative distances, directions, etc.) of the features on the sphere.
  Some maps don't necessarily try to do that at all.  Think of maps like the diagrams
  of subway lines in larger cities.  They are often simplified to show only what stops are
  on which lines and where the lines connect (usually but not always with some hints 
  about the directions of travel on different lines).  They are useful even though they 
  don't represent the actual geometry of those parts of the cities at all. 

  Page 3:  "While Claudius Ptolemy’s theories regarding the structure of the solar system 
  may not have turned out correct" -- It is sometimes argued that what the Greek astronomers 
  were trying to do was not to describe how the solar system actually was organized, but
  instead to provide a way to describe and predict the motions of the sun and the planets
  as seen from the earth.  This was sometimes called "saving the phenomena" -- providing
  a system that could predict what would be seen.  Ptolemy's system is actually pretty 
  good from that point of view!  It can be used to describe and predict planetary motions
  as seen from earth as accurately as one could wish.  

  
  Final Project: 

  Annotated Bibliography -- 8/10
   
  Presentation -- 28/35  

  Paper -- 45/55

  Total -- 81/100   (letter: B-)