Chris, 

     This is a mostly very good discussion of Egyptian mathematics and 
   a consideration of whether what they did constitutes algebra
   or not.  As you say, there are definitely differences of opinion,
   but I think that if you look at the solutions presented to some
   of the problems in the Rhind and Moscow papyri, then the 
   "rhetorical algebra" suggestion certainly seems plausible.  

   I have one comment about your description of the Egyptian
   method for multiplication, though.  Here's what they would 
   do to multiply 9 x 12 for instance.  I'll pick the factor 12 
   and start doubling:

                   1 x 12 = 12 <=
                   2 x 12 = 24
                   4 x 12 = 48
                   8 x 12 = 96 <=

   (They wouldn't need to double again, since 2 x 8 = 16 is 
   larger than 9.)  Now to make 9, you need 1 + 8, so mark
   those rows above.  Adding, 12 + 96 = 108, and that is 9 x 12.  
   As you can see, you really need to be making any multiple
   of 12 out of these steps of the repeated doubling to do 
   a general product n x 12.  So the Egyptians are really 
   relying on the fact that n can always be expressed as a
   sum of powers of 2, and in our language, that would be
   equivalent to looking at the binary or base-2 expansion
   of the other factor.  For instance 9 = (1001)_2 and 
   the 1's correspond exactly with the rows in the table
   above that are added to give the product 9 x 12.  
   The question of whether the Egyptians really understood
   that every integer could be represented in binary or 
   base 2 form is one of those semantic questions that 
   is hard to answer.  On the other hand, I think an example
   or two presented this way would make it clear that their whole 
   system of multiplication was based on the fact that it is 
   possible(!)
       

   Grade:  A-