Cody Wilkinson and Victoria Young -- Islamic Mathematics

   This is a good summary of some of the accomplishments of the Islamic mathematicians
   during the "Middle Ages."  I don't say "excellent," though, because a lot of it is
   rather superficial and you haven't provided as many details as you might have.
   For instance, you might have discussed more about Thabit Ibn Qurra's work on 
   "amicable pairs" of numbers that we considered briefly in class.  The discussion
   of the Alhambra could really use some illustrations like the ones you showed
   in your presentation.  You use a lot of space on generalities when the specifics
   would have been more interesting.  For instance, exactly how were some of the 
   tile work patterns created and what exactly were the "girih" tiles?

   The presentation was somewhat superficial too and you didn't always seem to have
   things organized very well.  I'll cut you some slack on that, though, since you
   did volunteer to go the first day of the presentations.  

   Specific Comments:

   1)  Page 2:  "Other Europeans came in seek of their wisdom and to bring back techniques 
       for irrigation, and agriculture."

       > "in seek of" is not correct -- I think you mean "in search of" or "seeking" 

   2)  Page 2:  "While the rest of the civilized world was sticking their hands in the sand, ... "
     
       > What does that mean?

   3)  Page 3:  " The concept of zero was a huge addition to math in general and is extremely 
       important in mathematics today."

       > The other, and actually *much more important,* advantage of the Hindu-Arabic numerals
       was the fact that they made most kinds of calculations much simpler to carry out, 
       especially by hand, but also with calculation aids like the abacus and other devices.
       If you think of the Babylonian base-60 numerals, then the use of extensive tables
       was required to carry out procedures like multiplication of one number by another.
       With a base-10 system, the memory burden is much reduced and almost all kids 
       today learn their "times tables" for multiplication by 2,3,4,5,6,7,8,9, ... with 
       little trouble.  Similarly, the Hindu-Arabic numerals are much simpler to use
       than Roman numerals or the Greek alphabetic numeral system for most calculations.

   4)  Page 5:  "His translation of Greek texts was a crucial development for medieval Europe 
       because once they were in Greek it made it easier for other scholars to then translate 
       the works into Latin, which was a common language."

       > I think you are misunderstanding something here.  The translators in the House
       of Wisdom translated works from the original Greek text into Arabic.  Having them
       in Greek was not that useful at the time because almost no one in Western Europe
       read or spoke Greek.  The study of ancient Greek only started again during the
       build-up into the Renaissance period.  The usual translation path for a lot of
       the mathematical texts was:  Greek ->  Arabic ->  Latin.


   5)  Page 5:  "he continued to work with algebraic integrals" 

       > I think this refers to Thabit's work on computing areas bounded by parabolas
       and straight lines.  This was a problem first considered by Archimedes in 
       the 3rd century BCE.  Archimedes' work, then Thabit's alternate version, are
       often seen as the precursors of the theory of the definite integral in calculus.
       But it's not really correct to talk about "algebraic integrals" there because
       they did not have the background about functions, partitions of intervals, Riemann
       sums, limits, etc. that go into the modern definition of the definite integral.
       They were essentially inventing something we can see is equivalent to our modern 
       definition of the integral to solve a particular problem.   

   6)  Pages 6 and 8:  "This was an attempt to explain and prove the Parallel Postulate, but due 
       to its difficulty, it was unsuccessful" 

       >  Yes, but in fact we know today that what he said was *incorrect* and 
       it is, in fact impossible to prove Euclid's Parallel Postulate from the others.
       It is independent of them, and there are other mathematical geometrical
       theories in which Euclid's 5th Postulate *does not hold* (!)  This was finally
       realized because of the work of Gauss, Bolyai, and Lobachevsky at the start
       of the 19th century.  This is not still an unsolved problem (as you say
       on page 8).  It is understood that the Parallel Postulate (Postulate V) is
       independent of the others and can neither be proved or disproved using them.

   7)  Page 10:  "they realized that this fear would be nearly impossible and take months 
       to complete so that method couldn’t possibly work"

       > What does that mean?  Did you mean "feat" instead of "fear"?

   8)  Pages 6 and 12:  "ratios of magnitudes, which resulted in the introduction of π and sqrt(2)"

       > I don't think it is correct to say that Omar Khayyam "introduced" these -- he
       was probably one of the first to treat them as numbers -- on the same footing as
       rational numbers, though.  
   
   Final Project Grade Computation 
 
   Bibliography:  10/10  

   Paper:         54/60

   Presentation:  27/30

   Total:         91/100