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\centerline{MATH 110--2 Algebra Through History}
\centerline{Information on Research Papers}
\centerline{October 15, 2019}
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{\it Assignment}
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As you know from the course syllabus and the schedule posted on the course homepage,
one of the final assignments for our course will be a medium-length research paper
on a topic related to our historical survey of algebra.  You should aim for about 6-8
pages in length (slightly longer or slightly shorter that this is also OK).  
As with any research paper of this kind, you should identify the sources
of information that you include that does not come from your own work (e.g. any direct 
quotations, any mathematical examples you find in your sources, ideas about 
interpreting the original sources, etc.).  You may do this with footnotes or 
endnotes as you prefer.  
You should {\it also} include a listing of all sources: books, articles, web pages,
etc. that you consult on the final page of your paper.  
I'm not picky about what form you use for the citations in your listing, 
but please pick one of the standard ones (e.g. MLA, APA, etc.) and follow 
it consistently.
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The timetable for the assignment is as follows:
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\item{$\bullet$}  The paper ``proposal:'' by 5:00pm on November 15 (at the latest), 
inform me which of the topics indicated below you would like to write on.  
{\it This is especially important if you want to design a topic of your
own; you will need to get my approval in that case.}
In several cases where there are a number of different directions a paper
could take, I'll also ask you to give me a preliminary description of which
aspects you would like to focus on at this time.  If there are any issues
with what you are proposing, I will get back to you early the following
week and we can discuss them in office hours.  
\item{$\bullet$} The final papers will be due no later than 5:00pm on
December 13 (the last day of class).  These will be submitted as Google 
docs or Word documents to:
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\centerline{\it jlittle@holycross.edu.}
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Early submission 
is also a possibility and that will gain the undying gratitude of your 
reader :)  

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Please get started thinking about which topic you would like to address 
early.  I don't have any problem if more than one of you want to write on
a particular topic, but we should try to spread out as much as possible 
so that there will not be too much demand for any particular book as
a source.
This will be especially important if you need to use interlibrary
loan to get sources not contained in our library's collection.  {\it Ms. Merolli,
our Science Librarian, will be more than happy to help out with searches
and obtaining sources if you need assistance.}  I can also
help you track things down in many cases, and I will be happy to 
make photocopies of sections of books in my personal collection if
you want to consult them.  

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{\it Some ideas about paper topics}
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\item{1.}  {\it Ancient Egyptian Mathematics.}  At almost the same
time as the Old Babylonian period in Mesopotamia, scribes in Middle Kingdom
Egypt were considering many quite algebraic questions.  For this paper topic,
you would research how the Egyptians dealt with multiplication and division
of numbers and how they worked with fractions.  Then you would discuss a selection of problems
from the principal surviving mathematical texts of this period:
the {\it Rhind and Moscow papyri}.  See George Joseph's book
``The Crest of the Peacock'' for a good discussion of this. There is also 
a brief discussion of Egyptian mathematics at the start of Chapter 1 in 
Katz and Parshall, but they don't think of what the Egyptians did 
as very ``algebraic;'' others might disagree(!)  
Katz's sourcebook {\it The Mathematics of Egypt, 
Mesopotamia, China, India, and Islam} contains other examples.  
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\item{2.} {\it More on Old Babylonian Mathematics.}
 There is quite a bit more known about both the sorts of problems
that were studied in Old Babylonian mathematics and the social context of 
the scribal schools where the tablets
we discussed in class were created than we had time to cover in class.
There are several directions that one might take here.  For instance,
\itemitem{(a)} The book {\it Mathematics in Ancient Iraq} by Eleanor Robson
is an excellent discussion of the social context and what we know about
the scribal schools.
\itemitem{(b)} The articles {\it Neither Sherlock Holmes nor Babylon: A Reassessment
of Plimpton 322}, Historia Mathematica 28 (2001), 167--206,
 and {\it Words and Pictures: New Light on Plimpton 322}, American Mathematical
Monthly, February 2002, 105--120, both 
by Eleanor Robson, discuss another very famous Old Babylonian tablet that we
mentioned briefly, and various modern attempts to account for the mathematics
it contains.  She also discusses the pitfalls of treating ancient texts
as ``mathematical puzzles'' to be solved with modern ideas and techniques
without taking account of what the creators could have been thinking.  
\itemitem{(c)}  There are many other Old Babylonian mathematical tablets that
have been deciphered.  Victor Katz's sourcebook {\it The Mathematics of Egypt, 
Mesopotamia, China, India, and Islam}, and J\"oran Friberg's {\it A Remarkable
Collection of Babylonian Mathematical Texts} would be good places to consult
to get an idea of the range of different things they did.  
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\item{3.}  {\it Algebraic Thought in India.}  We did not discuss anything
much about mathematics from the subcontinent of India, but there is a rich
tradition with much algebraic content there.  Moreover, the interconnections
between India, the Middle East, and Europe are only now becoming better 
understood.  Some good sources to consult here are Chapter 6 of {\it Taming
the Unknown}, Victor Katz's sourcebook {\it The Mathematics of Egypt, 
Mesopotamia, China, India, and Islam}, and George G. Joseph's {\it The 
Crest of the Peacock}.  There are several ways a paper here might develop:
\itemitem{(a)}  A survey of algebraic ideas in Indian mathematics
\itemitem{(b)}  A more in-depth treatment of a major figure like Brahmagupta
or Bhaskara II, or Arhyabata.
\itemitem{(c)} For students with a bit more mathematical background, a very
interesting topic is the question to what extent work on infinite series
in medieval Kerala (a state in south India) anticipated European work in 
calculus.  Katz and Parshall (in footnote 44 on page 131) mention that
there were Jesuit missionaries in India by the 17th century, and they 
say that ``there is no evidence from that time of the transmission of and
actual transplantation of western mathematical ideas there.''  In fact, 
other writers such as George G. Joseph in another book of his called 
{\it A Passage to Infinity} have asked whether the influence
might actually have {\it gone the other way}:  Might this Indian mathematics
have made it back to Europe by way of those very Jesuit missionaries and 
stimulated the development of European mathematics?  
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\item{4.}  {\it Archimedes' Quadrature of the Parabola}.  (This topic
is probably the most mathematically sophisticated of the lot -- only recommended
for those who are willing to ``dig deep'' into a piece of amazing mathematics.)
In the work mentioned above, Archimedes proves that the area of a segment of 
a parabola (the region between the parabola and one of its chords) is equal
to $4/3$ of the area of the triangle formed by the endpoints of the chord 
and a third point on the parabola called the {\it vertex} of the segment.
(The vertex is the point where the tangent line to the parabola is parallel
to the chord; it also happens to be point with the same $x$-coordinate
as the midpoint of the chord, if the parabola is $y = x^2$ in a coordinate
equation.)  For this topic, you could work through either one of the 
two arguments that Archimedes gives for this.  The first involves an 
ingenious method of slicing the segment up into infinitesimally thin
strips and ``weighing it on a balance.'' The second finds the area 
by filling up the region between the triangle and the parabola with
other triangles, then adding their areas with what turns out to be 
a {\it geometric series}.  (Note: Archimedes gives the second proof because
he says he knows that there are some questionable aspects of the first 
one -- it's far ahead of its time, in fact, and anticipates ideas from 
the integral calculus that were not formally developed until about 1800 
years after Archimedes' time.)
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\item{5.}  {\it Algebraic Thought in China.}   We did not discuss anything
much about mathematics from China, but there is a very rich
tradition with much algebraic content there.  Moreover, the interconnections
between China, India, the Middle East, and Europe are only now becoming better 
understood.  Some good sources to consult here are Chapter 5 of {\it Taming
the Unknown}, Victor Katz's sourcebook {\it The Mathematics of Egypt, 
Mesopotamia, China, India, and Islam}, and George G. Joseph's {\it The 
Crest of the Peacock}.  There are several ways a paper here might develop:
\itemitem{(a)}  A survey of algebraic ideas in Chinese mathematics
\itemitem{(b)}  A more detailed look at the Chinese Remainder Theorem 
and applications
\itemitem{(c)}  A more detailed look at Chinese methods for solving 
simultaneous systems of linear equations.  What they did there turns
out to be very close to the later matrix methods for these problems developed in 
Europe in the 1800's(!)
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\item{6.}  {\it Islamic Mathematics and the Translation Movement in 8th - 10th 
Century Baghdad.}  Victor Katz's sourcebook {\it The Mathematics of Egypt, 
Mesopotamia, China, India, and Islam} contains a number of translations
of original works from this and later periods.  
\itemitem{(a)} The methods developed
by Omar Khayyam for solving cubic equations would be an excellent topic.
\itemitem{(b)}  An interesting question is just why the translation of Greek scientific
and mathematical texts assumed such a large importance during the Abbasid 
period in Baghdad.  A related question is:  what was the nature and 
the role of the institution known as the {\it bayt al-hikmah} (House of 
Wisdom)?  The book {\it Greek Thought, Arabic 
Culture} by Dmitri Gutas is one of the most balanced and careful examinations 
of this question.  There are a number of other treatments as well, but modern 
tensions between the West and the Islamic world have tended to influence
scholarship on these issues too, with Western authors sometimes discounting
the importance of the work done in Baghdad, and Islamic authors claiming
too much importance for it(!)
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\item{7.}  {\it Mathematical Lives.}  If you want to work on a ``mathematical
biography'' of an important figure, the following would be interesting choices.
Try to bring out especially the connections with the history that
we have discussed.
\itemitem{(a)} Hypatia (and the end of the Alexandrian mathematical tradition)
\itemitem{(b)} Leonardo Pisano (also known as ``Fibonacci'')
\itemitem{(c)} Luca Pacioli (also known as the father of double-entry bookkeeping!)
\itemitem{(d)} Girolamo Cardano
\itemitem{(e)} Niccolo Tartaglia
\itemitem{(f)} Ludovico Ferrari (and quartic equations)
\itemitem{(g)} Francois Vi\`ete (Vieta)
\itemitem{(h)} \'Evariste Galois (and the problem of solution of general 
polynomial equations ``by radicals'')
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\item{8.}  {\it A topic of your own.}  If there is some topic that has occurred
to you that you don't see above, please put together a reasonably detailed description
of what you would like to write on and consult with me to get my approval before
you begin to work in earnest.   
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