*Extra credit.*
As you will see from the suggestions in the rest of the
guidelines, writing such a paper will involve quite a bit of
extra work (more than the final exam); if the finished
product is good, this will be rewarded with extra credit for
the course.

*What should such a
paper be like?* It should
concentrate on a fairly specific topic in the history of
Indian mathematics and describe it mathematically in a way
that's comprehensible to the intended audience. It should
include appropriate historical context for this topic and
some discussion of its relation to mathematics as your
audience knows it today. The following specific
characteristics are very important:

*Factual
correctness.* There is a lot
of misinformation and downright lies floating around on
the topic of ancient Indian science, which is largely a
function of the "information gap" described above: if
people don't have genuine data on a subject they feel
they need to know about, they tend to make stuff up. You
can go to otherwise quite respectable Web sites on the
history of mathematics and read that Brahmagupta was the
director of an observatory at Ujjain or that Aryabhata's
work involved a heliocentric theory, which is totally
unsupported by the sources. Be careful in your
assumptions and don't hesitate to ask one of us for
guidance.
*Focus on the
primary source.* General
histories of mathematics (the only source of information
on Indian math readily available to most readers) abound
in fluffy sound bites such as "Brahmagupta's ability to
address these subjects varies from pedestrian in
geometry to brilliant in algebra" and "At an early
period the Hindus exhibited great skill in calculating"
(actual quotes), but they tend to be skimpy on the
details. It is much more interesting and useful to
present something that a mathematician actually said and
then explain what it means, what seems to have inspired
it, and other interesting things about it, than to make
a lot of magisterial pronouncements.
*Mathematical
rigor.* Most of the Indian
mathematics we've encountered so far is not very
"advanced" by modern standards, and it should be fairly
easy to explain it to a high school or college audience.
But it is important to present your mathematical
explanation in a way that respects the modern
conventions of precision and rigor. E.g., you don't have
to discuss Indian algebra in terms of modern field
theory, but your presentation should not be
significantly more casual than the average textbook
exposition.

*Topic selection.*
For most final papers, it doesn't much matter if two or more
people choose the same topic; but for something that's meant
for a wider audience, it's good to avoid duplication. If you
spot a topic you like on the following list of suggestions,
please let me know as soon as possible that you'd like to
write about it (and supply a couple of sentences as to what
you find particularly interesting in it) so I can reserve it
for you. You are also strongly encouraged to investigate a
topic of your own choosing: just remember the rule that "the
broader the question, the harder the research." In other
words, while it should be feasible to write a good article
on the narrow topic of different values of pi in the
Sulbasutra, it is awfully hard to give a thorough
answer to "Where do we get our decimal numbers from?" in
anything less than a book. Some possible topics are:

- Squaring the circle and
circling the square in the Sulbasutra
- Rules for placing the
three sacrificial fires in the Sulbasutra
(lots of interesting historical context here!)
- Solving and classifying
quadratic equations in "manifest math" texts such as
Chapter IV of the Ganitasarasangraha
- Finding sines with
second-order interpolation
- Astronomical
applications of plane trigonometry
- Finding sums of series
(arithmetic or geometric or squares or cubes or piles)
- The algebraic
approximation to the sine function (we'll get to that
next week)
- Solving problems with
assumed numbers
- Permutations and
combinations: finding a given variation with a Meru
- Constructing magic
squares with rules for series and the pulverizer
- Computing the different
possible variations of a given magic square
- The cyclic method for
solving "square-nature" equations
- Geometry of projections
for the moon's illumination
- Computing approximate
chords and arcs in the Lilavati
- Differences between
"manifest" and "unmanifest" math

## Part II

Once you've decided to write
a final paper for HM137 and have picked a topic, here's some
advice on steps to follow in actually planning and writing
it.

*Looking at other
such papers.* Reasonably good
models for your ideal finished product appear in
historically-oriented articles in mathematics pedagogy
journals such as the following:

- The Morehead
Electronic Journal of Applications in Mathematics
(see
link)
- The
Mathematics Educator (see
link)
- Mathematics
Teacher (see
link)
- The American
Mathematical Monthly (This
journal and the two following ones are published by the
Mathematical Association of America: see
link)
- Mathematics
Magazine
- The College
Mathematics Journal

Some published historical
articles (rather on the high end of mathematical complexity)
are noted in issues of these journals available on reserve
for this class at the Sciences Library.

*Literature search.*
This step---checking through the existing corpus of relevant
published work for coverage of your chosen topic---is
crucial to most research papers, to avoid duplicating a
discovery that someone else has already published. The
requirements for a literature search in the case of an
expository paper for a non-specialist audience are much less
stringent: (most of) you are not attempting to unveil a
previously unknown discovery, but rather to make data from
obscure or inaccessible published materials more useful for
your target audience. In addition, many of the topics on the
list of suggestions are absolutely uncharted territory as
far as general exposition goes. Still, it is highly
desirable to do a basic literature search at least among
most of the journals listed above: it will warn you away
from any topics that have in fact been adequately covered,
it will alert you to related publications that may serve as
useful references, and it will bring home to you just how
small the corpus of popularly accessible writings on Indian
mathematics is! The MAA journals have an on-line search
feature at the site referenced above; you should also ask
the reference librarian at the Sciences Library if you are
uncertain how to do a subject search of a periodicals index.
(Find out about using Mathematical Reviews and
their subject classifications; you can also ask a friendly
math professor or TA about Math Reviews if the
librarian is busy.) Don't neglect the general textbooks on
history of mathematics available on reserve at the Sciences
Library, in order to get an idea of what the average reader
interested in math history might already know about Indian
mathematics.

*Finding source
materials.* As I said, the best
foundation for your paper is the primary source in which you
read about the topic, and your thoughtful explanation of how
to understand that source. But your readers will need other
information too: historical and biographical information to
put your discussion in its proper context, and references to
other published work where they can learn more about it. Put
your newfound knowledge about literature searches to use by
tracking your topic in research journals such as
Historia Mathematica and Archive for the
History of Exact Sciences. Hunt down citations in
articles or textbooks of works that look as though they may
have a bearing on your topic. BEWARE THE WORLD-WIDE WEB!
Internet reference materials in mathematics and science can
be a very useful resource (the Wolfram Research site, for
example, has a helpful definition and/or explanation of just
about any math topic you care to name), but their
reliability tends to fall off sharply in the humanities, and
website statements in such arcane fields as the history of
Indian exact sciences run the gamut from painstakingly
accurate to frankly delusional (sometimes within the same
paragraph). Be particularly wary of any biographical
assertions that you cannot confirm in the Dictionary
of Scientific Biography (available in all Brown
libraries) or in introductions or commentaries by
researchers working closely with original Sanskrit sources.

Above all, don't exhaust your
time and energy in hunting for sources independently; take
two or three good stabs at collecting background material
and then come and talk to me about where you've looked and
what you've found. Of course it's important to develop
independent research skills (and if you follow the advice
given above, they will get developed), but one of the most
important research skills is knowing when to ask an expert.
If Professor Pingree or I happen to know that the one
seminal research article on your topic appeared five years
ago in some obscure journal of which we have a copy but the
Brown library doesn't, it is silly to require you to find
that out on your own.

*Connecting with
your audience.* Once you have
enough data to say something informative about your topic,
you need to devote some serious thought to how and why you
should say it. What does a student or teacher of high school
or college mathematics already know that is relevant to your
topic, that might enable you to capture her or his
attention? After all, you yourself came into this class out
of some curiosity about the history of mathematics (however
ludicrous that idea may seem to you at 1 AM of the day the
paper's due), and if you can tap into that same inquiring
spirit in your reader, the communication will flow much
better. Try discussing your topic with a roommate or friend
to see what they seem to find interesting about it, and what
approach to explaining it best gets your point across.

*Keeping it
rigorous.* We in HM137, in keeping
with the style of our sources, have adopted a very comfy
approach to proof both in lectures and in homework, blithely
assuming whatever seems obvious and analyzing the rest just
strictly enough to be adequately convincing. You do not want
to scare teachers and students of modern mathematics (who
are all Greek by training, historically speaking) with this
no-ropes-no-helmet approach. While your readers will
appreciate some informal explanation, don't skimp on the
demands of formal definition and demonstration; the sample
papers in the journals on reserve will give a good idea of
what those demands entail. If you are unsure whether there
are any logical or terminological holes left in your
mathematical discussion, go find that friendly math
professor or TA, or another math grad student or advanced
undergraduate, and ask them to critique it; an invitation to
nitpick someone else's hard work is almost universally
irresistible, so you will probably get a good response.

*Cautionary
statement on publication.* If you
do all that, you will almost certainly have an excellent
paper that will not only get a good grade but will be
suitable for actual peer-reviewed publication. If you are
interested in taking that final step, be advised that it
will take a certain amount of time and effort beyond the end
of the semester in order to get something actually
conforming to the requirements for submission. Editors are
picky about electronic format, notes, diagrams, length,
pretty much everything; and while I will be happy to work
with you on any necessary revisions after mid-May, either in
person or long-distance, the process can drag on a bit.
What's more, even if you get every formatting detail right
in an excellent paper on a subject the editor wants to
cover, other factors may prevent him or her from accepting
it: a long publication backlog, prejudice against
undergraduate authors, an excess of high-quality
submissions, any number of reasons. If that should happen,
remember that (1) there are other editors, and (2) your
paper can be informally distributed with the other
non-published ones instead. Published or not published, the
important thing is that you'll have produced a quality work
of exposition on an understudied subject, and your efforts
therefore will not go to waste.