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.