As announced in the course syllabus, you may submit a 5-page paper to replace one exam grade in Math 36 this semester. Below, some topics from which you may choose in planning your work are summarized. These are only suggested topics. If there is some subject relating to calculus (or to some other area of mathematics) that you would prefer to write about, please discuss it with me and get my approval before you go to work. I will be able to suggest references to get you started there as well.
You can submit your work either as a traditional word-processed document, or as an .html web page containing about 5 pages worth of text if you like. I will be looking primarily for content in either case -- see comments below. If you go the .html route, while images and hyperlinks to other WWW pages are great to include, they alone will not be sufficient.
Papers may be submitted at any time up to Wednesday, December 8, 1999.
In working on this paper, you should follow the same procedures you would follow in preparing a research paper for any other course. Your grade will depend on the thoroughness of your research, the degree of independent thought about the subject revealed through your work, the organization of the paper, and the quality of your writing. Do not assume that because this is a mathematics paper writing does not matter! A word to the wise: I tend to be rather severe when I read papers with large numbers of misspellings, or many grammatical and other technical errors. The same thing happens with papers consisting largely of masses of undigested quotations from your sources, particularly when those quotations involve technical discussions of mathematics you do not understand fully. To avoid having this happen to you, take me up on the following offer: I will be happy to read a preliminary draft of your paper and give you comments. The Writing Workshop is also a good resource for help on organization and other more mechanical writing issues.
In the suggested topics below, all citations refer to the list of references at the end of this handout. These books are all on reserve in the Science Library in Swords Hall. There are many other histories of mathematics in the library which may be useful to you as well. Mr. Stankus, the science librarian, will be more than happy to assist you in finding other books on the history of mathematics if you need help.
At the end of the paper, include a full list of all the references you consulted. These references may come entirely or in part from the reserve list below; also include all sites on the Web you use to do your information gathering. For any Web sites you consult, please give a page title if you can identify one, and the complete URL in a reference like this: http://www.somewhere.edu/somepage.html.) Include foot- (or end-) notes identifying any direct quotations you use, or any statements of opinions which are not your own. Failure to follow these guidelines constitutes plagiarism, a serious academic offense, which will result in an automatic grade of F on this paper. If you have questions about the meaning of this, please consult with me. I will be glad to try to answer any questions you may have on these topics, and help you in the preparation of these papers.
I will also be happy to help you get started with .html if you have never used it before.
1. Bernhard Riemann.
For this paper, you will learn more about Bernhard Riemann, a profoundly original mathematician of the mid-19th century. Riemann is usually ranked (with Gauss and Euler) as one of the three greatest mathematicians of the modern era. His forte was his amazingly powerful intuitive grasp of new areas of mathematics. He was truly a pioneer and his work has inspired a tremendous amount of mathematical research to the present day. In this course, we have studied the definite integral and its applications by looking at the approximating sums for the integral (the Riemann sums). Riemann was the first to develop a truly satisfactory understanding of when a function would be integrable. What led Riemann to develop this general theory of integration? Why were the older approaches to integration viewed as inadequate? Give an example of a function that is not integrable by Riemann's definition. What other contributions did Riemann make to analysis? What contributions did Riemann make to the theory of numbers? What is the Riemann zeta function and why is it related to the distribution of prime numbers? (Hint: look at the Euler product formula for the zeta function and try to justify this.) What contributions did Riemann make to geometry? How did Einstein build on Riemann's work in his theory of relativity? Simmons's calculus book contains an excellent biographical sketch of Riemann. Kline's book is also a good reference for the mathematical questions. Bell's book contains much biographical information on Riemann.
2. The development of "rigor" in calculus.
By choosing this topic, you will certainly learn more about the concept of limit, the basic idea in all of calculus, and come to understand the historical development of the definitions of limit and derivative that are the beginning of the subject. The early work in calculus used a very intuitive, not to say cavalier, approach to the whole concept of limits. To illustrate this, you should consider one of Newton's examples of a derivative calculation (using implicit differentiation) reproduced on pages 324-325 of Eves. There is another similar example on page 198 of Cajori. Criticize these computations. What is really going on? Try to find out some of the historical criticisms and problems which led to the realization of the need for a more precise definition of limit expressed by Jean D'Alembert (see his paper in chapter V of Struik). What was the contribution of Karl Weierstrass which focused attention on the difference between continuity and differentiability? (Hint: look at the middle of page 425 in Cajori). Also look at the "snowflake" curve of Helge von Koch described on p. 452 of Eves. What are some of the remarkable properties of this curve? (The assertions to be demonstrated in the problem in Eves may all be proved using the formula for the sum of a geometric series. Prove them.) What is the meaning of the phrase "the arithmetization of analysis?" Do you think all this concern over definitions is justified? (On a slightly more technical level, chapter 40 of Kline also contains an overview of this question.)
3. Kepler's Laws.
(This topic is recommended mainly for students who have had some physics.) The first real "triumph" of the calculus was the proof by Newton that the three "laws" that Johannes Kepler had proposed to describe planetary motion could be derived from the Universal Law of Gravitation. That is, the three statements:
Kepler 1. The orbits of the planets are ellipses with the
sun at one focus,
Kepler 2. The line segment joining a planet to the sun sweeps
out equal areas in equal times,
Kepler 3. The square of the period of revolution of a planet
is proportional to the cube of the semimajor axis
of its elliptical orbit.
could all be deduced from Newton's Universal Law of Gravitation:
Newton: The magnitude of the force of attraction of two masses is proportional to the product of the masses, and inversely proportional to the square of the distance between them: F = GMm/r2, where G is the gravitational constant; M, m are the two masses and r = distance (The force is directed along the line segment joining the masses.)
For this paper, you would discuss (briefly) the history behind this discovery. (For "starters," who were Tycho Brahe, Johannes Kepler, and Isaac Newton? What part did each of these men play in the story?) Then, you would work through a modern adaptation of Newton's derivation as given in Simmons's book below, and give a presentation of the mathematics involved in your own words, filling in all details left for the reader. Finally, discuss briefly the historical impact of this discovery. How did man's conception of himself and his place in the world change as a result of this scientific work?
4. Newton vs. Leibniz
The fundamental theorem of calculus relating the operations of differentiation and integration was discovered at more or less the same time (the 1660's) by two different people: Isaac Newton in England, and Gottfried Wilhelm Leibniz in Germany. Friends and supporters of each of these two mathematicians soon noticed the similarities in the other's work and claimed that each had stolen ideas of the other. For this paper, you would investigate the history of the mathematical dispute that ensued, looking especially for answers to the following questions: How did Leibniz's approach to calculus differ from Newton's? Which approach do we follow more closely today? Do most historians of mathematics believe that plagiarism (in the form of theft of ideas of another and their presentation as one's own work) actually occurred? How did this fight influence the development of mathematics both on the European continent and in England?
Some of the calculus web sites accessible from the course home page have a lot of useful information. The following books are also on reserve for your use in the Science Library.