Below are some ideas for writing projects for which your math instructor might give you extra credit.

We continue to publish the best projects in The Right Angle, monthly newsletter of the Mathematics Department. You should feel free to contact the editor, Randy Schwartz, who can offer advice and sources of information such as those mentioned below (office LA-563, telephone/voicemail 462-4400 ext. 5290, e-mail rschwart@schoolcraft.edu).

In conjunction with the Focus Europe project this year at Schoolcraft College, we invite you to write about one of the following topics that highlight ties between mathematics and other facets of European culture, and help explain how modern Europe became pre-eminent in this field.

In writing your paper, you shouldn’t be satisfied with a general description of the topic. You should actually dig into it, explain what is involved, and preferably get into some actual mathematics using examples that help clarify it. Otherwise, what you and your readers learn will be superficial.

Also, remember that writing a research paper, even a brief one, doesn’t mean splicing together phrases and sentences that you find in other people’s books, articles, and websites. Of course, you’ll need to gather information and to list the sources where you found it, but you’ll also need to:

• analyze the information (break the facts and ideas down into their parts in order to see how they relate to one another)
• synthesize the information (put facts and ideas together in a systematic way in order to get a comprehensive understanding of the topic)
• organize and write the paper (figure out the best way, using your own words and understanding, to unfold what you’ve learned so that the readers get a clear understanding, too).

The Writing Fellows website at http://www.schoolcraft.edu/fellows/ has useful guidance in how to go about doing this.

1. The Mathematics of Trade and Commerce in Medieval Italy

   Capitalism arose in Italy before any other place in the world. As trade and commerce began to flourish there, how did the quantitative problems faced by merchants and bankers push mathematicians like Leonardo Fibonacci to adopt new techniques?

   • Joseph and Frances Gies, Leonard of Pisa and the New Mathematics of the Middle Ages (New York: Crowell, 1969).
   • Frank Swetz, Capitalism and Arithmetic: The New Math of the 15^th Century, Including the Full Text of the Treviso Arithmetic of 1478, Translated by David Eugene Smith (Chicago: Open Court, 1987)
   • Anthony Phillips, “The Romance of Double-Entry Bookkeeping”, http://www.ams.org/featurecolumn/archive/book1.html
   • Michael Moïssey Postan, Edwin Ernest Rich, and Edward Miller, eds., The Cambridge Economic History of Europe, Second edition (Cambridge, UK: Cambridge Univ. Press, 1963) [Bradner Library HC 240 .C312], vol. 3 (Economic Organization and Policies in the Middle Ages). See especially the chapter on “Organization of Trade” by Raymond de Roover, with its discussion of Italian cost accounting, double-entry bookkeeping, systems of weights and measures, and currency exchange.


2. The Ciphers of the Monks

   An almost-forgotten way of writing numbers was used in monasteries and among other small circles in medieval Europe. These numerals are described in a book by David A. King, The Ciphers of the Monks: A Forgotten Number-Notation of the Middle Ages (Stuttgart: Franz Steiner, 2001). How were various numbers written in this notation? How did this set of symbols arise and evolve? And why did the system not become widespread? In what ways was it more limited (or in some cases more useful) than Arabic or Roman numerals?

3. Mathematics in Church Architecture

   How did the builders and designers of medieval and Renaissance churches use practical geometry, involving ratios and circular arcs, in their work?

   • Hugh McCague, “A Mathematical Look at a Medieval Cathedral”, Math Horizons April 2003, pp. 11-15, 21.
   • Ken Williams, “The Pavements of the Cosmati”, The Mathematical Intelligencer 19:1 (Winter 1997), pp. 41-45.
   • Benno Artmann, “The Cloisters of Hauterine”, The Mathematical Intelligencer 13:2 (Spring 1991), pp. 44-49.


4. Geometric Perspective in Renaissance Art

   Going back to ancient Greek and Roman times, European artists and designers have used mathematics in a variety of ways. What methods were used by Renaissance artists when they invented concepts of visual perspective?

   • Erwin Panofsky, “Dürer as a Mathematician” in James R. Newman, ed., The World of Mathematics (Redmond, WA: Tempus, 1988) [Bradner Library QA 3 .W67 1988], Vol. 1, pp. 591-612.
   • Anthony Phillips, “Alberti’s Perspective Construction”, http://www.ams.org/featurecolumn/archive/alberti1.html
   • Judith Veronica Field, The Invention of Infinity: Mathematics and Art in the Renaissance (New York: Oxford University Press, 1997)
   • Judith Veronica Field, Piero Della Francesca: A Mathematician’s Art (New Haven: Yale University Press, 2005); review of this book by Anthony Phillips, http://www.ams.org/notices/200001/rev-phillips.pdf
   • Judith Veronica Field, “Piero Della Francesca and Painting as a Science”, http://halshs.archives-ouvertes.fr/view_by_stamp.php?label=REHSEIS&langue=fr&action_todo=view&id=halshs-00004274&version=3 (see 05 Field.tif.pdf near bottom of page)
   • Morris Kline, “Projective Geometry”, Scientific American 192:1 (January 1955), pp. 80-85; widely reprinted.
   • Jo Marchant, “Science and Art: A Leap of Faith”, Nature 446: 7135 (March 29, 2007), pp. 488-491.


5. Mathematics in European Warfare

   The use of mathematics and science to devise the most powerful gunnery, the most accurate rangefinding, and the strongest fortifications, utterly transformed the battlefields of early modern Europe. How did geometry, trigonometry, algebra, and calculus play a role in this new military technology?

   • Oxford University’s Museum of the History of Science, “The Geometry of War, 1500-1750”, http://www.mhs.ox.ac.uk/geometry/content.htm
   • C. W. Groetsch, “Tartaglia’s Inverse Problem in a Resistive Medium”, American Mathematical Monthly 103:6 (Aug-Sep 1996), pp. 546-551.


6. Mathematics in Revolutionary France

   The French Revolution that began in 1789 was inspired in part by the scientific revolution, and had a profound impact on mathematicians and scientists. Why did such mathematicians as Joseph Fourier, Gaspard Monge, and Alexandre Vandermonde become strong allies of the Revolution, some even accompanying Napoleon on his conquest of Egypt?

   • Carl Boyer, “Mathematicians of the French Revolution” in Frank Swetz, From Five Fingers to Infinity: A Journey Through the History of Mathematics (Chicago: Open Court, 1994), pp. 560-573
   • Ken Alder, The Measure of All Things: The Seven-Year Odyssey and Hidden Error that Transformed the World (New York: Free Press, 2002) [Bradner Library QB 291 .A43 2002]


7. The Marvelous Bernoullis of Switzerland

   The Bernoullis were an amazing Swiss family that included at least eight prominent mathematicians (one Daniel, two Jacobs, two Nicolauses, and three Johanns) during the 17^th and 18^th Centuries. What were their various contributions to math and science, and how did this aptitude get nurtured across three generations?

   • Brief biographies of all eight Bernoullis can be found in the MacTutor History of Mathematics Archive, http://www-history.mcs.st-andrews.ac.uk
   • James R. Newman, “Commentary on the Bernoullis” in his The World of Mathematics (Redmond, WA: Tempus, 1988) [Bradner Library QA 3 .W67 1988], Vol. 2, pp. 759-761, followed by Daniel Bernoulli’s “Kinetic Theory of Gases”.
   • Howard Eves, “The Bernoulli Family” in Frank Swetz, From Five Fingers to Infinity: A Journey Through the History of Mathematics (Chicago: Open Court, 1994), pp. 523-526, followed by William Dunham’s essay “The Bernoullis and the Harmonic Series”


8. The Principle of Maximum Goodness

   An interesting dialog between mathematics and philosophy was sparked by a series of natural observations in early modern Europe. In one set of observations, Pierre de Fermat and other mathematicians found that the laws of reflection and refraction of light can be deduced from the basic principle that a ray of light follows the path that minimizes the amount of time required for its travel (the “principle of least time”). Later, Pierre de Maupertuis showed that Newton’s laws of motion can be deduced from the basic principle that a process follows the path that minimizes the amount of energy required for its completion (“the principle of least action”). Maupertuis used this as a launchpad to formulate a more sweeping principle of “optimism” or maximum goodness: God created the best of all possible worlds, and the mathematics of optimization is the key to understanding not only nature but society, history, and morality. Voltaire and others argued strongly against this idea. What were the terms of this debate, and what were the mathematical underpinnings?

   • Ivar Ekeland, The Best of All Possible Worlds: Mathematics and Destiny (Chicago: University of Chicago Press, 2006)
   • Kevin Brown, “Stationary Paths” in Reflections on Relativity, http://www.mathpages.com/rr/s3-04/3-04.htm


9. Gauss and Humboldt: Two Lives Imagined

   Write a review of the novel by Daniel Kehlmann, Measuring the World (Vintage, 2007; 272 pp., $14.95 paperback). This imaginative tale is loosely based on the lives of mathematician Carl Friedrich Gauss and naturalist Alexander von Humboldt. While the Napoleonic Wars are raging in the early 1800’s, each of these scientists is trying to “measure the world”, but for different reasons: Humboldt in order to provide the first scientific description of South and Central America, and Gauss in order to prove that contrary to Euclid, parallel lines do meet. The meeting of these two minds, ultimately at a scientific congress in Berlin, makes for a witty, irreverent portrait of contrasting personalities that also brings in such characters as Jefferson, Daguerre, and a senile Kant.

10. Statistics and the European State

    “Statistics” originally meant the collection of data about states and localities. Beginning in the 18^th and 19^th Centuries, in Germany, England, France, and other countries, governments and other central administrative bodies tried to gather and analyze these data as a way to make their work more scientifically grounded. This “science of state” evolved into statistics as we know it today. Find out what kinds of data the original statisticians were interested in. How did they analyze it, summarize it, and visually display it in various ways?

    • Stephen M. Stigler, The History of Statistics: The Measurement of Uncertainty before 1900 (Cambridge, MA: Belknap Press of Harvard University Press, 1986), especially Part 2.
    • Ian Spence, “No Humble Pie: The Origins and Usage of a Statistical Chart”, Journal of Educational and Behavioral Statistics 30:4 (Winter 2005), pp. 353-368.
    • Sybilla Nikolow, “A. F. W. Crome’s Measurements of the ‘Strength of the State’: Statistical Representations in Central Europe around 1800”, History of Political Economy, Vol. 33, Annual Supplement, 2001, pp. 23-56.


11. Fourier’s Great Discovery

    The year 2007 marks the 200^th anniversary of the publication of Joseph Fourier’s great treatise on heat, in which he first described the technique that would come to be known as Fourier analysis. Write a paper on Fourier’s life and work. A good starting point is the online biography by O’Connor and Robertson (see below), which also lists many references. You should also try to focus on answering the following questions:

    • What was the impact on Fourier’s life and work due to the French Revolution and Napoleon’s actions in France and in Egypt?
    • What is Fourier analysis? How was this technique useful to Fourier in analyzing heat flow? What other applications to mathematics, engineering and physics has it seen in the past two centuries?

    Some helpful resources are:

    • Bochner, S., “Fourier Series Came First”, American Mathematical Monthly 86:3 (Mar. 1979), pp. 197-199.
    • O’Connor, John J., and Edmund F. Robertson, “Jean Baptiste Joseph Fourier”, http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Fourier.html
    • Shoucair, F. S., “Joseph Fourier’s Analytical Theory of Heat: A Legacy to Science and Engineering”, IEEE Transactions on Education 32 (1989), pp. 359-366.


12. Cauchy: A Prodigious Life in Mathematics

    May 23, 2007 was the 150^th anniversary of the death of Augustin Louis Cauchy, one of the most important and prolific mathematicians of all time. Born in Paris during the first year of the French Revolution, Cauchy would go on to write an astounding total of 789 mathematics papers— his collected works filled 27 volumes! He contributed to virtually every branch of mathematics and to many related fields in physics. Write a paper on Cauchy’s life and work. A good starting point is the online biography by O’Connor and Robertson (see below), which also lists many references. You should also try to focus on answering one or more of the following questions:

    • What effect did social and political events have on Cauchy’s life and work?
    • What is calculus, and how did its character change as a result of Cauchy’s work?
    • Summarize Cauchy’s work in the theory of matrices and their eigenvalues.

    Some helpful resources are:

    • Grabiner, Judith V., “Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus”, American Mathematical Monthly 90:3 (Mar. 1983), pp. 185-194.
    • Hawkins, Thomas “Cauchy and the Spectral Theory of Matrices”, Historia Mathematica 2:1 (Feb. 1975), pp. 1-29.
    • Nahin, Paul J., Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills (Princeton: Princeton University Press, 2006)
    • O’Connor, John J., and Edmund F. Robertson, “Augustin Louis Cauchy”, http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Cauchy.html


13. Trying to Understand the Shifting Natural Resources of Europe

    In the 1920’s, the Italian mathematician Vito Volterra was one of the first to formulate an important model of natural ecosystems. He wanted to be able to understand and predict the population fluctuations between a predator species and a prey species, like foxes and rabbits, or bass and sunfish, or ladybugs and aphids. The Lottka-Volterra model, as it is now known, is a system of two nonlinear differential equations, and it predicts that in many cases, the populations of predator and prey will mutually stabilize by cycling between two extremes. See if you can address these questions in your paper:

    • What practical problems prompted Volterra to formulate this model?
    • Explain the assumptions that were made and the variables and equations that were used to define the model.
    • Describe the different mathematical solutions of the equations. What ecological behavior is predicted by these solutions?
    • See if you can implement this model numerically or graphically with a graphing calculator or a computer.
    • What are some of the limitations of this model? How has it been extended to apply to other situations?

    Among the many good discussions are:

    • John Vandermeer, Elementary Mathematical Ecology (New York: Wiley & Sons, 1981), pp. 158-233.
    • C. Henry Edwards, Jr., and David E. Penney, Differential Equations and Boundary Value Problems: Computing and Modeling, Third edition (Upper Saddle River, NJ: Prentice-Hall, 2004), pp. 393-406. [This is the textbook used for Math 252 at Schoolcraft. Similar discussions are provided in other Differential Equations textbooks]
    • Judith R. Goodstein, The Volterra Chronicles: The Life and Times of an Extraordinary Mathematician, 1860-1940 (Providence, RI: American Mathematical Society, 2007).