Projects

This page will be modified during the semester.

Projects


A project is a short research paper. The project descriptions (below) are just hints to look at some significant individual or incident. You are to research the topic, using several references (library work will be necessary) and write it up (with a bibliography) making sure that you put the topic in its historical context, i.e., if the topic is about something Archimedes did, you should report on it as well as talk about Archimedes and his place in history. Remember that all topics have antecedents (things that led up to the subject) and consequences (things that the subject influenced).

A project is defined by two parameters, time period and theme. There are four time periods, I – before 1478, II – 1478 to 1800, III – 1800 to 1900 and IV – 1900 to June 16, 1947. There are several themes which are listed in the table below. To determine a project you need to select a time period and a theme, subject to the restrictions that your projects must be chosen from different time periods and you may not repeat a theme. After making your selections, look up, in the table below, the appropriate project numbers and then go to the project descriptions to choose a specific project. Throughout the semester I will be adding projects to this table. You will be able to view them on the webpage.


Theme\Time

I

II

III

IV

Women in Mathematics

1

2

3,4,5

6

Non-western Mathematics

7,8,11-16

9,17,18,19


10

Mathematics and War

20

21

22

23,24,25

Mathematical Institutions

26,27,28

29

30

31,32

Symbols and Notation

33,34

35,36

37,38

39

Who gets the credit?

40

41

42

43

The great mistakes


44,45

46-50

51

Paradigm Shifts

52

53

54

55

Potpourri

56,57

58

59

60,61






Project Chart

The Themes

One of the reasons to study history is to gain insight on issues that we have to deal with in the here and now. The historical data points shed light on how these issues came to be and how they were dealt with in the past; valuable information as we move forward. In order to avoid a “scatter-gun” approach to the study of the history of mathematics, I've selected a few “issues”, which I call themes, to concentrate on. My selection is fairly idiosyncratic, so I should say a few things about what they mean to me.


Project Descriptions


  1. Hypatia (370-415 A.D.)

  2. Maria Agnesi (1718-1799, Italian)

  3. Sophie Germain (1776-1831, French)

  4. Ada Augusta Lovelace (1815-1852, English)

  5. Sonya Kovalevskaya (1850-1891, Russian)

  6. Emmy Noether (1882-1935, German)

  7. Horner's method for finding approximate solutions of polynomial equations as found in Ch'in Chiu-shao's book “Su-shu Chiu-chang” (1247).

  8. The implications of finding Pascal's triangle in the 1303 text of Chu Shih-chieh, “Szu-yuen Yu-chien” (The Precious Mirror of the Four Elements).

  9. The effect on Chinese mathematics of the publication in China of Ricci and Hsu's translation of Euclid's Elements (“Chi-ho Yuan-pen” [1607]).

  10. The effect of the People's Cultural Revolution on Chinese mathematics.

  11. Brahmagupta and the “School” of Ujjain.

  12. The Golden Age of Bagdad during the reign of the first three caliphs.

  13. The life and works of Mohammed ibn Mûsâ al-Khowârizmî.

  14. The solution of the cubic equation by Omar Khayyam.

  15. Fibonnacci's role in the introduction of arabic numerals to the west.

  16. The role of astronomy in the development of Indian mathematics.

  17. The general nature of the literary activity in Mexico in the 16th Century with special reference to the need which produced the work of Juan Diez.

  18. The introduction of Western mathematics into the East in the 17th Century.

  19. The work, the general standing, and the influence of the Japanese scholars in the 17th Century.

  20. The death of Archimedes and his role in the protection of Syracuse.

  21. Tartaglia and his role in artillery science.

  22. Napolean hobnobbed with the a number of the great mathematicians of France, who were they and what role did they play in his campaigns?

  23. William Friedman spearheaded the effort to break the Japanese diplomatic code “purple”.

  24. Stanislaw Ulam and the Hydrogen Bomb.

  25. Why did Norbert Weiner stop doing applied mathematics?

  26. The Pythagorean Brotherhood.

  27. The Library at Alexandria.

  28. The earliest universities were established at Oxford, Paris and Bologna. What was the role of mathematics in their curricula?

  29. Why was the École Polytechnique founded and who were the mathematicians who taught there?

  30. Little effort was made to encourage the study of modern mathematics in the United States until Johns Hopkins University hired Prof. J.J. Sylvester.

  31. Why was the University of Göttingen considered the mathematical center of the world for the first third of the twentieth century?

  32. Trace the rise to prominence of the mathematics department at the University of Chicago.

  33. What is the derivation of the name of the trigonometric function sine?

  34. How were the Hindu-Arabic numerals introduced in the west?

  35. William Oughtred used over 150 mathematical symbols in his writings, many of his own invention. How many of these have survived in modern times?

  36. The invention of the symbolism for modern decimal fractions is usually attributed to the Belgian Simon Stevin.

  37. Who advocated the use of the solidus (“/”) to write fractions (as in 5/8)?

  38. For many centuries there has been a conflict between individual judgments, on the use of mathematical symbols. On the one side are those who, in geometry for instance, would employ hardly any mathematical symbols; on the other side are those who insist on the use of ideographs and pictographs almost to the exclusion of ordinary writing. Trace this conflict by examining various editions of Euclid's Elements that are written in English.

  39. What is a googol?

  40. Who discovered the Platonic solids?

  41. Who formulated L'Hospital's rule in calculus?

  42. Who invented Steiner Triple Systems?

  43. What did Pasch have to do with Pasch's Axiom in geometry?

  44. Saccheri's “proof” of Euclid's Parallel Postulate.

  45. Newton's classification of cubic curves.

  46. Legendre's several “proofs” of the Parallel Postulate in the various editions of his Elements of Geometry.

  47. Joseph Fourier's work on Fourier Series has been called “a classic example of physical insight leading to the right answer in spite of flagrantly wrong reasoning.”

  48. Cauchy's result on convergence of series that was fixed by Abel.

  49. Kempe's “proof” of the 4 Color Theorem.

  50. Riemann's incorrect statement of what he called “Dirichlet's Principle.”

  51. Wile's first “proof” of Fermat's Last Theorem.

  52. Why is Thales called “the Father of Mathematics”?

  53. Descarte's introduction of analytic geometry.

  54. Weierstrass and the crisis in the foundations of analysis.

  55. Gödel's Incompleteness Theorems.

  56. Who bought Archimedes' palmiset?

  57. What is Plimpton 322?

  58. Where did Newton get his data on the tides for the Principia?

  59. Compare the constructions of the natural numbers by Peano, Frege and Dedekind.

  60. What is the status of Hilbert's list of 23 problems?

  61. What was Edgar Allen Poe's contribution to mathematics?


These projects are meant to be interesting and enjoyable assignments, not chores, so choose your topics with care. Your report should satisfy the following constraints:


Notice that I have said nothing about length. A topic has a natural length; let your paper grow to that length. You are telling a story: follow the King's advice to the White Rabbit: "Begin at the beginning, and go on till you come to the end: then stop." However, in order to give you my sense of how much work should go into an assignment, let me say that I envision a project as a 3-5 page paper, the biographical paper as a 10-15 page paper and the topic paper as a 15-25 page paper.


Special Project for Graduate Students


All the graduate (those registered for Math 5010) students (and anyone else who may be interested) will work as a team for this project. That is, they should meet early in the term and break up the assignment into tasks for each team member (I will help facilitate this meeting if desired). The project is to write a detailed history of the UCD Math Department. This will entail finding out something about the Boulder Math Department before the “extension campus” in Denver was founded, determining who the full-time instructors were since the inception of the department (and some idea about the numbers of the honoraria – part time instructors) and their mathematical backgrounds. Interviews with current staff and possibly some recent retirees would be useful (Roxanne Byrne would be a particularly good person to interview.) The project is due at the end of the semester, but may be turned in as soon as it is finished. It must meet the standards of all the other projects.


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