### History Paper

**General Requirements:**

- This is a history of mathematics paper, so it must contain history about a mathematical person or subject.
- The paper should be 8 to 12 typed pages. However, if you can say something interesting in fewer than 8 pages, that is fine, so long as you say everything that should be said. (Be aware that final determination will be made by those evaluating your work.)
- Diagrams and pictures are allowed (but are not required). If images do appear, they must be relevant to the topic.
- The papers should be written in a formal style, in English, using proper grammar. There should be no slang expressions or contractions. The pronoun "I" should not appear (unless it is in a quote).
- You must have at least five references, including at least one primary or near-primary source. (That is, a document from the time you are writing about, or a translation of such a document.) If you are writing about a person, you can (for example) cite something written by that person, or a translation of it, if it was not originally written in English.

**Sample Papers:** Examples of past papers can be found on the UCSD Mathematics Department homepage.

**Groups:** Groups should consist of 1–3 people.

**Due Dates:** The history paper is **due at the end of Week 8.** Your paper will be submitted through TritonEd using the Turnitin system.

If your paper is submitted late, a penalty in the form of a 5% grade deduction will be applied for each day after the deadline. (This penalty will be applied only to the paper itself, not to your final course grade.)

**Turnitin and Academic Misconduct:** Your paper will be submitted through TritonEd, where we use an online "originality checking system" called Turnitin. Turnitin checks documents with text for potential plagiarism and reports matches with other student papers, institution papers, the internet, and periodicals. Plagiarism is a serious academic offense which can lead to severe sanctions.

The following policy, quoted from UCSD's Academic Integrity web site, applies to Math 163: "Students agree that by taking this course all required papers will be subject to submission for textual similarity review to Turnitin.com for the detection of plagiarism. All submitted papers will be included as source documents in the Turnitin.com reference database solely for the purpose of detecting plagiarism of such papers. Use of the Turnitin.com service is subject to the terms of use agreement posted on the Turnitin.com site."

Learn more about academic integrity and the consequences of academic misconduct at UCSD by visiting the Academic Integrity Office website.

**Turnitin and Citations:** Turnitin checks for "originality of content" and reports what percentage of the document is "not original". Text that you cite will be flagged as not original. This is fine, so long as you have properly cited them. (See below for information on how to site documents.) Plagiarism is passing off another person's ideas as your own. As long as you give proper credit to the original source, you are not committing an academic integrity violation.

When you submit your paper to Turnitin, you will be given a "similarity" score, which is the percentage of your document that is "similar" to other documents. For example, if your similarity score is 30%, then 30% of your paper appears in other documents. The similarity score is not an indication of plagiarism. It is an indication of originality. A large score does not mean anything was plagiarized, so long as sources are properly cited, and even a very small score could indicate an academic integrity violation, if citations are not given.

A large similarity score does not necessarily indicate plagiarism, but it does indicate a lack of originality. If 50% of your paper is cited from other works, then the paper cannot truly be said to be your own work.

**Topics:** The topic of your paper may be on any topic of your choice, but it must have something to do with the history of mathematics. It may be biographical or about a specific subject. Your topic does not have to be related to the topics or people discussed in class. If you are interested in a certain topic, but not sure if it is appropriate, you can ask your TA.

A comprehensive source for topics (including both subjects and people) can be found on the MacTutor History of Mathematics Archive webpage:

**References:** *At least five references must be used and cited in your document.* At least one of your references must be a good primary or near primary source. (See below for more on this topic.)

**Listing References:** A list of references should be included at the end of your paper. The list should only include references that are actually cited in the document. The list should be given in alphabetical order, by (first) author's last name. (If more than one source has the same author, then they should be listed in order of publication.) Here is an example containing several books related to Isaac Newton.

[1] González-Velasco, E. A. *Journey through Mathematics: Creative episodes in its history.* Springer, 2011.

[2] Kerr, Philip. *Dark Matter: The Private Life of Sir Isaac Newton: A Novel.* Broadway Books, 2003.

[3] Newton, Isaac. *Philosophiæ Naturalis Principia Mathematica.* (Latin) William Dawson & Sons, Ltd., London, 1687.

[4] Newton, Isaac. *Philosophiæ Naturalis Principia Mathematica: the Third edition (1726) with variant readings.* Assembled and ed. by

Alexandre Koyré and I Bernard Cohen with the assistance of Anne Whitman (Cambridge, MA, 1972, Harvard UP).

[5] Stephenson, Neal. *Quicksilver, Vol. I of the Baroque Cycle.* William Morrow of HarperCollins, 2003.

The book labeled [3] is a primary source, because it was written by Newton himself. The book labeled [4] is a near-primary source, because it is a translation of [3]. The books [2] and [5] are fictional works that include fictional versions of Isaac Newton. These should not be used in a history of mathematics paper of the type written for this class. The first book [1] is a history of mathematics textbook. It is not a primary or near-primary source, but it is a good way to find primary and near-primary sources, because it cites such sources.

The next example contains several articles related to Leonhard Euler.

[1] Euler, Leonhard. "Constructio linearum isochronarum in medio quocunque resistente." *Acta Eruditorum.* (1726) vol. 45, p. 361-363.

[2] Euler, Leonhard. "An essay on continued fractions." Translated from the Latin by B. F. Wyman and M. F. Wyman. *Math. Systems Theory 18* (1985), no. 4, 295–328.

[3] Euler, Leonhard. "Principles of the motion of fluids." English adaptation by Walter Pauls.

*Phys. D 237.*(2008), no. 14-17, 1840–1854.

[4] Sandifer, Ed. "How Euler Did It: Who proved e is Irrational?" MAA Online (2006). Retrieved 2010-06-18.

[5] Shanks, Daniel and Wrench, John W. "Calculation of Pi to 100,000 Decimals."

*Mathematics of Computation 16*(1962) 76–99.

[6] Swenson, Carl and Yandl, Andre. "An Alternative Definition of the Number e."

*The College Mathematics Journal.*(1993) vol. 24, no. 5,

pp. 458-461.

[7] Wei, Chun-Fu and Qi, Feng. "Several closed expressions for the Euler numbers."

*J. Inequal. Appl.*(2015) 2015:219.

In this case, [1] is a primary source, while [2] and [3] are near-primary sources. (Note that the first three articles are all by Euler, and so are listed in order of publication.) The articles [5] and [6] provide examples of sources with more than one author; so does [7], but this is a current research paper which is not appropriate for a history paper in this course.

Of special note, the article [4] is not from a journal; instead it is an article taken from *How Euler Did It*, which is an online MAA column, written by Ed Sandifer from 2003 to 2010. For an online source, like [4], you do not provide a URL. URL's are not static; that is, they are frequently changed, and so providing one may be of little value. The purpose of a bibliography is to provide the reader with enough information that they can find the reference on their own. For this reason, it is necessary to provide the title and author. It is also essential that you provide the date when the reference was retrieved. Online articles are often changed or updated, and it is important that the reader know which version of the article is being referenced.

**It is never appropriate to cite a website.** Websites (like Wikipedia) are dynamic and always changing. You should never list Wikipedia (or other websites) in your list of references. That does not mean these websites are not useful, as you can often use them to locate good sources. For example, the article [1] was found on <http://eulerarchive.maa.org/>, which is a very good resource for finding papers by Euler. If you find a paper on that website, you do not cite the website. Instead you cite the paper that you found on that website: Give the identifying information for the *paper*, not the website.

**Note:** All references should be given in English (excluding the title, if it is not written in English). If a title is not written in English, it must be transliterated to Latin script. (If the people grading your paper cannot identify your references, then they do not count as references.)

**Citing Sources:** When citing a source in your document, do so by referencing the number assigned to it (as in the paragraphs above.) The reference should be written in square brackets and should appear as part of the sentence.

A geometric definition of the number e was given by Swenson and Yandl in 1993 [2].

Sometimes it is helpful to provide a page number (or a range of page numbers). This should be done by adding the page number after a comma in the citation, as in the following example.

After providing a geometric definition of the number e, Swenson and Yandl proved that the derivative of the function e^x is e^x [2, p. 459].

For one page, use "p." and for multiple pages, use "pp." (As in, "[2, pp. 459-461]".)

Place the number (in square brackets) after the author's name or after referencing the source. Multiple sources may be referenced at the same time by including them all in the same set of square brackets, separated by commas.

Isaac Newton has become a part of modern culture, to the point that he even appears as a character in popular works of fiction [2, 5].

**Examples of Near-Primary Sources:** At least one of your references must be a primary or near-primary source. The following works are translations of primary sources, and can be used as near-primary sources. These are just *some examples* and it is not necessary to cite any of these.

__Translations of Historical Works__*The Rhind Mathematical Papyrus*by A.B. Chace (editor). (Egyptian mathematics.)*Euclid's Elements*by Heath.*The Medieval Latin Translation of the Data of Euclid*by Shuntaro Ito.*The Works of Archimedes*by Heath.*On Conic Sections Books I-III*by Apollonius (author) and Taliaferro (translator).*Introduction to Arithmetic*by Nicomachus.*The Algebra of Omar Khayyam*by Daoud Kasir. (Islamic mathematics.)*The Book on Games of Chance*by Jerome Cardan (aka Cardano).*Euclid Vindicatus*by Girolamo Saccheri. (Full title:*Euclides ab omni naevo vindicatus*, or*Euclid Freed of Every Flaw*.)*The Geometry*by Rene Descartes.*Fleeting Footsteps: Tracing the Conception of Arithmetic and Algebra in Ancient China*by Lam Lay Yong and

Ang Tian Se. [Includes a complete translation of*Sun Zi suanjing (The Mathematical Classic of Sun Zi)*.]

__Collections of Historical Texts__*The History of Mathematics: A Reader*by John Fauvel and Jeremy Gray (editors).*The Treasury of Mathematics*by H. O. Midonick.*A Source Book in Mathematics*by D. E. Smith.*A Source Book in Mathematics: 1200-1800*by Dirk Struik.

**Examples that are NOT Near-Primary Sources:** The following texts are considered "general interest" and do not count as primary or near-primary sources.

__Special Topics__*Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem*by S. Singh (1998).*Levels of Infinity: Selected Writings on Mathematics and Philosophy*by H. Weyl (author) and P. Pesic (editor) (2013).*Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics*by John Derbyshire (2003).

__Biographies__*Ada's Algorithm: How Lord Byron's Daughter Ada Lovelace Launched the Digital Age*by J. Essinger (2014).*Alfred Tarski: Life and Logic*by A.B. Feferman and S. Feferman (2008)*Alan Turing: The Enigma*by A. Hodges (1983). (Basis for the film*Imitation Game*.)*Stefan Banach: Remarkable Life, Brilliant Mathematics*by E. Jakimowicz and A. Miranowicz (2011).*Women in Mathematics*by L. M. Osen (1975).*Who Is Alexander Grothendieck?*by W. Scharlau (2011).*Emmy Noether: The Mother of Modern Algebra*by M.B.W. Trent (2008).

**Grading:** You will be graded on the following criteria.

*Originality:*Is the paper your own work or is there insufficient original content?*Appropriateness of Topic:*Is the topic related to the history of mathematics? This should not be a "survey paper" that describes the known results about a specific topic in mathematics. It must contain a discussion of the history of the topic. It should not be collection of theorems and proofs. Proofs should only be included if they are those given by historical figures, and then only if they are going to be featured in an historical discussion, such as contrasting or comparing them with other proofs of the same (or different!) results, or even simply discussing the style of the proof compared to current standards or practices.*Theme:*Is there a theme to the paper? Why has this topic been picked? A theme does not have to be controversial ("Isaac Newton made significant contributions to mathematics, and in this paper we will discuss some of his most interesting ones."), but it can be controversial ("Isaac Newton should get credit for inventing calculus, and in this paper we will demonstrate why.").*References:*Your bibliography must contain at least five references, and each of these must be cited in your paper. At least one of your references should be a primary or near-primary source. Your sources must meet the requirements described above.*Quality of Writing:*Is the paper readable? Is it easy to follow? Are there numerous typos, spelling errors, or grammar issues? The paper should be written as if you expect other people to read it and assess its quality, because they will. Would you be willing to submit it for publication in a scholarly journal? If not, then you probably shouldn't submit it for a grade.*Content:*Is there sufficient content? Is the topic treated in depth? Has the theme (from above) been addressed? Have points been made? Have conclusions been drawn? Is a compelling argument made? Have objectives been achieved?

It is necessarily true that significant elements of the grading process are subjective, and ultimately the determination of your grade on the history paper will be made by those who read it (your TAs and/or instructor).