What the course is about.

These concepts had also to be taught to students, and it was quickly discovered that students learn best when they learn something intriguing so 'fantasy' problems, having only indirect relationships to the real world soon found their way into textbooks- a herd of goats and chickens has 140 legs and 50 heads. How many of them are chickens? what is the length of an ellipse? a spider is chasing a fly in a rectangular room. What path should she follow?

 Another interesting problem is the transmission of knowledge over time and space. As old civilizations waned and new ones arose, how were mathematical concepts and techniques transmitted from culture to culture?

The course requires more reading and writing, but less theorem proving than other courses in the Department. The main objective is to gain an understanding of the discovery or invention of mathematical concepts by a close study of the original texts in which these concepts are thought to have first appeared. Since these texts were usually not written in English, this is not a realistic aim, but we shall at least try to get some of the flavour by using authentic translations, and not second--hand commentaries. Since there is far too much material available to cover all aspects, we concentrate on that stream of mathematics which includes numbers and calculus and leads from Baylonian clay tablets containing multiplication tables to the development of a rigorous theory of limits in 19th Century Europe.

The course assessment will be based on the following:

1. Explication of text, essay plus presentation 20%
2. Thematic essay 40%
3. Final examination 40%.

You will need to find the book from which the text is extracted, and read enough about your text to put it into its historical context. For this, you will also have to consult other library books and Internet sources. But this background material should occupy not more than a quarter of your essay. The main content should be a careful explanation of exactly what the author is trying to do and how he or she does it. Try to put yourself into the shoes of the author, not assuming any more knowledge or a cultural background that he or she did not have. When you employ modern notation or concepts in order to explain the text, make it clear that you are doing so.

 Here are some points to bear in mind when writing your explication:

1.  Sometimes it is important to discuss the actual document and how it was transmitted to us. This is especially important for documents from antiquity, or in cases that the original does not survive, or when the text is a translation.
2. Sometimes it is important to discuss biographical details of the authors, for example those which explain their mathematical development.
3. It is essential to discuss the mathematical significance of the text, for example how it is linked to earlier and later mathematics.
4. Discuss the historical, cultural, social or religious background of the author if this is relevant to the mathematics.
5. Use your own ideas wherever possible, especially when you do not agree with statements in the references.
6. Adequate referencing is essential. Whenever you make a statement that is based on something you have read, give a complete reference including page or section number. There are several reasons for this. First, it is essential for academic ethics. Second, it allows your readers to pursue a part of your work which interests them. Third, you will eventually need to go back and check something, so adequate referencing the first time round saves you time in the long run.

1.  A Babylonian tablet problem text which solves a quadratic.
2. A number theory problem of Diophantos.
3. An ancient Chinese text on solving a system of linear equations.
4. A text of the Arabic scholar Al-Khwarizmi on a quadratic equation with a geometric proof.
5. A text of Omar Khayyam on solving a cubic geometrically.
6. Two opposing accounts of a rivalry between Renaissance Italian mathematicians Tartaglia and Cardano on solving a cubic algebraically.
7. Descartes' solution of the general polynomial.

1. Calinger, Ronald (Ed) Classics of Mathematics.
2. Fauvel, John and Gray, Jeremy (Eds) The History of Mathematics: A Reader.
3. Midonick, Henrietta The Treasury of Mathematics (2 Vols.).
4. Struik, Dirk J. A Source book in Mathematics 1200-1800.
5. Thomas, Ivor Greek Mathematical Works, Vol. 2

 Remember that your audience has the text in front of them so there is no need to write it on the blackboard. Begin by explaining precisely what the author was trying to do, and then explain how he or she did it. This should occupy about half your talk. Your sources of information are the same as those mentioned in the section Explication of Text above.

 This is your major term paper for the course, counting for 40% of your final grade. You should choose a topic before the end of August and discuss your choice with me before you commence written work. Your topic must be different from your Explication of Text.

The paper is to be on a topic of your choice from the list below. This is meant to be an interesting and enjoyable assignment, not a chore, so choose a topic with care.

General points to bear in mind:

1. The essay is on the history of mathematics. It should be neither all history nor all mathematics but should contain a reasonably non-trivial piece of mathematics as well as the history and background of that mathematics.
2. Enough expository material should be included so as to make your paper self-contained.
3. You should use a variety of research materials and must give careful references to your sources. Usually, a source consists of a book or article which may refer to other sources. Give complete details!! For suitable formats, see any issue of the journals Archive for the History of the Exact Sciences or Historia Mathematica.
4. Your paper should include a Bibliography listing your sources and they should be cited in the body of your paper when appropriate. See the journals mentioned above for the correct method. Do not use Footnotes but refer in the body of the essay to specific page numbers or chapters of works (including web sites) listed in your Bibliography. Sometimes you may not be able to access the work referred to, which is cited in some secondary source. In that case, your reference in the body of the essay should say "cited in ..." and the Bibliography should include both primary and secondary source.
5. Your essay should not be just a paraphrase of other people's ideas, but should present your own point of view, or perhaps several opposing points of view with your reasoned arguments for supporting one of them. You are not expected to make any startling new contributions to human knowledge (though such would of course be welcome and suitably rewarded). However, you are expected to produce a coherent presentation of your own ideas and opinions, comparing and crticising the opinions of others, conjecturing how mathematicians may have been led to their discoveries, who may have influenced them, how they were affected by their social environment or personal background. You are especially expected to make a judicious choice of texts to quote and where necessary clarify them in modern terminology.
6. What you write should indicate that you understand what you are writing: indications to the contrary include quotes out of context, abstruse language "lifted" from elsewhere, and insufficient detail in a mathematical argument.
7. You are welcome to discuss the progress of your essay, and any difficulties you are having, with me at any time. I may be able to suggest new ideas and references. However, I will not comment on a draft of your completed essay. What you finally submit must be your own work.
8. The grading of your paper will be based on a number of factors, including: the historical and mathematical content; the significance, interest, accuracy, and completeness of the material; the accuracy, scope and significance of your references, and the sensitivity with which they are used and cited; and finally, the style in which it is written.

1. To give you a deeper understanding of mathematical concepts by learning how and why those concepts arose.
2. To add some humanity to your knowledge of mathematics.
3. To provide an overview of mathematics---so you can see how your various courses fit together and to see where they come from.
4. To learn how to use the library and internet.
5. To demonstrate that mathematics is part of our culture.
6. To indicate how you might use the history of mathematics in your own future career.
7. To improve your written and spoken communication skills.

•  Ideas on infinity: 300 B.C to 1300 A.D. or
• The concept of function in 18th Century France.

1. Probability and gambling.
2. The dispute between Newton and Leibniz.
3. Archimedes' work "On the Circle and Sphere".
4. Geometric solutions of polynomial equations.
5. Euclid's 5th Postulate.
6. The concept of infinity.
7. Calculus before Newton.
8. Complex numbers.
9. The concept of function.
10. Fractions.
11. Astronomy and Trigonometry.
12. Fermat's Last Theorem.
13. Mathematical Logic.
14. Real numbers.
15. Archimedes' Method and Descartes' Method
16. Secret code making and breaking.

1.  Carl B. Boyer and Uta C.Merzbach, A History of Mathematics
2. Howard Eves, An Introduction to the History of Mathematics.
3. Victor J. Katz A History of Mathematics

1. Ivor Grattan-Guiness Companion Encyclopaedia of the History and Philosophy of the Mathematical Sciences
2. Morris Kline Mathematical Thought from Ancient to Modern Times