Math 160 · Introduction to History of Mathematics

TuTh 11-12:30  9 Evans Hall
Final Exam Group 13
Final Exam:  Wednesday May 16,  8-11 am
(you should not enroll in this class if you cannot take the final exam at the scheduled time; there will be no make-up exams, neither Final nor Midterms)

Office Hours: Thursday 5-6:30
Prerequisites: Math 104, 110 and 113

Compared to the Midterms, I expect you to demonstrate on the Final a deeper knowledge of the subject which goes beyond knowing just a few names and work titles. Below is a detailed syllabus. Please spend several hours rereading the sources and the biograms of the mathematicians mentioned below. Use also your class notes (e.g., in the class I spent several hours talking about main characteristics of the Medieval period in the development of sciences and education).

Syllabus for the Final:

Egyptian, Babylonian, and Greek Mathematics --
the same requirements as for the two midterms, plus the contributions of Archimedes, Apollonius of Perge, Diophantus (complete knowledge of all integral solutions to the Pythagoras Equation x²+y²=z²; you should know these solutions), and Pappus (Pappus' Theorem); biograms from the St Andrews site are very useful source of information.

Arabic Mathematics  --  my handout, biograms of central figures

Medieval Europe:
the institution of university (its main characteristics, trivium, quadrivium)
Leonardo Pisano (Fibonacci)

16th Century Italy: solving equations of 3rd and 4th degree (my handout); main figures of the drama: Scipione del Ferro, Nicolo Fontana Tartaglia,
Girolamo Cardano, Lodovico Ferrari (biograms)

François Viète (biogram, you should be familiar with his achievements)

A circle of French scholars (the first half of the 17th Century): Girard Desargues (Desargues' Theorem), Marin Mersenne, Etienne Pascal, and his son Blaise Pascal (Pascal's Theorem), Pierre de Fermat, Renée Descartes (biograms;
you should be familiar with their achievements).

For Tuesday, May 1:

Read the following short account of Mathematics during the early islamic period that brought to the end Antiquity in the Middle East, Persia and North Africa.

One of the favorite problems of that period was finding roots of cubic equations. Such a solution was first found only in the Sixteenth Century in Italy. Here is a short and detailed discussion of  the solution.

The following account of the fates of the text of the Elements in the Middle Ages was written by one of the top experts in the subject.  Knowing how the transmission of Ancient mathematical knowledge was happening is crucial to our
understanding of the development of Mathematics from th e close of Antiquity until early modern times.

Continue, please, playing with Pappus, Desargues', and Pascal, Theorems using KSEG. I certainly will devote some time to them, as much as they reflect the continuity of geometric thought from Antiquity to 17-th Century Europe (read biograms of Pappus, Desargues and Pascal on the St Andrews site).

Download and unpack KSEG, a free interactive geometry software, (Linux, Windows), and read a short introduction kseg_help_en.html first. Installation of the Windows package is extremely simple: just unzip and click on kseg.exe in the unpacked folder.

Then dowload also the following zipped folder containing source files for KSEG which illustrate Theorems of Pappus, Pascal, and Desargue. Each of the files in that folder has an extension .seg, and can be simply opened inside of KSEG. Bring your notebook to th eclass on Tuesday.

What you should know and  prepare for the Midterm that will take place April 16:


You should be familiar with:

For Thursday, March 8: please go through Book I again and, for each proposition, write down its assertion (your statement must be formulated in modern mathematical language). Consider this to be your homework assignement.

Euclid's Elements -- an edition prepared by Richard Fitzpatrick who is professor of physics at the University of Texas. Print out the first 3 books (English text facing Greek original) and prepare Chapter I for our Thursday class. Please bring all three chapters to the class.

Your colleagues found this valuable link to the text of Hilbert's The Foundations of Geometry. It is not a scan but a well executed online re-edition in LaTeX (the full LaTeX sources are in a zip file at the bottom of the page. They are well worth studying as an example of how to use LaTeX). I corrected a few errors and replaced the fonts by a lot more attractive ones and it is -- my own PDF file of Hilbert's book.

We have entered the world of Greek Mathematics. Please print this out (rotate it clockwise for viewing), read pages 24-37, and also read the first four chapters from Ivor Thomas's Greek Mathematics, Vol. 1. If you have any general textbook of History of Mathematics, like , read what it has to say about the early stages of Greek Mathematics. Read also the following useful overviews prepared by Donald Allen:
The Saint Andrews site on History of Mathematics does not contain general introductions to Greek Mathematics, its strength being rather biograms of individual mathematicians, and discussions of problems that occupied ancient Greeks.

By now you shoud be thoroughly familiar with at leat the first four chapters of Neugebauer's short book The Exact Sciences in Antiquity. A high quality image of the September page from the calendar discussed by Neugebauer in Chapter 1 you can find here. Your familiarity with Chapters 2 and 3 will be tested on the Midterm that will take place Tuesday, February 21 but the Chapter 1 is useful in general, while very short Chapter 4 provides another useful summary of Egyptian Mathematics.

I also suggest that you now try working out on yor own line by line mathematical procedures in other problems of the sort we covered in class. Next Tuesday we will be finishing our encounter with Egyptian Mathematics by looking at some problems from Chapters 17 and 18 in Gillings, so be prepared.

This site contains a brief and apparently reliable collection of useful summaries about Egyptian history, language, and spiritual culture. Here you will find Egyptian "alphabet" (i.e., the list of monoconsonantal, also called monoliteral signs). Print out the whole document, study it, and carry it with you - it is only 10 pages long.

Now that I posted all the problems from the available papyri you are advised to print out the ones that are discussed by Gillings and you should be carrying them with you to the class.

On Thursday we will finish problem 34 and discuss problems 28, 29, 40, 79, and perhaps also 4 and 6 from
Papyrus Rhind. Your work with any problem should always start from identifying which groups of hieroglyphs correspond to which transliterated words.


Our sources

Papyrus Rhind: 1, 2, 3, 4, 5, 6, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53-54, 55, 56, 57, 58, 59, 60, 61b, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79

Moscow: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25

Lahun Papyrus (known also as Kahun Papyrus): UC32118B,
UC32134A, UC32134B, UC32159, UC32160, UC32161, UC32162, (introduction to Lahun mathematical texts)

Papyrus Berlin 6619:
1, 2, 3

Ostrakon Turin 57170

A very useful short modern
survey of  Egyptian Mathematics written by one of the most  knowledgeable persons in the field (print it out, study it, and bring it to the class)

A dictionary of words and phrases indispensable when studying Egyptian mathematical texts (print it out, study it, and bring it to the class)

A superb introduction to Egyptian Hieroglyphs (get it! - an exciting read, very highly recommended).  [ offers it with E. A. Wallis Budge's
An Egyptian Hieroglyphic Dictionary. Do not buy the latter - it is very unreliable.] The following page from it contains the list of transliterated letter-signs and tells you how to pronounce them.

Digital Egypt for Universities (contains a valuable section on Exact Sciences)

Further Reading

Required Texts:  We will begin from the following texts
Recommended Reading: Carl B. Boyer, A History of Mathematics, John Wiley & Sons
Syllabus: In an introductory course on History of Mathematics nothing can replace a first hand experience of working with original texts. We will focus on a number of such texts from various historical epochs. This will be supplemented by my lectures in which I will be presenting a panoramic overview of the development of Mathematics in its cultural perspective.

Comments: Students are expected to attend the class regularly.