Course Title: History of Mathematics

Math. 380 - Course Syllabus

Course Title: History of Mathematics

Instructor: Dr. J. L. Berggren

Class Times: MWF 9:30 - 10:20

Tutorial 1: Thursdays, 10:30 - 11:20

Tutorial 1: 11:30 - 12:20

Tutorials begin in the second week of classes

Location: Class in AQ5016

Tutorial 1 in AQ5018

Tutorial 2 in AQ5016

Office Hours (K 10525):

MW 10:30 - 11:20

Other times by appointment

Email berggren@sfu.ca

Office telephone: 291-3335

Prerequisites: Differential calculus (Math. 151) and linear algebra (Math. 232), as well as integral calculus (Math. 152) or euclidean geometry (Math. 113). The euclidean geometry course would be particularly useful, so if you have forgotten a lot of your high school geometry you should review it.

Calendar Description: The course is an account of the history of mathematics from ancient times through the development of calculus and the origins of modern algebra in the nineteenth century. I will emphasize developments that shaped the mathematics you studied in high school and the first two years of university.

Course Requirements	Percentage of final total for course
Weekly quizzes	20
Hour Exam	25
10 page paper	20
Homework	25
Participation (explained below)	10

The weekly quizzes will test your ongoing knowledge of the history of mathematics. You will write the quizzes in the first ten minutes of each Monday's lecture. If you are tardy for the quizzes on Monday you will not be given extra time to complete them.

The hour exam tests your ability to put together the facts you have learned into a short account of the development of different areas of mathematics at different times. It will have one part consisting of a few questions chosen from the ten minute quizzes and three short essay-type questions that are intended to have you write for ten or fifteen minutes on a particular topic. Since the development of writing skills is important in this course you will be required to write some short pieces for homework, which will be marked on the basics of writing as well as on content.

The ten page paper allows you the chance to do research on some aspect of the history of mathematics not covered in the lectures, and to apply the writing skills you have learned. E.g. you might want to go into more depth on a topic we only touched on, such as Egyptian fractions, or you might want to explore mathematics in one of the non-Western civilizations that we don't discuss in the course. The topic for the paper is due on June 22 and the paper itself is due on the last day of class. Two points are deducted for each weekday the paper is late (with an additional two points for any part of a weekend). If you see, in advance, some unavoidable reason why you cannot submit your paper on time you must consult me as soon as you know there's going to be a problem to see if you can get an extension.

Participation is measured by your attendance at lectures and tutorials, as well as coming to both of these — especially the tutorials - with questions to ask. I will record attendance at lectures by circulating a sign-up sheet at each lecture and the TA will explain to you how he will measure participation in tutorials.

Course goals: To impart an appreciation of how great achievements, ingenious instruments, and creative imagination led from the abacus–with many fascinating diversions–to the mathematics we learn today.

Course objectives: When you have finished this course you should be able to:

Recall the names and relative chronological order of important mathematicians, some major events of their lives, and their most important contributions.
Recognize the titles of important mathematical works and name their authors. You should be able to explain, in terms of historical development why the mathematicians and works are important.
Summarize the development of areas such as arithmetic, algebra, geometry, etc. in the major periods we study.
Describe the structure and functioning of mathematically-based instruments such as the sundial, astrolabe, proportional compass, and slide rule, and to understand why the instruments were important at the time.
Describe the changing and expanding concept of number from tally marks to complex numbers.
Give examples of significant historical applications of mathematics to astronomy, geography, timekeeping and everyday life.
Appreciate the creative aspects of mathematicians' work and how mathematicians have used both imagination and logic to advance the subject.

Teaching methods: Lectures and demonstrations, tutorial discussion, showing occasional videos.

Required Text: David M. Burton, The History of Mathematics: An Introduction (4^th ed'n)

Recommended reading: For biographies of mathematicians see the Dictionary of Scientific Biography in the Reference Area of the library. Other good books (though not required for the course) are: A. Aaboe Episodes in the Early History of Mathematics (for the Babylonian and Greek periods); R. Gillings, Mathematics in the Time of the Pharaohs (for the Egyptian period); J. L. Berggren, Episodes in the Mathematics of Medieval Islam (for the medieval Islamic period). J. L. Berggren, J. Borwein, P. Borwein, Pi: A Source Book. For the developments that led to Euclid’s Elements I recommned Benno Artmann, Euclid–the Creation of Mathematics.

Lecture topics and Readings (Please note: There are two holidays.)

Lecture	Topic(s)	Burton
1	Ancient numeration	1.2 and 1.3
2	Egyptian Mathematics	2.2 - 2.4
3	Babylonian Mathematics	2.5 - 6
4	Introduction to the Greek World — Guest Lecture	-------------
5	The monochord and mathemata of Thales and Pythagoras	3.1 - 3.3
6	Math as a problem driven activity — three famous prob’s	3.4
7	Victoria Day Holiday
8	Special curves in Greek mathematics	3.5
9	Visiting professor from Utrecht delivers a guest lecture
10	Euclid's Elements, I	4.1 - 4.3
11	Euclid's Elements, II	Finish above
12	Measuring the earth and cosmos in ancient Greece	4.4
13	The mathematics of Greek mapmaking	Reading B1
14	Greek astronomy and Ptolemy's Almagest	Reading A1
15	The genius of Archimedes, I	4.5
16	Archimedes' Method of the Mechanical Theorems	Video
17	Projections in Greek Mathematics, I: Conic sections and sundials; The Tower of the Winds	Apollonius
18	Projections in Greek Mathematics, II: The astrolabe	Reading B2
19	Diophantus and his Arithmetica: Fermat’s last theorem	5.2 and 5.3
20	Pappus and Greek Analysis	5.4
21	Al-Khwarizmi's Algebra and Hindu Reckoning	B3
22	Mathematics in medieval India and Islam	B4
23	Mathematics in medieval Islam (concluded)	B5
24	Leonardo Fibonacci's Liber Abaci	6.1 - 6.3
25	Canada Day Holiday
26	Mathematics in the early Renaissance	7.1 and 7.2
27	The solution of cubic and quartic equations	7.3 and 7.4
28	Mathematics at the time of Galileo	8.1
29	Mathematics in 16^th -century astronomy and cartography
30	Logarithms and the invention of the slide rule
31	Mathematics at the time of Descartes	Video
32	Descartes and his La Géometrie	8.2
33	The lives of Newton and Leibniz	8.3 and 8.4
34	Newton's contributions to mathematics
35	Leibniz's contributions and calculus controversies
36	One hour examination II	11.1 - 11.2
37	Calculus from Bernoullis to Cauchy	11.3
38	Complex numbers and beginings of modern algebra	11.4
39	Counting the infinite	12.2

In the above schedule items A1 and A2 refer to Chapter 3.6 and Chapter 4 (respectively) of the book by Aaboe (cited above), which is on reserve.

The item ‘Apollonius’ refers to the prefaces to the various books of Apollonius’s Conics, which are on Reserve.

Item B1 refers to a reprint from the book J. L. Berggren and A. Jones, Ptolemy’s Geography: An Annotated Translation of the Theoretical Chapters, which is on reserve.

Item B2 is something I have written for this course, and it is on reserve.

Items B3, 4, and 5 are Chapters 1, 2, and 4 of J. L. Berggren, Episodes in the Mathematics of Medieval Islam, which is on reserve.

Homework:

Due May 18, 2001

Read the objectives for the course and write a paragraph explaining your reasons for taking the course, how they relate to the course objectives, and which part of the course looks the most interesting to you.
Discuss one or more ways in which the Babylonian treatment and concept of numbers might be considered to have been more general than those of the Egyptians. ("Concept of numbers" refers to the sort of ‘things’ they recognized as being numbers, rather than the way they wrote numbers. For example a civilization that recognized the idea of irrational numbers might be said to have a broader concept of number than one that didn’t.)
pp. 16 - 18: 1 b; 2 a; 3 d; 5 c,e; 11 c; 12 b, c and p. 26: 1 b; 2 b; 3, 5

Due May 25, 2001

pp. 48 - 49: 2 b,d, 4 a, 16, 19 (for 8 loaves only), 22. pp. 57 -58: 1 a, c, 2 a, 3 - 5

pp. 67 - 68: 1, 3, 4, 11, 13 a. pp. 75 - 76 1, 2, 7, 8

Writing: Write a paragraph comparing Egyptian and Babylonian mathematics (other than numeration systems) from the point of view of content and level. In particular your paragraph should touch on their agebra and their geometry.

Due June 1, 2001

pp. 98 - 100: 1, 4, 6, 8, 12 a,b, 14 and pp. 122 - 124: 2, 4, 5, 8

Due June 8, 2001

pp. 131 - 133: 2, 3, 4 and pp. 161 - 163: 1, 8, 11

Writing: Write a half page on the mathematical achievements of Democritus of Abdera as described in The Dictionary of Scientific Biography.

Due June 15, 2001

pp. 175 - 176: 10, 13, 15, 16 a

Writing 1: How did Euclid's concept of number differ from that of the Babylonians? Cite relevant material from the books of the Elements dealing with number theory as well as relevant material from Babylonian mathematics to back up your claim.

Due June 22, 2001

Writing: View the video on Archimedes' Method of the Mechanical Theorems and write a half page on what you learned from it.

p. 185: 1, 2, 4 and pp. 199 - 200: 1 (a and b), 4 and 6.

Due June 29, 2001

Writing: Read Apollonius's own account of his Conics in T.L. Heath's History of Greek Mathematics, Vol. II, pp. 128 -133, and write a paragraph that answers the following questions:

To whom does Apollonius direct the letters?

What cities does he mention in the letters

What family members does he refer to in the letters? What were their names?

What other people does he mention in his letters?

In terms of level what is the main difference between the first four books and the last four?

Did Apollonius write the books all at once, or were there gaps between the times of their composition?
Question 1: Ptolemy's table of chords gives the value of 49; 45, 48 for the chord of 49°in a circle of radius 60. (This is the approximate latitude of Vancouver.) Convert this to a decimal fraction. A common ancient value for p is 3 ¹/₇. The diameter of the earth at the equator is approximately 12,756 k.m. Use the above data to find out the circumference of the earth at the 49^th parallel. You will be graded on the basis of how well you organize and explain your calculations. Lots of words are not necessary, but some thought is highly desirable.

Due July 6, 2001

pp. 220 - 221: 1, 4, 7, 12, 16 and p. 226: 1, 2(a)(d), 3, 4, 5

Due July 13, 2001

pp. 243 - 244: 2(b)(c), 3(b), 4, 5, 6, 7(a), 9, 11 and pp. 264 - 266: 1, 2, 6(a), 8, 11

p. 272: 1, 2(a), 4

Write a one-page essay in which you imagine yourself an eager mathematics student in the 12^th century and you have a chance to go study in Europe or the Islamic lands. Where would you go? You will be judged on the reasons you give for your choice that relate to the questions of who you could study with and what you could learn there.

Due July 20, 2001

pp. 302 0 304: 1(b), 3(a), 5, 7, 15 and p. 310: 1(a), 4, 6

Activity: Draw a map grid based on the ideas of Mercator for the part of the earth from 30S latitude to 60 degrees north latitude and 0° longitude to 180° longitude. Your grid lines, north to south and east to west, should be 10° apart.

Due July 27, 2001:

1(a), 2, 3, 5(a), 7, 8, 11, 12(d)

Writing: Write a page comparing and contrasting Viète and Napier from the point of view of their lives and their works. Among the questions you should consider are those concerning how they supported themselves, what their major interests (mathematical and otherwise) were, what they contributed to mathematics, and in what way(s) those contributions were historically important.

Activity: Make a pair of geometrical compasses of the kind that Galileo invented inscribed with scales to double the volumes of solid figures. Use the scales to find the diameter of a circle that has area twice that of a circle with diameter 12 cm and to find the distance from one corner to the diagonally opposite corner of a cube that has volume double that of a cube in which that dimension is 10 cm.

Due August 3, 2001

pp. 352 - 353: 2, 3, 5, 6, 8(a), 10(b), 11

Consult the book by Berggren, J. L., Borwein, J. and Borwein, P., Pi: A Source Book (on reserve in the library) to answer the following questions relevant to the history of mathematicians' struggle to discover the secrets of that number.
1. What is the difference between an algebraic number and a transcendental number? Is p transcendental or algebraic? Who proved it for the first time?