Math 105, History of Mathematics Spring 2017 Prof. D. Joyce, BP 322, 793-7421 Department of Mathematics and Computer Science Clark University

Euclid (ca. 300 BCE)
Museo_dell'Opera_del_Duomo_(Florence) 14th century, by Nino Pisano

Please bookmark this page, http://aleph0.clarku.edu/~djoyce/ma105/, so you can readily access it.

General information

• General description. We will explore some major themes in mathematics--calculation, number, geometry, algebra, infinity, formalism--and their historical development in various civilizations, ranging from the antiquity of Babylonia and Egypt through classical Greece, the Middle and Far East, and on to modern Europe. We will see how the earlier civilizations influenced or failed to influence later ones and how the concepts evolved in these various civilizations.

  The earliest civilizations have left only archaeological and limited historical evidence that requires substantial interpretation. We have many mathematical treatises from the later civilizations, but these are usually in a completed form which leave out the development of the concepts and the purposes for which the mathematics was developed. Thus, we will have to analyze the arguments given by historians of mathematics for their objectivity and completeness.

• Catalog text from Clark’s Academic Catalog
  Explores major themes—calculation, number, geometry, algebra, infinity—and their historical development in civilizations ranging from the antiquity of Babylonia and Egypt through classical Greece, the Middle and Far East and then modern Europe. Analyzes the tension between applications of mathematics and the tendency toward formalism. Emphasizes presentations and discussions. Fulfills the Historical Perspective.

• Course prerequisite :  The prerequisite for this course is an intense interest in mathematics. There are no other prerequisites for it other than a familiarity with plane geometry and algebra. Our study will reach just to the beginnings of calculus since we won't have time in one semester for more.

• Course goals.
  • Content goals:
    • follow the development of mathematics from early number systems to the invention of calculus
    • read and understand some historical mathematics
    • survey the development and use of methods of computation, some of which involve tools such as the abacus
    • study the mathematics of various different civilizations, their conception and use of mathematics, and how the historical conditions of those civilizations affected and were affected by mathematics
  • Historical perspective goals:
    • develop your capacity to understand the contemporary world in the larger framework of tradition and history
    • focus on the problems of interpreting the past and can also deal with the relationship between past and present
    • introduce students to the ways scholars think critically about the past, present and future
  • Other goals:
    • Develop your ability to present mathematics and history in spoken and written forms
    • Help you practice research skills
    • Satisfy, in part, your curiosity of how mathematics developed and how it fits into culture

• Course objectives.
  When you have finished this course you should be able to:
  • describe the development of various areas of mathematics within and across various civilizations
  • describe the changing character of mathematics over time and recognize the distinction between formal and intuitive mathematics
  • give examples of significant applications of mathematics to commerce, science, and general life, past and present
  • understand that history includes the interpretation the past, not just facts
  • better research historical questions and present your conclusions to others

  

• Course Hours. The class meets MWF 10:00–10:50 and M 12:00–12:50. We'll meet four hours a week so that there will be enough meeting times during the semester for all the students to give April class presentations in class.

• Textbook. A History of Mathematics, an Introduction by Victor J. Katz, Addison-Wesley, third edition, 2009. Addison-Wesley. Cloth, 992 pp. ISBN-10: 0321387007, ISBN-13: 9780321387004.

• Assignments, tests, and presentation/paper. You will do assignments every week or two from the text, and you'll take two tests (midterm and final). You will select, research, and present a topic of your choice. Your presentation will be a 15 to 20 minute class presentation accompanied by a 10 to 20 page paper.

• Course grade. 1/7 for assignments, 2/7 for each test, 2/7 for the presentation/paper.

Syllabus

The chapters refer to our textbook.

Class notes, quizzes, tests, homework assignments

The dates for the discussion topics and the assignments are tentative. They will change as the course progresses.

1. Wednesday, 18 Jan 2017.
   Welcome to the class! Course overview
   Egyptian numerals and arithmetic. Multiplication and division algorithms.
   Why teach history of math?

2. Friday, 20 Jan 2016.
   Progression of mathematics over the millennia. Mathematics in different places in the world.
   Discussion on what mathematics was used for in ancient times
   Egyptian unit fractions

3. Monday, 23 Jan 2017. (Two meetings)
   Assignment 1 due Friday.
   More on Egyptian mathematics
   • Egyptian fractions, arithmetic with fractions, the Egyptian 2/n table, areas of circles
   • Article on the Rhind Mathematical Papyrus at Mathground
   • On the Ancient Egyptian Value for Pi at The Number Warrior
   Introduction to Mesopotamian/Babylonian mathematics. Sexagesimal (base 60) system and cuneiform notation. Arithmetic, Babylonian multiplication table. Babylonian reciprocal table. Elementary geometry, the Pythagorean theorem, Plimpton 322 tablet

4. Wednesday, 25 Jan 2017
   • Clay tokens and the emergence of writing, 10,000 to 5,000 years ago in Mesopotamia
   • Denise Schmandt-Besserat's article Two precursors of writing: plain and complex tokens
   • Ivars Peterson's article From counting to writing at ScienceNews
   • Frank Swetz' article Mathematical Treasure: Mesopotamian accounting tokens at Mathematical Association of America
   • Mindy McAdams' Clay tokens and the origin of writing
   • Before Pythagoras: The Culture of Old Babylonian Mathematics 2010–2011 exhibition at NYU
   • Babynonian Mathematics at motivate.maths.org with commentary by Prof. Eleanor Robson
   • Babylonian square root algorithm

5. Friday, 27 Jan 2017
   Assignment 1 due. Answers.
   Assignment 2 due next Friday.
   • Introduction to Greek mathematics. Influence of Egyptian and Babylonian math. Used Egyptian unit fractions early, but later Babylonian base 60 as well.
   • Thales (624–547 BCE), , math attributions to him
   • Thales, Pythagoras (569–475 BCE), and others were said to have visited Egypt & Babylonia.
   • Informal versus formal mathematics

6. Monday, 30 Jan 2017. Two meetings.
   • Greek mathematics. timeline and maps
   • Numerals. Several types. Minoan and Mycenaean had symbols for 1, 10, 100, 1000, and 10000 like the Egyptian hieroglyphic symbols. Attic numerals, like Roman numerals but used Greek letters. Alphabetic numerals, 27 letters used.
   • Pythagorean school, the Pythagorean theorem, incommensurablity of diagonal and side of square, Pythagorean triples
   
   Three classical problems.
   • The Delian problem: doubling the cube
   • Quadrature of the circle, square circle, pi
   • Trisecting angles
   
   Zeno's paradoxes at the Stanford Encyclopedia of Philosophy

7. Wednesday, 1 Feb 2017
   Assignment 3 due Wednesday, February 8. Chapter 2, page 47, exercises 8, 9, 10, 11, and 13.
   
   • Hippocrates of Chios (fl. 450 BCE). First author of Elements, quadrature of lunes
   • Philosophy and logic. Plato, Aristotle, Chrysippus.
   • Number versus magnitude, discrete versus continuous, axioms
   • Euclid's Elements http://aleph0.clarku.edu/~djoyce/java/elements/elements.html and a quick trip of the Elements

8. Friday, 3 Feb 2017
   Assignment 2 due. Answers
   • Book I: Basic plane geometry through Euclid's proof of the Pythagorean theorem

9. Monday, 6 Feb 2017 (two meetings)
   Summary of the other books of the Elements including
   • the first few on plane geometry including the golden ratio and construction of a regular pentagon
   • Eudoxus' definition of ratio and proportion in Book V
   • Books VII through IX on number theory including primes and perfect numbers

10. Wednesday, 8 Feb 2017
    Assignment 3 due. Chapter 2, page 47, exercises 8, 9, 10, 11, and 13. Answers
    Assignment 4 due Monday, Feb 13. Chapter 3, page 90, exercises 6, 7, 14, 17, 19.
    • Books XI–XIII: Solid geometry, the method of exhaustion, constructions of regular polyhedra
    • Rational approximations to irrationals

11. Friday, 10 Feb 2017
    Archimedes: The law of the lever, approximation of pi

12. Monday, 13 Feb 2017, one meeting
    Archimedes: area of a parabolic segment and sums of series
    Short discussion of conic sections
    Astronomy before Ptolemy, Cosmology and astronomy

13. Wednesday, 15 Feb 2017
    Assignment 4 due. Chapter 3, page 90, exercises 6, 7, 14, 17, 19. Answers.
    Assignment 5 due next Monday. Chapter 3, page 91, exercises 27 and 36; and Chapter 4, page 127, exercises 1, 2, and 3.
    Select date for midterm. Selected Friday, March 3.
    Discussion of topics for paper and presentation. List of topics from two years ago:
    
    N. Iwamoto, NP Complete Problems
    B. Guo, Security and Cryptography
    S. Shrestha, Hamiltonian Paths & NP Completeness
    K. Schulz, Pascal's Arithmetic Triangle
    S. Deambrosi, Mathematics and Imperial Examinations in China
    J. Ullman, Mechanical Computations
    A. Joshi, Fibonacci's works
    C. Dollard, Newton's and Leibniz' derivatives
    N. Lew, Pacioli and Da Vinci on the golden ratio
    K. Stuck, Postmodernist math education
    L. Leung, L’Hôpital & his rule
    D. Scheff, Navier-Stokes equation
    M. Marrone, George Boole & logic
    A. Zhang, Geometric Brownian motion & finance
    R. Rasool, Cryptography in Islamic mathematics
    M. Lee, Game theory & business strategy
    Y. Qu, Jinzhang suanshu, the Nine Chapters
    E. Ainasse, Cubic equations in the 16th century
    R. Dutt, The financial crisis & the Black-Scholes equation
    B. Zhao, Math and Go (the board game)
    H. Nguyen, The fundamental theorem of calculus
    M. Blaifeder, Newton's mathematics
    T. Jiang, The development of set theory
    B. Tang, The Riemann Hypothesis
    T. Trinh, Geometry in architecture
    X. Ye, Development of calculus
    W. Shen, Hilbert spaces
    
    
    • Early trigonometry, History of Trigonometry
    • Ptolemy and the Almagest
    • Practical mathematics, Heron, Ptolemy's Geography

14. Friday, 17 Feb 2017
    Diophantus' Algebra

15. Monday, 20 Feb 2017
    Assignment 5 due. Chapter 3, page 91, exercises 27 and 36; and Chapter 4, page 127, exercises 1, 2, and 3. Answers.
    Assignment 6 due next Wednesday
    • Menelaus' spherical trig, Heron's optics
    • Eratosthenes' measurement of the circumference of the earth
    • Pappus' centroid theorem, commentators Theon and Hypatia
    • Outline of Mathematics in China. Number symbols, rod numerals, fractions

16. Wednesday, 22 Feb 2017
    • Nine Chapters: areas and volumes, the Pythagorean theorem, similar triangles
    • Algebra: simultaneous linear equations
    • Chinese remainder theorem

17. Friday, 24 Feb 2017
    • Indeterminate analysis and the Chinese remainder theorem, finding one
    • Solving polynomial equations.

18. Monday, 27 Feb 2017
    Outline of Mathematics in India
    Review for midterm.

19. Wednesday, 1 Mar 2017
    Assignment 6 due. Meet in Goddard Library.
    • Math 105 Lib Guide
    • Finding articles
    • MathSciNet

20. Friday, 3 Mar 2017
    Midterm. Chapters 1–5. Answers.
    
    March 6–10. Spring break.

21. Monday, 13 Mar 2017
    Assignment 7 due next Monday.
    • Outline of Mathematics in India
    • The Hindu-Arabic place-value system and arithmetic
    • Geometry
    • Equations and indeterminate analysis
    • Combinatorics
      Trigonometry, Aryabhata's trig table

22. Wednesday, 15 Mar 2017. Introduction to the mathematics of Islam. Combinatorics, arithmetic triangle, parallel postulate.

23. Friday, 17 Mar 2017. More on Islamic mathematics.

24. Monday, 20 Mar 2017.
    Assignment 7 due. Answers.
    Assignment 8 due next Wednesday.
    • Summary of combinations and binomial coefficients
    • Intro of math into Western Europe: translations from Arabic into Latin in the 12th and 13th centuries primarily at Toledo, Spain, and from Greek in the 13th century
    • Abraham bar Hiyya (fl. 1116), geometry
    • Fibonacci's contributions including transmission of concepts of Islamic mathematics and his own new research (fl. 1200-1230)
    • Summary of early mathematics in western Europe

25. Wednesday, 22 Mar 2017.
    • The mathematics of kinematics: velocity, the Merton mean speed theorem
    • The origins of calculus

26. Friday, 24 Mar 2017.
    • Oresme's Fundamental Theorem of Calculus
    • Changing quantities
    • Infinite series (infinite sums)

27. Monday, 27 Mar 2017
    • Mathematics at the turn of the 14th century
    • Mathematics in America, Africa, and the Pacific. Ethnomathematics
    • Symmetry, frieze groups, wallpaper groups. Abstract groups

28. Wednesday, 29 Mar 2017
    Assignment 8 due. Answers
    Algebra 1400–1600. Development of symbolic algebra

29. Friday, 31 Mar 2017
    Solution of the cubic equation in Italy, negative and complex numbers

30. Monday, 3 Apr 2017
    • Perspective in art, projective geometry, anamorphic projection
    • Cartography, Mercator projection, gnomonic projection
    • Copernicus and Kepler.
    • Logarithms

31. Wednesday, 5 Apr 2017
    • Citations, bibliography, and plagiarism, Clark's academic integrity policy, Internet Public Library's links
    • Galileo's study of motion and gravity

32. Friday, 7 Apr 2017
    • Theory of equations
    • Analytic geometry

• 2015 Midterm. Answers
• 2015 Final exam

This page is located on the web at

http://aleph0.clarku.edu/~djoyce/ma105/