• 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.

  See also Clark's Academic Catalog

• Prerequisites. 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 interpretion the past, not just facts
  • better research historical questions and present your conclusions to others
• Brief syllabus, Detailed syllabus
• Textbook. The text for the course is 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. Supplementary material will be included for those subjects of primary interest.
• Course Hours. MWF 10:00-10:50. BP 220.
• Office hours. Semester schedule. Humor in mathematics.
• Assignments, tests, and presentation/paper. You will do assignments every week or two from the text, and you'll take three tests, each test covering about 1/3 of the course. 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 120 page paper.
• Course grade. 1/5 for assignments, 1/5 for each test, 1/5 for the presentation/paper.

Class notes, quizzes, tests, homework assignments

• Wed, Sep 3. Welcome and survey of the course. Discuss Egyptian numerals, multiplication, and division.
• Fri, Sep 5. Egyptian unit fractions, arithmetic with fractions, the Egyptian 2/n table, linear equations and the method of false position, geometry.
• Mon, Sep 8. Sexagesimal (base 60) system and cuneiform notation, arithemetic, division by means of a Babylonian reciprocal table
• Wed, Sep 10. Asmt. 1 due on ancient Egyptian mathematics.
  Elementary geometry, the Pythagorean theorem, Plimpton 322 tablet, square roots, quadratic equations
• Fri, Sep 12. Greek alphabetic numbers. Thales, Thales' theorem. Pythagoras, Pythagorean arithmetic.
• Mon, Sep 15. Asmt. 2 due on ancient Babylonian mathematics
  Figurate numbers. Pythagorean theorem.
• Wed, Sep 17. Incommensurability of side and diagonal. Aristotle, Zeno's paradoxes, number (discrete) versus magnitude (continuous)
• Fri, Sep 19. Euclid's Euclid's Elements, Quick trip.
  Book I: Definitions, postulates, common notions, early propositions.
• Mon, Sep 22. More on Book I. Basic plane geometry, implications of the parallel postulate.
  Book II: geometric algebra, the "golden ratio"
  Book III: geometry of circles
• Wed, Sep 24.
  Book IV: Constructions of regular polygons
  Book V: Eudoxus' theory of ratios and proportions
  Book VI: Similarity of figures
• Fri, Sep 26. Asmt. 3 due on early Greek mathematics
  Books VII, VIII, and IX on number theory. Primes and perfect numbers.
• Mon, Sep 29.
  Books XI, XII, and XIII on solid geometry. Method of exhaustion. constructions of the five regular polyhedra
• Wed, Oct 1. Asmt. 4 due on the Elements
  Archimedes: law of the lever, approximation of pi
• Fri, Oct 3. Conic sections. Archimedes: area of section of a parabola, infinite sums done finitely with "Archimedian axiom". Archimedes' Cattle Problem
• Mon, Oct 6. Astronomy. Trigonometry: the chord as the trig function
• Wed, Oct 8. Trigonometry: Hipparchus, his supplementary and half-angle formulas, and his table of chords; Ptolemy, his theorem and his sum and difference formulas. The work of Heron.
  Hand out the first test from two years ago
• Fri, Oct 10.
• Oct 15. First test on chapters up through the chapter on mathematical metyhods in Hellenistic times
  

  This page is located on the web at

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

  David E. Joyce,