Course Schedule

The following syllabus is adapted from material provided by Victor Katz & Fred Richey during the 2009 MAA Mini-Course on Teaching the History of Mathematics.

The number corresponds to the class day

5. Euclid’s Elements - Book I and the Pythagorean Theorem (2.2.1)

6. Euclid’s Elements - Geometric algebra and the pentagon construction (2.2.2-2.2.3)

7. Euclid’s Elements - Ratio, proportion, and incommensurability (2.2.4, 2.2.6)

Alternate Schedule.

There will be 10 units in the course, lasting from 2 to 5 class periods each. The first and last date of each unit is indicated below, along with the text assignment and the additional readings for that unit.

1. Jan 20 – Jan 27 (4 days). Ancient Mathematics: We will see how and why ancient peoples solved various types of mathematical problems, putting these problems into the social context of the particular civilizations, especially Egypt, Mesopotamia, and China. We will consider who the people were who “did” mathematics and what their role was in the society. We will look at how ancient peoples calculated areas and volumes of various figures and try to understand how and why they arrived at their procedures. We will also study the origins of the quadratic formula in geometry and consider why solving quadratic equations was important in ancient Mesopotamia.

2. Jan 29 – Feb 3 (3 days). Greek Geometry: We will study Greek geometry, paying particular attention to the development of the proof process and how this relates to the position of mathematicians in Greek society.. We consider the idea of calculating areas and volumes by approximating them through the areas and 37 volumes of simpler figures and learn how this was accomplished and how the results were proved by Greek mathematicians. We will also look at Greek “geometric algebra” and consider its relationship to Mesopotamian algebra. We conclude this section with a brief look at the development of conic sections.

Text: Chapters 2, 3. Additional readings: Selected propositions from Euclid’s Elements and Archimedes’ Method.

3. Feb 5 – Feb 10 (2 days). Trigonometry and its Travels: The idea of a function grew up with the use of trigonometric ideas to measure the heavens. We will therefore study ancient astronomy and follow the growth and travels of trigonometry (both plane and spherical) from Greece and Egypt to India, then to China, to the Middle East, and finally back to Europe. We consider the importance of both astronomy and trigonometry in each of these civilizations.

Text: Sections 4.1, 4.2, 6.1, 6.2,. Additional readings: Selections from Ptolemy’s Almagest.

4. Feb 12 – Feb 26 (4 days). Islamic Algebra and Combinatorics: We consider the first true algebra text, written by al-Khwarizmi in the ninth century. We will consider the reasons for the study of algebra in Islam and then see how the Islamic mathematicians understood the subject and how they developed the laws of exponents and the basic principles of polynomial algebra. We will also consider the development of the idea of mathematical induction in the Islamic and Jewish cultures of the middle ages, especially in connection with combinatorial ideas.

Text: Sections 7.2, 7.3, 8.2, 8.3. Additional readings: Selections from al-Khwarizmi’s Algebra and Levi ben Gerson’s Maasei Hoshev.

5. Feb 29 – Mar 12 (7 days). Cubic Equations: Cubic equations were solved geometrically by Omar Khayyam in the eleventh century, through the use of nascent ideas of calculus by Sharaf al-Din al-Tusi in the twelfth century, and then algebraically by various Italian mathematicians in the sixteenth century. We will look at their methods and then see how the concept of a complex number grew out of the algebraic solution techniques. We also consider some of the reasons for the interest in solving equations of second and third degree in Europe in the Renaissance.

Text: Chapter 9; Section 11.2. Additional readings: Selections from Omar Khayyam’s Algebra and Gerolamo Cardano’s Ars Magna.

6. Apr 14 – Apr 19 (3 days). Analytic Geometry and the Beginnings of Calculus: Although curves were drawn in ancient times, a new era in their study began with the invention of analytic geometry in the seventeenth century in France. We consider the work of both Fermat and Descartes and the development of new techniques by them and others to deal with maxima and minima, techniques which eventually grew into our algorithms for derivatives. We will also look at the work of Fermat and others in working out algorithms for finding areas under certain curves.

Text: Sections 11.1, 12.1, 12.2 Additional readings: Selections from Fermat’s Introduction and Descartes’ Geometry.

7. Apr 21 – Apr 26 (3 days). The Invention of Calculus: Why are Newton and Leibniz considered the inventors of calculus? We look at the accomplishments of both men in the context of their times. We will see that, although Newton did not develop the calculus in order to develop his celestial mechanics, he did make use of it in working out the details of the Principia. And we will also consider the reasons for Leibniz’s interest in the subject and his use of it to solve numerous problems.

Text: Sections 12.3-12.6. Additional readings: Selections from Newton’s De Analysi and Principia and from Leibniz’s first papers on calculus.

8. Apr 22 – Apr 24 (2 days). Calculus in the Eighteenth Century: We consider developments in calculus in the eighteenth century, including the criticism of the foundations of calculus by Berkeley, the responses by Maclaurin and d’Alembert, and the continued development of calculus techniques by Johann Bernoulli, Leonhard Euler, and others. In particular, we will look at the problems that these mathematicians solved through the use of their new techniques.

Text: Chapter 13. Additional readings: Selections from Euler’s Introductio and Differential Calculus.

9. Apr 29 – May 6 (3 days). Rigor in the Calculus: The rigorous development of calculus took place in the nineteenth century. We look at Lagrange’s attempt to found calculus on the notion of a power series, at Cauchy’s notions of limits and continuity, and at the idea of a Fourier series and the reconsideration of the notion of a function. We will see how the changes in society wrought by the French Revolution impacted on the work of these men. We then deal with the arithmetization of analysis in the work of Dedekind, Cantor, and Weierstrass.

Text: Sections 14.4-14.5, 16.1-16.3. Additional readings: Cauchy’s first proof using epsilon and delta.

10. May 8 – May 13 (3 days). The Beginnings of Abstract Algebra: Solution techniques for cubic and quartic equations were investigated in detail in the eighteenth century at the same time as attempts were being made to solve equations of higher degree. These investigations were one of the sources of the idea of a group. We will consider other sources of this notion as well. We will see how this development led to the growth of abstraction in mathematics.