Math 629: History of Mathematics
Frank Sottile

• Opening Remarks:     Do begin to think about your second paper. It is due in 4 weeks, and you should decide what you will do, and perhaps start reading the text or sketch an outline. The deadline will arrive sooner than you think.
      Our reading is on geometry in the 17th and 18th centuries, and then some Calculus. (We are skipping projective geometry, a beautiful subject that drove my class to tears in the past; as there is a need to skip some topics, I have chosen to skip this one. I am sorry if you like projective geometry.) Fermat and most notably Descartes introduced what we now call the Cartesian coordinate system, and used algebraic formulas to describe geometric objects (initially plane curves). This melding of geometry and algebra is the subject of analytic geometry and leads into The Calculus, but it also allows the application of algebraic reasoning to geometry and geometric intuition to algebra. As it lays the foundations for The Calculus, we are familiar with some aspects of analytic geometry, and we all teach it. Fermat and Descartes figure prominently in this story.
      At the same time, and independently, geometry developed along some of the lines of the Greeks—without coordinates. A notable piece of this synthetic geometry was the truly new field of projective geometry, which arose in part from the study of perspective in painting. Arguably one of the most elegant topics is the projective plane and its duality of points and lines (which we are skipping).
      It takes a very long time for these two threads of geometry to come together (a key component is the introduction of homogeneous coordinates, which are not coordinates in the usual sense). By the end of the 19th century, they do in the field of algebraic geometry, which undergoes significant development in the 20th century (and is one of my research specialities), but we are getting ahead of ourselves.
      For most of us, Calculus is the first mathematical mountain that we scaled (were force-marched up?). Stillwell's treatment, and a reading next week, show its development in a different light than we learn in classes. It is worth noting that while Calculus looms large in the life of a student (many of us take 2-3 years of it), it only occupies a sliver of our textbook, which it trying to cover all of mathematics (up to about 1900). I hope that you do not mind all of the other bits that we are doing, as you try your hand at some of the different mathematics concepts that we humans have developed over the ages.
• Reading: Chapters 7 and 9 of Stillwell. (We will skip Chapter 8 on projective geometry).
• Assignment: Due Monday, February 28. (HW 6) Here is a .pdf and a LaTeX source of the assignment.
  To hand in: We are using Gradescope for homework submission.
  1. Find the parametrizations of exercises 7.4.1 and 7.4.2. Use the same method for y²=x²(x-1). What is different about the third "curve" ? (Hint: consider the origin)
  2. Carry out Descartes's claim by doing problems 7.5.1 and 7.5.2. Hint, they can both be parabolas, but will one will be horizontal and the other vertical.
  3. Try your hand at some heady calculations: Do 9.5.3 and 9.5.4 in Stillwell.
  4. And some more: Do 9.6.1, 9.6.2, and 9.6.3 in Stillwell, and you will one-up Leibniz!
     Partial fractions are typically not done at this depth in Calculus, at least at the Universities I have taught at in the US. When I taught at the University of Toronto, we were supposed to teach partial fractions, completely, and end up presenting the theorem that any rational function may be integrated in elementary terms. I doubt the students understood my presentation (fault on both sides).

Last modified: Sat Feb 12 14:08:56 CST 2022 by sottile