Math 629: History of Mathematics
Frank Sottile

• Opening Remarks: This week I am in Oacaxa, Mexico at Casa Matematica Oaxaca (CMO) attending the Congreso Nacional de Geometría Algebraica. I am also here to work with a former graduate student on several different papers, so a lot is going on. I will have your papers marked by next weekend, and likely will scan them and email them back to you.
      Do think about your second paper.
      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 last year; as there is a need to skip some topics, I have chosen to skip this one.) 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, and this is colourfully mentioned in the readings from Bell.
      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. Among the gems of this field is Desargues' Theorem, a nontrivial result about points and lines.
      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.
• Reading: Chapters 7 and 9 of Stillwell. (We will skip Chapter 8 on projective geometry)
  Bell's Chapters on Descaartes and Fermat. Note that his writings are by far less scholarly than Stillwell's.
• Assignment: Due Monday, March 5. (HW 8) Email a .pdf to Mehrzad Monzavi Mehrzad@math.tamu.edu.
  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. 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.