Math 629: History of Mathematics
Frank Sottile

• Opening Remarks: This week I am in Seoul, South Korea. This should not cause any problems, except that I may not mark your papers as quickly as we all would like.
  Next week is the lastg week before Srping break, so you will have 2 weeks for its homework. I will have information about your second paper by then. In the meantime, I read most papers on the 14 hour flight to Seoul and will start to mark them this week.
  Our reading is on geometry in the 17th and 18th centuries. 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.
      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, but we are getting ahead of ourselves.
• Reading: Chapters 7 and 8 of Stillwell. I have not found other satisfactory sources for this material.
• Assignment: Due Monday, March 6. (HW 8) To hand in: Email a .pdf to both Bennett Clayton bgclayton@math.tamu.edu and Frank Sottile. fjsteachmath@gmail.com
  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. Become acquainted with a great Theorem by doing exercises 8.3.1–8.3.4 in Stillwell.
  4. Problems 8.5.3 and 8.5.4 are an aid to beginning to understand the geometry of the projective plane. Do them. Note that the boundary of the Möbius band is a circle. Explain how attaching a disc (whose boundary is also a circle) to the Möbius band along their common circle gives the projective plane.
     Watch Vi Hart's entertaining video Wind and Mr Ug.