Study Guide for Chapter 15, Merzbach & Boyer

  1. How did two conic sections find applicability in science in the 1600’s?
  1. What did Galileo conclude about infinite quantities, in terms of being able to compare them with each other or with finite quantities?
  1. What was the basic idea behind Cavalieri’sGeometriaIndivisibilis in regards to calculating areas and volumes?
  1. What is Cavalieri’s Principle?
  1. What calculus-like theorems did Cavalieri prove in his book?
  1. We should pause to ask which ancient Greek mathematician’s work was being very closely duplicated by a number of 17th-Century mathematicians working on area and volume problems?
  1. What role did Marin Mersenne play in the mathematical community of the 17th Century?
  1. What was Descartes’ most famous and important book, and how is his Le géométrie related to it?
  1. There is a lot of discussion in the chapter on Descartes’ development of analytic geometry in Le géométrie. Make sure you understand the following:
  2. Rather than trying to reduce geometry to algebra, Descartes’ goal was to accomplish geometric constructions.
  3. This book contains the first really modern looking algebraic symbolism.
  4. How the quote on pp. 312-313 justifies the term analytic in analytic geometry.
  5. How Descartes’ development is different from our modern analytic geometry. No coordinates, no distance formula, no idea of slope, etc.
  6. Descartes did have a way of finding normals (and hence tangents) to curves, but it was more complicated than Fermat’s methods.
  7. The last of the three books or sections of Le géométrie contained Descartes’ work on theory of equations, which includes most of what you learn about solving polynomial equations in a college algebra class -- Descartes’ rule of signs, rational root theorem, etc.
  1. Fermat independently invented a form of analytic geometry. How did Fermat’s purposes or goals differ from those of Descartes?
  1. How were the two systems similar?
  1. What three “calculus” problems did Fermat develop some methods for?
  1. What problem in number theory did Fermat solve that was a special case of his famous “Last Theorem?”
  1. In what book did Fermat write about his “marvelous proof” of that theorem?
  1. Skim briefly over de Roberval and Desargues.
  1. What gave rise to Pascal’s work on probability, and who was his collaborator?
  1. Skim over the rest of the chapter, beginning on p. 337. Pause a little longer on page 341, where it discusses Hudde.