Study Guide for Math 306 Hourly 29Oct00 Compiled in class, Friday 22Oct00 Broad themes in Greek mathematics bearing on the ultimate discover of the calculus 1800 years later. The problem of measuring geometric quantities: Area in terms of length. Volume in terms of area and length. First by the method of dissection, later by exhaustion. Squaring a figure means finding a square of equal area. Know how to square a rectangle. (Understand Thales' theorems.) Square a parallelogram, a triangle, a trapezoid (and know why these were listed in this order!) Triangulated figures can be squared. (Know the area of a regular polygon, and its uses in approximating the area of a circle. Squaring curved figures can sometimes be solved easily, and not for other figures. Understand the Lune of Hipoocrates and note how the dissection proof (it is equal to the square on its lesser radius ...) depends on fact that "quarter circles are cool". Review the assigned reading on pi. Review how the Method of Exhaustion is used to 1. Establish that circle are "cool" (i.e. the area of two circles are in the proportion of the squares on their radii). 2. Show how Eudoxus' Theory of Proportions (what is it and what properties do proportions have? ) applies to (1) and conclude what pi meant to the ancients. 3. How does this differ from the way we understand pi today? (Review Weierstrass's defintition of a real number.) 4. Show how Exhaustion computes the area of a parabolic segment precisely. (What are you assuming for this? geometric series) Pause ... an aside on volumes: Be able to demonstrate that the volume of a triangular pyramid (a.k.a. tetrahedron) is 1/3 base x height. Steps: a. Use a shearing argument to "straighten" the pyramid out. b. Show how a triangular prism decomposes into 3 equal pyramids. c. Don't forget to conclude the theorem from a and b. Know how to apply Thales' Theorems 1. The diameter of a circle subtends a right angle from a periferal point. 2. The altitude of a right triangle dissects it into 3 similar triangles to 1. The Pythagorean Theorem 2. Squaring of a rectangle (geometric versus arithmetic mean) 3. ...more? Archimedes' Method 1. What is it? (Method of discovering area and volume formulas from slicing figures into "lamina" and comparing their areas (lengths) and how they balance according to a. The Law of the Lever (what is it?) b. The Law of the Centroid (what is it?) 2. We have studied three examples of this Method, two for computing the volume of spheres and one for the quadrature of the parabola. The Parabola 1. Draw it accurately. 2. Identify and lable all of its "vital statitistics". 3. What four properties were used by Archimedes. 4. Derive one or two of them from a modern defintion. 5. Know at least 2 more definitions of the parabola. Advice for the test: Understand the difference between being asked to 1. State a theorem 2. Name a theorem 3. Define a method 4. Name a method 5. Prove a theorem 6. Give an example illustrating such a proof. And don't answer the "wrong" question, by naming when asking to state etc. 7. An accurate picture is one from which the assertion it illustrates really looks true.