Math 305, Section 1 • Mathematics from a Historical Perspective • UNM, Fall 2008
Archimedes’ determination of circular area Question Set 5 Due: 10.8.2008
1. Refer the right triangle sketched on Fig. 4.3 on page 93 of Dunham’s text, and answer Items (a) through (c) below. (a) Explain what the length C of the triangle’s base represents, geometrically speaking. (b) Prove that the number C is not constructible. (Hint: start by assuming that it is, then arrive at a contradiction using the fact that π is transcendental.) (c) How did Archimedes managed to construct the base of the triangle in Fig. 4.3 using just a compass and a straight edge? Explain. 2. Present the proof by contradiction you did in Part (b) above as a Modus Tollens-type argument. That is write it as: [(A ⇒ B)∧∼ B] ⇒∼ A, and specify precisely what statements A, B, ∼ A and ∼ B say in your argument. 3. Explain what “double reductio ad absurdum” means, and state (precisely) two results of Archimedes whose proof is based on this construct. Pick one of those two results and explain how exactly did the aforementioned technique entered the proof in question. 4. Archimedes showed that a sphere can be contained in a cylinder in a 3 : 2 ratio. What is the precise meaning of this statement? 5. Your goal here is to determine Archimedes’ roughest upper bound estimate for π using a regular hexagon circumscribed about a circle of radius r.
(a) Calculate the perimeter of a circumscribed regular hexagon, assuming the inscribed circle has radius r. Show step-by-step work. Use the figure below (adding appropriate labels, etc.) to aid your explanation.
O r
(b) Calculate the upper estimate for π arising from the perimeter computation you did above. Of course, you need to show why whatever calculation you do below is such an upper estimate. (For example, you could use a string of inequalities like the ones used by Dunham to describe Archimedes’ lower bound estimates for π using inscribed polygons.)
6. You have a regular 2n-gon (a polygon with 2n sides) of area A circumscribed about a circle of area B. Give the exact perimeter of the regular polygon in terms of A and B. How long is each of the polygon’s sides? 7. What exactly was the Almagest, and how is it historically linked to Archimedes? 8. Dunham aptly identifies and characterizes four different phases in connection with the estimation of the value of π to varying degrees of accuracy. Write a summarized narrative of these four phases. You should be specific as far as dates, mathematicians, their work and contributions. You should also clearly indicate what distinguished each of these four phases from the remaining three. The emphasis should be on the mathematics. The length should be about one page. (over) Optional Questions
9. Explain how a regular 96-gon may be used to provide both upper and lower estimates for π a-la-Archimedes. Be precise. Do this without (!!!) calculating any actual estimates. 10. Beyond stating and proving results in pure mathematics, what other achievements was Archimedes known for? Name at least two; be very specific. 11. How were Eudoxus and Archimedes historically connected through mathematics? Be as specific as possible. 12. A certain sphere has volume equal to 57. Specify the dimensions of a cone whose volume fits exactly 4 times within the volume of the given sphere. Explain.