Study Guide: Quiz 8

Greek Mathematics after Euclid

Archimedes

1. Approximately when and where did Archimedes live? Name two areas of mathematics in which he made important contributions.
2. Explain Archimedes’ method for approximating π using inscribed and circumscribed regular polygons. Why does increasing the number of sides improve the approximation?

3. (This question is long and is therefore worth two quiz questions.) Recall that Archimedes’ goal in Measurement of the Circle was to find upper and lower bounds for the ratio of the circumference of a circle to its diameter (the ratio we now call π). He began with inscribed and circumscribed regular hexagons.

   1. Sketch a circle with an inscribed and a circumscribed regular hexagon.
   2. Using your sketch, explain why the perimeter of the inscribed hexagon is 6r, where r is the radius of the circle. Your explanation should carefully justify why the relevant triangles are equilateral. (Make sure to explain where you used the fact that the hexagon is inscribed.)
   3. What is the perimeter of the circumscribed hexagon, in terms of r? Compute it without trigonometry, as Archimedes did. (Make sure to explain where you used the fact that the hexagon is circumscribed.)
   4. Why was the hexagon a particularly convenient starting point for these calculations? Explain in terms of its geometry and the mathematical methods available to Archimedes.
   5. To improve his bounds, Archimedes doubled the number of sides: 6 → 12 → 24 → 48 → 96.
      • Explain why increasing the number of sides narrows the bounds for π.
      • Give a plausible reason why Archimedes stopped at 96 sides rather than continuing to 192.
      • Why did Archimedes double the number of sides rather than increase by 1 each time (6, 7, 8, …)? Answer in terms of the mathematical methods available to him.
4. Archimedes obtained the bounds 3 + 10/71 < π < 3 + 1/7 (in modern notation). Why is it significant that Archimedes established upper and lower bounds, rather than simply giving an approximation?
5. Archimedes showed that a circle has the same area as a certain right triangle. Describe this triangle in terms of the circle’s radius and circumference. Explain how this result relates the formulas for circumference and area.
6. A sphere of radius 3 is inscribed in a cylinder. Use Archimedes’ result about a sphere and its circumscribed cylinder to calculate the volume of the sphere, without using the modern formula for the volume of a sphere.
7. Describe how comparing the cross-sections of a hemisphere and of a cylinder with a cone removed explains Archimedes’ result about the sphere and its circumscribed cylinder. Include a sketch.
8. State Cavalieri’s principle. How can we use this principle to explain Archimedes’ result about the volumes of the sphere and the cylinder?
9. Describe the main idea of Archimedes’ Method. Did Archimedes consider The Method to be a way of producing rigorous mathematical proofs?
10. What is the Archimedes Palimpsest? Explain how the manuscript was reused and why its recovery matters for the history of mathematics.

11. The Stomachion is a 14-piece dissection puzzle known since antiquity. When the Palimpsest was reexamined in the early 2000s, scholars proposed an interpretation of what Archimedes was investigating.

    1. What did scholars propose Archimedes was trying to count?
    2. Why is this interpretation not completely certain?

Conics

12. What is the mean proportional (geometric mean) of two positive numbers? Calculate the mean proportional of 3 and 12.
13. Describe the semicircle construction of the mean proportional of two given segments. Explain why the construction works, using properties of right triangles. Sketch a figure.

14. A traditional account connects the early study of conic sections with one of the classical Greek construction problems, although the precise historical development is uncertain.

    1. Which classical problem is involved?
    2. Explain the mathematical path connecting this problem to conic sections, including the role of mean proportionals.
15. What types of curves can appear as the boundary of a flashlight beam on a wall? Explain geometrically why these curves arise.
16. Describe how the intersection of a plane with a cone can produce a circle, ellipse, parabola, or hyperbola. What determines which curve is obtained?
17. The side of a cone makes an angle α = 30° with its axis. A plane makes an angle β with the axis, where β = 20°, 30°, 60°, or 90°. Name the curve in each case. Assume the plane is positioned so that it intersects the cone in a nondegenerate conic.

Diophantus

18. Approximately when and where did Diophantus live? What kinds of mathematical problems appear in his Arithmetica? How does its style differ from Euclid’s Elements?
19. Fermat wrote his famous claim, now known as Fermat’s Last Theorem, in the margin of his copy of Diophantus’ Arithmetica. Paraphrase his claim. Beside which problem did he write it?