MAT336: History of Mathematics

Spring 2026

Study Guide: Quiz 8

Archimedes

Note: This is a study guide. The quiz will consist of two or three questions covering the material below. If you understand the ideas in the conceptual questions and can work through the computations, you will be well prepared.

Part 1: Quiz-Style Questions

Archimedes and π

  1. 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. His first step was to compute the perimeters of an inscribed and a circumscribed regular hexagon in a circle of radius 1.

    Sketch a circle of radius 1 with an inscribed and a circumscribed regular hexagon.

    1. Using your sketch, explain why the perimeter of the inscribed hexagon is 6. 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.)
    2. The side of the circumscribed hexagon is c. What is the perimeter of the circumscribed hexagon? (Make sure to explain where you used the fact that the hexagon is circumscribed.)
    3. Archimedes started with a hexagon, not a pentagon or a square or any other polygon. Give a mathematical reason why the hexagon allows an exact computation of the perimeter in a circle of radius 1.
    4. To improve his bounds, Archimedes doubled the number of sides: 6 → 12 → 24 → 48 → 96.
      1. Explain why increasing the number of sides narrows the bounds for π, and why this improves the approximation of the circumference.
      2. Give a plausible reason for why Archimedes stopped at 96 sides rather than continuing to 192.
      3. Why did Archimedes double the number of sides (6 → 12 → 24 → …) rather than increase by 1 each time (6, 7, 8, …)? Answer in terms of the mathematical methods available to him.

Archimedes’ Method

  1. Describe the main idea of Archimedes’ Method. Did Archimedes consider The Method to be a way of doing rigorous proofs?

Archimedes’ Favorite Result: The Sphere

    1. State the relationship between the volume of a sphere and the volume of its circumscribed cylinder.
    2. Describe the idea behind the argument (as discussed in class) that shows the relationship between these volumes. Illustrate your argument with the appropriate sketch.
    3. State Cavalieri’s principle. How does it enter Archimedes’ argument about the volumes of the sphere and the cylinder?

The Archimedes Palimpsest

    1. What is a palimpsest?
    2. What is the Archimedes Palimpsest?
    3. Name two works contained in it (describe their content; you do not need to recall exact titles).
    4. Why was the recovery of The Method significant?

The Stomachion

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

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

Conics and the Flashlight

    1. What classical Greek problem motivated the study of conic sections?
    2. What types of curves can appear as the boundary of a flashlight beam on a wall?
    3. Explain geometrically why these curves arise.

Part 2: Reflection Questions

  1. Archimedes asked that a sphere enclosed in a cylinder be carved on his tomb. Based on what you know about his result on the volume of the sphere, why do you think he considered this his most important discovery?
  2. The Stomachion was known only from brief references for almost two thousand years. When the Palimpsest was properly read, it turned out to contain a combinatorics problem — a field that only developed seriously in the 20th century. What does this suggest about how we recognize “important” mathematics?
  3. Archimedes used The Method — a non-rigorous heuristic — to discover results privately, while presenting only rigorous proofs publicly. Does this change how you think about the relationship between mathematical discovery and mathematical proof?

Quiz Problem Rubric

Points Criteria
3 Correct answer with reasoning/work shown
2 Partially correct with some reasoning shown
1 Correct answer without reasoning/work OR significant attempt with some understanding
0 Incorrect or blank

Notes