MAT336: History of Mathematics

Spring 2026

Study Guide: Quiz 7

Euclid’s Elements

Note: This is a study guide. The quiz will consist of three or four questions covering the material below. If you understand the ideas and facts in the non-computational questions and can work through the computational problems, you will be well prepared.

Key Themes to Understand

Suggestion to better understand Euclid: Play Euclidea — a game based on straightedge-and-compass constructions.

Part 1: Quiz-Style Questions

The Axiomatic Method and the Structure of the Elements

  1. What is the axiomatic method?
  2. In Euclid’s system, what are the three types of “starting points” (definitions, common notions, postulates)? Explain the role of each. Give one example of each.
  3. Write down the Elements’ definition of right angle. Postulate 4 states “all right angles are equal.” Why does the Elements need to state this explicitly when it seems obvious? Hint: Read carefully the definition of right angle.
  4. What does the parallel postulate state? Why is Postulate 5 (the parallel postulate) considered different from the first four postulates?
  5. What do Postulates 1–3 allow you to construct? Why does the Elements limit itself to these tools?

Proposition I.1 — Construction and Proof

  1. Explain the construction of an equilateral triangle on a given segment (Proposition I.1), as given in the Elements. Describe each construction step and justify it using the appropriate definitions, common notions, and postulates.
  2. Prove by the axiomatic method that the triangle constructed in Proposition I.1 is equilateral. Justify each step using definitions, common notions, and postulates.
  3. What gap exists in the proof of Proposition I.1 in the Elements? How is this gap addressed in modern geometry?
  4. The “all triangles are isosceles” argument follows valid-looking logical steps. Where does this argument fail to follow the axiomatic method?

Number Theory in the Elements

  1. Explain the key idea in the proof in the Elements that “prime numbers are more than any collection of primes.” Hint: What does the proof do with the product of primes plus one?
  2. Why doesn’t the Elements use the word “infinite” when discussing the result that prime numbers exceed any finite collection?
  3. In the proof that there are infinitely many primes, a new prime is obtained using three given primes. Find a new prime by the method in the Elements using 41 and 2 and another prime of your choice.

The Pythagorean Theorem

  1. The Pythagorean relationship appears in Mesopotamia, India, China, and Greece. What makes the treatment in the Elements different from those?
  2. Summarize the proof of the Pythagorean Theorem in the Elements (as discussed in class), highlighting the main steps and constructions.

Part 2: Reflection Questions

  1. What is the difference between actual and potential infinity in the context of ancient Greek mathematics?
  2. What are the consequences of treating postulates as truths vs. treating axioms as assumptions (modern view)?
  3. According to Proclus, Euclid told Ptolemy “there is no royal road to geometry.” What did he mean? Is there a “royal road” to geometry today?
  4. Morley’s trisector theorem (1899) and Euler’s polyhedral formula (1750s) can both be proven using only Euclidean-style reasoning. What does this tell you about whether the Elements “finished” geometry?
  5. Proposition IX.20 (about the infinitude of primes) is never cited again in the Elements. What might that say about its role?
  6. Some definitions in the Elements — for instance, “a point is that which has no part” — are never invoked in any proof, unlike, for instance, the definition of circle. What does this say about these definitions?

Part 3: Construction Questions

  1. In Proposition 14 of Book 2 of Euclid’s Elements, a construction for squaring a “rectilinear figure A” is given. Perform the construction in this GeoGebra activity starting with a rectangle BCDE (instead of a general rectilinear figure A). That is, in the fourth sentence of the proof (“Then, if BE equals ED, then that which was proposed is done, for a square BD has been constructed equal to the rectilinear figure A.”). Construct a square of side length ED, and check in GeoGebra using the Area tool that the areas of rectangle BCDE and the square of side ED are equal. Send a screenshot for credit.
  2. Consider a right triangle T whose legs have length 2 and 9. Divide T into polygonal pieces from which a square can be assembled. What is the side length of this square? Hint: First divide the triangle into pieces that can be reassembled into a rectangle. You can use cm or any other unit of your choice. “Make” your solution on paper or in GeoGebra. Explain how you obtained the solution and provide photos or screenshots of the pieces you made. Include measurements marked on your drawing. (If you complete this successfully, you will have produced a quadrature of a right triangle!)
  3. Given A, B, C on a line with B between A and C, and BC = 1, construct a segment of length √AB. Use straightedge and compass (or GeoGebra). Show and explain your steps. What proposition in Euclid’s Elements does this resemble? (Note: You can use AI for this last question, but make sure you check your answer. Here is a copy of Euclid’s Elements.)

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