MAT336: History of Mathematics

Spring 2026

Study Guide: Renaissance Mathematics and Review

Note: This is a study guide. The quiz will consist of three or four 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: Renaissance Mathematics

  1. Before the printing press, mathematical knowledge spread through hand-copied manuscripts. Describe two concrete consequences this had for how mathematics developed — and how printing changed them.
  2. Give one cause of mathematical change in the Renaissance and explain the specific mechanism of change.
  3. Why do Renaissance paintings often look more three-dimensional than medieval paintings? Give one mathematical reason.
  4. What is a vanishing point? Explain using floor tiles, ceiling beams, walls, or table edges.
  5. Why were algorithms with Hindu-Arabic numerals more efficient than those with Roman numerals? Give one reason connected to calculation, and one reason the system faced resistance in Europe.
  6. What was François Viète’s main contribution to algebraic notation? Explain the difference between solving one numerical problem and writing a general relation.
  7. In the Borromini corridor, from one column to the next, the ratio (column height) / (spacing to the next column) stays constant. Why is this the key geometric condition that makes the corridor appear longer than it is?
  8. What is a depressed cubic, and how was it used in solving cubic equations?
  9. Renaissance mathematicians treated x³ + ax = b and x³ = b + ax as different equations requiring separate methods. Why?
  10. When solving x³ = 15x + 4, Cardano’s formula involves √(−121) even though x = 4 is real. What was Bombelli’s contribution and why was it important for mathematics?

Part 2: Review — Euclid’s Elements

  1. What is the axiomatic method?
  2. In Euclid’s system, what are the three types of starting points — definitions, common notions, and postulates? Explain the role of each and give one example of each.
  3. Explain the key idea in the proof in the Elements that “prime numbers are more than any collection of primes.” What does the proof do with the product of primes plus one?
  4. The Pythagorean relationship appears in Mesopotamia, India, China, and Greece. What makes the treatment in the Elements different from those earlier appearances?

Part 3: Review — Archimedes and Conic Sections

  1. Archimedes’ goal in Measurement of the Circle was to find upper and lower bounds for π. Why did he double the number of sides (6 → 12 → 24 → …) rather than increase by one each time (6, 7, 8, …)? Answer in terms of the mathematical methods available to him.
  2. 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. What did scholars conclude Archimedes was trying to count, and why is this interpretation not completely certain?
  3. What are conic sections? Describe each one — ellipse, parabola, hyperbola, and circle — using the intersection of a plane and a cone.
  4. What types of curves can appear as the boundary of a flashlight beam on a wall? Explain geometrically why these curves arise.

Part 4: Review — China, Diophantus, Fermat

  1. What is the Nine Chapters on the Mathematical Art? Roughly when was it compiled, and what kind of mathematical text is it?
  2. Describe the rod numeral system. Is it additive, multiplicative, ciphered, positional, or some combination? Did it have a zero, and if so how was it represented?
  3. Problem II.8 of Diophantus’s Arithmetica asks: “Divide a given square into two squares.” Fermat read this problem in a 1670 edition and wrote a famous note in the margin. What did he write?
  4. State Fermat’s Last Theorem precisely. Who proved it and when?

Quiz Problem Rubric

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

Notes