## Day 26: Exam 2

### Coverage

This second exam will cover the material since the first exam.  That is, Greek and Hellenistic mathematics.  Some of the topics you should be familiar with are:
• geography of the Mediterranean littoral: Miletus, Samos, Athens, Alexandria, Syracuse, Pergamum;
• dates at which various centers flourished: Plato's Academy, Alexandria;
• location, date, sources and significance of the contributions of: Plato, Eudoxus, Theaetetus, Euclid, Archimedes, Apollonius;
• Greek numerical systems;
• 'fractions';
• Greek arithmetic and the abacus;
• arithmetical tables;
• Pythagorean number theory;
• incommensurability: side and diagonal arguments; classification of 'irrationals';
• numbers and magnitudes;
• ratio and proportion;
• axiomatic presentation of mathematics;
• Platonic mathematical philosophy: the nature of forms;
• the Platonic curriculum: use and purpose of mathematics;
• actual usage and purpose of mathematics in the classical world;
• Netz's analysis of classical mathematics and mathematicians;
• sources and transmission in the classical world: how mathematicians learned and corresponded;
• sources and transmission from the classical world to modern times;
• rhetoric and style in mathematics;
• general contents of the Elements;
• Euclid's definitions, common notions and postulates;
• plane geometry from Elements I;
• construction of regular solids in Elements XIII;
• number theory in the Elements;
• Archimedes: scope of work;
• Archimedes: main results;
• Archimedes: main techniques of proof;
• Archimedes on spheres, circles and parabolas;
• The Method and its connection to Archimedes' mechanical investigations;
• Apollonius: scope of work;
• Apollonius: style of proof;
• Apollonius on parabola, hyperbola and ellipse;
• properties of conics;
• anything else you can think of.

### Format

Possibly some simple arithmetical problems.

A selection of IDs and definitions.  Given a name, artifact or topic, identify and describe as fully as possible, including date, location, contents and significance as appropriate.
For example: Plato, Archimedes, Apollonius, the Library and Museum at Alexandria, acrophonic system, platonic solids, incommensurability of side and diagonal, Euclid's parallel postulate, method of exhaustion, Platonic Forms.

Some short answer questions.  These will be short discussion questions, often focussing on development of an idea or topic, or comparison of ideas.

Examples:  Discuss the use of diagrams in Greek geometry.
Discuss the Greek numeration systems.
Discuss the role of computation in Greek mathematics.
Discuss the idea of definitions in Greek mathematics.
Explain Platonic mathematical philosophy.
Discuss the role of patronage in Greek mathematics.
Compare Euclidean and Archimedean mathematics.  What do you see as the main concerns and interests of each?
Discuss Archimedes' use of exhaustion.
Compare Apollonian and Archimedean mathematics.  What do you see as the main concerns and interests of each?
Discuss Euclidean number theory?  How does this compare with Pythagorean number theory?
Discuss the concept of incommensurability in Greek mathematics.
Proof critiques.  I will give you one or two proofs from work we have studied and expect you to critique them, explaining the significance of the results, discussing the salient techniques employed and showing how a particular result fits into the larger corpus.
Proofs of alternate cases: I may give you one or two proofs that only treat specific cases and ask you to provide a proof of a different (specified) case.
Euclidean Proof: Prove a Euclidean Proposition from the Statement: I will choose one of Elements I, 1; I, 2; I, 35; VII, 19; IX, 20.

On to Day 27.

Up to Ancient and Classical Mathematics