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.
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.