Conjecture and proof

Pt. I. Proofs of impossibility, proofs of nonexistence

Proofs of irrationality

Elements of the theory of geometric constructions

Constructible regular polygons

Some basic facts about linear spaces and fields

Algebraic and transcendental numbers

Cauchy's functional equation

Geometric decompositions

Pt. II. Constructions, proofs of existence

Pigeonhole principle

Liouville numbers

Countable and uncountable sets

Isometries of Rn̂

Problem of invariant measures

Banach-Tarski paradox

Open and closed sets in R: the Cantor set

Peano curve

Borel sets

Diagonal method.