Mathematics-- the Music of Reason

algebraic algebraic geometry algebraic topology analysis angle Appendix applications arbitrary Archimedes areas of mathematics arithmetic axiomatic axioms bijection calculation called Cauchy Chapter circle classical coefficients commutative complex numbers concept congruence construction curve Dedekind defined definition denoted Descartes Diophantus element equation Euclid Euclid's Elements Euclidean Euclidean geometry Euler example exists fact Fermat Figure finite formula functions Galois Gauss Gaussian integers geometry greatest common divisor Greeks Hilbert idea infinite set isomorphism known length magnitudes mathe mathematical objects mathematicians method metric space multiplication natural numbers nested intervals nineteenth century non-Euclidean notation number theory obtained permutations plane Plato Poincaré polygons polynomial possible prime numbers problems proof properties proved rational numbers ratios real numbers relation satisfies segment sequence solution space square structure subgroup subset theorem topology triangle variables vector write