The Fourier analysis is applied to the deviations from divisibility, so that readers experience the first instances of mathematical dualities. Through this duality, we are able to see the congruence issues much better than solely dealing with them naively. The theory of quadratic binary equations yields the quadratic reciprocity law. Gauss’ eight proofs, save for the fifth, of it, as well as Legendre’s, are fully explained from various angles, which, as a whole, indicates the modern algebraic number theory and signifies the theory of primes in arithmetic progressions. We shall also be led inevitably to the history of the theory of algebraic equations, which will be presented with some concrete examples.