The number of representations of a number by various forms

We start with classical results due to Jacobi, Dirichlet and Lorenz which give the number of representations of a number by various quadratic forms in terms of divisor functions. We express the results in terms of products and series, then systematically dissect the products and series to obtain further results of similar type. In particular, we obtain results of Legendre and Ramanujan, as well as many others.