The science of growth is arithmetic in constitution. By drawing on the technical literature (from the 18th century onward), this monograph seeks to expose the deep combinatorial foundations of the theory. It contains ideas of mathematical logic, including the finitary concept of normal tree, and the feather diagram of a hierarchy. Central to the book is a proof of Birkhoff's descending chain condition, as it arises in a diagonalization of mathematical induction. At no point is the vexed axiomatic of choice used. Contingent on the above reductions are, for example, an acceptable Gentzen-type proof of freedom from contradiction for first-order arithmetic and combinatorial relations apparent at the centre of logic. These include an infinite product first studied by A. Cayley, and a subtle property of the Cantor normal form, contained in an exponential identity of G. Polya.