The automation of mathematical reasoning has been an important topic of research almost since computers were invented. The new technique of rippling, described here for the first time in book form, is designed to be an approach to mathematical reasoning that takes into account ideas of heuristics and searching. Rippling addresses the problem of combinatorial explosion which has proved a huge obstacle in the past, and the book offers a systematic and comprehensive introduction to this and to the wider subject of automated inductive theorem proving.