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There are more than a billion mechanical machines in our world of six billion humans. These machines are the source of both marvel and mayhem in the life of our planet. This book is about the evolution of these machines and the inv- tors and engineers who created them from the early Renaissance to the early 20th century. I have chosen two personalities who are icons of these two machine ages, Leonardo da Vinci [1452–1519] and Franz Reuleaux [1829– 1905], recognizing both the cadre of machine designers who in?uenced them as well as those who were in?uenced by the accomplishments of these two engineers. A major thesis of this book is that the evolution of machine design methodology, from the intuitive methods of the workshop to the math-based, engineering science analysis and synthesis of modern industrial design, was of equal achievement as the creation of the marvelous machines themselves. In the past two decades there has been increasing interest in rational me- ods of design from topology and optimization theories to genetic algorithms. In the teaching of design at the novitiate level, the importance of design - cles and iteration is emphasized. 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On the mathematics side of nonlinear dynamics, it is now recognized that spatio-temporal problems, hysteretic and time delay problems are the new frontiers in this field. Also the term complexity has been added to the lexicon of chaos theory to describe the dynamics of many interacting sub systems which can exhibit self organization and evolution. 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