It has been hard for me to escape the imprint of my early, strong, but scattered trains of thought. There was, at the beginning, little to go by; and I saw no clear way to go. This book is accordingly filled with internal tensions that are not, as yet, fully annealed. Subsequent writers may re-present the work, explaining it in a simpler way. Others may simply invert it. I mean by this that, by writing it backwards, from its found ends {practical machinable teeth) to its tentative beginnings (dimly perceived geometrical notions), one might conceivably write a manual, not on how to understand these kinds of gears, but on how to make them. Indeed a manual will need to be written. If this gearing is to be further investigated, evaluated and checked for applicability, prototypes will need to be made. I wish to say again however that my somewhat convoluted way of presenting these early ideas has been inevitable. It has simply not been possible to present a tidy set of explanations and rules without exploring first (and in a somewhat backwards-going direction) the complexities of the kinematic geometry. There remains, now in this book, a putting together of primitive geometric intuition, computer aided exploration of certain areas, geometric explanations of the discovered phenomena, and a loose sprinkling of a relevant algebra cementing the parts together.