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Of all the many types of machine elements which exist today, gears are among the most commonly used. The basic idea of a wheel with teeth is extremely simple, and dates back several thousand years. It is obvious to any observer that one gear drives another by means of the meshing teeth, and to the person who has never studied gears, it might seem that no further explanation is required. It may therefore come as a surprise to discover the large quantity of geometric theory that exists on the subject of gears, and to find that there is probably no branch of mechanical engineering where theory and practice are more closely linked. Enormous improvements have been made in the performance of gears during the last two hundred years or so, and this has been due principally to the careful attention given to the shape of the teeth. The theoretical shape of the tooth profile used in most modern gears is an involute. When precision gears are cut by modern gear-cutting machines, the accuracy with which the actual teeth conform to their theoretical shape is quite remarkable, and far exceeds the accuracy which is attained in the manufacture of most other types of machine elements. The first part of this book deals with spur gears, which are gears with teeth that are parallel to the gear axis. The second part describes helical gears, whose teeth form helices about the gear axis.