Two problems are treated in this paper, i.e., (1) given the shape of the milling cutter tooth or of the hob tooth, to find out what is the resulting shape of the worm or gear surface; (2) given the desired shape of a worm or gear tooth, what shape of the milling or hobbing tooth will produce it. These problems are fundamental to all cutter and hob design. Except for special cases, the solution of these problems obtained in the following treatise is believed to be new. The general equations which enable one to determine exactly the relation between the shape of the cutting tool and the shape of tooth produced are obtained in the present paper by going back to the fundamentals of the motion of a milling cutter or a hob and the motion of the worm, or gear part being cut. With these equations available it is possible to design precision tools for any worm or gear-cutting job, thus eliminating guesswork in the design, and possible delays, such as are involved in “cut-and-try” shop methods.