This section is from the book "The Tinman's Manual And Builder's And Mechanic's Handbook", by Isaac Ridler Butt. Also available from Amazon: The Tinman's Manual And Builder's And Mechanic's Handbook.
[Drawn for this work by L. W. Truesdell, Tinman, Owego, N. Y.
To enable machinists to enlarge or reduce machinery wheels without changing their respective motion.
First, describe two circles AB and CD the size of the largest wheels Which you wish to change to a large or small machine, with the centre P of the smaller circle CD on the circumference of the large one AB; then draw two lines LM and NO tangent to the circles AB and CD, and a line IK passing through their centres P and R; then if yon wish to reduce the machine, describe a circle the size you wish to reduce it to; if one-half, for example, have the centre Q one-half the distance from R to S and describe the circle EF, and on its circumference T as a centre, describe a circle GH, allowing their circumferences to touch the tangent lines LM and NO, which will make the circle EF one-half the size of the circle AB, and GH one-half the size of CD; therefore EF and GH are in the same proportion to each other as AB and CD.
If you wish to reduce one-third, have the centre Q one-third the distance from R to S; if one-fourth have the centre Q one-fourth the distance from R to S, and so on. This calculation may be applied beyond the centre R for enlarging machine wheels, which will enable you to make the alteration without changing their respective motion.