178 CHAPTER 9 9.2 Cogs and Millwrights Suppose that we wish to connect two shafts together with gears so that one runs at twice the speed of the other. Suppose further that the exact ratio of two to one may not be critical and some latitude around it is allowed. One way of doing this is to have a 20-toothed gear meshing with one of 40 teeth, giving a ratio of exactly two to one. This implies that a tooth on the larger wheel will always mesh with one particular tooth on the smaller. Looking at it from the point of view of the smaller wheel, any one of its teeth will always mesh with the same two diametrically opposite teeth on the larger wheel. This happens at every turn of the 40-toothed gear, and for every two turns of the smaller one. This point has been laboured so that the problem caused by one of the gears having an imperfect or badly formed tooth can be readily appreciated. The extra wear or damage will be confined to one or two teeth on the other gear and there is the possibility of jerky or erratic transmission. In order to overcome this localized wear an extra tooth could be added, a hunting cog, so that a ratio of 20:41 could be used instead. By this means wear will be distributed evenly among all teeth as only after 41 turns of the small wheel will the same teeth be meshing again. As it happens 41 is a prime number, but what is crucial is that the number of teeth should be prime to each other, as for example 21:38, and not that the number of teeth on any gear is necessarily prime. This even wear has been gained at the expense of losing the exact ratio of 2:1. The early millwrights, using the wooden gearing with inserted teeth used in wind- or water-driven machinery, recognized the advantage of this as they rarely needed an exact ratio of, say, 2:1. Exact ratios are possible where a train of gears is used, and to illustrate this we need to make two quite realistic and practical assumptions: the minimum number of teeth is 20, and the gear ratio should not exceed 6:1. We take as an example a required exact ratio of 360:1 and first note that 6 4 > 360 > 6 3 .