consisting of classes; suppose the number of terms in each of these classes is given. How shall we define the product of all these numbers? If we can frame our definition generally, it will be applicable whether
is finite or infinite. It is to be observed that the problem is to be able to deal with the case when
is infinite, not with the case when its members are. If
is not infinite, the method defined above is just as applicable when its members are infinite as when they are finite. It is the case when
is infinite, even though its members may be finite, that we have to find a way of dealing with.
The following method of defining multiplication generally is due to Dr Whitehead. It is explained and treated at length in Principia Mathematica, vol. I. * 80 ff., and vol. II. * 114.