(2) Much more subtle is the line of thought suggested by Professor Royce in the Supplementary Essay appended to his book, _The World and the Individual_, First Series. Professor Royce admits the indefinite regress as an inevitable consequence of the reduction of the world to terms in relation, but denies that it affects the soundness of the reduction. On the contrary, he regards it rather as a proof of the positive correctness of the interpretation of existence which gives rise to it. His argument, which is based upon the modern doctrine of infinite series, may be briefly summarised as follows:—It is a recognised characteristic of an infinite series (and of no others) that it can be adequately “represented” by a part of itself. That is to say, if you take any infinite series you please, you can always construct a second series such that it consists of a selection, and only of a selection, from the terms of the first series, and that every term is derived from and answers to the corresponding term of the first series according to a definite law. And this second series, as it is easy to prove, is itself infinite, and therefore capable of being itself represented adequately in a third series derived from it in the same manner as it was derived from the first, and so on indefinitely. For instance, let the first series be the infinite series of the natural integers 1, 2, 3, 4, ... then if, _e.g._, we construct a second series, 1^2, 2^2, 3^2 ... of the second powers of these integers, the terms of this second series are derived by a definite law from those of the first to which they correspond, and again they constitute a selection out of the terms of the first series. Every one of them is a term of the first series, but there are also terms of the first series which are not repeated in the second. Again, if we make a third series from the second in the same way as the second was made from the first, by taking the terms (1^2)^2, (2^2)^2, (3^2)^2, and so on, the terms of this third series fulfil the same conditions; they correspond according to a fixed law with the terms of the second, and are also themselves a selection from those terms. And thus we may go on without end to construct successive infinite series each of which “adequately represents” the preceding one. And we are led into this indefinite regress by the very attempt to carry out consistently a single definite principle of correspondence between our original infinite series and its first derivative. In constructing the first derived series in our illustration 1^2, 2^2, 3^2 ... we necessarily also construct the series (1^2)^2, (2^2)^2, (3^2)^2, ... and the other successive derivatives. Therefore Prof. Royce claims that any consistent attempt to make an orderly arrangement of the terms of an infinite whole _must_ lead to the indefinite repetition of itself. Hence that each term of every relation on analysis turns out itself to consist of terms in relation, is no