The sphere of change, however, one would answer, includes all nature, and science in its discoveries acts on the hypothesis that these steps of change may be infinitely divisible. Royce held to it firmly that any consistent attempt to make an orderly arrangement of the terms of an infinite whole must lead to the Indefinite Regress. And he further shows the connection with the fact that an infinite series can be adequately represented by a part of itself. In the Boyle Lecture, delivered in Oxford in 1908, on the properties of radium, two facts emerged which show that the Indefinite Regress is now recognised in science. First, that in the region of experiment we become aware of groups of elements allied to radium, which seem, in the number of individuals in their groups, to follow a simple arithmetical progression. Secondly, that radio-active elements lose in activity at a certain rate, which always represents an exact proportion of the mass which remains. The tremendous disintegrating force slackens in exact relation to the time which passes, so that the smaller the morsel the less the relative loss of mass. Here, then, is the Indefinite Regress. In the world of fact as well as of ideas we are dealing with aspects of Infinity.[25] II. INFINITE SERIES There are other aspects of Infinity which we can get at by studying series, and which in the conception of series of series give strength and point to the philosophic conception of an Absolute. Prof. C. Keyser develops this thought, and shows (in two recent articles, January and April 1909, in the _Hibbert Journal_) that certain theological dogmas, such as the doctrine of the Trinity, and certain attributes of the Divine Being, such as Omniscience and Omnipresence, are entirely conceivable by the human mind if regarded without the paralysing limitations of the Finite. He shows that in our mathematical formulæ which have to do with infinite series we have the exact replica of what to the lay, non-mathematical mind seem to be the paradoxes of the Athanasian Creed. He first shows that in a mathematical analogy points of view about an Infinite Being, even if partially discordant, may all be true if regard is had to His Infinity.[26] Further, he shows that certain assumptions, such as _the whole is greater than its part_, are inapplicable to Infinite Being. The conception of a Trinity in Unity in which “none is afore or after other, none is greater or less than another, but the whole three persons are co-eternal together and co-equal” is rationally conceivable by the mathematician who is familiar with the theory of manifolds.[27] We have, he shows, three infinite manifolds:-- E of the even integers. O of the odd ones. F of the fractions having integers for their terms. No two of these have a single element in common, yet the three together constitute one manifold M, that is exactly equal in wealth of elements to _each_ of its infinite components.