This is true even where we merely count a number of qualitatively equivalent units in order to ascertain their sum. It is their positive character of being qualitatively equivalent which makes it permissible in this case to take any one of them indifferently as first, any other as second, etc. Whenever you apply the numerical series to the arrangement in order of the qualitatively dissimilar, the nature of your material as related to the character of your special interest in it decides for you what you shall call first, second, third, etc. Footnote 89: As to the possibility of relations which are in this sense external to their terms, see B. Russell, _The Philosophy of Leibniz_, p. 130, and the articles by the same writer in _Mind_ for January and July 1901. Footnote 90: See the elaborate discussion of the relational scheme implied in any assertion of difference in Royce, _The World and the Individual_, Second Series, lect 2. Footnote 91: Bradley, _Appearance and Reality_, chap. 3. Compare also chap. 15, “Thought and Reality.” Footnote 92: The reader who desires further knowledge of the researches in the theory of Numbers upon which Prof. Royce’s doctrine is based, may profitably consult Dedekind, _Was sind und was sollen die Zahlen_, and Couturat, _L’Infini Mathématique_. Footnote 93: Professor Royce’s own illustration of the map of England executed upon a portion of the surface of the country is really a typical instance of a self-contradictory purpose. He argues that such a map, to be theoretically perfect, must contain a reduced facsimile of itself as part of the country mapped, and this again another, and so on indefinitely. But the whole force of the reasoning depends on overlooking the distinction between the surface of England as it is before the map is made, and the surface of England as altered by the presence of the map. Prof. Royce assumes that you set out to represent in the map a state of things which can in fact have no existence until after the map is made. The previous existence of the map at a certain spot is falsely taken to be one of the conditions to which the map-maker is to conform in executing it. Every one of the supposed “maps within the map” will thus involve distortion and misrepresentation of the district it proposes to map. It is as if Hamlet had chosen “Hamlet” as the subject of the “play within the play.” The professor’s illustration thus does less than justice to his theory. Footnote 94: