parts of a continuum, or as the relations amongst the units of a mere aggregate of particles), some corresponding detail, ɑ’, b’, c’, d’, could be identified in A’, in accordance with the system of projection used.
All this being understood, let us undertake to define a map that shall be in this sense perfect, but that shall be drawn subject to one special condition. It would seem as if, in case our map-drawing powers were perfect, we could draw our map wherever we chose to draw it. Let us, then, choose, for once, to draw it within and upon a part of the surface of the very region that is to be mapped. What would be the result of trying to carry out this one purpose? To fix our ideas, let us suppose, if you please, that a portion of the surface of England is very perfectly levelled and smoothed, and is then devoted to the production of our precise map of England. That in general, then, should be found upon the surface of England, map constructions which more or less roughly represent the whole of England, — all this has nothing puzzling about it. Any ordinary map of England spread out upon English ground would illustrate, in a way, such possession, by a part of the surface of England, of a resemblance to the whole. But now suppose that this our resemblance is to be made absolutely exact, in the sense previously defined. A map of England, contained within England, is to represent, down to the minutest detail, every contour and marking, natural or artificial, that occurs upon the surface of England. At once our imaginary case involves a new problem. This is now no longer the general problem of map making, but the nature of the internal meaning of our new purpose.
Absolute exactness of the representation of one object by another, with respect to contour, this, indeed, involves, as Mr. Bradley would say to us, the problem of identity in diversity; but it involves that problem only in a general way. Our map of England, contained in a portion of the surface of England, involves, however, a peculiar and infinite development of a special type of diversity within our map. For the map, in order to be complete, according to the rule given, will have to contain, as a part of itself, a representation of its own contour
