b have been in our discussion of series in general. If, comparing two of these subordinate systems (let us say A and B), we conceive also, in some comprehensive fashion, a series of intermediary systems that link A and B together, we conceive what the mathematicians would call the “series of transformations,” whereby we can, at least in our conceptions, if not in our observations, pass from one of these systems to the other. Thus, let A be, no longer, as in one of our earlier illustrations, a point in space, but a large solid body. And let B be this same body viewed by us at another time in another place, or else let it be another body of precisely the same shape and size as A, occupying another place. Let us suppose A and B compared together in one act of attention. Then we can conceive of a system of movements (consisting of translations from place to place, of rotations about one or another axis, or of a single translation followed by a single rotation), — a system whereby A could be brought to take precisely the place that B now occupies. Sometimes this serial system of movements can be actually observed. Or again, let A be the system of the characters, habits, and dispositions of the people of England just before the colonization of America; let B be the system of characters and habits and dispositions of the people of the North American Colonies at the middle of the Eighteenth Century. Then we can follow (although, in this case, only very inexactly) the series of transformations that English civilization early underwent in its passage to American soil. Other instances without limit could be named.
Between any two systems, A and B, there thus lie
