It must not be supposed, merely because series afford the most obvious application of the idea of likeness, that there are no other applications that are important. We have already mentioned maps, and we might extend our thoughts from this illustration to geometry generally. If the system of relations by which a geometry is applied to a certain set of terms can be brought fully into relations of likeness with a system applying to another set of terms, then the geometry of the two sets is indistinguishable from the mathematical point of view, i.e. all the propositions are the same, except for the fact that they are applied in one case to one set of terms and in the other to another. We may illustrate this by the relations of the sort that may be called "between," which we considered in Chapter IV. We there saw that, provided a three-term relation has certain formal logical properties, it will give rise to series, and may be called a "between-relation." Given any two points, we can use the between-relation to define the straight line determined by those two points; it consists of and together with all points , such that the between-relation holds between the three points , , in some order or other. It has been shown by O. Veblen that we may regard our whole space as the field of a three-term between-relation, and define our geometry by the properties we assign to our between-relation.[13] Now likeness is just as easily [Pg 58] definable between three-term relations as between two-term relations. If and ' are two between-relations, so that "" means " is between and with respect to ," we shall call a correlator of and ' if it has the field of ' for its converse domain, and is such that the relation holds between three terms when ' holds between their -correlates, and only then. And we shall say that is like ' when there is at least one correlator of with '. The reader can easily convince himself that, if is like ' in this sense, there can be no difference between the geometry generated by and that generated by '. [13]This does not apply to elliptic space, but only to spaces in which the straight line is an open series. Modern Mathematics, edited by J. W. A. Young, pp. 3-51 (monograph by O. Veblen on "The Foundations of Geometry").