then, that by adding, subtracting, or otherwise operating upon the numbers, you can reach results that will be valid regarding the objects that were to be counted. Again, a given plane curve can be made to correspond, point for point, with its own shadow, or with some other systematic projection of the curve as made upon a given surface. In this case, a great number of relationships between the points of the curve will remain true of the corresponding points of the projected curve. In the very familiar case of a map, the parts of the map correspond to the parts of the object represented, in a manner determined by a particular system of projection or of transformation of object into map.
But in consequence of the very general nature of this relation of correspondence, two complicated objects, or two collections of objects, may be made to correspond to one another, part for part, member for member, in wholly different ways. When you count objects, for instance, it makes no difference in what order you count them, or, in other words, in what order you make them correspond, object for object, to your number series. When you draw maps, you may use either Mercator’s projection, or some other plan of map-making. In any case, you can still get a definite correspondence of map and object, part for part, although, by varying the plan of projection followed, you may vary the way in which the correspondence used in any one case will prove useful in measuring distances, or in plotting courses on the map once drawn. Any sort of correspondence thus always fulfils one definite purpose, such as the purpose of counting, of map-drawing upon some special plan, or of
