The fourth dimension

88 THE FOURTH DIMENSION

of a circle about 0, the meeting point of the horizontal and vertical levels, which passes through 7, 1, and 5, 5, assert that all the triangles which are right-angled and have a hypothenuse whose square is 50 are represented by the points on this are.

Thus, each individual of a class being represented by a point, the whole class is represented by an assemblage of points forming a figure. Accepting this representation we can attach a definite and calculable significance to the expression, resemblance, or similarity between two individuals of the class represented, the difference being measured by the length of the line between two representative points, It is needless to multiply examples, or to show how, corresponding to different classes of triangles, we obtain different curves.

A representation of this kind in which an object, a thing in space, is represented as a point, and all its properties are left out, their effect remaining only in the relative position which the representative point bears to the representative points of the other objects, may be called, after the analogy of Sir William R. Hamilton’s hodograph, a “ Poiograph.”

Representations thus made have the character of natural objects; they have a determinate and definite character of their own. Any lack of completeness in them is probably due to a failure in point of completeness of those observations which form the ground of their construction.

Every system of classification is a poiograph. In Mendeléeff’s scheme of the elements, for instance, each element is represented by a point, and the relations between the elements are represented by the relations between the points.

So far I have simply brought into prominence processes and considerations with which we are all familiar, But