Thus we have obtained a figure consisting of the two points 7, 1, and 5, 5, representative of our two triangles. But we can go further, and, drawing an arc of a circle about O, 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 arc. 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 it is worth while to bring into the full light of our attention our habitual assumptions and processes. It often happens that we find there are two of them which have a bearing on each other, which, without this dragging into the light, we should have allowed to remain without mutual influence. There is a fact which it concerns us to take into account in discussing the theory of the poiograph. With respect to our knowledge of the world we are far from that condition which Laplace imagined when he asserted that an all-knowing mind could determine the future condition of every object, if he knew the co-ordinates of its particles in space, and their velocity at any particular moment. On the contrary, in the presence of any natural object, we have a great complexity of conditions before us, which we cannot reduce to position in space and date in time.