ABSTRACT
In a universe like the foregoing visual one, geometry is completely illustrated a great number of times. It is integrally embodied in the first view met: its points are the elementary data of this view, and congruence is the original relation N that we have named connection of couples of these data. Then, it is embodied again in each one of the following views, successively taking for points and congruence the elementary data of each view and the connection of their couples. We may place all these interpretations, which are as numerous as there are views, in one class: for they have the same relation N for congruence, and terms of the same sort for points.—Next, geometry is illustrated in the manifold l of the classes of sense data grouped by local similarity, namely perceived places. These classes are its points, and the relation N l of two couples of perceived places, whose members form two connected couples in every view, is the meaning of congruence for this geometry.—Again, geometry is illustrated in the manifold o of classes of sense data grouped by qualitative similarity, namely objects, taking these new classes for its points, and the relation N v of two couples of objects whose members form two connected couples in every view for congruence.—Geometry is illustrated also in each group v 1 of possible views having the same orientation (that is to say, coming under law VI): it takes these views for points, and for congruence the relation N v of two of their couples in which the appearances of any object have the perceived fields whose couples have the relation N l .—Finally, geometry is illustrated in each total group t 1 of the classes of views in which the object o 1 has its members in the same perceived field: points, then, are these classes, and congruence is the relation N t of two of their couples which “cut” every space v of parallel views into two couples of views having the relation N v .
