Manifolds. The idea of a manifold was first explicitly stated by Riemann; Grassmann had still earlier defined and investigated a particular kind of manifold.

Consider any number of things possessing any common property. That property may be possessed by different things in different modes: let each separate mode in which the property is possessed be called an element. The aggregate of all such elements is called the manifold of the property.

Any object which is specified as possessing a property in a given mode corresponds to an element in the manifold of that property. The element may be spoken of as representing the object or the object as representing the element according to convenience. All such objects may be conceived as equivalent in that they represent the same element of the manifold.

Various relations can be stated between one mode of a property and another mode; in other words, relations exist between two objects, whatever other properties they may possess, which possess this property in any two assigned modes. The relations will define how the objects necessarily differ in that they possess this property differently: they define the distinction between two sorts of the same property. These relations will be called relations between the various elements of the manifold of the property; and the axioms from which can be logically deduced the whole aggregate of such relations for all the elements of a given manifold are called the characteristics of the manifold.