MS 842 (1908) - A Neglected Argument - Early Drafts

G42

of rigid bodies in displacements which would be possible in four dimensions; and thus are not purely geometrical. The graphical characters are properties of rays. But it is impossible to distinguish a straight from a curved line, whether practically or theoretically, except by reference to some statical or dynamical (including optical) experiment. So that they, too, in the last analysis, are physical not purely geometrical properties. Topical characters of places are such as, being in no wise metrical or graphical, consist entirely in such relations of betweenness among the parts of a place as will remain

G43

unchanged, no matter how the place be continuously distorted, provided that every identical moveable and distortable place which at one instant has a given dimensionality (that of a point, of a line, of a surface, or of a space,) retains the same dimensionality at every instant. These alone are purely geometrical characters, and are not physical in any narrower sense.

Beside the four dimensionalities of places in our Space,

G44

(0 for a point, 1 for a line, 2 for a surface, 3 for a space), and beside the distinction between the ordinary and the singular characters of a place, P. The ordinary character of a character C of P consists

G41

The cyclosy of any place, L, is the least number of places in it, each of the dimensionality less by 1 than that of L, such that if all of them were occupied, no room would be left for a Line that could not shrink continuously in that

G68

dependent of the truth of such general axiom, and proceeds as if on the principle, which is not necessarily recognized, that there is in the particular class of objects to which

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