The axioms of descriptive geometry, by A.N. Whitehead.

8, 9]

DEDEKIND'S AXIOM

9

IX. If there exist three points not lying in the same line, there
exists a plane ABC such that there is a point D not lying in the
plane ABC.
Definition 7. If A, B, C, and D are four points not lying in the
same plane, they form a 'tetrahedron ' ABCD, whose 'faces' are the
interiors of the triangles ABC, BCD, CDA, DAB, whose 'vertices'
are the four points A, B, C, and D, and whose 'edges' are the
segments AB, BC, CD, DA, AC, BD. The points of faces, edges,
and vertices constitute the 'surface' of the tetrahedron.
Definition 8. If A, B, C, D are the vertices of a tetrahedron, the
space ABCD consists of all points collinear with any two points of the
faces of the tetrahedron.
X. If there exist four points, neither lying in the same line, nor
lying in the same plane, there exists a space ABCD, such that there is
no point E not collinear with two points of the space A BCD.
The above axioms of Veblen are equivalent to the axioms of Peano
which have been previously given. Both Peano and Veblen give an
axiom securing the Dedekind property (cf. ~ 9). Also Veblen gives an
axiom securing the 'Euclidean' property (cf. ~ 10).
9. Dedekind's original formulation* of his famous property applies
directly to the case of a descriptive line and is as follows:
"If all points of the straight line fall into two classes such that
every point of the first class lies to the left of every point of the second
class, then there exists one and only one point which produces this
division of all points into two classes, this severing of the straight line
into two portions."
It is of course to be understood that the dividing point itself
belongs to one of the two classes.
It follows immediately that the boundary of a triangle consists of
points in a compact closed order possessing the Dedekind property as
already formulated for closed seriest.
This definition may be repeated here to exhibit its essential independence of the special definition of projective segments upon which
the previous formulation rests.
Let A, B, C be any three points of a closed series. Then by
* Cf. his Continuity and Irrational Numbers, ch. III.; the quotation here is
from Beman's translation, Chicago, 1901.
t Cf. Proj. Geom. ~ 19 (a).