Theorems like this—Lines (or planes) which are parallel to a third are parallel to each other—follow at once.
This view of parallels leads therefore to no contradiction of Euclid’s Elements.
As immediate consequences we get the propositions:—
Every line meets a plane in one point, or it lies in it;
Every plane meets every other plane in a line;
Any two lines in the same plane meet.
§ 5. Aggregates of Geometrical Elements.—We have called points, lines and planes the elements of geometrical figures. We also say that an element of one kind contains one of the other if it lies in it or passes through it.
All the elements of one kind which are contained in one or two elements of a different kind form aggregates which have to be enumerated. They are the following:—
I. Of one dimension.
1. The row, or range, of points formed by all points in a line, which is called its base.