Euclid's Elements, Book XI, Definitions 6 through 8

Definitions 6 through 8

Def. 6. The inclination of a plane to a plane is the acute angle contained by the straight lines drawn at right angles to the intersection at the same point, one in each of the planes.

Def. 7. A plane is said to be similarly inclined to a plane as another is to another when the said angles of the inclinations equal one another.

Def. 8. Parallel planes are those which do not meet.

As the previous definition requires certain assumptions, so does definition 6. It assumes that any two such acute angles are equal, something Euclid does not prove but could have in the course of Book XI.

Definition 8 is analogous to definition I.23 for parallel lines in a plane. There is no proposition in Book XI which states that parallelism of planes is a transitive relation, but that is not difficult to prove given the rest of the propositions in the book. The first appearance of parallel planes is in proposition XI.14.

When two planes are not parallel, then, by this definition, they intersect. Proposition XI.3 proclaims that this intersection is a straight line.

Note that it is not defined when a line is parallel to a plane, but that would be when they don't meet.

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Book XI introduction