Definition:Prism/Height/Euclidean Variant
Definition
Although the height of a prism is generally understood to be the length of the perpendicular joining opposite faces, Euclid was inconsistent in his usage in The Elements.
In his Proposition $39$ of Book $\text{XI} $: Prisms of equal Height with Parallelogram and Triangle as Base, he defines the base of one prism as being one of the opposite parallel faces, but of the other he defines the base as being an arbitrary one of the parallelograms.
Having defined the base in this manner, the height is then defined as being the height of one of the opposite parallel faces whose base is the edge which intersects the base so defined.
Using this definition, the distance $h$ in the above diagram is the height of the prism $PQRSTU$.
In the words of Euclid:
- If there be two prisms of equal height, and one have a parallelogram as base and the other a triangle, and if the parallelogram be double the triangle, the prisms will be equal.