§ 80. These are the principal properties of lines and planes in space, but before we go on to their applications it will be well to define the word distance. In geometry distance means always “shortest distance”; viz. the distance of a point from a straight line, or from a plane, is the length of the perpendicular from the point to the line or plane. The distance between two non-intersecting lines is the length of their common perpendicular, there being but one. The distance between two parallel lines or between two parallel planes is the length of the common perpendicular between the lines or the planes.
§ 81. Parallelepipeds.—The rest of the book is devoted to the study of the parallelepiped. In Prop. 24 the possibility of such a solid is proved, viz.:—
Prop. 24. If a solid be contained by six planes two and two of which are parallel, the opposite planes are similar and equal parallelograms.
Euclid calls this solid henceforth a parallelepiped, though he never defines the word. Either face of it may be taken as base, and its distance from the opposite face as altitude.
Prop. 25. If a solid parallelepiped be cut by a plane parallel to two of its opposite planes, it divides the whole into two solids, the base of one of which shall be to the base of the other as the one solid is to the other.
This theorem corresponds to the theorem (VI. 1) that parallelograms between the same parallels are to one another as their bases. A similar analogy is to be observed among a number of the remaining propositions.
§ 82. After solving a few problems we come to
Prop. 28. If a solid parallelepiped be cut by a plane passing through the diagonals of two of the opposite planes, it shall be cut in two equal parts.
In the proof of this, as of several other propositions, Euclid neglects the difference between solids which are symmetrical like the right hand and the left.
Prop. 31. Solid parallelepipeds, which are upon equal bases, and of the same altitude, are equal to one another.