Book 11
28
A dodecahedron is a solid figure contained by twelve equal, equilateral, and equiangular pentagons.
BOOK XI. PROPOSITIONS.
PROPOSITION 1.
1
A part of a straight line cannot be in the plane of reference and a part in a plane more elevated.
1
For, if possible, let a part AB of the straight line ABC be in the plane of reference, and a part BC in a plane more elevated.
1
There will then be in the plane of reference some straight line continuous with AB in a straight line.
1
Let it be BD; therefore AB is a common segment of the two straight lines ABC, ABD: which is impossible, inasmuch as, if we describe a circle with centre B and distance AB, the diameters will cut off unequal circumferences of the circle.
1
Therefore a part of a straight line cannot be in the plane of reference, and a part in a plane more elevated. Q. E. D. 1. the plane of reference, τὸ ὑποκείμενον ἐπίπεδον, the plane laid down or assumed. 2. more elevated, μετεωροτέρῳ.
PROPOSITION 2.
2
If two straight lines cut one another, they are in one plane, and every triangle is in one plane.
2
For let the two straight lines AB, CD cut one another at the point E; I say that AB, CD are in one plane, and every triangle is in one plane.
2
For let points F, G be taken at random on EC, EB, let CB, FG be joined, and let FH, GK be drawn across; I say first that the triangle ECB is in one plane.
2
For, if part of the triangle ECB, either FHC or GBK, is in the plane of reference, and the rest in another, a part also of one of the straight lines EC, EB will be in the plane of reference, and a part in another.
2
But, if the part FCBG of the triangle ECB be in the plane of reference, and the rest in another, a part also of both the straight lines EC, EB will be in the plane of reference and a part in another: which was proved absurd. [XI. 1]
2
Therefore the triangle ECB is in one plane.
