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Proposition 1

A part of a straight line cannot be in the plane of reference and a part in plane more elevated.
For, if possible, let a part AB of the straight line ABC be in the plane of reference, and a part BC be in a plane more elevated.
Then there is in the plane of reference some straight line continuous with AB in a straight line. Let it be BD. Therefore AB is a common segment of the two straight lines ABC and ABD, which is impossible, since, if we describe a circle with center B and radius AB, then the diameters cut off unequal circumferences of the circle. |
Therefore, a part of a straight line cannot be in the plane of reference and a part in plane more elevated.
Q. E. D.

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This proposition is used in the proof of the next one as well as others in the last three books of the Elements..

Book XI Introduction - Proposition XI.2.

© 1996
D.E.Joyce
Clark University