PROPOSITION 1.

A part of a straight line cannot be in the plane of reference[1] and a part in a plane more elevated.[2]

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.

C D B A

There will then be 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, 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.
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.

Dependency Graph

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Required for

XI.2, XI.6, XI.8, XI.9, XI.10, XI.11, XI.12, XI.15, XI.16, XI.17, XI.18, XI.23, XI.24, XI.25, XI.26, XI.31, XI.32, XI.33, XI.34, XI.35, XI.36, XI.37, XI.38, XI.39, XII.3, XII.4, XII.5, XII.6, XII.7, XII.8, XII.9, XII.10, XII.11, XII.12, XII.13, XII.14, XII.15, XII.17, XII.18, XIII.13, XIII.14, XIII.16, XIII.17, XIII.18

See Commentary on Proposition XI.1.