Commentary on Proposition XI.2

Back to Proposition XI.2.

It must be admitted that the “proof” of this proposition is not of any value. For one thing, Euclid only takes certain triangles and a certain quadrilateral respectively forming part of the original triangle, and argues about these. But, for anything we are supposed to know, there may be some part of the triangle bounded (let us say) by some curve which is not in the same plane with the triangle.

We may agree with Simson that it would be preferable to enunciate the proposition as follows.

Two straight lines which intersect are in one plane, and three straight lines which intersect two and two are in one plane.

Adopting Smith and Bryant’s figure in preference to Simson’s, we suppose three straight lines PQ, RS, XY to intersect two and two in A, B, C.

R P A X B C Y Q S

Then Simson’s proof (adopted by Legendre also) proceeds thus.

Let any plane pass through the straight line PQ, and let this plane be turned about PQ (produced indefinitely) as axis until it passes through the point C.

Then, since the points A, C are in this plane, the straight line AC (and therefore the straight line RS produced indefinitely) lies wholly in the plane. [Simson's def.]

For the same reason, since the points B, C are in the plane, the straight line XY lies wholly in the plane.

Hence all three straight lines PQ, RS, XY (and of course any pair of them) lie in one plane.

But it has still to be proved that there is only one plane passing through the three straight lines.

This may be done, as in Mr Taylor’s Euclid, thus.

Suppose, if possible, that there are two different planes through A, B, C.

The straight lines BC, CA, AB then lie wholly in each of the two planes.

Now any straight line in one of the two planes must intersect at least two of the straight lines (produced if necessary); let it intersect two of them in $K$, $L$.

Then, since $K$, $L$ are also in the second plane; the line $KL$ lies wholly in that plane.

Hence every straight line in either of the planes lies wholly in the other also; and therefore the planes are coincident throughout their whole surface.

It follows from the above that

A plane is determined (i.e. uniquely determined) by any of the following data:

  1. by three straight lines meeting one another two and two,

  2. by three points not in a straight line,

  3. by two straight lines meeting one another,

  4. by a straight line and a point without it.