Commentary on Proposition XI.5

Back to Proposition XI.5.

It follows that, if a right angle be turned about one of the straight lines containing it the other will describe a plane.

At any point in a straight line it is possible to draw only one plane which is at right angles to the straight line.

One such plane can be found by taking any two planes through the given straight line, drawing perpendiculars to the straight line in the respective planes, e.g. BO, CO in the planes AOB, AOC, each perpendicular to AO, and then drawing a plane (BOC) through the perpendiculars.

A O B C C'

If there were another plane through O perpendicular to AO, it must meet the plane through AO and some perpendicular to it as OC in a straight line OC' different from OC.

Then, by XI. 4, AOC' is a right angle, and in the same plane with the right angle AOC: which is impossible.

Next, one plane and only one can be drawn through a point outside a straight line at right angles to that line.

Let P be the given point, AB the given straight line.

In the plane through P and AB, draw PO perpendicular to AB, and through O draw another straight line OQ at right angles to AB.

A P O Q B

Then the plane through OP, OQ is perpendicular to AB.

If there were another plane through P perpendicular to AB, either

  1. it would intersect AB at O but not pass through OQ, or

  2. it would intersect AB at a point different from O. In either case, an absurdity would result.