Commentary on Proposition XI.7¶
Back to Proposition XI.7.
It is true that this proposition, in the form in which Euclid enunciates it, is hardly necessary if the plane is defined as a surface such that, if any two points be taken in it, the straight line joining them lies wholly in the surface. But Euclid did not give this definition; and, moreover, Prop. 2 would be usefully supplemented by a proposition which should prove that two parallel straight lines determine a plane (i.e. one plane and one only) which also contains all the straight lines which join a point on one of the parallels to a point on the other. That there cannot be two planes through a pair of parallels would be proved in the same way as we prove that two or three intersecting straight lines cannot be in two different planes, inasmuch as each transversal lying in one of the two supposed planes through the parallels would lie wholly in the other also, so that the two supposed planes must coincide throughout (cf. note on Prop. 2 above).
But, whatever be the value of the proposition as it is, Simson seems to have spoilt it completely. He leaves out the construction of a plane through EGF, which, as Euclid says, must cut the plane containing the parallels in a straight line; and, instead, he says, “In the plane ABCD in which the parallels are draw the straight line EHF from E to F.” Now, although we can easily draw a straight line from E to F, to claim that we can draw it in the plane in which the parallels are is surely to assume the very result which is to be proved. All that we could properly say is that the straight line joining E to F is in some plane which contains the parallels; we do not know that there is no more than one such plane, or that the parallels determine a plane uniquely, without some such argument as that which Euclid gives.
Nor can I subscribe to the remarks in Simson’s note on the proposition. He says (1) “This proposition has been put into this book by some unskilful editor, as is evident from this, that straight lines which are drawn from one point to another in a plane are, in the preceding books, supposed to be in that plane; and if they were not, some demonstrations in which one straight line is supposed to meet another would not be conclusive. For instance, in Prop. 30, Book I., the straight line GK would not meet EF, if GK were not in the plane in which are the parallels AB, CD, and in which, by hypothesis, the straight line EF is.” But the subject-matter of Book I. and Book XI. is quite different; in Book I. everything is in one plane, and when Euclid, in defining parallels, says they are straight lines in the same plane etc., he only does so because he must, in order to exclude non-intersecting straight lines which are not parallel. Thus in I. 30 there is nothing wrong in assuming that there may be three parallels in one plane, and that the straight line GHK cuts all three.
But in Book XI. it becomes a question whether there can be more than one plane through parallel straight lines.
Simson goes on to say (2) “Besides, this 7th Proposition is demonstrated by the preceding 3rd; in which the very same thing which is proposed to be demonstrated in the 7th is twice assumed, viz., that the straight line drawn from one point to another in a plane is in that plane.” But there is nothing in Prop. 3 about a plane in which two parallel straight lines are; therefore there is no assumption of the result of Prop. 7. What is assumed is that, given two points in a plane, they can be joined by a straight line in the plane: a legitimate assumption.
Lastly, says Simson, “And the same thing is assumed in the preceding 6th Prop. in which the straight line which joins the points B, D that are in the plane to which AB and CD are at right angles is supposed to be in that plane.” Here again there is no question of a plane in which two parallels are; so that the criticism here, as with reference to Prop. 3, appears to rest on a misapprehension.