Welcome to MathOverflow

Question

0

Are XI.1 and XI.7 of Euclid’s Elements equivalent？

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

XI.7 If two straight lines are parallel and points are taken at random on each of them, then the straight line joining the points is in the same plane with the parallel straight lines

edited Jun 7 2011 at 9:25

asked Jun 7 2011 at 8:37

Yutao
1●1

3

I think you'll have more success if you fill out your question a little. Perhaps begin by reminding us of the statements you are interested in. – Simon Wadsley Jun 7 2011 at 8:45

Why not consult the Heath commentary on these two results? amazon.com/Thirteen-Books-Elements-Vol-10-13/dp/… – Gerald Edgar Jun 7 2011 at 14:45

3

I am pretty sure Euclid did not say "at random" – Igor Rivin Jun 7 2011 at 14:49

I think there's a better case that XI.7 is equivalent to XI.2: "If two straight lines cut one another, then they lie in one plane; and every triangle lies in one plane." It seems to me that XI.1 is saying something to the effect of "If one part of a line lies in a plane, so do the other parts of that line" -- which you really want to know before you say anything else. – Hugh Thomas Jun 7 2011 at 15:31

What is meant by "equivalent" here? In the given axiom system, both statements are true, so of course one is true iff the other is true. Do you mean to ask whether they remain equivalent in a weaker axiom system or whether they are "redundant" in some sense, i.e., that the truth of one follows obviously from the truth of the other? – Pete L. Clark Jun 7 2011 at 18:35

Are XI.1 and XI.7 of Euclid’s Elements equivalent？

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.