Is the proof of Proposition 2 in Book 1 of Euclid's elements a bit redundant?

Question

Here is the proposition together with a the proof :

Whats wrong with the following : Choose an arbitrary point A and another arbitrary one D. Now with center A describe a circle with radius BC and the intersection of the circle and the line AD is the required line segment.

Answer 1

I suspect that at this point all you can use in your proof is the postulates 1-5 and proposition 1. These does not that directly guarantee the existence of that point D you propose.

The point D is in fact guaranteed by proposition 1 that says that given a line AB (which is guaranteed by postulate 1) there is a equalateral triangle ABD.

Then a circle with radius BC centered at A is not either guaranteed directly. The only guaranteed circle AFAICS is a circle centered at one point and passing through another, for example the circle centered at B and with radius BC is guaranteed.

The problem one often face at this level is to remember not to use any facts that you know (even though it seems obvious) except only those provided by the postulates and previously proven propositions. This is harder than when covering new knowledge, because then you can use whatever you already "know".

Answer 2

I think that what's wrong here is your use of 'lifting the compass from the page without losing the distance between the "feet".' Euclid's compass could not do this (or was not assumed to be able to do this). The proof you've just read shows that it was safe to pretend that the compass could do this, because you could imitate it (via this proof) any time you needed to.

Euclid's assumptions about the geometry of the plane are remarkably weak from our modern point of view. He doesn't assume a priori anything about the compatability of the metric at distinct points. Indeed, there isn't actually a distance metric -- just a notion of congruence and between-ness. One of the great achievements of Eudoxus (I believe...check out a late chapter in Moise, "Elementary Geometry from an Advanced Standpoint" to be sure) was showing that the Euclidean axioms actually allowed you to construct, via clever tricks having to do with proportions, described geometrically, something that was essentially Dedekind cuts, i.e., to construct a function that actually produced a metric on Euclidean space, by first constructing the codomain, i.e., the reals. This is some serious work!