Elendil's sword & Isildur's bane

I was just listening to my recording of "Lord of the Rings: Return of the King" and an observation hit me regarding the re-forging of the "sword which was broken", Narsil. The elves reforged it for Aragorn, the heir of Elendil, it's previous owner, but only after it had been handed down broken for generations. It would seem that Isildur's failure to have the sword reforged earlier is part of his fall into evil ways due to the One Ring - it's as if the sword and the ring are enemies. To take up one is to forsake the other; the opposite being seen in Aragorn who takes up the reforged sword, renamed "Anduril" in "Fellowship of the Ring", and forsakes his claim to the evil One Ring. The sword and the ring are opposite symbols much in the way the circle and the cross are opposites to G. K. Chesterton in "The Everlasting Man": ....And though the symbol is of course only a coincidence, it is a coincidence that really does coincide. The mind of Asia can really be represented by a round 0, if not in the sense of a cypher at least of a circle.... It really is a curve that in one sense includes everything, and in another sense comes to nothing. In that sense it does confess, or rather boast, that all argument is an argument in a circle. And though the figure is but a symbol, we can see how sound is the symbolic sense that produces it, the parallel symbol of the Wheel of Buddha generally called the Swastika. The cross is a thing at right angles pointing boldly in opposite directions; but the Swastika is the same thing in the very act of returning to the recurrent curve. That crooked cross is in fact a cross turning into a wheel. Before we dismiss even these symbols as if they were arbitrary symbols, we must remember how intense was the imaginative instinct that produced them or selected them both in the east and the west. The cross has become something more than a historical memory; it does convey, almost as by a mathematical diagram, the truth about the real point at issue; the idea of a conflict stretching outwards into eternity. It is true, and even tautological, to say that the cross is the crux of the whole matter. In other words the cross, in fact as well as figure, does really stand for the idea of breaking out of the circle that is everything and nothing. It does escape from the circular argument by which everything begins and ends in the mind.... (G. K. Chesterton, "The Everlasting Man") His point and mine is that the shapes tell a lot about the function and nature of the objects: the cross or sword "boldly pointing in opposite directions", the circle or ring constrains and seeks to bind and trap. There is even a directional element in the name "Anduril" which means "flame of the West". Sauron's domain is in the East of Middle-earth. Is the fact that to GKC the cross symbolizes Western civilization and the circle represents the East somehow represented in Tolkien's devices of the Sword (West) and the Ring (East)? Who knows...but the sword really angers and scares Sauron when Aragorn shows it to him in the Palantir.