Tolkien as Calculus

From a rec.arts.books.tolkien posting dated 21 July 1995.

In an effort to compare the relative strengths of the Maiar, a recent poster to r.a.b.t. compared Sauron's strength to Gandalf's and the Balrog's by stating:

S>G and G=B implies B<S.

It's an intriguing way of stating the problem.But Gandalf the Grey, who fought the Balrog, wasn't as powerful as Gandalf the White. Also remember that we're talking about a Sauron who has invested much of his native power in the Ring, which has weakened him greatly while he is not in posssession of it; he is not as strong as he was with his original native power:

Gg < Gw

Sn = S + R

Sn > S

Now Gandalf was afraid of using the Ring, for fear it would conquer him; yet if he had used the Ring, he would have had enough power to defeat Sauron (Fellowship pp. 70-71 hardback):

Gg < R

Gg + R > S

But if the Balrog had arrived at the Bridge of Khazad-dum first it may have been possible that, though greatly weakened by Gandalf, it might have obtained the Ring. So, if the Balrog had been victorious,

Bv = B + R - Gg

would the Balrog have been able to overthrow a Sauron whose native power had been diminished by the loss of the Ring?:

B + R - Gg > Sn - R

And when Gandalf had returned from death, would he have assisted the Balrog, hoping that

(B + R - Gg) + Gw > Sn - R

then

Bv - 1/2(Sn-R) < Gw - 1/2(Sn-R) ?

Answers are due at the end of class next week. Be sure to show your proofs.

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