. . . that the Elves had destroyed it, as should have been done.'"
Gandalf to Frodo in "The Shadow of the Past"
You are correct that Gandalf said this to Frodo. There are two
explanations for it: Sauron was mistaken about what would happen if the Ring
perished, or Gandalf was mistaken about what Sauron believed.
I happen to believe the latter, that it is Gandalf who is mistaken, because
Sauron must have known how much of himself he invested in the Ring, and that he
and his works like Bara-dur and slaves like the Nazgul could not survive the
Ring's destruction. But Sauron was not necessarily aware of what had
happened to Isildur or the Ring, and may well have believed that the Ring was
lost forever. Certainly he was not aware that the Ring had been found
until he questioned Gollum. So even if Gandalf is not entirely correct,
he is mostly correct; Sauron did not know where to look for the Ring until he
questioned
Gollum.
________________________________________
"‘I think he was a silly little man,' said Councillor Tompkins. ‘Worthless, in fact; no use to Society at all.'
"‘Oh, I don't know,' said Atkins, who was nobody of importance, just a schoolmaster. ‘I am not so sure: it depends on what you mean by use .'
"‘No practical or economic use,' said Tompkins. . . .
. . .
"‘It is proving very useful indeed,' said the Second Voice. ‘As a holiday, and a refreshment. It is splendid for convalescence; and not only for that, for many it is the best introduction to the Mountains. It works wonders in some cases. I am sending more and more there. They seldom have to come back.'"
