Sweet are the uses of adversity
Duke Senior:
Sweet are the uses of adversity,
Which, like the toad, ugly and venomous,
Wears yet a precious jewel in his head;
And this our life, exempt from public haunt,
Finds tongues in trees, books in the running brooks,
Sermons in stones, and good in every thing.
The duke is describing the world view he's been forced to adopt now that he's been deposed and exiled by his villainous brother—this is the "adversity" for which he has found "sweet uses." By "uses," the duke means "profits." He compares his seeming suffering—for example, exposure to the elements—to an ugly toad, which legendarily had a "precious jewel" with healing qualities embedded in its temple. The jewel he discovers in his condition—its profit—is freedom from "public haunt," or society. The duke concludes that nature "speaks" more eloquently and truly than tongues, books, and sermons; stones turn out to be better company than courtiers.
The duke's metaphor now seems far-fetched, as it may also have seemed to Shakespeare's audience. Nevertheless, "sweet are the uses of adversity" survives as somewhat preciously sincere words of comfort, when not uttered sarcastically.