People wondered whether Hamlet was mad, not whether he was sane. The word itself -- derived from the Latin sanus and the French sain, and meaning originally of the body . . . "healthy, sound, not diseased" -- was not commonly used in the seventeenth century, when it first appeared. It is, indeed, used only once by Shakespeare, and perhaps unsurprisingly in Hamlet; and, also perhaps unsurprisingly, it is used by Polonius. Polonius wonders, like several other people in the play, whether Hamlet is "mad," a word used over two hundred times by Shakespeare; and used, with its cognate "madness," thirty-five times in Hamlet. The words "mad" and "madness" are bandied about in Hamlet (though not often by Hamlet himself) because they seem pertinent -- they seem to locate a problem without quite saying what the problem is. The madness resists definition -- no one is quite sure what it refers to, or indeed what Hamlet himself seems to be referring to. Characters in the play often don't understand what Hamlet is saying, but Hamlet reassures them that they do, at least, understand what they themselves are saying. "Mad," in the play, is a word for "puzzling." "Your noble son is mad," Polonius says helpfully to Gertrude and Claudius: "Mad call I it. For, to de-fine true madness, What is't but to be nothing else than mad? But let that go" (II.2.925). Polonius clearly has a problem about definition here; "true madness" is a strange phrase since madness is a form of dissembling. True madness, for example, could just mean acting. We might wonder whether Hamlet is pretending to wonder about the difference between selves on show in public and selves on show in private. Madness, in other words, tends to the theatrical, even when it is not good theater. Sanity tends the other way. Being mad, as Polonius suggests, can mean acting as if one were mad; being sane cannot mean acting as if one were sane.
The theatricality of madness is one clue that alerts us to the difficulties we have in imagining sanity. Hamlet's madness makes people suspicious, incites their curiosity, gets them talking. Even though it is an abstract word, madness is an abstraction we can visualize, we can picture how it performs. Sanity doesn't quite come to life for us in the same way: it has no drama. Like the "good" characters in literature, the sane don't have any memorable lines. They don't seem quite so real to us. Insofar as we can imagine them at all, they are featureless, bland, unremarkable.
What may seem striking about Hamlet to modern eyes and ears is that sanity is not invoked simply as a counter to madness, as a defining alternative. When the word turns up, it is used by Polonius to describe just how impressed he is by the inventiveness, the eloquent intelligence, of Hamlet's supposed madness.
Polonius: (aside) Though this be madness, yet there is method in't. -- Will you walk out of the air, my lord?
Hamlet: Into my grave?
Polonius: Indeed, that's out of the air. (aside) How pregnant sometimes his replies are! A happiness that often madness hits on, which reason and sanity could not so prosperously be delivered of.
Sanity, for Polonius, is a different way of speaking; madness is not worse than sanity, it just makes sanity sound dull. Madness is not opposed to sanity, it has a different method; even though method, of course, is usually associated with reason (even here Hamlet's madness is making Polonius himself more imaginative). Madness hits on things that sanity and reason can conceive of, but so "prosperously"; Hamlet's madness, though these would not be Polonius's words, is more poetic, more suggestive, more evocative, more flaunting of its verbal gifts and talents than mere sanity. Words can be delivered more or less prosperously; a happiness can be struck by madness that reason and sanity can diminish. Sanity tempers where madness excels. Both are "pregnant," promising the new life that is new words, but they deliver quite differently. It is a difference of quality but not of kind. The words of the mad are more prosperous than the words of the sane; and "prosperous" was then a more prosperous word than it is for us now, as it meant "bringing prosperity; favourable, propitious, auspicious" (Oxford English Dictionary). Prosperous words augured well for the future (even though Hamlet's prosperous words didn't, in fact, augur well for his). For Polonius, sanity and madness are two ways of being pregnant with words.
Polonius connects reason and sanity, an association that has become all too familiar to us, and suggests that compared with Hamlet's madness they are lacking in something. It is precisely what sanity may be lacking that Hamlet's madness makes Polonius wonder about (as though the mad expose the sane in the same way that the Fool exposes his Master). The replies of the mad are somehow more pregnant; the dialogue of the sane is poorer. And yet the mad make us suspicious, we can't rely on them to be telling us the truth. Whereas sanity is assumed to be morally good -- the folio has "sanctity" for "sanity" -- madness may be disreputable; and because it may be, because we can't ever be quite sure what the mad are really up to, it is. The mad don't let us take it for granted that we know where we are with them. Hamlet's madness is artful but duplicitous; Hamlet has good lines but a weak character, at least according to Polonius. Which, of course, immediately raises the question of what more there may be to a person's character than the words they speak. Polonius is so perturbed by Hamlet partly because he is so impressed by him.