Oddly, the word sheep’s-eye can be used as a verb to mean “to look amorously at someone”. (No, really.)
If you have the latest HH book in your collection, The Cabinet of Linguistic Curiosities, you might have come across this story already (alongside the tale of how flirting was once banned in the city of New York). But for the uninitiated, here’s the lowdown.
To cast or throw a sheep’s eye at someone has been an idiomatic (and thankfully not literal) way of romantically glancing at someone since the mid sixteenth century. Sheep’s eyes aren’t especially romantic of course (though there’s no accounting for taste), but that expression apparently alludes to the sheep’s dopey, wide-eyed appearance—the same appearance, incidentally, that’s at the root of the expression to look sheepish.
Three centuries later, in 1801 the poet Samuel Taylor Coleridge took that idiom as starting point and coined the verb sheep’s-eye, and romantic types have been sheep’s-eying each other ever since.
As for that “flirting in New York” thing, you’ll have to track the book down for that one...