The best way to keep out of trouble with these words is to check your desk dictionary before you use them. We say desk dictionary, not unabridged dictionary, because a desk dictionary has the present-day mainstream meanings. An unabridged includes all meanings, and that is the problem.

The meanings we are concerned with are "of acute discernment; keen" (you might even say "shrewd") — the chief modern meaning of perspicacious — and "acuteness of judgment or insight" — the chief modern meaning of perspicacity. The modern meanings of perspicuous and perspicuity are, respectively, "clear to the understanding; plain, lucid" and "clarity, lucidity."

According to most usage commentators, the chief mistake is substitution of perspicuity for perspicacity (the confusion has been going on since 1662) and of perspicuous for perspicacious. The latter is a little tricky, because perspicuous was used in the sense of perspicacious in 1584, two years before it was used in its current mainstream meaning and 56 years before perspicacious is attested in English at all. However, the trend of development since the 16th and 17th centuries has been toward differentiation, and if you follow the distinctions described here, you will not go wrong.

Here are some examples from our citation file showing the words in action:

As always, Didion is scrupulous in her research, discerning in her observations, and eloquent as she scours the outer world for keys to inner conflicts, and, consequently, her insights into California's psyche are perspicacious and arresting. — Donna Seaman, Booklist, July 2003

Samuel Johnson's jurisdiction over the English language was never simply a matter of his poetic skill as a definer of meaning, nor of his compendious way with examples, though these elements of his talent continue to make one feel larger and smaller at the same time. His special feature lay in how he brought such moral perspicacity to the task, how he made it angelic. — Andrew O'Hagan, New York Review of Books, April 27, 2006

The whole is less than the sum of its parts and does not add up to either a perspicuous account or a judicious analysis. — Steven Marcus, New York Times Book Review, March 31, 1996

. . . Bishop Lowth, Hugh Blair and Lindley Murray plead eloquently for purity, precision, and perspicuity. — Bertil Sundby et al., A Dictionary of English Normative Grammar, 1991