In ancient Greek literature (most notably in Homer’s Iliad), Achilles is a powerful warrior whose only weak spot is his heel. This is the origin of the modern phrase Achilles’ heel, which refers to the lone point of vulnerability in an otherwise powerful or self-assured person or thing.
The heel in the expression belongs to Achilles, so Achilles is possessive. We usually make ancient names ending in s possessive by adding only an apostrophe (instead of adding ‘s). And Achilles is of course a name, so the A is usually capitalized. Not everyone treats the phrase this way, though. For instance, the British Guardian newspaper, which takes a minimalist approach to apostrophes and capitals, uses achilles heel—with a lowercase a and no apostrophe. Other publications omit the apostrophe but capitalize the A.
As President Bashar al-Assad’s government falters, Syria is becoming Iran’s Achilles’ heel. [New York Times]
Most actors want to be loved—it’s the Achilles’ heel of the profession—but Stanwyck seems to have been after something else: respect. [Wall Street Journal]
The arrogance and intolerance of the atheists, exemplified by Prof Dawkins, is their Achilles’ heel. [Telegraph]
At 1 per cent of GDP, Quebec’s budget deficit this year isn’t as ominous as Ontario’s. But high debt remains its Achilles heel. [Globe and Mail]
The NHS was long the Conservative party’s achilles heel. [Guardian]
Authenticity may well be Romney’s Achilles heel. [Sydney Morning Herald]