An adjective brings an article with it. What's the general rule?

I often notice that the presence of an adjective causes an article to appear where, in the absence of the adjective, there would not be an article. For example: here's a quote from William Dunham's Euler the Master of Us All:

Hiding their secrets with an embarrassing ease, the integers provide a worthy challenge for the greatest of mathematicians.

Is this correct, appropriate, and what's the general rule governing this?

Solution 1:

Many uncountables can be treated as countable when a specific example, or a specific kind, is meant.

That specificity can come from a relative clause:

the beauty that fades ...

or from a possessive phrase:

the beauty of a red rose ...

or from an adjective

a fleeting beauty ...

Having said that, the last (adjective) case is different from the others, because it is not specific (and so does not take the definite article 'the') but only partly specified (a fleeting beauty, rather than any other kind) so it takes 'a'. It is a bit literary.