Final answer:
The correct sentence that uses parentheses to enclose nonessential information is the one about David Hume. The enclosed text provides extra details that are not necessary to the meaning of the sentence.
Step-by-step explanation:
Based on the information provided from the style manual, parentheses are used to enclose nonessential information, which is supplementary and does not change the overall meaning of the sentence when removed. In the given options, the correct sentence that uses parentheses to provide additional, nonessential information is: 'David Hume (the eighteenth-century Scottish philosopher and historian) is widely regarded as brilliant today but was never able to get an academic position during his lifetime.' This sentence remains grammatically correct and meaningful without the information within the parentheses. The other sentences either include information that is essential to the sentence, or use parentheses in a way that the sentence becomes grammatically incorrect without the parenthetical information.