These words refer to unusual mental agility or perceptiveness. Keen suggests both these attributes, adding to them a vigorous and forceful ability to grapple with complex or obscure problem: a keen mind for fine distinctions. Sometimes, by analogy with good vision, the word may suggest an ability to observe details and see them as part of a larger pattern: a keen understanding of the problems facing the conservation movement. Acute suggests a finely honed sensitivity or receptivity to nuances that might escape others; it might also imply a high-keyed state or nervous attention that is not sustainable for long: an acute awareness of the slightest ambiguity in each statement made by his opponent; an acute alertness, heightened by the strange silence in the enemy trenches.

Penetrating relates more to the vigorous agility suggested by keen than the high-keyed attunement implicit in acute . It stresses the ability to see the root causes underlying details, where a keen mind might see only the surface details , however clearly. The word may also suggest a brusque eagerness to get to basic principles, regardless of possible injury to the feelings of others: a penetrating analysis of the play’s weakness that was unsparing in the harshness of its criticism. Shrewd , by contrast, suggests practical wisdom that does not necessarily look deeply into things at all but is wily and conscious of its own self-interest: a shrewd idea of how far he could go in criticizing the existing régime. Occasionally the word can be used without this overtone of self-interest, but the word still suggests cleverness rather than the impatient intensity of penetrating : a shrewd estimate of the materials the job would require.

Astute suggests a thorough and profound understanding, stemming from a scholarly or experienced mind that is in full command of a given field: an astute assessment of the strengths and weaknesses of the plans for reorganizing the department: an astute evaluation of the gaps in our knowledge of how life evolved. Sharp , by contrast, suggests a mind generally well-endowed but not necessarily well –grounded in a given field. Its very informality suggests the practical cleverness inherent in shrewd , but without that word’s suggestion of self-interest: a sharp mind for figures; a sharp awareness of social niceties.

Perspicacious is the most formal of these words; it stresses intensity of perception, without being very rich in other connotations: a perspicacious remark that illuminated the whole problem for all of them.