The ae at the front of aesthetic is a Greek diphthong that travelled through Latin and came out the other side as a two-letter spelling English never quite digested. The Greek is aisthetikos, "perceptive, of or for perception by the senses," from aisthanesthai, "to perceive, to feel." American spelling has largely simplified it to esthetic; British spelling keeps the ae, and the word remains a good example of how a classical spelling can survive centuries of use without ever matching the way anyone says it.
The route into English is unusually recent and indirect. English took it in 1798 from German Ästhetisch or French esthétique, not from Greek directly, which is why it arrived already carrying a German philosophical argument. Kant used it in the strict sense of the science of sense perception; Alexander Baumgarten had used it to mean criticism of taste; and Baumgarten's looser sense won out in English from the 1830s. Walter Pater's use of it in 1868 for the art-for-art's-sake movement finished the job of pulling it away from philosophy.
aesthetic (n.) 1798, from German Ästhetisch (mid-18c.) or French esthétique (which is from German), ultimately from Greek aisthetikos "of or for perception by the senses, perceptive," of things, "perceptible," from aisthanesthai "to perceive (by the senses or by the mind), to feel," from PIE *awis-dh-yo- , from root *au- "to perceive." Popularized in English by translations of Kant and used originally in the classically correct sense "science which treats of the conditions of sensuous perception" [OED]. Kant had tried to reclaim the word after Alexander Baumgarten had taken it in German to mean "criticism of taste" (1750s), but Baumgarten's sense attained popularity in English c. 1830s (despite scholarly resistance) and freed the word from philosophy. Walter Pater used it (1868) to describe the late 19c. movement that advocated "art for art's sake," which further blurred the sense. [Whewell had proposed callesthetics for "the science of the perception of the beautiful."] As an adjective by 1798 "of or pertaining to sensual perception;" 1821 as "of or characterized by appreciation of the beautiful." Related: Aesthetically . also from 1798
aesthetic came in through Greek, and its spelling follows that family's conventions.
How that family writes these sounds
| /iː/ or /ɛ/ in an old learned word | ae or oe — American spelling simplifies both to eaesthetic, amoeba, oesophagus |
What gives it away on the page
| ae, oe | Greek diphthongs through Latin. American spelling often simplifies them to e.aesthetic, amoeba, oesophagus |