The Cambridge guide to English Usage is available for free online! I just looked it up and this is what it says about h.
A single rule resolves all such queries: a is used before words beginning with a consonant, and an before those beginning with a vowel. This is straightforwardly applied in a doctor, a receptionist and an astronaut, an engineer. But note that the rule depends on the sound not the spelling. We write a union, a unique gift and a once-in-a-lifetime experience because the words following the article actually begin with a consonant sound (the “y” sound in the first two cases, and the “w” sound in the third). The same principle makes it an hour, an honor, and an honest man. The word following the indefinite article begins with a vowel sound.
Despite all that, certain words beginning with h are made exceptions by some writers and speakers. Theywould preface hotel and heroic with an rather than a, despite pronouncing the h at the start of those words. Other polysyllabic words beginning with h will be given the same treatment, especially if their first syllable is unstressed. In both American and British English the words historic, historical and historian are the most frequent of these exceptional cases, but the tendency goes further in Britain, by the evidence of matching databases (LOB and Brown corpora).
They show that British writers use an to preface adjectives such as habitual, hereditary, heroic, horrific, hypothetical, hysterical (and their adverbs) as well as the noun hotel. There are far fewer examples in the American data, and the only distinctive case is herb, which is commonly pronounced without h in the US (though not in the
UK or elsewhere). The King James bible (1611) records the use of an with other monosyllabic words, as in an host and an house, though they are supposed to go with h-less pronunciations, formerly much more common.
Over the centuries h has been an uncertain quantity at the beginnings of words in many European languages. Most words beginning with h lost it as they passed from Latin into French and Italian. The Latin word hora meaning “hour” became French heure (pronounced “err,” with no h sound) and also the Italian ora, without an h even in the spelling. English retains an h in the spelling of hour but not in the pronunciation. The process also shows up in the contrasting pronunciations of heir (an early English loan from French) and hereditary (a Renaissance borrowing direct from Latin), which embody the same Latin stem. Spelling pronunciation has revived the hin some French loanwords like heritage and historian (those well used in English writing); while others such as hour, heir, hono(u)r are h-less, in keeping with French pronunciation. Classical loanwords (apart from honorary, honorarium, honorific) have settled on pronunciations with the h sounded; and they complement the many basic Anglo-Saxon words such as here, how, him and hair, home, honey in which h is pronounced. (See further under h.)
Nowadays the silent h persists in only a handful of French loanwords (heir, honest, hono(u)r, hour and their derivatives), and these need to be preceded by an. The h of other loans like heroic, historical and hypothesis may have been silent or varied in earlier times, leaving uncertainty as to whether an was required or not. But their pronunciation is no longer variable and provides no phonetic justification for an. Its use with them is a stylistic nicety, lending historical nuances to discourse in which tradition dies hard.