Heighth is no error
It is a misunderstanding that the spelling or pronunciation of heighth is an illiterate and uneducated error. Although many wrongly consider it such, history is not on their side, nor are the better dictionaries.
Despite how in particular over the last century the heighth spelling has come to be stigmatized, heighth is a perfectly legitimate word of ancient lineage. It was used not only by Shakespeare and Milton, but even by Charles Dickens, who wrote considerably later than the first two.
The spelling that is no longer used is hight, although it was once common. Interestingly, Shakespeare variously employed not only height and heighth but also hight, depending on the work:
•1591 Shaks. Two Gentlemen from Verona, ɪᴠ. iv. 169 — I know she is about my height.
•1594 Shaks. Richard III, ɪ. iii. 41 — I feare our happinesse is at the height.
•circa 1600 Shaks. Sonnet xxxii — Exceeded by the hight of happier men.
•1606 Shaks. Anthony & Cleopatra ɪɪɪ. x. 21 — Anthony..Leauing the Fight in heighth, flyes after her.
•1606 Shaks. Troilus & Cressida, ᴠ. i. 3 — Let vs Feast him to the hight.
•1613 Shaks. Henry VIII, ɪ. ii. 214 — By day and night Hee’s Traytor to th’ height.
Editions of Anthony & Cleopatra with normalized spellings now generally read:
The noble ruin of her magic, Antony,
Claps on his sea-wing and, like a doting mallard
Leaving the fight in height, flies after her.
I never saw an action of such shame.
Experience, manhood, honor, ne’er before
Did violate so itself.
The thing to understand, though, is that this is not some spelling that died out when the reign of Elizabeth I came to an end. It remained in use by writers of unimpeachable integrity up through the 19th century and sometimes even into the early 20th.
All three forms of the word — height, heighth, hight — were long used, but in truth, only the last of those has completely fallen away. For while the OED no longer admits hight as a modern spelling, it does present both height and heighth as admissible variants, with -th listed second. In its rather long etymology section on this word, it observes