Glosses of Full Fathom Five
According to the OED, there are a number of definitions of the word "fathom" which might shed light on its use in Ariel's song:
1. A measure of length; the length of a forearm; a cubit.
2. The embracing in arms.
3. fig. Breadth of comprehension, grasp of intellect; ability. Obs. exc. arch.
First, we have the fathom as it functions on a literal level: as a unit of length or, in this case, depth. Five fathoms would apparently have been deep enough to render Alonso lost. The second definition allows the word "fathom" to function as imagery -- the sea has embraced Alonso, and taken him in permanently. The final definition of a fathom as a unit of thought or the range of comprehension suggests that the magnitude of the loss is beyond what Ferdinand can process -- it is five times more than he can fathom. This phrase is thereby pulling triple duty, painting a rich picture of associations at the beginning of Ariel's song.
Beginning the song with such excessive alliteration adds to the sense of magic inherent in the song -- it is sung by an unseen, mystical voice, and the repetitive sounds give it the feel of an incantation. There is excessive assonance as well, in pairings such as fathom/father and thy/lies. This mastery of language is associated elsewhere in the play with other sorts of power, with magic, and with the voice of Shakespeare himself.
The word "fathom" occurs only one more time in The Tempest, during Prospero's famous monologue relinquishing power over the island: "I'll break my staff, / Bury it certain fathoms in the earth, / and deeper than did ever plummet sound / I'll drown my book." Thus within the play, we can see a direct correlation between the vast distance of a fathom as a distance beyond reach, and the loss of power.
Although "fathom" appears many times across the body of Shakespeare's work both as a literal measurement and an allusion to understanding, there is only one other case where the specific measure of five fathoms appears: in Mercutio's famous "Queen Mab" monologue of R&J 1.4.~85:
And then dreams he of cutting foreign throats,
Of breaches, ambuscadoes, Spanish blades,
Of healths five fathom deep; and then anon
Drums in his ear, at which he starts and wakes,
And being thus frighted, swears a prayer or two
And sleeps again.
The "healths" described here probably refer, according to the OED, to "toasts drunk in a person's honour." The five fathoms, then, is intended to conjur the image of vast quantities of liquor as part of the soldier's dream of glory, which is nevertheless frightening enough to prompt him to say prayer before returning to sleep. The measurement of five fathoms in both cases occurs in a dream-like, magical setting that is both precious and somewhat scary. Since the R&J reference occurs in the description of Queen Mab, we can deduce that there is an association in the mind of the author between that particular depth and the world of spirits and fairies.
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