Was "book" to "beek" as "foot" is to "feet"?

Whenever you find an O (or some other back vowel like A or U) in one form of an English word and an E (or some other front vowel like Æ or I) in the corresponding place in another, you have two suspects to interrogate.

1. If the two words are not from the same language, but from two separate Indo-European languages, like Latin and Greek (e.g, ped-al from Latin and pod-iatrist from Greek, both roots meaning 'foot' — the p ~ f and d ~ t alternations are Grimm's Law in action), then what you're seeing is E–O Ablaut. Proto-Indo-European often alternated between an "E-grade" and an "O-Grade" form for morphology, and various daughter languages inherited various words. Sort of like the distribution of family furniture when the parents die. That's not what happened here, however.

2. If the two words are from English, and not borrowed, then what you're seeing is Umlaut. This term refers to changing a back rounded vowel [o, u] to the corresponding front rounded vowel [œ, y] in anticipation of a front vowel [i, e] in the next syllable. This is a common Germanic feature, and is still productive in German, where there are special vowel symbols (ü, ö, ä, called U-umlaut, O-umlaut, and A-umlaut) that represent this phenomenon, and these sounds. Umlaut is the guilty suspect.

In English, this happened to many normal plurals because of the E in the regular plural suffix -es. That E was pronounced in Old English and Middle English, but not in Modern English; however the root vowel had been changed already and is maintained in some, but not most, of the nouns.

The original Old English (or possibly Proto-, West, or Low Germanic) of goose/geese (in Modern English [gus/gis]) was [go:s/go:ses]. There were several steps in the derivation:

• starting words: [go:s/go:ses]
• final s going silent ⇒ [go:s/go:se]
• fronting the o: to œ: by umlaut ⇒ [go:s/gœ:se]
• derounding œ to e ⇒ [go:s/ge:se]
• final e going silent ⇒ [go:s/ge:s]
• Great Vowel Shift raising all long vowels one notch ⇒ [gu:s/gi:s]
• long vowels become short ⇒ [gus/gis]
• ending words: [gus/gis]

Edit: In this medium, where writing and typography has to express speech and sounds, I use italics and boldface like this:

• I use plain italics only for citing examples and titles. Never for emphasis.
• I use boldface for emphasis. These are words that would be LOUD in my speech.
• I use bold italics for technical terms, usually with capitals, and links if I have them.
• I also use bold italics in examples to point out individual parts that get mentioned in the text.
• Like all linguists, I use
  • [ˌskwɛɹ'bɹækɨts] (square brackets) for phonetic data,
    distinguished from
  • /'slæʃəz/ (slashes) for phonemic transcriptions.

Here is a quote from the Old English dictionary:

bóc [] f (béc/béc) 1. a book, a document, register, catalog; 1a. a legal document, (1) a bill of divorce; (2) a charter; (3) a title deed; (4) conveyance; 2. a book, volume, literary work, pages; main division of a work;

Let's now look at:

fót [] m (-es/fét) 1. a foot; 2. the foot, the foot of a man

So, I guess yes, we can say that at least the word from which modern book evolved had plural in the same form as the word from which modern 'foot' evolved. I guess, as early of Middle English period, "book" lost its irregular plural, since Chaucer used bookes.

It's also interesting to notice that fót actually had an alternative form fótes, but this form has not survived.

As a rule, morphological irregularities of Modern English are remnants of previously regular processes of Old English.

So, if English had been a more conservative language, we would have retained beech as the plural of book. Moreover, we would have had book as the past tense of bake (like took and shook from take and shake).

German, which is more morphologically conservative than English, still has vowel alternation in both these places. The plural of Buch “book” is Bücher, and the past tense of back- “bake” is buk (though a regular form, backte, is emerging, just like English baked).

According to the Online Etymology Dictionary this is an instance of I-Mutation

I-MUTATION (also known as "i-umlaut") is the raising and fronting of a root vowel in anticipation of "i" or "y" sound in a suffix.

Other examples of noun plurals that exhibit I-Mutation are

man → men
tooth → teeth
goose → geese
louse → lice
mouse → mice

I-mutation is caused by the very human habit of laziness: taking the shortest distance between two points. The plural of man in ancient West Germanic, the ancestor of Old English, used to be a word something like *manniz. The speakers "cheated" on the first vowel in the word to be in position for the second vowel. It's the same thing you do with doing. It doesn't change the meaning of the word to do so.

So, it comes from the old Saxon roots of English. Book, being a relatively new and uncommon noun compared to foot, probably wasn't shortened so commonly.