I was wondering about this for a while now. Could anyone explain this phenomenon or is it just "English quirks"?
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The letters we think of as vowels 'a, e, i, o, u' are commonly associated with (at least) two different actual vowel sounds. These are so deeply embedded into the minds of English speakers that most speakers won't stop to think that in contemporary English there is no phonetic relationship between these vowels at all.
In British English RP we can observe the following relationships between these letters and some of the sounds they commonly represent:
Notice that the sounds on the left are relatively short, and the ones on the right are longer. These symbols from British RP use a colon to represent length, or they use two symbols. You might also notice that the right hand sounds are actually the names of the letters involved. So the letter 'A' is actually pronounced /eɪ/. We can hear these different vowels being represented by these letters in the following pairs of words:
You will notice that one way that we represent the longer vowel sounds associated with these letters in the orthography is to put an 'e' after a single consonant at the end of a word.
This is what we see in the word invoke. The purpose of the E following the K in this word is to show that the O represents /əʊ/ and not /ɒ/.
Now in the word invoke we see this convention interacting with a different one. The letter C in English more often than not represents a /k/ when followed by the written vowels A, O and U as in the words cat, cot and cut. However, when followed by an E, I or Y it often represents an /s/ sound as in the words cent, cistern or cynic. Of course, we don't see this happening with the letter K, which always represents the hard sound /k/. We can observe this contrast, therefore, in pairs of words such as mace and make or lice and like
In the word invoke we require the E to show that the O is representing a long /əʊ/ and not the short /ɒ/. If we spelled the word as either invoc or invok then a reader unfamiliar with the word would assume it rhymed with the word wok. Now if we just put an E after invoc so it was spelled invoce then an uninitiated reader would assume that the word rhymed with dose, because of the convention for using C to represent an /s/ sound when followed by E. In order to maintain the hard /k/ required, we need to use the letter K here.
Notice that in the word invocation, the C is followed by an A and so can therefore be used to represent the /k/ sound. The pattern that we see above is, of course, also seen in the other pairs of words listed in the question.
Actually, invocation comes from the Latin invocatio(n), from the verb invocare, via Old French; invoke comes from French invoquer, from the Latin invocare.
in the first case the word derives from a word containing ca, which is maintained in the English word; in the latter case, the word derives from a French work containing que, which has been changed in ke.
There are some words that derives from a word in another language that contains que. In some cases, the que part is changed in ke; in other cases, the que part is kept in English too. For example, conquest derives from the Old French conquest(e), which comes from the Latin conquirere; conquer derives from Old French conquerre, which comes from the Latin conquirere.
Actually, taking a look at their etymologies, you'd be expecting "invoke" to be spelt "invoce"!
invoke: late 15c., from M.Fr. envoquer (12c.), from L. invocare "call upon, implore," from in- "upon" (see in- (2)) + vocare "to call,"
Invocation: late 14c., "petition (to God or a god) for aid or comfort; invocation, prayer;" also "a summoning of evil spirits," from O.Fr. invocation (12c.), from L. invocationem, noun of action from pp. stem of invocare
So, they've got the exact same root, but why is "invoke" spelt "invoke", or "invocation" spelt "invocation"?
My conjecture is that "invoce" was deemed easily confused in pronunciation (Invose?) as it is followed by an "e", and so, to reaffirm its "k" sound, they wrote it as "invoke" instead of "invoce". "Invocation" is followed by an "a", so change of spelling was not necessary.
And by the way, yes, that's the correct word.
For "invocation," your cognate should be "vocal." The spelling of "invoke" was likely a convention adopted in spite of the etymology.