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Although the word "chief" in American English has become widely
associated with indigenous Americans, the word is actually much older
than the white people's arrival in America. It is derived from the
Latin word "kaput," which meant "head."
Like most English words derived from Latin, "chief" came to us by way
of French, which explains why it is so similar to "chef," which is
short for "chef de cuisine," meaning "head of the kitchen."
Here are some sources of information on the etymology that may be helpful:
Chief (American Heritage Dictionary)
Indo-European roots (American Heritage Dictionary)
Chief (Online Etymology Dictionary)
Chef (American Heritage Dictionary)
"Sheik" has no etymological connection with "chief." Not only do the
words come from different language families, they also had different
original meanings. Rather than coming from Latin, it comes from an
Arabic word for "old man." The words' similarity in meaning is an
interesting coincidence, though.
Sheik (American Heritage Dictionary)
Semitic Roots (American Heritage Dictionary)
Sheik (Online Etymology Dictionary)
These etymologies are confirmed in the Merriam-Webster dictionary,
although I can't give you the URL for a specific word because of the
way the dictionary is set up.
I have been able to find no indication that there are any American
Indian words for "chief" that are similar in pronunciation. There are
excellent lists of American Indian language resources here:
North American languages
Native American Culture
Checking through some of the dictionaries, here are a few of the
Indian words I found for "chief":
Klallam -- si?ám? (boss or important person)
Cheyenne -- vehoo'o (chiefs)
Navaho -- bih-keh-he (war chief)
Cayuga -- hahsenowá:neh or hoyá:neh
I hope you find this information helpful.