Oh boy....WELL, part 1 of the answer and in my opinion, the most important part is I'm afraid that you have a bit of a misunderstanding of how Hangul works. It is an alphabet....just like ours. It IS NOT a series of characters like chinese. Saying that Hangul is comprised of characters is akin to saying that English is comprised of characters. Also, saying that the ye in yedo and the ye in muye are the same "character" is kind of like saying that the words food and good share the od "character." It just so happens that those two words use the same letters, specifically....they use the null character ㅇ and the ㅖ character. ACTUALLY, the WORD 예 means "instance." (as in for instance or an instance of something)
Oh yeah....and the fact that 예 means instance, does not mean that the translation for yedo is "Instance way." Yedo is a separate word that just so happens to share some LETTERS.
Ok, now on to your real question. The word 무예 specifically means "Military Arts." You are right about the translations of mudo and musul. What is confusing to most people is that we aren't REALLY talking about Korean or Hangul here....
Whoa! Right? Yes, these words are not Korean words. They are Hanmun (or Hanja), meaning that they are words borrowed from Chinese. When you take a Chinese word and write it in Chinese, but do it IN KOREA, it is pronounced differently and is called Hanmun or Hanja. NOW, those words have been increasginly often written in hangul so that more people can recognize them. The average Korean child will learn about 16,000 chinese characters and their korean pronunciations before leaving high school. This is needed to survive in Korea. Many newspapers are still speckled with hanmun. MANY MANY academic books are written almost exclusively in hanmun (that means that it can be read by a Korean OR a Chinese person, but would be pronounced differently). In fact, when I took my cat to the vet in Korea, every book on his shelf was written in Hanmun (or chinese).
The words mudo, musul, and muye are really hanmun, which are used so frequently that the Korean have started writing them in Hangul to simplify things. They are gradually getting rid of the chinese characters so that more people can read things...then taking those words and making them a part of the normal Korean language. So Mudo is actually....武通. Look familiar? First character of Moo duk kwan and the last character of soo bahk do. Musul and Muye are the same, they have chinese equivalents.
More importantly, UNLIKE Hangul, those words can be broken down into their root words. SO, 武通 is actually the character 武 or military and the character 通 or way put together to make martial arts or martial way, whatever you prefer. In hangul, that isn't always true. You can't take yedo and break it apart into instance and way.
SO, the Muyedobotongji is not actually one really long Korean word. It is actually 武藝圖譜通志. Now that CAN be broken down into individual words related to each chinese character. Unfortunately, I can't get to my chinese dictionaries at work...so I can't translate it from here and I don't remember the whole thing. But portions of it have been borrowed and made a part of the Korean language - such as muye.
Ok, I have to get back to work - does that answer the question?
Incidentally 銳刀 means sharp sword.
