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Bow and arrow are one of the oldest tools of hunt and war. They were known to almost all civilizations since before the written history and are still in use today although not in such extent. Read about history of bow and arrow.
Crossbow is a younger sibling of a bow and arrow. It is easier to use and more precise with less effort but slower. That is why it replaced bow and arrow in battlefields until it was itself replaced with firearms.
Compound bow is one of the most modern designs of a bow. It uses more complex physics to give more power to the arrow and allows an archer to hold the arrow in full draw without too much strength.
Japanese archery is an old tradition which is perfected for centuries. It appeared as the military class took the power over the country and it lasted until firearms appeared and became more effective.