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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.
During the long history of China, archery held a very important place in the life of its citizens. It was one of the Six Noble Arts and a method of choosing officers in the army, a method of hunting and an art itself. Read more about history of Chinese archery.
Composite bow, although a bow if a pretty complex design, is old weapon and many civilizations knew about it and used it above others. It can be made to be much smaller and stronger but it could last for a shorter time if exposed to humidity.
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.