有 and 机 are some of the first characters we learn in Chinese, but only recently did I find out that they mean "organic" when put together. Can anyone explain why this is?
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As you may know, "Organic" in your case is originally a chemical term and its noun-"Organ" means "a collection of tissues joined in structural unit to serve a common function" from Wikipedia. Once the chemists incorrectly believed that "organic compound" could only be produced by lifes. I think this is how "organic" comes from "organ". I don't find the exact origin of the word "有机" in Chinese, but I believe it originates as "organic" originates in English, from chemistry.
机 originally means
Some other meanings extend from the original one.
In recent times, after Chinese touched the sciences from the west, some pioneering scientists tranlated the terms in sciences. In general, they translated them in a liberal way.I think this is why “有机" means "organic", as you have "organic" from "organ" in English.
I never gave thought about how "有机" came to mean "organic" before.
I search in a dictionary give the following meaning for "机": 有生命的生物体器官的作用 (The functionality of organs in lifeforms). 《现代汉语词典》 the following meaning: 生活机能 (ability to live). "有机" is formed from this meaning of "机". But the usage of this meaning of "机" is very limited, probably only in "有机" "无机" "机能", so it should be save to remember these words as a whole.
BTW: I don't know how "机" got this meaning.