For instance, とじる and しめる. とじる means an activity or a show has finished so
"the door is closed". This word is used in ending or closing something like
傘をとじる, 本をとじる.

しめる means to tie the bag tight with a string. So this word is used as "keep
something away" or "Don't let something out". For example, 窓をしめる, 鍋の蓋をしめる.

I am wondering what's the history of 衰える.

Any help is appreciated.
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nuances history
衰えるという言葉の由来は何ですか?
The main meaning of 衰える is "to go into a decline"; it's used with someone or
something which was _once strong or prosperous_. It's typically used with
things like a falling country, a human being (physically or socially), a
typhoon.

I could not find the etymology of 衰える on the net, but [this word existed 1000
years
ago](https://kobun.weblio.jp/content/%E3%81%8A%E3%81%A8%E3%82%8D%E3%81%B5). A
large dictionary about archaic Japanese may have some speculation about the
history, but I doubt it will help you understand the nuance. (I suspect you
are expecting something interesting [like
this](https://en.wiktionary.org/wiki/%E3%81%82%E3%82%8A%E3%81%8C%E3%81%A8%E3%81%86#Etymology),
but the etymology of 衰える is probably not that interesting or intuitive...)
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## The etymology: historical derivation

衰【おとろ】える comes from earlier 衰【おとろ】へる. This is the regular shift to
下一段活用【しもいちだんかつよう】 conjugation of earlier 下二段活用【しもにだんかつよう】 verb 衰【おとろ】ふ. For
more about this verb conjugation pattern, see [the Japanese Wikipedia
article](https://ja.wikipedia.org/wiki/%E4%B8%8B%E4%BA%8C%E6%AE%B5%E6%B4%BB%E7%94%A8),
or more briefly, [the English Wiktionary
entry](https://en.wiktionary.org/wiki/%E4%B8%8B%E4%BA%8C%E6%AE%B5%E6%B4%BB%E7%94%A8#Japanese).

衰【おとろ】ふ itself derives from 劣【おと】る in the less-common //o// variant of the
未然形【みぜんけい】 (irrealis form) + suffix ふ that indicates repetition, ongoing
action, or continued change.

So:

  * 劣【おと】る: "to decline in money or amount, to make a loss; to be lesser, to be inferior" →
  * おとろ + ふ → 衰【おとろ】ふ: "to be continuously lesser: to become lesser, to become weaker, to be in ongoing decline" →
  * 衰【おとろ】へる: regular 下一段 shift
  * 衰【おとろ】える: regular shift where mid-word //h//'s disappear

### Sources

  * Shogakukan's 国語大辞典: dead tree version
  * Daijisen and Daijirin: entries for 衰える available online via Kotobank [here](https://kotobank.jp/word/%E8%A1%B0%E3%81%88%E3%82%8B-453573)
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