"gay" 與 "cheerful" 的分別

Cheerful = happy, behaving in a way that shows you are happy or tending to be happy most of the time (adjective form of “cheer”)
Cheer originates from old French word, “chiere” - face. The original sense was “face”, hence “expression, mood”.
We tend to use “cheerful” to describe one’s mood, manner or disposition.
You are in a cheerful mood this morning as shown on your smiling face. Last night, you were in a bad mood.

Gay = happy, merry
It originates from the old French word “gai” - happy
Before 60s, gay could be used to describe one’s lifestyle which is happy and full of fun and laughter.

"Gay Science has overtones of a light-hearted defiance of convention; it suggests Nietzsche's "immoralism" and his "revaluation of values."
~ a statement by the author.

I think there is a subtle difference between “cheerful” and “gay”. In order to understand why Kaufmann used “gay”, not “cheerful”, it is advisable to read Nietzsche’s works. The words “Gay Science” translate the German title “Die fröhliche Wissenschaft”. No one, presumably, is going to be misled by the more recent associations of the word “gay”(homosexual) – it simply means joyful, light-hearted, and above all, lacking in solemnity. The Chinese translation 《快樂的科學》is quite misleading. The word “Wissenschaft”, unlike the English word “science” in its modern use, does not mean simply the natural science like physics, chemistry, biology … etc. It means any organized study or body of knowledge, including history, philosophy, criticism and generally what we call ‘the humanities” and that is often what Nietzsche has in mind. In the title, it translates a phrase, “gai saber”, as Nietzsche writes on his title page, “gaya scienza”, which referred to the art of songs. “Wissenschaft” means to convey “a new image and ideal of free spirits”.

題外:
Nietzsche's criticism of morality and Christianity is well known. His famous phrase is “God is dead”. Whether you buy his doctrine or not 仁者見仁, 智者見智. He died insane.

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這是個好問題!

文章正正指出1950年前gay這個字的意思就是cheerful(快樂/愉悅)，但由於1969年後同性戀組織使用了這個gay字作為名詞(當時用意上亦可能是想指出性取向不同的人可快樂開放自由地生活)，現今在翻譯或名稱上都會忌諱/避免用gay字以免讀者或驟眼看來以為是「同性戀」的主題。