Why was 非 chosen for 非洲 (Africa) and 亚 chosen for 亚洲 (Asia)?
I can't find a particular reason to use these characters.
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亚洲 is short for 亚细亚洲 while 非洲 is short for 阿非利加洲.
See the article "各大洲名称由来".
亞洲 (亚洲) is short for 亞細亞洲 (亚细亚洲, Asia) which can be found as early as 坤輿萬國全圖 published in 1602 (萬曆三十年) , mainly by Italian Jesuit priest Matteo Ricci or 利瑪竇 (Lì Mǎdòu), 6 Oct 1552 - 11 May 1610, who is believed to coin this transliteration into Chinese characters.
非洲 is from 阿非利加洲 (Africa). The above work, however, gives a different name for Africa: 利未亞 (Libya) for some historical reason.