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Appears in Japanese in 1872 in the 代微積拾級訳解(Dai Biseki Jikkyū Yakkai, literally “Translation and Explication of Algebraic and Differential Calculus”), a Japanese translation by 福田理軒 (Fukuda Riken) and 福田半(Fukuda Han) of the 1851 book Elements of Analytical Geometry and of the Differential and Integral Calculus by Elias Loomis. According to the foreword, the translators referenced an earlier Chinese translation from 1859, so the term likely derives originally from Chinese拋物線/抛物线(pāowùxiàn) (see that entry for further details).
According to other Japanese sources, analyzed as a compound of 抛物(hōbutsu, literally “throwing thing”, perhaps in reference to the etymology of parabola) + 線(sen, “line”). See also related term 放物面(hōbutsumen, “paraboloid”).
Note the shift from earlier spelling 抛物線, now considered an archaic alternative, to 放物線 in modern usage. 放(hō, “to release, to set free”) is a daiyōji replacing 抛(hō, “to throw, to hurl”).