comparative of high (adj.), Old English hierra (West Saxon), hera (Anglian). Higher education is attested by 1839.

The French distinguish l'instruction secondaire, which includes what we term a liberal education, from l'instruction supérieure, which denotes professional education; but I do not think the corresponding English phrases are used with this distinction. [William Whewell, "Of a Liberal Education in General," 1850]

Higher-up (n.) "one in a superior post" is from 1905, American English.

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Definitions of higher from WordNet

higher (adj.)
of education beyond the secondary level;
higher education
higher learning
higher (adj.)
advanced in complexity or elaboration;
higher mathematics
higher finance