cap (n.)youdaoicibaDictYouDict
late Old English cæppe "hood, head-covering, cape," from Late Latin cappa "a cape, hooded cloak" (source of Spanish capa, Old North French cape, French chape), possibly a shortened from capitulare "headdress," from Latin caput "head" (see head (n.)).

Meaning "women's head covering" is early 13c. in English; extended to men late 14c. Figurative thinking cap is from 1839 (considering cap is 1650s). Of cap-like coverings on the ends of anything (such as hub-cap) from mid-15c. Meaning "contraceptive device" is first recorded 1916. That of "cap-shaped piece of copper lined with gunpowder and used to ignite a firearm" is c. 1826; extended to paper version used in toy pistols, 1872 (cap-pistol is from 1879).

The Late Latin word apparently originally meant "a woman's head-covering," but the sense was transferred to "hood of a cloak," then to "cloak" itself, though the various senses co-existed. Old English took in two forms of the Late Latin word, one meaning "head-covering," the other "ecclesiastical dress" (see cape (n.1)). In most Romance languages, a diminutive of Late Latin cappa has become the usual word for "head-covering" (such as French chapeau).
think (v.)youdaoicibaDictYouDict
Old English þencan "imagine, conceive in the mind; consider, meditate, remember; intend, wish, desire" (past tense þohte, past participle geþoht), probably originally "cause to appear to oneself," from Proto-Germanic *thankjan (cognates: Old Frisian thinka, Old Saxon thenkian, Old High German denchen, German denken, Old Norse þekkja, Gothic þagkjan).

Old English þencan is the causative form of the distinct Old English verb þyncan "to seem, to appear" (past tense þuhte, past participle geþuht), from Proto-Germanic *thunkjan (cognates: German dünken, däuchte). Both are from PIE *tong- "to think, feel" which also is the root of thought and thank.

The two Old English words converged in Middle English and þyncan "to seem" was absorbed, except for its preservation in archaic methinks "it seems to me." As a noun, "act of prolonged thinking," from 1834. The figurative thinking cap is attested from 1839.