late 14c., "originating in the root or ground;" of body parts or fluids, "vital to life," from Latin radicalis "of or having roots," from Latin radix (genitive radicis) "root" (from PIE root *wrād- "branch, root"). The basic sense of the word in all meanings is "pertaining or relating to a root or roots," hence "thoroughgoing, extreme."
The figurative meaning "going to the origin, essential" is from 1650s. The political sense of "reformist" is by 1817, of the extreme section of the British Liberal party (radical reform had been a current phrase since 1786), via the notion of "change from the roots" (see radical (n.)). The meaning "unconventional" is from 1921. U.S. youth slang use is from 1983, from 1970s surfer slang meaning "at the limits of control."
The mathematical radical sign, placed before any quantity to denote that its root is to be extracted, is from 1680s; the sign itself is a modification of the letter -r-. Radical chic is attested from 1970; popularized, if not coined, by Tom Wolfe. Radical empiricism was coined 1897 by William James (see empiricism).
1630s, in philology, "root part of a word, primitive verbal element serving as the root of inflected or derivative words," from radical (adj.) Political sense of "extremist, person who holds radical principles, one who pursues a theory to its furthest limit" is by 1802; chemical sense is by 1816.
In the political sense, in which the word has been most used, a Radical is one who aims at thorough reform in government from a liberal or democratic point of view, or desires the establishment of what he regards as abstract principles of right and justice, by the most direct and uncompromising methods. ... The name Radical is often applied as one of reproach to the members of a party by their opponents. [Century Dictionary, 1895]