Etymology
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rhyme (n.)

"agreement in terminal sounds of words or metrical lines," a 16c. attempt to restore a classical spelling to Middle English ryme, rime (c. 1200) "measure, meter, rhythm," later "rhymed verse" (mid-13c.), from Old French rime (fem.), which is related to Old Provençal rim (masc.), earlier *ritme, from Latin rithmus, from Greek rhythmos "measured flow or movement, rhythm; proportion, symmetry; arrangement, order; form, shape, wise, manner; soul, disposition," related to rhein "to flow" (from PIE root *sreu- "to flow").

The persistence of rime, the older form of the word, perhaps is due to popular association with Old English rim "number" (from PIE root *re- "to reason, count"). The intermediate form rhime was frequent until late 18c.

In Medieval Latin, rithmus was used for accentual, as opposed to quantitative, verse, and accentual verse usually was rhymed, hence the sense shift. In prosody, specifically the quality of agreement in end-sounds such that the last stressed vowel, and any sounds after it, are the same, and preceding sounds differ. 

Verse was invented as an aid to memory. Later it was preserved to increase pleasure by the spectacle of difficulty overcome. That it should still survive in dramatic art is a vestige of barbarism. [Stendhal "de l'Amour," 1822]

The sense of "a piece of poetry in which consonance of end-sounds is observed" is from 1610s. From 1650s as "word that rhymes with another." The phrase rhyme or reason "good sense" (chiefly used in the negative) is from late 15c. (see reason (n.)). Rhyme scheme "ordered pattern of end-rhymes in metrical composition" is attested from 1931. Rhyme royal (1841) is a stanza of seven 10-syllable lines rhymed a-b-a-b-b-c-c.

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rhyme (v.)

a modern spelling variant or replacement of Middle English rime, rimen, from Old French rimer, from rime "verse" (see rhyme (n.)). The Middle English word is attested from late 12c. as "poetic measure, meter," from c. 1300 as "agreement in terminal sounds of words or metrical lines; a rhyming song or ballad."

 The spelling shifted from mid-17c. by influence of rhythm and Latin rhythmus, from the same Greek source, and the intermediate form rhime is frequent for a while (Dryden and Steele have rhime; Pope and Scott rhyme). Related: Rhymed; rhyming; rhymer (Middle English rimer, early 15c., from rime, also from Anglo-French rimour, Old French rimeur).

The poetaster's rhyming dictionary is attested from 1775 (in John Walker's introduction to his "Dictionary of the English Language, Answering at once the Purposes of Rhyming, Spelling, and Pronouncing. On a Plan Not Hitherto Attempted"). The phrase rhyming slang for the Cockney disguised speech in which a word is replaced by a phrase which rhymes with it is attested from 1859 (the thing itself described by 1851). Especially if the rhyming word is then omitted, which seals the reference from the uninitiated: Richard, in rhyming slang "a girl" (a couple of likely Richards), short for Richard the Third, chosen to rhyme with bird "girl."

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off-rhyme (n.)

"partial or near rhyme," 1938, from off (prep.) + rhyme (n.).

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ottava rima 

form of versification, 1820, Italian, "eight-lined stanza," literally "eighth rhyme," from ottava "eighth" (see octave + rhyme (n.)). A stanza of eight 11-syllable lines, rhymed a b a b a b c c, but in the Byronic variety the lines are typically 10-syllable English heroics.

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rhythm (n.)

 16c. spelling variant or attempted classical correction of Middle English rime "measure, meter, rhythm," also "agreement in end-sounds of words or metrical lines, rhyme; a rhyming poem" (12c.), from Old French rime "verse," from Latin rhythmus "movement in time," from Greek rhythmos "measured flow or movement, rhythm; proportion, symmetry; arrangement, order; form, shape, wise, manner; soul, disposition," related to rhein "to flow" (from PIE root *sreu- "to flow"). Compare rhyme.

The word rhyme has no connection with the word rhythm, nor is rhyme necessary to accentual verse. Nevertheless, rhyme was usually present. On the other hand, in classical Greek metrical poetry, rhymes, if not accidental, were never an essential element of metrical verse structure. [Henry Osborn Taylor, "The Classical Heritage of the Middle Ages," 1911]

The spelling fluctuated 16c.-17c., rithme and ri'me also being used. From 1550s as "metrical movement, movement in time characterized by equality of measures and alteration of stress and relaxation." By 1776 as "regular succession of beats or accents in music."

The rhythm method in reference to birth control is attested from 1936. Rhythm and blues, U.S. music style, is from 1949 (first in Billboard magazine).

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*re- 

*rē-, Proto-Indo-European root meaning "to reason, count;" a variant of PIE root *ar-, also arə-, "to fit together." 

It forms all or part of: Alfred; arraign; arithmetic; Conrad; dread; Eldred; Ethelred; hatred; hundred; kindred; logarithm; Ralph; rate (n.) "estimated value or worth;" rathskeller; ratify; ratio; ration; read; reason; rede; rhyme; riddle (n.1) "word-game;" rite; ritual.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit radh- "to succeed, accomplish;" Greek arithmos "number, amount;" Latin reri "to consider, confirm, ratify," ritus "rite, religious custom;" Old Church Slavonic raditi "to take thought, attend to;" Old Irish im-radim "to deliberate, consider;" Old English rædan "to advise, counsel, persuade; read;" Old English, Old High German rim "number;" Old Irish rim "number," dorimu "I count."

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free verse (n.)

1869; Englishing of vers libre.

But it is possible that excessive devotion to rhyme has thickened the modern ear. The rejection of rhyme is not a leap at facility; on the contrary, it imposes a much severer strain upon the language. When the comforting echo of rhyme is removed, success or failure in the choice of words, in the sentence structure, in the order, is at once more apparent. Rhyme removed, the poet is at once held up to the standards of prose. Rhyme removed, much ethereal music leaps up from the word, music which has hitherto chirped unnoticed in the expanse of prose. [T.S. Eliot, "Reflections on Vers libre," New Statesman, 3 March 1917]
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rock-a-bye 

phrase in nursery rhyme sleeping-songs, by 1805; see rock (v.1).  Compare lullaby.

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homoioteleuton (n.)

 in general, "the repetition of endings in words, rhyme and near rhyme," but also, in palaeography, a form of scribal error which occurs "when two words/phrases/lines end with the same sequence of letters. The scribe, having finished copying the first, skips to the second, omitting all intervening words" [Robert B. Waltz, "The Encyclopedia of New Testament Textual Criticism," 2013]; Greek, literally "same ending;" see homo- (1) "the same" + telos.

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terza rima (n.)
1819, Italian, literally "third rhyme." Dante's measure: a-b-a-b-c-b-c-d-c-, etc.
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