Etymology
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John Hancock 

colloquial for "signature," 1903 (sometimes, through some unexplainable error, John Henry), from the Boston merchant and rebel (1736-1793), signer of the U.S. Declaration of Independence. The extended sense is from his signing that dangerous document first or most flamboyantly.

John Hancock, president of Congress, was the first to sign the Declaration of Independence, writing his name in large, plain letters, and saying: "There; John Bull can read my name without spectacles. Now let him double the price on my head, for this is my defiance." [Hélène Adeline Guerber, "The Story of the Thirteen Colonies," New York, 1898]

The family name is attested from 1276 in Yorkshire, a diminutive (see cock (n.1)) of Hann, a very common given name in 13c. Yorkshire as a pet form of Henry or John.

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Tartar 

mid-14c. (implied in Tartary, "the land of the Tartars"), from Medieval Latin Tartarus, from Persian Tatar, first used 13c. in reference to the hordes of Ghengis Khan (1202-1227), said to be ultimately from Tata, a name of the Mongols for themselves. Form in European languages probably influenced by Latin Tartarus "hell" (e.g. letter of St. Louis of France, 1270: "In the present danger of the Tartars either we shall push them back into the Tartarus whence they are come, or they will bring us all into heaven").

The historical word for what now are called in ethnological works Tatars. A Turkic people, their native region was east of the Caspian Sea. Ghengis' horde was a mix of Tatars, Mongols, Turks, etc. Used figuratively for "savage, rough, irascible person" (1660s). To catch a Tartar "get hold of what cannot be controlled" is recorded from 1660s; original sense not preserved, but probably from some military story similar to the old battlefield joke:

Irish soldier (shouting from within the brush): I've captured one of the enemy.
Captain: Excellent! Bring him here.
Soldier: He won't come.
Captain: Well, then, you come here.
Soldier: I would, but he won't let me.

Among the adjectival forms that have been used are Tartarian (16c.), Tartarous (Ben Jonson), Tartarean (17c.); Byron's Tartarly (1821) is a nonce-word (but a good one). Tartar sauce is attested by 1855, from French sauce tartare.

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Hollerith (adj.)
1890, in reference to a punch-card system used in a mechanical tabulator and later for data processing in in the earliest computers, from name of U.S. inventor Herman Hollerith (1860-1929), who designed the system. For a time, in mid-20c. it sometimes was used figuratively in reference to modern society viewed as a processing machine.
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Miami 
place name in U.S.; the one in Florida is of unknown origin, attested in Spanish as Maymi (1566), Mayaimi (1575). The one in Ohio is from the Miami, native people there, attested from 18c., apparently from a native word /myaamiwa "downstream person."
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Black Death (n.)

"bubonic/pneumonic plague epidemic of 1347-51 in Europe," a modern name, introduced in English 1823 by Elizabeth Penrose's history of England. The contemporary 14c. name for it in most European languages was something like "the great dying" or simply "the plague;" in English it was the pestilence (or, looking back after its return in 1361-2, the first pestilence).

The term "Black Death" first turns up in 16c. Swedish and Danish chronicles, but it is used in reference to a visitation of plague in Iceland (which had been spared in the earlier outbreaks) in 1402-3 that carried off much of the population there. The exact sense of "black" is not clear. The term appears in English translations of the Scandinavian works from 1750s. It was picked up in German c. 1770 and applied to the earlier outbreak and was taken from there into English in that sense.

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Braxton Hicks 

in reference to uterine contractions in pregnancy, 1905, from the name of English obstetrician John Braxton Hicks, who described them in 1872.

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Trent 
river in England, a Celtic name, perhaps "great wanderer," in reference to its flooding. The city in Italy (Italian Trento) is Roman Tridentum, in reference to the triple-peaked mountain nearby. The great ecumenical council there was held from 1543-63.
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Cassegrain (adj.)
also Cassegrainian, in reference to a type of reflecting telescope, 1813, named for 17c. French priest and teacher Laurent Cassegrain, who described it in a journal article in 1672.
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Lamaze (adj.)
in reference to a method of childbirth technique, 1957, named for French obstetrician Dr. Fernand Lamaze (1891-1957), who promoted his methods of "psycho-prophylaxis," a form of childbirth preparation he had studied in the Soviet Union, in the West in the early 1950s.
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Ariel 

1382, in the Wyclif Bible, a word taken untranslated from the Vulgate, from Greek ariel (Septuagint), from Hebrew ariel; in later Bibles, translated as "altar."

(Gesenius would here translate "fire-hearth of God," after Arab. arr; elsewhere in O.T. the same word occurs as a man's name, and appellation of Jerusalem, where it is taken as = "lion of God.") Ariel in T. Heywood and Milton is the name of an angel, in Shakespeare of "an Ayrie spirit"; in Astron. of one of the satellites of Uranus. [OED]

As the name of a species of gazelle found in the Middle East, 1832, from Arabic aryil, variant of ayyil "stag." The Uranian satellite was discovered in 1851.

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