Etymology
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*ghos-ti- 

Proto-Indo-European root meaning "stranger, guest, host," properly "someone with whom one has reciprocal duties of hospitality," representing "a mutual exchange relationship highly important to ancient Indo-European society" [Watkins]. But as strangers are potential enemies as well as guests, the word has a forked path.

The word ghos-ti- was thus the central expression of the guest-host relationship, a mutual exchange relationship highly important to ancient Indo-European society. A guest-friendship was a bond of trust between two people that was accompanied by ritualized gift-giving and created an obligation of mutual hospitality and friendship that, once established, could continue in perpetuity and be renewed years later by the same parties or their descendants. [Calvert Watkins, "American Heritage Dictionary of Indo-European Roots"]

It forms all or part of: Euxine; guest; hospice; hospitable; hospital; hospitality; hospodar; host (n.1) "person who receives guests;" host (n.2) "multitude;" hostage; hostel; hostile; hostility; hostler; hotel; Xenia; xeno-; xenon.

It is the hypothetical source of/evidence for its existence is provided by: Greek xenos "guest, host, stranger;" Latin hostis, in earlier use "a stranger," in classical use "an enemy," hospes "host;" Old Church Slavonic gosti "guest, friend," gospodi "lord, master;" Old English gæst, "chance comer, a stranger."

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*sawel- 

*sāwel-, Proto-Indo-European root meaning "the sun." According to Watkins, the *-el- in it originally was a suffix, and there was an alternative form *s(u)wen-, with suffix *-en-, hence the two forms represented by Latin sol, English sun.

It forms all or part of: anthelion; aphelion; girasole; heliacal; helio-; heliotrope; helium; insolate; insolation; parasol; parhelion; perihelion; Sol; solar; solarium; solstice; south; southern; sun; Sunday.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit suryah, Avestan hvar "sun, light, heavens;" Greek hēlios; Latin sol "the sun, sunlight;" Lithuanian saulė, Old Church Slavonic slunice; Gothic sauil, Old English sol "sun;" Old English swegl "sky, heavens, the sun;" Welsh haul, Old Cornish heuul, Breton heol "sun;" Old Irish suil "eye;" Avestan xueng "sun;" Old Irish fur-sunnud "lighting up;" Old English sunne German Sonne, Gothic sunno "the sun."

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rat-race (n.)

also rat race, 1934 in reference to aviation training, from rat (n.) + race (n.1).

A rat race is ... a simple game of "follow the leader" in fighter planes. The leader does everything he can think of — Immelmanns, loops, snap rolls, and turns, always turns, tighter and tighter. [Popular Science, May 1941]

In the 1930s actual rat races of some sort are frequently mentioned among popular carnival and gambling attractions. Meaning "fiercely competitive struggle," especially to maintain one's position in work or life is by 1939. Rat-run is from 1870 in the sense of "maze-like passages by which rats move about their territory," but originally and usually in a derogatory transferred sense.

[Matthew] Milton was not, at the period we write of [c. 1811], at all in the ring ; for in the following March he was steward of a rat-race, held at a public-house in Shepherd's-market, where four of these "varmin," decorated with different coloured ribands, were started for a sweepstakes, round the clubroom, before a host of sportsmen. ["Sporting Incidents at Home and Abroad," The Sporting Review, May 1848]
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*g(a)lag- 
also *g(a)lakt-, Proto-Indo-European root meaning "milk."

It forms all or part of: ablactation; cafe au lait; galactic; galaxy; lactate (v.); lactate (n.); lactation; lacteal; lactescence; lactic; lactivorous; lacto-; lactose; latte; lettuce.

It is the hypothetical source of/evidence for its existence is provided by: Latin lac (genitive lactis) "milk;" Greek gala (genitive galaktos), "milk;" Armenian dialectal kaxc' "milk." The initial "g" probably was lost in Latin by dissimilation. This and the separate root *melg-, account for words for "milk" in most of the Indo-European languages. The absence of a common word for it is considered a mystery.
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-i- 
a "connective" element in many words formed with Latin or Greek suffixes, now often felt as part of them (as in -iac, -iacal, -ial, -ian, -ify, -ity, etc.). Properly it forms no proper part of the suffix but is often the stem-vowel of the initial word in the Latin compounds (genial from genius), or a modified form of it. As such forms were very common, -i- was used merely connectively or euphonically in some Latin compounds (uniformis) and in later words made from Latin components in English or French (centennial, editorial).

The Greek equivalent is -o-, which also became an active connective in English, but they now are used indifferently with elements from either language.
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*wegh- 

Proto-Indo-European root meaning "to go, move, transport in a vehicle."

The root wegh-, "to convey, especially by wheeled vehicle," is found in virtually every branch of Indo-European, including now Anatolian. The root, as well as other widely represented roots such as aks- and nobh-, attests to the presence of the wheel — and vehicles using it — at the time Proto-Indo-European was spoken. [Watkins, p. 96]

It forms all or part of: always; away; convection; convey; convex; convoy; deviate; devious; envoy; evection; earwig; foy; graywacke; impervious; invective; inveigh; invoice; Norway; obviate; obvious; ochlocracy; ogee; pervious; previous; provection; quadrivium; thalweg; trivia; trivial; trivium; vector; vehemence; vehement; vehicle; vex; via; viaduct; viatic; viaticum; vogue; voyage; wacke; wag; waggish; wagon; wain; wall-eyed; wave (n.); way; wee; weigh; weight; wey; wiggle.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit vahati "carries, conveys," vahitram, vahanam "vessel, ship;" Avestan vazaiti "he leads, draws;" Greek okhos "carriage, chariot;" Latin vehere "to carry, convey," vehiculum "carriage, chariot;" Old Church Slavonic vesti "to carry, convey," vozŭ "carriage, chariot;" Russian povozka "small sled;" Lithuanian vežu, vežti "to carry, convey," važis "a small sled;" Old Irish fecht "campaign, journey," fen "carriage, cart;" Welsh gwain "carriage, cart;" Old English wegan "to carry;" Old Norse vegr, Old High German weg "way;" Middle Dutch wagen "wagon."

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*gene- 

*genə-, also *gen-, Proto-Indo-European root meaning "give birth, beget," with derivatives referring to procreation and familial and tribal groups.

It forms all or part of: Antigone; autogenous; benign; cognate; congener; congenial; congenital; connate; cosmogony; cryogenic; degenerate; engender; engine; epigone; eugenics; -gen; gendarme; gender; gene; genealogy; general; generate; generation; generic; generous; genesis; -genesis; genial; -genic; genital; genitive; genius; genocide; genotype; genre; gens; gent; genteel; gentile; gentle; gentry; genuine; genus; -geny; germ; german (adj.) "of the same parents or grandparents;" germane; germinal; germinate; germination; gingerly; gonad; gono-; gonorrhea; heterogeneous; homogeneous; homogenize; homogenous; impregnate; indigenous; ingenious; ingenuous; innate; jaunty; kermes; kin; kindergarten; kindred; king; kind (n.) "class, sort, variety;" kind (adj.) "friendly, deliberately doing good to others;" Kriss Kringle; malign; miscegenation; nada; naive; nascent; natal; Natalie; nation; native; nature; nee; neonate; Noel; oncogene; ontogeny; photogenic; phylogeny; pregnant (adj.1) "with child;" primogenitor; primogeniture; progenitor; progeny; puisne; puny; renaissance; theogony; wunderkind.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit janati "begets, bears," janah "offspring, child, person," janman- "birth, origin," jatah "born;" Avestan zizanenti "they bear;" Greek gignesthai "to become, happen," genos "race, kind," gonos "birth, offspring, stock;" Latin gignere "to beget," gnasci "to be born," genus (genitive generis) "race, stock, kind; family, birth, descent, origin," genius "procreative divinity, inborn tutelary spirit, innate quality," ingenium "inborn character," possibly germen "shoot, bud, embryo, germ;" Lithuanian gentis "kinsmen;" Gothic kuni "race;" Old English cennan "beget, create," gecynd "kind, nature, race;" Old High German kind "child;" Old Irish ro-genar "I was born;" Welsh geni "to be born;" Armenian cnanim "I bear, I am born."

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