Etymology
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Words related to *tag-

attain (v.)
c. 1300, "succeed in reaching, come so near as to touch," from ataign-, stem of Old French ataindre "to come up to, reach, attain, endeavor, strive" (11c., Modern French atteindre), from Vulgar Latin *attangere, corresponding to Latin attingere "to touch; arrive at," from ad "to" (see ad-) + tangere "to touch," from PIE root *tag- "to touch, handle." Latin attingere had a wide range of meanings, including "to attack, to strike, to appropriate, to manage," all somehow suggested by the literal sense "to touch." Related: Attained; attaining.
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contact (n.)

1620s, "action, state, or condition of touching," from Latin contactus "a touching" (especially "a touching of something unclean, contamination"), from past participle of contingere "to touch, seize," from assimilated form of com "with, together" (see con-) + tangere "to touch," from PIE root *tag- "to touch, handle."

The figurative sense of "a connection, communication" is attested from 1818. The meaning "a person who can be called upon for assistance" is attested by 1931. As a call to the person about to spin an aircraft propeller to signal that the ignition is switched on, contact was in use by 1913.

To make contact (1860) originally was in reference to electrical circuits. Contact lens " thin artificial lens placed directly on the surface of the eye to correct visual defects" is first recorded 1888, in a translation of an article published in Zurich in 1887 by A. Eugen Fick; contacts for "contact lenses" is from 1957. Contact sport, for one involving bodily contact, is attested from 1922.

contaminate (v.)

early 15c., "infect with a disease, defile," from Latin contaminatus, past participle of contaminare "to defile, to corrupt, to deteriorate by mingling," originally "to bring into contact," from contamen "contact; pollution," from assimilated form of com "with, together" (see con-) + *tag-, base of tangere "to touch" (from PIE root *tag- "to touch, handle"). Related: Contaminant (1934).

entire (adj.)

mid-14c., of things, "whole, intact," from Old French entier "whole, unbroken, intact, complete," from Latin integrum "completeness" (nominative integer; see integer). Related: Entireness.

intact (adj.)
mid-15c., from Latin intactus "untouched, uninjured; undefiled, chaste; unsubdued," from in- "not" (see in- (1)) + tactus, past participle of tangere "to touch," from PIE root *tag- "to touch, handle."
integer (n.)

"a whole number" (as opposed to a fraction), 1570s, from noun use of Latin integer (adj.) "intact, whole, complete," figuratively, "untainted, upright," literally "untouched," from in- "not" (see in- (1)) + root of tangere "to touch" (from PIE root *tag- "to touch, handle"). The word was used earlier in English as an adjective in the Latin sense, "whole, entire" (c. 1500).

integrate (v.)
1630s, "to render (something) whole, bring together the parts of," from Latin integratus, past participle of integrare "make whole," from integer "whole, complete," figuratively, "untainted, upright," literally "untouched," from in- "not" (see in- (1)) + root of tangere "to touch," from PIE root *tag- "to touch, handle."

The meaning "put together parts or elements and combine them into a whole" is from 1802. The "racially desegregate" sense (1940) probably is a back-formation from integration. Related: Integrated; integrating.
integrity (n.)
Origin and meaning of integrity

c. 1400, integrite, "innocence, blamelessness; chastity, purity," from Old French integrité and directly from Latin integritatem (nominative integritas) "soundness, wholeness, completeness," figuratively "purity, correctness, blamelessness," from integer "whole" (see integer).

The sense of "wholeness, perfect condition" is attested from mid-15c.; that of "soundness of moral principle and character; entire uprightness or fidelity, especially in regard to truth and fair dealing" is by 1540s.

noli me tangere 
late 14c., "type of facial ulcer, lupus," Latin, literally "touch me not," from noli, imperative of nolle "to be unwilling" + me (see me) + tangere "to touch" (from PIE root *tag- "to touch, handle"). Used over the years of various persons or things that must not be touched, especially "picture of Jesus as he appeared to Mary Magdalene" (1670s, see John xx.17) and "plant of the genus Impatiens" (1560s, so called because the ripe seed pods burst when touched).
tact (n.)
1650s, "sense of touch or feeling" (with an isolated instance, tacþe from c. 1200), from Latin tactus "a touch, handling, sense of touch," from root of tangere "to touch," from PIE root *tag- "to touch, handle." Meaning "sense of discernment in action or conduct, diplomacy, fine intuitive mental perception" first recorded 1804, from development in French cognate tact. The Latin figurative sense was "influence, effect."