Etymology
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segue (n.)
1740, an instruction in musical scores, from Italian segue, literally "now follows," meaning to play into the following movement without a break, third person singular of seguire "to follow," from Latin sequi "to follow," from PIE root *sekw- (1) "to follow." Extended noun sense of "transition without a break" is from 1937; the verb in this sense is first recorded 1958.
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Segway 
trademark name (Segway Inc., Bedford, New Hampshire, U.S.), in use from 2001; according to the company, chosen for similarity to segue on notion of "a smooth transition from one place to another," with probably influence of way (n.).
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*sekw- (1)

Proto-Indo-European root meaning "to follow."

It forms all or part of: associate; association; consequence; consequent; dissociate; ensue; execute; extrinsic; intrinsic; obsequious; persecute; persecution; prosecute; pursue; second (adj.) "next after first;" second (n.) "one-sixtieth of a minute;" sect; secundine; segue; sequacious; sequel; sequence; sequester; sociable; social; society; socio-; subsequent; sue; suit; suite; suitor; tocsin.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit sacate "accompanies, follows;" Avestan hacaiti, Greek hepesthai "to follow;" Latin sequi "to follow, come after," secundus "second, the following;" Lithuanian seku, sekti "to follow;" Old Irish sechim "I follow."

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