trademark name (Segway Inc., Bedford, New Hampshire, U.S.), in use from 2001; according to the company, chosen for similarity to segue (q.v.) on notion of "a smooth transition from one place to another," with probable influence of way (n.).
1740, an instruction in musical scores, from Italian segue, "now follows," a direction 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").
The extended noun sense of "transition without a break" is from 1937; the verb in this sense is recorded by 1958.
Old English weg "road, path; course of travel; room, space, freedom of movement;" also, figuratively, "course of life" especially, in plural, "habits of life" as regards moral, ethical, or spiritual choices, from Proto-Germanic *wega- "course of travel, way" (source also of Old Saxon, Dutch weg, Old Norse vegr, Old Frisian wei, Old High German weg, German Weg, Gothic wigs "way"), from PIE root *wegh- "to go, move, transport in a vehicle."
From c. 1300 as "manner in which something occurs." Adverbial constructions attested since Middle English include this way "in this direction," that way "in that direction," both from late 15c.; out of the way "remote" (c. 1300). In the way "so placed as to impede" is from 1560s.
From the "course of life" sense comes way of life (c. 1600), get (or have) one's way (1590s), have it (one's) way (1709). From the "course of travel" sense comes the figurative go separate ways (1837); one way or (the) other (1550s); have it both ways (1847); and the figurative sense of come a long way (1922).
Adverbial phrase all the way "completely, to conclusion" is by 1915; sexual sense implied by 1924. Make way is from c. 1200. Ways and means "resources at a person's disposal" is attested from early 15c. (with mean (n.)). Way out "means of exit" is from 1926. Encouragement phrase way to go is short for that's the way to go.
<a href="https://www.etymonline.com/word/Segway">Etymology of Segway by etymonline</a>
Harper, D. (n.d.). Etymology of Segway. Online Etymology Dictionary. Retrieved $(datetime), from https://www.etymonline.com/word/Segway