With thanks to the editorial staff of Rugby World:
Originally, there were only two Rugby Positions - forwards and backs. The third position “full back” (of which there were three) was added when the laws of the game were drafted in 1870. This position evolved to a single fullback and two halfway backs (halfway between the forwards and the full back). These became the halfbacks.
Early rugby was the province of the forwards with the backs primarily to recover the ball when it got out of the scrum. Generally, the backs/fullback were to fall on the ball if the opposition scrum won the ball.
The Cardiff (Wales) team in 1878 developed a play that included a short pass to one of the halfbacks who would then charge up the field. This player became the “fly half”, shortened from “flying half back”. The Cardiff team also developed a style of play from a reorganized scrum that included short passes among the forwards and longer passes among the backs. Three players were taken from the scrum and added to the backline. Since these three players were positioned between the fullback and the halfbacks, they were called the “three-quarter backs” (not to be confused with the gridiron position of the same name). These players evolved to the center and the wing.
So, towards the end nineteenth century the players were configured into the scrum, a row of halfbacks, a row of quarterbacks, and the full back.
Purely by accident, Cardiff has the honor of adding the fourth player to the three-quarters. During a very tough match, a player was brought up from the second side to play center when the first side center was unavailable. The replacement player played an outstanding game. When the first side center became available, rather then drop the replacement center; they essentially just added him to the backs, creating an additional center.
Now the development of the term “5/8”s gets a little confusing, so I will quote directly from the web-page source:
“The New Zealanders were quick to see the advantage of having a fourth player in the three-quarters. Their solution was to change the standard rugby positions by pulling a forward out of the pack and put him between the halfback and the three-quarters. Their problem was what did they call the new position. Legend has it that consent was reached by deciding that the half back was 4/8ths and the three-quarters 6/8ths, so therefore the new position must be a 5/8ths, a name that has continued to this day in that country. When fly half play developed they introduced the first 5/8th and the second 5/8th.”