early 15c. (implied in diagonally), "extending as a line from one angle to another not adjacent," from Old French diagonal, from Latin diagonalis, from diagonus "slanting line," from Greek diagonios "from angle to angle," from dia "across, through" (see dia-) + gōnia "angle, corner" (from PIE root *genu- (1) "knee; angle").
As a noun, from 1570s, "straight line drawn from one angle to or through another not adjacent, in a plane or solid figure." In chess, "a line of squares running diagonally across a board."
"half a poetic line," 1570s, from French hémistiche (16c.) or directly from Late Latin hemistichium, from Greek hēmistikhion "half-line, half-verse," from hēmi- "half" (see hemi-) + stikhos "row, line of verse," from PIE *stigho-, suffixed form of root *steigh- "to stride, step, rise" (see stair). Related: Hemistichal.
"British rule in India," 1859, from Hindi raj "rule, dominion, kingdom" (from PIE root *reg- "move in a straight line," with derivatives meaning "to direct in a straight line," thus "to lead, rule").