late 13c., "a step in walking," also "rate of motion; the space traveled by the foot in one completed movement in walking," from Old French pas "a step, pace, trace," and directly from Latin passus, passum "a step, pace, stride," noun use of past participle of pandere "to stretch (the leg), spread out," probably from PIE *pat-no-, nasalized variant form of root *pete- "to spread."
It also was, from late 14c., a lineal measurement of vague and variable extent, representing the space naturally traversed by the adult human foot in walking. In some places and situations it was reckoned as the distance from the place where either foot is taken up, in walking, to that where the same foot is set down again (a great pace), usually 5 feet or a little less. The pace of a single step (military pace) is about 2.5 feet.
To keep pace (with) "maintain the same speed, advance at an equal rate" is from 1580s. Pace-setter "one who establishes trends in fashion," is by 1895; it also had literal meanings.
It is customary for the contractor to employ some expert as a pace setter. A man who can thin an acre of beets a day commands as high as $2.00 per day as a pace setter. The other employees are paid in the proportion their work bears to that of the pace setter. The weak, lazy and unskillful get the smallest wage. Besides that the contractor runs a commissary department and feeds the gang. They sleep in tents or in the shade of trees near where they work. [report on Oxnard, Calif., beet harvesting in "The Louisiana Planter and Sugar Manufacturer," May 13, 1899]
"with the leave of, by the permission of," 1863, from Latin pace, ablative of pax "peace," as in pace tua "with all deference to you;" from PIE root *pag- "to fasten." "Used chiefly as a courteous or ironical apology for a contradiction or difference of opinion" [OED]. It is sometimes misused as though it means "according to" instead of the opposite.
1510s, "to walk at a steady rate," from pace (n.). Meaning "to measure by pacing" is from 1570s. That of "to set the pace for" (another) is from 1886. Related: Paced; pacing.