degree (n.) Look up degree at Dictionary.com
early 13c., from Old French degré (12c.) "a step (of a stair), pace, degree (of relationship), academic degree; rank, status, position," said to be from Vulgar Latin *degradus "a step," from Latin de- "down" (see de-) + gradus "a step; a step climbed;" figuratively "a step toward something, a degree of something rising by stages" (see grade (n.)).

Most modern senses date from Middle English, from notion of a hierarchy of steps. Meaning "a grade of crime" is 1670s; that of "a unit of temperature" is from 1727. The division of the circle into 360 degrees was known in Babylon and Egypt. It is perhaps from the daily motion of the sun through the zodiac in the course of a year.