Here is an interesting article about the history of the enlisted chevrons. It comes from an article based on the US Air Force's perspective, but it applies to all of the US service branches.:
The American chevron is not a new idea. For thousand of years the military, ecclesiastical and civil authorities have used some outward symbol to identify rank and function in society. In the U.S. military, noncommissioned officer rank insignia evolved over the past 150 years from a mishmash of epaulets, sashes, cockades and stripes to today's limited set of stylized and standardized chevrons. Prior to 1872, documentation standards were almost nonexistent. A general order from the War Department dated March 27, 1821, documented the first firm reference to U.S. soldiers wearing chevrons. Today, the chevron represents a pay grade, not a specific trade.
Originally, officers also wore chevrons, but this practice began to phase out in 1829. Despite this 10-year use of chevrons by officers, most people think only of enlisted grades when chevrons are mentioned.
The direction a chevron points alternated through the years. Originally, they pointed down, and on some uniforms, covered almost the entire width of the arm. In 1847, the point reversed to an "up" position, which lasted until 1851. Service chevrons, commonly called "hash marks" or "service stripes," were established by George Washington to show completion of three years service. After the American Revolution, they fell into disuse and it wasn't until 1832 before the idea was reinstituted. They have been authorized in one form or another ever since.
Distinctive chevrons became cumbersome. Specific designs often depicted a trade skill and each branch required individual colors. For example, in 1919, the Medical Department had seven different chevrons that no other branch used. In 1903, a sergeant might have worn four different chevrons, depending upon which uniform he wore. The overwhelming problems of pay, grade, titles, and allowances caused Congress in 1920 to consolidate all ranks into seven pay grades. This broke the historic practice of authorizing each and every position and listing the pay of each job throughout the Army. The change drastically affected chevron design.
Stopping the use of branch and specialty chevrons died hard despite the official War Department policy. Private manufacturers made old specialty designs with the new blue background prescribed for the new chevrons. Unauthorized chevrons were common and these improvised sleeve insignia were even sold in some post exchanges. Throughout the 1920s and 1930s, the War Department fought a losing battle against the specialty chevrons. The most prevalent of the unauthorized specialty chevrons were those worn by Army Air Corps members, with the winged propeller.
The Air Force won its independence Sept. 18, 1947, as a full partner with the Army and the Navy when the National Security Act of 1947 became law. There was a time of transition following the new status given the Air Force. The chevrons retained the "Army look." Enlisted personnel were still "soldiers" until 1950, when they became "airmen" to distinguish them from "soldiers" or "sailors."
9 March 1948 - There is no documented official rationale for the design of the present USAF enlisted chevrons, except the minutes of a meeting held at the Pentagon on 9 March 1948, chaired by General Hoyt S. Vandenberg, Air Force Chief of Staff. These minutes reveal that chevron designs were sampled at Bolling Air Force Base and the style used today was selected by 55% of 150 airmen polled. General Vandenberg therefore approved the choice of the enlisted majority.
Whoever designed the stripes might have been trying to combine the shoulder patch worn by members of the Army Air Force (AAF) during World War II and the insignia used on aircraft. The patch featured wings with a pierced star in the center while the aircraft insignia was a star with two bars. The stripes might be the bars from the aircraft insignia slanted gracefully upward to suggest wings. The silver grey color contrasts with the blue uniform and might suggest clouds against blue sky.