KEVIN WHITE, the mayor of Boston, finally delivered on one of the promises he made in the 1979 campaign when he announced last week that he would not run for a fifth term. It's a measure of the cynicism Mr. White inspired in recent years that the betting in this most political of cities was that he would run; Rupert Murdoch's Boston Herald even ran a "White Will Run" headline. And in old-fashioned Boston, Mr. White announced his non-candidacy in his own modern way: on a five- minute commercial, at the end of local newscasts, on a tape made in New York under conditions of top secrecy.

In recent years, Kevin White has been known as a kind of big-city political boss, exacting campaign contributions from city employees and contractors, and with top aides implicated in scandal. That stemmed not from tradition, but from a conscious decision by Mr. White in 1976 to build his own political machine --"to accomplish John Lindsay's goals with Richard Daley's methods," in the words of Rep. Barney Frank (D-Mass.), a onetime White aide. Before that, Mr. White was better known as one of the "'60s mayors" --the young men who came to power in big cities across the nation, with the image of John Kennedy fresh in their minds and the potential of glittering new federal programs available to executives with an aptitude for coming up with what federal administrators wanted and, when they were needed, the right political connections. First elected in 1967, at age 38, Mr. White is now the last of the '60s mayors.

He can claim a lot of accomplishments for the city. He stimulated downtown development, and not just the erection of a few high-rise buildings: he can point also to the imaginative Faneuil Hall and Quincy Market redevelopments, which have been a model for the nation. Boston is one of the world's greatest centers of high-tech industry--something that seems inevitable today, but seemed by no means obvious in the down- at-the-heels, bedraggled, unimaginative Boston of the early 1960s. Mayor White, with his little city halls, his summer festivals, his flair for promotion, helped to create a sense of excitement about Boston that served the city well.

Kevin White's early success in office hurt him when he sought promotions. He lost a race for governor in 1970, and Edward Kennedy kept him from being George McGovern's running mate in 1972. Urban programs became less glamorous and less well-funded as the years went on. Even as he clung to office, he showed signs of boredom and its companion, cynicism. His political career has charted the rise and fall of the hopes and ambitions we associate with the big cities in the 1960s.