In the fall of 1965, a season that brought movies as distinct as “Alphaville” and “Thunderball” to the screen, Pauline Kael came to dinner at Sidney Lumet’s apartment, in New York. Lumet was then a prolific young director, having just finished shooting his tenth feature, “The Group,” for United Artists. Kael was a small-time movie critic who had recently arrived from Northern California. Her hardcover début, “I Lost It at the Movies,” had appeared that spring, to critical and popular acclaim, but she had never been on staff at any publication, and had only recently begun to write for major magazines. Lumet liked Kael’s work. Over the previous few weeks, he had allowed her on his set as a reporter, hoping she would learn something about shooting technique. Also present that night was the caricaturist Al Hirschfeld, and after a few drinks—actually, after quite a lot of drinks—Hirschfeld and Kael started quibbling about the uses of movie criticism. Finally, Hirschfeld asked her point-blank what she thought critics were good for. Kael gestured toward Lumet. “My job,” she said, “is to show him which way to go.” The evening ended soon afterward. Lumet later explained, “I thought, This is a very dangerous person.”