The officer’s goal is to form a rapport with the person and seize upon the one emotional chord that will get him or her to climb down from the edge. “You have to understand and extend yourself because your obvious goal is to save someone’s life,” Inspector Lukach said. “So if you have to give a little, you give a little. That’s the sacrifice you make.”

The mental gymnastics can go on for hours, and do not always pay off.

On a cold day this past winter, Detective Taylor was talking to a psychiatric patient who had squeezed through a sixth-floor bathroom window at Bellevue Hospital Center. The man’s toes barely fit on a building lip below, so he mostly clung to the window ledge by his fingers. He told the detective that he had killed somebody a few years back and could no longer live with the guilt.

“O.K., we all make mistakes,” Detective Taylor said he told him. “That doesn’t mean you should take your life. We’re all human beings. None of us are perfect.”

“Why don’t you just push me? Why don’t you just end it for me?” the man goaded the detective, who recounted his words.

“That’s not my purpose for being here,” Detective Taylor gently told him.

For nearly three hours, Detective Taylor leaned out a seventh-floor window, talking, buying time, as other officers cut away window glass to create an opening large enough to make a grab. Detective Taylor sensed the man was ready to come in. He was shirtless and cold; his muscles quivered. He asked for a blanket, the detective recalled.

“Fatigue set in,” he said. “He was extending his arms to me, but I couldn’t reach him. At that point, he panicked a little bit, and that’s when he kind of groaned and said, ‘O.K.,’ and he left — fell.” Detective Taylor, who has worked in emergency services for 12 years, spoke in a low voice, pausing pensively between words.