In the meantime, he set out a new rule that “the database can be queried” only with permission from the surveillance court, but allowed an exception “in the case of a true emergency.” He did not define what would constitute such an emergency or who would determine whether a situation qualified. Nor did he clarify whether the court would have to approve each time a new telephone number was searched or each time a new target was searched. But some program supporters expressed concern that it could take too long.

He also limited the scope of searches, allowing analysts to study data two layers removed from the target, instead of three. Intelligence officials have accepted such a change because the amount of data expands so vastly three layers out that it becomes less useful.

The details matter, and may become clearer in coming days. But for a president who came to office promising to end what he considered the excesses of the new security state, Mr. Obama’s speech on Friday was as much about the larger question of faith. Rather than throw out the programs at issue, he hoped to convince the public that they are being run appropriately.