The absolute magnitude and perihelion distributions of long-period comets are derived, using data from the Lincoln Near-Earth Asteroid Research (LINEAR) survey. The results are surprising in three ways. First, the flux of comets through the inner solar system is much lower than some previous estimates. Second, the expected rise in comet numbers to larger perihelia is not seen. Third, the number of comets per unit absolute magnitude does not significantly rise to fainter magnitudes. These results imply that the Oort Cloud contains many fewer comets than some previous estimates, that small long-period comets collide with the Earth too infrequently to be a plausible source of Tunguska-style impacts, and that some physical process must have prevented small icy planetesimals from reaching the Oort Cloud, or have rendered them unobservable. A tight limit is placed on the space density of interstellar comets, but the predicted space density is lower still. The number of long-period comets that will be discovered by telescopes such as SkyMapper, the Panoramic Survey Telescope and Rapid Response System (Pan-STARRS), and the Large Synoptic Survey Telescope (LSST) is predicted, and the optimum observing strategy is discussed.