In 1661, 1532, 1402, 1145, 891, and 243 great comets appeared—that of 1402 being bright enough to be seen at noonday. A period of 129 years would conciliate all these appearances, and should have brought back the comet in 1789 or 1790 (other circumstances agreeing). That no such comet was observed about that time is no proof that it did not return, since, owing to the situation of its orbit, had the perihelion passage taken place in July it might have escaped observation. We come now, however, to a class of comets of short period, respecting whose return there is no doubt, inasmuch as two at least of them have been identified as having performed successive revolutions round the sun; have had their return predicted already several times; and have on each occasion scrupulously kept to their appointments. The first of these is the comet of Encke, so called from Professor Encke of Berlin, who first ascertained its periodical return. It revolves in an ellipse of great eccentricity (though not comparable to that of Halley’s), the plane of which is inclined at an angle of about 13° 22′ to the plane of the ecliptic, and in the short period of 1,211 days, or about 3⅓ years. This remarkable discovery was made on the occasion of its fourth recorded appearance, in 1819. From the ellipse then calculated by Encke, its return in 1822 was predicted by him, and observed at Paramata, in New South Wales, by M. Rümker, being invisible in Europe: since which it has been repredicted and reobserved in all the principal observatories, both in the Northern and Southern Hemispheres, as a phenomenon of regular occurrence. Another comet of short period is that of _Biela_, so called from M. Biela of Josephstadt, who first arrived at this interesting conclusion on the occasion of its appearance in 1826. It is considered to be identical with comets which appeared in 1772, 1805, etc., and describes its very eccentric ellipse about the sun in 2,410 days, or about 6¾ years; and in a plane inclined 12° 34′ to the ecliptic. It appeared again, according to the prediction, in 1832 and in 1846.