The comet arrived at its perihelion on the thirteenth of March, only twenty-three days from the time assigned by Clairaut. It appeared very round, with a brilliant nucleus, well distinguished from the surrounding nebulosity. It had, however, no appearance of a tail. It became lost in the sun, as it approached its perihelion, and emerged again, on the other side of the sun, on the first of April. Its exhibiting an appearance, so inferior to what it presented on some of its previous returns, is partly accounted for by its being seen by the European astronomers under peculiarly disadvantageous circumstances, being almost always within the twilight, and in the most unfavorable situations. In the southern hemisphere, however, the circumstances for observing it were more favorable, and there it exhibited a tail varying from ten to forty-seven degrees in length. In my next Letter I will give you some particulars respecting the late return of Halley's comet. LETTER XXVI. COMETS, CONTINUED. "Incensed with indignation, Satan stood Unterrified, and like a comet burned, That fires the length of Ophiucus huge In the Arctic sky, and from his horrid train Shakes pestilence and war."--_Milton._ AMONG other great results which have marked the history of Halley's comet, it has itself been a criterion of the existing state of the mathematical and astronomical sciences. We have just seen how far the knowledge of the great laws of physical astronomy, and of the higher mathematics, enabled the astronomers of 1682 and 1759, respectively, to deal with this wonderful body; and let us now see what higher advantages were possessed by the astronomers of 1835. During this last interval of seventy-six years, the science of mathematics, in its most profound and refined branches, has made prodigious advances, more especially in its application to the laws of the celestial motions, as exemplified in the 'Mecanique Celeste' of La Place. The methods of investigation have acquired greater simplicity, and have likewise become more general and comprehensive; and mechanical science, in the largest sense of that term, now embraces in its formularies the most complicated motions, and the most minute effects of the mutual influences of the various members of our system. You will probably find it difficult to comprehend, how such hidden facts can be disclosed by formularies, consisting of _a_'s and _b_'s, and _x_'s and _y_'s, and other algebraic symbols; nor will it be easy to give you a clear idea of this subject, without a more extensive acquaintance than you have formed with algebraic investigations; but you can easily understand that even an equation expressed in numbers may be so changed in its form, by adding, subtracting, multiplying and dividing, as to express some new truth at every transformation. Some idea of this may be formed by the simplest example. Take the following: 3+4=7. This equation expresses the fact,