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 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 knowl edge 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 28 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, that three added to fcur ist equal to seven. By multiplying all the terms by 2, we obtain a new equation, in which 6+8=14. This expresses a new truth; and by varying the form, by similar operations, an indefinite number of separate truths may be elicited from the simple fundamental expression. I will add another illustration, which involves a little more algebra, but not, I think, more than you can understand; or, if it does, you will please pass over it to the next paragraph. According to a rule of arithmetical progression, the sum of all the terms is equal to half the sum of the extremes multiplied into the number of terms. Calling the sum of the terms s, the first term a, the last h, and the number of terms n, and we have in(a+h)=s; or n(a+h)=2s; or a+h= or a=2h; or h= -a. These are only a few of the changes which may be made in the original expression, still preserving the equality between the quantities on the left hand and those on the right; yet each of these transformations expresses a new truth, indicating distinct and (as might be the case) before unknown relations between the several quantities of which the whole expression is composed. The last, for exam ple, shows us that the last term in an arithmetical series is always equal to twice the sum of the whole series divided by the number of terms and diminished by the first term. In analytical formularies, as expressions of this kind are called, the value of a single unknown quantity is sometimes given in a very complicated expression, consisting of known quantities; but before we can ascertain their united value, we must reduce them, by actually performing all the additions, subtractions, multiplications, divisions, raising to powers, and 28 extracting roots, which are denoted by the symbols. This makes the actual calculations derived from such formularies immensely laborious. We have already had an instance of this in the calculations made by Lalande and Madame Lepaute, from formularies furnished by Clairaut. The analytical formularies, contained in such works as La Place's Mecanique Celeste,' exhibit to the eye of the mathematician a record of all the evolutions of the bodies of the solar system in ages past, and of all the changes they must undergo in ages to come. Such has been the result of the combination of transcendent mathematical genius and unexampled labor and perseverance, for the last century. The learned societies established in various centres of civilization have more especially directed their attention to the advancement of physical astronomy, and have stimulated the spirit of inquiry by a succession of prizes, offered for the solutions of problems arising out of the difficulties which were progressively developed by the advancement of astronomical knowledge. Among these questions, the determination of the return of comets, and the disturbances which they experience in their course, by the action of the planets near which they happen to pass, hold a prominent place. In 1826, the French Institute offered a prize for the determination of the exact time of the return of Halley's comet to its perihelion in 1835. M. Pontecoulant aspired to the honor. "After calculations," says he, "of which those alone who have engaged in such researches can estimate the extent and appreciate the fastidious monotony, I arrived at a result which satisfied all the conditions proposed by the Institute. I determined the perturbations of Halley's comet, by taking into account the simultaneous actions of Jupiter, Saturn, Uranus, and the Earth, and I then fixed its return to its perihelion for the seventh of November." Subsequently to this, however, M. Pontecoulant made some further researches, which led him to correct the former result; and he afterwards altered the time to November fourteenth. It actually came to its perihelion on the sixteenth, within two days of the time assigned. Nothing can convince us more fully of the complete mastery which astronomers have at last acquired over these erratic bodies, than to read in the Edinburgh Review for April, 1835, the paragraph containing the final results of all the labors and anticipations of astronomers, matured as they were, in readiness for the approaching visitant, and then to compare the prediction with the event, as we saw it fulfilled a few months af terwards. The paragraph was as follows: "On the whole, it may be considered as tolerably certain, that the comet will become visible in every part of Europe about the latter end of August, or beginning of September, next. It will most probably be distinguishable by the naked eye, like a star of the first magnitude, but with a duller light than that of a planet, and surrounded with a pale nebulosity, which will slightly impair its splendor. On the night of the seventh of October, the comet will approach the well-known con stellation of the Great Bear; and between that and the eleventh, it will pass directly through the seven conspicuous stars of that constellation, (the Dipper.) Tow ards the end of November, the comet will plunge among the rays of the sun, and disappear, and will not issue from them, on the other side, until the end of December." Let us now see how far the actual appearances corresponded to these predictions. The comet was first discovered from the observatory at Rome, on the morning of the fifth of August; by Professor Struve, at Dorpat, on the twentieth; in England and France, on the twenty-third; and at Yale College, by Professor Loomis and myself, on the thirty-first. On the morning of that day, between two and three o'clock, in obedience to the directions which the great minds that had marked out its path among the stars had prescribed, we directed Clarke's telescope (a noble instrument, belonging to Yale College) towards the northeastern quarter of the heavens, and lo! there was the wanderer so long foretold,—a dim speck of fog on the confines of creation. It came on slowly, from night to night, increasing constantly in magnitude and brightness, but did not become distinctly visible to the naked eye until the twenty-second of September. For a month, therefore, astronomers enjoyed this interesting spectacle before it exhibited itself to the world at large. From this time it moved rapidly along the northern sky, until, about the tenth of October, it traversed the constellation of the Great Bear, passing a little above, instead of "through" the seven conspicuous stars constituting the Dipper. At this time it had a lengthened train, and became, as you doubtless remember, an object of universal interest. Early in November, the comet ran down to the sun, and was lost in his beams; but on the morning of December thirty-first, I again obtained, through Clarke's telescope, a distinct view of it on the other side of the sun, a moment before the morning dawn. This return of Halley's comet was an astronomical event of transcendent importance. It was the chronicler of ages, and carried us, by a few steps, up to the origin of time. If a gallant ship, which has sailed round the globe, and commanded successively the admiration of many great cities, diverse in language and customs, is invested with a peculiar interest, what interest must attach to one that has made the circuit of the solar system, and fixed the gaze of successive worlds! So intimate, moreover, is the bond which binds together all truths in one indissoluble chain, that the establishment of one great truth often confirms a multitude of others, equally important. Thus the re turn of Halley's comet, in exact conformity with the predictions of astronomers, established the truth of all those principles by which those predictions were made. It afforded most triumphant proof of the doctrine of universal gravitation, and of course of the received |