Theory of the Motion of the Heavenly Bodies Moving about the Sun in Conic Sections
In 1801 Piazzi discovered the "planet" Ceres and later that year, Ceres having been lost, Gauss computed its orbit from three observations by Piazzi. Gauss explained his methods in the book Theoria motus (1809). This work was translated by Charles Henry Davis, given the English title Theory of the Motion of the Heavenly Bodies Moving about the Sun in Conic Sections and published by Little, Brown and Company, Boston, in 1857. Below we give the Preface from this book.
Theory of the Motion of the Heavenly Bodies Moving about the Sun in Conic Sections: A translation of Gauss's 'Theoria motus', by Charles Henry Davis.
After the laws of planetary motion were discovered, the genius of Kepler was not without resources for deriving from observations the elements of motion of individual planets. Tycho Brahe, by whom practical astronomy had been carried to a degree of perfection before unknown, had observed all the planets through a long series of years with the greatest care, and with so much perseverance, that there remained to Kepler, the most worthy inheritor of such a repository, the trouble only of selecting what might seem suited to any special purpose. The mean motions of the planets already determined with great precision by means of very ancient observations diminished not a little this labour.
Astronomers who, subsequently to Kepler, endeavoured to determine still more accurately the orbits of the planets with the aid of more recent or better observations, enjoyed the same or even greater facilities. For the problem was no longer to deduce elements wholly unknown, but only slightly to correct those already known, and to define them within narrower limits.
The principle of universal gravitation discovered by the illustrious Newton opened a field entirely new, and showed that all the heavenly bodies, at least those the motions of which are regulated by the attraction of the sun, must necessarily, conform to the same laws, with a slight modification only, by which Kepler had found the five planets to be governed. Kepler, relying upon the evidence of observations, had announced that the orbit of every planet is an ellipse, in which the areas are described uniformly about the sun occupying one focus of the ellipse, and in such a manner that in different ellipses the times of revolution are in the sesquialteral ratio of the semi-major axes. On the other hand, Newton, starting from the principle of universal gravitation, demonstrated à priori that all bodies controlled by the attractive force of the sun must move in conic sections, of which the planets present one form to us, namely, ellipses, while the remaining forms, parabolas and hyperbolas, must be regarded as being equally possible, provided there may be bodies encountering the force of the sun with the requisite velocity; that the sun must always occupy one focus of the conic section; that the areas which the same body describes in different times about the sun are proportional to those times; and finally, that the areas described about the sun by different bodies, in equal times, are in the sub-duplicate ratio of the semi-parameters of the orbits: the latter of these laws, identical in elliptic motion with the last law of Kepler, extends to the parabolic and hyperbolic motion, to which Kepler's law cannot be applied, because the revolutions are wanting. The clue was now discovered by following which it became possible to enter the hitherto inaccessible labyrinth of the motions of the comets. And this was so successful that the single hypothesis, that their orbits were parabolas, sufficed to explain the motions of all the comets which had been accurately observed. Thus the system of universal gravitation had paved the way to new and most brilliant triumphs in analysis; and the comets, up to that time wholly unmanageable, or soon breaking from the restraints to which they seemed to be subjected, having now submitted to control, and being transformed from enemies to guests, moved on in the paths marked out by the calculus, scrupulously conforming to the same eternal laws that govern the planets.
In determining the parabolic orbits of comets from observation, difficulties arose far greater than in determining the elliptic orbits of planets, and principally from this source, that comets, seen for a brief interval, did not afford a choice of observations particularly suited to a given object: but the geometer was compelled to employ those which happened to be furnished him, so that it became necessary to make use of special methods seldom applied in planetary calculations. The great Newton himself, the first geometer of his age, did not disguise the difficulty of the problem: as might have been expected, he came out of this contest also the victor. Since the time of Newton, many geometers have laboured zealously on the same problem, with various success, of course, but still in such a manner as to leave but little to be desired at the present time.
The truth, however, is not to be overlooked that in this problem the difficulty is very fortunately lessened by the knowledge of one element of the conic section, since the major-axis is put equal to infinity by the very assumption of the parabolic orbit. For, all parabolas, if position is neglected, differ among themselves only by the greater or less distance of the vertex from the focus; while conic sections, generally considered, admit of infinitely greater variety. There existed, in point of fact, no sufficient reason why it should be taken for granted that the paths of comets are exactly parabolic: on the contrary, it must be regarded as in the highest degree improbable that nature should ever have favoured such an hypothesis. Since, nevertheless, it was known, that the phenomena of a heavenly body moving in an ellipse or hyperbola, the major-axis of which is very great relatively to the parameter, differs very little near the perihelion from the motion in a parabola of which the vertex is at the same distance from the focus; and that this difference becomes the more inconsiderable the greater the ratio of the axis to the parameter: and since, moreover, experience had shown that between the observed motion and the motion computed in the parabolic orbit, there remained differences scarcely ever greater than those which might safely be attributed to errors of observation (errors quite considerable in most cases): astronomers have thought proper to retain the parabola, and very properly, because there are no means whatever of ascertaining satisfactorily what, if any, are the differences from a parabola. We must except the celebrated comet of Halley, which, describing a very elongated ellipse and frequently observed at its return to the perihelion, revealed to us its periodic time; but then the major-axis being thus known, the computation of the remaining elements is to be considered as hardly more difficult than the determination of the parabolic orbit. And we must not omit to mention that astronomers, in the case of some other comets observed for a somewhat longer time, have attempted to determine the deviation from a parabola. However, all the methods either proposed or used for this object, rest upon the assumption that the variation from a parabola is inconsiderable, and hence in the trials referred to, the parabola itself, previously computed, furnished an approximate idea of the several elements (except the major-axis, or the time of revolution depending on it), to be corrected by only slight changes. Besides, it must be acknowledged, that the whole of these trials hardly served in any case to settle anything with certainty, if, perhaps, the comet of the year 1770 is excepted.
As soon as it was ascertained that the motion of the new planet, discovered in 1781, could not be reconciled with the parabolic hypothesis, astronomers undertook to adapt a circular orbit to it, which is a matter of simple and very easy calculation. By a happy accident the orbit of this planet had but a small eccentricity, in consequence of which the elements resulting from the circular hypothesis sufficed at least for an approximation on which could be based the determination of the elliptic elements. There was a concurrence of several other very favourable circumstances. For, the slow motion of the planet, and the very small inclination of the orbit to the plane of the ecliptic, not only rendered the calculations much more simple, and allowed the use of special methods not suited to other cases; but they removed the apprehension, lest the planet, lost in the rays of the sun, should subsequently elude the search of observers, (an apprehension which some astronomers might have felt, especially if its light had been less brilliant); so that the more accurate determination of the orbit might be safely deferred, until a selection could be made from observations more frequent and more remote, such as seemed best fitted for the end in view.
Thus, in every case in which it was necessary to deduce the orbits of heavenly bodies from observations, there existed advantages not to be despised, suggesting, or at any rate permitting, the application of special methods; of which advantages the chief one was, that by means of hypothetical assumptions an approximate knowledge of some elements could be obtained before the computation of the elliptic elements was commenced. Notwithstanding this, it seems somewhat strange that the general problem, -
To determine the orbit of a heavenly body, without any hypothetical assumption, from observations not embracing a great period of time, and not allowing a selection with a view to the application of special methods,was almost wholly neglected up to the beginning of the present century; or, at least, not treated by any one in a manner worthy of its importance; since it assuredly commended itself to mathematicians by its difficulty and elegance, even if its great utility in practice were not apparent. An opinion had universally prevailed that a complete determination from observations embracing a short interval of time was impossible, - an ill-founded opinion, - for it is now clearly shown that the orbit of a heavenly body may be determined quite nearly from good observations embracing only a few days; and this without any hypothetical assumption.
Some ideas occurred to me in the month of September of the year 1801, engaged at the time on a very different subject, which seemed to point to the solution of the great problem of which I have spoken. Under such circumstances we not infrequently, for fear of being too much led away by an attractive investigation, suffer the associations of ideas, which, more attentively considered, might have proved most fruitful in results, to be lost from neglect. And the same fate might have befallen these conceptions, had they not happily occurred at the most propitious moment for their preservation and encouragement that could have been selected. For just about this time the report of the new planet, discovered on the first day of January of that year with the telescope at Palermo, was the subject of universal conversation; and soon afterwards the observations made by that distinguished astronomer Piazzi from the above date to the eleventh of February were published. Nowhere in the annals of astronomy do we meet with so great an opportunity, and a greater one could hardly be imagined, for showing most strikingly, the value of this problem, than in this crisis and urgent necessity, when all hope of discovering in the heavens this planetary atom, among innumerable small stars after the lapse of nearly a year, rested solely upon a sufficiently approximate knowledge of its orbit to be based upon these very few observations. Could I ever have found a more seasonable opportunity to test the practical value of my conceptions, than now in employing them for the determination of the orbit of the planet Ceres, which during these forty-one days had described a geocentric arc of only three degrees, and after the lapse of a year must be looked for in a region of the heavens very remote from that in which it was last seen? This first application of the method was made in the month of October, 1801, and the first clear night, when the planet was sought for [by von Zach on 7 December 1801] as directed by the numbers deduced from it, restored the fugitive to observation. Three other new planets, subsequently discovered, furnished new opportunities for examining and verifying the efficiency and generality of the method.
Several astronomers wished me to publish the methods employed in these calculations immediately after the second discovery of Ceres; but many things - other occupations, the desire of treating the subject more fully at some subsequent period, and, especially, the hope that a further prosecution of this investigation would raise various parts of the solution to a greater degree of generality, simplicity, and elegance, - prevented my complying at the time with these friendly solicitations. I was not disappointed in this expectation, and have no cause to regret the delay. For, the methods first employed have undergone so many and such great changes, that scarcely any trace of resemblance remains between the method in which the orbit of Ceres was first computed, and the form given in this work. Although it would be foreign to my purpose, to narrate in detail all the steps by which these investigations have been gradually perfected, still, in several instances, particularly when the problem was one of more importance than usual, I have thought that the earlier methods ought not to be wholly suppressed. But in this work, besides the solutions of the principal problems, I have given many things which, during the long time I have been engaged upon the motions of the heavenly bodies in conic sections, struck me as worthy of attention, either on account of their analytical elegance, or more especially on account of their practical utility. But in every case I have devoted greater care both to the subjects and methods which are peculiar to myself, touching lightly and so far only as the connection seemed to require, on those previously known.
The whole work is divided into two parts. In the First Book are developed the relations between the quantities on which the motion of the heavenly bodies about the sun, according to the laws of Kepler, depends; the two first sections comprise those relations in which one place only is considered, and the third and fourth sections those in which the relations between several places are considered. The two latter contain an explanation of the common methods, and also, and more particularly, of other methods, greatly preferable to them in practice if I am not mistaken, by means of which we pass from the known elements to the phenomena; the former treat of many most important problems which prepare the way to inverse processes. Since these very phenomena result from a certain artificial and intricate complication of the elements, the nature of this texture must be thoroughly examined before we can undertake with hope of success to disentangle the threads and to resolve the fabric into its constituent parts. Accordingly, in the First Book, the means and appliances are provided, by means of which, in the second, this difficult task is accomplished; the chief part of the labour, therefore, consists in this, that these means should be properly collected together, should be suitably arranged, and directed to the proposed end. The more important problems are, for the most part, illustrated by appropriate examples, taken, wherever it was possible, from actual observations. In this way not only is the efficacy of the methods more fully established and their use more clearly shown, but also, care, I hope, has been taken that inexperienced computers should not be deterred from the study of these subjects, which undoubtedly constitute the richest and most attractive part of theoretical astronomy.
Göttingen, March 28, 1809.
Last Updated July 2022