and the two together are sufficient for the complete study of the motion of any system of bodies, in either direction. And here comes in the facility afforded by M. Poinsot's conception— referred to under the head of Statics. By substituting for the areas or momentum of the geometers, the couples engendered by the proposed forces, a philoso- phical completeness is given to the theory, and a concrete value, and proper dynamic direction, to what was before a simple geometrical expression of a part of the fundamental equations of motion.
Laplace elicited from the theory of areas -p,..,, that dynamic property which he called the tuVJ'''^ invariaoie plane, the consideration oi which is highly important in celestial mechanics. It is in the study of astronomy that the importance fully appears of the determination of a plane, whose direction is unaffected by the mutual action of different bodies in our own solar system; for we thus obtain a point of reference, a neces- sarily fixed term of comparison, by which to estimate the variations of the heavenly bodies. We are far from having I. L 146 POSITIVE PHILOSOPHY.
yet attained precision in the determination of tlie situation of this plane; but this does not impair the character of the theorem in its relation to rational Mechanics. Again, we are indebted to Poinsot, who, by simj^lifying, has once more extended the i:)rocess to which his method is applied: and he has rej)aired an important omission made by Laplace, in takiug into the account the smaller areas described by satellites, and their rotation, and that^ of the sun itself; whereas Laplace has attended only to the larger areas de- scribed by the planets in their course round the sun.
^. Finally, there are Euler's theorems of the inertia moment oj inertia, and the principal axes, which are among the most important general PrinciDal axes I'^sults of rational Mechanics. By means of these we are able to arrive at a complete analysis of the motion of rotation. — By means of all the theorems just touched upon, we are j)ut in the direct way to determine the entire motion of any body, or system of bodies whatever. — Besides them, geometers have discovered some which are less general, but, though by no means in- dispensable, yet very important from the simplification they introduce into special researches. Students will re- cognize their functions, when their mere names are pre- sented; which is all that our space allows: — I refer to the theorem of the conservation of active forces,- — singularly important in its apj^licatious to industrial Mechanics: the theorem, impropei'ly called the princvple of the least action, as old as Ptolemy, who observed that reflected light tates the shortest way from one point to another, — an observa- tion which was the basis of Maupertuis' discovery of this proj^erty: and lastly, a theorem not usually classed with the foregoing, yet worthy of no less esteem, — the theorem of the co-existence of small oscillations, of Daniel Beruouilli. This discovery is as important in its physical as its logical bearings; and it explains a multitude of facts which, clearly known, could not be referred to their principles. It consists in showing that the infinitely small oscillations caused by the return of any system of forces to a state of stable equilibrium coexist without interference, and can be treated separately.
This reference to the principal general theorems hitherto CONCLUSION. 147 discovered in Rational Mechanics concludes our review of the second branch of Concrete Mathematics.
As for our review of the whole science, I r 1 ' wish I could better have communicated my own profound sense of the nature of this immense and admirable science, which, the necessary basis of the whole of Positive Philosophy, constitutes the most unquestionable proof of the compass of the human intellect. But I hoj^e tliat those who have not the misfortune to be wholly igno- rant of this fundamental science may, according to the process of thought which I have indicated, attain some clear idea of its philosophical character.
To preserve complete the philosophical arrangement of Mathematics in its present state, I ought to consider here a third branch of Concrete Mathematics; — the application of analysis to therm ological phenomena, according to the discoveries of Fourier. But, to avoid too great a bi-each of customary arrangement, I have reserved the subject, and shall place Thermology among the departments of Physics.
Mathematical philosophy being now completely character- ized, we shall proceed to examine its application to the study of Natural Phenomena, in their various orders, ranked ac- cording to their degree of simplicity. By this character alone can they cast light back again upon the science which explains them; and under this character alone can they be suitably estimated. According to the natural order laid down at the beginning, we now proceed to that class of phenomena with which Mathematics is most concerned, — the phenomena of Astronomy.
148 BOOK II.
GENERAL VIEW.
T,, T T is easy to describe clearly tlie character -L of astronomical science, from its being thoroughly separated, in onr time, from all theological and metajjliysical influence. Looting at the simple facts of the case, it is evident that though three of our senses take cognizance of distant objects, only one of the three per- ceives the stars. The blind could know nothing of them; and we who see, after all our preparation, know nothing of stars hidden by distance, except by induction. Of all objects, the planets are those which appear to us under the least varied aspect. We see hoAV we may determine their forms, their distances, their l)ulk, and their motions, but we can never know anything of their chemical or niineralogical structure; and, much less, that of organized beings living on their surface. We may obtain positive knowledge of their geometrical and mechanical phenomena; but all physical, chemical, physiological, and social researches, for which our powers fit us on our own earth, are out of the question in regard to the planets. Whatever knowledge is obtainable by means of the sense of Sight, we may hope to attain with regard to the stars, whether we at j)resent see the method or not; and whatever knowledge requires the aid of other senses, we must at once exclude from our expectations, in spite of any appearances to the contrary. As to questions about which we are uncertain whether they finally depend on Sight or not, — we must patiently wait, DEFINITION AND LIMITATIONS OF ASTRONOMY. 149 for an ascertainment of their character, before we can settle whether they are applicable to the stars or not. The only case in which this rule will be pronounced too sevei'e is that of questions of temperatures. The mathematical thermology created by Fourier may tempt us to hope that, as he has estimated the temperature of the sj^ace in which we move, we may in time ascertain the mean temperature of the heavenly bodies: but I regard this order of facts as for ever excluded from our recognition. We cau never learn their internal constitution, nor, in regard to some of them, how heat is absorbed by their atmosphere. Newton's attempt to estimate the temperature of the comet of 1680 at its perihelion could accomplish nothing more, even with the science of our day, than show what would be the tem- perature of our globe in the circumstances of that comet. We may therefore define Astronomy as the ^ j,.. science by which we discover the laws of the geometrical and mechanical phenomena pi'esented by the heavenly bodies.
It is desirable to add a limitation which is important, though not of primary necessity. Restriction. The part of the science which we command from what we may call the Solar point of view is distinct, and evidently capable of being made complete and satis- factory; while that which is regarded from the Universal point of view is in its infancy to us now, and must ever be illimitable to our successors of the remotest generations. Men will never compass in their conceptions the whole of the stars. The difference is very striking now to us who find a perfect knowledge of the solar system at our com- mand, while we have not obtained the first and most simple element in sidereal astronomy — the determination of the stellar intervals. Wliatever may be the ultimate progress of our knowledge in certain portions of the larger field, it will leave us always at an immeasurable distance from understanding the universe.
Throughout the whole range of science, there exists a constant and necessary harmony between our needs and our knowledge. We shall find this to be true everywhere. The fact is, we need to know only what, in some way or other, acts upon us; and the influence which acts upon us 150 POSITIVE PHILOSOPHY.
becomes, in turn, our means of knowledge. This is evi- dently and remarkably true in regard to Astronomy. It is of the highest importance to its to know the laws of the solar system: and we have attained great j^recision with regard to them; but, if the knowledge of the starry universe is forbidden to us, it is clear that it is of no real consequence to us, except as a gratification of our curiosity. The interior mechanism of eadi solar system is essentially independent of the mutual action of distant suns; as it may well be, considering the distance of these suns from each other, in comparison with the distance of planets from their suns. Our tables of astronomical events, constructed in advance, proceed on the supposition of there being no other system than our own; and they agree with our direct observations, precisely and necessarily. This is our proper field; and we must remember that it is so. We must keep carefully apart the idea of the solar system and that of the universe, and be always assured that our only true interest is in the former. Within this boundary alone is astro- nomy the supreme and positive science that we have deter- mined it to be; and, in fact, the innumerable stars that are scattered through space serve us scientifically only as providing positions which may be called fixed, with which we may compare the interior movements of our system.
We shall find, as we proceed through the ^OTatlon ^^' whole gradation of the science, that the more complex the science, the more various are the means of exploration; whereas, it does not at all follow, as we shall see, that the completeness of the knowledge ob- tained is in any proportion to the abundance of our means. Our knowledge of astronomy is more perfect than that of any of the sciences which follow it; yet in none are our means of exploration so few.
The means of exploration are three: — direct observation; observation by experiment; and observation by comparison. In the first case, we look at the iihenomenon before our eyes; in the second, we see how it is modified by artificial circumstances to which we have subjected it; and in the third, we contemplate a series of analagous cases, in which the phenomenon is more and more simplified. It is only MEANS OF ASTRONOMICAL INVESTIGATION. 151 in the case of organized bodies, whose phenomena are extremely difficult of access, that all the three methods can be employed; and it is evident that in astronomy we can use only the first. Experiment is, of course, impossible; and comparison could take place only if we were familiar with abundance of solar systems, which is equally out of the question. Even simple observation is reduced to the use of one sense, — that of sight alone. And again, even this sense is very little used. Reasoning bears a greater proportion to observation here, than in any science that follows it; and hence its high intellectual dignity. To measure angles and compute times are the only methods by which we can discover the laws of the heavenly bodies; and they are enough. The few incoherent sensations con- concerned would be, of themselves, very insignificant; they could not teach us the figure of the earth, nor the path of a planet. They are combined and rendered serviceable by long-drawn and complex reasonings; so that we might truly say that the phenomena, however real, j,,, are constructed by our understanding. The simplicity of the phenomena to lie studied, and the diffi- culty of getting at them, constitute, by their conibiuatiou, the eminently mathematical character of the science of astronomy. On the one hand, the perpetual necessity of deducing from a small number of direct measures, whether angular or horary, quantities which are not themselves immediately observable, renders the use of abstract mathe- matics indispensable; while, on the other hand, astro- nomical questions being always, in themselves, problems of geometry, or else of mechanics, must fall into the depart- ment of concrete mathematics. Again, the regularity of astronomical forms admits of geometrical treatment; and the simplicity of astronomical movements admits of me- chanical treatment with a very high degree of precision. There is perhaps no analytical process, no geometrical or mechanical doctrine, which is not employed in astronomical researches, and many of them have as yet had no other aim. Considering the simple nature of astronomical investi- gations, and the easy application to them of mathematical means, it is evident why astronomy is, by common consent, placed at the head of the natural sciences. It deserves 152 POSITIVE PHILOSOPHY.
tliis place, first, by the perfection of its scientific character; and, next, by the preponderant importance of the laws which it discloses.
Passing over, for the present, its utility in the measure- ment of time, the exact description of the globe, and the perfecting of navigation, which are not circumstances that could determine its rank, we may just observe that it affords an instance of the necessity of the loftiest scientific speculations to the satisfaction of the most ordinary wants. Hipparchus began to apply astronomical theory to the find- ing the longitude at sea. A prodigious amount of geome- metrical science has gone to improve our tables of longitude up to their present point; and if we cannot now get within half-a-dozen miles of a true estimate in the seas under the line, it is for want of more science still.
Those who say that science consists in an accumulation of observed facts may here see how imperfect is their account of the matter. The Chaldeans and Egyptians collected facts from observation of the VV lien it be- i i j. xi x ■ i came a science ^©^'^^^^s; but there was no astronomical science till the early Greek philosoj^hers re- ferred the diurnal movement to geometrical laws. The aim of astronomical researches was to establish what would be the state of the sky at some future time; and no accu- mulation of facts could effect this, till the facts were made the basis of reasonings. Till the rising of the sun, or of some star, could be accurately predicted, as to time and place, there was no astronomical science. Its whole j^rogress since has been by introducing more and more certainty and precision into its predictions, and in using smaller and smaller data from direct observation for a more and more distant prevision. No part of natural philosophy manifests more strikingly the truth of the axiom that all science has prevision for its end: an axiom which separates science from erudition, which relates the events of the past, with- out any regard to the future.
However impossible may be the aim to a '«ino'le law I'educe the phenomena of the respective sciences to a single law, supreme in each, this should be the aim of pliilosophers, as it is only the imperfection of our knowledge which prevents its accomplishment. The perfection of a science is in exact propor- tion to its approach to this consummation; and, according- to this test, astronomy distances all other sciences. Suj)- posing it to relate to our solar system alone, the point is attained; for the single general law of gravitation compre- hends the whole of its phenomena. It is to this that we must recur when we wish to show what we mean by the explanation of a phenomenon, without any inquiry into its. first or final cause; and it is here that we learn the true character and conditions of scientific hypothesis, — no other science having applied this powerful instrument so ex- tensively or so usefully. After having exhibited these great general properties of astronomical philosophy, I shall apply them to perfect the philosophical character of the other principal sciences.
Regarding astronomical science, apart from its method, and with a view to the natui'al laws which it discloses, its pre-eminence is no less incontestable. I have always ad- mired, as a stroke of philosophic genius, Newton's title of his treatise on Celestial Mechanics, — ' The Mathematical Principles of Natural Philosophy; ' for it would be im- jDossible to indicate with a more energetic conciseness that the general laws of astronomical phenomena are the basis of all our real knowledge.
We may see at a glance that astronomy is. independent of all the natural sciences, de- qH^q^. sciences pending on Mathematics alone: and though, philosojjhically speaking, we put Mathematics at the head of the whole series, we practically regard it less as a natural science of itself (from the paucity of phenomena which it presents to observation) than as the repository of principles by which the natural sciences are intei'preted and investi- gated. Philosophically speaking, astronomy depends on Mathematics alone, owing nothing to Physics, Chemistry, or Physiology, which Avere either undiscovered, or lost in theological and metaphysical confusion, while astronomy was a true science in the hands of the ancient geometers. But the phenomena of the other sciences are dependent, naturally as well as systematically, on astronomical facts, and can be perfectly sti;died only through astronomy. We cannot thoroughly undex'stand any terrestrial phenomenon 154 POSITIVE PHILOSOPHY.
withoiat considering what our globe is, and what part it bears in the solar system, as its situation and motions affect the conditions of everything upon it; and what would become of our physical, chemical, and j^hysiological ideas, without consideration of the law of gravitation? In the remotest case of all, that of Social i^heuomena, it is certain that changes in the distance of the earth from the sun, and consequently in the duration of the year, in the obliquity of the ecliptic, etc., which in astronomy would merely modify some coefficients, would largely affect or completely destroy our social development. It is no exaggeration to say that Social physics would be an impossible science, if geometers had not shown us that the perturbations of our solar system can never be more than gradual and restricted oscillations round a mean condition which is invariable. If astronomical conditions wei'e liable to indefinite varia- tions, the human existence which depends upon them could never be reduced to laws.
Not less important is the influence of astronomical science on our own intelligence. It has done much more than relieve us from superstitious terrors and absurd notions about comets and eclipses, — notions which, as Laplace observed, would spring up again immediately if our astronomy were forgotten. This science has done much more for our understandings than that. It has done more than any other pursviit — simply because it is the most scientific of all — to expose and destroy the doctrine of final causes, which is generally regarded by the moderns as the basis of every religious system, though it is in fact a con- sequence and not a cause. The knowledge of the motion of the earth has overthrown the very foundation of the doctrine, which supposed the universe to be subordinated to our globe, and therefore to Man. Since Newton's time, the development of celestial Mechanics has dej^rived theo- logical philosophy of its principal intellectual office, by proving that the order maiiitained throughout our system and the whole universe is by the simple gravitation of its parts. If we took an a priori view, we should say that, as we exist, our system must be such as to admit of our existence; and one necessary condition of this is such a degree of stability in our system as we actually find. This TWO GREAT DIVISIONS OF THE SCIENCE. 155 stability we scientifically perceive to he a simple conse- quence of mechanical laws woi'king among the incidents of our system, — the extremely small planetary bodies in their relation to the larger sun; the small eccentricity of their orbits, and moderate inclination of their planes; which in- cidents, again, are necessary consequences of the mode of formation of the entire system. The stability by virtue of which we hold our existence is not found in the case of comets, whose perturbations are not only great, but liable to indefinite increase; and their being inhabited is incon- ceivable. Thus, the doctrine of final causes would be reduced to the truism that there are no inhabited bodies in our system but those which are habitable. This brings us back to the principle of the conditions of existence, which is the true positive transformation of the doctrine of final causes, and of far superior sco2:)e and profit in every way.
We have next to consider the divisions of p,- • • the science. These arise immediately out of ^^^^ science the fact, now familiar to us, that astronomical phenomena are either geometrical or mechanical. They are Celestial Geometry, which is still called Astronomy, from its having possessed a scientific character before the other; and Celestial Mechanics, of which Newton was the immortal founder. Thougli our business is with our own system, the same division extends to Sidereal astronomy, supposing that kind of exploration to be within our „,., power. As before, we see geometry to be Geometrv.
more simple in its phenomena than me- chanics, and that mechanics is dependent on geometry, without reciprocity. In fact, men were successfully in- quiring into the forms and sizes of the heavenly bodies, and studying their geometrical laws, before anything was known of the forces which changed their positions. Whereas, the province of Celestial Mechanics is to analyse „,., the motions of the stars, in order to refer Mechanics them, by the rules of Rational Mechanics, to the elementary motions regulated by a universal and in- variable mathematical law; — thence, again, departing to I^erfect the knowledge of real motions by scientifically determining them a priori, taking from observation the 156 POSITIVE PHILOSOPHY.
necessary data — the fewest possible — for tlie calculations of general mechanics. This is the link by which astronomy and physics are connected, and connected so closely that some great phenomena render the transition almost in- sensible; as in the theory of the Tides. But it is evident that the whole reality of celestial mechanics consists in its having issued from the exact knowledge of true movements, furnished by celestial geometry. It was for want of this point of departure that all attempts before the time of Newton, even Descartes', however valuable in other ways, failed to establish systems of celestial mechanics. This division of the science into two parts has therefore nothing arbitrary in it, nor even scholastic: it is derived from the nature of the science, and is at once historical and dog- matic. As for the subdivisions, we need not trouble our- selves with them now.
In regard to the point of view from which the science should be regarded, Lacaille thought it would simplify matters extremely to place his observer on the surface of the sun. And so it would, if the thing could be done in accor- dance with positive knowledge; but undoubtedly the solar station should be the ultimate and not the original one, under a rational system of astronomical study. And when, as in the case of this work, the object is the analysis of the scientific method, and the observation of the logical filia- tion of the leading scientific ideas, it matters less to obtain a clearer exposition of general resvilts than to adhere to the positive method.
I suppose my readers to be well acquainted with the two fundamental facts of the diurnal and annual rotation of our globe, as data without which nothing could be clearly understood of the essential methods and general results of astronomical science. I am not giving a treatise on astronomy, nor even a summaiy; but a series of philo- sojjhical considerations upon the different parts of the science, in which any extended special exposition would be misplaced.
We must first see what methods of observation astro- nomers need, and are possessed of.
157 CHAPTER II.
METHODS OF STUDY OP ASTRONOMY.
SECTION I.
INSTRUMENTS, ALL astronomical observation is, as we ^,,.
nave seen, compreuenaea m the measure- ment of times and. of angles. The two considerations concerned in attaining the great perfection we have reached are the perfecting the instruments, and the appli- cation by theory of certain corrections, without which their precision woulcl be misleading.
The observation of shadows was the first ^i i resource ot astronomers, wlien the rectinnear