Comte - The positive philosophy. Vol. 1

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as those masses; or, in other words, that the forces are proportional to the masses, as we have seen them, under the third law of motion, to be proportional to the velocities. All phenomena, such as the communication of motion by collision, or in any other way, have tended to confirm the supposition of this new kind of proportion. It evidently results from this, that when we have to compare forces which impress different velocities on unequal masses, each must be measured according to the product of the mass upon which it acts by the corresponding velocity. This product is called by geometers quantity of motion; and it determines the percussion, of a body, and also the pressure that a body may exercise against any fixed obstacle to its motion.

Proceeding to the second dynamical case, we see that the characteristic difficulty of this order of questions consists in the way of estimating the connection of the different bodies of the system, in virtue of which their mutual reactions will necessarily affect the motions which each would take if alone; and we can have no a priori knowledge of what the alterations will be. In the case of the pendulum, for instance, the particles nearest the point of suspension, and those furthest from it, must react on each other by their connection,—the one moving faster and the other slower than if they had been free; and no established dynamic principle exists revealing the law which determines these reactions. Geometers naturally began by laying down a principle for each particular case; and many were the principles thus offered, which turned out to be only remarkable theorems furnished simultaneously by fundamental dynamic equations. Lagrange has given us, in his “Analytical Mechanics,” the general history of this series of labours: and very interesting it is, as a study of the progressive march of the human intellect. This method of proceeding continued till the time of D’Alembert, who put an end to all these isolated researches by seeing how to compute the reactions of the bodies of a system in virtue of their connection, and establishing the fundamental equations of the motion of any system. By the aid of the great principle which bears his name, he made questions of motion merge in simple questions of equilibrium. The principle is simply this. In the case supposed, the natural motion clearly divides itself into two,—the one which subsists and the one which has been destroyed. By D’Alembert’s view, all these last, or, in other words, all the motions that have been lost or gained by the different bodies of the system by their reaction, necessarily balance each other, under the conditions of the connection which characterizes the proposed system. James

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Bernouilli saw this with regard to the particular case of the pendulum; and he was led by it to form an equation adapted to determine the centre of oscillation of the most simple system of weight. But he extended the resource no further; and what he did detracts nothing from the credit of D’Alembert’s conception, the excellence of which consists in its entire generality.

In D’Alembert’s hands the principle seemed to have a purely logical character. But its germ may be recognized in the second law of motion, established by Newton, under the name of the equality of reaction and action. They are in fact, the same, with regard to two bodies only acting upon each other in the line which connects them. The one is the greatest possible generalization of the other; and this way of regarding it brings out its true nature, by giving it the physical character which D’Alembert did not impress upon it. Henceforth therefore we recognize in it the second law of motion, extended to any number of bodies connected in any manner.

We see how every dynamical question is thus convertible into one of Statics, by forming, in each case, equations of equilibrium between the destroyed motions. But then comes the difficulty of making out what the destroyed motions are. In endeavouring to get rid of the embarrassing consideration of the quantities of motion lost or gained Euler, above others, has supplied us with the method most suitable for use,—that of attributing to each body a quantity of motion equal and contrary to that which it exhibits, it being evident that if such equal and contrary motion could be imposed upon it, equilibrium would be the result. This method contemplates only the primitive and the actual motions which are the true elements of the dynamic problem,—the given find the unknown; and it is under this method that D’Alembert’s principle is habitually conceived of. Questions of motion being thus reduced to questions of equilibrium, the next step is to combine D’Alembert’s principle with that of virtual velocities. This is the combination proposed by Lagrange, and developed in his “Analytical Mechanics,” which has carried up the science of abstract Mechanics to the highest degree of logical perfec- tion,—that is, to a rigorous unity. All questions that it can comprehend are brought under a single principle, through which the solution of any problem whatever offers only analytical difficulties.

D’Alembert immediately applied his principle to the case of flu- ids—liquid and gaseous, which evidently admit of its use as well as solids, their peculiar conditions being considered. The result was our

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obtaining general equations of the motion of fluids, wholly unknown before. The principal of virtual velocities rendered this perfectly easy, and again left nothing to be desired, in regard to concrete considerations, and presented none but analytical difficulties. We must admit however that our actual knowledge obtained under this theory is extremely imperfect, owing to insurmountable difficulties in the integrations required. If it was so in questions of pure Statics, much more must it be so in the more complex dynamical questions. The problem of the flow of a gravitating liquid through a given orifice, simple as it appears, has never yet been resolved. To simplify as far as they could, geometers have had recourse to Daniel Bernouilli’s hypothesis of the parallelism of sections, which admits of our considering motion in regard to horizontal laminae instead of particle by particle. But this method of considering each horizontal laming of a liquid as moving altogether, and talking the place of the following, is evidently contrary to the fact in almost all cases. The lateral motions are wholly abstracted, and their sensible existence imposes on us the necessity of studying the motion of each particle. We must then consider the science of hydrodynamics as being still in its infancy, even with regard to liquids, and much more with retard to gases. Yet, as the fundamental equations of the motions of fluids are irreversibly established, it is clear that what remains to be accomplished is in the direction of mathematical analysis alone.

Such is the Method of Rational Mechanics. As for the great theoretical results of the science,—the principal general properties of equilibrium and motion thus far discovered,—they were at first, taken for real principles, each being destined to furnish the solution of a certain order of new problems in .Mechanics. As the systematic character of the science has come out however, these supposed principles have shown themselves to be mere theorems,—necessary results of the fundamental theories of abstract Statics and Dynamics.

Of these theorems, two belong to Statics. The most remarkable is that discovered by Torricelli with regard to the equilibrium of heavy bodies. It consists in this; that when any system of heavy bodies is in a situation of equilibrium, its centre of gravity is necessarily placed at the lowest or highest possible point, in comparison with all the positions it might take under any other situation of the system.—Maupertuis afterwards by his working out of his Law of Repose, gave a large generalization to this theorem of Torricelli’s, which at once became a mere particular case under that law, Torricelli’s applying merely to cases of ter-

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restrial gravitation. while that of Maupertuis extends throughout the whole sphere of the great natural attractive forces.

The other general property relating to equilibrium may be regarded as a necessary complement of the former. It consists in the fundamental distinction between the cases of stability and instability of equilibrium. There being no such thing in nature as abstract repose, the term is applied here to that state of stable equilibrium which exists where the centre of gravity is placed as low as possible, while unstable equilibrium is that which is popularly called equilibrium; and it exists when the centre of gravity is placed as high as possible. Maupertuis’s theorem consisted in this,—that the situation of equilibrium of any system is always that in which the sum of vires vivae (active forces) is a maximum or a minimum; mud the one under notice developed by Lagrange, consists in this,—that in any system equilibrium is stable or unstable according as the sum of vires vivae is a maximum or a minimum Observation teaches the facts in the most simple cases; but it requires a large theory to exhibit to geometers that the distinction is equally applicable to the most compound systems.

Proceeding to the theorems relative to dynamics, the most direct way of establishing them is that used by Lagrange,—exhibiting them as immediate consequences of the general equation of dynamics, deduced from the combination of D’Alembert’s principle with the principle of virtual velocities. The first theorem is that of the conservation of the motion of the centre of gravity, discovered by Newton. Newton showed that the mutual action of the bodies of any system, whether of attraction, impulsion, or any of the notion other,—regard being had to the constant equality between action and reaction,—can not in any way affect the state of the centre of gravity; so that if there were no accelerating forces besides, and if the exterior forces of the system were educed to instantaneous forces, the centre of gravity would remain immovable, or would move uniformly in a right line. D’Alembert generalized this property, and exhibited it in such a form that every case in which the motion of the centre of gravity has to be considered may be treated as that of a single molecule. It is seldom that we form an idea of the entire theoretical generality of such great results as those of rational Mechanics. We think of them as relating to inorganic bodies, or as otherwise circumscribed j but we cannot too carefully remember that they apply to all phenomena whatever; and in virtue of this universality alone are the basis of all real science.

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The second general theorem of dynamics is the principle of areas, the first perception of which is attributable to Kepler. In its simplest form it is this, that if the accelerating force of any molecule tends constantly towards a fixed point, the vector radius of the moving body describes equal areas in equal times round the fixed point; so that the area described at the end of any time increases in proportion to the time: and the reciprocal fact is clear,—that the evidence of the areas and the times proves the action upon the body of a force directed towards the fixed point. This discovery of Kepler’s is the more remarkable for having been made before dynamics had been really created by Galileo. Its importance in astronomy we shall see hereafter. But though, in its simplest form, it is one of the bases of celestial Mechanics, it is, in fact, only the simplest particular case of the great general theorem of areas, exhibited in the middle of the last century by D’Arcy, Daniel Bernouilli, and Euler. Kepler’s discovery related only to the motion of a point, while the later one refers to the motion of any system of bodies, acting on each other in any manner whatever; which constitutes a case, not only more complex, but different, on account of the mutual actions involved. It yields proof, however, that though the area described by the vector radius of each molecule may be altered by reciprocal actions, the sum of the areas described will remain invariable in a given time, and will increase therefore in proportion to the time. As the theorem of the centre of gravity determines all that relates to motions of translation, this determines all that relates to motions of rotation: 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 concep- tion—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 philosophical 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 that dynamic property which he called the invariable plane, the consideration of 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 necessarily fixed term of comparison, by which to estimate the variations of the heavenly

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bodies. We are far from having yet attained precision in the determination of the 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 simplifying, has once more extended the process to which his method is applied: and he has repaired an important omission made by Laplace, in taking into the account the smaller areas described by satellites, and their rotation, and that of the sun itself; whereas Replace 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 moment of inertia, and the principal axes, which are among the most important general results 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 put 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 indispensable, vet very important from the simplification they introduce into special researches. Students will recognize their functions, when their mere names are presented, which is all that our space allows:—I refer to the theorem of the conservation of active forces,— singularly important in its applications to industrial Mechanics: the theorem, improperly called the principle of the least action, as old as Ptolemy, who observed that reflected light takes the shortest way from one point to another,—an observation which was the basis of Maupertuis’ discovery of this property: and lastly, a theorem not usually classed with the foregoing, yet worthy of no less esteem,—the theorem of the coexistence of small oscillations, of Daniel Bernouilli. 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 discovered in Rational Mechanics concludes our review of the second branch of Concrete Mathematics.

As for our review of the whole science, I 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 com-

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pass of the human intellect. But I hope that those who have not the misfortune to be wholly ignorant 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 thermological phenomena, according to the discoveries of Fourier. But, to avoid too great a breach of customary arrangement, I have reserved the subject, and shall place Thermology among the departments of Physics. Mathematical philosophy being now completely characterized, we shall proceed to examine its application to the study of Natural Phenomena, in their various orders, ranked according 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.

Book II: Astronomy

Chapter I General View

It is easy to describe clearly the character of astronomical science, from its being thoroughly separated, in our time, from all theological and metaphysical influence. Looking 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 perceives 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 how we may determine their forms, their distances, their bulk, and their motions, but we can never know anything of their chemical or mineralogical 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 present 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, 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 severe is that of questions of temperatures. The mathematical thermology

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created by Fourier may tempt us to hope that, as he has estimated the temperature of the space 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 can 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 temperature of our globe in the circumstances of that comet. We may therefore define Astronomy as the science by which we discover the laws of the geometrical and mechanical phenomena presented by the heavenly bodies.

It is desirable to add a limitation which is important, though not of primary necessity. 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 satisfactory; 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 command, while we have not obtained the first and most simple element in sidereal astronomy—the determination of the stellar intervals. Whatever 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 becomes, in turn, our means of knowledge. This is evidently and remarkably true in regard to Astronomy. It is of the highest importance to us to know the laws of the solar system: and we have attained great precision 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 each 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

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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 astronomy the supreme and positive science that we have determined 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 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 obtained 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 phenomenon 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 analogous cases, in which the phenomenon is more and more simplified. It is only 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 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 are constructed by our understanding. The simplicity of the phenomena to be studied, and