the place of the condition imposed by the third co-ordi- nate. This last co-ordinate then becomes a determinate function of the two others, they remaining independent of each other. Thus, there will be a certain equation between the three variable co-ordinates which will be permanent, 112 POSITIVE PHILOSOPHY.
and which will be the only one, in order to correspond to the precise degree of in determination in the position of the point.
Determin<ation ^^ ^^^® expression of Surfaces by Equations, of Surfaces by ^^*^ again in the exj)ression of Equations by Equations, and Surfaces, the same conception is pursued as of Equations j^ ^.he analytical geometry of two dimensions, by Surfaces. -^^ ^^^ ^^.^^ ^^^^^ ^-^^ equation will be the analytical definition of the proposed surface, since it must be verified for all the points of this surface, and for them only. If the surface viudergoes any change, the equation must, as in the case of changing lines, be modified accord- ingly. All geometrical phenomena relating to surfaces may be translated by certain equivalent analytical conditions, proper to equations of three variables: and it is in the establishment and interpretation of this harmony that the science of analytical geometry of three dimensions essentially consists. In the second and converse case, every equation of three variables may, in general, be represented geometri- cally by a determinate surface, defined by the characteristic property that the co-ordinates of all its points always pre- serve the mutual relation exhibited in this equation.
Thus we see in this application the complement of the original idea of Descartes; and it is enough to say this, as every one can extend to surfaces the other considerations which have been indicated with regard to lines. I will only add that the superiority of the rectilinear system of co-ordinates becomes more evident in analytical geometry of three dimensions than in that of two, on account of the geometrical complication which would follow the choice of any other.
Curves of I^ determining Curves of double curvature, double — which is the last elementary point of view curvature. Qf analytical geometry of three dimensions, — the same principle is employed. According to it, it is clear that when a point is reqiaired to be situated upon some certain curve, a single co-ordinate is enough to deter- mine its position completely, by the intersection of this curve with the surface resulting from this co-ordinate. The two other co-ordinates of the point must thus be regarded as functions necessarily determinate, and distinct IMPERFECTIONS OF ANALYTICAL GEOMETRY. 113 from the first. Consequently, every line, considered in space, is represented analytically no longer by a single equation, but by a system of two equations between tlie three co-ordinates of any one of its points. It is evident, indeed, from another point of view, that the equations which, considered sej^arately, express a certain surface, must in combination present the line sought as the inter- section of two determinate surfaces. As for the difficulty occasioned by the infinity of the number of couples of equations, through the infinity of couples of sui'faces which can enter the same system of co-ordinates, and by which the line sought may be hidden under endless algebraical disguises, it must be got rid of by giving up the facilities resulting from such a variety of geometrical constructions. It is sufficient in fact, to obtain from the analytical system established for a certain line, the system corresponding to a single couple of surfaces uniformly generated, and which will not vary except when the line itself shall change. Such is a natural use of this kind of geometrical combina- tion, which thus affords us a certain means of recognizing the identity of lines in spite of the extensive diversity of their equations.
Analytical Geometry still presents some Imperfections imperfections on the side both of geometry of Analytical and of analysis. Geometry.
In regard to Geometry, the equations can as yet represent only entire geometrical loci, and not determinate portions of those loci. Yet it is necessary, occasionally, to be able to express analytically a part of a line or surface, or even a dis- continuous line or surface, composed of a series of sections belonging to distinct geometrical figures. Some progress has been made in supplying means for this purpose, to which our analytical geometry is inapplicable; hxit the method introduced by M. Fourier, in his labours on discon- tinuous functions, is too complicated to be at present intro- duced into our established system.
In regard to analysis, we are so far from ^. ^. i-"" ■),• TP ii-T J in perfections having a complete command or analytical ^^ Analysis geometry, that we cannot furnish anything like an adequate geometrical representation of analytical processes. This is not an imperfection in science, but iu- I. I 114 POSITIVE PHILOSOPHY.
lierent in the very nature of the subject. As Analysis is much more general than geometry, it is of course impos- sible to find among geometrical phenomena a concrete representation of all the laws expressed by analysis: but there is another evil which is due to our own imperfect conceptions; that, in our representations of equations of two or of three variables by lines or surfaces, we regard only the real solutions of equations, without noticing any imaginary ones. Yet these last should, in their general course, be as capable of representation as the first. Hence the graphic representation of the equation is always im- perfect; and it fails altogether when the equation admits of only imaginary solutions. This brings after it, in ana- lytical geometry of two or three dimensions, many incon- veniences of less consequence, arising from the want of correspondence between various analytical modifications and any geometrical phenomena.
We have now seen what Analytical Geometry is. By this science we determine what is the analytical expression of such or such a geometrical phenomenon belonging to lines or surfaces: and, reciprocally, we ascertain the geo- metrical interpretation of such or such an analytical con- sideration. It would be interesting now to consider the most important general questions which would exemplify the manner in which geometers have actually established this beautiful harmony: but such a review is not neces- sary to the purpose of this Work, and would occupy too much space. We have seen what is the character of gene- rality and simplicity inherent iii the science of Geometry. We must now proceed to ascertain v/hat is the true philo- so])hical character of the immense and more complex science of Rational Mechanics.
115 CHAPTER IV.
RATIONAL MECHANICS.
MECHANICAL phenomena are by their j., , ^ -• 1 "^ T Its nature, nature more particular, more comphcated, and more concrete than geometrical phenomena. Therefore they come after geometry in our survey; and therefore must they be pronounced to be more difficult to study, and, as yet, more imperfect. Greometrical questions are always completely independent of Mechanics, while mechanical questions are closely involved with geometrical considerations, — the form of bodies necessarily influencing the phenomena of motion and equilibrium. The simplest change in the form of a body may enhance immeasurably the difficulties of the mechanical problem relating to it, as we see in the question of the mutual gravitation of two bodies, as a result of that of all their molecules; a question which can be completely resolved only by supposing the bodies to be spherical; and thus, the chief difficulty arises out of the geometrical part of the circumstances.
Our tendency to look for the essences of things, instead of studying concrete facts, enters disastrously into the study of Mechanics. "VVe found something of it in geo- metry; but it appears in an aggravated form in Mechanics, from the greater comjilexity of the science. We encounter a perpetual confusion between the abstract and the con- crete points of view; between the logical and the physical; between the ai'tificial conceptions necessary to help us to general laws of equilibrium and motion, and the natural facts furnished by observation, which must form the basis of the science. Great as is the gain of applying Mathe- matical analysis to Mechanics, it has set us back in some respects. The tendency to a priori suppositions, drawn by us from analysis where Newton wisely had recourse to ol)servation, has made our expositions of the science less 116 POSITIVE PHILOSOPHY.
clear tliau those of Newton's days. Inestimable as mathe- matical analysis is for carrying the science on and upwards, there must first be a basis of facts to employ it upon; and Laplace and others were therefore wrong in attempting to prove the elementary law of the composition of forces by analytical demonstration. Even if the science of Mechanics could be constructed on an analytical basis, it is not easy to see how such a science could ever be applied to the actual study of nature. In fact, that which constitutes the reality of Mechanics is that the science is founded on some general facts, furnished by observation, of which we can give no explanation whatever. Our business now is to jjoint out exactly the philosophical character of the science, distinguishing the abstract from the concrete point of view, and separating the experimental department from the logical.
T,,, We have nothing to do here with the causes or modes of production ot motion, but only with the motion itself. Thus, as we are not treating of Physics, but of Mechanics, forces are only motions pro- duced or tending to be produced; and two forces which move a body with the same velocity in the same direction are regarded as identical, whether they proceed from muscular contractions in an animal, or from a gravitation towards a centre, or from collision with another body, or from the elasticity of a fluid. This is now practically understood; but we hear too much still of the old meta- physical language about forces, and the like; and it would be wise to suit our terms to our positive ])liilosophy. J. } • f The bvisiness of Rational Mechanics is to determine how a given l)ody will be affected by any different forces whatever, acting together, when we know what motion would be j^roduced by any one of them acting alone: or, taking it the other way, what are the simple motions whose combination would occasion a known compound motion. This statement shows precisely what are the data and what the unknown parts of every me- chanical question. The science has nothing to do with the action of a single force; for this is, by the terms of tlie statement, supposed to be known. It is concerned solely with the combination of forces whether there results from SCOPE OF RATIONAL MECHANICS. 117 that combination a motion to be studied, or a state of equilibrium, whose conditions have to be described.
The two general questions, the one direct, the other inverse, which constitute the science, are equivalent in importance, as regards their application. Simple motions are a matter of observation, and their combined operation can be understood only through a theory: and again, the compound result being a matter of observation, the simple constituent motions can be ascertained only by reasoning. When we see a heavy body falling obliquely, we know what would be its two simple movements if acted upon separately by the forces to which it is subject, — the direc- tion and uniform velocity which would be caused by the impulsion alone; and again, the acceleration of the vertical motion by its weight alone. The problem is to discover thence the different circumstances of the compound move- ment produced by the combination of the two, — to deter- mine the path of the body, its velocity at each moment, the time occupied in falling; and we might add to the two given forces the resistance of the medium, if its law was known. The best example of the inverse problem is found in celestial mechanics, where we have to determine the forces which carry the planets round the sun, and the satellites round the planets. We know immediately only the compound movement: Kepler's laws give us the characteristics of the movement; and then we have to go liack to the elementary forces by which the heavenly bodies are suj^posed to be impelled, to correspond with the observed result: and these forces once iinderstood, the converse of the question can be managed by geometers, who could never have mastered it in any other way.
Such being the destination of Mechanics, we must now notice its fundamental principles, after clearing the ground by a preparatory observation.
In ancient times, men conceived of matter,,,, 1. •., 11 i- -J. 1 • Matter not inas bemg passive or inert, — all activity being ^^.^ • ^^ Plivsics produced by some external agency, — either of supernatural beings or some metaphysical entities. Now that science enables us to view things more truly, we are aware that there is some movement or activity, more or less, in all bodies whatever. The difference is merelv of 118 POSITIVE PHILOSOPHY.
degree between what men call hrute matter and animated beings. Moreover, science shows iis that there are not different kinds of matter, but that the elements are the same in the most primitive and the most highly organized. If we knew of any substance which had nothing but weight, we could not deny activity even to that; for in falling it is as active as the globe itself, — attracting the earth's particles pi'ecisely as much as its own particles are attracted by the earth. Looking through the whole range of substances, up to those of the highest organization, we find everywhere a spontaneous activity, very various, and at most, in some cases, peculiar; though physiologists are more and more disposed to regard the most peculiar as a modification of antecedent kinds. However this may be, it would be purely absurd now to regard any portion of matter whatever as inert, as a matter of fact, or under the ^ head of Physics. But in Mechanics it must in Mechanicfe^ ^® ^*^ regarded, because we cannot establish any general proposition upon the abstract laws of equilibrium or motion withovit putting out of the question all interference with them by other and inherent forces. What we have to beware of is mixing up this logical supposition with the old notion of actual inertia. Field of Ra- As for how this is to be done, — we must tional Me- remember what has been just said, — that in chanics. Mechanics, we have nothing to do with the origin or different nature of forces; and they are all one while their mechanical operation is uniform. It is impos- sible to conceive of any substance as devoid of weight, for instance; yet geometers have logically to treat of bodies as without an inherent power of attraction. They treat of this power as an external force; that is, it is to them simply a force; and it does not matter to them whether it is inherent or external, — whether it is attraction or impul- sion,— while it is the fall of the body that they have to study. And so on, through the whole range of properties of bodies. When we have so abstracted natural properties, in our logical view, as to have befoi'e us an unmixed case of the action of certain forces, and have ascertained their laws, — then we can pass from abstract to concrete Mechanics, and restore to bodies their natural active pro- THE THREE LAWS OF MOTION. 119 perties, and interpret their action by what we have learned of the laws of motion and equilibrium. This restoration is so difficult to effect, — the transition from the abstract to the concrete in Mecha,nics is so difficult, — that, while its theo- retical domain is unbounded, its practical apjilication is singularly limited. In fact, the application of rational mechanics is limited (accurately speaking) not only to celestial phenomena, but to those of our own solar system. One would suppose that the single property of weight was manageable enough; and that of a given form intelligible enough: but there are such complications of physical cir- cumstances,— as the resistance of media, friction, etc., — even if bodies are conceived of as in a fluid state, that their mechanical phenomena cannot be estimated with any accuracy. And when we proceed to electrical and chemical, and especially to physiological phenomena, we are yet more baffled. General gravitation affords us the only simple and determinate law; and even there we are pei'plexed, when we come to regard certain secondary actions. It may be doubted whether questions of terrestrial mechanics will ever admit — restricted as our means are — of a study at once purely rational and precisely accordant with the general laws of abstract mechanics, — though the knowledge of these laws, primarily indisjjensible, may often lead us to frequent and valuable indications and suggestions.
Bodies being supjjosed inert, the general facts, or latvs of motion to which they are, i?^ ^^^'"^ subject, are tliree; all results of observation.
The first is that law discovered by Kepler, t i- • wliicn is inaptly called the laiv oj inertia. According to it, all motion is rectilinear and uniform; that is, any body impelled by a single foi'ce will move in a right line, and with an invariable velocity. Instead of resorting to the old ways of pronouncing or imagining why it must be so, the Positive Philosophy instructs us to recognise the simple fact that it is so; that, through the whole range of nature, bodies move in a right line, and with a uniform velocity, when impelled by a single force.
The second law we owe to Newton. It is Law of equa- that of the constant equality of action and lity of action reaction; that is, whenever one body is moved ^"•^^ reaction.
120. POSITIVE PHILOSOPHY.
by another, the reaction is such that the second loses pre- cisely as much motion, in proportion to its masses, as the first gains. Whether the movement proceeds from impul- sion or attraction, is, of course, of no consequence. NeAvton treated this general fact as a matter of observation, and most geometers have done the same; so that there has been less fruitless search into the vhy with regard to this second law than to the first.
Law of co-ex- The third fundamental law of motion istence of involves the principle of the independence or motions. coexistence of motions,-w\iich. leads immediately to what is commonly called the composition of forces. Galileo is, strictly speaking, the true discoverer of this law, though he did not regard it precisely under the form in which it is presented here: — that any motion common to all the bodies of any system whatever does not aft'ect the particular motions of these bodies with regard to each other; which motions proceed as if the system were motionless. Speak- ing strictly, we must conceive that all the points of the system describe at the same time parallel and equal straight lines, and consider that this general motion, whatever may be its velocity and direction will not affect relative motions. No a priori considerations can enter here. There is no seeing why the fact should be so, and therefore no antici- pating that it wovdd be so. On the contrary, when Galileo stated this law, he was assailed by a host of objections that his fact was logically impossible. Philosophers were ready with plenty of a priori reasons that it could not be true: and the fact was not unanimously admitted till men had quitted the logical for the physical point of view. We now find, however, that no proposition in the whole range of natural philosophy is founded on observations so simple, so various, so multiplied, so easy of verification. In fact, the whole economy of the universe would be over- thrown, from end to end, but for this law. A ship impelled smoothly, without rolling and pitching, has everything going on within it just the same as if it were at rest; and, in the same way, but on the grandest scale, the great globe itself rushes through space, without its motion at all affect- ing the movements going on on its surface. As we all know, it was ignorance of this third law of motion which THIRD LAW OF MOTIOX. 121 was the main obstacle to the estabhshment of the Coper- nican theory. The Copernicans struggled to get rid of the insurmountable objections to which their doctrine was liable by vain metaphysical subtleties, till Galileo cleared up the difficulty. Since his time, the movement of the globe has been considered an all-sufficient confirmation of the law. Laplace points out to us that if the motion of the globe affected the movements on it, the effect could not be uniform, but must vary with the diversities of their direction, and of the angle that each direction would make with that of the earth: whereas, we know how invariable is, for instance, the oscillating movement of the pendulum, whatever may be its direction in comparison with that of the travelling globe.
It may be as well to point out that rotary motion does not enter into this case at all, but only translation, because the latter is the only motion which can be, in degree and direction, absolutely common to all the parts of a system. In a rotatiug system, for instance, all the i:)arts are not at an equal distance from the centre of rotation. When the interior of a ship is affected, it is by the rolling and pitch- ing, which are rotary movements. We may carry a watch any distance without affecting its interior movements; but it will not bear whirling.
And, again, the forward motion of the globe could be discovered by no other means than astronomical observation; whereas, the changes which occur on the surface of the earth, produced by the inequality of the centrifugal force at its different points, are sufficient evidence of its rotation, independently of all astronomical considerations whatever.
The law or rule of the composition of forces, which is involved in the general fact just stated, is, in fact, identical with it. It is only another way of expressing the same law. If a single impulsion describes a parallelogram of forces, as the scientific term is, the effect of a second will be to describe the diagonal of the jiarallelogram. This is nothing more than an application of the law of the inde- pendence of forces; since the motion of any body along a straight line is in no way disturbed by a general motion which carries away, parallel with itself, the whole of this right line along any other right line whatever. This con- 122 POSITIVE PHILOSOPHY.
sideration leads immediately to the geometrical construction expressed by the rule of the parallelogram of forces. And thus it appears that this fundamental theorem of Rational Mechanics is a true natural law; or, at least, a direct appli- cation of one of the greatest natural laws. And this is the best account to give of it, instead of looking to logic for a fallacious a priori deduction of it. Any analytical demon- stration, too, must suppose certain portions of the case to be evident; and to talk of a thing being evident is to refer back to nature, and to depend on observation of nature.
It is worthy of remark that those who wish to make a separate law of the comjjosition of forces, in order to avoid introducingthe third law into the prolegomena of Mechanics, and to dispense with it in the exjjosition of Statics, are brought back to it when entering upon the study of Dynamics. Upon this alone can be based the important law of the proportion of forces to velocities. The rela- tions of forces may be determined either by a statical or dynamical j^rocedure. No purpose is answei-ed by the transposition of the general fact of the independence of forces to the dynamical department of the science: it is equally necessary for the statical; and a world of meta- physical confusion is saved by laying it down as the broad basis that in fact it is.
These three laws are the experimental basis of the science of Mechanics. From them the mind may proceed to the logical construction of the science, without further reference to the external world. At least, so it appears to me; though I am far from assigning any a priori reasons why more laws may not be hereafter discovered, if these three should prove to be incomplete. There cannot, in the nature of things, be many more; and I would rather incur the inconvenience of the introduction of one or two, than run any risk of surrendering the positive character of the science and overstraining its logical considerations. We cannot however conceive of any case which is not met by these three laws of Kepler, of Newton, and of Gralileo; and their expression is so precise, that they can be immediately treated in the form of analytical equations easily obtained. As for the most extensive, important, and difficult part of the science, the mechanics of varied motion or continuous DIVISIONS OF RATIONAL MECHANICS. 123 forces, we can perceive the possibility of reducing it to elementary Mechanics by the application of the infinitesimal method. For each infinitely small point of time, we must substitute a uniform motion in the place of a varied one, whence will immediately result the differential equa- tions relative to these varied motions. We may hereafter see what results have been obtained in regard to the abstract laws of equilibrium and motion. Meantime, we see that the whole science is founded on the combination of the three physical laws just established; and here lies the distinct boundary between the physical and the logical parts of the science.