As for its divisions, the first and most im- -p^^.^ Primary portant is into Statics and Dynamics; that divisions, is, into questions relating to equilibrium and Statics and questions relating to motion. Statics are the -IJ^'NAMics. easiest to treat, because we abstract from them the element of time, which must enter into Dynamical questions, and complicate them. The whole of Statics corresponds to the very small portion of Dynamics which relates to the theory of uniform motions. This division corresponds well with the facts of human education in this science. The fine researches of Archimedes show us that the ancients, though far from having obtained any complete system of rational Statics, had acquired much essential knowledge of equili- brium— both of solids and fluids — while as yet wholly without the most rudimentary knowledge of Dynamics. G-alileo, in fact, created that department of the science.
The next division is that of Solids from Secondary Fluids. This division is generally placed divisions, first, but it is unquestionable that the laws Solids and of statics and dynamics must enter into the -ttmda.
study of solids and fluids, that of fluids requiring the addition of one more consideration, — variability of form. This however is a consideration which introduces the neces- sity of treating separately the molecules of which fluids are composed, and fluids as systems comj^osed of an infinity of distinct forces. A new order of researches is introduced into Statics, relative to the form of the system in a state of equilibrium; but in Dynamics the questions are still more difiicult to deal with. The importance and difficulty of the 124 rOSITIVE PHILOSOPHY.
researches under this division cannot be exaggerated. Their complication places even the easiest cases beyond our reach, except by the aid of extremely precarious hypotheses. We must admit the vast necessary difficulty of hydrostatics, and yet more of hydrodynamics, in comparison with statics and dynamics, properly so called, which are in fact far more advanced.
Much of the difficulty arises from the mathematical statement of the question differing from the natural facts. Mathematical fluids have no adhesion between their par- ticles; whereas natural fluids have, more or less; and many natural phenomena are due to this adherence, small tliough it be in comparison with that of solids. Thus, the result of an observation of the quantity of a given fluid which Avill run out of a given oi'ifice will differ widely from the result of the mathematical calculation of what it should be. Though the case of solids is easier, yet there perplexity may be introduced by the disrupting action of forces, of which abstraction must be made in the mathe- matical question. The theory of the rupture of solids, initiated by Galileo, Huyghens, and Leibnitz, is still in a very imperfect and precarious state, great as are the pains which have been taken with it, and much needed as it is. Not so much needed however as the mechanics of fluids, because it does not affect questions of celestial mechanics; and in this highest department alone can we, as I said before, see the complete application of rational mechanics.
There is a gap left between these two studies, which should be pointed out, though it is of secondary importance. We want a Mechanics of semi-fluids, or semi-solids, — as of sand, in relation to solids, and gelatinous conditions of fluids. Some considerations have been offered with regard to these " imperfect fluids," as they are called; but their true theory has never been established in any direct and general manner.
Such is the general view of the philosophical character of Rational Mechanics. We must now take a philosophical view of the composition of the science, in order to see how this great second department of Concrete Mathematics has attained the theoretical perfection in which it appears in the works of Lagrange, who has rendered all its j^ossible METHODS OF STATICS. 125 abstract questions capable of an analytical solution, like those of geometry. We must first take a view of Statics, and then proceed to Dynamics.
SECTION I.
STATICS.
There are two ways of treating Rational Converse Mechanics, according as Statics are regarded metliods of directly, or as a particular case of Dynamics. treatment.
By the first method we have to discover a principle of equilibrivim so general as to be applicable to the conditions of eqviilibrium of all systems of possible forces. By the second method, we reverse the process, — ascertaining what motion would result from the simultaneous action of any proposed differing forces, and then determining what rela- tions of these forces would render motion null.
The first method was the only one possible First method. in the early days of science; for, as I have Statics by said before, Glalileo was the creator of the itself, science of Dynamics. Archimedes, the founder of Statics, established the condition of equilibrium of two weights suspended at the ends of a straight lever; that is, he showed that the weights must be in an inverse ratio to their distances from the fulcrum of the lever. He endea- voured to refer to this principle the relations of equilibrium proper to other systems of forces; but the principle of the lever is not in itself general enough for such application. The various devices by which it was attempted to extend the process, and to supply the remaining deficiences, were relinquished when the establishment of Dynamics permitted the use of the second method, — of seeking gecond the conditions of equilibrium through the method, laws of the composition of forces. It is by Statics this last method that Yarignon discovered through Dy- the theory of the equilibrium of a system "^""^s- of forces applied upon a single point; and that D'Alembert afterwards established, for the first time, the equations of equilibrium of any system of forces applied to the diiferent 126 POSITIVE PHILOSOPHY.
points of a solid body of an invariable form. At this day, this is the method universally employed. At the first glance, it does not appear the most rational, — Dynamics being more complicated than Statics, and precedence being natural to the simpler. It would, in fact, be more philoso- phical to refer dynamics to statics, as has since been done; but we may observe that it is only the most elementary part of dynamics, the theory of uniform motions, that we are concerned with in treating statics as a particular case of dynamics. The comj^licated considerations of varied motions do not enter into the process at all.
The easiest method of applying the theory of uniform motions to statical questions is through the view that, when forces are in equilibrio, each of them, taken singly, may be regaixled as destroying the effect of all the others together. Thus, the thing to be done is to show that any one of the forces of the system is equal, and directly opposed, to the resulting force of all the rest. The only difficulty here is in determining the resultant force; that is, in mutually conapounding the given forces. Here comes in the aid of the third great law of motion, and having compounded the two first forces, we can deduce the com- position of any number of forces.
After having established the elementary laws of the composition of force, geometers, before applying them to the investigation of the conditions of equilibrium, usually subject them to an important transformation, which, with- out being indispensable, is of eminent utility, in an analytical view, from the extreme simplification which it introduces into the algebraical expression of the conditions ,,, ^ of equilibrium. The transformation consists in what is called the theory of Moments, the essential property of which is to reduce, analytically, all the laws of the composition of forces to simple additions and subtractions. Without going into an examination of this theory, it is necessary simply to say that it considers statics as a particular case of elementary dynamics, and that its value is in the simplicity which it gives to the analytical part of the process of investigation into the con- ditions of equilibrium. Simple, however, as may be the operation, and great as may be the practical advantage LAGRANGE'S DEVELOPMENT OF RATIONAL MECHANICS. 127 gained tlirotigli the treatment of statics as a particular case of elementary Dynamics, it would be satisfactory to return, if we could, to the method of the,,^, ^ ancients, — to leave Dynamics on one side, ^^ ^j^^ method and proceed directly to the investigation of the laws of equilibrium regarded by itself, by means of a direct general principle of equilibrium. Geometers strove after this as soon as the general equations of equilibrium were discovered by the dynamic method. But a higher motive than even the desire to place statics in a more philosophical position impelled them to establish a direct Statical method: and this it was which caused Lagrange to carry up the whole science of Rational Mechanics to the philosophical perfection which it now enjoys.
D'Alembert made a discoveiy (to be treated of here- after), by the help of which all investigation of the motion of any body or system might be converted at once into a question of equilibrium. This amounts, in fact, to a vast generalization of the second fundamental law of motion; and it has served for a century past as a permanent basis for the solution of all great dynamical questions; and it must be so applied more and more, from its high merits of simplification in the most difficult investigations. Still, it is clear that this method compels a return into statics; and Statics as independent of Dynamics, which are altogether derived from Statics. A science must be imj^erfectly laid down, as long as it is necessary thus to jiass backwards and forwards between its two departments. In order to estab- lish the necessary unity, and to j^rovide scope for D'Alem- bert's principle, a complete reconstitution of Rational Mechanics was indispensable. Lagrange effected this in his admirable treatise on " Analytical Mechanics," the lead- ing conception of which must be the basis of all future labours of geometers upon the laws of equilibrium and motion, as we have seen that the great idea of Descartes is with regard to geometrical speculations.
The principle of Virtual Velocities, — the ■.- + i one which Lagrange selected from among the cities'^ ^^ °" proj^erties of equilibrium, — had been dis- covered by Galileo in the case of two forces, as a general property manifested by the equilibrium of all machines.
128 POSITIVE PHILOSOPHY.
John Beniouilli extended it to any number of forces, com- posing any system. Varignon afterwards expressly pointed out the universal use that might be made of it in Statics. The combination of it with D'Alembert's principle led Lagrange to conceive of the whole of Rational Mechanics as deduced from a single fundamental theorem, and to give it that rigorous unity which is the highest philosophical perfection of a science.
The clearest idea of the system of virtual velocities may be obtained by considering the simple case of two forces, w^hich was that presented by Galileo. We suppose two forces balancing each other by the aid of any instrument whatever. If we suppose that the system should assume an infinitely small motion, the forces are, with regard to each other, in an inverse ratio to the spaces traversed by their points of application in the path of their directions. These spaces are called virtual velocities, in distinction from the real velocities which would take place if the equilibrium did not exist. In this primitive state, the principle, easily verified with regard to all known machines, offers great practical utility; for it permits us to obtain with ease the mathematical condition of equilibrium of any machine whatever, whether its constitution is known or not. If we give the name of virtual momentum (or simply of momentum in its primitive sense) to the product of each force by its virtual velocity, — a product which in fact then measures the effort of the force to move the machine, — we may greatly simplify the statement of the principle in merely saying that, in this case, the momentum of the two forces must be equal and of opposite signs, that there may be equilibrium, and that the positive or negative sign of each momentum is determined according to that of the virtual velocity, which will be considered positive or nega- tive according as, by the supposed motion, the projection of the point of application would be found to fall upon the direction of the force or upon its prolongation. This abridged expression of the principle of virtual velocities is especially iiseful for the statement of this princijjle in a general manner, with regard to any system of forces whatever. It is simply this: that the algebraic si;m of the virtual moments of all forces, estimated according to the jpreced- VIRTUAL VELOCITIES. 129 iug rule, must be null to cause equilibrium: and this con- dition must exist distinctly witli regard to all the elemen- tary motions which the system might assume in virtue of the forces by which it is aniinated. In the equation, con- taining this principle, furnished by Lagrange, the whole of Rational Mechanics may be considered to be implicitly comprehended.
While the theorem of vii'tual velocities was conceived of only as a general property of equilibrium, it could be veri- fied by observing its constant conformity with the ordinary laws of equilibrium, otherwise obtained, of which it was a summary, useful by its simplicity and uniformity. But, if it was to be a fundamental principle, a basis of the whole science, it must be underived, or at least caj^able of being presented in its preliminary propositions as a matter of observation. This was done by Lagrange, by his ingenious demonstration through a system of pulleys. He exhibited the theorem of virtual velocities very easily by imagining a single weight which, by means of pulleys suitably con- structed, replaces simultaneously all the forces of the system. Many other demonstrations have been furnished; but, while more com^^licated, they are not logically superior. From the philosophical jjoiut of view it is clear that this general theorem, being a necessary consequence of the fundamental laws of motion, can be deduced in various ways, and becomes practically the point of departure of the whole of Rational Mechanics. A perfect unity having been established by this principle, we need not look for any others; and we may rest assured that Lagrange has carried the co-ordination of the science as far as it can go. The only possible object would be to simplify the analytical researches to which the science is now reduced; and nothinsr can be conceived more admirable for this purpose than Lagrange's adaptation of the ])rinciple of virtual velocities to the uniform application of mathematical analysis.
Striking as is the philosophical eminence of this prin- ciple, there are difficulties enough in its use to prevent its being considered elementary, so far as to j^reclude the con- sideration of any other in a course of dogmatic teaching. It is for this reason that I have referred to the dynamic method, properly so called, which is the only one in general I. K 130 POSITIVE PHILOSOPHY.
use at present. All other considerations must however be only provisional. Lagrange's method is at pi'esenttoo new; but it is impossible that it should for ever remain in the hands of a small number of geometers, who alone shall be able to make use of its admirable properties. It must become as popular in the mathematical world as the great geometrical conception of Descartes: and this general pro- gress would be almost accomplished if the fundamental ideas of transcendental analysis were as widely spread as they ought to be.
The greatest acquisition, since the regene- Counles ration of the science by Lagrange, is the conception of M. Poinsot, — the theory of Couples, which appears to me to be far from being sufficiently valued by the greater number of geometers. These Couples, or systems of parallel forces, equal and contrary, had been merely remarked before the time of M. Poinsot, as a sort of paradox in Statics. He seized upon this idea, and made it the subject of an extended and original theory relating to the transformation, composition, and use of these singular groups, which he has shown to be endowed with properties remarkable for their generality and simplicity. He used the dynamic method in his study of the conditions of equilibrium: but he presented it, by the aid of his theory of couples, in a new and simplified aspect. But his con- ception will do more for dynamics than for statics; and it has hardly yet entered upon its chief office. Its value will be appreciated when it is fonnd to render the notion of the movements of rotation as natural, as familiar, and almost as simple as that of forward movement or translation. Share of eqiia- ^^^^ more consideration should, I think, be tions in pro- adverted to before we quit the subject of ducing equili- statics as a whole. When we study the nature bruuii. q£ ^j-^g equations which express the conditions of equilibrium of any system of forces, it seems to me not enough to establish that the sum of these equations is in- dispensable for equilibrium. I think the fiu'ther statement is necessary, — in what degree each contributes to the result. It is clear that each equation must destroy some one of the possible motions that the body would make in virtue of existing forces; so that the whole of the equations must EQUATIONS OF EQUILIBRIUM. 131 produce equilibrium by leaving an impossibility for the body to move in any way whatever. Now the natural state of things is for movement to consist of rotation and translation. Either of tliese may exist without the other; Ijut the cases ai-e so extremely rare of their being found apart, that the verification of either is regarded by geo- meters as the strongest presumption of the existence of the other. Thus, when the rotation of the sun uj^on its axis was established, every geometer concluded that it had also a progressive motion, carrying all its planets Avith it, before astronomers had produced any evidence that such was actually the case. In the same way we conclude that certain planets, travelling in their orbits, rotate round their axes, though the fact has not yet been verified. Some equations must therefore tend to destroy all progres- sive motion, and others all motion of rotation. How many equations of each kind must there be?
It is clear that, to keep a body motionless, it must be hindered from moving according to three axes in different planes — commonly supposed to be perpendicular to each other. If a body cannot move from north to south, nor the reverse; nor from east to west, nor the reverse; nor up, nor down, it is clear that it cannot move at all. Movement in any intermediate direction might be conceived of as partial progression in one of these, and is therefore impos- sible. On the other hand, we cannot reckon fewer than three independent elementary motions; for the body might move in the direction of one of the axes, without having any translation in the direction of either of the others. Thus we see that, in a general way, three equations are necessary, and three are sufficient to establish the absence of translation; each being specially adapted to destroy one of the three progressive motions of which the body is capable. The same view presents itself with regard to the other motion, — of rotation. The mechanical conception is more complicated; but it is true, as in the simpler case, that motion is possible in only three directions, — in three co-ordinated planes, or round three axes. Three equations are necessary and sufficient here also; and thus we have six which are indis2:)ensable and sxifficient to stop all motion whatever.
132 POSITIVE PHILOSOPHY.
When, instead of supposing any system of forces what- ever as the subject of the question, we particularize any, we get rid of more or fewer possible movements. Havmg excluded these, we may exclude also their corresponding equations, retaining only those which relate to the possible motions that remain. Thus, instead of having to deal with six equations necessary to equilibrium, there may be only three, or two, or even one, which it will be easy enough to obtain in each case. These remarhs may be extended to any restrictions upon motion, whether resulting from the special constitution of the system of forces, or from any other kind of control, affecting the body under notice. If, for instance, the body were fastened to a point, so that it could freely rotate but not advance, three equations would siiffice: and again, if it is fastened to two fixed points, two equations are enough; and even one, if these two fixed points are so placed as to prevent the body from moving on the axis between them. Finally, its being attached to thi*ee fixed points, not in a right line, will prevent its moving at all, and establish equilibrium without any condition, what- ever may be the forces of the system. The spirit of this analysis is entirely independent of any method by which the equations of equilibrium will have been obtained: but the different general methods ai'e far from being equally suitable to the application of this rule. The one which is best adapted to it is, undoubtedly, the Statical one, pro- perly so called, founded, as has been shown, on the prin- ciple of virtual velocities. In fact, one of the chai*acteristic properties of this principle is the pei-fect precision with which it analyses the phenomena of equilibrium, by dis- tinctly considering each of the elementary motions per- mitted by the forces of the system, and furnishing im- mediately an equation of equilibrium specially relating to this motion.
Connection of When we conje to the inquiry how tlie concrete geometers apply the principles of abstract with the ab- Mechanics to the properties of real bodies, we stract question, j^^^gt state that the only complete application yet accomplished is in the question of terrestrial gravity. Now, this is a subject which cannot, logically, be treated under the head of Mechanics, as it belongs to Physics. It CONNECTIOX OF COXCRETE AND ABSTRACT QUESTION. 133 is sufficient to explain that the statical study of terrestrial gravity becomes convertible into that of centres of gravity; and that all confusion between the two dej^artments of research would be avoided if we accustomed ourselves to class the theory of centres of gravity among the questions of pure geometry. In seeking the centre of gravity as (according to the logical denomination of the ancient geoineters) the centre of mean distances, we remove all traces of the mechanical origin of the Cjuestion, and convert it into this jiroblem of general geometry: — Given, any system of points disposed in a determinate way with regard to each other, to find a point whose distance to any plane shall be a mean between the distances of all the given points to the same plane. — The abstraction of all considera- tion of gravity is an assistance in every way. The simple geometrical idea is precisely what we want in most of the principal theories of Rational Mechanics, and especially when we contemplate the great dynamic properties of the centre of mean distances; in which study the idea of gravity becomes a mere encumbrance and perplexity. It is true that, by proceeding thus, we exclude the question from the domain of Mechanics, to place it in that of Geometry. I should have so classed it but for an un- willingness to break in upon established customs. How- ever it may be as to the matter of arrangement, it is highly important for us not to misapprehend the true nature of the cjuestion. — The integral calculus offers the means of surmounting those difficulties in determining the centre of gravity which are imposed by the conditions of the ques- tion. But, the integrations in this case being more com- plicated than those to which they are analogous, — those of quadratures and cubatures, — their precise solution is, owing to the extreme imperfection of the integral Calcuhis, much more rarely obtained. It is a matter of high importance, however, to be able to introduce the consideration of the centre of gravity into general theories of analytical mechanics.
Such is, then, the relation of terrestrial gravitation to the science of abstract Statics. As for universal gravita- tion, no complete study has yet been made of it, except in regard to spherical bodies. What we know of the law of 134 POSITIVE PHILOSOPEIY.
gravitation would easily enable us to compute the mutual attraction of all known bodies, if tlie conditions of each body were understood by us; but this is not the case. For instance, we know nothing of the law of density in the interior of the heavenly bodies. It is still true that the primitive theorems of Newton on the attraction of sj^herical bodies are the most useful part of our knowledge in this direction.