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 the three co-ordinates of any one of its points. It is evident, indeed, from another point of view, that the equations which, considered separately, express a certain surface, must in combination present the line sought as the intersection 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 surfaces 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 combination, 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 discontinuous 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; but the method introduced by M. Fourier, in his labours on discontinuous functions, is too complicated to be at present introduced into our established system. In regard to analysis, we are so far from having a complete command of analytical of Analysis. Imperfections geometry, that we cannot furnish anything like an adequate geometrical representation of analytical processes. This is not an imperfection in science, but in herent in the very nature of the subject. As Analysis is much more general than geometry, it is of course impossible 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 imperfect; and it fails altogether when the equation admits of only imaginary solutions. This brings after it, in analytical geometry of two or three dimensions, many inconveniences 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 geometrical interpretation of such or such an analytical consideration. 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 necessary to the purpose of this Work, and would occupy too much space. We have seen what is the character of generality and simplicity inherent in the science of GEOMETRY. We must now proceed to ascertain what is the true philosophical character of the immense and more complex science of RATIONAL MECHANICS. CHAPTER IV. RATIONAL MECHANICS. MECHANICAL phenomena are by their Its nature. nature more particular, more complicated, 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. Geometrical 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. We found something of it in geometry; but it appears in an aggravated form in Mechanics, from the greater complexity of the science. We encounter a perpetual confusion between the abstract and the concrete points of view; between the logical and the physical; between the artificial 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 Mathematical analysis to Mechanics, it has set us back in some respects. The tendency to à priori suppositions, drawn by us from analysis where Newton wisely had recourse to observation, has made our expositions of the science less clear than those of Newton's days. Inestimable as mathematical 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 analvtical 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 point 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. We have nothing to do here with the causes Its characters. or modes of production of motion, but only with the motion itself. Thus, as we are not treating of Physics, but of Mechanics, forces are only motions produced 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 metaphysical language about forces, and the like; and it would be wise to suit our terms to our positive philosophy. The business of Rational Mechanics is to Its object. determine how a given body will be affected by any different forces whatever, acting together, when we know what motion would be produced 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 mechanical question. The science has nothing to do with the action of a single force; for this is, by the terms of the 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 direction 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 movement produced by the combination of the two, to determine 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 back to the elementary forces by which the heavenly bodies are supposed to be impelled, to correspond with the observed result: and these forces once understood, 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 Matter not in as being passive or inert, all activity being ert in Physics. 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 merely of |