General (or Analytical) geometry being entirely founded upon the transformation of geometrical considerations into equivalent analytical considerations, we must begin with examining directly and in a thorough manner the beautiful conception by which Descartes has established in a uniform manner the constant possibility of such a co-relation. Besides its own extreme importance as a means of highly perfecting geometrical science, or, rather, of establishing the whole of it on rational bases, the philosophical study of this admirable conception must have so much the greater interest in our eyes from its characterizing with perfect clearness the general method to be employed in organizing the relations of the abstract to the concrete in mathematics, by the analytical representation of natural phenomena. There is no conception, in the whole philosophy of mathematics which better deserves to fix all our attention.

ANALYTICAL REPRESENTATION OF FIGURES.

In order to succeed in expressing all imaginable geometrical phenomena by simple analytical relations, we must evidently, in the first place, establish a general method for representing analytically the subjects themselves in which these phenomena are found, that is, the lines or the surfaces to be considered. The subject being thus habitually considered in a purely analytical point of view, we see how it is thenceforth possible to conceive in the same manner the various accidents of which it is susceptible.

In order to organize the representation of geometrical figures by analytical equations, we must previously surmount a fundamental difficulty; that of reducing the general elements of the various conceptions of geometry to simply numerical ideas; in a word, that of substituting in geometry pure considerations of quantity for all considerations of quality.

Reduction of Figure to Position. For this purpose let us observe, in the first place, that all geometrical ideas relate necessarily to these three universal categories: the magnitude, the figure, and the position of the extensions to be considered. As to the first, there is evidently no difficulty; it enters at once into the ideas of numbers. With relation to the second, it must be remarked that it will always admit of being reduced to the third. For the figure of a body evidently results from the mutual position of the different points of which it is composed, so that the idea of position necessarily comprehends that of figure, and every circumstance of figure can be translated by a circumstance of position. It is in this way, in fact, that the human mind has proceeded in order to arrive at the analytical representation of geometrical figures, their conception relating directly only to positions. All the elementary difficulty is then properly reduced to that of referring ideas of situation to ideas of magnitude. Such is the direct destination of the preliminary conception upon which Descartes has established the general system of analytical geometry.

His philosophical labour, in this relation, has consisted simply in the entire generalization of an elementary operation, which we may regard as natural to the human mind, since it is performed spontaneously, so to say, in all minds, even the most uncultivated. Thus, when we have to indicate the situation of an object without directly pointing it out, the method which we always adopt, and evidently the only one which can be employed, consists in referring that object to others which are known, by assigning the magnitude of the various geometrical elements, by which we conceive it connected with the known objects. These elements constitute what Descartes, and after him all geometers, have called the co-ordinates of each point considered. They are necessarily two in number, if it is known in advance in what plane the point is situated; and three, if it may be found indifferently in any region of space. As many different constructions as can be imagined for determining the position of a point, whether on a plane or in space, so many distinct systems of co-ordinates may be conceived; they are consequently susceptible of being multiplied to infinity. But, whatever may be the system adopted, we shall always have reduced the ideas of situation to simple ideas of magnitude, so that we will consider the change in the position of a point as produced by mere numerical variations in the values of its co-ordinates.

Determination of the Position of a Point. Considering at first only the least complicated case, that of plane geometry, it is in this way that we usually determine the position of a point on a plane, by its distances from two fixed right lines considered as known, which are called axes, and which are commonly supposed to be perpendicular to each other. This system is that most frequently adopted, because of its simplicity; but geometers employ occasionally an infinity of others. Thus the position of a point on a plane may be determined, 1°, by its distances from two fixed points; or, 2°, by its distance from a single fixed point, and the direction of that distance, estimated by the greater or less angle which it makes with a fixed right line, which constitutes the system of what are called polar co-ordinates, the most frequently used after the system first mentioned; or, 3°, by the angles which the right lines drawn from the variable point to two fixed points make with the right line which joins these last; or, 4°, by the distances from that point to a fixed right line and a fixed point, &c. In a word, there is no geometrical figure whatever from which it is not possible to deduce a certain system of co-ordinates more or less susceptible of being employed.

A general observation, which it is important to make in this connexion, is, that every system of co-ordinates is equivalent to determining a point, in plane geometry, by the intersection of two lines, each of which is subjected to certain fixed conditions of determination; a single one of these conditions remaining variable, sometimes the one, sometimes the other, according to the system considered. We could not, indeed, conceive any other means of constructing a point than to mark it by the meeting of two lines. Thus, in the most common system, that of rectilinear co-ordinates, properly so called, the point is determined by the intersection of two right lines, each of which remains constantly parallel to a fixed axis, at a greater or less distance from it; in the polar system, the position of the point is marked by the meeting of a circle, of variable radius and fixed centre, with a movable right line compelled to turn about this centre: in other systems, the required point might be designated by the intersection of two circles, or of any other two lines, &c. In a word, to assign the value of one of the co-ordinates of a point in any system whatever, is always necessarily equivalent to determining a certain line on which that point must be situated. The geometers of antiquity had already made this essential remark, which served as the base of their method of geometrical loci, of which they made so happy a use to direct their researches in the resolution of determinate problems, in considering separately the influence of each of the two conditions by which was defined each point constituting the object, direct or indirect, of the proposed question. It was the general systematization of this method which was the immediate motive of the labours of Descartes, which led him to create analytical geometry.

After having clearly established this preliminary conception—by means of which ideas of position, and thence, implicitly, all elementary geometrical conceptions are capable of being reduced to simple numerical considerations—it is easy to form a direct conception, in its entire generality, of the great original idea of Descartes, relative to the analytical representation of geometrical figures: it is this which forms the special object of this chapter. I will continue to consider at first, for more facility, only geometry of two dimensions, which alone was treated by Descartes; and will afterwards examine separately, under the same point of view, the theory of surfaces and curves of double curvature.

PLANE CURVES.

Expression of Lines by Equations. In accordance with the manner of expressing analytically the position of a point on a plane, it can be easily established that, by whatever property any line may be defined, that definition always admits of being replaced by a corresponding equation between the two variable co-ordinates of the point which describes this line; an equation which will be thenceforth the analytical representation of the proposed line, every phenomenon of which will be translated by a certain algebraic modification of its equation. Thus, if we suppose that a point moves on a plane without its course being in any manner determined, we shall evidently have to regard its co-ordinates, to whatever system they may belong, as two variables entirely independent of one another. But if, on the contrary, this point is compelled to describe a certain line, we shall necessarily be compelled to conceive that its co-ordinates, in all the positions which it can take, retain a certain permanent and precise relation to each other, which is consequently susceptible of being expressed by a suitable equation; which will become the very clear and very rigorous analytical definition of the line under consideration, since it will express an algebraical property belonging exclusively to the co-ordinates of all the points of this line. It is clear, indeed, that when a point is not subjected to any condition, its situation is not determined except in giving at once its two co-ordinates, independently of each other; while, when the point must continue upon a defined line, a single co-ordinate is sufficient for completely fixing its position. The second co-ordinate is then a determinate function of the first; or, in other words, there must exist between them a certain equation, of a nature corresponding to that of the line on which the point is compelled to remain. In a word, each of the co-ordinates of a point requiring it to be situated on a certain line, we conceive reciprocally that the condition, on the part of a point, of having to belong to a line defined in any manner whatever, is equivalent to assigning the value of one of the two co-ordinates; which is found in that case to be entirely dependent on the other. The analytical relation which expresses this dependence may be more or less difficult to discover, but it must evidently be always conceived to exist, even in the cases in which our present means may be insufficient to make it known. It is by this simple consideration that we may demonstrate, in an entirely general manner—independently of the particular verifications on which this fundamental conception is ordinarily established for each special definition of a line—the necessity of the analytical representation of lines by equations.

Expression of Equations by Lines. Taking up again the same reflections in the inverse direction, we could show as easily the geometrical necessity of the representation of every equation of two variables, in a determinate system of co-ordinates, by a certain line; of which such a relation would be, in the absence of any other known property, a very characteristic definition, the scientific destination of which will be to fix the attention directly upon the general course of the solutions of the equation, which will thus be noted in the most striking and the most simple manner. This picturing of equations is one of the most important fundamental advantages of analytical geometry, which has thereby reacted in the highest degree upon the general perfecting of analysis itself; not only by assigning to purely abstract researches a clearly determined object and an inexhaustible career, but, in a still more direct relation, by furnishing a new philosophical medium for analytical meditation which could not be replaced by any other. In fact, the purely algebraic discussion of an equation undoubtedly makes known its solutions in the most precise manner, but in considering them only one by one, so that in this way no general view of them could be obtained, except as the final result of a long and laborious series of numerical comparisons. On the other hand, the geometrical locus of the equation, being only designed to represent distinctly and with perfect clearness the summing up of all these comparisons, permits it to be directly considered, without paying any attention to the details which have furnished it. It can thereby suggest to our mind general analytical views, which we should have arrived at with much difficulty in any other manner, for want of a means of clearly characterizing their object. It is evident, for example, that the simple inspection of the logarithmic curve, or of the curve y = sin. x, makes us perceive much more distinctly the general manner of the variations of logarithms with respect to their numbers, or of sines with respect to their arcs, than could the most attentive study of a table of logarithms or of natural sines. It is well known that this method has become entirely elementary at the present day, and that it is employed whenever it is desired to get a clear idea of the general character of the law which reigns in a series of precise observations of any kind whatever.

Any Change in the Line causes a Change in the Equation. Returning to the representation of lines by equations, which is our principal object, we see that this representation is, by its nature, so faithful, that the line could not experience any modification, however slight it might be, without causing a corresponding change in the equation. This perfect exactitude even gives rise oftentimes to special difficulties; for since, in our system of analytical geometry, the mere displacements of lines affect the equations, as well as their real variations in magnitude or form, we should be liable to confound them with one another in our analytical expressions, if geometers had not discovered an ingenious method designed expressly to always distinguish them. This method is founded on this principle, that although it is impossible to change analytically at will the position of a line with respect to the axes of the co-ordinates, we can change in any manner whatever the situation of the axes themselves, which evidently amounts to the same; then, by the aid of the very simple general formula by which this transformation of the axes is produced, it becomes easy to discover whether two different equations are the analytical expressions of only the same line differently situated, or refer to truly distinct geometrical loci; since, in the former case, one of them will pass into the other by suitably changing the axes or the other constants of the system of co-ordinates employed. It must, moreover, be remarked on this subject, that general inconveniences of this nature seem to be absolutely inevitable in analytical geometry; for, since the ideas of position are, as we have seen, the only geometrical ideas immediately reducible to numerical considerations, and the conceptions of figure cannot be thus reduced, except by seeing in them relations of situation, it is impossible for analysis to escape confounding, at first, the phenomena of figure with simple phenomena of position, which alone are directly expressed by the equations.

Every Definition of a Line is an Equation. In order to complete the philosophical explanation of the fundamental conception which serves as the base of analytical geometry, I think that I should here indicate a new general consideration, which seems to me particularly well adapted for putting in the clearest point of view this necessary representation of lines by equations with two variables. It consists in this, that not only, as we have shown, must every defined line necessarily give rise to a certain equation between the two co-ordinates of any one of its points, but, still farther, every definition of a line may be regarded as being already of itself an equation of that line in a suitable system of co-ordinates.

It is easy to establish this principle, first making a preliminary logical distinction with respect to different kinds of definitions. The rigorously indispensable condition of every definition is that of distinguishing the object defined from all others, by assigning to it a property which belongs to it exclusively. But this end may be generally attained in two very different ways; either by a definition which is simply characteristic, that is, indicative of a property which, although truly exclusive, does not make known the mode of generation of the object; or by a definition which is really explanatory, that is, which characterizes the object by a property which expresses one of its modes of generation. For example, in considering the circle as the line, which, under the same contour, contains the greatest area, we have evidently a definition of the first kind; while in choosing the property of its having all its points equally distant from a fixed point, we have a definition of the second kind. It is, besides, evident, as a general principle, that even when any object whatever is known at first only by a characteristic definition, we ought, nevertheless, to regard it as susceptible of explanatory definitions, which the farther study of the object would necessarily lead us to discover.

This being premised, it is clear that the general observation above made, which represents every definition of a line as being necessarily an equation of that line in a certain system of co-ordinates, cannot apply to definitions which are simply characteristic; it is to be understood only of definitions which are truly explanatory. But, in considering only this class, the principle is easy to prove. In fact, it is evidently impossible to define the generation of a line without specifying a certain relation between the two simple motions of translation or of rotation, into which the motion of the point which describes it will be decomposed at each instant. Now if we form the most general conception of what constitutes a system of co-ordinates, and admit all possible systems, it is clear that such a relation will be nothing else but the equation of the proposed line, in a system of co-ordinates of a nature corresponding to that of the mode of generation considered. Thus, for example, the common definition of the circle may evidently be regarded as being immediately the polar equation of this curve, taking the centre of the circle for the pole. In the same way, the elementary definition of the ellipse or of the hyperbola—as being the curve generated by a point which moves in such a manner that the sum or the difference of its distances from two fixed points remains constant—gives at once, for either the one or the other curve, the equation y + x = c, taking for the system of co-ordinates that in which the position of a point would be determined by its distances from two fixed points, and choosing for these poles the two given foci. In like manner, the common definition of any cycloid would furnish directly, for that curve, the equation y = mx; adopting as the co-ordinates of each point the arc which it marks upon a circle of invariable radius, measuring from the point of contact of that circle with a fixed line, and the rectilinear distance from that point of contact to a certain origin taken on that right line. We can make analogous and equally easy verifications with respect to the customary definitions of spirals, of epicycloids, &c. We shall constantly find that there exists a certain system of co-ordinates, in which we immediately obtain a very simple equation of the proposed line, by merely writing algebraically the condition imposed by the mode of generation considered.

Besides its direct importance as a means of rendering perfectly apparent the necessary representation of every line by an equation, the preceding consideration seems to me to possess a true scientific utility, in characterizing with precision the principal general difficulty which occurs in the actual establishment of these equations, and in consequently furnishing an interesting indication with respect to the course to be pursued in inquiries of this kind, which, by their nature, could not admit of complete and invariable rules. In fact, since any definition whatever of a line, at least among those which indicate a mode of generation, furnishes directly the equation of that line in a certain system of co-ordinates, or, rather, of itself constitutes that equation, it follows that the difficulty which we often experience in discovering the equation of a curve, by means of certain of its characteristic properties, a difficulty which is sometimes very great, must proceed essentially only from the commonly imposed condition of expressing this curve analytically by the aid of a designated system of co-ordinates, instead of admitting indifferently all possible systems. These different systems cannot be regarded in analytical geometry as being all equally suitable; for various reasons, the most important of which will be hereafter discussed, geometers think that curves should almost always be referred, as far as is possible, to rectilinear co-ordinates, properly so called. Now we see, from what precedes, that in many cases these particular co-ordinates will not be those with reference to which the equation of the curve will be found to be directly established by the proposed definition. The principal difficulty presented by the formation of the equation of a line really consists, then, in general, in a certain transformation of co-ordinates. It is undoubtedly true that this consideration does not subject the establishment of these equations to a truly complete general method, the success of which is always certain; which, from the very nature of the subject, is evidently chimerical: but such a view may throw much useful light upon the course which it is proper to adopt, in order to arrive at the end proposed. Thus, after having in the first place formed the preparatory equation, which is spontaneously derived from the definition which we are considering, it will be necessary, in order to obtain the equation belonging to the system of co-ordinates which must be finally admitted, to endeavour to express in a function of these last co-ordinates those which naturally correspond to the given mode of generation. It is upon this last labour that it is evidently impossible to give invariable and precise precepts. We can only say that we shall have so many more resources in this matter as we shall know more of true analytical geometry, that is, as we shall know the algebraical expression of a greater number of different algebraical phenomena.

CHOICE OF CO-ORDINATES.

In order to complete the philosophical exposition of the conception which serves as the base of analytical geometry, I have yet to notice the considerations relating to the choice of the system of co-ordinates which is in general the most suitable. They will give the rational explanation of the preference unanimously accorded to the ordinary rectilinear system; a preference which has hitherto been rather the effect of an empirical sentiment of the superiority of this system, than the exact result of a direct and thorough analysis.

Two different Points of View. In order to decide clearly between all the different systems of co-ordinates, it is indispensable to distinguish with care the two general points of view, the converse of one another, which belong to analytical geometry; namely, the relation of algebra to geometry, founded upon the representation of lines by equations; and, reciprocally, the relation of geometry to algebra, founded on the representation of equations by lines.

It is evident that in every investigation of general geometry these two fundamental points of view are of necessity always found combined, since we have always to pass alternately, and at insensible intervals, so to say, from geometrical to analytical considerations, and from analytical to geometrical considerations. But the necessity of here temporarily separating them is none the less real; for the answer to the question of method which we are examining is, in fact, as we shall see presently, very far from being the same in both these relations, so that without this distinction we could not form any clear idea of it.

1. Representation of Lines by Equations. Under the first point of view—the representation of lines by equations—the only reason which could lead us to prefer one system of co-ordinates to another would be the greater simplicity of the equation of each line, and greater facility in arriving at it. Now it is easy to see that there does not exist, and could not be expected to exist, any system of co-ordinates deserving in that respect a constant preference over all others. In fact, we have above remarked that for each geometrical definition proposed we can conceive a system of co-ordinates in which the equation of the line is obtained at once, and is necessarily found to be also very simple; and this system, moreover, inevitably varies with the nature of the characteristic property under consideration. The rectilinear system could not, therefore, be constantly the most advantageous for this object, although it may often be very favourable; there is probably no system which, in certain particular cases, should not be preferred to it, as well as to every other.

2. Representation of Equations by Lines. It is by no means so, however, under the second point of view. We can, indeed, easily establish, as a general principle, that the ordinary rectilinear system must necessarily be better adapted than any other to the representation of equations by the corresponding geometrical loci; that is to say, that this representation is constantly more simple and more faithful in it than in any other.

Let us consider, for this object, that, since every system of co-ordinates consists in determining a point by the intersection of two lines, the system adapted to furnish the most suitable geometrical loci must be that in which these two lines are the simplest possible; a consideration which confines our choice to the rectilinear system. In truth, there is evidently an infinite number of systems which deserve that name, that is to say, which employ only right lines to determine points, besides the ordinary system which assigns the distances from two fixed lines as co-ordinates; such, for example, would be that in which the co-ordinates of each point should be the two angles which the right lines, which go from that point to two fixed points, make with the right line, which joins these last points: so that this first consideration is not rigorously sufficient to explain the preference unanimously given to the common system. But in examining in a more thorough manner the nature of every system of co-ordinates, we also perceive that each of the two lines, whose meeting determines the point considered, must necessarily offer at every instant, among its different conditions of determination, a single variable condition, which gives rise to the corresponding co-ordinate, all the rest being fixed, and constituting the axes of the system, taking this term in its most extended mathematical acceptation. The variation is indispensable, in order that we may be able to consider all possible positions; and the fixity is no less so, in order that there may exist means of comparison. Thus, in all rectilinear systems, each of the two right lines will be subjected to a fixed condition, and the ordinate will result from the variable condition.

Superiority of rectilinear Co-ordinates. From these considerations it is evident, as a general principle, that the most favourable system for the construction of geometrical loci will necessarily be that in which the variable condition of each right line shall be the simplest possible; the fixed condition being left free to be made complex, if necessary to attain that object. Now, of all possible manners of determining two movable right lines, the easiest to follow geometrically is certainly that in which, the direction of each right line remaining invariable, it only approaches or recedes, more or less, to or from a constant axis. It would be, for example, evidently more difficult to figure to one's self clearly the changes of place of a point which is determined by the intersection of two right lines, which each turn around a fixed point, making a greater or smaller angle with a certain axis, as in the system of co-ordinates previously noticed. Such is the true general explanation of the fundamental property possessed by the common rectilinear system, of being better adapted than any other to the geometrical representation of equations, inasmuch as it is that one in which it is the easiest to conceive the change of place of a point resulting from the change in the value of its co-ordinates. In order to feel clearly all the force of this consideration, it would be sufficient to carefully compare this system with the polar system, in which this geometrical image, so simple and so easy to follow, of two right lines moving parallel, each one of them, to its corresponding axis, is replaced by the complicated picture of an infinite series of concentric circles, cut by a right line compelled to turn about a fixed point. It is, moreover, easy to conceive in advance what must be the extreme importance to analytical geometry of a property so profoundly elementary, which, for that reason, must be recurring at every instant, and take a progressively increasing value in all labours of this kind.

Perpendicularity of the Axes. In pursuing farther the consideration which demonstrates the superiority of the ordinary system of co-ordinates over any other as to the representation of equations, we may also take notice of the utility for this object of the common usage of taking the two axes perpendicular to each other, whenever possible, rather than with any other inclination. As regards the representation of lines by equations, this secondary circumstance is no more universally proper than we have seen the general nature of the system to be; since, according to the particular occasion, any other inclination of the axes may deserve our preference in that respect. But, in the inverse point of view, it is easy to see that rectangular axes constantly permit us to represent equations in a more simple and even more faithful manner; for, with oblique axes, space being divided by them into regions which no longer have a perfect identity, it follows that, if the geometrical locus of the equation extends into all these regions at once, there will be presented, by reason merely of this inequality of the angles, differences of figure which do not correspond to any analytical diversity, and will necessarily alter the rigorous exactness of the representation, by being confounded with the proper results of the algebraic comparisons. For example, an equation like: x^m + y^m = c, which, by its perfect symmetry, should evidently give a curve composed of four identical quarters, will be represented, on the contrary, if we take axes not rectangular, by a geometric locus, the four parts of which will be unequal. It is plain that the only means of avoiding all inconveniences of this kind is to suppose the angle of the two axes to be a right angle.

The preceding discussion clearly shows that, although the ordinary system of rectilinear co-ordinates has no constant superiority over all others in one of the two fundamental points of view which are continually combined in analytical geometry, yet as, on the other hand, it is not constantly inferior, its necessary and absolute greater aptitude for the representation of equations must cause it to generally receive the preference; although it may evidently happen, in some particular cases, that the necessity of simplifying equations and of obtaining them more easily may determine geometers to adopt a less perfect system. The rectilinear system is, therefore, the one by means of which are ordinarily constructed the most essential theories of general geometry, intended to express analytically the most important geometrical phenomena. When it is thought necessary to choose some other, the polar system is almost always the one which is fixed upon, this system being of a nature sufficiently opposite to that of the rectilinear system to cause the equations, which are too complicated with respect to the latter, to become, in general, sufficiently simple with respect to the other. Polar co-ordinates, moreover, have often the advantage of admitting of a more direct and natural concrete signification; as is the case in mechanics, for the geometrical questions to which the theory of circular movement gives rise, and in almost all the cases of celestial geometry.

In order to simplify the exposition, we have thus far considered the fundamental conception of analytical geometry only with respect to plane curves, the general study of which was the only object of the great philosophical renovation produced by Descartes. To complete this important explanation, we have now to show summarily how this elementary idea was extended by Clairaut, about a century afterwards, to the general study of surfaces and curves of double curvature. The considerations which have been already given will permit me to limit myself on this subject to the rapid examination of what is strictly peculiar to this new case.

Determination of a Point in Space. The complete analytical determination of a point in space evidently requires the values of three co-ordinates to be assigned; as, for example, in the system which is generally adopted, and which corresponds to the rectilinear system of plane geometry, distances from the point to three fixed planes, usually perpendicular to one another; which presents the point as the intersection of three planes whose direction is invariable. We might also employ the distances from the movable point to three fixed points, which would determine it by the intersection of three spheres with a common centre. In like manner, the position of a point would be defined by giving its distance from a fixed point, and the direction of that distance, by means of the two angles which this right line makes with two invariable axes; this is the polar system of geometry of three dimensions; the point is then constructed by the intersection of a sphere having a fixed centre, with two right cones with circular bases, whose axes and common summit do not change. In a word, there is evidently, in this case at least, the same infinite variety among the various possible systems of co-ordinates which we have already observed in geometry of two dimensions. In general, we have to conceive a point as being always determined by the intersection of any three surfaces whatever, as it was in the former case by that of two lines: each of these three surfaces has, in like manner, all its conditions of determination constant, excepting one, which gives rise to the corresponding co-ordinates, whose peculiar geometrical influence is thus to constrain the point to be situated upon that surface.

This being premised, it is clear that if the three co-ordinates of a point are entirely independent of one another, that point can take successively all possible positions in space. But if the point is compelled to remain upon a certain surface defined in any manner whatever, then two co-ordinates are evidently sufficient for determining its situation at each instant, since the proposed surface will take the place of the condition imposed by the third co-ordinate. We must then, in this case, under the analytical point of view, necessarily conceive this last co-ordinate as a determinate function of the two others, these latter remaining perfectly independent of each other. Thus there will be a certain equation between the three variable co-ordinates, which will be permanent, and which will be the only one, in order to correspond to the precise degree of indetermination in the position of the point.

Expression of Surfaces by Equations. This equation, more or less easy to be discovered, but always possible, will be the analytical definition of the proposed surface, since it must be verified for all the points of that surface, and for them alone. If the surface undergoes any change whatever, even a simple change of place, the equation must undergo a more or less serious corresponding modification. In a word, all geometrical phenomena relating to surfaces will admit of being translated by certain equivalent analytical conditions appropriate to equations of three variables; and in the establishment and interpretation of this general and necessary harmony will essentially consist the science of analytical geometry of three dimensions.

Expression of Equations by Surfaces. Considering next this fundamental conception in the inverse point of view, we see in the same manner that every equation of three variables may, in general, be represented geometrically by a determinate surface, primitively defined by the very characteristic property, that the co-ordinates of all its points always retain the mutual relation enunciated in this equation. This geometrical locus will evidently change, for the same equation, according to the system of co-ordinates which may serve for the construction of this representation. In adopting, for example, the rectilinear system, it is clear that in the equation between the three variables, x, y, z, every particular value attributed to z will give an equation between at x and y, the geometrical locus of which will be a certain line situated in a plane parallel to the plane of x and y, and at a distance from this last equal to the value of z; so that the complete geometrical locus will present itself as composed of an infinite series of lines superimposed in a series of parallel planes (excepting the interruptions which may exist), and will consequently form a veritable surface. It would be the same in considering any other system of co-ordinates, although the geometrical construction of the equation becomes more difficult to follow.

Such is the elementary conception, the complement of the original idea of Descartes, on which is founded general geometry relative to surfaces. It would be useless to take up here directly the other considerations which have been above indicated, with respect to lines, and which any one can easily extend to surfaces; whether to show that every definition of a surface by any method of generation whatever is really a direct equation of that surface in a certain system of co-ordinates, or to determine among all the different systems of possible co-ordinates that one which is generally the most convenient. I will only add, on this last point, that the necessary superiority of the ordinary rectilinear system, as to the representation of equations, is evidently still more marked in analytical geometry of three dimensions than in that of two, because of the incomparably greater geometrical complication which would result from the choice of any other system. This can be verified in the most striking manner by considering the polar system in particular, which is the most employed after the ordinary rectilinear system, for surfaces as well as for plane curves, and for the same reasons.

In order to complete the general exposition of the fundamental conception relative to the analytical study of surfaces, a philosophical examination should be made of a final improvement of the highest importance, which Monge has introduced into the very elements of this theory, for the classification of surfaces in natural families, established according to the mode of generation, and expressed algebraically by common differential equations, or by finite equations containing arbitrary functions.

CURVES OF DOUBLE CURVATURE.

Let us now consider the last elementary point of view of analytical geometry of three dimensions; that relating to the algebraic representation of curves considered in space, in the most general manner. In continuing to follow the principle which has been constantly employed, that of the degree of indetermination of the geometrical locus, corresponding to the degree of independence of the variables, it is evident, as a general principle, that when a point is required to be situated upon some certain curve, a single co-ordinate is enough for completely determining its position, by the intersection of this curve with the surface which results from this co-ordinate. Thus, in this case, the two other co-ordinates of the point must be conceived as functions necessarily determinate and distinct from the first. It follows that every line, considered in space, is then represented analytically, no longer by a single equation, but by the system of two equations between the three co-ordinates of any one of its points. It is clear, indeed, from another point of view, that since each of these equations, considered separately, expresses a certain surface, their combination presents the proposed line as the intersection of two determinate surfaces. Such is the most general manner of conceiving the algebraic representation of a line in analytical geometry of three dimensions. This conception is commonly considered in too restricted a manner, when we confine ourselves to considering a line as determined by the system of its two projections upon two of the co-ordinate planes; a system characterized, analytically, by this peculiarity, that each of the two equations of the line then contains only two of the three co-ordinates, instead of simultaneously including the three variables. This consideration, which consists in regarding the line as the intersection of two cylindrical surfaces parallel to two of the three axes of the co-ordinates, besides the inconvenience of being confined to the ordinary rectilinear system, has the fault, if we strictly confine ourselves to it, of introducing useless difficulties into the analytical representation of lines, since the combination of these two cylinders would evidently not be always the most suitable for forming the equations of a line. Thus, considering this fundamental notion in its entire generality, it will be necessary in each case to choose, from among the infinite number of couples of surfaces, the intersection of which might produce the proposed curve, that one which will lend itself the best to the establishment of equations, as being composed of the best known surfaces. Thus, if the problem is to express analytically a circle in space, it will evidently be preferable to consider it as the intersection of a sphere and a plane, rather than as proceeding from any other combination of surfaces which could equally produce it.

In truth, this manner of conceiving the representation of lines by equations, in analytical geometry of three dimensions, produces, by its nature, a necessary inconvenience, that of a certain analytical confusion, consisting in this: that the same line may thus be expressed, with the same system of co-ordinates, by an infinite number of different couples of equations, on account of the infinite number of couples of surfaces which can form it; a circumstance which may cause some difficulties in recognizing this line under all the algebraical disguises of which it admits. But there exists a very simple method for causing this inconvenience to disappear; it consists in giving up the facilities which result from this variety of geometrical constructions. It suffices, in fact, whatever may be the analytical system primitively established for a certain line, to be able to deduce from it the system corresponding to a single couple of surfaces uniformly generated; as, for example, to that of the two cylindrical surfaces which project the proposed line upon two of the co-ordinate planes; surfaces which will evidently be always identical, in whatever manner the line may have been obtained, and which will not vary except when that line itself shall change. Now, in choosing this fixed system, which is actually the most simple, we shall generally be able to deduce from the primitive equations those which correspond to them in this special construction, by transforming them, by two successive eliminations, into two equations, each containing only two of the variable co-ordinates, and thereby corresponding to the two surfaces of projection. Such is really the principal destination of this sort of geometrical combination, which thus offers to us an invariable and certain means of recognizing the identity of lines in spite of the diversity of their equations, which is sometimes very great.

IMPERFECTIONS OF ANALYTICAL GEOMETRY.

Having now considered the fundamental conception of analytical geometry under its principal elementary aspects, it is proper, in order to make the sketch complete, to notice here the general imperfections yet presented by this conception with respect to both geometry and to analysis.

Relatively to geometry, we must remark that the equations are as yet adapted to represent only entire geometrical loci, and not at all determinate portions of those loci. It would, however, be necessary, in some circumstances, to be able to express analytically a part of a line or of a surface, or even a discontinuous line or surface, composed of a series of sections belonging to distinct geometrical figures, such as the contour of a polygon, or the surface of a polyhedron. Thermology, especially, often gives rise to such considerations, to which our present analytical geometry is necessarily inapplicable. The labours of M. Fourier on discontinuous functions have, however, begun to fill up this great gap, and have thereby introduced a new and essential improvement into the fundamental conception of Descartes. But this manner of representing heterogeneous or partial figures, being founded on the employment of trigonometrical series proceeding according to the sines of an infinite series of multiple arcs, or on the use of certain definite integrals equivalent to those series, and the general integral of which is unknown, presents as yet too much complication to admit of being immediately introduced into the system of analytical geometry.

Relatively to analysis, we must begin by observing that our inability to conceive a geometrical representation of equations containing four, five, or more variables, analogous to those representations which all equations of two or of three variables admit, must not be viewed as an imperfection of our system of analytical geometry, for it evidently belongs to the very nature of the subject. Analysis being necessarily more general than geometry, since it relates to all possible phenomena, it would be very unphilosophical to desire always to find among geometrical phenomena alone a concrete representation of all the laws which analysis can express.

There exists, however, another imperfection of less importance, which must really be viewed as proceeding from the manner in which we conceive analytical geometry. It consists in the evident incompleteness of our present representation of equations of two or of three variables by lines or surfaces, inasmuch as in the construction of the geometric locus we pay regard only to the real solutions of equations, without at all noticing any imaginary solutions. The general course of these last should, however, by its nature, be quite as susceptible as that of the others of a geometrical representation. It follows from this omission that the graphic picture of the equation is constantly imperfect, and sometimes even so much so that there is no geometric representation at all when the equation admits of only imaginary solutions. But, even in this last case, we evidently ought to be able to distinguish between equations as different in themselves as these, for example,

x² + y² + 1 = 0, x⁶ + y⁴ + 1 = 0, y² + e^x = 0.

We know, moreover, that this principal imperfection often brings with it, in analytical geometry of two or of three dimensions, a number of secondary inconveniences, arising from several analytical modifications not corresponding to any geometrical phenomena.

Our philosophical exposition of the fundamental conception of analytical geometry shows us clearly that this science consists essentially in determining what is the general analytical expression of such or such a geometrical phenomenon belonging to lines or to surfaces; and, reciprocally, in discovering the geometrical interpretation of such or such an analytical consideration. A detailed examination of the most important general questions would show us how geometers have succeeded in actually establishing this beautiful harmony, and in thus imprinting on geometrical science, regarded as a whole, its present eminently perfect character of rationality and of simplicity.

THE END.