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 geometricallocusof 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 confoundedwith the proper results of the algebraic comparisons. For example, an equation like:xm+ym=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 geometriclocus, 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 andnatural 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 toplane 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 ofsurfacesandcurves 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 therectilinearsystem 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 thepolarsystem 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 tozwill give an equation between atxandy, the geometrical locus of which will be a certain line situatedin a plane parallel to the plane ofxandy, and at a distance from this last equal to the value ofz; 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 fundamentalconception 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.
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 twoprojectionsupon 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 whichprojectthe 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.
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 adiscontinuousline 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 observingthat 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 therealsolutions of equations, without at all noticing anyimaginarysolutions. 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+ex= 0.
We know, moreover, that this principal imperfection often brings with it, in analytical geometry of two or ofthree 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.
Note.—The author devotes the two following chapters of his course to the more detailed examination of Analytical Geometry of two and of three dimensions; but his subsequent publication of a separate work upon this branch of mathematics has been thought to render unnecessary the reproduction of these two chapters in the present volume.
Note.—The author devotes the two following chapters of his course to the more detailed examination of Analytical Geometry of two and of three dimensions; but his subsequent publication of a separate work upon this branch of mathematics has been thought to render unnecessary the reproduction of these two chapters in the present volume.
THE END.
FOOTNOTES:[1]The investigation of the mathematical phenomena of the laws of heat by Baron Fourier has led to the establishment, in an entirely direct manner, of Thermological equations. This great discovery tends to elevate our philosophical hopes as to the future extensions of the legitimate applications of mathematical analysis, and renders it proper, in the opinion of author, to regardThermologyas a third principal branch of concrete mathematics.[2]The translator has felt justified in employing this very convenient word (for which our language has no precise equivalent) as an English one, in its most extended sense, in spite of its being often popularly confounded with its Differential and Integral department.[3]With the view of increasing as much as possible the resources and the extent (now so insufficient) of mathematical analysis, geometers count this last couple of functions among the analytical elements. Although this inscription is strictly legitimate, it is important to remark that circular functions are not exactly in the same situation as the other abstract elementary functions. There is this very essential difference, that the functions of the four first couples are at the same time simple and abstract, while the circular functions, which may manifest each character in succession, according to the point of view under which they are considered and the manner in which they are employed, never present these two properties simultaneously.Some other concrete functions may be usefully introduced into the number of analytical elements, certain conditions being fulfilled. It is thus, for example, that the labours of M. Legendre and of M. Jacobi onellipticalfunctions have truly enlarged the field of analysis; and the same is true of some definite integrals obtained by M. Fourier in the theory of heat.[4]Suppose, for example, that a question gives the following equation between an unknown magnitude x, and two known magnitudes,aandb,x3+ 3ax= 2b,as is the case in the problem of the trisection of an angle. We see at once that the dependence betweenxon the one side, andabon the other, is completely determined; but, so long as the equation preserves its primitive form, we do not at all perceive in what manner the unknown quantity is derived from the data. This must be discovered, however, before we can think of determining its value. Such is the object of the algebraic part of the solution. When, by a series of transformations which have successively rendered that derivation more and more apparent, we have arrived at presenting the proposed equation under the formx= ∛(b+ √(b2+a3)) + ∛(b- √(b2+a3)),the work ofalgebrais finished; and even if we could not perform the arithmetical operations indicated by that formula, we would nevertheless have obtained a knowledge very real, and often very important. The work ofarithmeticwill now consist in taking that formula for its starting point, and finding the numberxwhen the values of the numbersaandbare given.[5]I have thought that I ought to specially notice this definition, because it serves as the basis of the opinion which many intelligent persons, unacquainted with mathematical science, form of its abstract part, without considering that at the time of this definition mathematical analysis was not sufficiently developed to enable the general character of each of its principal parts to be properly apprehended, which explains why Newton could at that time propose a definition which at the present day he would certainly reject.[6]This is less strictly true in the English system of numeration than in the French, since "twenty-one" is our more usual mode of expressing this number.[7]Simple as may seem, for example, the equationax+bx=cx,we do not yet know how to resolve it, which may give some idea of the extreme imperfection of this part of algebra.[8]The same error was afterward committed, in the infancy of the infinitesimal calculus, in relation to the integration of differential equations.[9]The fundamental principle on which reposes the theory of equations, and which is so frequently applied in all mathematical analysis—the decomposition of algebraic, rational, and entire functions, of any degree whatever, into factors of the first degree—is never employed except for functions of a single variable, without any one having examined if it ought to be extended to functions of several variables. The general impossibility of such a decomposition is demonstrated by the author in detail, but more properly belongs to a special treatise.[10]The only important case of this class which has thus far been completely treated is the general integration oflinearequations of any order whatever, with constant coefficients. Even this case finally depends on the algebraic resolution of equations of a degree equal to the order of differentiation.[11]Leibnitz had already considered the comparison of one curve with an other infinitely near to it, calling it "Differentiatio de curva in curvam." But this comparison had no analogy with the conception of Lagrange, the curves of Leibnitz being embraced in the same general equation, from which they were deduced by the simple change of an arbitrary constant.[12]I propose hereafter to develop this new consideration, in a special work upon theCalculus of Variations, intended to present this hyper-transcendental analysis in a new point of view, which I think adapted to extend its general range.[13]Lacroix has justly criticised the expression ofsolid, commonly used by geometers to designate avolume. It is certain, in fact, that when we wish to consider separately a certain portion of indefinite space, conceived as gaseous, we mentally solidify its exterior envelope, so that alineand asurfaceare habitually, to our minds, just assolidas avolume. It may also be remarked that most generally, in order that bodies may penetrate one another with more facility, we are obliged to imagine the interior of thevolumesto be hollow, which renders still more sensible the impropriety of the wordsolid.
FOOTNOTES:
[1]The investigation of the mathematical phenomena of the laws of heat by Baron Fourier has led to the establishment, in an entirely direct manner, of Thermological equations. This great discovery tends to elevate our philosophical hopes as to the future extensions of the legitimate applications of mathematical analysis, and renders it proper, in the opinion of author, to regardThermologyas a third principal branch of concrete mathematics.
[1]The investigation of the mathematical phenomena of the laws of heat by Baron Fourier has led to the establishment, in an entirely direct manner, of Thermological equations. This great discovery tends to elevate our philosophical hopes as to the future extensions of the legitimate applications of mathematical analysis, and renders it proper, in the opinion of author, to regardThermologyas a third principal branch of concrete mathematics.
[2]The translator has felt justified in employing this very convenient word (for which our language has no precise equivalent) as an English one, in its most extended sense, in spite of its being often popularly confounded with its Differential and Integral department.
[2]The translator has felt justified in employing this very convenient word (for which our language has no precise equivalent) as an English one, in its most extended sense, in spite of its being often popularly confounded with its Differential and Integral department.
[3]With the view of increasing as much as possible the resources and the extent (now so insufficient) of mathematical analysis, geometers count this last couple of functions among the analytical elements. Although this inscription is strictly legitimate, it is important to remark that circular functions are not exactly in the same situation as the other abstract elementary functions. There is this very essential difference, that the functions of the four first couples are at the same time simple and abstract, while the circular functions, which may manifest each character in succession, according to the point of view under which they are considered and the manner in which they are employed, never present these two properties simultaneously.Some other concrete functions may be usefully introduced into the number of analytical elements, certain conditions being fulfilled. It is thus, for example, that the labours of M. Legendre and of M. Jacobi onellipticalfunctions have truly enlarged the field of analysis; and the same is true of some definite integrals obtained by M. Fourier in the theory of heat.
[3]With the view of increasing as much as possible the resources and the extent (now so insufficient) of mathematical analysis, geometers count this last couple of functions among the analytical elements. Although this inscription is strictly legitimate, it is important to remark that circular functions are not exactly in the same situation as the other abstract elementary functions. There is this very essential difference, that the functions of the four first couples are at the same time simple and abstract, while the circular functions, which may manifest each character in succession, according to the point of view under which they are considered and the manner in which they are employed, never present these two properties simultaneously.
Some other concrete functions may be usefully introduced into the number of analytical elements, certain conditions being fulfilled. It is thus, for example, that the labours of M. Legendre and of M. Jacobi onellipticalfunctions have truly enlarged the field of analysis; and the same is true of some definite integrals obtained by M. Fourier in the theory of heat.
[4]Suppose, for example, that a question gives the following equation between an unknown magnitude x, and two known magnitudes,aandb,x3+ 3ax= 2b,as is the case in the problem of the trisection of an angle. We see at once that the dependence betweenxon the one side, andabon the other, is completely determined; but, so long as the equation preserves its primitive form, we do not at all perceive in what manner the unknown quantity is derived from the data. This must be discovered, however, before we can think of determining its value. Such is the object of the algebraic part of the solution. When, by a series of transformations which have successively rendered that derivation more and more apparent, we have arrived at presenting the proposed equation under the formx= ∛(b+ √(b2+a3)) + ∛(b- √(b2+a3)),the work ofalgebrais finished; and even if we could not perform the arithmetical operations indicated by that formula, we would nevertheless have obtained a knowledge very real, and often very important. The work ofarithmeticwill now consist in taking that formula for its starting point, and finding the numberxwhen the values of the numbersaandbare given.
[4]Suppose, for example, that a question gives the following equation between an unknown magnitude x, and two known magnitudes,aandb,
x3+ 3ax= 2b,
as is the case in the problem of the trisection of an angle. We see at once that the dependence betweenxon the one side, andabon the other, is completely determined; but, so long as the equation preserves its primitive form, we do not at all perceive in what manner the unknown quantity is derived from the data. This must be discovered, however, before we can think of determining its value. Such is the object of the algebraic part of the solution. When, by a series of transformations which have successively rendered that derivation more and more apparent, we have arrived at presenting the proposed equation under the form
x= ∛(b+ √(b2+a3)) + ∛(b- √(b2+a3)),
the work ofalgebrais finished; and even if we could not perform the arithmetical operations indicated by that formula, we would nevertheless have obtained a knowledge very real, and often very important. The work ofarithmeticwill now consist in taking that formula for its starting point, and finding the numberxwhen the values of the numbersaandbare given.
[5]I have thought that I ought to specially notice this definition, because it serves as the basis of the opinion which many intelligent persons, unacquainted with mathematical science, form of its abstract part, without considering that at the time of this definition mathematical analysis was not sufficiently developed to enable the general character of each of its principal parts to be properly apprehended, which explains why Newton could at that time propose a definition which at the present day he would certainly reject.
[5]I have thought that I ought to specially notice this definition, because it serves as the basis of the opinion which many intelligent persons, unacquainted with mathematical science, form of its abstract part, without considering that at the time of this definition mathematical analysis was not sufficiently developed to enable the general character of each of its principal parts to be properly apprehended, which explains why Newton could at that time propose a definition which at the present day he would certainly reject.
[6]This is less strictly true in the English system of numeration than in the French, since "twenty-one" is our more usual mode of expressing this number.
[6]This is less strictly true in the English system of numeration than in the French, since "twenty-one" is our more usual mode of expressing this number.
[7]Simple as may seem, for example, the equationax+bx=cx,we do not yet know how to resolve it, which may give some idea of the extreme imperfection of this part of algebra.
[7]Simple as may seem, for example, the equation
ax+bx=cx,
we do not yet know how to resolve it, which may give some idea of the extreme imperfection of this part of algebra.
[8]The same error was afterward committed, in the infancy of the infinitesimal calculus, in relation to the integration of differential equations.
[8]The same error was afterward committed, in the infancy of the infinitesimal calculus, in relation to the integration of differential equations.
[9]The fundamental principle on which reposes the theory of equations, and which is so frequently applied in all mathematical analysis—the decomposition of algebraic, rational, and entire functions, of any degree whatever, into factors of the first degree—is never employed except for functions of a single variable, without any one having examined if it ought to be extended to functions of several variables. The general impossibility of such a decomposition is demonstrated by the author in detail, but more properly belongs to a special treatise.
[9]The fundamental principle on which reposes the theory of equations, and which is so frequently applied in all mathematical analysis—the decomposition of algebraic, rational, and entire functions, of any degree whatever, into factors of the first degree—is never employed except for functions of a single variable, without any one having examined if it ought to be extended to functions of several variables. The general impossibility of such a decomposition is demonstrated by the author in detail, but more properly belongs to a special treatise.
[10]The only important case of this class which has thus far been completely treated is the general integration oflinearequations of any order whatever, with constant coefficients. Even this case finally depends on the algebraic resolution of equations of a degree equal to the order of differentiation.
[10]The only important case of this class which has thus far been completely treated is the general integration oflinearequations of any order whatever, with constant coefficients. Even this case finally depends on the algebraic resolution of equations of a degree equal to the order of differentiation.
[11]Leibnitz had already considered the comparison of one curve with an other infinitely near to it, calling it "Differentiatio de curva in curvam." But this comparison had no analogy with the conception of Lagrange, the curves of Leibnitz being embraced in the same general equation, from which they were deduced by the simple change of an arbitrary constant.
[11]Leibnitz had already considered the comparison of one curve with an other infinitely near to it, calling it "Differentiatio de curva in curvam." But this comparison had no analogy with the conception of Lagrange, the curves of Leibnitz being embraced in the same general equation, from which they were deduced by the simple change of an arbitrary constant.
[12]I propose hereafter to develop this new consideration, in a special work upon theCalculus of Variations, intended to present this hyper-transcendental analysis in a new point of view, which I think adapted to extend its general range.
[12]I propose hereafter to develop this new consideration, in a special work upon theCalculus of Variations, intended to present this hyper-transcendental analysis in a new point of view, which I think adapted to extend its general range.
[13]Lacroix has justly criticised the expression ofsolid, commonly used by geometers to designate avolume. It is certain, in fact, that when we wish to consider separately a certain portion of indefinite space, conceived as gaseous, we mentally solidify its exterior envelope, so that alineand asurfaceare habitually, to our minds, just assolidas avolume. It may also be remarked that most generally, in order that bodies may penetrate one another with more facility, we are obliged to imagine the interior of thevolumesto be hollow, which renders still more sensible the impropriety of the wordsolid.
[13]Lacroix has justly criticised the expression ofsolid, commonly used by geometers to designate avolume. It is certain, in fact, that when we wish to consider separately a certain portion of indefinite space, conceived as gaseous, we mentally solidify its exterior envelope, so that alineand asurfaceare habitually, to our minds, just assolidas avolume. It may also be remarked that most generally, in order that bodies may penetrate one another with more facility, we are obliged to imagine the interior of thevolumesto be hollow, which renders still more sensible the impropriety of the wordsolid.