The philosophy of mathematics · Auguste Comte
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.
Text from Project Gutenberg, public domain in the United States.