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 + ex = 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.
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.