Nor is this comparison between descriptive geometry and algebra al together useless : li ndieseseieueesareinti.uately connected. There is no construction of descriptive geometry, but what may be reduced to an analysis; and when questions require no more than three unknown quantities, each analysis may be looked upon as the record of a spectacle in geometry.
It is much to be wished, that these two sciences were studied together : descriptive geometry would carry that evi dence which is its peculiar characteristic, into the most com plicated analytical operations; while on the other hand alge braical analyses would give to geometry, that generality which it stands in need of "The principle upon which we ground the theory of pro jections is convenient for describing the position of a point in space or that of an indefinite or terminated right line, and, consequently. for representing the form and position of a body terminated either with plain faces, rectilinear arretes, or the apices of solid angles; fbr when once we are acquainted with the position of all its arretes and of the apices of all its angles, the body itself is entirely known. But were all bodies bounded, either by an uniformly curved surface. all whose points were governed by the same law, as in spheres, or by an unconnected assemblage of several parts of differently carved surfaces, as in a body turned on a lathe ; this prin ciple would not only be inconvenient, impracticable, and destitute of the advantage of forming an idea of the shape, but would also be insufficient through want of variety.
" For instance : it is easy to perceive that this principle by itself, would be inconvenient and impracticable, if we wished to describe all the points of a curved surface; because it would be necessary not only to indicate each of them, as well by its horizontal, as its vertical projection. but also to have the two projections of the same point united together, in order to avoid a combination of the horizontal projection of one point with the vertical projection of another ; and the most ready mode of thus uniting these projections, being to join them by one perpendicular right line to the line of intersec tion of the two planes of projection, the draught would become surcharged with a prodigious number of lines, and cause a eonffision, which increase in proportion as we would aim at accuracy and precision.
" We shall now prove this method to be insufficient, and destitute of the requisite copiousness.
" Amongst the vast variety of differently-curved surfimees, there are some which extend only through a finite and cir cumscribed portion of space, and whose projections arc limited, as to extent, in every direction ; as iii the case of a sphere, the extent of whose projection on a plane is reduced to that of a circle, having its circumference equal to that of the sphere ; and we must allow the plane, on which the projection falls, to be of dimensions sufficient to receive it. But all cylindric surfaces are as indefinite in a certain direc tion, as the right line by which they are generated ; and the plane itself, the most simple of all surfaces, is indefinite in two ways. There are likewise a great number of surfaces, whose protuberant particles (nappes) shoot at once into all the regions of space. Now, as the planes on which projec tions are received, are unavoidably of a limited extent. this mode of describing the nature of a curved surflmce, had we no other than that of the two projections of each of the points by which it passes, could be only applicable to those of which the points of the surface correspond to the size of the planes of projection ; all beyond this, could neither be expressed nor known : consequently, this mode would be insufficient. Lastly, it would want variety, because we could not deduce from it anything relative either to tangent planes to the sur face, nor to its normals, nor to its two curvatures in each point, nor to its lines of inflection, nor to its returning arretes, nor to its multiplied lines and points, nor, in a word, to any of the affections necessary to be considered in operating on a curved surface.
" It is therefore necessary, that we should have recourse to some new principle, not only compatible with the former, hut also capable of supplying its place, whenever it becomes in itself insufficient for our purpose. It is this new principle that we are now about to lay down.