"Every curved surface may lie considered as generated by the movement of a curved line, either inflexible in form when its position is changed, or variable both in form and situation. As the universality of this proposition may render it difficult of comprehension, we shall explain it by some film i iar examples.
"Cylindric surfaces may lie generated in two principal ways, viz., either by the movement of a right line, which keeps constantly parallel to a given line whilst in motion, yet inclining always towards a given curve ; or by the movement of the curve itself, taken in the foregoing instance as the con ductor, which so moves, as that while from one point it inclines towards a given line, all its other points may describe parallels to this line. In both these modes of origin, the generating line, which is a right line in the first case, and a curve, of whatever description, in the second, invariably retains its form ; only changing its position in space.
"Conical surfaces have, in like manner, two principal modes of being generated. First, they may be considered as generated by an indefinite right line, which being forced to pass always upon a given point, moves so as to lean constantly towards a given curve, that directs its movement. The only point through which it always passes the right line, is the centre of the surfitee, improperly called its apex or head. In this mode, the genemting line still preserves its identity, never ceasing to he a right line.
Conic surfaces may also be generated in another way, which, tir greater plainness, we shall here apply only to those with circular bases. The surfaces may be considered as bounded by the circumference of a circle, moving with its plane always parallel to itself, and its centre upon a right line passing through the apex ; its radius being, in every inevement, proportionate to the distance of the centre from the apex. 1 Jere it is evident, that it' in its motion, the plane of the circle tends towards the apex at' the surihee, the radius of time circle will decease, till, in passing the apex, it will be an absolute nullity, afier which it will again increase indefi nitely, in proportion as the plane, having passed the apex. is withdrawn tither• and farther from it. In this second mode of generating, the circumference of the circle, which is the generating curve, nut only changes its position, but its form also, at every motion ; for, changing its radius, it conse quently varies both in curvature and extent.
Let us take a third example.
A circular surface may be generated by the movement of a plane curve, turning about a right line ; drawn in any direction upon its plane. In this way. we find the generating curve inflexible in form, but changeable in position. We may also see it generated by the circumference of a circle, moving with its centre always on the axis, and its plane being pe• pendieular to this axis, the radius will be unitbrinly equal to the distance of the point in which the plane of the circle cuts the axis, from that in which it cuts a given curve in space. lien. the generating curve changes both in form and position.
From these three examples, we may perceive that all curved surfaces may he generated by the movement of cer tain curved lines, and that there are none of which the form and position may not be accurately described from an exact and complete definition of its generation. This new prin ciple forms a complement to the method of projections; and in proceeding, we shall have frequent occasion to be con vinced of its simplicity and copiousness.
It is not, therefore, by merely giving the projections of individual puhas, through •hieh tt curved surfiice passes, that we tire enabled to determine its form and position ; but by being able to construct tor any point the generating curve, according to the tbrm and position it would have in passing such point. And here we may remark, 1. That as every curved surthce may be generated in an infinite number of different ways, it must depend upon the dexterity and knoe ledge of the operator, to make choice, among all the possible generations, of such as Will require the most simple curve, and least complex consideratious. 2. That long expe rience has taught us, instead of considering only one gene•a ting principle of a curved surtitee. as prescribed by the laws of mutioe and of the change of form in its generation, it is frequently more simple to take two generating principles, and to indicate for each point the construction of the two genci.atiug curves.