" The other part of painting may be properly called the trade : its object is a correct execution of the conceptions of the former. Ilere nothing is arbitrary ; all is foreseen, by the help of sound reasoning, as the necessary result of deter. initiate subjects and given circumstances. When the form and position of an object are ascertained ; when its nature, and the number and positions of all the bodies by which it may be illumined. whether by direct light or reflected rays, are understood ; when the position of the eye of the spectator is determined ; and when, in a word, every circumstance that can influence the vision, is well established and known, the tint of each if' the points of the visible surface is abso lutely determined. Whatever relates to the colour of this tint, or its brightness, depends on the position of the plane tangent in this point, with respect to the illuminating bodies and to the eye of the spectator. This may be discovered by mere reasoning, and when so determined should be applied with accuracy. Every diminution, or exaggeration, will change the appearances, alter the firms, and produce an effect quite diffi-rent from that intended by the artist.
"I am aware that the rapidity of execution, which is often necessary, rarely admits of the use of a method which deprives the genius of all corporeal succours, and leaves it to the exercise of its faculties alone. as well as that it is much more easy for the painter to look at objects, to observe their tints, and to imitate theni: but were he accustomed to con sider the positions I if tangent planes, and the two curvatures of surfaces in each of their points (curvatures which will farm the subject of subsequent lessons) he would not fail to derive from this material method, a more advantageous result : it would enable hini to supply effects which the omission of some circumstances had prevented him from producing. and to suppress others which had arisen front extraneous incidents.
" In conclusion. we may remark. that the vague expres sions, such as fiats, which painters are in the constant habit of using, are standing evidences of the need in which they are of more accurate knowledge, and of deeper reflection.
Besides its utility in the arts, the knowledge of planes tangent and normals to curved surfaces, is one of the most fertile methods employed in descriptive geometry for the solution of questions, which it would be very difficult to resolve by any other process, as will appear f?om the follow ing examples.
"The general mode of determining the plane. tangent to a curved surface, consists, as we have already remarked, in conceiving at the point of contact, the tangents with two different generating curves, which would pass through this point. and in constructing the plane that would pass through
these two right lines. In some particular cases. in order to shorten the construction, the strict letter of this mode is departed from but an equivalent is always adopted.
" As to the construction of the normal, we shall not dwell upon it particularly, as it reduces itself merely to that of a right line perpendicular to the tangent plane, which is suffi ciently understood.
"First Question.—From a supposed point on a cylindric surfice, of which the horizontal projection is given, to draw a tangent plane to that surface.
Sittion.-1 iyure 1:2. Let A a, a b, represent the hori zontal and vertical projections of the given right line, to which the generator of the cylindric surface must be parallel; let E P D be the given curve in the horizontal plane, on which the generator must constantly rest, and which may be con sidered as the outline of the cylindric surface; lastly, let c be the given horizontal projection of the point supposed on the cylindric surface, from which the tangent plane must be drawn.
" Next, from the supposed point on the surface, whose horizontal projection is in c. imagine the generating right line in the position it would have, if it passed through that point : this generator being a straight line, will be its own tangent, and consequently one of the two right lines by which the position of the tangent plane will be determined ; also, it will be parallel to the given right line: its two projections, therefore, will be respectively parallel to A n and a b: then, if from the point c, an indefinite line be drawn, parallel to A 13, as E F. we sled] have the horizontal projection of the generator. To obtain its vertical projection, we must sup pose the generator to he produced upon the cylindric surface, till it meet the horizontal plane, which it can only do in a point, that will be at once on the projection E F, and on the curve E P D, and consequently the intersection of these two lines: thus the point will be determined by producing E F till it cut some part of the curve E P D.
Two cases here present themselves : either the line E F will C111 the outline of the cylinder in a single point, or it will cut it in several points. In examining these two cases sepa rately, we shall suppose, in the first instance, that to what ever length the line E F may he produced, it shall cut the curve E e D only in the single point D.