" The preceding observations relate to curved surfaces generally ; yet we must except from them, whatever regards cylindric, conic, and developable surfaces ; in the contact w ith the plane is not reduced to a single point, but extends the whole length of an indefinite line, which loses itself in the generator, in one of its positkus. The property of a plane touching one only of these surfaces, would be equivalent to two conditions, since it would subject it to pass along, a right line ; and there would only remain one CC/11114AM' to be disposed of, viz., to make it pass through a given point. It were in vain, therefore, to propose to draw a plane. that should be at one time tangent to two of these surthees, much less to three of them, unless there were some peculiar circumstances which should render these conditions compatible.
" It may net be altogether useless, before we proceed farther, to illustrate by a few examples, the necessity there is for drawing planes tangent to curved surfaces through points taken from the outside of them. The first of these examples is selected from the construction of fortifieations.
" In treating of the general principles of l'o•tilleatitm it is taken flu• granted, first, that, in every direction, the ground by which the place is surrounded, at least w ithin the reach of cannon-shot, is flat, and free from every eminence that might be converted to advantage by at besieging army. This hypothesis loving nettled, the draught of the place is next determined, with its hair-1110011S, covered ways, and ad vaneed works ; the bearings of the various parts of the fortifications upon each other are then marked out. so that they may all contribute, in the most efficacious manner, to their mutual and reciprocal defence. But, in order to apply these prin ciples to cases where the surrounding country presents some height, of which besiegers Might take advantage, and from which it is requisite that the fortification should be nude to defile, a new consideration presents itself. If there be only a single eminence, two points should be fixed upon in the place, through which might be conceived a plane tangent to the height, from which it is desirable to defile : this tangent plane is denominated defiling plane ; and all the parts of the fortification must receive the same relief above such plane. as they would have had above the horizontal plane. had the country been quite level : by this means, they all acquire, relatively upon each other, and collectively upon the neigh bouring height, a command equal to what they would other wise have possessed over the flat CI mntry : and the fortifira tion will possess the same advantages as in the first case.
As to the choice of the two points, through which the defiling plane ought to pass, it mnst be conformable to the two following conditions: 1st, That the angle formed by the plane with the horizon, be the least possible, in order that, the platforms having less slope, the service of defence may be attended with fewer impediments ; 2dly, That the relief tif the fortification abuse the natural ground, he likewise as little as possible, that its construe tion may require less labour, and be attended with less expense.
" Should there be two heights in the environs of the place, from which the fortification should defile, the defiling plane must be tangent to the surfaces of both, at the same time : and to determine its position, there is but one disposable condition ; and it is to lie disposed of, by choosing in tin.. place, a point, which the plane may pass, as nearly confbrinable as possible to the conditions prescribed in the first Case.
The second example we shall take, is from painting.
"The surfaces of bodies, especially when polished, present brilliant points, whose lustre may be compared to that of the luminous body by which they are enlightened. The bright ness of these points is greater, and their extent more con fined, in proportion as the surfaces are more polished. When the surfaces are unpolished, the brilliant points have much less lustre, and occupy a greater portion of the surface.
" In every surface. the position of the bright point is determined by the following condition ; that the incidental ray of light, and the reflected ray, directed to the eye of the spectator, be in the same plane, perpendicular to the plane tangent in this point, and make equal angles with it; for the shining point of the surface acts as a mirror, and reflects upon the eye a portion of the image of the luminous object. The determination of this point demands the utmost pre cision : for be the design never so correct, or the apparent contours traced with mathematical nicety, the least mistake committed in fixing the position of the brilliant point would be productive of the most palpable errors in the appearance of the shapes. We will give a single, but very striking case in proof.