" Solution .—Fig ure 11. Let A be the horizontal projection of the apex of the angle sought for, and A n that of one of its sides, so that the other side may be represented by A E. Conceive the vertical plane of to pass along A tt ; and having drawn the vertical indefinite line A a, through the point A, any point, as d, may be taken at pleasure, as the vertical projection of the apex of the angle observed. If from the point d, the right line dB be drawn, so as to make with the horizontal line, an angle dB A. equal to that made by the first side with the horizon, the point B will be the coincidence of this side with the horizontal plane. Also, if from the point d, the line d c lie drawn, so as to make with the horizontal line an angle. dc A, equal to that made by the second side with the horizon ; and if from the point A, as from a centre. with the radius A c, the indefinite are of a circle, c E I', be described, the second side can meet the horizontal plane only in the points of the are C E F. It remains, therefore, only to ascertain the distance between this point and some other, as B.
"Now this latter distance is in the plane of the angle observed. If, therefore, the right line d a be drawn, so as that the angle D d B may be equal to the angle observed, and if d c be carried from d to n, the right line n 13 will be equal to such distance.
" Therefore, taking a as a centre, and, at an interval equal to c D. describing the are of a circle, the point E, 13 here it will cut the first, will be the point of coincidence of the second side with the horizontal plane; consequently, the right line A E will be the horizontal projection of snob side, and the angle a A E that of the angle observed.
'lite nine preceding questions barely convey an idea of the method of projections; they are inadequate to a display of all the resources: but in proportion as we rise to more general considerations, we shall take care to introduce such operations as will he most conducive to this object.
" Of planes tangent to enrced snifuces, and of nmonals.
"There is no curved surface but what, may be generated in several ways, by the inovement of curved lines: there fore, if from any point of a surfitee, two generating lines be supposed to spring in the position they would naturally have in passing each other through such point, and if the tangents be supposed in this point, to each of the two generators, the plane described by such two tangents is the tangent plane. The point of the surface in which the two generators cut each other, and is at. the same titne common to the two tangents and to the tangent plane, is the point of contact between the sur•tce and the plane.
The right line drawn through the point of contact. per pendicularly to the tangent plane. is said to be normal to the surthce. It is perpendicular to the ground of the surface,
because the direction of such ground coincides, in every part, with that of the tangent plane, which may be considered as its prolongation.
" A knowledge of tangents and of normals to curved sur faces, is very useful in a great number of arts; in many, it is indispensable. We -shall here adduce only a single example of each ease, selected in architecture and painting.
"The several portions which compose vaults of hewn stone, are called missoirs, and the farces on which two con tiguous voussoirs touch each other. are denominated joints, whether the voussoirs form but a single course, or whether they be comprised in two successive courses, `•The position of the joints of vaults is subject to several conditions, which must necessarily be complied with, and which we shall demonstrate in succession, in the sequel of this discourse; but at present we must confine our attention to the object more immediately before us.
" One of the conditions required in the position of joints, is, that they he all perpendicular to each other, and to the surthce of the vault. Any material deviation from this rule, not only destroys the general symmetry of the structure, but also diminishes the firmness and durability of the vault. For instance, if one of the joints be made oblique to the surface of the vault, one of the two continuous voussoirs will form an obtuse angle, and the other, an acute one; and in the reaction which these voussoirs would exert against each other, the two angles would present an unequal resistance, whence, in consequence of the fragility of the materials, the acute angle would bilge, and spoil the shape of the vault, as well as endanger the edifice. The reduction of vaults into voussoirs, therefore, absolutely requires a knowledge of planes tangent and normals to the curved surface of the arch, " Let us take another example, from an art, which, at first view, seems to require a much less rigid attention to this rule.
" Painting is generally considered as consisting of two parts. The one is, properly the art; its object is to excite in the spectator a determinate emotion, to create in him a given idea, or to place him in a situation the most favourable for receiving a certain impression ; it supposes in the artist it great knowledge of philosophy ; exacts, on his part, a most intimate acquaintance with the nature of thing-4, the mode in which they aGet us, and the move. ments, even involuntary, by which such affection manifests itself. This can only he the result of a very relined educa tion, such as no one receives, and such as we are tar from giving to young artists : it is governed by no general rule, but is subject solely to genius.