projection of the proposed point. Then, as points in space are the boundaries of lines, se ... their projections similarly form lines, by whose means their projection is obtained; and by the projections of points lying in curves of any description, the projections of those curve are obtained.

1112. For obvious reasons, surfaces cannot be similarly represented; but if we suppos the surface to be represented, covered by a system of lines, according to some determinati law, then these lines projected on each of the two planes will, by their boundaries, enable us to project the surface in a rigorous and satisfactory manner.

1113. There are, however, some surfaces which may be more simply represented; for a plane is completely defined by the straight lines in which it intersects the two planes of projection, which lines are called the traces of the plane. So a sphere is completely defined by the two projections of its centre and the great circle which limits the projections of its points. So also a cylinder is defined by its intersection (or trace) with one of the planes of projection and by the two projections of one of its ends; and a cone by its intersection with one of the planes of projection and the two projections of its summit.

1114. Monge, before mentioned, Hachette, Vallée, and Leroi, are the most systematic writers on this subject, whose immediate application to architecture, and to the mechanical arts, and most especially to engineering, is very extensive; in consequence, indeed, of which it is considered of so much importance in France, as to form one of the principal departments of study in the Polytechnic School of Paris. A sufficient general idea of it for the architectural student may be obtained in a small work of Le Croix, entitled, Complément des Elemens de Geometrie. In the following pages, and occasionally in other parts of this work, we shall detail all those points of it which are connected more immediately with our subject, inasmuch as we do not think it necessary to involve the reader in a mass of scientific matter connected therewith, which we are certain he would never find necessary in the practice of the art whereon we are engaged.

1115. In order to comprehend the method of tracing geometrically the projections of all sorts of objects, we must observe,-I. That the visible faces only of solids are to be expressed. II. That the surfaces which enclose solids are of two sorts, rectilinear and curved. These, however, may be divided into three classes, - 1st. Those included by plane surfaces, as prisms, pyramids, and, generally, similar sorts of figures used in building. 2d. Those included by surfaces whereof some are plane and others with a simple curvature, as cylinders, cones, or parts of them, and the voussoirs of arches. 3d. Solids enclosed by one or several surfaces of double flexure, as the sphere, spheroids, and the voussoirs of arches on circular planes.

1116. First class, or solids with plane surfaces. - The plane surfaces by which these solids are bounded form at their junction edges or arrisses, which may be represented by right lines.

1117. And it is useful to observe in respect of solids that there are three sorts of angles formed by them. First, those arising from the meeting of the lines which bound the faces of a solid. Second, those which result from the concurrence of several faces whose edges unite and form the summit of an angle: thus a solid angle is composed of as many plane angles as there are planes uniting at the point, recollecting however that their number must be at least three. Third, the angles of the planes, which is that formed by two of the faces of a solid. A cube enclosed by six square equal planes comprises twelve rectilineal edges or arrisses and eight solid angles.

1118. Pyramids are solids standing on any polygonal bases, their planes or faces being triangular and meeting in a point at the top, where they form a solid angle.

1119. Prisms, like pyramids, may be placed on all sorts of polygonal bases, but they rise on every side of the base in parallelograms instead of triangles, thus having throughout similar form and thickness.

1120. Though, strictly speaking, pyramids and prisms are polyhedrons, the latter term is only applied to those solids whose faces forming polygons may each be considered as the base of a separate pyramid.

1121. In all solids with plane surfaces the arrisses terminate in solid angles formed by several of these surfaces, which unite with one another; whence, in order to find the projection of the right lines which represent those arrisses, all that we require to know is the position of the solid angles where they meet; and as a solid angle is generally composed of several plane angles, a single solid angle will determine the extremity of all the arrisses by which it is formed.

1122. Second class: solids terminated by plane and curved surfaces. Some of these, as cones for instance, exhibit merely a point and two surfaces, one curved and the other flat. The meeting of these surfaces forms a circular or elliptical arris common to both. The projection of an entire cone requires several points for the curvature which forms its base, but a single point only is necessary to determine its summit. This solid may be considered as a pyramid with an elliptic or circular base; and to facilitate its projection a polygon is inscribed in the ellipsis or circle, which serves as its base.

1123. If the cone is truncated or cut off, the curves which produce the sections.

polygons may in like manner be inscribed in

1124. Cylinders may be considered as prisms whose bases are formed by circles, ellipses, or other curves, and their projections may be obtained in a similar manner: that is, by inscribing polygons in the curves which form their bases.

1125. Third class: solids whose surfaces have a double curvature.-A solid of this sort may be enclosed in a single surface, as a sphere or spheroid.

1126. As these bodies present neither angles nor lines, they can only be represented by the apparent curve which seems to bound their superficies. This curve may be determined by tangents parallel to a line drawn from the centre of the solid perpendicularly to the plane of projection.

1127. If these solids are truncated or cut by planes, we must, after having traced the curves which represent them entire, inscribe polygons in each curve produced by the sections, in order to proceed as directed for cones and cylinders.

1128. To obtain a clear notion of the combination of several pieces, as, for instance, of a vault, we must imagine the bodies themselves annihilated, and that nothing remains but the arrisses or edges which form the extremes of the surfaces of the voussoirs. The whole assemblage of material lines which would result from this consideration being considered transparent would project upon a plane perpendicular to the rays of light, traces defining all these edges that we have supposed material, some foreshortened, and others of the same size. These will form the outlines of the vault, whence follow the subjoined remarks. I. That in order, on a plane, to obtain the projection of a right line representing the arris of any solid body, we must on such plane let fall verticals from each of its extremities.

II. That if the arris be parallel to the plane of the drawing, the line which represents its projection is the same size as the original.

III. That if it be oblique, its representation will be shorter than the original line. IV. That perpendiculars by means of which the projection is made being parallel to each other, the line projected cannot be longer than the line it represents.

V. That in order to represent an arris or edge perpendicular to the plane of projection, a mere point marks it because it coincides in the length with the perpendiculars of projection.

VI. That the measure of the obliquity of an arris or edge will be found by verticals let fall from its extremities.

1129. In conducting all the operations relative to projections, they are referable to two planes, whereof one is horizontal and the other vertical.

PROJECTION OF RIGHT LINES.

1130. The projection of a line AB (fig. 459.) perpendicular to a horizontal plane is ex

pressed on such plane by a point K, and by the lines ab, a'b', equal to the original on verical planes, whatever their direction.

1131. An inclined line CD (fig. 460.) is represented on an horizontal or a vertical plane by cd, c'd', shorter than the line itself, except on a vertical plane, parallel to its projection, on the horizontal plane c'd", where it is equal to the original CD.

1132. An inclined line EF (fig. 461.) moveable on its extremity E, may, by preserving the same inclination in respect of the plane on which it lies, have its projection successively in all the radii of the circle Ef, determined by the perpendicular let fall from the point F. 1133. Two lines GH, IK (fig. 462.), whereof one is parallel to an horizontal plane and the other inclined, may have the same projection m, n, upon such plane. Upon a vertica!

plane perpendicular to mn, the projection of the line GH will be a point g; and that of tl inclined line IK, the vertical ik, which measures the inclination of that line. Lastly, or vertical plane parallel to mn, the projection i'k' and g'h' will be parallel and equal to tl original lines.

PROJECTION OF SURFACES.

on

1134. What has been said in respect of right lines projected on vertical and horizont. planes may be applied to plane surfaces; thus, from the surface ABCD (fig. 463.), parallel to an horizontal plane, results the projection abcd of the same size and form. An inclined surface EFGH may have, though longer, the same projection as the level one ABCD, if the lines of projection AE, BF, DH, CG are in the same direction.

1135. The level surface ABCD would have for projection on vertical planes the right lines ab, b'c', because that surface is in the same plane as the lines of projection.

1136. The inclined surface EFGH will give on vertical planes the foreshortened figure hgef of that surface; and upon the other the simple line fq, which shows the profile of its inclination, because this plane is parallel to the side of the inclined surface.

d

Fig. 463.

JC

PROJECTION OF CURVED LINES.

1137. Curved lines not having their points in the same direction occupy a space which brings them under the laws of those of surfaces. The projection of a curve on a plane parallel to the surface in which it lies (fig. 464.) is similar to the curve.

Fig. 464.

Fig. 465.

Fig. 466.

Fig. 467.

B

1138. If the plane of projection be not parallel, a foreshortened curve is the result, on account of its obliquity to the surface (fig. 465.).

1139. If the curve be perpendicular to the plane of projection, we shall have a line representing the profile of the surface in which it is comprised; that is to say, a right line if the surface lie in the same plane (fig. 466.), and a curved line if the surface be curved (fig. 467.).

1140. In order to describe the projection of the curved line ABC (fig. 467.), if the surface in which it lies is curved, and it is not perpendicular to the plane of projection, a polygon must be inscribed in the curve, and from each of the angles of such polygon a perpendicular must be let fall, and parallels made to the chords which subtend the arcs. But it is to be observed, that this line having a double flexure, we must further inscribe a polygon in the curvature which forms the plane abc of the surface wherein the curved line lies.

1141. The combination and developement of all the parts which compose the curved surfaces of vaults being susceptible of representation upon vertical and horizontal planes by right or curve lines terminating their surfaces, if what has been above stated be thoroughly understood, it will not be difficult to trace their projections for practical purposes, whatever their situation and direction in vaults or other surfaces.

PROJECTION OF SOLIDS.

1142. The projections of a cube ABCDEFGH placed parallel to two planes, one borizontal and the other vertical, are squares whose sides represent faces perpendicular to these planes (fig. 468.), which are represented by corresponding small letters.

1143. If we suppose the cube to move on an axis, so that two of its opposite faces remain perpendicular to the planes (fig. 469.), its projection on each will be a rectangle, whose length will vary in proportion to the difference between the side and the diagonal of the square. The motion of the opposite arrisses will, on the contrary, produce a rectangle whose width will be constant in all the dimensions contained of the image of the perfect square to the exact period when the two arrisses unite in a single right line. 1144. A cylinder (fig. 470.) stands perpendicularly on an horizontal plane, and on such

Fig. 470.

Fig. 471.

plane its projection ADBC is shown, being thereon represented by a circle, and upon a vertical plane by the rectangle gedh.

1145. The projection of an inclined cylinder (fig. 471.) is shown on a vertical an horizontal plane.

1146. In fig. 472. we have the representation of a cube doubly inclined, so that th diagonal from the angle B to the angle G is upright. The projection produced by thi position upon an horizontal plane is a regular hexagon acbefg, and upon a vertical plane th rectangle Bege whose diagonal Bg is upright; but as the effect of perspective changes tn effect of the cube and its projections, it is represented geometrically in fig. 473.

1147. In figures 474. and 475. a pyramid and cone are represented with their pro jections on horizontal and vertical planes.

1148. Fig. 476. represents a ball or sphere with its projections upon two planes, on

Fig. 474.

Fig. 475.

Fig. 476.

vertical and the other horizontal, wherein is to be remarked the perfection of this solic seeing that its projection on a plane is always a circle whenever the plane is parallel to th circular base formed by the contact of the tangents.

DEVELOPEMENT OF SOLIDS WHOSE SURFACES ARE PLANE.

1149. We have already observed that solids are only distinguished by their apparer. faces, and that in those which have plane surfaces, their faces unite so as to form solid angle We have also observed that at least three plane angles are necessary to form a solid angle. whence it is manifest that the most simple of all the solids is a pyramid with a triangula base, which is formed by four triangles, whereof three are united in the angles at its apex. (Fig. 477.)

1150. The developement of this solid is obtained by placing on the sides of the bas Fig. 479.

Fig. 477.

Fig. 481.

the three triangles whose faces are inclined (fig. 478.); by which we obtain a figu composed of four triangles. To cut this out in paper, for instance, or any other flexib material, after bending it on the lines ab, be, ac, which form the triangle at the base, t three triangles are turned up so as to unite in the summit.

DEVELOPEMENT OF REGULAR POLYHEDRONS.

1151. The solid just described formed of four equal equilateral triangles, as we ha seen, is the simplest of the five regular polyhedrons, and is called a tetrahedron, from being composed of four similar faces. The others are —