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plane perpendicular to mn, the projection of the line GH will be a point g; and that of the inclined line IK, the vertical ik, which measures the inclination of that line. Lastly, on a vertical plane parallel to mn, the projection i'k' and g'h' will be parallel and equal to the original lines.
prior Ection of surfaces.
1134. What has been said in respect of right lines projected on vertical and horizontal planes may be applied to plane surfaces; thus, from the surface ABCD (fig. 463.), parallel to an horizontal plane, results the projection abcd 9f the same size and form. An inclined surface o 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 fo, which shows the profile of its inclination, because this plane is parallel to the side of the inclined sur- Fig. 465. face.
projection or curved LiNEs.
1137. Curve 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.
1138. If the plane of projection be not parallel, a foreshortened curve is the result, on account of its obliquity with 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 curve 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 horizontal and the other vertical, are squares whose sides represent faces perpendicular to these planes (fig. 468.), which are represented by corresponding small letters.
Fig. 468. Fig. 469. 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 1145. The projection of an inclined cylinder (fig. 471.) is shown on a vertical and horizontal plane. 1146. In fig. 472. we have the representation of a cube doubly inclined, so that the diagonal from the angle B to the angle G is upright. The projection produced by this position upon an horizontal plane is a regular hexagon acbefo, and upon a vertical plane the rectangle Begc whose diagonal Bg is upright; but as the effect of perspective changes the 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 projectins on horizontal and vertical planes. 1148. Fig. 476. represents a ball or sphere with its projections upon two planes, one
vertical and the other horizontal, wherein is to be remarked the perfection of this solid, seeing that its projection on a plane is always a circle whenever the plane is parallel to the circular base formed by the contact of the tangents.
1149. We have already observed that solids are only distinguished by their apparent faces, and that in those which have plane surfaces, their faces unite so as to form solid angles. 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 triangular 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 base,
the three triangles whose faces are inclined (fig. 478.); by which we obtain a figure composed of four triangles. To cut this out in paper, for instance, or any other flexible material, after bending it on the lines ab, be, ac, which form the triangle at the base, the three triangles are turned up so as to unite in the summit.
1151. The solid just described formed of four equal equilateral triangles, as we have seen, is the simplest of the five regular polyhedrons, and is called a tetrahedron, from its being composed of four similar faces. The others are —
The herahedron, or cube whose faces are six in number; The octahedron, whose faces are eight equilateral triangles; The dodecahedron, whose faces are twelve regular pentagons; The icosahedron, consisting of twenty equilateral triangles. These five regular polyhedrons are represented by the figures 477. 479, 480, 481, and 482., and their developement by the figures 478. 483, 484, 485, and 486.
Fig. 485. Fig. 484. Fig. 485.
1152. The surfaces of these developements are so arranged as to be capable of being united by moving them on the lines by which they are joined. 1153. It is here proper to remark, that the equilateral triangle, the square, and the pentagon, are the only figures which will form regular polyhedrons whose angles and sides are equal; but by cutting in a regular method the solid angles of these polyhedrons, others regularly symmetrical may be formed whose sides will be formed of two similar figures. Thus, by cutting in a regular way the angles of a tetrahedron, we obtain a polyhedron of eight faces, composed of four hexagons and four equilateral triangles. Similarly operating on the cube, we shall have six octagons, connected by eight equilateral triangles, forming a polyhedron of fourteen faces. 1154. The same operation being performed on the octahedron also gives a figure of fourteen faces, whereof eight are octagons and six are squares. 1155. The dodecahedron so cut produces twelve pentagons united by twenty hexagons, and having thirty-two sides. This last, from some points of view, so approaches the figure of the sphere, that, at a little distance, it looks almost spherical.
1156. The other solids whose surfaces are plane, whereof mention has already been made, are pyramids and prisms, partaking of the tetrahedron and cube; of the former, inasmuch as their sides above the base are formed by triangles which approach each other so as together to form the solid angle which is the summit of the pyramid; of the latter, because their faces, which rise above the base, are formed by rectangles or parallelograms which preserve the same distance from each other, but differ, from their rising on a polygonal base and being undetermined as to height.
1157. This species may be regular or irregular, they may have their axes perpendicular or inclined, they may be truncated or cut in a direction either parallel or inclined to their bases.
1158. The developement of a pyramid or right prism, whose base and height are given, is not attended with difficulty. The operation is by raising on each side of the base a triangle equal in height to the inclined face, as in the pyramidal figures 487. and 488., and a rectangle equal to the perpendicular height if it be a prism.
DEVELOPEMENT of AN oblique PYRAM ID.
1159. If the pyramid be oblique, as in fig. 489., wherein the length of the sides of each triangle can only be represented by foreshortening them in a vertical or horizontal projection, a third operation is necessary, and that is founded on a principle common to all projections; viz. that the length of an inclined line projected or foreshortened on a plane, depends upon the difference of the perpendicular elongation of its extremities from the plane, whence in all cases a rectangular triangle, whose vertical and horizontal projections give two sides, the third, which is the hypothen use, joining them, will express the length of the foreshortened line.
Pig. 488. t’ Fig. 492. 1160. In the application of this rule to the oblique pyramid of fig. 489., the position of the point P (fig. 490.) must be shown on the plan or horizontal projection answering to the apex of the pyramid, and from this point perpendicular to the face CD on the same side the perpendicular PG must be drawn. Then from the point P as a centre describe the arcs Bb, Ce, which will transfer upon PG the horizontal projections of the inclined arrisses AP, EP, and DP; and raising the perpendicular PS equal to the height of the apex P of the pyramid above the plane of projection, draw the lines Sa, Sb, Se, which will give the real lengths of all the edges or arrisses of the pyramid. 1161. We may then obtain the triangles which form the developement of this pyramid, by describing from C as a centre with the radius Sc, the arc ig, and from the point D another arc intersecting the other in F. Drawing the lines CF, DF, the triangles CFD will be the developement of the side DC. To obtain that answering to BC, from the points F and C with Sb and Bc as radii, describe arcs intersecting in B' and draw B'F and CB': the triangle FCB will be the developement of the face answering to the side Bc. 1162. We shall find the triangle FA'B', by using the lengths SA and BA to find the points B' and F, which will determine the triangle corresponding to the face AB, and lastly the triangles FDE' and FE"A" corresponding to the faces DE, AE by using the lengths Sb, DE and SA, AE. The whole developement AEDE"A"F, A'B', CBA being bent on the lines B'Fc F, CD, DF, and EF will form the inclined figure represented in fig. 489. 1163. If this pyramid be truncated by the plane mn, parallel to the base, the contour resulting from the section may be traced on the developement by producing Pm from F to a, and drawing the lines ab, be, cd, de and ea" parallel to A'B', BC, CD, DE' and E'A". 1164. But if the plane of the section be perpendicular to the axis, as mo, from the point F with a radius equal to Po describe an arc of a circle, in which inscribe the polygon ab"c"de"a". Then the polygon oqmq'o' is the plane of the section induced by the line mo.
1165. In a right prism, the faces being all perpendicular to the bases which terminate the solid, the developements are rectangles, consisting of all these faces joined together and enclosed by two parallel right lines equal to the contours of the bases. 1166. When a prism is inclined, the faces form different angles with the lines of the contours of the bases, whence results a developement whose extremities are terminated by lines forming portions of polygons. 1167. We must first begin by tracing the profile of the prism parallel to its degree of inclination (fig. 491.). Having drawn the line Ce, which represents the inclined axis of the prism in the direction of its length, and the lines AD, bd, to show the surfaces by which it is terminated, describe on such axis the polygon which forms the plane of the prism h, i, k, l, m perpendicular to the axis. Producing the sides kl, hn parallel to the axis to meet the lines AD, ba, they will give the four arrisses of the prism, answering to the angles h, n, k, l; and the line Ce which loses itself in the axis will give the arrisses im. 1168. It must be observed, that in this profile the sides of the polygon h, i, k, l, m give the width of the faces round the prism, and the lines Ab, Ce, Da their length. From this profile follows the horizontal projection (fig. 492.) wherein the lengthened polygons repre