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.

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

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

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 acbefg, 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

Fig. 474.

Fig. 475.

Fig. 476.

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.

DEVELOPEMENT OF SOLIDS WHOSE SURFACES ARE PLANE.

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, bc, ac, which form the triangle at the base, the 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 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.

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 PYRAMID.

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 hypothenuse, joining them, will express the length of the foreshortened

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, Cc, 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, Sc, 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'FcF, 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, bc, cd, de and ea" parallel to A'B', B'C, 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"d'e"a". Then the polygon oqmq'o' is the plane of the section induced by the line mo.

DEVELOPEMENT OF RIGHT AND OBLIQUE PRISMS.

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 Cc, 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, bd, 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, Cc, Dd their length. From this profile follows the horizontal projection (fig. 492.) wherein the lengthened polygons repre

sent the bases of the prism. In order to obtain the developement of this inclined prism, so that being bent up it may form the figure, from the middle of Cc, fig. 491. a perpendicular o, p, q must be raised, produced to l, l', fig. 493.; on this line must be transferred the widths of the faces shown by the polygon h, i, k, l, m, n, of fig. 491. in l, k, i, h, n, m, l', fig. 493. through these points parallel to the axis, lines are to be drawn, upon which qD of fig. 491. must be laid from 1 to E, from k to D, and from l' to E', fig. 493. ; pC, fig. 491., must be laid from i to C, and from m to F in fig. 493.

o A, fig. 491., is to be laid from h to B and from n to A, fig. 493., which will give the contour of the developement of the upper part by drawing the lines ED, DCB, BA, AFE', fig. 492.

To obtain the contour of the base, qd of fig. 491. must be transferred from 1 to q, from k to d, and from l' to e', fig. 493.

pe from fig. 491. from i to c and from m to f (fig. 493.); lastly, ob of fig. 16. must

be transferred from h to b and from n to a (fig. 493.) and drawing the lines ed, bed, ba, and afe', the contour will be obtained.

1169. The developement will be completed by drawing on the faces BA and ba, elongated polygons similar to ABCDEF and abcdef of fig. 491. and of the same size.

DEVELOPEMENT OF RIGHT AND OBLIQUE CYLINDERS.

1170. Cylinders may be considered as prisms whose bases are formed by polygons of an infinite number of sides. Thus, graphically, the developement of a right cylinder is obtained, by a rectangle of the same height, having in its other direction the circumference of the circle which serves as its base measured by a greater or less number of equal parts. 1171. But if the cylinder is oblique (fig. 494.), we must take the same measures as for

the prism, and consider the inclination of it. Having described centrally on its axis the circle or ellipsis which forms its perpendicular thickness in respect of the axis, the circumference must be divided into an equal number of equal parts, as, for instance, twelve, beginning from the diameter and drawing from the points of division the parallels to the axis HA, bi, ck, dl, em, fm, GO, which will give the projection of the bases and the general developement.

1172. For the projection of the bases on an horizontal plane, it is necessary that from the points where the parallels meet the line of the base HO, indefinite perpendiculars should be let fall, and after having drawn the line H', O', parallel to HO upon these perpendiculars above and below the parallel, must be transferred the size of the ordinates of