The hexahedron, 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, 48, 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.

DEVELOPEMENT OF PYRAMIDS AND PRISMS.

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 pro jection, 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 line

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 ares 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, 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 Be 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'e 'de"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 for ning 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

seat 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 3, p, q must be raised, produced to 1, 1, 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.

A. fig. 491., is to be laid from A 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 / 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. 491. must be transferred from h to b and from a to a (fig. 493.) and drawing the lines ed, bed, ba, and afé, the contour will be obtained.

1169. The developement will be completed by drawing on the faces B A 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 should be divided into an even 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, ek, 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 hould 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

the circle or ellipsis traced on the axis of the cylinder, that is, pl and p10 to 'l, and i'10: 92 and 99 in k2 and k'q, &c. In order to avoid unnecessary repetition, the figs. 494, 495, 496. are similarly figured, and will by inspection indicate the corresponding lines. 1173. In the last figure the line E'E' is the approximate developement of the circumference of the circles which follow the section DE perpendicular to the axis of the cylinder, divided into 12 equal parts, fig. 494. For which purpose there have been transferred upon this line on each side of the point D, six of the divisions of the circle, and through these points have been drawn an equal number of indefinite parallels to the lines traced upon the cylinder in fig. 494. then taking the point D' as correspondent to D, the length of these lines is determined by transferring to each of them their relative dimensions, measured from DE in AG for the superior surface of the cylinder, and from DE to HO for

the base.

1174. In respect of the two elliptical surfaces which terminate this solid, what has been above stated, on the manner of describing a curve by means of ordinates, will render further explanation on that point needless.

DEVELOPEMENT OF RIGHT AND OBLIQUE CONES.

1175. The reasoning which has been used in respect of cylinders and prisms, is applicable to cones and pyramids.

1176. In right pyramids, with regular and symmetrical bases, the edges or arrisses from the base to the apex are equal, and the sides of the polygon on which they stand being equal, their developement must be composed of similar isosceles triangles, which in their union will form throughout, part of a regular polygon, inscribed in a circle whose inclined sides will be the radii. Thus, in considering the base of the cone A B (fig. 497.) as a

regular polygon of an infinite number of sides, its developement becomes a sector of a circle A"B" BC" (fig. 498.) whose radius is equal to the side AC of the cone, and the arc equal to the circumference of the circle which is its base.

1177. Upon this may be traced the developement of the curves which would result from the cone cut according to the lines DI, EF, and GH, which are the ellipsis, the parabola, and the hyperbola. For this purpose the circumference of the base of the cone must be divided into equal parts; from each point lines must be drawn to the centre C, representing in this case the apex of the cone. Having transferred, by means of parallels, to FF, the divisions of the semi-circumference AFB of the plan, upon the line A'B', forming the base of the vertical projection of the cone (fig. 497.) to the points 1'2', FS', and 4', which, because of the uniformity of the curvature of the circle will also represent the divisions on the plan marked 8, 7 F', 6, and 5; from the summit C' in the elevation of the cone, the lines C'1', C2, C'F, C's', C'4' are to be drawn, cutting the plans DI, EF, and GH of the ellipse, of the parabola, and of the hyperbola; then by the assistance of these intersections their figures may be drawn on the plan, the first in D'p'I'p"; the second in FE'F'; the third in H'GH".

1178. To obtain the points of the circumference of the ellipse upon the developement (fig. 498.), from the points n, o, p, q, r of the line DI (fig. 497.), draw parallels to the base for the purpose of transferring their heights upon C'B' at the points 1, 2, 3, 4, 5. Then transfer C'D upon the developement, in C"n"", C"o"", C"p"", C"q", C"r""; and in the same order below, C''n"""', C''o'""', C'p""", C'q'"', Cr"""; and CI from C" in I" and I"".

The

curve passing through these points will be the developement of the circumference of the ellipse indicated in fig. 497. by the right line DI, which is its great axis.

1179. For the parabola (fig. 499.) on the side C'A' (fig. 497.), draw bg and ah; then transfer CE on the developement in C"E"; Cg from C" to b'" and b'''; C ́h, from C'_to a" and """; and through the points F", a"", b', E", b'''', a'''', F' ', trace a curve, which will be the developement of the parabola shown in fig. 497. by the line EF.

1180. For the hyperbola, having drawn through the points m and i, the parallels me, if, transfer CG from C" to G", and from C" to G" of the developement, Ce from C" to m and m"", C'ƒ from C'' to i''' and i''''; and after having transferred 3H ́ and 6H ́ of the plan to the circumference of the developement, from 3 to H, and from 6 to H, by the aid of the points H'", i", m'"', G” and H'''', i''', m'''', G''', draw two curves, of which each will be the developement of one half of the hyperbola represented by the right lines GH and H'H", figs. 497. and 500., and by fig. 501.

1181. The mode of finding the developement of an oblique cone, shown in figs. 502, 5ʊ3,

504, 505. differs, as follows, from the preceding. 1. From the position of the apex C upon the plan 503., determined by a vertical let fall from such apex in fig. 502. 2. Because the line DI of this figure, being parallel to the base, gives for the plan a circle instead of an ellipsis. 3. Because in finding the lengthened extent of the right lines drawn from the apex of the cone to the circumference of the base, divided into equal parts, fig. 504. is introduced to bring them together in order to avoid confusion, these lines being all of a different size on account of the obliquity of the cone. In this figure the line CC' shows the perpendicular height of the apex of the cone above the plan; so that by transferring from each side the projections of these lines taken on the plan from the point C to the circumference, we shall have CA", C1, C2, CF", C3, C4, CB', on one side, and CA', C8, C7, CF, C6, C5, and CB" on the other; lastly, from the point C drawing lines to all these points, they will give the edges or arrisses of the inscribed pyramid, by which the developement in fig. 505. is obtained. Having obtained the point C" representing the apex, a line is to be drawn through it equal to CA" (fig. 504.); then with one of the divisions of the base taken on the plan, such as Al, it must be laid from the point A of the developement of the section; then taking C'1 of fig. 504., describe from the point C" another are which will cross the former, and will determine the point 1 of the developement. Continuing the operations with the constant length Al and the different lengths C2, CF', C3, &c., taken from fig. 504. and transferred to C2, C"F, C3, &c. of the developement, the necessary points will be obtained for tracing the curve B"AB"", representing the contour of the oblique base of the cone.

1182. We obtain the developement of the circle shown by the line DI of fig. 502. parallel to that of the base AB, by drawing another line I'D D'I' (fig. 504.) at the same distance from the summit C, cutting all the oblique lines that have served for the preceding developement; and on one side, CD", Cn, Ĉo, Cp, Cq, Cr, CI", must be carried to fig. 505., from C" to D', n, o, p, q, r, and on the other from C" to n, o, p, q, r, and I", on fig. 505. The curve line passing through these points will be the developement of this circle.

1183. To trace upon the developement the parabola and hyperbola shown by the lines EF, G3 of fig. 502., from the points Eba, Gmi draw parallels to the base AB, which, transferred to fig. 504., will indicate upon corresponding lines the real distance of these points from the apex C. which are to be laid in fig. 505. from C" to E, b, a, b and a for