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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 1, 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 / 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.

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oA, 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 1 to q, from k to d, and from l' to e', fig. 493.

pc 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, bcd, 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

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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

the circle or ellipsis traced on the axis of the cylinder, that is, pl and p10 to i'l, and '10: q2 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 DG 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

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regular polygon of an infinite number of sides, its developement becomes a sector of a circle A"B"B"C" (fig. 498.) whose radius is equal to the side A'C' 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 C1, C2, C'F, C'3', 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

HIGH".

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"""', C"r"""; 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 C'E on the developement in C"E"; C'g from C" to b"" and b"""; C'h, from C" to a"" and a"""; and through the points F", a"", "", 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 C'G from C" to G", and from C" to G"" of the developement, C'e from C" to m"" and m'"", C'f from C" to i"" and ""; 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""", "", 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, 503,

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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 arc 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 C"2, 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, Co, 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

the parabola; and from C" to G, m, i on one side, and on the other to G, m, i, for the hyperbola. Each of these is represented in figs. 506. and 507.

DEVELOPEMENT OF BODIES OR SOLIDS WHOSE SURFACES HAVE A DOUBLE CURVATURE.

1184. The developement of the sphere and other bodies whose surface has a double flexure would be impossible, unless we considered them as consisting of a great number of plane faces or of simple curvatures, as the cylinder and the cone. Thus a sphere or spheroid may be considered, I. As a polyhedron of a great number of plane faces formed by truncated pyramids whose base is a polygon, as fig. 508. II. By truncated cones, forming zones, as in fig. 509. III. By

parts of cylinders cut in gores, forming flat sides that diminish in width, shown by fig. 510.

1185. In reducing the sphere or spheroid to a polyhedron with flat faces, the developement may be accomplished in two ways, which differ only by the manner in which the faces are developed.

1186. The most simple method of dividing the sphere to reduce it to a polyhedron is that of parallel circles crossed by others perpendicular to them, and intersecting in two opposite points, as in the common geographical globes. If, instead of the circle, the polygons are supposed of the same number of sides, a polyhedron a will be the result, similar to that represented by fig. 508., whose half ADB shows the geometrical elevation, and AEB the plan of it.

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1187. For the developement, produce A1, 12, 23, so as to meet the produced axis CP in order to obtain the summits P, q, r, D of the truncated pyramids which form the semi-polyhedron ADB; then from the points P, q, r, with the radii PA, P1, ql, q2, r2, r3 and D3, describe the indefinite arcs AB', 1b', 1b", 2f', 2f", 3g', and 3g, upon which, after having transferred the divisions of the demi-polygons AEB, 1e6"", 2e'5"", 3e", 4", from all the transferred points, as A, 4', 5', 6', 7', 8', 9', B', for each truncated pyramid draw lines to the summits PqrD, and other lines which will form inscribed polygons in each of the arcs AB', lb', lb", &c. These lines will represent for each band or zone the faces of the truncated pyramids whereof they are part.

1188. We may arrive at the same developement by raising upon the middle of each side of the polygon AEB indefinite perpendiculars, upon which must be laid the height of the faces of the elevation in 1, 2, 3, 4; through which points draw parallels to the base, upon which transfer the widths of each of the faces taken on the plan, whereby trapezia will be formed, and triangles similar to those found in the first developement, but ranged in another manner. This last developement, which is called in gores, is more suitable for geographical globes; the other method, for the formation of the centres, moulds, and the like, of spherical vaults.

1189. The developement of the sphere by conic zones (fig. 509.) is obtained by the same process as that by truncated pyramids, the only difference being, that the developement of the arrisses AB, 1b, 2f, 3g, are arcs of circles described from the summits of cones, instead of being polygons.

1190. The developement of the sphere reduced to portions of cylinders cut in gores (fig. 510.) is conducted in the second manner, but instead of joining with lines the points 6, h, i, k, d, (fig. 508.) they must be united by a curve. This last method is useful in

drawing the caissons or pannels in spherical or spheroidal vaults.

OF THE ANGLES OF PLANES OR SURFACES BY WHICH SOLIDS ARE BOUNDED.

1191. In considering the formation of solids, we have already noticed three sorts of angles, viz. plane angles, solid angles, and the angles of planes. The two first have been

treated of in the preceding sections, and we have now to speak of the third, which must not be confounded with plane angles. Of these last, we have explained that they are formed by the lines or arrisses which bound the faces of a solid; but the angles of planes, whereof we are about to speak, are those formed by the meeting of two surfaces joining in an edge.

L

D

1192. The inclination of one plane ALDE to another ALCB (fig. 511.) is measured by two perpendiculars FG, FH raised upon each of these planes from the same point F of the line or arris AL formed by their union.

A

F

H

B

C

1193. It is to be observed, that this angle is the greatest of all those formed by lines drawn from the point F upon these two planes; for the lines FG, FH being perpendicular to AL, common to both the planes, they will be the shortest that can be drawn from the point F to the sides ED, BC, which we suppose parallel to AL; thus their distance GH will be throughout the same, whilst the lines FI, FK will be so much the longer as they extend beyond the perpendiculars FG, FH, and we shall always have KI equal to GH, and consequently the angle IFK so much smaller than GFH as it is more distant.

Fig. 511.

1194. Thus, let a rectangular surface be folded perpendicularly to one of its sides so that the contours of the parts separated by the fold may fall exactly on each other. If we raise one of them, so as to move it on the fold as on a hinge, and so as to make it form all degrees of angles, we shall see that each of the central extremities of the moveable part is always in a plane perpendicular to the part that is fixed.

1195. This property of lines moving in a perpendicular plane, furnishes a simple method of finding the angles of planes of all sorts of solids whose vertical and horizontal projections or whose developements are known.

1196. Thus, in order to find the angles formed by the tetrahedron or pyramid on a triangular base (fig. 477.), we must for the angles of the base with the sides, let fall from the angles ABC perpendiculars to the sides ac, cb, and ab, which meet at the centre of the base in D. It is manifest from what has just been said on this subject, that if the three triangles are made to move, their angles at the summit A, B, C will not be the vertical planes shown by the lines AD, DB, DC, and that they will meet at the extremity of the vertical, passing through the intersection of these planes at the point D. Thus we obtain for each side a rectangular triangle, wherein two sides are known, namely, for the side cb, the hypothenuse ed, and the side e D. Thus raising from the point D an indefinite perpendicular, if from the point e with eB for a radius an arc is described cutting the perpendicular in d, and the line de be drawn, the angle de D will be that sought, and will be the same for the three sides if the polyhedron be regular; otherwise, if it is not, the operation must be repeated for each.

1197. These angles may be obtained with great accuracy by taking de, or its equal e B, for the whole sine; then de eD:: sine sine 19° 28', whose complement 70° 32′ will, if the polyhedron be regular, be the angle sought. In this case, all the sides being equal, and each being capable of serving as base, the angles throughout are equal. In respect of the cube (figs. 479. and 483.) whose faces are composed of equal squares, and whose angles are all right angles, it is evident that no other angles can enter into their combination with each other.

1198. To obtain the angle formed by the faces of the octahedron (fig. 480.) from the points C and D with a distance equal to a vertical dropped upon the base of one of the triangles of its developement (fig. 484.), describe ares crossing each other in F; and the angle CFD will be equal to that formed by the faces of the polyhedron, and will be found by trigonometry to be 70° 32'. In the dodecahedron (fig. 481.), the angle formed by the faces will be found by drawing upon its projection the lines DA, and producing the side B to E, determined by an arc made from the point D with a radius equal to BA. The angle sought will be found to be 108 degrees.

1199. For the icosahedron (fig. 482.), draw the parallels Aa, Bb, Cc, and after having made be parallel and equal to BC, with a radius equal thereto, describe an arc cutting in a the parallel drawn from the point A; the angle abc will be equal to that formed by the sides of the polygon, which by trigonometry is found to be 108 degrees, as in the dodecahedron.

1200. For the pyramid with a quadrangular base (fig. 487.) the angle of each face with the base is equal to PAB or PBA, because this figure, which represents its vertical projection, is in a plane parallel to that within which will be found the perpendiculars dropped from the summit on the lateral faces of the base.

1201. In order to obtain the angles which the inclined sides form with one another, draw upon the developement (fig. 488.) the line ED, which, because the triangles PEC, PCD are equal and isoceles, will be perpendicular to the line PC, representing one of the arrisses which are formed. Then from the point D with a radius equal to DF, and

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