CARPENTRY. 





ATI 



ing plane; CB is termed the intersection, AB the 

 director, DB the directing line, and BE the co-inter, 

 section. 



It tin- object, the section of which is required, be 

 a pyramid or cow, and a plane be supposed to be 

 drawn through the vertex parallel to the base, until 

 it Mitu-scct the projecting plane, the line of intersec- 

 tion is called the vanishing line. 



The inclination of the sectional plane and the 

 intersection, and the seat of the inclination of the ob- 

 ject, the section of which is required, being given, 

 the director AB may be any line parallel to the seat 

 of inclination. To determine the directing plane 

 and th;- sectional plane, proceed in Fig. >J. thus : Let 

 ABC be the original plane, BC the intersection, AB 

 tlu- director. From any point C in BC, draw CA 

 perpendicular to BC, and Ag perpendicular to CA. 

 Make the angle ACg equal to the inclination of the 

 sectional plane. Draw AD perpendicular to AB, 

 eqinl to Ag join DB, and ABD is the directing 

 plane. From B, with the distance BC, describe an 

 arc at F ; and from D, with the distance gC, de- 

 scribe another arc, cutting the former at F. Draw 

 BF, and DBF is the sectional plane. 



In the projection of solids in general, the nature 

 of the solid, or its elementary construction, must be 

 considered. Suppose the solid to be a pyramid or 

 cone, and a plane to pass through its axis until it 

 meet the directing plane, the original plane, and the 

 plane of projection. Place the height of the pyra- 

 mid on the line where this plane meets the directing 

 plane, and let this line be called the director of the 

 axis. Let a line be supposed to be drawn from the 

 upper end of the director of the axis to the vertex of 

 the pyramid or cone ; this line will be parallel to the 

 original plane, and in the same plane with the line 

 formed on the original plane, by the aforesaid plane 

 passing through the axis. Suppose any number of 

 planes to pass through this line, to any number of 

 points in the base of the solid, to cut the original 

 plane, the directing plane, and the sectional plane ; 

 the intersections of the several planes thus drawn, will 

 form parallel lines on the original plane, converging 

 lines from the intersections of the parallel lines to the 

 summit of the directing axis, and lines will also be 

 formed on the sectional plane, meeting the parallels 

 on the original plane at the intersection at one of their 

 extremities, and the converging lines at the points 

 where they meet the directing line at the other ex- 

 tremity. The planes which thus generate these three 

 sets of lines, will cut the surface of the solid in straight 

 lines passing through its vertex. Again, if there be 

 taken another directing axis, and another set of planes 

 be supposed to pass through the line which extends 

 between the apex of this axis and that of the pyra- 

 mid or cone, and through the several points taken be- 

 fore in the base of the object, three other sets of lines 

 will be formed in the same manner, and will cut the 

 sides of the pyramid or cone in the same intersec- 

 tions as at first : also the last set of lines in the plane 

 of projection will intersect the former in the very 

 same points, as lines drawn from the several points in 

 the base of rlw object before taken to its vertex. If 

 the section required be that of a prism, draw the 

 plane parallel to the inclination of the prism, instead 



of planet interacting each other on the ln.c joining Contruc- 

 the apex of the pyramid or cone, and the top of the "*' 

 pex line. The following problem*, in particular 

 examples, wi'l illustrate these general description* and 

 definitions. 



PHOII. XI. To find the keclion of a pyramid, PLAT* 

 placed with one of its sidci parallel to the mtertec- C * V|1 - 

 lion. ' 



Let the parallelogram IFGH be the bate of the 

 pyramid, with the sides FI and GH parallel to CB ; 

 draw the diagonals GI and Fil intersecting at K, 

 and produce FG and Hi to the inters ctiou at O 

 and Q, and draw KP parallel to them, meeting the 

 said intersection at P ; make Bry, B/>, Bo, respec- 

 tively equal to BQ, BP, BO ; produce FI and GH to 

 cut AB at M and N, and draw KL parallel to them, 

 cutting AB at L ; draw LR, making an angle 

 with AB equal to the angle which the axis or line 

 from K to the vertex makes with the pbne of the 

 base ; make LR 'qual to the length of the said 

 axis ; draw RD parallel to AB, cutting BD at 

 draw TDU parallel to BE, then LR will be the di. 

 rector of the axis, and TU the vanishing line ; draw 

 M/H, L/, MM towards R, cutting BD at n, I, m ; 

 also draw mif y Ik, nhg parallel to BE; make L/C- 

 equal LK ; draw pic, and produce it till it meet 

 TU at S ; draw ogf S and q h i S, and fg h i will 

 be the projection of FGH[, or the section of the py- 

 ramid, the base of which is FGHI. The reason why 

 mif, /, nhg are drawn parallel, is, because that 

 any section made by a plane through ar.y line on the 

 original plane parallel to the intersection, will cut the 

 sectional plane also parallel to the intersection. 



PROB. XII. To find the section of a pyramid, when pig. fl* 

 the base stands at oblique angles to the intersec- 



ton. 



Let the rectangle FGHI be the base ; produce 

 IH and FG to Q and S ; FI and GH to N and P 

 in the intersecting line ; through K, draw LR pa- 

 rallel to FG or IH, cutting AB at L, and CB at 

 R ; also through K, draw KM parallel to the inter- 

 section CB, cutting AB at M ; draw Mm, making 

 the same angle with AB as the axis makes with the 

 original plane. Draw m k parallel to BE ; make mft 

 equal to MK ; draw KO parallel to FN, cutting 

 CBat O; make Bn, Bo, B/>, Bg, Br, BJ each re- 

 spectively equal to BN, BO, BP, BQ, BR, BS; 

 draw rk and ok; draw Lu parallel to Mm, and 

 make L equal to the length of the axis of the py- 

 ramid ; draw D parallel to AB, cutting BD at D ; 

 draw VW parallel to BE ; produce rk and ok to V 

 and W ; draw sgfV, q h iV, also p hgW, n ifW : 

 Then wiling h i be the section of the pyramid re. 

 quired. 



From what has been now said, Figs. 4. and 5. will Fig. 4, . . 

 be understood, and may be described in the same 

 words as in Fig. 1 ; only it is to be observed, that 

 the different points in a cone are formed in the same 

 manner as the angles, and as many as may be thought 

 necessary for tracing the curve. Fig. 4. shows the 

 section, when the axis of the cone stands perpendicu- 

 lar to the plane of the base ; and Fig. 5. for a cone 

 which has its axis inclined to the base. 



PROB. XIII. Given the seat D of the vertex, 

 the intersection AC of a triangular plane, also 



