518 



CARPENTRY. 



tive 

 Carpentry 



Construe- a sectional line oi> the given plane, and the angle 

 which the common side makes with the line of inter- 

 section of the prism. 



Produce the line of concourse BA, and the sec- 

 tional line CD, till they meet in E ; make the 

 angle BEF equal to the angle which the section line 

 is to make with the line of concourse ; from any 

 point F in EF draw FG perpendicular to EB, cut- 

 ting EB in G, and GH perpendicular to EC, cut- 

 ting EC at I ; from E, with the distance EF, de- 

 scribe an arc cutting GH at H, and join EH ; in 

 the boundary of the base take any point k ; draw k I 

 parallel to EF, cutting AB, or AB produced at /; 

 draw I in parallel to the side BC of the plane, cut- 

 ting CD at m ; draw m n parallel to EH ; make m n 

 equal to Ik, and n will be in the boundary of the 

 section required : In the same manner all other ne- 

 cessary points are to be found. 



Examples. In Fig4. the base being a rectangle, the 

 determination of the point n corresponding to one of 

 the angles is sufficient ; for being joined to the point 

 C in the opposite side, two contiguous sides of the 

 section will then be formed : The whole section will 

 be inclosed by drawing the other two sides parallel 

 to these sides. 



In Fig. 5. the base consists of straight lines, and 

 therefore the ordinates are taken from the angles, and 

 the points found in the section joined by straight lines. 



In Fig. 6. the base is wholly a curve, excepting 

 the line of concourse ; therefore drawing a curve 

 through the points found in the section completes 

 the boundary or inclosure. 



Fig. 6. 



PLATE 

 CXVI. 



Fijf. 9- 



CONIC SECTIONS. 



PROB. VIII. To find the section of a cone cut by 

 p. XVI * a plane at any given inclination to the axis, or in a 

 Fig. 7, 8. ^. ^^ pos j t ; on to t } ie s ides of the cone, Fig. 7- and 8. 

 Let ABC be a section of the cone through the 

 axis, ABD the half base, and let MN be the position 

 of the axis of the section, in respect to the side AB 

 of the cone ; now, to find any point in the curve, 

 take any point D in the circumference of the base ; 

 draw DE perpendicular to the diameter AB, cut- 

 ting it at E, and EC cutting MN at g; draw gh 

 parallel to AB, and g i perpendicular to MN ; make 

 E/ equal to ED ; draw /AC, make gi equal to 

 gh, and i is a point in the'curve. In like manner all 

 other points k and I are to be obtained, and thus the 

 curve may be completed by tracing it through the 

 points. 



In Fig. 7. MN, the axis of the section, cuts the 

 other side BC of the cone ; and the section produced 

 from this position is an ellipse. In Fig. 8. the axis 

 MN is parallel to the side CB of the cone ; and the 

 section produced is therefore a parabola. In the same 

 manner the hyperbola may be found ; it is only fixing 

 upon the position of the axis. 



CUNIOIDAL SECTIONS. 



PROB. IX. To find the section of a cunioid cut 

 by a plane perpendicular to the axal triangle. 



Let ABC be the axal triangle, ADB the half 

 base, MN the axis of the section. In the circumfe- 



rence of the base take any point D; draw DE per- Construe, 

 pendicular to AB, cutting it at E ; join EC, cutting tive 

 MN at/; draw/g perpendicular to MN ; make/g ^JC2' 

 equal to ED ; and g, h, i, k, &c. are obtained. 



PROB. X. To find the section of a cunioid, cut PLATE 

 obliquely to the axal section ABC, through two CXVI. 

 given points X, Y, on the sides of the axal triangle, ft S- la 

 and any point on the surface, the seat of which being 

 given on the axal section at Z. 



Draw CZD, cutting the diameter of the base at 

 D ; draw DE perpendicular to AB, cutting the 

 circumference at E ; join XY, and draw FZG pa- 

 rallel to XY, cutting AC and BC at F and G ; draw 

 FH perpendicular to FG ; produce YX to K; make 

 KH equal to DE, and draw HI parallel to KY or 

 FG. To find any point in the curve, from any point 

 L in the circumference of the base, draw LM per- 

 pendicular to AB, cutting AB at M ; join MC, 

 cutting FG at o, and XY at n ; make JrLp equal to 

 Fo; join pn; make Kg equal to ML, and draw qr 

 parallel to XY, cutting p n at r : then r is a point in 

 the curve. In the same manner all other points s,> 

 &c. are found. 



The following is a general method of finding the 

 sections of all solids, the surfaces of which are formed 

 by straight lines constituted according to any given 

 law. When cut by a plane, the intersection and incli- 

 nation of which are given to the plane of the base of 

 the solid ; also the inclination of the axis, and the seat 

 on the said base, in position to the intersection. 



In the consideration of this subject, there are three 

 angular planes concerned. Two of these are perpen- 

 dicular to each other ; and the third joining the other 

 two, forms a right angled solid angle* the last plane 

 being the hypothenuse, and the former two the legs. 

 The base of the object, the section of which is requi- 

 red, is given on one of the legs, and this plane is rai- 

 led the original plane, and all lines drawn in it what- 

 ever are called original lines. The hypothenusal 

 plane is that which forms the section, and is called 

 the plane of projection, or sectional plane. The third 

 plane, which is the remaining leg, is called the di- 

 recting plane. The directing plane is always paral- 

 lel to the axis, or principal elevated line of the solid. 

 The section of the solid is likewise called the pro- 

 jection. In finding the projection or section, the 

 sides of the solid angle are all extended on the same 

 plane. The line of concourse of the original and 

 projecting planes is called intersection. As the two 

 sides and hypothenuse are in contiguity when spread 

 out on a plane, the side of the original plane, and the 

 projecting plane which join each other, will be sepa- 

 rated, that is, what has been called the intersec- 

 tion ; to distinguish the one from the other, the 

 side of the original plane which meets the project- 

 ing plane is called the intersection, and the side 

 of the projecting plane which meets the base the 

 co-intersection. The intersection of the vertical plane 

 with the original plane is called the director ; and 

 the intersection of the plane of projection, and the 

 directing plane, is called the directing line. Thus in 

 Plate CXVII. Fig?. 1, 2,3,4, ABC is denominated PLATE 

 the original plane, ABD, the directing plane, DBE CXVII^ 

 the sectional plane, or plane of projection, or project- **S- Ji 2 > 



is * 



