CARPENTRY. 



521 



PtATI 



L XVI 1 1. 



Comtruo at follows: Take any convenient straight line for 

 the primary abscisga as a base, and describe the re- 

 gulating section ; describe the seat of the envelope 

 in position to the primary abscissa ; divide the re- 

 gulating arc into any number of equal parts ; extend 

 the parts on any convenient straight line for the se- 

 condary abscissa ; draw lines through the points of di- 

 vision in the regulating arc, at right angles to the pri- 

 mary abscissa, so as to cut both sides of the seat of 

 the envelope; through the points of division, in the 

 secondary abscissa, draw lines at right angles to the 

 aid abscissa; then taking the ordinates of the pri- 

 mary abscissa and placing them from the secondary 

 abscissa on the indefinite perpendiculars, their extre- 

 mities will give the points through which the curves 

 or sides of the envelope will pass. 



Examples. Let A BCD, Fig. 5. No. 1, represent the 

 quarter of a cylinder ABE, a section at right angles 

 to the axis projected on a plane, orthographically, 

 with its axis parallel to the said plane ; let the right 

 angled triangle A By represent a part to be covered. 

 In this case, AB and B/ form a right angle ; make 

 ABC, No. 2, equal to AB/, No. 1 ; then ABC will 

 be the seat of the envelope, AB the primary abscissa, 

 and al.so a side of the seat of the envelope. Make 

 A BD the regulating section of the quadrantal cy- 

 li:idir ; divide the regulating arc BD into any 

 number of equal parts ; from the points of division 

 draw lines perpendicular to AB; continue these lines 

 to meet AC ; take the equal parts of the arc BD 

 and extend them on the secondary abscissa BE ; 

 draw indefinite perpendiculars to BE ; take all the 

 ordinati s from the seat of the envelope, beginning with 

 BC, and place them respectively on the indefinite 

 perpendiculars from the secondary abscissa BE, and 

 through the extremities draw the curve CE, then 

 will BEC be the envelope required. 



Let the obtuse angled triangle, d ef, No. 1. re- 

 present the part to be covered : make AFC, No. 3. 

 <-qual to d e"f t and AFC is the seat of the envelope : 

 produce Cr to B, and draw AB, forming a right 

 angle with BC. Upon AB describe the regulating 

 section ABD : divide the arc BD into any number 

 of equal parts : from the points of division draw 

 perpendiculars to AB : produce the perpendiculars 

 to meet AC : extend the parts of the arc BD upon 

 BE : from the points of division draw the indefinite 

 perpendiculars : take the ordinates from the seat of 

 the envelope contained between AB and AF, begin- 

 nii.g the first with BF, and apply them respectively 

 as ordinates to BE. Again, take the ordinates in 

 the seat of the envelope contained between AB and 

 AC, beginning with BC, and apply them respec- 

 tively as ordinates to BE. Through the extremities 

 of the first set of ordii.ates, draw the curve FE ; and 

 through the extremities of the second set, draw the 

 curve CE ; then will FCE be the envelope required. 

 Again, let the acute angled triangle, g h i, No. 1. 

 represent the part to be covered : it may be shown 

 in the same words, and by the same letters of refer- 

 ence, that FCE, No. 4. will be the envelope for the 

 surface g h i, No. 1. 



In the same manner, it may be shown, that FEHC, 

 Fig. 6. No. 2. is the envelope for that part efgc. 

 Fig. 6. No. 1. of the semicylinder ABCD, BE be- 



VOL. V. PART II. 



f. . 



ing the tretch out of BD A, and AFCG the teat of 

 fp C e 



Nothing can be more useful in geometrical lines ^ 

 than the method of describing the envelope of any 

 portion of a cylinder. By this method we air en- 

 abled to describe the soffits of the intradogct of any 

 arched aperture ; to find the exact situation of a line 

 on the curved surface of any rib line that would stand 

 vertically over any given line as a seat, whatever he 

 the form of the line which constitutes the seat. The 

 covering of domes with boards is exactly the same a 

 in Fig. 5. No. 4-. The dome, even though spherical, 

 may be supposed to be constituted of a number of 

 cylindric pieces, each forming an isosceles triangle on 

 the base. In this case, the part to be covered wiH 

 be contained between two oblique planes, making 

 equal angles with the axis of the cylinder : the qua- 

 drant ABD may be considered as a section of the 

 dome through the axis ; and the envelope CEF one 

 of the boards, which, when found, is a mould for all 

 the rest. Having a clear understanding of these e- 

 veral diagrams, it cannot be difficult to form an idea 

 how to cover any portion of a cylinder, whether 

 bounded by parallel planes or by any curved section 

 whatever. The falling moulds of the handrails of 

 stairs are to be considered as the envelope of certain 

 portions of cylinders. 



PKOB. XVIII. To cover the surface of a se- PLAT* 

 micylindroid, contained between two parallel planes, CXVUL 

 given the regulating section and the seat of the en- 

 velope. 



Let the regulating section be ABC, and let CDEF F'fr 7 - 

 be the seat of the envelope : divide the regulating 

 arc ABC into any number of equal arcs : extend 

 these arcs on the straight line AH: draw line* 

 through the divisions of the arc ABC, at right an- 

 gles to AC, producing them till they meet EF. 

 Through the points of division in AH draw indefi- 

 nite ordinates : take the respective ordinates contain- 

 ed between AC and DC, beginning with AD, and 

 apply them successively from AD towards H ; and 

 through their extremities draw the curve DH. la 

 like manner, take the ordinates contained between 

 AC and EF, and apply them in the same order on 

 the indefinite ordinates of AH, and draw the curve 

 EG ; then will DEGH be the envelope required. 



PROB. XIX. To cover the portion of the sur- Fig. 8. 

 face of a setnicylinder contained between two other 

 cylindrical surfaces, which have their axes passing 

 through that of the semicylinder, and to the plane of 

 the said semicylinder, Fig. 8. 



The method of proceeding with this is exactly the 

 same as in the former cases, and may be expressed 

 in the same words. It may only be observed, that 

 as the seat is alike on both sides, by finding one half 

 of the envelope the other will be given. 



PR->B. XX. To find the envelope for the frus- Fig. j. 

 turn of a sernicone, Fig. 9. Let ABCD be the seat 

 of the envelope ; AEB the end of the frustum or re- 

 gulating section : continue AD and BC till they meet 

 in F : from F, with the radius FA, describe the arc 

 AH ; and with the radius FD describe the arc DG : 

 make the arc AH equal to the arc AEB, and draw 

 HGF: then will ADGH be the envelope required. 

 For, suppose the semicircle AEB turned upon 

 3 u 



