522 



CARPENTRY. 



PlATE 

 CXVI1I. 



Fi. 10. 



Construe- AB at a right angle with ABCD, and the envelope 



tive ADGH bent and turned upon AD, so that the arc 



"7* AH may coincide in all points with the semicircle 



AEB, and HG with BC, the arc AH will have 



AB for its seat, and the arc DG will have DC for 



its seat : therefore ADGH will be the envelope for 



the seat ABCD, which is the axal section of the 



frustum. 



PROB. XXI. To find the envelope for the por- 

 tion of a semicone terminated by two cylindrical sur- 

 faces, which have their axes perpendicular to the plane 

 of the semicone, and passing through the axis of the 

 said seniicone, Fig. 10. 



Let ABCDEFA be the seat of the envelope : 

 join AC ; and draw Gti a tangent to the arc FED, 

 cutting AF and CD produced at G and H ; then, 

 by the last problem, find the envelope A I KG A to 

 cover the frustum of the cone, whose seat is 

 ACHGA : then having marked the equal parts on 

 the quadrant AL, and the same parts on the arc 

 AI, which will be double in number to that contain- 

 ed in AL, draw lines through the points of division 

 in AL, at right angles to AC. From the cutting 

 points in AC, draw lines to the centre M of the 

 cone : likewise through the points of division in the 

 circular arc A I, draw lines again to M: from the 

 points in the arc AB, and from those in the arc FE 

 intersected by the lines drawn to M, draw lines pa- 

 rallel to AC, to meet AM : from the points of meet- 

 ing describe arcs from M as a centre, so as to inter- 

 cept the first, second, third, &c. drawn from the di- 

 visions in AB to the point M, taken in the same or- 

 der from A towards B, and from F towards E : 

 through the points thus intercepted, draw the curves 

 AN and FO ; and AFON will be the half of the 

 envelope. Make the other half NOPI similar by 

 the same method, or any other; and AFOPINA 

 will be the whole envelope required. 



PROB. XXII. To find the envelope of the frus- 

 tum of a semicunioid, Fig. 11. 



Let ABCE be the seat of the surface to be cover- 

 ed ; let DEC, the section of the lesser end, be a semi- 

 circle, that of the greater end a semi ellipse, and each 

 of the same altitude, or NE equal MF. Produce AD 

 and BC to mt-et in G. To find any point t, in the enve- 

 lope, proceed thus : In the semicircumference DEC, 

 take any point p very near to the extremity D of the 

 diameter DC ; draw pq perpendicular to DC, cut- 

 ting it in q ; join qG ; draw GH perpendicular to 

 AG ; on GH make G* equal to qp ; extend the 

 arc Dp; from D, with th : s extension as a radius, 

 describe an arc at t ; from s, with the distance Gq, 

 describe another arc, cutting the former at t. In 

 like manner, by taking the point v very near to p, 

 draw viv perpendicular to DC. cutting DC at w ; 

 join Gtv; make Go equal to 'on \ extend the arc 

 pv, and from t, with this extension, describe an arc 

 at y ; from o, with thr distance Grv, describe an- 

 other arc, cutting the former at y and ij will be ano- 

 ther point in the curve. In this manner all succeed- 

 ing points to the centre j will be found ; so that 

 ~Qt'y t &c. will be the curve, which will correspond 

 with the semicircle DEC ; join st and oy> and pro- 

 duce them to u and z\ also produce Gq and Grv to 

 meet AB at n and x ', make tu equal to q n, and y z 



Kg. H. 



equal to rvx\ then draw the curve Dly, &c. to J, and Construe- 

 the curve Auz, &c. to the middle, and this will com- tive 

 plete one half of the envelope ; the other half being J^P en 

 joined upon the same base, and made equal and similar, 

 will complete the whole envelope ADKLA. 



To find the envelope of a part of a semicunioid, P LAT * 

 contained between two cylindric surfaces, having S^ 

 their axes perpendicular to the triangle passing through ' 

 the axis of the cunioid, and intersecting that of the 

 cumoid, Fig. 12. 



Find the curve line nopq, which corresponds to 

 the semicircumference ot the semicircle, as in the last 

 problem, the straight line aiklm being a tangent, 

 and parallel to the chord on the concave side ; make 

 all the distances bf, eg, dh, &c. respectively equal to 

 in, ko. Ip, mg, tVc. ; draw the curve efgh, and it 

 will be the edge of the envelope, which will have 

 enopq, &c. for its seat ; and \ffs be made equal to 

 nv, gt equal to ow, hu equal to qy, and the curve 

 rstu, &c. being drawn, will give the other edge of 

 the envelope, which will have the arc vivxy, &c. for 

 its seat. 



These two last problems are exceedingly useful in 

 all kinds of arched work that is circular on the plan, 

 the intradoses of the arches splaying on the sides and 

 level at the crown, particularly in sash work : but 

 it is to be regretted, that no accurate method for find- 

 ing the envelope has yet been discovered ; every at- 

 tempt to greater accuracy than the above has been 

 foiled. There is one method, however, that will 

 give the envelope, or veneer, exactly true, by the use 

 of a centre, and describing the lines on its surface 

 the same manner as in plans, then applying the equal 

 distances from the seats from a graduated pole, which 

 is the vertex of the cunioid, that is, by making the 

 distances equal to pq, vw, &c. Fig. 11. or by draw- 

 ing the equal ordinates on the ends of the centre, 

 and joining the level lines, and taking all the distances 

 from their seats. 



BOOK III. 



Constructive Carpentry. 



CONSTRUCTIVE Carpentry shows the principles Construct 

 and mode of combining individual pieces of timber tive car- 

 into one complete frame, which shall form an essential pentry. 

 part of a building. 



The knowledge of this configuration depends on 

 the properties of the several solids which are to con- 

 stitute the whole, and must be sought for in the prin- 

 ciples of Stereography, which we have already illus- 

 trated. The angular ribs of domes, groins, roofs, 

 &c. ; the art of hand railing, sashes, and archivolts 

 with circular heads upon circular plans, depend prin- 

 cipally on the sections of solids. By the coverings 

 of bodies, the under sides of groined ribs and the front 

 ribs of niches are formed to their plans ; the falling 

 moulds of winding stairs are adapted to the spiral 

 line of the steps, and the archivolts of arches to the 

 plans o'f the apertures. By the angles which the ar- 

 rises make with the adjacent sectional lines, we are 

 enabled to cut prismatic or pyramidal bodies., so as 



