gQ ls ^ ie centre to receive the side 6 times, d seven times, andon , to *, which is the centre to draw the circle to receive® side twelve times.”
ag r 118 problem should have been classed in practical geometry,in tb' laS n ° re ^ erence whatever to carpentry. It is only truedes • hexa S on an< i dodecagon, and is very incorrect for the, cri ption of polygons upon a given straight line, whichm It ^ ewer sides than six. In his diagram, he shows thethod of describing the hip from the common rib being6 5? 5 but the text contains no description of it.hr, ^ re * s a dome, whose plan is a hexagon, and showsfo i° ^' v *de a circle into any number of parts: divide one-rth of the circle into the number of parts you would havecircle, as, 1, 2, 3, 4, 5, 6, and always take four of them,op.jd the backing of the curve-line hips, lay down the planle hip at the angle, as a; then take the distance 1, 2, atbv •° tn ’ tac k * n a na 'h then shift the hip-mould, and markingo^i'*’ as 12, 34, 56, 78, 9, 10, will show the wood to come<jjy. . this description, Mr. Pain proposes to find thei>ot' Sl ° n ^ a c i r< d e i nt0 any number of parts ; but as he doesat r i IIlent i on any fixed ratio of these parts, they may be takent 0 " eas ure, without rule; but allowing that he had neglectedthis condition, under which the circle was to beatl( j ea > and suppose that he meant the parts to be equal,rul e 0 h e found by a general method ; there is no regularIp performing this problem but by an approximation.W e ® allow that the quadrant can be divided into equal parts,t; 0leg Uat also allow that the whole circle will contain fouriot 0 tlat number of parts ; but the division of the quadrantcirc -ml parts is equally impossible with that of the wholeonly ’ ^ le ru le is therefore absurd, and consequently it isone absurdity by another, and wastingla st r 8 ^ me to no purpose. In this problem, as in theis f’ e "as also neglected to inform his reader how the hipth e ^ lou gb it is sufficiently clear on the plate to presentea °f forming it to any intelligent person,the op l- e “ is a dome on a circular plan ; a and b show® Ion °f the horizontal rib.”
the tL “' s a dome on an elliptic plan ; the centres for
the pi. U a °f the horizontal ribs d d, are a a, b b, c c, d d;lines f, C<! that rib on the plan is found by dropping dota skyi| 0 .? rn „ t ^ le sections d d: c c on the top, is designed forl‘l a £, ^ These descriptions are unintelligible.
^ardst* II - F l 0 ure 1) “ shows the method for cutting theParts as ° Cover the dome; divide the dome into as manythe e ,j y°tt think it will take boards, and draw lines to cutthat i a 0 eac h board, and where they meet the centre line,°he i tlc , e , cen t re for the edge of each board. This is drawn? es crib e( j l a ^ 00t- ” This has already been shown andb 0ard 3 as p . van > but he is deficient in not representing theev er, tho ^ as ^ one 5 the description of the former, how-v figure 9 ?^ ort ’ I s mu ch clearer than that of the latter,bat t], e , > Is taken from the Practical House Carpenter,l | e pj a * es<a 'iption is to be found neither in the text nor on£® a der t 0 g Tjb.^ast therefore be left to the sagacity of thebe ftiethodp 1 *' ou k We suppose the diagram to representb® Joint s o °* C0 7 erin g a dome by bending the boards witha ®ttiod h a n , Vert * ca l planes passing through the axis. Thisathor S; 0 f a , rea dy been shown by several of the preceding, a d diff er ' v *om Price and Langley are the most accurate,° ar <l, h ut os b as to the mode of ascertaining the form of thest^^ley C ^t me to the same result at last.
•gm ,. it-—, “nvi j.ixcc, penuiin uue operation uy
^'aateg. <llnates ’ whereas Langley and Pain do it by curvedr a ®y as t]' 1 „° ne ) met l 10c l ascertains the form with as much accu-e< lUi res other, but the operation with straight ordinatesch less trouble than the other. Pain even shows
many more lines than are sufficient, which superabundancemakes his diagram much less intelligible to the understandingof his reader. The concentric arcs, which he has used asordinates, are erroneous in principle, though the use of themdoes not affect the practice, as we shall here show.
Figure 3. “ Let A b c be half the section of a dome passingthrough its vertical axis; divide the curve A b into as manyequal parts as the number of boards of which the coveringis to consist; through each point in the circumference drawa line to the centre C; and through each of the same pointsdraw another line at right angles to the respective radii, andproduce them upwards, as also the axis C b, so as to cut eachof the tangents from the several points in the curve; theneach of these tangents, so limited, are the radii of the suc-cessive ordinates; the tangent at the bottom is of infinitelength, the next is limited, and the succeeding ones becomegradually shorter and shorter, till the tangent and the arcbecome nearly of one length. So that the ordinates of theboard, exhibited in Figure 4, are arcs of radii, respectivelyequal to the tangents; consequently, the bottom ordinate ofthe board is a straight line, the next is the arc of a circleof a very flat curvature, the next is an arc of greater curva-ture, and so each arc becoming quicker and quicker in itscurvature, till they reach the summit of the board, which isthe last centre.”
The most eligible method in practice, founded upon evi-dent principles, is to suppose the dome to become an equi-lateral and equiangular polygon, and suppose the axal sectionABC, Figure 5, perpendicular to one of the sides of the dometo be given, and the curve A b to be divided into equal parts,and suppose the parts to be extended upon b c produced;then, if lines be drawn through the divisions of the curve,and through the points of division in c b, produced perpen-dicular to the said b c ; and if b d be drawn perpendicularto b C, equal to half the width of a board, and d c joined,and the ordinates produced so as to meet d c ; then, if thelines parallel to b d, contained within the triangle b d c besuccessively taken towards c, and applied on the perpen-diculars from and on each side of C d produced; then if curvesbe drawn through the points at the extremities, they will ter-minate the edges of a board, which will accurately cover aside of the polygonal dome.
Let us now suppose the number of sides of this dome tobe very great, then the sides will vary only in a very smalldegree, either from the inscribing or circumscribing sphere,and this variation will decrease as the number of sides isincreased, and the excess or defect would become insensible ;and if we suppose the sides to be infinite, the board whichcovers the side of the polygonal dome will accurately coverthe inscribing or circumscribing spherical dome. This lattermethod is founded upon principle, but that of forming thebottom ends of the boards into curves is totally destitute of it.
Figure 6, from Pain ’s British Palladio, which he says “ isa pentagon to be covered with a domical roof. To find thecurve of the boarding, divide the girth or curve of the rib onthe back into as many parts as you please, as here into four,and draw them to the base-line of the rib e a, as 7, 8, 9;stretch out a b, the middle of the side, and 9 9, 8 8, 7 7,parallel thereto; then take one of the divisions on the girthof the rib, and set off from e to 7, 8, 9, b, and where thatcuts the lines 7 7, 8 8, 9 9, b, there tack in nails, and benda thin slip to the nails, and mark from e to b, as that curvedirects; this will be one edge of the covering: prick thesemarks on the other side of the line a b, and proceed as before;then will the covering or boarding be complete. The cover-ing or boarding of (Figure 7) is found in the same man-ner, which is very plain to inspection; the girth of” the