ROOF. 



points either from the roof itfelf, or other immoveable places. 

 There are three points for this purpofe ; if the rafters have a 

 tie-beam below, that is, at the vertex, and at the two ex- 

 tremities of the rafters, the triangle being immoveable at 

 the angles, every force applied there tends either to com- 

 prefs or extend the fides of the frame without tranfverfe 

 ftrain. 



Examples. — Let it be required to divide each of the rafters 

 into three equal lengths, in order to fupport two purlins on 

 each fide; this may be done, as mjig. 3, by pieces, C E, 

 C D, A G, A F, reaching from the two lower angles Cand A, 

 and to the oppofite fides of the rafters A B and B C, inter- 

 fering each other at H and I, and halved upon each other 

 at thefe interfeftions : this mode prevents the rafters from fag- 

 gino-, but does not afford any fupport to the tie-beam. The 

 meeting of fo many braces at the fame point, too, gives little 

 opportunity of making the ends entirely fecure, even though 

 afiifted by iron ftraps. Another mode may be by intro- 

 ducing a king-poft, B K, Jig. 4, to which the ftrutts D H, 

 EI, FM, G L, may be firmly joggled at H, 1, M, L, 

 and mortifed to the rafter at D, E, F, G : this method keeps 

 up the middle of the beam, but when the roof is low, and 

 the fpan great, the ftrutts, D H and F M, require them- 

 felves to he fupported, and are much too oblique to prevent 

 a change of figure. 



Another method may be as mjig. 5, with the king-poft 

 in the middle, as before, two queen-pofts under the rafters 

 at E and F, two ftrutts, H E and I F, joggled to the 

 bottom of the king-poft at H and I, and to the top of the 

 queen-pofts at E and F ; and in order to fecure the points 

 N and P, two other braces, N Q, P R, are joggled to the 

 bottom of the queen-pofts at Q and R, and mortifed into 

 the rafters at trie upper end. This conftruftion fupports 

 the tie-beam in three different points, and each of the rafters 

 in two. The timbers are much ihorter than thofe of the 

 preceding ; but fo many joggles are certainly an objection to 

 this method, as the fhrinking of the timber muft be very 

 confiderable in three breadths, which would allow the roof 

 to defcend. When the fpan is great, and the points to be 

 fupported many, an excellent method may be as injg. 6, 

 where there are two arches of caft-iron or good Englifh oak 

 introduced, which abutt on the king-poft, and at the other 

 extremity at the ends of the beam. The rafters and the beam 

 by this mode may be fupported by as many equidiftant 

 points as we pleafe. 



When the tie-beam is removed from the bottom, as in 

 fig. 7, and no fixed points are to be found from below, a lon- 

 gitudinal trufs may be conftrufted, the end of which is 

 lliewn at A B, and the manner of framing it v^fg. 9, the 

 two ends being fuppofed to be firmly fixed into the gables ; 

 but where the length is great, the form of Jig. 10, with a 

 parabolic arch, would be much better : by this method, the 

 rafters will be kept nearly in the fame plane, and all lateral 

 prefiure from the walls will be removed ; for it is evident 

 that if the ridge-tree is fupported, there can be no motion 

 downwards in the direction of the rafters, the whole roof 

 being hung to this longitudinal frame. 



Prop. XI. 



If a roof be conftrufted with two equal rafters, A M, 

 CM, {Jig. 8.) and if a tie extend from the bottom of each 

 rafter to an intermediate point in the oppofite rafter, and the 

 ties halved together at their interfeftion B, forming with 

 the rafters a quadrilateral M D B E at the vertex, and two 

 triangles A D B and C E B ; then if M D is equal to M E, 

 and if C P reprefent the direction and quantity of force 



Therefore L K : 

 S. 



on the wall at C, the force tending to break the rafters at 

 „ ,„. S.PCKxS.DMExLKxDM 

 D and E . - s.xcKxR 



For complete the parallelogram PCLK; make M N equal 

 to C L, and draw N O parallel, and M O perpendicular to 

 A M. Now the triangles C B E and A B D may be looked 

 upon as folid levers, ( at leait with regard to forces applied 

 to the angles,) moveable round B. Then the force C P will 

 communicate the force C L to the rafter, and C L is the 

 power ailing obliquely at M, upon the rafter A M : then 

 becaule N O is parallel, and O M perpendicular to A M, 

 O M is the force tending to break the rafter at D ; O N 

 that pufhing it towards A : let M N be confidered radius, 

 then O M will be the fine of the angle D M N, or D M E ; 

 for produce AMtoQ, and the angle N M Q will be the 

 fupplement of the vertical angle DME, therefore the fine 

 of N M Q, equal to the fine of the angle M N O, is the 

 fame with the fine of N M A ; then by trigonometry, 



L K:L C:: S.L C K : S . C KL orS.PCK 

 N M or LC:MO::R:S.DME = S.MNO 

 MO::S.LCKx R:S.PCKx 

 DME. 



„ .,» S.PCKxS.DMExLK , 



Hence M O = T _. „ D — — = the 



b . L C lv x R 



force afting perpendicular to A M at M, but the force tend- 

 ing to break the rafter at D, is as the lever D M multiplied 



, . S.PCKxS.DMExLKxDM 



into this force; thatis = c T ,-, Jr — n . 



S . LC K x R 



Corol. 1. — Hence, if the angle D M E is a right angle, 

 the force tending to break the rafter at D will be 



S.PCKxLKxDM 



S.LCK 

 Corol. 2. — Hence the rafters of every roof of this con- 

 ftruftion muft fag in a greater or lefs degree, by the aftion 

 of the rafters againft each other at the point M, that is, 

 they will be bent into curves concave on the upper edges ; 

 but if a diagonal conneft the two vertical points M and B, 

 this change of figure will be prevented. 



Prop. XII. 



To remove the lateral preffure of a roof without any in- 

 termediate beam, brace, or ftrutt. 



Let AB, BC, (PiiftXLV. Jig. 1. N° 2.) be two 

 rafters, and let there be conftrufted a ftrong wall-plate 

 D E F G, N° I, firmly bolted together at the angles; then 

 if the roof is to be gable-ended, after having fixed the 

 rafters to a common ridge-tree, let two curves be made 

 of caft-iron, or good Englifh oak, of a parabolic form, 

 and let into the rafters, either on the upper or under furface, 

 and firmly fecured to them by bolts or nails, and at their 

 lower extremities to the angles of the wall-plates, the vertex 

 of each curve meeting the ridge-piece on each fide of it, or 

 nearly, as may be found convenient. One half of the plan 

 N° I. exhibits the form for the execution of a gable-ended 

 roof, and the other for a hiped roof. The two fides, laid 

 in piano for each form, are fhewn in N 3 and 4; at 

 HIKE and M N O P, H L and D G reprefent the fame 

 wall-plate ; D G, N O, and E F another wall-plate, I K 

 and M P meeting the ridge on each fide of it : but it mult 

 be obferved, when the roof is to be hipt, that the ridge-tree 

 muft be very ftrong, as the compreflure will be very great, 

 the hip-rafters afting like powerful braces at the extremities 

 of it. Hence it is evident that the wall-plates aft as the tie- 

 beams 



