N A K
226
NAK
are braced by others, or themselves are bracers ; whence thearea is extended less widely, as by the paucity of the raftersit extends from wall to wall by the rafters s Sf, QP, u y, lit,e s and z Sf : but so long as there is occasion for only fourrafters, the construction may be similar to Figure 4, or whenof only three, as in Figure 5, which is the most simple formof any.
“ But even if there were occasion ( Figure 1) the raftersx E, w v, (and the remaining shorter ones that terminate inthe wall) might be produced to an equal length with theothers, and would support the corresponding rafters parallelto the principal beams, in which are fixed x e, v TV, and Y u;and thus the ends of these, continued, would either be sup-ported by the beams lying upon the wall, or even by others,as far as the area is extended, paralleFto m i, and p Q ; andso on, to any distance that the work may require. For inthe continued area,the numberof the rafters being augmented,the weight may be increased to as much as the walls are ableto sustain ; for rafters may be found that will not give way,nor break under any weight the walls will carry.
“ But although we are able to proceed safely in this regularproportion, it is from calculation that we must discover theweight which such a piece of framing will bear. For if itappears, from the quantity of weight laid on, that the rafterswill everywhere bear it, then as much of the weight as anysituation, or place of the area, has laid upon it, being in likeproportion also laid on another, it becomes regularly andfairly distributed over the whole; and hence is the best judg-ment formed of what may safely be done. The calculationwhen the number of timbers is few, is pretty easy ; but morelaborious where there are many.
“But I shall show by what method this may be effected ;each rafter of the continued framing, joined to its fellow, issubjected to this calculation ; perplexed in a manner, it istrue, from their number, but producing the result much morerapidly where there are few rafters.
“ Calculation accommodated to the raftering now ex-plained.
“That the calculation delivered in the following synopsismay be more clearly understood, it is to be observed, as isindeed sufficiently indicated to the eye, that some of therafters are longer than others ; and that the longer are aboutone and a half of the shorter ones. On this account we shalldesignate the weight of each of the longer rafters by x ; andthat of each of the shorter by | T.
“ But the weight upon every one of the joints of therafters, will be indicated by the letter written on that jointin the diagram.
“Let it be moreover observed, since it may tend to eluci-date our subject, that there are always four points which, onaccount of their similarity of situation in regard to the wholeframing, equally determine the weight laid on them : thesewe shall always designate by the same marks, lest the num-ber of the symbols should increase, and the calculation bethereby perplexed. Hence we shall have four a’s, similarlysituated and equally loaded, the same number of b’s, and soon with the others. Whence it will be, for example (toNo. 2, or the following equation) b = 1 t + § C + A a ; inthe same way, also, if the point, b, of the rafter, ab, thussubjected, be said to sustain as much weight as is laid uponit; besides that firmness which is necessary to prevent thewood giving way from the weight it carries, as this equationdoes not involve it; so much then is ^t (one-half of theweight lying on A B, one of the longer rafters) ; then ■§ c(two-thirds of the weight of the same a b, situated on thepoint c ; seeing that in C, A, it is understood to be cut inthree different ways by the points) ; then J A (one-third of
. -iY)ien ce
the weight of the same rafter a b is placed in A.)also there will be (to No. 21.) w = J t + ^ v : ^ er , e ]abtdiscover that in the point w of the rafter l m, the ofplaced upon it is as much as is J T (one-half of the w ’ el ® n e)»the short rafter, which is also a third of the long er f t i,ethen ^ v (one-half of the weight, which is placed ° ve j ) ; c li,middle point, v, of the rafter so lying upon this one) iby similarity, enables us to obtain 25 in the first“ The equation 26, and those which follow, arefrom the preceding, which we have cited by numbers ^ g( , e g!for those equations ; whence the manner of the whole p 1 ^may be more clearly deduced. These partly ser r andabbreviating fractions, as often as the numerato g jdenominators may be divided by the same common ‘ t j,ebut their most essential use is, to reduce and exp Iapreceding equations, by substituting the value of ever ^ v iiicl'vidual symbol, and explaining it by other marks, ° n unl l) efaccount, as often as any symbols are expunged, the gofl) eof those which remain is sensibly lessened, till * :lS eS ge&one of the symbols unknown from the beginning, is eX j. a f t e>' 9 ’by the known quantity t, the weight of the longer , e iffconsidered simply by itself; and then, by examine -pinSstep of the preceding equations, the values of the resymbols are also found first of the unknown weigh*;*- ^ j C“ For example, when there is (No. 1.) a = \ x + f .fio' 1 /(on account of a being found on both sides of *he cQ t |, e reit is manifest, that on taking away •§ A on both S1 . j. bo** 1will remain jA=|x + $C; that is (by multiply 1 ?^^ fi 1 '*sides by 3) A = x + c; and thus we have (N®* j)equation, as derived from the first equation (N°-which we have cited. i T +
“By similarity, since it may be (No. 2.) B == a | s+ A A ; and (by No. 26) it may be A = § T + C > a |C= £ X + 1 c ; B will be(=ix+fc + $A = i** (»*
I x + J c) = X + c. Whence we have this equa ^ Ji)>x + c) No. 27, as derived from No. 2 and 26. nmanner for the others.
“But, at no time, as we have said, is there * n01 ® eS ol* 9 ^abbreviation of the fraction, as a reduction to le sSa common divisor. As when (No. 64) we have •
_ 6912 x + 672 g + 552 i + 2790 ,, the n al1
n ~ 2907 = 3 x 969 ’ an
members can be divided by 3 ; we have No. 65 ( as ^ erl
+ 1841 + "
No. 64) the equation n =and so for others.
2304 x + 224 G696
of
ation 9 u '{ e *’
,uu — j thes e
equations are deduced singly, as we have deduce ^ s yiicP'. e r
“But as all these reductions and abbrevia'u --^ e i
we shall here briefly express them all in a con tin t j j6indicating those antecedent equations upon n't 1 *depend.
“ Synopsis of ihis Calculation.— Fig ureNo.
1. A = iT + fA + |0,
2. B = lT + fC+iA.
3. C=iT + fG + ^I.
4. D = 1 T + f I + l Q.
5. E=|T + fB + |D.
6. f = ^t+|d-(-|b.
7. g=^i + |e-(-|l
8. h=£t + $l + l E.
9,i=1t + |n + |p.
10. IC = | T + | P + l N.
11. L = It +. § xv+ f x.