M E C

156

MEC

for the same reason, c u is also in a state of compression; foreither a or b may be considered as the point that is impelledor withheld. Therefore d a and d b are stretched, and areresisting with attractive forces, d C and c b are compressed,and are resisting with repulsive forces, db is also actingwith repulsive forces, being compressed in like manner : andthus the support of the prop, combined with the firmnessof d c, puts the frame adbc into the condition of the twoframes in Figure 18 and Figure 19. Therefore the externalforce at a is really in equilibrio with an attracting forceacting in the direction a d, and a repulsive force acting inthe direction A k. And since all the connecting forces aremutual and equal, the point d is pulled or drawn in thedirection d A. The condition of the point b is similar tothat of A, and d is also drawn in the direction D b. Thusthe point d, being urged by the forces in the directions D aand d b, presses the beam d c on the prop, and the prop resistsin the opposite direction. Therefore the line D C is the dia-gonal of the parallelogram, whose sides have the proportionof the forces which connect d with A and B. This is theprinciple on which the rest of our investigation proceeds.We may take DC as the representation and measure of theirjoint effect. Therefore draw c H, c G, parallel to da, db.Draw h L, GO, parallel to C A C B, cutting A E, B F, in L and o,and cutting da, db, ini and m. Complete the parallelo-grams ilka, monb; then D G and A i are the equal andopposite forces which connect A and d ; for gd = ch, = aiIn like manner d h and b m are the forces which connectD and b.

The external force at a is in immediate equilibrio with thecombined forces, connecting A with d and with c. AI is oneof them ; therefore ak is the other; and A L is the compoundforce with which the external force at. a is in immediateequilibrium. This external force is therefore equal andopposite to A l. In like manner, the external force at b isequal and opposite to bo ; and A l is to b o as the externalforce at a to the external force at b. The prop C resists with| forces equal to those which are propagated to it from thej points d, a, and c. Therefore it resists with forces C H, CG,equal and opposite to D G, D H ; and it resists the compres-sions K a, n b, with equal and opposite forces C p, C n.Draw p b, no, parallel to A D, b d, and draw c 6 Q, cop: itis plain thaty»cn5 is a parallelogram equal to icail, andthat C b is equal to A L. In like manner, C o is equal to B o.Now the forces C p, C h, exerted by the prop, compose theforce C b and Cn, CG compose the force c o. These twoforces ci, Co, are equal and parallel to A l and B o; andtherefore they are equal and opposite to the external forcesacting at A and B. But they are (primitively) equal andopposite to the pressures (or at least the compounds of thepressures) exerted on the prop, by the forces propagatedto c from A, d, and b. Therefore the pressures exerted onthe prop are Jhe same as if the external forces were appliedthere in the same directions as they are applied to a and b.Now if we make C v, C z, equal to C b and C o, and completethe parallelogram cvyz; it is plain that the force YCis in equilibrio with b c and o c. Therefore the pressuresat A, C, and B, are such as would balance if applied toone point.

Lastly, in order to determine their proportions, draw C Sand c K perpendicular to da and D b. Also draw Ad, nfperpendicular to C Q and C P; and draw C g, C i perpendicularto A E, B F.

The triangles C P K and bp f are similar, havinga common angle, p, and a right angle at it and f Inlike manner the triangles C Q S and A Q d are similar,j Also the triangles C H B, C G S, are similar, by reason

of the equal angles at h and G, and the right angl e3b and s. Hence we obtain the following analogies :

C 0 J

:cp = on : p B =

C G : P B

C P !

: c B =

pb:/b

C R

;cs =

ch : CG

c s

: CQ =

a d : a Q

CQ :

: c 6 = AQ:p6:LK =

AQ :ch

Therefore, by equality,

Co :ci = a d :/b

or, b o : a l = c g : c *

ortion";

That is, the external forces are reciprocally prop 0 "to the perpendiculars drawn from the prop on theof their direction. . egg

This proposition is fertile in consequences, and fu rnmany useful instructions to the artist. The strains L A < g (C y, that are excited, occur in many, if not in all fram in » gjcarpentry, whether for edifices or engines, and are the souof their efficacy. It is also evident, that the doctrine O .^ oa 'transverse strength of timber is contained in this propo 01 ^for every piece of timber may be considered as an asseffl^ ^of parts, connected by forces acting in the direction °. ; n glines which join the strained points on the matter i ^between those points, and also act on the rest of theexciting those lateral forces which produce the inflexib 1the whole. Thus it appears that this proposition co ^ fl9 [the principles which direct the artist to frame thepowerful levers; to secure uprights by shores or bra c ’ ofby ties and ropes ; to secure scaffoldings for the erec jj#spires, and many other most delicate problems of hie art ' jpsalso learns, from this proposition, how to ascertain j )6

X&

that are produced, without his intention, by pieces

vvhic 11

afl 0vel

intended for other offices, and which, by their tran 0 n jaction, put liis work in hazard. In short, this propo 01

the key to the science of his art.

There is a proposition which has been called in q ue ® 1 ljjofseveral very intelligent persons ; and they say that Phas demonstrated, in his Science des Ingeneurs, that ft .^piyfirmly fixed at both ends is not twice as strong as when jjilying on the props, and that its strength is increased [jot 1the proportion of 2 to 3 ; and they support this deterW^ ^by a list of experiments recited by Belidor , which ag rcisely with it. Belidor also says, that Pitot had , or iti eS , :results in his experiments. These are respectable aut # ni»but Belidor ’s reasoning is anything but demonstratehis experiments are described in such an imperfect ^ ^lifdthat we cannot build much on them. It is not said t |„i0manner the battens were secured at the ends, any f artthat it was by chevalets. If by this word is meant 3 gwe cannot conceive how they were employed; buthis term sometimes used for a wedge, or key. If 116 r6were wedged in the holes, their resistance tobe made what we please : they may be loose, andresist little more than when simply laid on the prop 3,may be (and probably were) wedged very fast, an j

or crippled. _ . g ; x

Figure 24. — Let l m be a long beam divided ' nt0 ^ at u gparts, in the points D, B, A, C, E, and firmly suppo'tf , ntC, m ; let it be cut through at A, and have compass J

■re'

id * 3 jft

and c ; let fb,gc, be two equal uprights,resting o^^ e 5 u jbut without any connection ; let A H be a similai a^ eX te a tpiece, to be occasionally applied at the seam A ; fa 3t . ta thread, or wire, age, over the piece G c, and m ^a, g, and e ; do the same on the other side of A. ^ ^jll ^weight be laid on at A, the wires A F d, A. G ’ctu re ’ .strained, and may be broken. In the instant oxmay suppose their strains to be represented by Ag ofComplete the parallelogram, and A a is the mag nI