CS:CQ= A A Q CQ: c b =AQ:pb:LK=AQ:C n Therefore, by equality, co:cb Ad: fn or, 13 0 : A L = CY:C1: That is, the external forces are reciprocally proportional to the perpendiculars drawn from the prop on the lines of their direction.
This proposition is fertile in consequences, and furnishes many useful instructions to the artist. The strains L A, o B, C 1', that are excited, occur in many, if not in all Ira filings of carpentry, whether for edifices or engines. and are the sources of their efficacy. It is also evident, that the doctrine of the transverse strength of timber is contained in this proposition ; for every piece of timber may be considered as au assemblage of parts, connected by forces acting in the direction of the lines which join the strained points on the matter lying between those points, and also act on the rest of the matter, exciting those lateral forces which produce the inflexibility of the whole. Thus it appears that this proposition contains the principles which direct the artist to frame the most powerful levers ; to secure uprights by shores or braces, or by ties and ropes ; to secure scaffoldings for the erection of spires, and many other most delicate problems of his art. l le also learns, from this proposition, how to ascertain the strains that are produced, without his intention, by pieces which he intended for other offices, and which, by their transverse action, put his work in hazard. In short, this proposition is the key to the science of his art.
There is a proposition which has been called in question by several very intelligent persons ; and they say that Belidur has demonstrated, in his Science des Ie.geueurc, that a Leant firmly fixed at both ends is not twice as strong as when simply lying on the props, and that its strength is increased only in the proportion of 2 to 3 ; and they support this determination by a list of experiments recited by Belidor, which agree pre cisely with it. Belidor also says, that Pitot had the same results in his experiments. These are respectable authorities:
but Belidor's reasoning is anything but demonstration ; and his experiments are described in such an imperfect manner, that we cannot build much on them. It is not said in what manner the battens were secured at the ends, any farther than that it was by chevalets. If by this word is meant a trestle, we cannot conceive how they were employed ; but we see this term sometimes used for a eerie, or key. If the battens were wedged in the holes, their resistance to fracture may be made what we please : they may be loose, and therefore resist little more than when simply laid on the props. They may be (and probably were) wedged very fast, and bruised or crippled.
Figure 24.—Let L m be a long beam divided into six equal parts, in the points D, n, A, C, E, and firmly supported at L, 13, C, ; let it be cut through at A, and have compass joints at a and c ; let F B, 0 c, be two equal uprights, resting on It and c, but without any connection ; let A 11 be a similar and equal piece, to be occasionally applied at the seam A ; then extend a thread, or wire, A U a, over the piece o c, and made fist at A, 0, a ; do the same on the other side of A. Now, if a weight be laid on at A, the wires A F D, A 0 E, will be strained, and may be broken. In the instant of fracture, we may suppose their strains to be represented by A f and A g. Complete the parallelogram, and A a is the magnitude of the weight. It is plain that nothing is concerned here but the cohesion of the m Tres; for the beam is sawn through at A, and its parts are perfectly moveable round B and c.
Instead of this process, apply the piece A n below A, and i keep it there by straining the same wire, B n c, over it: lay on a weight, w hich must press down the ends of B A and C A, and cause the piece A n to strain the wire B H c. In the instant of fracture of the same wire, its resistances, u b and n c must be equal to A f and A g, and the weight h n, which breaks them, must be equal to A a.