On the Resistance of Solids. 417 
Then, the weight W will act with the leverage AD; and D 
will be the centre of motion. Now it is evident, that when the 
weight at A causes the beam to move round the centre D, the 
upper side at B will be extended and the lower side at C com- 
pressed (art. 10); and the strain on any part above or below 
the neutral line will be directly as its distance from that line 5 
and the extension or compression will be as the strain (arts. 6 
and 11). 
Also, as the compressed part is the fulerum which supports 
the lever till the extended part is torn asunder; and bodies are ex- 
tended and compressed in equal degrees by equal forces (art. 7) 5 
therefore, the neutral line must divide the section into two equal 
and similar parts, because the forces on each side of the neutral 
line must be equal*. 
Put / = the length of the beam AD; 
@ = the distance of the neutral line from the upper side 5 
y = the breadth of the beam ; 
Jf = the cohesive force of an unit of the area ; 
ya = a variable part of the section of fracture ; 
and x = its distance from the neutral line. 
Then the force of any variable part of the section is as its di- 
stance from the neutral line, or a: x::f: £; and as its area 
a 3 which being multiplied 
yt; hence, its whole force is = — 
Be 
pus = = the 
fluxion of the force exerted by the part of the beam above the 
neutral line. But the forces on each side of the neutral line are 
equal ; therefore, in the case of equilibrium, we have 
Fluent of pine | LW; or flu, “¥"* — w. 
la 
by its distance from the centre of motion, gives’ 
* Most of the writers on the strength of materials have considered the 
point of support C, as the fulcrum to the extending forces, but the support 
is a fulcrum only in respect to a force at E;—this mistake arises from the 
absurd method of demonstrating the properties of the lever, by supposing 
it to be an inflexible line; had the properties of the lever been sought for 
in the lever itself, this could never have happened. 
In the natural order of science, the resistance of beams should occupy 
the place that is now assigned to the doctrine of the lever, as its properties 
are merely so many corollaries which naturally fow from propositions in the 
doctrine of the resistance of beams. For the strains excited in a beam may 
be investigated directly by, means of the properties of the parallelogram of 
forces, without referring at all to those of the lever; and the properties of 
a beam considered as a lever, may be deduced from the strains excited in 
the beam, which appears to be the only legitimate mode of demonstrating 
its properties. See art. Carpentry, p.167. Supplem. Encyclo. Brit. 3d ed. 
1811; or art. Carpentry, p. 633, New Supp. 1817. 
Vol, 50, No, 236, Dec, 1817, Dd 14, When 
