eo 
TABLE III. 
Absolute cohesive Strength of Wood drawn in a direc- 
tion at right angles to the fibres. 
Teak, 4 818 lbs 
American White Pine, 757 
Norway Fir, 648 
Beech, - 615 
English Oak, 098 
Canada Oak, 588 
Pitch Pine, i 588 
Elm, ; 509 
Ash, 359 
II. Of the IRerneeiae Seon Gf Mare, tals. To 
apply the hypothesis of Galileo to the case of a 
transverse strain, we shall suppose in the first 
place, the substance to have the form of a pris- 
matic beam, that it is firmly inserted at one end 
into a fixed support, lies in a horizontal position, 
and is acted upon by a weight that presses at 
the end that is not fixed; that the fracture takes 
place at the point of support, beginning at the 
upper side, on which the weight presses, and ter- 
minating at the other. The beam then, its fibres 
being by hypothesis inflexible and inextensible, 
will turn around the latter point in a vertical 
plane. At the instant of fracture, the two forces 
that act are in equilibrio with each other, their 
respective moments of rotation must be dhanebne 
equal. 
When a beam, lying in a horizontal position, 
rests upon two props, and is broken by a weight 
placed at an equal distance from the two props, 
we may consider the laws of its strength as in- 
cluded in the general case of a beam supported 
at one end only; for if, according to the hypo- 
thesis, it break without bending, we may con- 
ceive it to be formed of two beams, each inserted 
in a firm support at the place of fracture, and 
acted upon at each end by a force equal to half 
the weight that just breaks it, but which is 
directed upwards instead of downwards. This 
force, which is equal to half the weight, will act 
at a distance which is equal to half the length ; 
its effort is therefore equal to no more than a 
fourth part of the effort of the same weight, ap- 
plied to the same beam, if supported at one end 
only; and as this effort must be just equal, at 
thé instant of breaking, to the transverse 
strength of the beam, the latter will be four 
times as strong as when supported at one end 
only. 
In the case of a beam lying horizontally, and 
firmly fixed at each end, the resistance will be 
equal to that of a single beam supported by two 
props, added to those of two beams fixed at one 
end only; for it is obvious, that three fractures 
must be produced, one in the middle, and one at 
each end; at the first, the resistance will be 
equal to that of the supported beam, or four 
times as great as that of the same beam, if sup- 
ported at one end only. The resistance in each 
of the latter cases, will be that of a beam of half 
the length fixed at one end only, or one fourth 
of the last resistance; the whole resistance will 
i 
STRANGTH OF MATERIALS. 
therefore be six times as great as that of the 
same beam when fixed at one end only. 
When the weight that tends to break a beam 
is not accumulated at a single point, but is uni- 
formly distributed over its whole length; its 
effort is diminished to the half of what it exerts, 
when, in the case of a beam fixed at one end, it 
acts at the opposite extremity ; or when, in the 
case of a beam supported or fixed at both ends, 
it acts in the middle. 
We have hitherto omitted the action of the 
weight of the beam itself. In small beams in- 
deed, their own weights are of little importance, 
and need hardly be taken into account in the 
experiments; but in large beams this is not the 
case, aS may easily be seen. The weight which 
breaks a beam is made up of its own weight, and 
the weight which is applied for the purpose: the 
former acts at the centre of gravity of the beam, 
which in prismatic beams is in the middle of 
their length. 
We shall now recapitulate the results of our 
hypothesis, and state what discrepancies have 
been observed between them, and the inferences 
from actual experiment. 
(1.) In any prismatic beam whatsoever, the 
strength is directly proportioned, to the area of 
its section, and to the distance of its centre of 
gravity from the point where the fracture ter- 
minates; and inversely, to the length of the 
beam. 
(2.) The strengths of beams lying in a horizon- 
tal position, when fixed at one end only ; when 
supported by a prop at each end; and when 
firmly fixed at both ends, are as the numbers 
1:4: 6,—that is to say, a beam firmly fixed at 
both ends, is six times, a beam merely supported 
at both ends, four times as strong as when it is 
fixed at one end only. 
These several inferences from the hypothesis 
agree within all usual limits, with the results of 
experiments. The discrepancies are, that the 
strengths increase in a ratio a little greater than 
the square of the depth, in rectangular beams ; 
and decrease rather more rapidly than the in- 
verse ratio of the length. 
The second of the above propositions admits of 
the following cases :— 
(a) In beams of the same material, with equal 
and similar sections and unequal lengths, the 
strengths are inversely proportioned to the 
lengths. 
The lengths being equal : 
(6) In rectangular beams of the same materials 
the strengths are proportioned to the product of 
the breadth by the square of the depth ; 
(c) In square beams, the strengths are propor- 
tioned to the cubes of the sides of the square sec- 
tions ; 
(d) In solid cylindric beams, the strengths are 
proportioned to the cubes of the radii ; 
(e) In hollow cylindric beams, having the same 
quantities of material distributed around cylin- 
