508 
RESISTANCE. 
that both the resistance and the weight will 
he increased in the same ratio as the base ; 
and hence it appears that ail cylinders of the 
fame matter and length, whatever their bases 
tti'e, have an equal resistance, when vertically 
suspended. 
Eut it the length of the cylinder is increas- 
ed without increasing its base, its weight is 
increased, while the resistance or strength 
continues unaltered ; consequently the length- 
ening has the eff. ct of weakening it, or in- 
creases its tendency to break. 
Hence, to find the greatest length a cylinder 
of any matter may have, when it just breaks 
with the addition of another given weight, we 
need only take any cylinder of the same matter, 
and fasten to it the least weight that is just suf- 
ficient to break it ; and then consider how much 
it must be lengthened, so that the weight of the 
part added, together with the given weight, 
may be just equal to that ■weight, and the thing 
is done. Thus, let l denote the first length of 
the cylinder, c its weight, g the given weight 
the lengthened cylinder is to bear, and to the 
least weight that breaks the cylinder /, also -v the 
length sought ; then as l \ x \\ c [ = the 
weight of the longest cylinder sought ; and this, 
together with the given weight g, must be equal 
to c, together with the weight iv ; hence then 
CK . , , c -j- iv — o- 
— — j- g ~ c -j- iv ; therefore x — - — 
l c 
l — the whole length of the cylinder sought. 
If the cylinder must just break with its own 
weight, then is g — 0, and in that case x = 
c • j ■ iu , g 
— l is the whole length that just breaks by 
its own weight. By this means Galileo found 
that a copper wire, and of consequence any 
other cylinder of copper, might be extended to 
4801 fathoms of 6 feet each. 
If the cylinder is fixed by one end into a wall, 
with the axis horizontally ; the force to break 
it, and its resistance to fracture, will here be 
both different; as both the weight to cause the 
fracture, and the resistance of the fibres to op- 
pose it, are combined with the effects of the le- 
ver ; for the weight to cause the fracture, whe- 
ther of the weight of the beam alone, or com- 
bined with an additional weight hung to it, is to 
be supposed collected into the centre of gravity, 
where it is considered as acting by a lever equal 
to the distance of that centre beyond the face 
of the wall where the cylinder or other prism is 
fixed ; and then the product of the said whole 
weight and distance, will be the momentum or 
force to break the prism. Again, the resistance 
of the fibres may be supposed collected into the 
centre of the transverse section, and all acting 
there at the end of a lever equal to the vertical 
semidiameter of the section, the lowest point of 
that diameter being immoveable, and about 
which the whole diameter turns when the prism 
breaks ; and hence the product of the adhesive 
force of the fibres, multiplied by the said semi- 
diameter, will be the momentum of resistance, 
and must be equal to the former momentum 
when the prism just breaks. 
Hence, to find the length a prism will bear, 
fixed so horizontally, before it breaks, either by 
its own weight, or by the addition of any ad- 
ventitious weight ; take any length of such a 
prism, and load it with weights till it just breaks. 
Then, put 
/ =. the length of this prism, 
c z=. its weight, 
to — the weight that breaks it, 
a — distance of weight to 
g — any given weight to be borne, 
d — its distance, 
x = the length required to break. 
Then / 
c * — the weight of the prism 
r, and — X i 
— its momentum ; also 
dg — the momentum of the weighty; therefore 
— — -J- dg is the momentum of the prism x and 
its added weight. In like manner \cl -j- a™ is 
that of the former or short prism, and the weight 
that broke it ; consequently ~ -j- dg — \cl -f- 
. atv -4- id — dg . , 
aiu, and x — y' — - X 2 / is the 
length sought, that just breaks with the weight 
g at the distance d. If this weight g is nothing, 
then x — \/ - X 2 / is the length of 
c 
the prism that just breaks with its own weight. 
If two prisms of the same matter, having 
their bases and lengths in the same propor- 
tion, are suspended horizontally ; it is evi- 
dent that the greater has more weight than 
the lesser, both on account of its length, and 
of its base ; but it has less resistance on ac- 
count of its length, considered as a longer 
arm of a lever, and has only more resistance 
on account of its base ; therefore it exceeds 
the lesser in its momentum more than it does 
in its resistance, and consequently it must 
break more easily. 
Hence appears the reason why, in making 
small machines and models, people are apt 
to be mistaken as to the resistance and 
strength of certain horizontal pieces, when 
they come to execute their designs in large, 
by observing the same proportions as in the 
small. 
When the prism, fixed vertically, is just 
about to break, there is an equilibrium be- 
tween its positive and relative weight; and 
consequently those two opposite powers are 
to each other reciprocally as the arms of the 
lever to which they are applied, that is, as 
half the diameter to half the axis of the 
prism. On the other hand, the resistance 
of a body is always equal to the greatest 
weight which it will just sustain in a vertical 
position, that is, to its absolute weight. 
Therefore, substituting the absolute weight 
for the resistance, it appears, that the abso- 
lute weight of a body, suspended horizon- 
tally, is to its relative weight, as the distance 
of its centre of gravity from the fixed point 
or axis of motion, is to the distance of the 
centre of gravity of its base from the same. 
The discovery of this important truth, at 
least of an equivalent to it, and to which this 
is reducible, we owe to Galileo. On this 
system of resistance of that author, Mariotte 
made an ingenious remark, which gave birth 
to a new system. Galileo supposes that 
where the body breaks, all the fibres break 
at once ; so that the body always resists with 
its whole absolute force, or the whole force 
that all its fibres have in the place where it 
breaks. But Mariotte, finding that all 
bodies, even glass itself, bend before they 
break, shews that fibres are to be considered 
as so many little bent springs, which never 
exert their whole force till stretched to a 
certain point, and never break till entirely 
unbent. Hence those nearest the fulcrum of 
the lever, or lowest point of the fracture, are 
stretched less than those farther off, and con- 
sequently employ a less part of their fore*, 
and break later. 
This consideration only takes place in the 
horizontal situation of the body ; in the ver- 
tical, the fibres of the base all break at once ; 
so that the absolute weight of the body must 
exceed the united resistance of all its fibres ; 
a greater weight is therefore required here 
than in the horizontal situation ; that is, a 
greater weight i, required to overcome their 
united resistance, than to overcome their 
several resistances one after another. See 
Timber, strength of 
Resistance of fluids, is the force with 
which bodies, moving in fluid mediums, are 
impeded and retarded in their motion. 
A body moving in a fluid is resisted from 
two causes. The first of these is the cohesion 
of the parts of the fluid. For a body, in its 
motion, separating the parts of a fluid, must 
overcome the force with which those parts 
cohere. The second is the inertia or in- 
activity of matter, by which a certain force 
is required to move the particles from their 
places in order to let the body pass. 
The retardation from the first cause is al- 
ways the same in the same space, whatever 
the velocity may be, the body remaining the 
same ; that is, the resistance is as the space 
run through in the same time ; but the ve- 
iocity is also in the same ratio of the space 
run over in the same time ; and therefore the 
resistance from this cause, is as the velocity 
itself. 
The resistance from tire second cause, 
when a body moves through the same fluid 
with different velocities, is as the square of 
the velocity. For, first, the resistance in- 
creases according to the number of particles 
or quantity of the fluid struck in the same 
time ; which number must be as the space 
run through in that time, that is, as the ve- 
locity : but the resistance also increases in 
proportion to the force with which the body 
strikes against every part; which force is 
also as the velocity of the body, so as to be 
double with a double velocity, and triple 
with a triple one, &c. ; therefore, on both 
these accounts, the resistance is as the ve- 
locity multiplied by the velocity, or as the 
square of the velocity. Upon the whole 
therefore, on account of both causes, viz. the 
tenacity and inertia of the fluid, the body is 
resisted partly as the velocity and partly as 
the square of the velocity. 
But when the same body moves through 
different fluids with the same velocity, the 
resistance from the second cause follows the 
proportion of the matter to be removed in 
the same time, which is as the density of the- 
fluid. 
Hence therefore, if d denotes the density 
of the fluid, 
body, 
v the velocity of the 
and a and b constant co- 
efficients : 
then udv 2 + bv will be proportional to the 
whole resistance to the same body, moving 
with different velocities, in the same direction, 
through fluids of different densities, but of 
the same tenacity. 
But to take in the consideration of differ- 
.ent tenacities of fluids; if t denotes the te- 
nacity, or the cohesion of the parts of the 
fluid, then adv 2 -(- btv will be as the said 
whole resistance. 
Indeed ttie quantity of resistance from the 
cohesion of the parts of fluids, except in gluti- 
