MAGAZINE OF SCIENCE AND ART. 
237 
pertv of the circle may, at first sight, appear, it lies 
at the foundation of all those contrivances by which the 
power of man is increased; for it is an immediate con¬ 
sequence of this property that the whole fabric of me¬ 
chanical science rests, whether for raising the heaviest 
weights, for pumping water, or in short for performing 
any work. The lever, the wheel and axle, (which is 
only a kind of perpetual lever) and the other mechani¬ 
cal powers depend on this property, to which Archimedes 
referred when he said that “ if ho had only a pivot or 
fulcrum whereon he might rest his machinery, he could 
move the Earth.” 
20. Viewing a machine in its simplest form, the 
problem which first presents itself is the determination 
of the power which by its means won Id be capable of 
supporting a given weight. There is one great prin¬ 
ciple which applies equally to all machines, and by 
which the power which is capable of snstaining a given 
weight “ in equilibrio” is determined. This is called 
the principle of “ virtual velocities,” and it is one of 
very great importance, as it applies to all questions re¬ 
specting equilibrium, and also includes the resolution 
of many problems relating to bodies in motion. In 
order that a given power may be capable of sustaining 
a given weight, it is only necessary that it should have, 
to the weight the same ratio as the velocity of the 
weight has to the power, or to express the same condi¬ 
tion in other words, “the power multiplied by the 
space through which it moves in the vertical direction, 
must be equal to the weight multiplied by the spate 
through which it moves in the vertical, direction.” 
21. This simple principle, abating friction, will apply 
to all machines however complicated they may be. and 
hence the advantage gained by a machine is represented 
by the velocity of the power divided by the velocity of 
the^ weight raised. In order to illustrate this principle, 
which may be called the “ golden rule” of mechanics, 
the diagram (Fig. 1) represents the section of a solid 
body moving on a fixed axis at C. Let A C, C D be the 
radii, at the extremities of which.the power and weight 
act. Then A CD may be considered as a lever, the ful¬ 
crum of which is at C ; and since the power P and the 
weight W, being suspended by cords, act perpendicularly 
to A C, we shall have the following equation: 
P’s vel A C 
-- = -- and P x AC = W x DC. 
lY’s vel D C 
Supposing therefore that the length of the radius AC be 
10 feet, and that of CD be 1 foot, then a power of lOlbs. 
weight suspended 'at P, will sustain a weight of lOOlbs. 
at V. 
22. As this principle is the most important in the 
whole range of mechanical science, and may be said to 
form its basis, it is desirable to illustrate it’ by another 
method. If two weights in equilibria in (as m Fig. 2) 
at the extremities A and B of a bar or lever supported 
on an axis a, passing through its centre of gravity, be 
made to oscillate gently through a small space, it is 
evident that the spaces passed through by the two ends 
of the bar will be directly as their distances from the 
axis, for the angles A a m and. Ban being equal, the 
arcs A m and B n are as their radii a A and a B. For 
instance, if the weight B be 12 pounds suspended at 3 
inches from a, its moment may be expressed by the 
number 36; and it will be balanced by a weight of 6 
pounds, 6 inches from a, because its moment is'also 36. 
© Q 
23. This is the most simple aspect in which power 
nnd weight can he viewed in a machine, and according 
to this principle the power will sustain the weight in 
equilibrio, provided, that the number of pounds weight 
of the power, multiplied by tho number of feet in its 
distance from tho centre, shall be equal to the number 
of pounds in the weight, multiplied by the number of 
feet in its distance from the centre. This important 
rinciple may be said to include the whole science of 
oth statics and dynamics, and its most striking results 
are spread over departments of physical science far 
beyond the necessary limits of this brief and imperfect 
memoir; the principle of virtual velocities, it may 
however be remarked, depends also upon a property of 
the circle ; it follows that the effort of any force to turn 
a body round an axis, is to be measured by multiplying 
the force by the perpendicular from the axis in its di¬ 
rection. The product thus obtained is called the 
41 moment” of the force ropnd the axis, but ibis must not 
be confounded with “ momentum." 
24. The “moment” of power is its intensity or 
weight multiplied into its leverage or distance from the 
axis, and its “momentum” is its intensity or weight 
multiplied into its velocity. It is important to observe 
that any two forces applied to a body supported on an 
axis and tending to turn it round, will be in equili¬ 
brium when the “moments” of the two forces are 
equal. In like manner, if the moment ho doubled or 
halved, or increased or decreased in any proportion, the 
efficacy^ of the force in turning the body round the 
axis will be doubled or halved, or increased or de¬ 
creased in exactly the same proportion. For instance, 
if a weight of 12 pounds, situated at a distance of 1 foot 
from the axis, bo brought 6 inches nearer that axis, its 
moment is reduced one half; and in order to produce 
equilibrium, the moment of the weight of 12 pounds 
on the other side of the axis must bo hal ved also, either 
by reducing its weight to C pounds, or by bringing the 
12 pounds six inches nearer the axis, as it wilL be found 
that each of these multiplied into the distance will 
equal 6; for 12 x £ — 6: and 6x1=6. It is 
scarcely necessary to sar, that if the sum of tho “ mo¬ 
ments” of the forces which tend to turn a body round 
in one direction be greater than the sum of the moments 
of the forces which tend to turn it in the opposite di¬ 
rection, the body will move round its centre in the 
direction of the former. 
24. However self-evident these propositions may ap¬ 
pear to mechanics and others, their results are neither 
few nor unimportant. Engineering and mechanical 
science is eminently a science of facts and of figures; 
nothing in it must be taken for granted, and every cal¬ 
culation made in reference to it must be founded on 
correct data, otherwise it will be worse than useless. 
