multiplied by half the difference of the diameters of the thinner and
thicker parts of the axle.

A wheel and axle constructed in this manner is equivalent to an
ordinary one, in which the wheel has the same diameter, and whose axle
has a diameter equal to half the difference of the diameters of the
thicker and thinner parts. The power of the machine is expressed by the
proportion which the diameter of the wheel bears to half the difference
of these diameters; and therefore this power, when the diameter of the
wheel is given, does not, as in the ordinary wheel and axle, depend
on the smallness of the axle, but on the smallness of the difference
of the thinner and thicker parts of it. The axle may, therefore,
be constructed of such a thickness as to give it all the requisite
strength, and yet the difference of the diameters of its different
parts may be so small as to give it all the requisite power.

(255.) It often happens that a varying weight is to be raised, or
resistance overcome by a uniform power. If, in such a case, the weight
be raised by a rope coiled upon a uniform axle, the action of the
power would not be uniform, but would vary with the weight. It is,
however, in most cases desirable or necessary that the weight or
resistance, even though it vary, shall be moved uniformly. This will
be accomplished if by any means the leverage of the weight is made
to increase in the same proportion as the weight diminishes, and to
diminish in the same proportion as the weight increases: for in that
case the moment of the weight will never vary, whatever it gains by the
increase of weight being lost by the diminished leverage, and whatever
it loses by the diminished weight being gained by the increased
leverage. An axle, the surface of which is curved in such a manner,
that the thickness on which the rope is coiled continually increased
or diminishes in the same proportion as the weight or resistance
diminishes or increases, will produce this effect.

It is obvious that all that has been said respecting a variable
weight or resistance, is also applicable to a variable power, which,
therefore, may, by the same means, be made to produce a uniform effect.
An instance of this occurs in a watch, which is moved by a spiral
spring. When the watch has been wound up, this spring acts with its
greatest intensity, and as the watch goes down, the elastic force of
the spring gradually loses its energy. This spring is connected by a
chain with an axle of varying thickness, called a _fusee_. When the
spring is at its greatest intensity, the chain acts upon the thinnest
part of the fusee, and as it is uncoiled it acts upon a part of the
fusee which is continually increasing in thickness, the spring at the
same time losing its elastic power in exactly the same proportion. A
representation of the fusee, and the cylindrical box which contains
the spring, is given in _fig. 98._, and of the spring itself in
_fig. 99._

(256.) When great power is required, wheels and axles may be combined
in a manner analogous to a compound system of levers, explained
in (246.) In this case the power acts on the circumference of the
first wheel, and its effect is transmitted to the circumference
of the first axle. That circumference is placed in connection with
the circumference of the second wheel, and the effect is thereby
transmitted to the circumference of the second axle, and so on. It
is obvious from what was proved in (248.), that the power of such a
combination of wheels and axles will be found by multiplying together
the powers of the several wheels of which it is composed. It is
sometimes convenient to compute this power by numbers expressing the
proportions of the circumferences or diameters of the several wheels,
to the circumferences or diameters of the several axles respectively.
This computation is made by first multiplying the numbers together
which express the circumferences or diameters of the wheels, and then
multiplying together the numbers which express the circumferences or
diameters of the several axles. The proportion of the two products
will express the power of the machine. Thus, if the circumferences or
diameters be as the numbers 10, 14, and 15, their product will be 2100;
and if the circumferences or diameters of the axles be expressed by the
numbers 3, 4, and 5, their product will be 60, and the power of the
machine will be expressed by the proportion of 2100 and 60, or 35 to 1.

[Illustration: _H. Adlard, sc._

_London, Pubd. by Longman & Co._]

(257.) The manner in which the circumferences of the axles act upon
the circumferences of the wheels in compound wheel-work is various.
Sometimes a strap or cord is applied to a groove in the circumference
of the axle, and carried round a similar groove in the circumference
of the succeeding wheel. The friction of this cord or strap with the
groove is sufficient to prevent its sliding and to communicate the
force from the axle to the wheel, or _vice versa_. This method of
connecting wheel-work is represented in _fig. 100._

Numerous examples of wheels and axles driven by straps or cords occur
in machinery applied to almost every department of the arts and
manufactures. In the turning lathe, the wheel worked by the treddle is
connected with the mandrel by a catgut cord passing through grooves
in the wheel and axle. In all great factories, revolving shafts are
carried along the apartments, on which, at certain intervals, straps
are attached passing round their circumferences and carried round
the wheels which give motion to the several machines. If the wheels,
connected by straps or cords, are required to revolve in the same
direction, these cords are arranged as in _fig. 100._; but if they
are required to revolve in contrary directions, they are applied as in
_fig. 101._

One of the chief advantages of the method of transmitting motion
between wheels and axles by straps or cords, is that the wheel and
axle may be placed at any distance from each other which may be found
convenient, and may be made to turn either in the same or contrary
directions.

(258.) When the circumference of the wheel acts immediately on the
circumference of the succeeding axle, some means must necessarily be
adopted to prevent the wheel from moving in contact with the axle
without compelling the latter to turn. If the surfaces of both were
perfectly smooth, so that all friction were removed, it is obvious that
either would slide over the surface of the other, without communicating
motion to it. But, on the other hand, if there were any asperities,
however small, upon these surfaces, they would become mutually inserted
among each other, and neither the wheel nor axle could move without
causing the asperities with which its edge is studded to encounter
those asperities which project from the surface of the other; and
thus, until these projections should be broken off, both wheel and
axle must be moved at the same time. It is on this account that if the
surfaces of the wheels and axles are by any means rendered rough, and
pressed together with sufficient force, the motion of either will turn
the other, provided the load or resistance be not greater than the
force necessary to break off these small projections which produce the
friction.

In cases where great power is not required, motion is communicated in
this way through a train of wheel-work, by rendering the surface of the
wheel and axle rough, either by facing them with buff leather, or with
wood cut across the grain. This method is sometimes used in spinning
machinery, where one large buffed wheel, placed in a horizontal
position, revolves in contact with several small buffed rollers, each
roller communicating motion to a spindle. The position of the wheel W,
and the rollers R R, &c., are represented in _fig. 102._ Each
roller can be thrown out of contact with the wheel, and restored to it
at pleasure.

The communication of motion between wheels and axles by friction has
the advantage of great smoothness and evenness, and of proceeding with
little noise; but this method can only be used in cases where the
resistance is not very considerable, and therefore is seldom adopted in
works on a large scale. Dr. Gregory mentions an instance of a saw mill
at Southampton, where the wheels act upon each other by the contact of
the end grain of wood. The machinery makes very little noise, and wears
very well, having been used not less than 20 years.

(259.) The most usual method of transmitting motion through a train of
wheel-work is by the formation of teeth upon their circumferences, so
that these indentures of each wheel fall between the corresponding ones
of that in which it works, and ensure the action so long as the strain
is not so great as to fracture the tooth.

In the formation of teeth very minute attention must be given to their
figure, in order that the motion may be communicated from wheel to
wheel with smoothness and uniformity. This can only be accomplished
by shaping the teeth according to curves of a peculiar kind, which
mathematicians have invented, and assigned rules for drawing. The ill
consequences of neglecting this will be very apparent, by considering
the nature of the action which would be produced if the teeth were
formed of square projecting pins, as in _fig. 103._ When the
tooth A comes into contact with B, it acts obliquely upon it, and,
as it moves, the corner of B slides upon the plane surface of A in
such a manner as to produce much friction, and to grind away the side
of A and the end of B. As they approach the position C D, they
sustain a jolt the moment their surfaces come into full contact; and
after passing the position of C D, the same scraping and grinding
effect is produced in the opposite direction, until by the revolution
of the wheels the teeth become disengaged. These effects are avoided by
giving to the teeth the curved forms represented in _fig. 104._
By such means the surfaces of the teeth roll upon each other with very
inconsiderable friction, and the direction in which the pressure is
excited is always that of a line M N, touching the two wheels, and
at right angles to the radii. Thus the pressure being always the same,
and acting with the same leverage, produces a uniform effect.

(260.) When wheels work together, their teeth must necessarily be of
the same size, and therefore the proportion of their circumferences may
always be estimated by the number of teeth which they carry. Hence it
follows, that in computing the power of compound wheel-work, the number
of teeth may always be used to express the circumferences respectively,
or the diameters which are proportional to these circumferences. When
teeth are raised upon an axle, it is generally called a _pinion_, and
in that case the teeth are called _leaves_. The rule for computing the
train of wheel-work given in (256.) will be expressed as follows: when
the wheel and axle carry teeth, multiply together the number of teeth
in each of the wheels, and next the number of leaves in each of the
pinions; the proportion of the two products will express the power of
the machine. If some of the wheels and axles carry teeth, and others
not, this computation may be made by using for those circumferences
which do not bear teeth the number of teeth which would fill them.
_Fig. 105._ represents a train of three wheels and pinions. The
wheel F which bears the power, and the axle which bears the weight,
have no teeth; but it is easy to find the number of teeth which they
would carry.

(261.) It is evident that each pinion revolves much more frequently in
a given time than the wheel which it drives. Thus, if the pinion C be
furnished with ten teeth, and the wheel E, which it drives, have sixty
teeth, the pinion C must turn six times, in order to turn the wheel
E once round. The velocities of revolution of every wheel and pinion
which work in one another will therefore have the same proportion as
their number of teeth taken in a reverse order, and by this means the
relative velocity of wheels and pinions may be determined according to
any proposed rate.

Wheel-work, like all other machinery, is used to transmit and modify
force in every department of the arts and manufactures; but it is also
used in cases where motion alone, and not force, is the object to be
attained. The most remarkable example of this occurs in watch and
clock-work, where the object is merely to produce uniform motions of
rotation, having certain proportions, and without any regard to the
elevation of weights, or the overcoming of resistances.

(262.) A _crane_ is an example of combination of wheel-work used for
the purpose of raising or lowering great weights. _Fig. 106._
represents a machine of this kind. A B is a strong vertical beam,
resting on a pivot, and secured in its position by beams in the floor.
It is capable, however, of turning on its axis, being confined between
rollers attached to the beams and fixed in the floor. C D is a
projecting arm called a _gib_, formed of beams which are mortised into
A B. The wheel-work is mounted in two cast-iron crosses, bolted on
each side of the beams, one of which appears at E F G H.
The winch at which the power is applied is at I. This carries a pinion
immediately behind H. This pinion works in a wheel K, which carries
another pinion upon its axle. This last pinion works in a larger wheel
L, which carries upon its axis a barrel M, on which a chain or rope
is coiled. The chain passes over a pulley D at the top of the gib. At
the end of the chain a hook O is attached, to support the weight W.
During the elevation of the weight it is convenient that its recoil
should be hindered in case of any occasional suspension of the power.
This is accomplished by a ratchet wheel attached to the barrel M, as
explained in (253.); but when the weight W is to be lowered, the catch
must be removed from this ratchet wheel. In this case the too rapid
descent of the weight is in some cases checked by pressure excited on
some part of the wheel-work, so as to produce sufficient friction to
retard the descent in any required degree, or even to suspend it, if
necessary. The vertical beam at B resting on a pivot, and being fixed
between rollers, allows the gib to be turned round in any direction; so
that a weight raised from one side of the crane may be carried round,
and deposited on another side, at any distance within the range of the
gib. Thus, if a crane be placed upon a wharf near a vessel, weights may
be raised, and when elevated, the gib may be turned round so as to let
them descend into the hold.

The power of this machine may be computed upon the principles already
explained. The magnitude of the circle, in which the power at I moves,
may be determined by the radius of the winch, and therefore the number
of teeth which a wheel of that size would carry may be found. In
like manner we may determine the number of leaves in a pinion whose
magnitude would be equal to the barrel M. Let the first number be
multiplied by the number of teeth in the wheel K, and that product
by the number of teeth in the wheel L. Next let the number of leaves
in the pinion H be multiplied by the number of leaves in the pinion
attached to the axle of the wheel K, and let that product be multiplied
by the number of leaves in a pinion, whose diameter is equal to that of
the barrel M. These two products will express the power of the machine.

(263.) Toothed wheels are of three kinds, distinguished by the position
which the teeth bear with respect to the axis of the wheel. When they
are raised upon the edge of the wheel as in _fig. 105._, they are
called _spur wheels_, or _spur gear_. When they are raised parallel to
the axis, as in _fig. 107._, it is called a _crown wheel_. When
the teeth are raised on a surface inclined to the plane of the wheel,
as in _fig. 108._, they are called _bevelled wheels_.

[Illustration: _H. Adlard, sc._

_London, Pubd. by Longman & Co._]

If a motion round one axis is to be communicated to another axis
parallel to it, spur gear is generally used. Thus, in _fig. 105._,
the three axes are parallel to each other. If a motion round one
axis is to be communicated to another at right angles to it, a crown
wheel, working in a spur pinion, as in _fig. 107._, will serve.
Or the same object may be obtained by two bevelled wheels, as in
_fig. 108._

If a motion round one axis is required to be communicated to another
inclined to it at any proposed angle, two bevelled wheels can always be
used. In _fig. 109._ let A B and A C be the two axles;
two bevelled wheels, such as D E and E F, on these axles will
transmit the motion or rotation from one to the other, and the relative
velocity may, as usual, be regulated by the proportional magnitude of
the wheels.

(264.) In order to equalise the wear of the teeth of a wheel and
pinion, which work in one another, it is necessary that every leaf
of the pinion should work in succession through every tooth of the
wheel, and not continually act upon the same set of teeth. If the
teeth could be accurately shaped according to mathematical principles,
and the materials of which they are formed be perfectly uniform, this
precaution would be less necessary; but as slight inequalities, both
of material and form, must necessarily exist, the effects of these
should be as far as possible equalised, by distributing them through
every part of the wheel. For this purpose it is usual, especially
in mill-work, where considerable force is used, so to regulate the
proportion of the number of teeth in the wheel and pinion, that the
same leaf of the pinion shall not be engaged twice with any one tooth
of the wheel, until after the action of a number of teeth, expressed
by the product of the number of teeth in the wheel and pinion. Let us
suppose that the pinion contains ten leaves, which we shall denominate
by the numbers 1, 2, 3, &c., and that the wheel contains 60 teeth
similarly denominated. At the commencement of the motion suppose the
leaf 1 of the pinion engages the tooth 1 of the wheel; then after
one revolution the leaf 1 of the pinion will engage the tooth 11 of
the wheel, and after two revolutions the leaf 1 of the pinion will
engage the tooth 21 of the wheel; and in like manner, after 3, 4, and
5 revolutions of the pinion, the leaf 1 will engage successively the
teeth 31, 41, and 51 of the wheel. After the sixth revolution, the
leaf 1 of the pinion will again engage the tooth 1 of the wheel. Thus
it is evident, that in the case here supposed the leaf 1 of the pinion
will continually be engaged with the teeth 1, 11, 21, 31, 41, and 51
of the wheel, and no others. The like may be said of every leaf of the
pinion. Thus the leaf 2 of the pinion will be successively engaged with
the teeth 2, 12, 22, 32, 42, and 52 of the wheel, and no others. Any
accidental inequalities of these teeth will therefore continually act
upon each other, until the circumference of the wheel be divided into
parts of ten teeth each, unequally worn. This effect would be avoided
by giving either the wheel or pinion one tooth more or one tooth less.
Thus, suppose the wheel, instead of having sixty teeth, had sixty-one,
then after six revolutions of the pinion the leaf 1 of the pinion would
be engaged with the tooth 61 of the wheel; and after one revolution of
the wheel, the leaf 2 of the pinion would be engaged with the tooth 1
of the wheel. Thus, during the first revolution of the wheel the leaf
1 of the pinion would be successively engaged with the teeth 1, 11,
21, 31, 41, 51, and 61 of the wheel: at the commencement of the second
revolution of the wheel the leaf 2 of the pinion would be engaged with
the tooth 1 of the wheel; and during the second revolution of the wheel
the leaf 1 of the pinion would be successively engaged with the teeth
10, 20, 30, 40, 50, and 60 of the wheel. In the same manner it may be
shown, that in the third revolution of the wheel the leaf 1 of the
pinion would be successively engaged with the teeth 9, 19, 29, 39, 49,
and 59 of the wheel: during the fourth revolution of the wheel the
leaf 1 of the pinion would be successively engaged with the teeth 8,
18, 28, 38, 48, and 58 of the wheel. By continuing this reasoning it
will appear, that during the tenth revolution of the wheel the leaf
1 of the pinion will be engaged successively with the teeth 2, 12,
22, 32, 42, and 52 of the wheel. At the commencement of the eleventh
revolution of the wheel the leaf 1 of the pinion will be engaged with
the tooth 1 of the wheel, as at the beginning of the motion. It is
evident, therefore, that during the first ten revolutions of the wheel
each leaf of the pinion has been successively engaged with every tooth
of the wheel, and that during these ten revolutions the pinion has
revolved sixty-one times. Thus the leaves of the pinion have acted six
hundred and ten times upon the teeth of the wheel, before two teeth can
have acted twice upon each other.

The odd tooth which produces this effect is called by millwrights the
_hunting cog_.

(265.) The most familiar case in which wheel-work is used to produce
and regulate motion merely, without any reference to weights to be
raised or resistances to be overcome, is that of chronometers. In watch
and clock work the object is to cause a wheel to revolve with a uniform
velocity, and at a certain rate. The motion of this wheel is indicated
by an index or hand placed upon its axis, and carried round with it.
In proportion to the length of the hand the circle over which its
extremity plays is enlarged, and its motion becomes more perceptible.
This circle is divided, so that very small fractions of a revolution
of the hand may be accurately observed. In most chronometers it is
required to give motion to two hands, and sometimes to three. These
motions proceed at different rates, according to the subdivisions of
time generally adopted. One wheel revolves in a minute, bearing a
hand which plays round a circle divided into sixty equal parts; the
motion of the hand over each part indicating one second, and a complete
revolution of the hand being performed in one minute. Another wheel
revolves once, while the former revolves sixty times; consequently the
hand carried by this wheel revolves once in sixty minutes, or one hour.
The circle on which it plays is, like the former, divided into sixty
equal parts, and the motion of the hand over each division is performed
in one minute. This is generally called the _minute hand_, and the
former the _second hand_.

A third wheel revolves once, while that which carries the minute hand
revolves twelve times; consequently this last wheel, which carries
the _hour hand_, revolves at a rate twelve times less than that of
the minute hand, and therefore seven hundred and twenty times less
than the second hand. We shall now endeavour to explain the manner in
which these motions are produced and regulated. Let A, B, C, D, E,
_fig. 110._, represent a train of wheels, and _a_, _b_, _c_, _d_
represent their pinions, _e_ being a cylinder on the axis of the wheel
E, round which a rope is coiled, sustaining a weight W. Let the effect
of this weight transmitted through the train of wheels be opposed by
a power P acting upon the wheel A, and let this power be supposed
to be of such a nature as to cause the weight W to descend with a
uniform velocity, and at any proposed rate. The wheel E carries on its
circumference eighty-four teeth. The wheel D carries eighty teeth;
the wheel C is also furnished with eighty teeth, and the wheel B with
seventy-five. The pinions _d_ and _c_ are each furnished with twelve
leaves, and the pinions _b_ and _a_ with ten.

If the power at P be so regulated as to allow the wheel A to revolve
once in a minute, with a uniform velocity, a hand attached to the axis
of this wheel will serve as the _second hand_. The pinion _a_ carrying
ten teeth must revolve seven times and a half to produce one revolution
of B, consequently fifteen revolutions of the wheel A will produce two
revolutions of the wheel B; the wheel B, therefore, revolves twice in
fifteen minutes. The pinion _b_ must revolve eight times to produce
one revolution of the wheel C, and therefore the wheel C must revolve
once in four quarters of an hour, or in one hour. If a hand be attached
to the axis of this wheel, it will have the motion necessary for the
minute hand. The pinion _c_ must revolve six times and two thirds to
produce one revolution of the wheel D, and therefore this wheel must
revolve once in six hours and two thirds. The pinion _d_ revolves seven
times for one revolution of the wheel E, and therefore the wheel E will
revolve once in forty-six hours and two thirds.

On the axis of the wheel C a second pinion may be placed, furnished
with seven leaves, which may lead a wheel of eighty-four teeth, so
that this wheel shall turn once during twelve turns of the wheel C. If
a hand be fixed upon the axis, this hand will revolve once for twelve
revolutions of the minute hand fixed upon the axis of the wheel C;
that is, it will revolve once in twelve hours. If it play upon a dial
divided into twelve equal parts, it will move over each part in an
hour, and will serve the purpose of the hour hand of the chronometer.

We have here supposed that the second hand, the minute hand, and the
hour hand move on separate dials. This, however, is not necessary. The
axis of the hour hand is commonly a tube, inclosing within it that of
the minute hand, so that the same dial serves for both. The second
hand, however, is generally furnished with a separate dial.

(266.) We shall now explain the manner in which a power is applied
to the wheel A, so as to regulate and equalise the effect of the
weight W. Suppose the wheel A furnished with thirty teeth, as in
_fig. 111._; if nothing check the motion, the weight W would
descend with an accelerated velocity, and would communicate an
accelerated motion to the wheel A. This effect, however, is interrupted
by the following contrivance:--L M is a pendulum vibrating on the
centre L, and so regulated that the time of its oscillation is one
second. The pallets I and K are connected with the pendulum, so as to
oscillate with it. In the position of the pendulum represented in the
figure, the pallet I stops the motion of the wheel A, and entirely
suspends the action of the weight W, _fig. 110._, so that for
a moment the entire machine is motionless. The weight M, however,
falls by its gravity towards the lowest position, and disengages the
pallet I from the tooth of the wheel. The weight W begins then to take
effect, and the wheel A turns from A towards B. Meanwhile the pendulum
M oscillates to the other side, and the pallet K falls under a tooth
of the wheel A, and checks for a moment its further motion. On the
returning vibration the pallet K becomes again disengaged, and allows
the tooth of the wheel to escape, and by the influence of the weight W
another tooth passes before the motion of the wheel A is again checked
by the interposition of the pallet I.

From this explanation it will appear that, in two vibrations of the
pendulum, one tooth of the wheel A passes the pallet I, and therefore,
if the wheel A be furnished with 30 teeth, it will be allowed to make
one revolution during 60 vibrations of the pendulum. If, therefore, the
pendulum be regulated so as to vibrate seconds, this wheel will revolve
once in a minute. From the action of the pallets in checking the motion
of the wheel A, and allowing its teeth alternately to _escape_, this
has been called the _escapement_ wheel; and the wheel and pallets
together are generally called the _escapement_, or _’scapement_.

We have already explained, that by reason of the friction on the
points of support, and other causes, the swing of the pendulum would
gradually diminish, and its vibration at length cease. This, however,
is prevented by the action of the teeth of the scapement wheel upon the
pallets, which is just sufficient to communicate that quantity of force
to the pendulum which is necessary to counteract the retarding effects,
and to maintain its motion. It thus appears, that although the effect
of the gravity of the weight W in giving motion to the machine is at
intervals suspended, yet this part of the force is not lost, being,
during these intervals, employed in giving to the pendulum all that
motion which it would lose by the resistances to which it is inevitably
exposed.

In stationary clocks, and in other cases in which the bulk of the
machine is not an objection, a descending weight is used as the
moving power. But in watches and portable chronometers, this would be
attended with evident inconvenience. In such cases, a spiral spring,
called the _mainspring_, is the moving power. The manner in which this
spring communicates rotation to an axis, and the ingenious method of
equalising the effect of its variable elasticity by giving to it a
leverage, which increases as the elastic force diminishes, have been
already explained. (255.)

A similar objection lies against the use of a pendulum in portable
chronometers. A spiral spring of a similar kind, but infinitely
more delicate, called a _hair spring_, is substituted in its place.
This spring is connected with a nicely-balanced wheel, called _the
balance wheel_, which plays in pivots. When this wheel is turned to
a certain extent in one direction, the hair spring is coiled up, and
its elasticity causes the wheel to recoil, and return to a position
in which the energy of the spring acts in the opposite direction.
The balance wheel then returns, and continually vibrates in the same
manner. The axis of this wheel is furnished with pallets similar to
those of the pendulum, which are alternately engaged with the teeth of
a crown wheel, which takes the place of the scapement wheel already
described.

A general view of the work of a common watch is represented in
_fig. 111._ _bis._ A is the balance wheel bearing pallets _p_
_p_ upon its axis; C is the crown wheel, whose teeth are suffered to
escape alternately by those pallets in the manner already described
in the scapement of a clock. On the axis of the crown wheel is placed
a pinion _d_, which drives another crown wheel K. On the axis of this
is placed the pinion _c_, which plays in the teeth of the third wheel
L. The pinion _b_ on the axis of L is engaged with the wheel M, called
the centre wheel. The axle of this wheel is carried up through the
centre of the dial. A pinion _a_ is placed upon it, which works in
the great wheel N. On this wheel the mainspring immediately acts.
O P is the mainspring stripped of its barrel. The axis of the
wheel M passing through the centre of the dial is squared at the end
to receive the minute hand. A second pinion Q is placed upon this
axle which drives a wheel T. On the axle of this wheel a pinion _g_
is placed, which drives the hour wheel V. This wheel is placed upon a
tubular axis, which incloses within it the axis of the wheel M. This
tubular axis passing through the centre of the dial, carries the hour
hand. The wheels A, B, C, D, E, _fig. 110._, correspond to the wheels
C, K, L, M, N, _fig. 112._; and the pinions _a_, _b_, _c_, _d_,
_e_, _fig. 109._, correspond to the pinions _d_, _c_, _b_, _a_,
_fig. 111_. From what has already been explained of these wheels,
it will be obvious that the wheel M, _fig. 111._, revolves once
in an hour, causing the minute hand to move round the dial once in
that time. This wheel at the same time turns the pinion Q which leads
the wheel T. This wheel again turns the pinion _g_ which leads the
hour wheel V. The leaves and teeth of these pinions and wheels are
proportioned, as already explained, so that the wheel V revolves once
during twelve revolutions of the wheel M. The hour hand, therefore,
which is carried by the tubular axle of the wheel V, moves once round
the dial in twelve hours.

Our object here has not been to give a detailed account of watch and
clock work, a subject for which we must refer the reader to the proper
department of this work. Such a general account has only been attempted
as may explain how tooth and pinion work may be applied to regulate
motion.

[Illustration: _H. Adlard, sc._

_London, Pubd. by Longman & Co._]

CHAP. XV.

OF THE PULLEY.

(267.) The next class of simple machines, which present themselves
to our attention, is that which we have called the _cord_. If a rope
were perfectly flexible, and were capable of being bent over a sharp
edge, and of moving upon it without friction, we should be enabled by
its means to make a force in any one direction overcome resistance, or
communicate motion in any other direction. Thus if P, _fig. 112._,
be such an edge, a perfectly flexible rope passing over it would be
capable of transmitting a force S F to a resistance Q R,
so as to support or overcome R, or by a motion in the direction of
S F to produce another motion in the direction R Q. But as
no materials of which ropes can be constructed can give them perfect
flexibility, and as in proportion to the strength by which they are
enabled to transmit force their rigidity increases, it is necessary,
in practice, to adopt means to remove or mitigate those effects which
attend imperfect flexibility, and which would otherwise render cords
practically inapplicable as machines.

When a cord is used to transmit a force from one direction to another,
its stiffness renders some force necessary in bending it over the
angle P, which the two directions form; and if the angle be sharp,
the exertion of such a force may be attended with the rupture of the
cord. If, instead of bending the rope at one point over a single angle,
the change of direction were produced by successively deflecting it
over several angles, each of which would be less sharp than a single
one could be, the force requisite for the deflection, as well as the
liability of rupturing the cord, would be considerably diminished. But
this end will be still more perfectly attained if the deflection of the
cord be produced by bending it over the surface of a curve.

If a rope were applied only to sustain, and not to move a weight,
this would be sufficient to remove the inconveniences arising from
its rigidity. But when motion is to be produced, the rope, in passing
over the curved surface, would be subject to excessive friction, and
consequently to rapid wear. This inconvenience is removed by causing
the surface on which the rope runs to move with it, so that no more
friction is produced than would arise from the curved surface rolling
upon the rope.

(268.) All these ends are attained by the common pulley, which consists
of a wheel called a _sheave_, fixed in a block and turning on a pivot.
A groove is formed in the edge of the wheel in which the rope runs,
the wheel revolving with it. Such an apparatus is represented in
_fig. 113._

We shall, for the present, omit the consideration of that part of the
effects of the stiffness and friction of the machine which is not
removed by the contrivance just explained, and shall consider the rope
as perfectly flexible and moving without friction.

From the definition of a flexible cord, it follows, that its tension,
or the force by which it is stretched throughout its entire length,
must be uniform. From this principle, and this alone, all the
mechanical properties of pulleys may be derived.

Although, as already explained, the whole mechanical efficacy of this
machine depends on the qualities of the cord, and not on those of the
block and sheave, which are only introduced to remove the accidental
effects of stiffness and friction; yet it has been usual to give the
name pulley to the block and sheave, and a combination of blocks,
sheaves, and ropes is called a _tackle_.

(269.) When the rope passes over a single wheel, which is fixed in
its position, as in _fig. 113._, the machine is called a _fixed
pulley_. Since the tension of the cord is uniform throughout its
length, it follows, that in this machine the power and weight are
equal. For the weight stretches that part of the cord which is between
the weight and pulley, and the power stretches that part between the
power and the pulley. And since the tension throughout the whole length
is the same, the weight must be equal to the power.

Hence it appears that no mechanical advantage is gained by this
machine. Nevertheless, there is scarcely any engine, simple or complex,
attended with more convenience. In the application of power, whether
of men or animals, or arising from natural forces, there are always
some directions in which it may be exerted to much greater convenience
and advantage than others, and in many cases the exertion of these
powers is limited to a single direction. A machine, therefore, which
enables us to give the most advantageous direction to the moving power,
whatever be the direction of the resistance opposed to it, contributes
as much practical convenience as one which enables a small power to
balance or overcome a great weight. In directing the power against the
resistance, it is often necessary to use two fixed pulleys. Thus, in
elevating a weight A, _fig. 114._, to the summit of a building,
by the strength of a horse moving below, two fixed pulleys B and C may
be used. The rope is carried from A over the pulley B; and, passing
downwards, is brought under C, and finally drawn by the animal on
the horizontal plane. In the same manner sails are spread, and flags
hoisted on the yards and masts of a ship, by sailors pulling a rope on
the deck.

By means of the fixed pulley a man may raise himself to a considerable
height, or descend to any proposed depth. If he be placed in a chair
or bucket attached to one end of a rope which is carried over a fixed
pulley, by laying hold of this rope on the other side, as represented
in _fig. 115._, he may, at will, descend to a depth equal to half
of the entire length of the rope, by continually yielding rope on the
one side, and depressing the bucket or chair by his weight on the
other. Fire-escapes have been constructed on this principle, the fixed
pulley being attached to some part of the building.

(270.) A _single moveable pulley_ is represented in _fig. 116._
A cord is carried from a fixed point F, and passing through a block
B, attached to a weight W, passes over a fixed pulley C, the power
being applied at P. We shall first suppose the parts of the cord on
each side the wheel B to be parallel; in this case, the whole weight W
being sustained by the parts of the cords B C and B F, and
these parts being equally stretched (268.), each must sustain half the
weight, which is therefore the tension of the cord. This tension is
resisted by the power at P, which must, therefore, be equal to half the
weight. In this machine, therefore, the weight is twice the power.

(271.) If the parts of the cord B C and B F be not parallel,
as in _fig. 117._, a greater power than half the weight is
therefore necessary to sustain it. To determine the power necessary
to support a given weight, in this case take the line B A in the
vertical direction, consisting of as many inches as the weight consists
of ounces; from A draw A D parallel to B C, and A E
parallel to B F; the force of the weight represented by A B
will be equivalent to two forces represented by B D and B E.
(74.) The number of inches in these lines respectively will represent
the number of ounces which are equivalent to the tensions of the parts
B F and B C of the cord. But as these tensions are equal,
B D and B E must be equal, and each will express the amount
of the power P, which stretches the cord at P C.

It is evident that the four lines, A E, E B, B D, and
D A, are equal. And as each of them represents the power, the
weight which is represented by A B must be less than twice the
power which is represented by A E and E B taken together. It
follows, therefore, that as parts of the ropes which support the weight
depart from parallelism the machine becomes less and less efficacious;
and there are certain obliquities at which the equilibrating power
would be much greater than the weight.

(272.) The mechanical power of pulleys admits of being almost
indefinitely increased by combination. Systems of pulleys may be
divided into two classes; those in which a single rope is used, and
those which consist of several distinct ropes. _Fig. 118._ and
_119._ represent two systems of pulleys, each having a single rope.
The weight is in each case attached to a moveable block, B, in which
are fixed two or more wheels; A is a fixed block, and the rope is
successively passed over the wheels above and below, and, after passing
over the last wheel above, is attached to the power. The tension of
that part of the cord to which the power is attached is produced by
the power, and therefore equivalent to it, and the same tension must
extend throughout its whole length. The weight is sustained by all
those parts of the cord which pass from the lower block, and as the
force which stretches them all is the same, viz. that of the power,
the effect of the weight must be equally distributed among them, their
directions being supposed to be parallel. It will be evident, from
this reasoning, that the weight will be as many times greater than the
power as the number of cords which support the lower block. Thus, if
there be six cords, each cord will support a sixth part of the weight,
that is, the weight will be six times the tension of the cord, or six
times the power. In _fig. 118._ the cord is represented as being
finally attached to a hook on the upper block. But it may be carried
over an additional wheel fixed in that block, and finally attached
to a hook in the lower block, as in _fig. 119._, by which one
will be added to the power of the machine, the number of cords at
the lower block being increased by one. In the system represented in
_fig. 118._ the wheels are placed in the blocks one above the
other; in _fig. 119._ they are placed side by side. In all systems
of pulleys of this class, the weight of the lower block is to be
considered as a part of the weight to be raised, and in estimating the
power of the machine, this should always be attended to.

(273.) When the power of the machine, and therefore the number of
wheels, is considerable, some difficulty arises in the arrangement of
the wheels and cords. The celebrated Smeaton contrived a tackle, which
takes its name from him, in which there are ten wheels in each block:
five large wheels placed side by side, and five smaller ones similarly
placed above them in the lower block, and below them in the upper.
_Fig. 120._ represents Smeaton’s blocks without the rope. The
wheels are marked with the numbers 1, 2, 3, &c., in the order in which
the rope is to be passed over them. As in this pulley 20 distinct parts
of the rope support the lower block, the weight, including the lower
block, will be 20 times the equilibrating power.

(274.) In all these systems of pulleys, every wheel has a separate
axle, and there is a distinct wheel for every turn of the rope at each
block. Each wheel is attended with friction on its axle, and also with
friction between the sheave and block. The machine is by this means
robbed of a great part of its efficacy, since, to overcome the friction
alone, a considerable power is in most cases necessary.

An ingenious contrivance has been suggested, by which all the advantage
of a large number of wheels may be obtained without the multiplied
friction of distinct sheaves and axles. To comprehend the excellence
of this contrivance, it will be necessary to consider the rate at
which the rope passes over the several wheels of such a system, as
_fig. 118._ If one foot of the rope G F pass over the
pulley F, two feet must pass over the pulley E, because the distance
between F and E being shortened one foot, the total length of the rope
G F E must be shortened two feet. These two feet of rope
must pass in the direction E D, and the wheel D, rising one foot,
three feet of rope must consequently pass over it. These three feet of
rope passing in the direction D C, and the rope D C being
also shortened one foot by the ascent of the lower block, four feet of
rope must pass over the wheel C. In the same way it may be shown that
five feet must pass over B, and six feet over A. Thus, whatever be
the number of wheels in the upper and lower blocks, the parts of the
rope which pass in the same time over the wheels in the lower block
are in the proportion of the odd numbers 1, 3, 5, &c.; and those which
pass over the wheels in the upper block in the same time, are as the
even numbers 2, 4, 6, &c. If the wheels were all of equal size, as in
_fig. 119._, they would revolve with velocities proportional to
the rate at which the rope passes over them. So that, while the first
wheel below revolves once, the first wheel above will revolve twice;
the second wheel below three times; the second wheel above, four times,
and so on. If, however, the wheels differed in size in proportion to
the quantity of rope which must pass over them, they would evidently
revolve in the same time. Thus, if the first wheel above were twice the
size of the first wheel below, one revolution would throw off twice the
quantity of rope. Again, if the second wheel below were thrice the size
of the first wheel below, it would throw off in one revolution thrice
the quantity of rope, and so on. Wheels thus proportioned, revolving
in exactly the same time, might be all placed on one axle, and would
partake of one common motion, or, what is to the same effect, several
grooves might be cut upon the face of one solid wheel, with diameters
in the proportion of the odd numbers 1, 3, and 5, &c., for the lower
pulley, and corresponding grooves on the face of another solid wheel
represented by the even numbers 2, 4, 6, &c., for the upper pulley. The
rope being passed successively over the grooves of such wheels, would
be thrown off exactly in the same manner as if every groove were upon a
separate wheel, and every wheel revolved independently of the others.
Such is White’s pulley, represented in _fig. 121._

The advantage of this machine, when accurately constructed, is very
considerable. The friction, even when great resistances are to be
opposed, is very trifling; but, on the other hand, it has corresponding
disadvantages which greatly circumscribe its practical utility. In the
workmanship of the grooves great difficulty is found in giving them
the exact proportions. In doing which, the thickness of the rope must
be accurately allowed for; and consequently it follows, that the same
pulley can never act except with a rope of a particular diameter. A
very slight deviation from the true proportion of the grooves will
cause the rope to be unequally stretched, and will throw on some parts
of it an undue proportion of the weight, while other parts become
nearly, and sometimes altogether slack. Besides these defects, the rope
is so liable to derangement by being thrown out of the grooves, that
the pulley can scarcely be considered portable.

For these and other reasons, this machine, ingenious as it
unquestionably is, has never been extensively used.

(275.) In the several systems of pulleys just explained, the hook to
which the fixed block is attached supports the entire of both the power
and weight. When the machine is in equilibrium, the power only supports
so much of the weight as is equal to the tension of the cord, all the
remainder of the weight being thrown on the fixed point, according to
what was observed in (225.)

If the power be moved so as to raise the weight, it will move with a
velocity as many times greater than that of the weight as the weight
itself is greater than the power. Thus in _fig. 118._ if the
weight attached to the lower block ascend one foot, six feet of line
will pass over the pulley A, according to what has been already proved.
Thus, the power will descend through six feet, while the weight rises
one foot. But, in this case, the weight is six times the power. All the
observations in (226.) will therefore be applicable to the cases of
great weights raised by small powers by means of the system of pulleys
just described.

(276.) When two or more ropes are used, pulleys may be combined in
various ways so as to produce any degree of mechanical effect. If
to any of the systems already described a single moveable pulley
be added, the power of the machine would be doubled. In this case,
the second rope is attached to the hook of the lower block, as in
_fig. 122._, and being carried through a moveable pulley
attached to the weight, it is finally brought up to a fixed point. The
tension of the second cord is equal to half the weight (270.); and
therefore the power P, by means of the first cord, will have only half
the tension which it would have if the weight were attached to the
lower block. A moveable pulley thus applied is called a _runner_.

[Illustration: _C. Varley, del._     _H. Adlard, sc._

_London, Pubd. by Longman & Co._]

(277.) Two systems of pulleys, called _Spanish bartons_, having
each two ropes, are represented in _fig. 123._ The tension of
the rope P A B C in the first system is equal to
the power; and therefore the parts B A and B C support a
portion of the weight equal to twice the power. The rope E A
supports the tensions of A P and A B; and therefore the
tension of A E D is twice the power. Thus, the united
tensions of the ropes which support the pulley B is four times the
power, which is therefore the amount of the weight. In the second
system, the rope P A D is stretched by the power. The rope
A E B C acts against the united tensions A P and
A D; and therefore the tension of A E or E B is twice
the power. Thus, the weight acts against three tensions; two of which
are equal to twice the power, and the remaining one is equal to the
power. The weight is therefore equal to five times the power.

A single rope may be so arranged with one moveable pulley as to support
a weight equal to three times the power. In _fig. 124._ this
arrangement is represented, where the numbers sufficiently indicate the
tension of the rope, and the proportion of the weight and power. In
_fig. 125._ another method of producing the same effect with two
ropes is represented.

(278.) If several single moveable pulleys be made successively to act
upon each other, the effect is doubled by every additional pulley:
such a system as this is represented in _fig. 126._ The tension
of the first rope is equal to the power; the second rope acts against
twice the tension of the first, and therefore it is stretched with
a force equal to twice the power: the third rope acts against twice
this tension, and therefore it is stretched with a force equal
to four times the power, and so on. In the system represented in
_fig. 126._ there are three ropes, and the weight is eight times
the power. Another rope would render it sixteen times the power, and so
on.

In this system, it is obvious that the ropes will require to have
different degrees of strength, since the tension to which they are
subject increases in a double proportion from the power to the weight.

(279.) If each of the ropes, instead of being attached to fixed points
at the top, are carried over fixed pulleys, and attached to the several
moveable pulleys respectively, as in _fig. 127._, the power of
the machine will be greatly increased; for in that case the forces
which stretch the successive ropes increase in a treble instead of a
double proportion, as will be evident by attending to the numbers which
express the tensions in the figure. One rope would render the weight
three times the power, two ropes nine times, three ropes twenty-seven
times, and so on. An arrangement of pulleys is represented in _fig.
128._, by which each rope, instead of being finally attached to a fixed
point, as in _fig. 126._, is attached to the weight. The weight
is in this case supported by three ropes; one stretched with a force
equal to the power; another with a force equal to twice the power;
and a third with a force equal to four times the power. The weight is
therefore, in this case, seven times the power.

(280.) If the ropes, instead of being attached to the weight, pass
through wheels, as in _fig. 129._, and are finally attached to the
pulleys above, the power of the machine will be considerably increased.
In the system here represented the weight is twenty-six times the power.

(281.) In considering these several combinations of pulleys, we have
omitted to estimate the effects produced by the weights of the sheaves
and blocks. Without entering into the details of this computation,
it may be observed generally, that in the systems represented in
_figs. 126._, _127._ the weight of the wheel and blocks acts
against the power; but that in _figs. 128._ and _129._ they
assist the powers in supporting the weight. In the systems represented
in _fig. 123._ the weight of the pulleys, to a certain extent,
neutralise each other.

(282.) It will in all cases be found, that that quantity by which the
weight exceeds the power is supported by fixed points; and therefore,
although it be commonly stated that a small power supports a great
weight, yet in the pulley, as in all other machines, the power supports
no more of the weight than is exactly equal to its own amount. It
will not be necessary to establish this in each of the examples which
have been given: having explained it in one instance, the student
will find no difficulty in applying the same reasoning to others. In
_fig. 126._, the fixed pulley sustains a force equal to twice the
power, and by it the power giving tension to the first rope sustains a
part of the weight equal to itself. The first hook sustains a portion
of the weight equal to the tension of the first string, or to the
power. The second hook sustains a force equal to twice the power; and
the third hook sustains a force equal to four times the power. The
three hooks therefore sustain a portion of the weight equal to seven
times the power; and the weight itself being eight times the power, it
is evident that the part of the weight which remains to be supported by
the power is equal to the power itself.

(283.) When a weight is raised by any of the systems of pulleys which
have been last described, the proportion between the velocity of
the weight and the velocity of the power, so frequently noticed in
other machines, will always be observed. In the system of pulleys
represented in _fig. 126._ the weight being eight times the power,
the velocity of the power will be eight times that of the weight. If
the power be moved through eight feet, that part of the rope between
the fixed pulley and the first moveable pulley will be shortened by
eight feet. And since the two parts which lie above the first moveable
pulley must be equally shortened, each will be diminished by four feet;
therefore the first pulley will rise through four feet while the power
moves through eight feet. In the same way it may be shown, that while
the first pulley moves through four feet, the second moves through two;
and while the second moves through two, the third, to which the weight
is attached, is raised through one foot. While the power, therefore, is
carried through eight feet, the weight is moved through one foot.

By reasoning similar to this, it may be shown that the space through
which the power is moved in every case is as many times greater than
the height through which the weight is raised, as the weight is greater
than the power.

(284.) From its portable form, cheapness of construction, and the
facility with which it may be applied in almost every situation,
the pulley is one of the most useful of the simple machines. The
mechanical advantage, however, which it appears in theory to possess
is considerably diminished in practice, owing to the stiffness of the
cordage, and the friction of the wheels and blocks. By this means it
is computed that in most cases so great a proportion as two thirds of
the power is lost. The pulley is much used in building, where weights
are to be elevated to great heights. But its most extensive application
is found in the rigging of ships, where almost every motion is
accomplished by its means.

(285.) In all the examples of pulleys, we have supposed the parts of
the rope sustaining the weight and each of the moveable pulleys to be
parallel to each other. If they be subject to considerable obliquity,
the relative tensions of the different ropes must be estimated
according to the principle applied in (271.)

CHAP. XVI.

ON THE INCLINED PLANE, WEDGE, AND SCREW.

(286.) The inclined plane is the most simple of all machines. It is
a hard plane surface forming some angle with a horizontal plane,
that angle not being a right angle. When a weight is placed on such
a plane, a two-fold effect is produced. A part of the effect of the
weight is resisted by the plane, and produces a pressure upon it; and
the remainder urges the weight down the plane, and would produce a
pressure against any surface resisting its motion placed in a direction
perpendicular to the plane (131.)

Let A B, _fig. 130._, be such a plane, B C its
horizontal base, A C its height, and A B C its angle
of elevation. Let W be a weight placed upon it. This weight acts in
the vertical direction W D, and is equivalent to two forces,
W F perpendicular to the plane, and W E directed down the
plane (74.) If a plane be placed at right angles to the inclined
plane below W, it will resist the descent of the weight, and sustain
a pressure expressed by W E. Thus, the weight W resting in the
corner, instead of producing one pressure in the direction W D,
will produce two pressures, one expressed by W F upon the inclined
plane, and the other expressed by W E upon the resisting plane.
These pressures respectively have the same proportion to the entire
weight as W F and W E have to W D, or as D E and
W E have to W D, because D E is equal to W F. Now
the triangle W E D is in all respects similar to the triangle
A B C, the one differing from the other only in the scale on
which it is constructed. Therefore, the three lines A C, C B,
and B A, are in the same proportion to each other as the lines
W E, E D, and W D. Hence, A B has to A C the
same proportion as the whole weight has to the pressure directed toward
B, and A B has to B C the same proportion as the whole
weight has to the pressure on the inclined plane.

We have here supposed the weight to be sustained upon the inclined
plane by a hard plane fixed at right angles to it. But the power
necessary to sustain the weight will be the same in whatever way it is
applied, provided it act in the direction of the plane. Thus, a cord
may be attached to the weight, and stretched towards A, or the hands of
men may be applied to the weight below it, so as to resist its descent
towards B. But in whatever way it be applied, the amount of the power
will be determined in the same manner. Suppose the weight to consist
of as many pounds as there are inches in A B, then the power
requisite to sustain it upon the plane will consist of as many pounds
as there are inches in A C, and the pressure on the plane will
amount to as many pounds as there are inches in B C.

From what has been stated it may easily be inferred that the less the
elevation of the plane is, the less will be the power requisite to
sustain a given weight upon it, and the greater will be the pressure
upon it. Suppose the inclined plane A B to turn upon a hinge
at B, and to be depressed so that its angle of elevation shall be
diminished, it is evident that as this angle decreases the height of
the plane decreases, and its base increases. Thus, when it takes the
position B A′, the height A′ C′ is less than the former
height A C, while the base B C′ is greater than the former
base B C. The power requisite to support the weight upon the plane
in the position B A′ is represented by A′ C′, and is as much
less than the power requisite to sustain it upon the plane A B,
as the height A′ C′ is less than the height A C. On the
other hand, the pressure upon the plane in the position B A′ is
as much greater than the pressure upon the plane B A, as the base
B C′ is greater than the base B C.

(287.) The power of an inclined plane, considered as a machine, is
therefore estimated by the proportion which its length bears to its
height. This power is always increased by diminishing the elevation of
the plane.

Roads which are not level may be regarded as inclined planes, and
loads drawn upon them in carriages, considered in reference to the
powers which impel them, are subject to all the conditions which have
been established for inclined planes. The inclination of the road is
estimated by the height corresponding to some proposed length. Thus it
is said to rise one foot in fifteen, one foot in twenty, &c., meaning
that if fifteen or twenty feet of the road be taken as the length of
an inclined plane, such as A B, the corresponding height will be
one foot. Or the same may be expressed thus: that if fifteen or twenty
feet be measured upon the road, the difference of the levels of the two
extremities of the distance measured is one foot. According to this
method of estimating the inclination of roads, the power requisite to
sustain a load upon them (setting aside the effect of friction), is
always proportional to that elevation. Thus, if a road rise one foot in
twenty, a power of one ton will be sufficient to sustain twenty tons,
and so on.

On a horizontal plane the only resistance which the power has
to overcome is the friction of the load with the plane, and the
consideration of this being for the present omitted, a weight once put
in motion would continue moving for ever, without any further action of
the power. But if the plane be inclined, the power will be expended in
raising the weight through the perpendicular height of the plane. Thus,
in a road which rises one foot in ten, the power is expended in raising
the weight through one perpendicular foot for every ten feet of the
road over which it is moved. As the expenditure of power depends upon
the rate at which the weight is raised perpendicularly, it is evident
that the greater the inclination of the road is, the slower the motion
must be with the same force. If the energy of the power be such as to
raise the weight at the rate of one foot per minute, the weight may be
moved in each minute through that length of the road which corresponds
to a rise of one foot. Thus, if two roads rise one at the rate of a
foot in fifteen feet, and the other at the rate of one foot in twenty
feet, the same expenditure of power will move the weight through
fifteen feet of the one, and twenty feet of the other at the same rate.

From such considerations as these, it will readily appear that it may
often be more expedient to carry a road through a circuitous route
than to continue it in the most direct course; for though the measured
length of road may be considerably greater than in the former case, yet
more may be gained in speed with the same expenditure of power than is
lost by the increase of distance. By attending to these circumstances,
modern road-makers have greatly facilitated and expedited the
intercourse between distant places.

(288.) If the power act obliquely to the plane, it will have a twofold
effect; a part being expended in supporting or drawing the weight,
and a part in diminishing or increasing the pressure upon the plane.
Let W P, _fig. 130._, be the power. This will be equivalent
to two forces, W F′, perpendicular to the plane, and W E′
in the direction of the plane. (74.) In order that the power should
sustain the weight, it is necessary that that part W E′ of the
power which acts in the direction of the plane should be equal to that
part W E, _fig. 130._, of the weight which acts down the
plane. The other part W F′ of the power acting perpendicular to
the plane is immediately opposed to that part W F of the weight
which produces pressure. The pressure upon the plane will therefore
be diminished by the amount of W F′. The amount of the power
which will equilibrate with the weight may, in this case, be found
as follows. Take W E′ equal to W E, and draw E′ P
perpendicular to the plane, and meeting the direction of the power.
The proportion of the power to the weight will be that of W P to
W D. And the proportion of the pressure to the weight will be that
of the difference between W F and W F′ to W D. If the
amount of the power have a less proportion to the weight than W P
has to W D, it will not support the body on the plane, but will
allow it to descend. And if it have a greater proportion, it will draw
the weight up the plane towards A.

(289.) It sometimes happens that a weight upon one inclined plane is
raised or supported by another weight upon another inclined plane.
Thus, if A B and A B′, _fig. 131._, be two inclined
planes forming an angle at A, and W W′ be two weights placed
upon these planes, and connected by a cord passing over a pulley at
A, the one weight will either sustain the other, or one will descend,
drawing the other up. To determine the circumstances under which these
effects will ensue, draw the lines W D and W′ D′ in the
vertical direction, and take upon them as many inches as there are
ounces in the weights respectively. W D and W′ D′ being the
lengths thus taken, and therefore representing the weights, the lines
W E and W′ E′ will represent the effects of these weights
respectively down the planes. If W E and W′ E′ be equal, the
weights will sustain each other without motion. But if W E be
greater than W′ E′, the weight W will descend, drawing the weight
W′ up. And if W′ E′ be greater than W E, the weight W′ will
descend, drawing the weight W up. In every case the lines W F and
W′ F′ will represent the pressures upon the planes respectively.

It is not necessary, for the effect just described, that the inclined
planes should, as represented in the figure, form an angle with each
other. They may be parallel, or in any other position, the rope being
carried over a sufficient number of wheels placed so as to give it the
necessary deflection. This method of moving loads is frequently applied
in great public works where rail-roads are used. Loaded waggons descend
one inclined plane, while other waggons, either empty or so loaded as
to permit the descent of those with which they are connected, are drawn
up the other.

(290.) In the application of the inclined plane which we have hitherto
noticed, the machine itself is supposed to be fixed in its position,
while the weight or load is moved upon it. But it frequently happens
that resistances are to be overcome which do not admit of being thus
moved. In such cases, instead of moving the load upon the planes,
the plane is to be moved under or against the load. Let D E,
_fig. 132._, be a heavy beam secured in a vertical position
between guides F G and H I, so that it is free to move
upwards and downwards, but not laterally. Let A B C be an
inclined plane, the extremity of which is placed beneath the end of
the beam. A force applied to the back of this plane A C, in the
direction C B, will urge the plane under the beam so as to raise
the beam to the position represented in _fig. 133._ Thus, while
the inclined plane is moved through the distance C B, the beam is
raised through the height C A.

(291.) When the inclined plane is applied in this manner, it is called
a _wedge_. And if the power applied to the back were a continued
pressure, its proportion to the weight would be that of A C to
C B. It follows, therefore, that the more acute the angle B is,
the more powerful will be the wedge.

In some cases, the wedge is formed of two inclined planes, placed base
to base, as represented in _fig. 134._ The theoretical estimation
of the power of this machine is not applicable in practice with any
degree of accuracy. This is in part owing to the enormous proportion
which the friction in most cases bears to the theoretical value of
the power, but still more to the nature of the power generally used.
The force of a blow is of a nature so wholly different from continued
forces, such as the pressure of weights, or the resistance offered by
the cohesion of bodies, that it admits of no numerical comparison with
them. Hence we cannot properly state the proportion which the force
of a blow bears to the amount of a weight or resistance. The wedge is
almost invariably urged by percussion; while the resistances which it
has to overcome are as constantly forces of the other kind. Although,
however, no exact numerical comparison can be made, yet it may be
stated in a general way that the wedge is more and more powerful as its
angle is more acute.

[Illustration: _C. Varley, del._     _H. Adlard, sc._

_London, Pubd. by Longman & Co._]

In the arts and manufactures, wedges are used where enormous force
is to be exerted through a very small space. Thus it is resorted to
for splitting masses of timber or stone. Ships are raised in docks
by wedges driven under their keels. The wedge is the principal agent
in the oil-mill. The seeds from which the oil is to be extracted are
introduced into hair bags, and placed between planes of hard wood.
Wedges inserted between the bags are driven by allowing heavy beams to
fall on them. The pressure thus excited is so intense, that the seeds
in the bags are formed into a mass nearly as solid as wood. Instances
have occurred in which the wedge has been used to restore a tottering
edifice to its perpendicular position.

All cutting and piercing instruments, such as knives, razors, scissors,
chisels, &c., nails, pins, needles, awls, &c. are wedges. The angle
of the wedge, in these cases, is more or less acute, according to
the purpose to which it is to be applied. In determining this, two
things are to be considered--the mechanical power, which is increased
by diminishing the angle of the wedge; and the strength of the tool,
which is always diminished by the same cause. There is, therefore,
a practical limit to the increase of the power, and that degree of
sharpness only is to be given to the tool which is consistent with the
strength requisite for the purpose to which it is to be applied. In
tools intended for cutting wood, the angle is generally about 30°. For
iron it is from 50° to 60°; and for brass, from 80° to 90°. Tools which
act by pressure may be made more acute than those which are driven by a
blow; and in general the softer and more yielding the substance to be
divided is, and the less the power required to act upon it, the more
acute the wedge may be constructed.

In many cases the utility of the wedge depends on that which is
entirely omitted in its theory, viz. the friction which arises between
its surface and the substance which it divides. This is the case when
pins, bolts, or nails are used for binding the parts of structures
together; in which case, were it not for the friction, they would
recoil from their places, and fail to produce the desired effect. Even
when the wedge is used as a mechanical engine, the presence of friction
is absolutely indispensable to its practical utility. The power, as
has already been stated, generally acts by successive blows, and is
therefore subject to constant intermission, and but for the friction
the wedge would recoil between the intervals of the blows with as much
force as it had been driven forward. Thus the object of the labour
would be continually frustrated. The friction in this case is of the
same use as a ratchet wheel, but is much more necessary, as the power
applied to the wedge is more liable to intermission than in the cases
where ratchet wheels are generally used.

(292.) When a road directly ascends the side of a hill, it is to be
considered as an inclined plane; but it will not lose its mechanical
character, if, instead of directly ascending towards the top of the
hill, it winds successively round it, and gradually ascends so as after
several revolutions to reach the top. In the same manner a path may be
conceived to surround a pillar by which the ascent may be facilitated
upon the principle of the inclined plane. Winding stairs constructed in
the interior of great columns partake of this character; for although
the ascent be produced by successive steps, yet if a floor could
be made sufficiently rough to prevent the feet from slipping, the
ascent would be accomplished with equal facility. In such a case the
winding path would be equivalent to an inclined plane, bent into such
a form as to accommodate it to the peculiar circumstances in which it
would be required to be used. It will not be difficult to trace the
resemblance between such an adaptation of the inclined plane and the
appearances presented by the thread of a _screw_: and it may hence be
easily understood that a screw is nothing more than an inclined plane
constructed upon the surface of a cylinder.

This will, perhaps, be more apparent by the following contrivance:
Let A B, _fig. 135._, be a common round ruler, and let
C D E be a piece of white paper cut in the form of an
inclined plane, whose height C D is equal to the length of the
ruler A B, and let the edge C E of the paper be marked with
a broad black line: let the edge C D be applied to the ruler
A B, and being attached thereto, let the paper be rolled round
the ruler; the ruler will then present the appearance of a screw,
_fig. 136._ the thread of the screw being marked by the black
line C E, winding continually round the ruler. Let D F,
_fig. 135._, be equal to the circumference of the ruler, and draw
F G parallel to D C, and G H parallel to D E, the
part C G F D of the paper will exactly surround the
ruler once: the part C G will form one convolution of the thread,
and may be considered as the length of one inclined plane surrounding
the cylinder, C H being the corresponding height, and G H
the base. The power of the screw does not, as in the ordinary cases of
the inclined plane, act parallel to the plane or thread, but at right
angles to the length of the cylinder A B, or, what is to the same
effect, parallel to the base H G; therefore the proportion of the
power to the weight will be, according to principles already explained,
the same as that of C H to the space through which the power
moves parallel to H G in one revolution of the screw. H C is
evidently the distance between the successive positions of the thread
as it winds round the cylinder; and it appears from what has been just
stated, that the less this distance is, or, in other words, the finer
the thread is, the more powerful the machine will be.

(293.) In the application of the screw the weight or resistance is
not, as in the inclined plane and wedge, placed upon the surface of
the plane or thread. The power is usually transmitted by causing the
screw to move in a concave cylinder, on the interior surface of which
a spiral cavity is cut, corresponding exactly to the thread of the
screw, and in which the thread will move by turning round the screw
continually in the same direction. This hollow cylinder is usually
called the _nut_ or _concave screw_. The screw surrounded by its
spiral thread is represented in _fig. 137._; and a section of the
same playing in the nut is represented in _fig. 138._

There are several ways in which the effect of the power may be conveyed
to the resistance by this apparatus.

First, let us suppose that the nut A B is fixed. If the screw be
continually turned on its axis, by a lever E F inserted in one
end of it, it will be moved in the direction C D, advancing every
revolution through a space equal to the distance between two contiguous
threads. By turning the lever in an opposite direction, the screw will
be moved in the direction D C.

If the screw be fixed, so as to be incapable either of moving
longitudinally or revolving on its axis, the nut A B may be turned
upon the screw by a lever, and will move on the screw towards C or
towards D, according to the direction in which the lever is turned.

In the former case we have supposed the nut to be absolutely
immoveable, and in the latter case the screw to be absolutely
immoveable. It may happen, however, that the nut, though capable of
revolving, is incapable of moving longitudinally; and that the screw,
though incapable of revolving, is capable of moving longitudinally. In
that case, by turning the nut A B upon the screw by the lever, the
screw will be urged in the direction C D or D C, according to
the way in which the nut is turned.

The apparatus may, on the contrary, be so arranged, that the nut,
though incapable of revolving, is capable of moving longitudinally;
and the screw, though capable of revolving, is incapable of moving
longitudinally. In this case, by turning the screw in the one direction
or in the other, the nut A B will be urged in the direction
C D or D C.

All these various arrangements may be observed in different
applications to the machine.

(294.) A screw may be cut upon a cylinder by placing the cylinder in
a turning lathe, and giving it a rotatory motion upon its axis. The
cutting point is then presented to the cylinder, and moved in the
direction of its length, at such a rate as to be carried through the
distance between the intended thread, while the cylinder revolves
once. The relative motions of the cutting point and the cylinder being
preserved with perfect uniformity, the thread will be cut from one end
to the other. The shape of the threads may be either square, as in
_fig. 137._, or triangular, as in _fig. 139._

(295.) The screw is generally used in cases where severe pressure
is to be excited through small spaces; it is therefore the agent in
most presses. In _fig. 140._, the nut is fixed, and by turning
the lever, which passes through the head of the screw, a pressure is
excited upon any substance placed upon the plate immediately under
the end of the screw. In _fig. 141._, the screw is incapable of
revolving, but is capable of advancing in the direction of its length.
On the other hand, the nut is capable of revolving, but does not
advance in the direction of the screw. When the nut is turned by means
of the screw inserted in it, the screw advances in the direction of its
length, and urges the board which is attached to it upwards, so as to
press any substance placed between it and the fixed board above.

In cases where liquids or juices are to be expressed from solid bodies,
the screw is the agent generally employed. It is also used in coining,
where the impression of a die is to be made upon a piece of metal, and
in the same way in producing the impression of a seal upon wax or other
substance adapted to receive it. When soft and light materials, such
as cotton, are to be reduced to a convenient bulk for transportation,
the screw is used to compress them, and they are thus reduced into hard
dense masses. In printing, the paper is urged by a severe and sudden
pressure upon the types, by means of a screw.

(296.) As the mechanical power of the screw depends upon the relative
magnitude of the circumference through which the power revolves, and
the distance between the threads, it is evident, that, to increase
the efficacy of the machine, we must either increase the length
of the lever by which the power acts, or diminish the magnitude of
the thread. Although there is no limit in theory to the increase of
the mechanical efficacy by these means, yet practical inconvenience
arises which effectually prevents that increase being carried beyond
a certain extent. If the lever by which the power acts be increased,
the same difficulty arises as was already explained in the wheel and
axle (254.); the space through which the power should act would be so
unwieldy, that its application would become impracticable. If, on the
other hand, the power of the machine be increased by diminishing the
size of the thread, the strength of the thread will be so diminished,
that a slight resistance will tear it from the cylinder. The cases
in which it is necessary to increase the power of the machine, being
those in which the greatest resistances are to be overcome, the object
will evidently be defeated, if the means chosen to increase that power
deprive the machine of the strength which is necessary to sustain the
force to which it is to be submitted.

(297.) These inconveniences are removed by a contrivance of Mr. Hunter,
which, while it gives to the machine all the requisite strength and
compactness, allows it to have an almost unlimited degree of mechanical
efficacy.

This contrivance consists in the use of two screws, the threads of
which may have any strength and magnitude, but which have a very small
difference of breadth. While the working point is urged forward by
that which has the greater thread, it is drawn back by that which has
the less; so that during each revolution of the screw, instead of
being advanced through a space equal to the magnitude of either of
the threads, it moves through a space equal to their difference. The
mechanical power of such a machine will be the same as that of a single
screw having a thread, whose magnitude is equal to the difference of
the magnitudes of the two threads just mentioned.

Thus, without inconveniently increasing the sweep of the power, on the
one hand, or, on the other, diminishing the thread until the necessary
strength is lost, the machine will acquire an efficacy limited by
nothing but the smallness of the difference between the two threads.

This principle was first applied in the manner represented in
_fig. 142._ A is the greater thread, playing in the fixed nut; B
is the lesser thread, cut upon a smaller cylinder, and playing in a
concave screw, cut within the greater cylinder. During every revolution
of the screw, the cylinder A descends through a space equal to the
distance between its threads. At the same time the smaller cylinder
B ascends through a space equal to the distance between the threads
cut upon it: the effect is, that the board D descends through a space
equal to the difference between the threads upon A and the threads upon
B, and the machine has a power proportionate to the smallness of this
difference.

Thus, suppose the screw A has twenty threads in an inch, while the
screw B has twenty-one; during one revolution, the screw A will
descend through a space equal to the 20th part of an inch. If, during
this motion, the screw B did not turn within A, the board D would be
advanced through the 20th of an inch; but because the hollow screw
within A turns upon B, the screw B will, relatively to A, be raised in
one revolution through a space equal to the 21st part of an inch. Thus,
while the board D is depressed through the 20th of an inch by the screw
A, it is raised through the 21st of an inch by the screw B. It is,
therefore, on the whole, depressed through a space equal to the excess
of the 20th of an inch above the 21st of an inch, that is, through the
420th of an inch.

The power of this machine will, therefore, be expressed by the number
of times the 420th of an inch is contained in the circumference through
which the power moves.

(298.) In the practical application of this principle at present the
arrangement is somewhat different. The two threads are usually cut
on different parts of the same cylinder. If nuts be supposed to be
placed upon these, which are capable of moving in the direction of
the length, but not of revolving, it is evident that by turning the
screw once round, each nut will be advanced through a space equal to
the breadth of the respective threads. By this means the two nuts
will either approach each other, or mutually recede, according to the
direction in which the screw is turned, through a space equal to the
difference of the breadth of the threads, and they will exert a force
either in compressing or extending any substance placed between them,
proportionate to the smallness of that difference.

(299.) A toothed wheel is sometimes used instead of a nut, so that
the same quality by which the revolution of the screw urges the nut
forward is applied to make the wheel revolve. The screw is in this
case called an endless screw, because its action upon the wheel may be
continued without limit. This application of the screw is represented
in _fig. 143._ P is the winch to which the power is applied; and
its effect at the circumference of the wheel is estimated in the same
manner as the effect of the screw upon the nut. This effect is to be
considered as a power acting upon the circumference of the wheel; and
its proportion to the weight or resistance is to be calculated in the
same manner as the proportion of the power to the weight in the wheel
and axle.

(300.) We have hitherto considered the screw as an engine used to
overcome great resistances. It is also eminently useful in several
departments of experimental science, for the measurement of very
minute motions and spaces, the magnitude of which could scarcely be
ascertained by any other means. The very slow motion which may be
imparted to the end of a screw, by a very considerable motion in the
power, renders it peculiarly well adapted for this purpose. To explain
the manner in which it is applied--suppose a screw to be so cut as
to have fifty threads in an inch, each revolution of the screw will
advance its point through the fiftieth part of an inch. Now, suppose
the head of the screw to be a circle, whose diameter is an inch, the
circumference of the head will be something more than three inches:
this may be easily divided into a hundred equal parts distinctly
visible. If a fixed index be presented to this graduated circumference,
the hundredth part of a revolution of the screw may be observed, by
noting the passage of one division of the head under the index. Since
one entire revolution of the head moves the point through the fiftieth
of an inch, one division will correspond to the five thousandth of an
inch. In order to observe the motion of the point of the screw in this
case, a fine wire is attached to it, which is carried across the field
of view of a powerful microscope, by which the motion is so magnified
as to be distinctly perceptible.

A screw used for such purposes is called a _micrometer screw_. Such an
apparatus is usually attached to the limbs of graduated instruments,
for the purposes of astronomical and other observation. Without the
aid of this apparatus, no observation could be taken with greater
accuracy than the amount of the smallest division upon the limb. Thus,
if an instrument for measuring angles were divided into small arcs of
one minute, and an angle were observed which brought the index of the
instrument to some point between two divisions, we could only conclude
that the observed angle must consist of a certain number of degrees and
minutes, together with an additional number of seconds, which would
be unknown, inasmuch as there would be no means of ascertaining the
fraction of a minute between the index and the adjacent division of
the instrument. But if a screw be provided, the point of which moves
through a space equal to one division of the instrument, with sixty
revolutions of the head, and that the head itself be divided into
one hundred equal parts, each complete revolution of the screw will
correspond to the sixtieth part of a minute, or to one second, and each
division on the head of the screw will correspond to the hundredth part
of a second. The index being attached to this screw, let the head
be turned until the index be moved from its observed position to the
adjacent division of the limb. The number of complete revolutions of
the screw necessary to accomplish this will be the number of seconds;
and the number of parts of a revolution over the complete number of
revolutions will be the hundredth parts of a second necessary to be
added to the degrees and minutes primarily observed.

It is not, however, only to such instruments that the micrometer screw
is applicable; any spaces whatever may be measured by it. An instance
of its mechanical application may be mentioned in a steel-yard,
an instrument for ascertaining the amount of weights by a given
weight, sliding on a long graduated arm of a lever. The distance from
the fulcrum, at which this weight counterpoises the weight to be
ascertained, serves as a measure to the amount of that weight. When the
sliding weight happens to be placed between two divisions of the arm, a
micrometer screw is used to ascertain the fraction of the division.

Hunter’s screw, already described, seems to be well adapted to
micrometrical purposes; since the motion of the point may be rendered
indefinitely slow, without requiring an exquisitely fine thread, such
as in the single screw would be necessary.

CHAP. XVII.

ON THE REGULATION AND ACCUMULATION OF FORCE.

(301.) It is frequently indispensable, and always desirable, that the
operation of a machine should be regular and uniform. Sudden changes
in its velocity, and desultory variations in the effective energy
of its power, are often injurious or destructive to the apparatus
itself, and when applied to manufactures never fail to produce
unevenness in the work. To invent methods for insuring the regular
motion of machinery, by removing those causes of inequality which may
be avoided, and by compensating others, has therefore been a problem to
which much attention and ingenuity have been directed. This is chiefly
accomplished by controlling, and, as it were, measuring out the power
according to the exigencies of the machine, and causing its effective
energy to be always commensurate with the resistance which it has to
overcome.

[Illustration: _C. Varley, del._     _H. Adlard, sc._

_London, Pubd. by Longman & Co._]

Irregularity in the motion of machinery may proceed from one or more
of the following causes:--1. irregularity in the prime mover; 2.
occasional variation in the amount of the load or resistance; and, 3.
because, in the various positions which the parts of the machine assume
during its motion, the power may not be transmitted with equal effect
to the working point.

The energy of the prime mover is seldom if ever regular. The force of
water varies with the copiousness of the stream. The power which impels
the windmill is proverbially capricious. The pressure of steam varies
with the intensity of the furnace. Animal power, the result of will,
temper, and health is difficult of control. Human labour is most of
all unmanageable; hence no machine works so irregularly as one which
is manipulated. In some cases the moving force is subject, by the very
conditions of its existence, to constant variation, as in the example
of a spring, which gradually loses its energy as it recoils. (255.) In
many instances the prime mover is liable to regular intermission, and
is actually suspended for certain intervals of time. This is the case
in the single acting steam-engine, where the pressure of the steam
urges the descent of the piston, but is suspended during its ascent.

The load or resistance to which the machine is applied is not less
fluctuating. In mills there are a multiplicity of parts which are
severally liable to be occasionally disengaged, and to have their
operation suspended. In large factories for spinning, weaving,
printing, &c. a great number of separate spinning machines, looms,
presses, or other engines, are usually worked by one common mover, such
as a water-wheel or steam-engine. In these cases the number of machines
employed from time to time necessarily varies with the fluctuating
demand for the articles produced, and from other causes. Under such
circumstances the velocity with which every part of the machinery is
moved would suffer corresponding changes, increasing its rapidity with
every augmentation of the moving power or diminution of the resistance,
or being retarded in its speed by the contrary circumstances.

But even when the prime mover and the resistance are both regular, or
rendered so by proper contrivances, still it will rarely happen that
the machine by which the energy of the one is transmitted to the other
conveys this with unimpaired effect in all the phases of its operation.
To give a general notion of this cause of inequality to those who have
not been familiar with machinery would not be easy, without having
recourse to an example. For the present we shall merely state, that the
several moving parts of every machine assume in succession a variety of
positions; that at regular periods they return to their first position,
and again undergo the same succession of changes. In the different
positions through which they are carried in every period of motion,
the efficacy of the machine to transmit the power to the resistance
is different, and thus the effective energy of the machine in acting
upon the resistance would be subject to continual fluctuation. This
will be more clearly understood when we come to explain the methods of
counteracting the defect or equalising the action of the power upon the
resistance.

Such are the chief causes of the inequalities incidental to the
motion of machinery, and we now propose to describe a few of the many
ingenious contrivances which the skill of engineers has produced to
remove the consequent inconveniences.

(302.) Setting aside, for the present, the last cause of inequality,
and considering the machinery, whatever it be, to transmit the power to
the resistance without irregular interruption, it is evident that every
contrivance, having for its object to render the velocity uniform,
can only accomplish this by causing the variations of the power and
resistance to be proportionate to each other. This may be done either
by increasing or diminishing the power as the resistance increases or
diminishes; or by increasing or diminishing the resistance as the power
increases or diminishes.

According to the facilities or convenience presented by the peculiar
circumstances of the case either of these methods is adopted.

The contrivances for effecting this are called _regulators_. Most
regulators act upon that part of the machine which commands the supply
of the power by means of levers, or some other mechanical contrivance,
so as to check the quantity of the moving principle conveyed to the
machine when the velocity has a tendency to increase; and, on the
other hand, to increase that supply upon any undue abatement of its
speed. In a water-mill this is done by acting upon the shuttle; in a
wind-mill, by an adjustment of the sail-cloth; and in a steam-engine,
by opening or closing, in a greater or less degree, the valve by which
the cylinder is supplied with steam.

(303.) Of all the contrivances for regulating machinery, that which is
best known and most commonly used is the _governor_. This regulator,
which had been long in use in mill-work and other machinery, has of
late years attracted more general notice by its beautiful adaptation
in the steam-engines of Watt. It consists of heavy balls B B,
_fig. 144._, attached to the extremities of rods B F.
These rods play upon a joint at E, passing through a mortise in the
vertical stem D D′. At F they are united by joints to the short
rods F H, which are again connected by joints at H to a ring
which slides upon the vertical shaft D D′. From this description
it will be apparent that when the balls B are drawn from the axis,
their upper arms E F are caused to increase their divergence in
the same manner as the blades of a scissors are opened by separating
the handles. These, acting upon the ring by means of the short links
F H, draw it down the vertical axis from D towards E. A contrary
effect is produced when the balls B are brought closer to the axis, and
the divergence of the rods B E diminished. A horizontal wheel W is
attached to the vertical axis D D′, having a groove to receive a
rope or strap upon its rim. This strap passes round the wheel or axis
by which motion is transmitted to the machinery to be regulated, so
that the spindle or shaft D D′ will always be made to revolve with
a speed proportionate to that of the machinery.

As the shaft D D′ revolves, the balls B are carried round it
with a circular motion, and consequently acquire a centrifugal force
which causes them to recede from the axle, and therefore to depress
the ring H. On the edge or rim of this ring is formed a groove, which
is embraced by the prongs of a fork I, at the extremity of one arm of
a lever whose fulcrum is at G. The extremity K of the other arm is
connected by some means with the part of the machine which supplies
the power. In the present instance we shall suppose it a steam-engine,
in which case the rod K I communicates with a flat circular valve
V, placed in the principal steam-pipe, and so arranged that, when K is
elevated as far as by their divergence the balls B have power over it,
the passage of the pipe will be closed by the valve V, and the passage
of steam entirely stopped; and, on the other hand, when the balls
subside to their lowest position, the valve will be presented with its
edge in the direction of the tube, so as to intercept no part of the
steam.

The property which renders this instrument so admirably adapted to
the purpose to which it is applied is, that when the divergence of
the balls is not very considerable, they must always revolve with the
same velocity, whether they move at a greater or lesser distance from
the vertical axis. If any circumstance increases that velocity, the
balls instantly recede from the axis, and closing the valve V, check
the supply of steam, and thereby diminishing the speed of the motion,
restore the machine to its former rate. If, on the contrary, that
fixed velocity be diminished, the centrifugal force being no longer
sufficient to support the balls, they descend towards the axle, open
the valve V, and, increasing the supply of steam, restore the proper
velocity of the machine.

When the governor is applied to a water-wheel it is made to act upon
the shuttle through which the water flows, and controls its quantity as
effectually, and upon the same principle, as has just been explained in
reference to the steam-engine. When applied to a windmill it regulates
the sail-cloth so as to diminish the efficacy of the power upon the
arms as the force of the wind increases, or _vice versâ_.

In cases where the resistance admits of easy and convenient change, the
governor may act so as to accommodate it to the varying energy of the
power. This is often done in corn-mills, where it acts upon the shuttle
which metes out the corn to the millstones. When the power which drives
the mill increases, a proportionally increased feed of corn is given
to the stones, so that the resistance being varied in the ratio of the
power, the same velocity will be maintained.

(304.) In some cases the centrifugal force of the revolving balls is
not sufficiently great to control the power or the resistance, and
regulators of a different kind must be resorted to. The following
contrivance is called the _water-regulator_:--

A common pump is worked by the machine, whose motion is to be
regulated, and water is thus raised and discharged into a cistern.
It is allowed to flow from this cistern through a pipe of a given
magnitude. When the water is pumped up with the same velocity as it is
discharged by this pipe, it is evident that the level of the water in
the cistern will be stationary, since it receives from the pump the
exact quantity which it discharges from the pipe. But if the pump
throw in more water in a given time than is discharged by the pipe,
the cistern will begin to be filled, and the level of the water will
rise. If, on the other hand, the supply from the pump be less than
the discharge from the pipe, the level of the water in the cistern
will subside. Since the rate at which water is supplied from the
pump will always be proportional to the velocity of the machine, it
follows that every fluctuation in this velocity will be indicated by
the rising or subsiding of the level of the water in the cistern, and
that level never can remain stationary, except at that exact velocity
which supplies the quantity of water discharged by the pipe. This pipe
may be constructed so as by an adjustment to discharge the water at
any required rate; and thus the cistern may be adapted to indicate a
constant velocity of any proposed amount.

If the cistern were constantly watched by an attendant, the velocity of
the machine might be abated by regulating the power when the level of
the water is observed to rise, or increased when it falls; but this is
much more effectually and regularly performed by causing the surface
of the water itself to perform the duty. A float or large hollow metal
ball is placed upon the surface of the water in the cistern. This ball
is connected with a lever acting upon some part of the machinery, which
controls the power or regulates the amount of resistance, as already
explained in the case of the governor. When the level of the water
rises, the buoyancy of the ball causes it to rise also with a force
equal to the difference between its own weight and the weight of as
much water as it displaces. By enlarging the floating ball, a force may
be obtained sufficiently great to move those parts of the machinery
which act upon the power or resistance, and thus either to diminish
the supply of the moving principle or to increase the amount of the
resistance, and thereby retard the motion and reduce the velocity to
its proper limit. When the level of the water in the cistern falls,
the floating ball being no longer supported on the liquid surface,
descends with the force of its own weight, and producing an effect upon
the power or resistance contrary to the former, increases the effective
energy of the one, or diminishes that of the other, until the velocity
proper to the machine be restored.

The sensibility of these regulators is increased by making the surface
of water in the cistern as small as possible; for then a small change
in the rate at which the water is supplied by the pump will produce a
considerable change in the level of the water in the cistern.

Instead of using a float, the cistern itself may be suspended from
the lever which controls the supply of the power, and in this case a
sliding weight may be placed on the other arm, so that it will balance
the cistern when it contains that quantity of water which corresponds
to the fixed level already explained. If the quantity of water in
the cistern be increased by an undue velocity of the machine, the
weight of the cistern will preponderate, draw down the arm of the
lever, and check the supply of the power. If, on the other hand, the
supply of water be too small, the cistern will no longer balance the
counterpoise, the arm by which it is suspended will be raised, and the
energy of the power will be increased.

(305.) In the steam-engine the self-regulating principle is carried
to an astonishing pitch of perfection. The machine itself raises in
due quantity the cold water necessary to condense the steam. It pumps
off the hot water produced by the steam, which has been cooled, and
lodges it in a reservoir for the supply of the boiler. It carries from
this reservoir exactly that quantity of water which is necessary to
supply the wants of the boiler, and lodges it therein according as it
is required. It breathes the boiler of redundant steam, and preserves
that which remains fit, both in quantity and quality, for the use of
the engine. It blows its own fire, maintaining its intensity, and
increasing or diminishing it according to the quantity of steam which
it is necessary to raise; so that when much work is expected from the
engine, the fire is proportionally brisk and vivid. It breaks and
prepares its own fuel, and scatters it upon the bars at proper times
and in due quantity. It opens and closes its several valves at the
proper moments, works its own pumps, turns its own wheels, and is only
not alive. Among so many beautiful examples of the self-regulating
principle, it is difficult to select. We shall, however, mention one or
two, and for others refer the reader to our treatise on this subject.[3]

[3] Lardner on the Steam-Engine, Steam-Navigation, Roads, and Railways.
8th edition. 1851.

It is necessary in this machine that the water in the boiler be
maintained constantly at the same level, and, therefore, that as
much be supplied, from time to time, as is consumed by evaporation.
A pump which is wrought by the engine itself supplies a cistern C,
_fig. 145._, with hot water. At the bottom of this cistern is a
valve V opening into a tube which descends into the boiler. This valve
is connected by a wire with the arm of a lever on the fulcrum D, the
other arm E of which is also connected by a wire with a stone float F,
which is partially immersed in the water of the boiler, and is balanced
by a sliding weight A. The weight A only counterpoises the stone float
F by the aid of its buoyance in the water; for if the water be removed,
the stone F will preponderate, and raise the weight A. When the water
in the boiler is at its proper level, the length of the wire connecting
the valve V with the lever is so adjusted that this valve shall be
closed, the wire at the same time being fully extended. When, by
evaporation, the water in the boiler begins to be diminished, the level
falls, and the stone weight F, being no longer supported, overcomes
the counterpoise A, raises the arm of the lever, and, pulling the
wire, opens the valve V. The water in the cistern C then flows through
the tube into the boiler, and continues to flow until the level be so
raised that the stone weight F is again elevated, the valve V closed,
and the further supply of water from the cistern C suspended.

In order to render the operation of this apparatus easily
intelligible, we have here supposed an imperfection which does not
exist. According to what has just been stated, the level of the water
in the boiler descends from its proper height, and subsequently returns
to it. But, in fact, this does not happen. The float F and valve V
adjust themselves, so that a constant supply of water passes through
the valve, which proceeds exactly at the same rate as that at which the
water in the boiler is consumed.

(306.) In the same machine there occurs a singularly happy example of
self-adjustment, in the method by which the strength of the fire is
regulated. The governor regulates the supply of steam to the engine,
and proportions it to the work to be done. With this work, therefore,
the demands upon the boiler increase or diminish, and with these
demands the production of steam in the boiler ought to vary. In fact,
the rate at which steam is generated in the boiler, ought to be equal
to that at which it is consumed in the engine, otherwise one of two
effects must ensue: either the boiler will fail to supply the engine
with steam, or steam will accumulate in the boiler, being produced in
undue quantity, and, escaping at the safety valve, will thus be wasted.
It is, therefore, necessary to control the agent which generates the
steam, namely, the fire, and to vary its intensity from time to time,
proportioning it to the demands of the engine. To accomplish this,
the following contrivance has been adopted:--Let T, _fig. 146._,
be a tube inserted in the top of the boiler, and descending nearly
to the bottom. The pressure of the steam confined in the boiler,
acting upon the surface of the water, forces it to a certain height
in the tube T. A weight F, half immersed in the water in the tube, is
suspended by a chain, which passes over the wheels P P′, and is
balanced by a metal plate D, in the same manner as the stone float,
_fig. 145._, is balanced by the weight A. The plate D passes
through the mouth of the flue E as it issues finally from the boiler;
so that when the plate D falls it stops the flue, suspending thereby
the draught of air through the furnace, mitigating the intensity of
the fire, and checking the production of steam. If, on the contrary,
the plate D be drawn up, the draught is increased, the fire is rendered
more active, and the production of steam in the boiler is stimulated.
Now, suppose that the boiler produces steam faster than the engine
consumes it, either because the load on the engine has been diminished,
and, therefore, its consumption of steam reduced, or because the
fire has become too intense; the consequence is, that the steam,
beginning to accumulate in the boiler, will press upon the surface of
the water with increased force, and the water will be raised in the
tube T. The weight F will, therefore, be lifted, and the plate D will
descend, diminish, or stop the draught, mitigate the fire, and retard
the production of steam, and will continue to do so until the rate
at which steam is produced shall be commensurate to the wants of the
engine. If, on the other hand, the production of steam be inadequate
to the exigency of the machine, either because of an increased load,
or of the insufficient force of the fire, the steam in the boiler will
lose its elasticity, and the surface of the water not sustaining its
wonted pressure, the water in the tube T will fall; consequently the
weight F will descend, and the plate D will be raised. The flue being
thus opened, the draught will be increased, and the fire rendered
more intense. Thus the production of steam becomes more rapid, and is
rendered sufficiently abundant for the purposes of the engine. This
apparatus is called the _self-acting damper_.

(307.) When a perfectly uniform rate of motion has not been attained,
it is often necessary to indicate small variations of velocity. The
following contrivance, called a _tachometer_[4], has been invented
to accomplish this. A cup, _fig. 147._, is filled to the level
C D with quicksilver, and is attached to a spindle, which is
whirled by the machine in the same manner as the governor already
described. It is well known that the centrifugal force produced by this
whirling motion will cause the mercury to recede from the centre and
rise upon the sides of the cup, so that its surface will assume the
concave appearance represented in _fig. 148._ In this case the
centre of the surface will obviously have fallen below its original
level, _fig. 147._, and the edges will have risen above that
level. As this effect is produced by the velocity of the machine, so
it is proportionate to that velocity, and subject to corresponding
variations. Any method of rendering visible small changes in the
central level of the surface of the quicksilver will indicate minute
variations in the velocity of the machine.

[4] From the Greek words _tachos_ speed, and _metron_ measure.

A glass tube A, open at both ends, and expanding at one extremity into
a bell B, is immersed with its wider end in the mercury, the surface
of which will stand at the same level in the bell B, and in the cup
C D. The tube is so suspended as to be unconnected with the cup.
This tube is then filled to a certain height A, with spirits tinged
with some colouring matter, to render it easily observable. When the
cup is whirled by the machine to which it is attached, the level of
the quicksilver in the bell falls, leaving more space for the spirits,
which, therefore, descends in the tube. As the motion is continued,
every change of velocity causes a corresponding change in the level
of the mercury, and, therefore, also in the level A of the spirits.
It will be observed, that, in consequence of the capacity of the bell
B being much greater than that of the tube A, a very small change in
the level of the quicksilver in the bell will produce a considerable
change in the height of the spirits in the tube. Thus this ingenious
instrument becomes a very delicate indicator of variations in the
motion of machinery.

(308.) The governor, and other methods of regulating the motion of
machinery which have been just described, are adapted principally to
cases in which the proportion of the resistance to the load is subject
to certain fluctuations or gradual changes, or at least to cases in
which the resistance is not at any time entirely withdrawn, nor the
energy of the power actually suspended. Circumstances, however,
frequently occur in which, while the power remains in full activity,
the resistance is at intervals suddenly removed and as suddenly again
returns. On the other hand, cases also present themselves, in which,
while the resistance is continued, the impelling power is subject to
intermission at regular periods. In the former case, the machine would
be driven with a ruinous rapidity during those periods at which it is
relieved from its load, and on the return of the load every part would
suffer a violent strain, from its endeavour to retain the velocity
which it had acquired, and the speedy destruction of the engine could
not fail to ensue. In the latter case, the motion would be greatly
retarded or entirely suspended during those periods at which the moving
power is deprived of its activity, and, consequently, the motion which
it would communicate would be so irregular as to be useless for the
purposes of manufactures.

It is also frequently desirable, by means of a weak but continued
power, to produce a severe but instantaneous effect. Thus a blow may
be required to be given by the muscular action of a man’s arm with a
force to which, unaided by mechanical contrivance, its strength would
be entirely inadequate.

In all these cases, it is evident that the object to be attained is,
an effectual method of accumulating the energy of the power so as
to make it available after the action by which it has been produced
has ceased. Thus, in the case in which the load is at periodical
intervals withdrawn from the machine, if the force of the power could
be imparted to something by which it would be preserved, so as to be
brought against the load when it again returned, the inconvenience
would be removed. In like manner, in the case where the power itself
is subject to intermission, if a part of the force which it exerts in
its intervals of action could be accumulated and preserved, it might
be brought to bear upon the machine during its periods of suspension.
By the same means of accumulating force, the strength of an infant,
by repeated efforts, might produce effects which would be vainly
attempted by the single and momentary action of the strongest man.

(309.) The property of inertia, explained and illustrated in the third
and fourth chapters of this volume furnishes an easy and effectual
method of accomplishing this. A mass of matter retains, by virtue of
its inertia, the whole of any force which may have been given to it,
except that part of which friction and the atmospheric resistance
deprives it. By contrivances which are well known and present no
difficulty, the part of the moving force thus lost may be rendered
comparatively small, and the moving mass may be regarded as retaining
nearly the whole of the force impressed upon it. To render this method
of accumulating force fully intelligible, let us first imagine a
polished level plane on which a heavy globe of metal, also polished, is
placed. It is evident that the globe will remain at rest on any part of
the plane without a tendency to move in any direction. As the friction
is nearly removed by the polish of the surfaces, the globe will be
easily moved by the least force applied to it. Suppose a slight impulse
given to it, which will cause it to move at the rate of one foot in
a second. Setting aside the effects of friction, it will continue to
move at this rate for any length of time. The same impulse repeated
will increase its speed to two feet per second. A third impulse to
three feet, and so on. Thus 10,000 repetitions of the impulse will
cause it to move at the rate of 10,000 feet per second. If the body to
which these impulses were communicated were a cannon ball, it might,
by a constant repetition of the impelling force, be at length made to
move with as much force as if it were projected from the most powerful
piece of ordnance. The force with which the ball in such a case would
strike a building might be sufficient to reduce it to ruins, and yet
such force would be nothing more than the accumulation of a number
of weak efforts not beyond the power of a child to exert, which are
stored up, and preserved, as it were, by the moving mass, and thereby
brought to bear, at the same moment, upon the point to which the force
is directed. It is the sum of a number of actions exerted successively,
and, during a long interval, brought into operation at one and the same
moment.

But the case which is here supposed cannot actually occur; because
we have not usually any practical means of moving a body for any
considerable time in the same direction without much friction, and
without encountering numerous obstacles which would impede its
progress. It is not, however, essential to the effect which is to be
produced, that the motion should be in a straight line. If a leaden
weight be attached to the end of a light rod or cord, and be whirled by
the force of the arm in a circle, it will gradually acquire increased
speed and force, and at length may receive an impetus which would
cause it to penetrate a piece of board as effectually as if it were
discharged from a musket.

The force of a hammer or sledge depends partly on its weight, but much
more on the principle just explained. Were it allowed merely to fall
by the force of its weight upon the head of a nail, or upon a bar of
heated iron which is to be flattened, an inconsiderable effect would be
produced. But when it is wielded by the arm of a man, it receives at
every moment of its motion increased force, which is finally expended
in a single instant on the head of the nail, or on the bar of iron.

The effects of flails in threshing, of clubs, whips, canes, and
instruments for striking, axes, hatchets, cleavers, and all instruments
which cut by a blow, depend on the same principle, and are similarly
explained.

The bow-string which impels the arrow does not produce its effect at
once. It continues to act upon the shaft until it resumes its straight
position, and then the arrow takes flight with the force accumulated
during the continuance of the action of the string, from the moment it
was disengaged from the finger of the bow-man.

Fire-arms themselves act upon a similar principle, as also the air-gun
and steam-gun. In these instruments the ball is placed in a tube, and
suddenly exposed to the pressure of a highly elastic fluid, either
produced by explosion as in fire-arms, by previous condensation as in
the air-gun, or by the evaporation of highly heated liquids as in the
steam-gun. But in every case this pressure continues to act upon it
until it leaves the mouth of the tube, and then it departs with the
whole force communicated to it during its passage along the tube.

(310.) From all these considerations it will easily be perceived that
a mass of inert matter may be regarded as a magazine in which force
may be deposited and accumulated, to be used in any way which may be
necessary. For many reasons, which will be sufficiently obvious, the
form commonly given to the mass of matter used for this purpose in
machinery is that of a wheel, in the rim of which it is principally
collected. Conceive a massive ring of metal, _fig. 149._,
connected with a central box or nave by light spokes, and turning on
an axis with little friction. Such an apparatus is called a fly-wheel.
If any force be applied to it, with that force (making some slight
deduction for friction) it will move, and will continue to move until
some obstacle be opposed to its motion, which will receive from it a
part of the force it has acquired. The uses of this apparatus will be
easily understood by examples of its application.

Suppose that a heavy stamper or hammer is to be raised to a certain
height, and thence to be allowed to fall, and that the power used
for this purpose is a water-wheel. While the stamper ascends, the
power of the wheel is nearly balanced by its weight, and the motion
of the machine is slow. But the moment the stamper is disengaged and
allowed to fall, the power of the wheel, having no resistance, nor any
object on which to expend itself, suddenly accelerates the machine,
which moves with a speed proportioned to the amount of the power,
until it again engages the stamper, when its velocity is as suddenly
checked. Every part suffers a strain, and the machine moves again
slowly until it discharges its load, when it is again accelerated,
and so on. In this case, besides the certainty of injury and wear, and
the probability of fracture from the sudden and frequent changes of
velocity, nearly the whole force exerted by the power in the intervals
between the commencement of each descent of the stamper and the next
ascent is lost. These defects are removed by a fly-wheel. When the
stamper is discharged, the energy of the power is expended in moving
the wheel, which, by reason of its great mass, will not receive an
undue velocity. In the interval between the descent and ascent of the
stamper, the force of the power is lodged in the heavy rim of the
fly-wheel. When the stamper is again taken up by the machine, this
force is brought to bear upon it, combined with the immediate power
of the water-wheel, and the stamper is elevated with nearly the same
velocity as that with which the machine moved in the interval of its
descent.

(311.) In many cases, when the moving power is not subject to
variation, the efficacy of the machine to transmit it to the working
point is subject to continual change. The several parts of every
machine have certain periods of motion, in which they pass through a
variety of positions, to which they continually return after stated
intervals. In these different positions the effect of the power
transmitted to the working point is different; and cases even occur
in which this effect is altogether annihilated, and the machine is
brought into a predicament in which the power loses all influence
over the weight. In such cases the aid of a fly-wheel is effectual
and indispensable. In those phases of the machine, which are most
favourable to the transmission of force, the fly-wheel shares the
effect of the power with the load, and retaining the force thus
received directs it upon the load at the moments when the transmission
of power by the machine is either feeble or altogether suspended. These
general observations will, perhaps, be more clearly apprehended by an
example of an application of the fly-wheel, in a case such as those now
alluded to.

Let A B C D E F, _fig. 150._, be a
_crank_, which is a double winch ((252.) and _fig. 89._), by
which an axle, A B E F, is to be turned. Attached
to the middle of C D by a joint is a rod, which is connected
with a beam, worked with an alternate motion on a centre, like
the brake of a pump, and driven by any constant power, such as a
steam-engine. The bar C D is to be carried with a circular motion
round the axis A E. Let the machine, viewed in the direction
A B E F of the axis, be conceived to be represented
in _fig. 151._, where A represents the centre round which the
motion is to be produced, and G the point where the connecting rod
G H is attached to the arm of the crank. The circle through which
G is to be urged by the rod is represented by the dotted line. In
the position represented in _fig. 151._, the rod acting in the
direction H G has its full power to turn the crank G A round
the centre A. As the crank comes into the position represented in
_fig. 152._, this power is diminished, and when the point G comes
immediately below A, as in _fig. 153._, the force in the direction
H G has no effect in turning the crank round A, but, on the
contrary, is entirely expended in pulling the crank in the direction
A G, and, therefore, only acts upon the pivots or gudgeons which
support the axle. At this crisis of the motion, therefore, the whole
effective energy of the power is annihilated.

After the crank has passed to the position represented in
_fig. 154._, the direction of the force which acts upon the
connecting rod is changed, and now the crank is drawn upward in the
direction G H. In this position the moving force has some efficacy
to produce rotation round A, which efficacy continually increases
until the crank attains the position shown in _fig. 155._, when
its power is greatest. Passing from this position its efficacy is
continually diminished, until the point G comes immediately above the
axis A, _fig. 156._ Here again the power loses all its efficacy
to turn the axle. The force in the direction G H or H G can
obviously produce no other effect than a strain upon the pivots or
gudgeons.

In the critical situations represented in _fig. 153._, and
_fig. 156._, the machine would be incapable of moving, were
the immediate force of the power the only impelling principle. But
having been previously in motion by virtue of the inertia of its
various parts, it has a tendency to continue in motion; and if the
resistance of the load and the effects of friction be not too great,
this disposition to preserve its state of motion will extricate the
machine from the dilemma in which it is involved in the cases just
mentioned, by the peculiar arrangement of its parts. In many cases,
however, the force thus acquired during the phases of the machine, in
which the power is active, is insufficient to carry it through the
dead points (_fig. 153._ and _fig. 156._); and in all cases
the motion would be very unequal, being continually retarded as it
approached these points, and continually accelerated after it passed
them. A fly-wheel attached to the axis A, or to some other part of the
machinery, will effectually remove this defect. When the crank assumes
the positions in _fig. 151._ and _fig. 155._, the power is
in full play upon it, and a share of the effect is imparted to the
massive rim of the fly-wheel. When the crank gets into the predicament
exhibited in _fig. 153._ and _fig. 156._, the momentum which
the fly-wheel received when the crank acted with most advantage,
immediately extricates the machine, and, carrying the crank beyond the
dead point, brings the power again to bear upon it.

The astonishing effects of a fly-wheel, as an accumulator of force,
have led some into the error of supposing that such an apparatus
increases the actual power of a machine. It is hoped, however, that
after what has been explained respecting the inertia of matter and the
true effects of machines, the reader will not be liable to a similar
mistake. On the contrary, as a fly cannot act without friction, and as
the amount of the friction, like that of inertia, is in proportion to
the weight, a portion of the actual moving force must unavoidably be
lost by the use of a fly. In cases, however, where a fly is properly
applied this loss of power is inconsiderable, compared with the
advantageous distribution of what remains.

[Illustration: _C. Varley, del._     _H. Adlard, sc._

_London, Pubd. by Longman & Co._]

As an accumulator of force, a fly can never have more force than has
been applied to put it in motion. In this respect it is analogous to
an elastic spring, or the force of condensed air, or any other power
which derives its existence from causes purely mechanical. In bending
a spring a gradual expenditure of power is necessary. On the recoil
this power is exerted in a much shorter time than that consumed in its
production, but its total amount is not altered. Air is condensed by a
succession of manual efforts, one of which alone would be incapable of
projecting a leaden ball with any considerable force, and all of which
could not be immediately applied to the ball at the same instant. But