of tills axis, depends chiefly the pcrfecfion of 
the instrument. 
5- The pivots which form the axis or ful- 
crum, should be in a straight line, acd at right 
angles to the beam. 6. '1 lie arms should be 
as long as possible, relatively to their thick- 
ness, and the purposes for which they are in- 
tended ; as the longer they are, the more sen- 
sible is the balance. 
They should also be made as stiff and in- 
flexible as possible ; for if the beam is too 
weak, it will bend, and become untrue. 7. 
The rings, or the piece on which the axis 
bears, should be hard and well polished, pa- 
rallel to each other, and of an oval form, that 
the axis may always keep its proper bearing, 
or remain always at the lowest point. 
Very delicate balances are not only useful 
in nice experiments, but are likewise much 
more expeditious than others in common 
weighing. If a pair of scales with a certain 
load is barely sensible to one-tenth of a grain, 
it will require a considerable time to ascer- 
tain the weight to that degree of accuracy, 
because the turn must be observed several 
times over, and is very small. But if no 
greater accuracy was required, and scales 
were used which would turn with one-hun- 
dredth of a grain, a tenth of a grain more or 
less would make so great a difference in the 
turn, that it would be seen immediately. 
The statera, or Roman steel-yard, is a lever 
of the first kind, and is used for finding the 
weights of different bodies, by one single 
i weight placed at different distances from the 
prop or centre of motion D (fig. 7.). For the 
; shorter arm 1 )G is of such a weight as ex- 
actly to counterpoise the longer arm DX. If 
this arm is divided into as many equal parts 
;j as it will contain, each equal to GD, the 
| single weight P (which we may suppose to be 
one pound) will serve for weighing any thing 
; as heavy as itself, or. as man) times heavier 
as there are divisions in the arm DX, or any 
5 quantity between its own weight and that 
I quantity. As for example, if P is one pound, 
; and placed at the first division 1 in the arm 
DX, it will balance one pound in the scale at 
W; if it is removed to the second division at 
2, it will balance two pounds in the scale; if 
to the third three pounds; and so on to the 
l end of the arm DX. If any of these integral 
divisions is subdivided into as many equal 
parts as a pound contains ounces, and the 
weight P is placed at any of these subdivi- 
sions, so as to counterpoise what Is in the 
scale, the pounds and odd ounces therein will 
by that means Jae ascertained. 
The wheel and axle is a machine much 
used, and is made in a variety of forms. It 
consists of a wheel with an axle fixed to it, 
so as to turn round with it; the power being 
applied at the circumference of the wheel, 
and the weight to be raised is fastened to a 
rope which coils round the axle. 
AB (tig. 9.) is a wheel and CD an axle 
fixed to it, and which moves round with it. 
It the rope which goes round the wheel is 
pulled, and the wheel turned once round, it 
is evident that as much rope will be drawn 
off as the circumference of the wheel; but 
while the wheel turns once round, the axle 
turns once round ; and consequently the rope 
by which the weight is suspended, will wind 
once round the axis, and the weight will be 
raised through a space equal to the circum- 
ference of the axis, 
MECHANICS. 
i he velocity of the power, therefore, will 
be to that ot the weight, as the circumference 
of the wheel to that of the axis. 
'1 hat the power and the weight may be in 
equilibrio, therefore, the power must be to the 
weight as the circumference of tiie wheel to 
that of the axis. 
It is proved by geometry that the circum- 
ferences ot dilferefit circles bear the same pro- 
portion to each other as their respective dia- 
meters do; consequently the power is to the 
weight, as the diameter also of the axis to 
that of the wheel. 
Thus, suppose the diameter of the wheel 
to be eight inches, and the diameter of the 
axis to be one inch; then one ounce acting 
as the power P, will balance eight ounces as 
a weight W ; and a small additional force will 
cause the wheel to turn with its axis, and 
raise the weight ; and for every inch which 
the weight rises, the power will fall eight 
inches. 
1 he wheel and axis may be considered as a 
kind of perpetual lever, of which the fulcrum 
is the centre of the axis, and the long and 
short arms are the diameter of the wheel and 
the diameter of the axis. See fig. 10. 
From this it is evident, that the larger the 
wheel, and the smaller the axis, the stronger 
is the power of this machine; but then the 
weight must rise slower in proportion. 
A capstan is a cylinder of wood, with holes 
in it, into which are put bars, or levers, to 
turn it round ; these are like the spokes of a 
wheel without the rim. Sometimes the axis 
is turned by a winch fastened to it, which in 
this respect serves for a wheel ; and is more 
powerful in proportion to the largeness of 
the circle it describes, compared with the dia- 
meter of the axle. 
When the parts of th* axis differ in thick- 
ness, and weights are suspended at the differ- 
ent parts, they may be sustained by one and 
the same power applied to the circumference 
of the wheel ; provided the product arising 
from the multiplication of the power into the 
diameter of the wheel, is equal to the sum of 
the products arising from the multiplication 
of the several weights into the diameters of 
those parts of the axis from which they are 
suspended. 
In considering the theory of the wheel and 
axle, we have supposed the rope that goes 
round the axle to have no sensible thickness; 
but as in practice this cannot be the case, if 
it is a thick rope, or if there are several folds 
of it round the axis, you must measure to the 
middle of the outside rope, to obtain the dia- 
meter of the axis, for the distance of the 
weight from the centre is increased by the 
coiling up of the rope. 
It teeth are cut in the circumference of a 
wheel, and if they work in the teeth of ano- 
ther wheel of the same size, as fig. 11, it is 
evident that both the wheels will revolve in 
the same time; and the weight appended to 
the axle of the wheel B, will be raised in the 
same time as if the axle had been fixed to the 
wheel A. But if the teeth of the second 
wheel are made to work in teeth made in the 
axle of the first, as at fig. 12, as every part of 
the circumference of the second wheel is ap- 
plied successively to the circumference of the 
axle of the first, and as the former is much 
greater than the latter, it is evident that the 
first wheel must go round as many times 
ns ' 
more than the second, as the circumference 
of the second wheel exceeds that of the first 
axle. 
In order to a balance here, (he power 
must be to the weight, as the product of the 
circumferences, or diameters of the two axles 
multiplied together, is to the circumferences' 
or diameters of the two wheels. 
This will become sufficiently clear, if it is 
considered as a compound lever, which was 
explained above. Instead of a combination 
of two wheels, three or four wheels may 
work in each other, or any number; and by 
thus increasing the number of wheels, or by 
proportioning the wheels to the axis, any de- 
gree of power may be acquired. 
To this sort of engine belong all cranes for 
raising great weights; and in this case the 
wheel may have cogs all round it, instead of 
handles; and a small lanthorn, or trundle, 
may be made to work in the cogs, and be 
turned by a winch, which will make the 
power of the engine to exceed the power of 
the man who works it, as much asthe number 
of revolutions of the winch exceeds those of 
the axle CD, fig. 9, when multiplied by the 
excess of the length of the winch above the 
length of the semiduuneter of the axle, added 
to the semidiameter or half-thickness of the 
rope K, by which the weight is drawn up. 
d ims, suppose the diameter of the rope and 
axle taken together to be 13 inches, and 
consequently half their diameter to be 6% 
inches, so that the weight W will hang at 6^ 
inches perpendicular distance frqm below the 
centre of the axle. Now, let us suppose the 
wheel AB, which is fixed on the axle, to 
have 80 cogs, and to be turned by means of 
a winch 6^ inches long, lixed on the axle of a 
trundle of eight staves, or rounds, working in 
the cogs of the wheel ; here it is plain, that 
the winch and trundle would make ten revo- 
lutions for one of the wheel A B, and its axis 
C D, on which the rope K winds in raising 
the weight W ; and the winch being no- 
longer than the sum of the semi diameters of. 
the great axle and rope, the trundle could, 
have no more power on the wheel than a man 
could have by pulling it round by the edge,., 
because the winch would have no greater 
velocity than the edge of the wheel has, which 
we here suppose to be ten times as great as 
the velocity of the rising weight ; so that, in 
this case, the power gained would be as 10 is 
to 1. But if the length of the winches 13 
inches, the power gained will be as 20 to I ^ 
if 19 J incites (which is long enough for anv 
man to work by), the power gained will be as- 
30 to 1 ; that is, a man could raise 30 times 
as much by such an engine, as lie could do by 
his natural strength without it, because the 
velocity of the handle of the winch would be. 
30 times as great as the velocity of the rising 
weight; the absolute force of any engine be- 
ing in the proportion of the velocity of the 
power, to the velocity of the weight raised by 
it- But then, just as much power or advan- 
tage as is gained by the engine, so much time 
is iost in. working it, which is common in all 
mechanical cases whatever. 
! In this sort of machines it is requisite t<y 
have a ratchet wheel on the end of the axle 
C, with a catch to fall into its teeth; which 
will at any time support the weight, and keep 
it from descending, if the person who turns 
the handle should, through inadvertence or 
carelessness, quit his hold while the weigte is 
