length be raised to the fulcrum S, and the beam will rest indifferently
in any position. If more weight be then added, the centre of gravity
will be raised above the fulcrum, and the beam will turn over.
Lastly, if the fulcrum S, _fig. 189._, is above the line joining
the two points of support, as any additional weights placed in the
scales will be referred to the point W, in the line joining A and B,
if the weight of the beam be doubled by such added weights, and the
centre of gravity be consequently raised to _g_, W _g_ will become
equal to half of W G. But _g p_, being greater than one half
of G P, the end of the beam B will rise until _g p_ becomes
such as to be equal, when multiplied by the whole increased weight of
the beam, to P B, multiplied by the small weight, which we suppose
to have been placed as in the preceding examples, in the scale.
From what has been said it will be seen that there are three positions
of the fulcrum which influence the sensibility of the balance: first,
when the fulcrum and the points of support are in a right line, when
the sensibility of the balance will remain the same, though the weight
with which the beam is loaded should be varied: secondly, when the
fulcrum is below the line joining the two points of support, in which
case the sensibility of the balance will be increased by additional
weights, until at length the centre of gravity is raised above the
fulcrum, when the beam will turn over; and, thirdly, when the fulcrum
is above the line joining the two points of support, in which case the
sensibility of the balance will be diminished as the weight with which
the beam is loaded is increased.
The sensibility of a balance, as here defined, is the angular deviation
of the beam occasioned by placing an additional constant small weight
in one of the scales; but it is frequently expressed by the proportion
which such small additional weight bears to the weight of the beam
and its load, and sometimes to the weight the value of which is to be
determined.
This proportion, however, will evidently vary with different weights,
except in the case where the centre of gravity of the beam is in the
line joining the points supporting the scales, the fulcrum being
above this line, and it is therefore necessary, in every other case,
when speaking of the sensibility of the balance, to designate the
weight with which it is loaded: thus, if a balance has a troy pound in
each scale, and the horizontality of the beam varies a certain small
quantity, just perceptible on the addition of one hundredth of a grain,
we say that the balance is sensible to 1/1152000 part of its load with
a pound in each scale, or that it will determine the weight of a troy
pound within 1/576000 part of the whole.
The nearer the centre of gravity of a balance is to its fulcrum
the slower will be the oscillations of the beam. The number of
oscillations, therefore, made by the beam in a given time (a minute
for example), affords the most accurate method of judging of the
sensibility of the balance, which will be the greater as the
oscillations are fewer.
Balances of the most perfect kind, and of such only it is our present
object to treat, are usually furnished with adjustments, by means of
which the length of the arms, or the distances of the fulcrum from the
points of support, may be equalised, and the fulcrum and the two points
of support be placed in a right line; but these adjustments, as will
hereafter be seen, are not absolutely necessary.
The beam is variously constructed, according to the purposes to
which the balance is to be applied. Sometimes it is made of a rod of
solid steel; sometimes of two hollow cones joined at their bases;
and, in some balances, the beam is a frame in the form of a rhombus:
the principal object in all, however, is to combine strength and
inflexibility with lightness.
A balance of the best kind, made by Troughton, is so contrived as to
be contained, when not in use, in a drawer below the case; and when
in use, it is protected from any disturbance from currents of air, by
being enclosed in the case above the drawer, the back and front of
which are of plate glass. There are doors in the sides, through which
the scale-pans are loaded, and there is a door at the top through which
the beam may be taken out.
A strong brass pillar, in the centre of the box, supports a square
piece, on the front and back of which rise two arches, nearly
semicircular, on which are fixed two horizontal planes of agate,
intended to support the fulcrum. Within the pillar is a cylindrical
tube, which slides up and down by means of a handle on the outside
of the case. To the top of this interior tube is fixed an arch, the
terminations of which pass beneath and outside of the two arches before
described. These terminations are formed into Y _s_, destined to
receive the ends of the fulcrum, which are made cylindrical for this
purpose, when the interior tube is elevated in order to relieve the
axis when the balance is not in use. On depressing the interior tube,
the Y _s_ quit the axis, and leave it in its proper position on the
agate planes. The beam is about eighteen inches long, and is formed of
two hollow cones of brass, joined at their bases. The thickness of the
brass does not exceed 0·02 of an inch, but by means of circular rings
driven into the cones at intervals they are rendered almost inflexible.
Across the middle of the beam passes a cylinder of steel, the lower
side of which is formed into an edge, having an angle of about thirty
degrees, which, being hardened and well polished, constitutes the
fulcrum, and rests upon the agate planes for the length of about 0·05
of an inch.
Each point of suspension is formed of an axis having two sharp concave
edges, upon which rest at right angles two other sharp concave
edges formed in the spur-shaped piece to which the strings carrying
the scale-pan are attached. The two points are adjustable, the one
horizontally, for the purpose of equalising the arms of the beam, and
the other vertically, for bringing the points of suspension and the
fulcrum into a right line.
Such is the form of Troughton’s balance: we shall now give the
description of a balance as constructed by Mr. Robinson of Devonshire
Street, Portland Place:--
The beam of this balance is only ten inches long. It is a frame of
bell-metal in the form of a rhombus. The fulcrum is an equilateral
triangular prism of steel one inch in length; but the edge on which
the beam vibrates is formed to an angle of 120°, in order to prevent
any injury from the weight with which it may be loaded. The chief
peculiarity in this balance consists in the knife-edge which forms
the fulcrum bearing upon an agate plane throughout its whole length,
whereas we have seen in the balance before described that the whole
weight is supported by portions only of the knife-edge, amounting
together to one tenth of an inch. The supports for the scales are
knife-edges each six tenths of an inch long. These are each furnished
with two pressing screws, by means of which they may be made parallel
to the central knife-edge.
Each end of the beam is sprung obliquely upwards and towards the
middle, so as to form a spring through which a pushing screw passes,
which serves to vary the distance of the point of support from the
fulcrum, and, at the same time, by its oblique action to raise or
depress it, so as to furnish a means of bringing the points of support
and the fulcrum into a right line.
A piece of wire, four inches long, on which a screw is cut, proceeds
from the middle of the beam downwards. This is pointed to serve as
an index, and a small brass ball moves on the screw, by changing the
situation of which the place of the centre of gravity may be varied at
pleasure.
The fulcrum, as before remarked, rests upon an agate plane throughout
its whole length, and the scale-pans are attached to planes of agate
which rest upon the knife-edges forming the points of support. This
method of supporting the scale-pans, we have reason to believe, is
due to Mr. Cavendish. Upon the lower half of the pillar to which the
agate plane is fixed, a tube slides up and down by means of a lever
which passes to the outside of the case. From the top of this tube
arms proceed obliquely towards the ends of the balance, serving to
support a horizontal piece, carrying at each extremity two sets of Y
_s_, one a little above the other. The upper Y _s_ are destined to
receive the agate planes to which the scale-pans are attached, and thus
to relieve the knife-edges from their pressure; the lower to receive
the knife-edges which, form the points of support, consequently these
latter Y _s_, when in action, sustain the whole beam.
When the lever is freed from a notch in which it is lodged, a spring
is allowed to act upon the tube we have mentioned, and to elevate it.
The upper Y _s_ first meet the agate planes carrying the scale-pans
and free them from the knife-edges. The lower Y _s_ then come into
action and raise the whole beam, elevating the central knife-edge above
the agate plane. This is the usual state of the balance when not in
use: when it is to be brought into action, the reverse of what we
have described takes place. On pressing down the lever, the central
knife-edge first meets the agate plane, and afterwards the two agate
planes carrying the scale-pans are deposited upon their supporting
knife-edges.
A balance of this construction was employed by the writer of this
article in adjusting the national standard pound. With a pound troy in
each scale, the addition of one hundredth of a grain caused the index
to vary one division, equal to one tenth of an inch, and Mr. Robinson
adjusts these balances so that with one thousand grains in each scale,
the index varies perceptibly on the addition of one thousandth of a
grain, or of one-millionth part of the weight to be determined.
It may not be uninteresting to subjoin, from the Philosophical
Transactions for 1826, the description of a balance perhaps the most
sensible that has yet been made, constructed for verifying the national
standard bushel. The author says,--
“The weight of the bushel measure, together with the 80 lbs. of
water it should contain, was about 250 lbs.; and as I could find
no balance capable of determining so large a weight with sufficient
accuracy, I was under the necessity of constructing one for this
express purpose.
“I first tried cast iron; but though the beam was made as light as was
consistent with the requisite degree of strength, the inertia of such
a mass appeared to be so considerable, that much time must have been
lost before the balance would have answered to the small differences I
wished to ascertain. Lightness was a property essentially necessary,
and bulk was very desirable, in order to preclude such errors as might
arise from the beam being partially affected by sudden alterations of
temperature. I therefore determined to employ wood, a material in which
the requisites I sought were combined. The beam was made of a plank
of mahogany, about 7O inches long, 22 inches wide, and 2-1/4 thick,
tapering from the middle to the extremities. An opening was cut in the
centre, and strong blocks screwed to each side of the plank, to form a
bearing for the back of a knife-edge which passed through the centre.
Blocks were also screwed to each side at the extremities of the beam on
which rested the backs of the knife-edges for supporting the pans. The
opening in the centre was made sufficiently large to admit the support
hereafter to be described, upon which the knife-edge rested.
“In all beams which I have seen, with the exception of those made by
Mr. Robinson, the whole weight is sustained by short portions at the
extremities of the knife-edge; and the weight being thus thrown upon a
few points, the knife-edge becomes more liable to change its figure and
to suffer injury.
“To remedy this defect, the central knife-edge of the beam I am
describing was made 6 inches, and the two others 5 inches long. They
were triangular prisms with equal sides of three fourths of an inch,
very carefully finished, and the edges ultimately formed to an angle of
120°.
“Each knife-edge was screwed to a thick plate of brass, the surfaces in
contact having been previously ground together; and these plates were
screwed to the beam, the knife-edges being placed in the same plane,
and as nearly equidistant and parallel to each other as could be done
by construction.
“The support upon which the central knife-edge rested throughout its
whole length was formed of a plate of polished hard steel, screwed to
a block of cast iron. This block was passed through the opening before
mentioned in the centre of the beam, and properly attached to a frame
of cast iron.
“The stirrups to which the scales were hooked rested upon plates of
polished steel to which they were attached, and the under surfaces of
which were formed by careful grinding into cylindrical segments. These
were in contact with the knife-edges their whole length, and were
known to be in their proper position by the correspondence of their
extremities with those of the knife-edges. A well imagined contrivance
was applied by Mr. Bate for raising the beam when loaded, in order
to prevent unnecessary wear of the knife-edge, and for the purpose of
adjusting the place of the centre of gravity, when the beam was loaded
with the weight required to be determined, a screw carrying a movable
ball projected vertically from the middle of die beam.
“The performance of this balance fully equalled my expectations. With
two hundred and fifty pounds in each scale, the addition of a single
grain occasioned an immediate variation in the index of one twentieth
of an inch, the radius being fifty inches.”
From the preceding account it appears that this balance is sensible to
1/1750000 part of the weight which was to be determined.
We shall now describe the method to be pursued in adjusting a balance.
1. To bring the points of suspension and the fulcrum into a right line.
Make the vibrations of the balance very slow by moving the weight which
influences the centre of gravity, and bring the beam into a horizontal
position, by means of small bits of paper thrown into the scales. Then
load the scales with nearly the greatest weight the beam is fitted to
carry. If the vibrations are performed in the same time as before, no
further adjustment is necessary; but if the beam vibrates quicker,
or if it oversets, cause it to vibrate in the same time as at first,
by moving the adjusting weight, and note the distance through which
the weight has passed. Move the weight then in the contrary direction
through double this distance, and then produce the former slow motion
by means of the screw acting vertically on the point of support. Repeat
this operation until the adjustment is perfect.
2. To make the arms of the beam of an equal length.
Put weights in the scales as before; bring the beam as nearly as
possible to a horizontal position, and note the division at which the
index stands; unhook the scales, and transfer them with their weights
to the other ends of the beam, when, if the index points to the same
division, the arms are of an equal length; but if not, bring the index
to the division which had been noted, by placing small weights in one
or the other scale. Take away half these weights, and bring the index
again to the observed division by the adjusting screw, which acts
horizontally on the point of support. If the scale-pans are known to be
of the same weight, it will not be necessary to change the scales, but
merely to transfer the weights from one scale-pan to the other.
_Of the Use of the Balance._
Though we have described the method of adjusting the balance, these
adjustments, as we have before remarked, may be dispensed with.
Indeed, in all delicate scientific operations, it is advisable never
to rely upon adjustments, which, after every care has been employed in
effecting them, can only be considered as approximations to the truth.
We shall, therefore, now describe the best method of ascertaining the
weight of a body, and which does not depend on the accuracy of these
adjustments.
Having levelled the case which contains the balance, and thrown the
beam out of action, place a weight in each scale-pan nearly equal to
the weight which is to be determined. Lower the beam very gently till
it is in action, and by means of the adjustment for raising or lowering
the centre of gravity, cause the beam to vibrate very slowly. Remove
these weights, and place the substance, the weight of which is to be
determined in one of the scale-pans; carefully counterpoise it by means
of any convenient substances put into the other scale-pan, and observe
the division at which the index stands; remove the body, the weight of
which is to be ascertained, and substitute standard weights for it so
as to bring the index to the same division as before. These weights
will be equal to the weight of the body.
If it be required to compare two weights together which are intended
to be equal, and to ascertain their difference, if any, the method
of proceeding will be nearly the same. The standard weight is to be
carefully counterpoised, and the division at which the index stands,
noted. And now it will be convenient to add in either of the scales
some small weight, such as one or two hundredths of a grain, and mark
the number of divisions passed over in consequence by the index, by
which the value of one division of the scale will be known. This should
be repeated a few times, and the mean taken for greater certainty.
Having noted the division at which the index rests, the standard
weight is to be removed, and the weight which is to be compared with
it substituted for it. The index is then again to be noted, and the
difference between this and the former indication will give the
difference between the weights in parts of a grain.
If the balance is adjusted so as to be very sensible, it will be long
before it comes to a state of rest. It may, therefore, sometimes be
advisable to take the mean of the extent of the vibrations of the index
as the point where it would rest, and this may be repeated several
times for greater accuracy. It must, however, be remembered, that it is
not safe to do this when the extent of the vibrations is beyond one or
two divisions of the scale; but with this limitation it is, perhaps, as
good a method as can be pursued.
Many precautions are necessary to ensure a satisfactory result. The
weights should never be touched by the hand; for not only would
this oxydate the weight, but by raising its temperature it would
appear lighter, when placed in the scale-pan, than it should do, in
consequence of the ascent of the heated air. For the larger weights a
wooden fork or tongs, according to the form of the weight, should be
employed; and for the smaller, a pair of forceps made of copper will be
found the most convenient. This metal possessing sufficient elasticity
to open the forceps on their being released from pressure, and yet not
opposing a resistance sufficient to interfere with that delicacy of
touch which is desirable in such operations.
_Of Weights._
It must be obvious, that the excellence of the balance would be of
little use, unless the weights employed were equally to be depended
upon. The weights may either be accurately adjusted, or the difference
between each weight and the standard may be determined, and,
consequently, its true value ascertained. It has been already shown how
the latter may be effected, in the instructions which have been given
for comparing two weights together; and we shall now show the readiest
mode of adjusting weights to an exact equality with a given standard.
The material of the weight may be either brass or platina, and its form
may be cylindrical: the diameter being nearly twice the height. A small
spherical knob is screwed into the centre, a space being left under
the screw to receive the portions of fine wire used in the adjustment.
It will be convenient to form a cavity in the bottom of each weight to
receive the knob of the weight upon which it may be placed.
Each weight is now to be compared with the standard, and should it
be too heavy, it is to be reduced till it becomes in a very small
degree too light, when the amount of the deficiency is to be carefully
determined.
Some very fine silver wire is now to be taken, and the weight of three
or four feet of it ascertained. From this it will be known what length
of the wire is equal to the error of the weight to be adjusted; and
this length being cut off is to be enclosed under the screw. To guard
against any possible error, it will be advisable before the screw
is firmly fixed in its place, again to compare the weight with the
standard.
The most approved method of making weights expressing the decimal parts
of a grain, is to determine, as before, with great care, the weight of
a certain length of fine wire, and then to cut off such portions as are
equal to the weights required.
Before we conclude this article we shall give a description, from the
Annals of Philosophy for 1825, of “a very sensible balance,” used by
the late Dr. Black:--
“A thin piece of fir wood, not thicker than a shilling, and a foot
long, three tenths of an inch broad in the middle, and one tenth and
a half at each end, is divided by transverse lines into twenty parts;
that is, ten parts on each side of the middle. These are the principal
divisions, and each of them is subdivided into halves and quarters.
Across the middle is fixed one of the smallest needles I could procure,
to serve as an axis, and it is fixed in its place by means of a little
sealing wax. The numeration of the divisions is from the middle to each
end of the beam. The fulcrum is a bit of plate brass, the middle of
which lies flat on my table when I use the balance, and the two ends
are bent up to a right angle so as to stand upright. These two ends
are ground at the same time on a flat hone, that the extreme surfaces
of them may be in the same plane; and their distance is such that the
needle, when laid across them, rests on them at a small distance from
the sides of the beam. They rise above the surface of the table only
one tenth and a half or two tenths of an inch, so that the beam is very
limited in its play. See _fig. 190._
“The weights I use are one globule of gold, which weighs one grain, and
two or three others which weigh one tenth of a grain each; and also
a number of small rings of fine brass wire, made in the manner first
mentioned by Mr. Lewis, by appending a weight to the wire, and coiling
it with the tension of that weight round a thicker brass wire in a
close spiral, after which, the extremity of the spiral being tied hard
with waxed thread, I put the covered wire into a vice, and applying
a sharp knife, which is struck with a hammer, I cut through a great
number of the coils at one stroke, and find them as exactly equal to
one another as can be desired. Those I use happen to be the 1/30 part
of a grain each, or 300 of them weigh ten grains; but I have others
much lighter.
“You will perceive that by means of these weights placed on different
parts of the beam, I can learn the weight of any little mass from one
grain, or a little more, to the 1/1200 of a grain. For if the thing to
be weighed weighs one grain, it will, when placed on one extremity of
the beam, counterpoise the large gold weight at the other extremity.
If it weighs half a grain it will counterpoise the heavy gold weight
placed at 5. If it weigh 6/10 of a grain, you must place the heavy
gold weight at 5, and one of the lighter ones at the extremity to
counterpoise it, and if it weighs only one or two, or three or four
hundredths of a grain, it will be counterpoised by one of the small
gold weights placed at the first or second, or third or fourth
division. If, on the contrary, it weighs one grain and a fraction, it
will be counterpoised by the heavy gold weight at the extremity, and
one or more of the lighter ones placed in some other part of the beam.
“This beam has served me hitherto for every purpose; but had I occasion
for a more delicate one, I could make it easily by taking a much
thinner and lighter slip of wood, and grinding the needle to give it an
edge. It would also be easy to make it carry small scales of paper for
particular purposes.”