Chapter XIII: Part 13

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.”

The writer of this article has used a balance of this kind, and finds that it is sensible to 1/1000 of a grain when loaded with ten grains. It is necessary, however, where accuracy is required, to employ a scale-pan. This may be made of thin card paper, shaped as in _fig. 191._

A thread is to be passed through the two ends, by tightening which they may be brought near each other.

The most convenient weights for this beam appear to be two of one grain each, and one of one tenth of a grain. They should be made of straight wire; and if the beam be notched at the divisions, they may be lodged in these notches very conveniently. Ten divisions on each side of the middle will be sufficient. The weight of the scale-pan must first be carefully ascertained, in order that it may be deducted from the weight, afterwards determined, of the scale-pan and the substance it may contain.

If the scale-pan be placed at the tenth division of the beam, it is evident that by means of the two grain weights, a greater weight cannot be determined than one grain and nine tenths; but if the scale-pan be placed at any other division of the beam, the resulting apparent weight must be increased by multiplying it by ten, and dividing by the number of the division at which the scale-pan is placed; and in this manner it is evident that if the scale-pan be placed at the division numbered 1, a weight amounting to nineteen grains may be determined.

We have been tempted to describe this little apparatus, because it is extremely simple in its construction, may be easily made, and may be very usefully employed on many occasions where extreme accuracy is not necessary.

_Description of the Steelyard._

The steelyard is a lever, having unequal arms; and in its most simple form it is so arranged, that one weight alone serves to determine a great variety of others, by sliding it along the longer arm of the lever, and thus varying its distance from the fulcrum.

It has been demonstrated, chapter xiii., that in the lever the proportion of the power to the weight will be always the same as that of their distances from the fulcrum, taken in a reverse order; consequently, when a constant weight is used, and an equilibrium established by sliding this weight on the longer arm of the lever, the relative weight of the substance weighed, to the constant weight, will be in the same proportion as the distance of the constant weight from the fulcrum is to the length of the shorter arm.

Thus, suppose the length of the shorter arm, or the distance of the fulcrum from the point from which the weight to be determined is suspended, to be one inch; let the longer arm of the lever be divided into parts of one inch each, beginning at the fulcrum. Now let the constant weight be equal to one pound, and let the steelyard be so constructed that the shorter arm shall be sufficiently heavy to counterpoise the longer when the bar is unloaded. Then suppose a substance, the weight of which is five pounds, to be suspended from the shorter arm. It will be found that when the constant weight is placed at the distance of five inches from the fulcrum, the weights will be in equilibrium, and the bar consequently horizontal. In this steelyard, therefore, the distance of each inch from the fulcrum indicates a weight of one pound. An instrument of this form was used by the Romans, and it is usually described as the Roman statera or steelyard. A representation of it is given at _fig. 192._

The steelyard is in very general use for the coarser purposes of commerce, but constructed differently from that which we have described. The beam with the scales or hooks is seldom in equilibrium upon the point F, when the weight P is removed; but the longer arm usually preponderates, and the commencement of the graduations, therefore, is not at F, but at some point between B and F. The common steelyard, which we have represented at _fig. 193._, is usually furnished with two points, from either of which the substance, the weight of which is to be determined, may be suspended. The value of the divisions is in this case increased in proportion as the length of the shorter arm is decreased. Thus, in the steelyard which we have described, if there be a second point of suspension at the distance of half an inch from the fulcrum, each division of the longer arm will indicate two pounds instead of one, and these divisions are usually marked upon the opposite edge of the steelyard, which is made to turn over.

This instrument is very convenient, because it requires but one weight; and the pressure on the fulcrum is less than in the balance, when the substance to be weighed is heavier than the constant weight. But, on the contrary, when the constant weight exceeds the substance to be weighed, the pressure on the fulcrum is greater in the steelyard than in the balance, and the balance is, therefore, preferable in determining small weights. There is also an advantage in the balance, because the subdivision of weights can be effected with a greater degree of precision than the subdivision of the arm of the steelyard.

_C. Paul’s Steelyard._

A steelyard has been constructed by Mr. C. Paul, inspector of weights and measures at Geneva, which is much to be preferred to that in common use. Mr. C. Paul states, that steelyards have two advantages over balances: 1. That their axis of suspension is not loaded with any other weight than that of the merchandise, the constant weight of the apparatus itself excepted; while the axis of the balance, besides the weight of the instrument, sustains a weight double to that of the merchandise. 2. The use of the balance requires a considerable assortment of weights, which causes a proportional increase in the price of the apparatus, independently of the chances of error which it multiplies, and of the time employed in producing an equilibrium.

1. In C. Paul’s steelyard the centres of the movement of suspension, or the two constant centres, are placed on the exact line of the divisions of the beam; an elevation almost imperceptible in the axis of the beam, destined to compensate for the very slight flexion of the bar, alone excepted.

2. The apparatus, by the construction of the beam, is balanced below its centre of motion, so that when no weight is suspended the beam naturally remains horizontal, and resumes that position when removed from it, as also when the steelyard is loaded, and the weight is at the division which ought to show how much the merchandise weighs. The horizontal situation in this steelyard, as well as in the others, is known by means of the tongue which rises vertically above the axis of suspension.

3. It may be discovered, that the steelyard is deranged if, when not loaded, the beam does not remain horizontal.

4. The advantage of a great and a small side (which in the other augments the extent of their power of weighing) is supplied by a very simple process, which accomplishes the same end with some additional advantages. This process is to employ on the same division different weights. The numbers of the divisions on the bar, point out the degree of heaviness expressed by the corresponding weights. For example, when the large weight of the large steelyard weighs 16 lbs., each division it passes over on the bar is equivalent to a pound; the small weight, weighing sixteen times less than the large one, will represent on each of these divisions the sixteenth part of a pound, or one ounce; and the opposite face of the bar is marked by pounds at each sixteenth division. In this construction, therefore, we have the advantage of being able, by employing both weights at once, to ascertain, for example, almost within an ounce, the weight of 500 pounds of merchandise. It will be sufficient to add what is indicated by the small weight in ounces, to that of the large one in pounds, after an equilibrium has been obtained by the position of the two weights, viz. the large one placed at the next pound below its real weight, and the small one at the division which determines the number of ounces to be added.

5. As the beam is graduated only on one edge, it may have the form of a thin bar, which renders it much less susceptible of being bent by the action of the weight, and affords room for making the figures more visible on both the faces.

6. In these steelyards the disposition of the axes is not only such that the beam represents a mathematical lever without weight, but in the principle of its division, the interval between every two divisions is a determined and aliquot part of the distance between the two fixed points of suspension; and each of the two weights employed has for its absolute weight the unity of the weight it represents, multiplied by the number of the divisions contained in the interval between the two centres of motion.

Thus, supposing the arms of the steelyard divided in such a manner that ten divisions are exactly contained in the distance between the two constant centres of motion, a weight to express the pounds on each division of the beam must really weigh ten pounds; that to point out the ounces on the same divisions must weigh ten ounces, &c. So that the same steelyard may be adapted to any system of measures whatever, and in particular to the decimal system, by varying the absolute heaviness of the weights, and their relation with each other.

But to trace out, in a few words, the advantages of the steelyards constructed by C. Paul for commercial purposes, we shall only observe,--

1. That the buyer and seller are certain of the correctness of the instrument, if the beam remains horizontal when it is unloaded and in its usual position. 2. That these steelyards have one suspension less than the old ones, and are so much more simple. 3. That by these means we obtain, with the greatest facility, by employing two weights, the exact weight of merchandise, with all the approximation that can be desired, and even with a greater precision than that given by common balances. There are few of these which, when loaded with 500 pounds at each end, give decided indication of an ounce variation; and the steelyards of C. Paul possess that advantage, and cost one half less than balances of equal dominion. 4. In the last place, we may verify at pleasure the justness of the weights, by the transposition which their ratio to each other will permit; for example, by observing whether, when the weight of one pound is brought back one division, and the weight of one ounce carried forward sixteen divisions, the equilibrium still remains.

It is on this simple and advantageous principle that C. Paul has constructed his universal steelyard. It serves for weighing in the usual manner, and according to any system of weights, all ponderable bodies to the precision of half a grain in the weight of a hundred ounces; that is to say, of a ten-thousandth part. It is employed, besides, for ascertaining the specific gravity of solids, of liquids, and of air, by processes extremely simple, and which do not require many subdivisions in the weights.

We think the description above given will be sufficiently intelligible without a representation of this instrument. An account of its application to the determination of specific gravities will be found in vol. iii. of the Philosophical Magazine.

_The Chinese Steelyard._

This instrument is used in China and the East Indies for weighing gems, precious metals, &c. The beam is a small rod of ivory, about a foot in length. Upon this are three lines of divisions, marked by fine silver studs, all beginning from the end of the beam, whence the first is extended 8 inches, the second 6-1/2, and the third 8-1/2. The first is European weight, and the other two Chinese. At the other end of the beam hangs a round scale, and at three several distances from this end are holes, through which pass so many fine strings, serving as different points of suspension. The first distance makes 1-3/5 inches, the second 3-1/5, or double the former, and the third 4-4/5, or triple the same. The instrument, when used, is held by one of the strings, and a sealed weight of about 1-1/4 oz. troy, is slid upon the beam until an equilibrium is produced; the weight of the body is then indicated by the graduated scale above mentioned.

_The Danish Balance._

The Danish balance is a straight bar or lever, having a heavy weight fixed to one end, and a hook or scale-pan to receive the substance, the weight of which is to be determined, suspended from the other end. The fulcrum is moveable, and is made to slide upon the bar, till the beam rests in a horizontal position, when the place of the fulcrum indicates the weight required. In order to construct a balance of this kind, let the distance of the centre of gravity from that point to which the substance to be weighed is suspended be found by experiment, when the beam is unloaded. Multiply this distance by the weight of the whole apparatus, and divide the product by the weight of the apparatus increased by the weight of the body. This will give the distance from the point of suspension, at which the fulcrum being placed, the whole will be in equilibrio: for example, supposing the distance of the centre of gravity from the point of suspension to be 10 inches, and the weight of the whole apparatus to be ten pounds; suppose, also, it were required to mark the divisions which should indicate weights of one, two, or three pounds, &c. First, for the place of the division indicating one pound we have (10 × 10)/(10 + 1) = 100/(10 + 1) = 9-1/11 inches, the place of the division marking one pound. For two pounds we have 100/(10 + 2) = 8-1/3 inches, the place of the division indicating two pounds; and for three pounds 100/(10 + 3) = 7-9/13 inches for the place of the division indicating three pounds, and so on.

This balance is subject to the inconvenience of the divisions becoming much shorter as the weight increases. The distance between the divisions indicating one and two pounds being, in the example we have given, about seven tenths of an inch, whilst that between 20 and 21 pounds is only one tenth of an inch; consequently a very small error in the place of the divisions indicating the larger weights would occasion very inaccurate results. The Danish balance is represented at _fig. 194._

_The Bent Lever Balance._

This instrument is represented at _fig. 195._ The weight at C, is fixed at the end of the bent lever A B C, which is supported by its axis B on the pillar I H. A scale-pan E, is suspended from the other end of the lever at A. Through the centre of motion B draw the horizontal line K B G, upon which, from A and C let fall the perpendiculars A K and C D. Then if B K and B D are reciprocally proportional to the weights at A and C, they will be in equilibrio, but if not, the weight C will move upwards or downwards along the arc F G till that ratio is obtained. If the lever be so bent that when A coincides with the line G K, C coincides with the vertical B H, then as C moves from F to G, its momentum will increase while that of the weight in the scale-pan E will decrease. Hence the weight in E, corresponding to different positions of the balance, may be expressed on the graduated arc F G.

_Brady’s Balance, or Weighing Apparatus._

This partakes of the properties both of the bent lever balance and of the steelyard. It is represented, at _fig. 196._ A B C is a frame of cast iron having a great part of its weight towards A. F is a fulcrum, and E a moveable suspender, having a scale and hook at its lower extremity. E K G are three distinct places, to which the suspender E may be applied, and to which belong respectively the three graduated scales of division expressing weights, _f_ C, _c d_, and _a b_. When the scale and suspender are applied at G, the apparatus is in equilibrio, with the edge A B horizontal, and the suspender cuts the zero on the scale _a b_. Now, any substance, the weight of which is to be ascertained, being put into the scale, the whole apparatus turns about F, and the part towards B descends till the equilibrium is again established, when the weight of the body is read off from the scale _a b_, which registers to ounces and extends to two pounds. If the weight of the body exceed two pounds, and be less than eleven pounds, the suspender is placed at K; and when the scale is empty, the number 2 is found to the right of the index of the suspender. If now weights exceeding two pounds be placed in the scale, the whole again turns about F, and the weight of the body is shown on the graduated arc _c d_, which extends to eleven pounds, and registers to every two ounces.

If the weight of the body exceed eleven pounds, the suspender is hung on at E, and the weights are ascertained in the same manner on the scale _f_ C to thirty pounds, the subdivisions being on this scale quarters of pounds. The same principles would obviously apply to weights greater or less than the above. To prevent mistake, the three points of support G, K, E, are numbered 1, 2, 3; and the corresponding arcs are respectively numbered in the same manner. When the hook is used instead of the scale, the latter is turned upwards, there being a joint at _m_ for that purpose.

_The Weighing Machine for Turnpike Roads._

This machine is for the purpose of ascertaining the weight of heavy bodies, such as wheel carriages. It consists of a wooden platform placed over a pit made in the line of the road, and which contains the machinery. The pit is walled withinside, and the platform is fitted to the walls of the pit, but without touching them, and it is therefore at liberty to move freely up and down. The platform is supported by levers placed beneath it, and is exactly level with the surface of the road, so that a carriage is easily drawn on it, the wheels being upon the platform whilst the horses are upon the solid ground beyond it. The construction of this machine will be readily understood by reference to _fig. 197._, in which the platform is supposed to be transparent so as to allow of the levers being seen below it.

A, B, C, D, represent four levers tending towards the centre of the platform, and each moveable on its fulcrum at A, B, C, D; the fulcrum of each rests upon a piece securely fixed in the corner of the pit. The platform is supported upon the cross pins _a_, _b_, _c_, _d_, by means of pieces of iron which project from it near its corners, and which are represented in the plate by the short dark lines crossing the pins _a_, _b_, _c_, _d_. The four levers are connected under the centre of the platform, but not so as to prevent their free motion, and are supported by a long lever at the point F, the fulcrum of which rests upon a piece of masonry at E: the end of this last lever passes below the surface of the road into the turnpike house, and is there attached to one arm of a balance, or, as in Salmon’s patent weighing machine, to a strap passing round a cylinder which winds up a small weight round a spiral, and indicates, by means of an index, the weight placed upon the platform.

_London, Pubd. by Longman & Co._]

Suppose the distance from A to F to be ten times as great as that from A to _a_, then a force of one pound applied beneath F would balance ten pounds applied at _a_, or upon the platform. Again: let the distance from E to G be also ten times greater than the distance from the fulcrum E to F; then a force of one pound applied to raise up the end of the lever G would counterpoise a weight of ten pounds placed upon F. Now, as we gain ten times the power by the first levers, and ten times more by the lever E G, it follows, that a force of one pound tending to elevate G, would balance 100 lbs. placed on the platform; so that if the end of the lever G be attached to one arm of a balance, a weight of 10 lbs. placed in a scale suspended from the other arm, will express the value of 1000 lbs. placed upon the platform. The levers are counterpoised, when the platform is not loaded, by a weight H applied to the end of the last lever, continued beyond the fulcrum for that purpose.

_Of Instruments for weighing by means of a Spring._

The spring is well adapted to the construction of a weighing machine, from the property it possesses of yielding in proportion to the force impressed, and consequently giving a scale of equal parts for equal additions of weight. It is liable, however, to suffer injury, unless the steel of which it is composed be very well tempered, from a want of perfect elasticity, and, consequently, from not returning to its original place after it has been forcibly compressed. This, however, must be considered to arise, in a great measure, from imperfection of workmanship, or of the material employed, or from its having been subjected to too great a force.

_The Spring Steelyard._

The little instrument known by this name is in very general use, and is particularly convenient where great accuracy is not necessary, as a spring which will ascertain weights from one pound to fifty, is contained in a cylinder only 4 inches long and 3/4 inch diameter.

This instrument is represented at _fig. 198._ It consists of a tube of iron, of the dimensions just stated, closed at the bottom, to which is attached an iron hook for supporting the substance to be weighed; a rod of iron _a b_, four tenths of an inch wide and one tenth thick, is firmly fixed in the circular plate _c d_, which slides smoothly in the iron tube.

A strong steel spring is also fastened to this plate, and passed round the rod _a b_ without touching it, and without coming in contact with the interior of the cylindrical tube. The tube is closed at the top by a circular piece of iron through which the piece _a b_ passes.

Upon the face of _a b_ the weight is expressed by divisions, each of which indicates one pound, and five of such divisions in the instrument now before us occupy two tenths of an inch. The divisions, notwithstanding, are of sufficient size to enable them to be subdivided by the eye.

To use this instrument, the substance to be weighed is suspended by the hook, the instrument being held by a ring passing through the rod at the other end. The spring then suffers a compression proportionate to the weight, and the number of pounds is indicated by the division on the rod which is cut by the top of the cylindrical tube.

_Salter’s improved Spring Balance._

A very neat form of the instrument last described has been recently brought before the public by Mr. Salter, under the name of the Improved Spring Balance. It is represented at _fig. 199._ The spring is contained in the upper half of a cylinder behind the brass plate forming the face of the instrument; and the rod is fixed to the lower extremity of the spring, which is consequently extended, instead of being compressed, by the application of the weight. The divisions, each indicating half a pound, are engraved upon the face of the brass plate, and are pointed out by an index attached to the rod.

_Marriott’s Patent Dial Weighing Machine._

The exterior of this instrument is represented at _fig. 200._, and the interior at _fig. 201._ A B C is a shallow brass box, having a solid piece as represented at A, to which the spring D E F is firmly fixed by a nut at D. The other end of the spring at F is pinned to the brass piece G H, to the part of which at G is also fixed the iron racked plate I. A screw L serves as a stop to keep this rack in its place. The teeth of the rack fit into those of the pinion M, the axis of which passes through the centre of the dial-plate, and carries an index which points out the weight. The brass piece G H is merely a plate where it passes over the spring, and the tail piece H, to which the weight is suspended, passes through an opening in the side of the box.

_Of the Dynamometer._

This is an important instrument in mechanics, calculated to measure the muscular strength exerted by men and animals. It consists essentially of a spring steelyard, such as that we first described. This is sometimes employed alone, and sometimes in combination with various levers, which allow of the spring being made more delicate, and consequently increase the extent of the divisions indicating the weight.

The first instrument of this kind appears to have been invented by Mr. Graham, but it was too bulky and inconvenient for use. M. le Roy made one of a more simple construction. It consisted of a metal tube, about a foot long, placed vertically upon a stand, and containing in the inside a spiral spring, having above it a graduated rod terminating in a globe. This rod entered the tube more or less in proportion to the force applied to the globe, and the divisions indicated the quantity of this force. Therefore, when a man pressed upon the globe with all his strength, the divisions upon the rod showed the number of pounds weight to which it was equal.

An instrument of this kind for determining the force of a blow struck by a man with his fist was lately exhibited at the National Repository. It was fixed to a wall, from which it projected horizontally. In place of the globe there was a cushion to receive the blow, and as the suddenness with which the spring returned rendered it impossible to read the division upon the rod, another rod similarly divided was forced in by the plate forming the basis of the cushion, and remained stationary when the spring returned. The common spring steelyard, however, which we first described, is in principle the same as M. le Roy’s dynamometer, and is much more conveniently constructed for the purpose we are considering. The ring at one end may be fixed to an immovable object, and the hook at the other attached to a man, or to an animal, and the extent to which the graduated rod is drawn out of the cylinder shows at once the force which is applied. Though this is perhaps the best, and certainly the most simple dynamometer, others have been contrived, which are, however, but modifications of the spring steelyard. One of these is represented at _fig. 202._ The spiral spring acts in the manner before described, but its divisions are increased in size, and therefore rendered more perceptible by means of a rack fixed to the plate, acting against the spiral spring, the teeth of which move a pinion upon which the arm I is fixed, pointing to the graduated arc K.

Another dynamometer has been invented by Mr. Salmon; it is represented at _fig. 203._ and is a combination of levers with the spring. By means of these levers a much more delicate spring, and which is therefore more sensible, may be employed than in the dynamometer last described.

The manner in which these levers and spring act will be readily understood by an inspection of the figure. Like the weighing machine for carriages, the fulcrum of each lever is at one end, and the force is diminished in passing to the spring, in the ratio of the length of its arms. The spring moves a pinion by means of a rack, upon which pinion a hand is placed, indicating by divisions upon a circular dial-plate, the amount of the force employed.

The spring used in this machine is calculated to weigh only about 50 lbs. instead of about 5 cwt., as in the last described; but by means of the levers which intervene between it and the force applied, it will serve to estimate a force equal to 6 cwt., and might obviously be made to go to a much greater extent, by varying the ratio of the length of the arms of the levers.

ON COMPENSATION PENDULUMS.

(336.) It is said of Galileo that, when very young, he observed a lamp suspended from the roof of a church at Pisa, swinging backwards and forwards with a pendulous motion. This, if it had been remarked at all by an uneducated mind, would, most probably, have been passed by as a common occurrence, unworthy of the slightest notice; but to the mind imbued with science no incident is insignificant; and a circumstance apparently the most trivial, when subjected to the giant force of expanded intellect, may become of immense importance to the improvement and to the well-being of man. The fall of an apple, it is said, suggested to Newton the theory of gravitation, and his powerful mind speedily extended to all creation that great law which brings an apple to the ground. The swinging of a lamp in a church at Pisa, viewed by the piercing intellect of Galileo, gave rise to an instrument which affords the most perfect measure of time, which serves to determine the figure of the earth, and which is inseparably connected with all the refinements of modern astronomy.

The properties of the pendulum, and the manner in which it serves to measure time, have been fully explained in chapter xi.; and if a substance could be found not susceptible of any change in its dimensions from a change of temperature, nothing more would be necessary, as the centre of oscillation would always remain at the same distance from the point of suspension. As every known substance, however, expands with heat, and contracts with cold, the length of the pendulum will vary with every alteration of temperature, and thus the time of its vibration will suffer a corresponding change. The effect of a difference of temperature of 25°, or that which usually occurs between winter and summer, would occasion a clock furnished with a pendulum having an iron rod to gain or lose six seconds in twenty-four hours.

It became, then, highly important to discover some means of counteracting this variation to which the length of the pendulum was liable, or, in other words, to devise a method by which the centre of oscillation should, under every change of temperature, remain at the same distance from the point of suspension: happily, the difference in the rate of expansion of different metals presented a ready means of effecting this.

Graham, in the year 1715, made several experiments to ascertain the relative expansions of various metals, with a view of availing himself of the difference of the expansions of two or more of them when opposed to each other, to construct a compensating pendulum. But the difference he found was so small, that he gave up all hope of being able to accomplish his object in that way. Knowing, however, that mercury was much more affected by a given change of temperature than any other substance, he saw that if the mercury could be made to ascend while the rod of the pendulum became longer, and _vice versâ_, the centre of oscillation might always be kept at the same distance from the point of suspension. This idea happily gave birth to the mercurial pendulum, which is now in very general use.

_London, Pubd. by Longman & Co._]

In the mean time, Graham’s suggestion excited the ingenuity of Harrison, originally a carpenter at Barton in Lincolnshire, who, in 1726, produced a pendulum formed of parallel brass and steel rods, known by the name of the gridiron pendulum.

In the mercurial pendulum, the bob or weight is the material affording the compensation; but in the gridiron pendulum the object is attained by the greater expansion of the brass rods, which raise the bob upwards towards the point of suspension as much as the steel rods elongate downwards.

In the present article, we shall describe such compensation pendulums as appear to us likely to answer best in practice; and we trust we shall be able to simplify the subject so as to render a knowledge of mathematics in the construction of this important instrument unnecessary.

The following table contains the linear expansion of various substances in parts of their length, occasioned by a change of temperature amounting to one degree. We have taken the liberty of extracting it from a very valuable paper by F. Bailey, Esq., on the mercurial compensation pendulum, published in the Memoirs of the Astronomical Society of London for 1824.

TABLE I.

_Linear Expansion of various Substances for One Degree of Fahrenheit’s Thermometer._

+----------------------+-------------+-----------------------+
| Substances. | Expansions. | Authors. |
+----------------------+-------------+-----------------------+
|White Deal, { | ·0000022685 | Captain Kater. |
| { | ·0000028444 | Dr. Struve. |
|English Flint Glass, | ·0000047887 | Dulong and Petit. |
|Iron (cast), { | ·0000061700 | General Roy. |
| { | ·0000065668 | Dulong and Petit. |
|Iron (wire), | ·0000068613 | Lavoisier and L. |
|Iron (bar), | ·0000069844 | Hasslar. |
|Steel (rod), | ·0000063596 | General Roy. |
| | | {Commissioners of |
|Brass, | ·0000104400 | {Weights and Measures |
| | | {--mean of several |
| | | {experiments. |
|Lead, | ·0000159259 | Smeaton. |
|Zinc, | ·0000163426 | Ditto. |
|Zinc (hammered), | ·0000172685 | Ditto. |
|Mercury _in bulk_, | ·00010010 | Dulong and Petit. |
+----------------------+-------------+-----------------------+

From this table it is easy to determine the length of a rod of any substance the expansion of which shall be equal to that of a rod of given length of any other substance.

The lengths of such rods will be inversely proportionate to their expansions. If, therefore, we divide the lesser expansion by the greater (supposing the rod the length of which is given to be made of the lesser expansible material), and multiply the given length by this quotient, we shall have the required length of a rod, the expansion of which will be equal to that of the rod given. For example:--The expansion of a rod of steel being, from the above table, ·0000063596, and that of brass, ·0000104400; if it were required to determine the length of a rod of brass which should expand as much as a rod of steel of 39 inches in length, we have ·0000063596/·0000104400 = ·6091, which, multiplied by 39, gives 23·75 inches for the length of brass required.

We shall here, in order to facilitate calculation, give the ratio of the lengths of such substances as may be employed in the construction of compensation pendulums.

TABLE II.

+---------------------------------------------------+
| Steel rod and brass compensation, as 1: ·6091 |
| Iron wire rod and lead compensation, ·4308 |
| Steel rod and lead compensation, ·3993 |
| Iron wire rod and zinc compensation, ·3973 |
| Steel rod and zinc compensation, ·3682 |
| Glass rod and lead compensation, ·3007 |
| Glass rod and zinc compensation, ·2773 |
| Deal rod and lead compensation, ·1427 |
| Deal rod and zinc compensation, ·1313 |
| Steel rod and mercury in a steel cylinder, ·0728 |
| Steel rod and mercury in a glass cylinder, ·0703 |
| Glass rod and mercury in a glass cylinder, ·0529 |
+---------------------------------------------------+

It is evident that in this table the decimals express the length of a rod of the compensating material, the expansion of which is equal to that of a pendulum rod whose length is unity.

As we are not aware of the existence of any work which contains instructions that might enable an artist or an amateur to make a compensation pendulum, we shall endeavour to give such detailed information as may free the subject from every difficulty.

The pendulum of a clock is generally suspended by a spring, fixed to its upper extremity, and passing through a slit made in a piece which is called the cock of the pendulum. The point of suspension is, therefore, that part of the spring which meets the lower surface of the cock. Now the distance of the centre of oscillation of the pendulum from this point may be varied in two ways; the one by drawing up the spring through this slit, and the other by raising the bob of the pendulum. Either of these methods may be practised in the compensation pendulum, but the former is subject to objections from which the latter is exempt.

Suppose it were required to compensate a pendulum of 39 inches in length, of steel, by means of the expansion of a brass rod. Here, referring to _fig. 204._, we have S C 39 inches (which is to remain constant) of steel; the pendulum spring, passing through the cock at S, is attached to another rod of steel, which is fixed to the cross piece R A at A. The other end of the cross piece at R is fastened to a brass rod, the lower extremity of which is fixed to the cock of the pendulum at B. Now the brass rod B R must expand upwards, as much as the steel rod A C expands downwards; and the length of the brass must be such as to effect this, leaving 39 inches of the steel rod below the cock of the pendulum.

Let us first try 80 inches of steel. Multiplying this by ·6091, we have 48·73 inches for the length of brass, which compensates 80 inches of steel. But as 48·73 inches of the steel, equal in length to the brass, would in this case be above the cock of the pendulum, it would leave only 31·27 inches below it, instead of 39 inches.

Let us now try 100 inches of steel. This, multiplied as before by ·6091, gives 60·91 inches, according to the expansions which we have used, for the length of the brass rod, and leaves 39·09 inches below the cock of the pendulum, which is sufficiently near for our present purpose.

From what has been said we may perceive that the total length of the material of which the pendulum rod is composed must be always equal to the length of the pendulum added to the length of the compensation.

In this instance we have effected our object, by drawing the pendulum-spring through the slit; but we will now show how the same thing may be done by moving the bob of the pendulum. At _fig. 205._, let S C, as before, be equal to 39 inches. Let the steel rod S D turn off at right angles at D, and let a rod of brass B R, of 61 inches in length, ascend perpendicularly from this cross piece to R. To the upper part of the brass rod fix another cross piece R A, and from the extremity A let a steel rod descend to E, bending it as in the figure till it reaches C. Now the total length of the pieces of steel expanding downwards is equal to S D, D F, and F C (amounting together to 39 inches), to which must be added a length of steel equal to that of the brass rod B R, (61 inches), making together 100 inches of steel as before, the expansion of which downwards is compensated by that of the brass rod, of 61 inches in length, expanding upwards.

This form, however, is evidently inconvenient, from the great length of brass and steel which is carried above the cock of the pendulum; but it is the same thing whether the brass and steel be each in one piece, or divided into several, provided the pieces of steel be all so arranged as to expand downwards, and those of brass upwards. Thus, at _fig. 206._, the portions of steel expanding downwards are together equal, as before, to 100 inches, and the two brass pieces expanding upwards are together equal to 61 inches. So that, in fact, the two last forms of compensation which we have described differ in no respect from each other in principle, but only in the arrangement of the materials. The last is the half of the gridiron pendulum, the remaining bars being merely duplicates of those we have described, and serving no other purpose but to form a secure frame-work.

_Harrison’s Gridiron Pendulum._

After what has been said, little more is necessary than to give a representation of this pendulum. This is done at _fig. 207._, in which the darker lines represent the steel rods, and the lighter those of brass. The central rod is fixed at its lower extremity to the middle of the third cross piece from the bottom, and passes freely through holes in the cross pieces which are above, whilst the other rods are secured near their extremities to the cross pieces by pins passing through them. In order to render the whole more secure, the bars pass freely through holes made in two other cross pieces, the extremities of which are fixed to the exterior steel wires. As different kinds of the same metal vary in their rate of expansion, the pendulum when finished may be found upon trial to be not duly compensated. In this case one or more of the cross pieces is shifted higher or lower upon the bars, and secured by pins passed through fresh holes.

_Troughton’s Tubular Pendulum._

This is an admirable modification of Harrison’s gridiron pendulum. It is represented at _fig. 208._, where it may be seen that it has the appearance of a simple pendulum, as the whole compensation is concealed within a tube six tenths of an inch in diameter.