WATCHWORK

WATCHWORK. Our intention in this article does not extend to the manual practice of this art, nor even to all the parts of the machine. We mean to consider the most important and difficult part of the construction, namely, the method of applying the maintaining power of the wheels to the regulator of the motion, so as not to hurt its power of regulation. Our observations would have come with more propriety under the title SCAPEMENT, that being the name given by our artists to this part of the construction. Indeed they were intended for that article, which had been unaccountably omitted in the body of the Dictionary under the words CLOCK and WATCH. But the bad health and occupations of the person who had engaged to write the article, have obliged us to defer it to the last opportunity which the alphabetical arrangement affords us; and, even now, the same causes unfortunately prevent the author from treating the subject in the manner he intended, and which it well deserves. But we trust that, from the account which is here given, the reader, who is conversant in mathematical philosophy, will perceive the justice of the conclusions, and that an intelligent artist will have no hesitation in acceding to the propriety of the maxims of construction deduced from them.

Watch-
work.

The regulator of a clock or watch is a pendulum or a balance. Without this check to the motion of the wheels, impelled by a weight or a spring, the machine would run down with a motion rapidly accelerating, till friction and the resistance of the air induced a sort of uniformity, as they do in a kitchen jack. But if a pendulum be so put in the way of this motion, that only one tooth of a wheel can pass it at each vibration, the revolution of the wheels will depend on the vibration of the pendulum. This has long been observed to have a certain consistency, inasmuch that the astronomers of the East employed pendulums in measuring the times of their observations, patiently counting their vibrations during the phases of an eclipse or the transits of the stars, and renewing them by a little push with the finger when they became too small. Gassendi, Riccioli, and others, in more recent times, followed this example. The celebrated physician Sanctorius is the first person who is mentioned as having applied them as regulators of clock movements. Machines, however, called clocks, with a train of toothed wheels leading round an index of hours, had been contrived long before. The earliest of which we have any account is that of Richard of Wallingford, Abbot of St Alban's, in 1326 (A). It appears to have been regulated by a fly like a kitchen jack*. Not long after this Giacomo Donati made one at Padua, which had a motus successorius, a hobbling or trotting motion; from which expression it seems probable that it was regulated by some alternate movement. We cannot think that this was a pendulum, because, once it was introduced, it never could have been supplanted by a balance. The alternate motion of a pendulum, and its seeming uniformity, are among the most familiar observations of common life; and it is surprising that they were not more early thought of for regulating time measurers. The alternate motions of the old balance is one of the most far-fetched means that can be imagined, and might pass for the invention of a very reflecting mind, while a pendulum only requires to be drawn aside from the plumb-line, to make it vibrate with regularity. The balance must be put in motion by the clock, and that motion must be stopped, and the contrary motion induced; and we must know that the same force and the same checks will produce uniform oscillations. All this must be previously known before we can think of it as a regulator; yet so it is that clocks, regulated by a balance, were long used, and very common through Europe, before Galileo proposed the pendulum, about the year 1600. Pendulum clocks then came into general use, and were found to be greatly preferable to balance clocks as accurate measurers of time. Mathematicians saw that their vibrations had some regular

regular dependance on uniform gravity, and in their writings we meet with many attempts to determine the time and demonstrate the isochronism of the vibrations. It is amusing to read these attempts. We wonder at the awkwardness and insufficiency of the explanation given of the motions of pendulums, even by men of acknowledged eminence. Merfennus carried on a most useful correspondence with all the mathematicians of Europe, and was the means of making them acquainted with each other; nay, he was himself well conversant in the science; yet one cannot but smile at his reasonings on this subject. Standing on the shoulders of our predecessors, we look around us, in great satisfaction with our own powers of observation, not thinking how we are raised up, or that we are trading with the stock left us by the diligent and sagacious philosophers of the 17th century (a). Riccioli, Gassendus, and Galileo, made similar attempts to explain the motion of pendulums; but without success. This honour was reserved for Mr Huyghens, the most elegant of modern geometers. He had succeeded in 1656 or 1657 in adapting the machinery of a clock to the maintaining of the vibrations of a pendulum. Charmed with the accuracy of its performance, he began to investigate with scrupulous attention the theory of its motion. By the most ingenious and elegant application of geometry to mechanical problems, he demonstrated that the wider vibrations of a pendulum employed more time than the narrower, and that the time of a semicircular vibration is to that of a very small one nearly as 34 to 29; and, aided by a new department of geometrical science invented by himself, namely, the evolution of curves, he shewed how to make a pendulum swing in a cycloid, and that its vibrations in this curve are all performed in equal times, whatever be their extent.

But before this time, Dr Hooke, the most ingenious and inventive mechanician of his age, had discovered the great accuracy of pendulum clocks, having found that the manner in which they had been employed had obscured their real merit. They had been made to vibrate in very large arches, the only motion that could be given them by the contrivances then known; and in 1656 he invented another method, and made a clock which moved with astonishing regularity. Using a heavy pendulum, and making it swing in very small arches, the clocks so constructed were found to excel Mr Huyghens's cycloidal pendulums; and those who were unfriendly to Huyghens had a sort of triumph on the occasion. But this was the result of ignorance. Mr Huyghens had shewn, that the error of $\backslash frac\{1\}\{100\}$ of an inch, in the formation of the parts which produced the cycloidal motion, caused a greater irregularity of vibration than a circular vibration could do, although it should extend five or six degrees on each side of the perpendicular. It has been found that the unavoidable inaccuracies, even of the best artists, in the cycloidal construction, make the performance much inferior to that of a common pendulum vibrating in arches which

do not exceed three or four degrees from the perpendicular. Such clocks alone are now made, and they exceed all expectation.

We have said that a pendulum needed only to be removed from the perpendicular, and then let go, in order to vibrate and measure time. Hence it might seem, that nothing is wanted but a machinery connected with the pendulum as to keep a register, as it were, of the vibration. It could not be difficult to contrive a method of doing this; but more is wanted. The air must be displaced by the pendulum. This requires some force, and must therefore employ some part of the momentum of the pendulum. The pivot on which it swings occasions friction—the thread, or thin piece of metal by which it is hung, in order to avoid this friction, occasions some expenditure of force by its want of perfect flexibility or elasticity. These, and other causes, make the vibrations grow more and more narrow by degrees, till at last the pendulum is brought to rest. We must therefore have a contrivance in the wheelwork which will restore to the pendulum the small portion of force which it loses in every vibration. The action of the wheels therefore may be called a maintaining power, because it keeps up the vibrations.

But we now see that this may affect the regularity of vibration. If it be supposed that the action of gravity renders all the vibrations isochronous, we must grant that the additional impulsion by the wheels will destroy that isochronism, unless it be so applied that the sum total of this impulsion and the force of gravity may vary so with the situation of the pendulum, as still to give a series of forces, or a law of variation, perfectly similar to that of gravity. This cannot be effected, unless we know both the law which regulates the action of gravity, producing isochronism of vibration, and the intensity of the force to be derived from the wheels in every situation of the pendulum.

The necessary requisite for the isochronous motion of the pendulum is, that the force which urges it toward the perpendicular, be proportional to its distance from it (see DYNAMICS, n^o 103. Cor. 7. Suppl.); and therefore, since pendulums swinging in small circular arches are sensibly isochronous, we must infer that such is the law by which the accelerating action of gravity on them is really accommodated to every situation in those arches.

It will greatly conduce to the better understanding of the effect of the maintaining power, if the reader keep in continual view the chief circumstances of a motion of this kind. Therefore let $ACa$ (fig. 1.) represent the arch passed over by the pendulum, stretched out into a straight line. Let $C$ be its middle point, when the pendulum hangs perpendicular, and $A$ and $a$ be the extremities of the oscillation. Let $AD$ be drawn perpendicular to $AC$, to represent the accelerating action of gravity on the pendulum when it is at $A$. Draw the straight line $DCd$, and $ad$, perpendicular to $Aa$. About $C$, as a centre, describe the semicircle $AFHa$. Through

Through any points B, K, $k$, $b$, &c. of A $a$, draw the perpendiculars BFE, KLM, &c. cutting both the straight line and the semicircle. Then,

1. The actions of gravity on the pendulum, when in the situations B, K, &c. by which it is urged toward C, are proportional to, and may be represented by, the ordinates BE, KL, $b\; e$, $k\; l$, &c. to the straight line DC $d$.

2. The velocities acquired at B, K, &c. by the acceleration along AB, AK, &c. are proportional to the ordinates BF, KM, &c. to the semicircle AH $a$; and, therefore, the velocity with which the pendulum passes through the middle point C, is to its velocity in any other point B, as CH to BF.

3. The times of describing the parts AB, BK, KC, &c. of the whole arch of oscillation, are proportional to, and may be represented by, the arches AF, FM, MH, &c. of the semicircle.

4. If one pendulum describe the arch represented by AC $a$, and another describe the arch KC $k$, they will describe them in equal times, and their maximum velocities (viz. their velocities in the middle point), are proportional to AC and KC; that is, the velocities in the middle point are proportional to the width of the oscillations.

The same proportions are true with respect to the motions outwards from C. That is, when the pendulum describes CA, with the initial velocity CH, its velocity at K is reduced to KM by the retarding action of gravity. It is reduced to BF at B, and to nothing at A; and the times of describing CK, KB, BA, CA, are as HM, HF, HA. Another pendulum setting out from C, with the initial velocity CO, reaches only to K, CK being = CO. Also the times are equal.—If we consider the whole oscillation as performed in the direction A $a$, the forces AD, BE, KL accelerate the pendulum, and the similar forces $a\; d$, $b\; e$, $k\; l$, on the other side, retard it. The contrary happens in the next oscillation aCA.

5. The areas DABE, DAKL, &c. are proportional to the squares of the velocities acquired by moving along AB, AK, &c. or to the diminution of the squares of the velocities sustained by moving outwards along BA or KA, &c.

The consideration of this figure will enable the reader (even though not a mathematician) to form some notion of the effect of any proposed application of a maintaining power by means of wheelwork: For, knowing the weight of the pendulum, we know the accelerating action of that weight in any particular situation A of the pendulum. We also know what addition or subtraction we produce on the pendulum in that situation by the wheelwork. Suppose it is an addition of pressure equal to a certain number of grains. We can make AD to D' as the first to the last; and then A' will be the whole force urging the pendulum toward C. Doing the same for every point of AC, we obtain a line $A\text{'}\; a\text{'}$, which is a new scale of forces, and the space DC', comprehended between the two scales CD and C', will express the addition made to the square of the velocity in passing along AC by the joint action of gravity and the maintaining power. Also, by drawing a line $x\; x\text{'}$ perpendicular to AC, making the space C $x\; x\text{'}$ equal to CAD, the point $x\text{'}$ will be the limit of the oscillation outward from C, where the initial

velocity HC is extinguished. If the line $x\; x\text{'}$ cut the same circle in $x\text{'}$, one-half the arch A will nearly express the contraction made in the time of the outward oscillation by the maintaining power. An accurate determination of this last circumstance is operose, and even difficult; but this solution is not far from the truth, and will greatly assist our judgment of the effect of any proposal, even though $x\; x\text{'}$ be drawn only by the judgement of the eye, making the area left out as nearly equal to the area taken in as we can estimate by inspection. This is said from experience.

Since the motion of a pendulum or balance is alternate, while the pressure of the wheels is constantly in one direction, it is plain that some art must be used to accommodate the one to the other. When a tooth of the wheel has given the balance a motion in one direction, it must quit it, that it may get an impulsion in the opposite direction. The balance or pendulum thus escaping from the tooth of the wheel, or the tooth escaping from the balance, has given to the general contrivance the name of SCAPEMENT among our artists, from the French word échappement. We proceed, therefore, to consider this subject more particularly, first considering the scapements which are peculiarly suited to the small vibrations of pendulums, and then those which must produce much wider vibrations in balances. This, with some other circumstances, render the scapements for pendulums and balances very different.

THE scapement which has been in use for clocks and watches ever since their first appearance in Europe, is extremely simple, and its mode of operation is too obvious to need much explanation. In fig. 2. XY represents a horizontal axis, to which the pendulum P is attached by a slender rod, or otherwise. This axis has two leaves C and D attached to it, one near each end, and not in the same plane, but so that when the pendulum hangs perpendicularly, and at rest, the piece C spreads a few degrees to the right hand, and D as much to the left. They commonly make an angle of 70, 80, or 90 degrees. These two pieces are called PALLETS. AFB represents a wheel, turning round on a perpendicular axis EO, in the order of the letters AFB. The teeth of this wheel are cut into the form of the teeth of a saw, leaning forward, in the direction of the motion of the rim. As they somewhat resemble the points of an old fashioned royal diadem, this wheel has got the name of the CROWN WHEEL. In watches it is often called the balance wheel. The number of teeth is generally odd; so that when one of them B is pressing on a pallet D, the opposite pallet C is in the space between two teeth A and I. The figure represents the pendulum at the extremity of its excursion to the right hand, the tooth A having just escaped from the pallet C, and the tooth B having just dropped on the pallet D. It is plain, that as the pendulum now moves over to the left, in the arch PG, the tooth B continues to press on the pallet D, and thus accelerates the pendulum, both during its descent along the arch PH, and its ascent along the arch HG. It is no less evident, that when the pallet D, by turning round the axis XY, raises its point above the plane of the wheel, the tooth B escapes from it, and I drops on the pallet C, which is now nearly perpendicular. I presses C to the right,

and accelerates the motion of the pendulum along the arch GP. Nothing can be more obvious than this action of the wheel in maintaining the vibrations of the pendulum. We can easily perceive, also, that when the pendulum is hanging perpendicularly in the line XH, the tooth B, by pressing on the pallet D, will force the pendulum a little way to the left of the perpendicular, and will force it so much the farther as the pendulum is lighter; and, if it be sufficiently light, it will be forced so far from the perpendicular that the tooth B will escape, and then I will catch on C, and force the pendulum back to P, where the whole operation will be repeated. The same effect will be produced in a more remarkable degree, if the rod of the pendulum be continued through the axis XY, and a ball Q put on the other end to balance P. And, indeed, this is the contrivance which was first applied to clocks all over Europe, before the application of the pendulum. They were balance clocks. The force of the wheel was of a certain magnitude, and therefore able, during its action on a pallet, to communicate a certain quantity of motion and velocity to the balls of the balance. When the tooth B escapes from the pallet D, the balls are then moving with a certain velocity and momentum. In this condition, the balance is checked by the tooth I catching on the pallet C. But it is not instantly stopped. It continues its motion a little to the left, and the pallet C forces the tooth I a little backward. But it cannot force it so far as to escape over the top of the tooth I; because all the momentum of the balance was generated by the force of the tooth B; and the tooth I is equally powerful. Besides, when I catches on C, and C continues its motion to the left, its lower point applies to the face of the tooth I, which now acts on the balance by a long and powerful lever, and soon stops its farther motion in that direction, and now, continuing to press on C, it urges the balance in the opposite direction.

Thus we see that in a escapement of this kind, the motion of the wheel must be very hobbling and unequal, making a great step forward, and a short step backward, at every beat. This has occasioned the contrivance to get the name of the RECOILING ESCAPEMENT, the recoiling pallets. This hobbling motion is very observable in the wheel of an alarm.

Thus have we obtained two principles of regulation. The first and most obvious, as well as the most perfect, is the natural isochronous vibration of a pendulum. The only use of the wheelwork here, besides regulating the vibrations, is to give a gentle impulsion to the pendulum, by means of the pallet, in order to compensate friction, &c. and thus maintain the vibrations in their primitive magnitude. But there is no such native motion in a balance, to which the motion of the wheels must accommodate itself. The wheels, urged by a determined pressure, and acting through a determined space (the face of the pallet), must generate a certain determined velocity in the balance; and therefore the time of the oscillation is also determined, both during the progressive and the retrograde motion of the wheel. The actions being similar, and through equal spaces, in every oscillation, they must employ the same time. Therefore a balance, moved in this manner, must be isochronous, and a regulator for a time-keeper.

By thus employing a balance, the horizontal position

of the axis XY is unnecessary. Accordingly, the old clocks had this axis perpendicular, by which means the whole weight of the balance rested on the point of the pivot Y or X, according as the balance PQ was placed above or below. By making the supporting pivot of hard steel, and very sharp, friction was greatly diminished. Nay, it was entirely removed from this part of the machine by suspending the balance by a thread at the end X, instead of allowing it to rest on the point of the pivot Y.

As the balance regulator of the motion admits of every position of the machine, those clocks were made in an infinite variety of fanciful forms, especially in Germany, a country famous for mechanical contrivances. They were made of all sizes, from that of a great steeple clock, to that of an ornament for a lady's toilet. The substitution of a spring in place of a weight, as a first mover of the wheel-work, was a most ingenious thought. It was very gradual. We have seen, in the Emperor's museum at Brussels, an old (perhaps the first) spring clock, the spring of which was an old sword blade, from the point of which a catgut was wound round the barrel of the first wheel. Some ingenious German substituted the spiral spring, which both took less room, and produced more revolutions of the first wheel.

When clocks had been reduced to such small sizes, the wish to make them portable was very natural; and the means of accomplishing this were obvious, namely, a farther reduction of their size. This was accomplished very early; and thus we obtained pocket watches, moved by a spiral spring, and regulated by a balance with the recoiling escapement, which is still in use for common watches. The hobbling motion of the crown wheel is very easily seen in all of them.

It is very uncertain who first substituted a pendulum in place of the balance (CLOCK, Encyc.) Huyghens, as we have already observed, was the first who investigated the motions of pendulums with success, and his book De Horologio Oscillatorio may be considered as the elements of refined mechanics, and the source of all the improvements that have been made in the construction of escapements. But it is certain that Dr Hooke had employed a pendulum for the regulation of a clock many years before the publication of the abovementioned treatise, and he claims the merit of the invention of the only proper method of employing it. We imagine therefore that Dr Hooke's invention was nothing more than a escapement for a pendulum making small vibrations, without making use of the opposite motions of the two sides of the crown wheel. Dr Hooke had contrived some escapement more proper for pendulums than the recoiling pallets, because certainly those might be employed, and are actually employed as a escapement for pendulum clocks to this day, although they are indeed very ill adapted to the purpose. He had not only remarked the great superiority of such pendulum clocks as were made before Huyghens's publication of the cycloidal pendulum over the balance clocks, but had also seen their defects, arising from the light pendulums and wide arches of vibration, and invented a escapement of the nature of those now employed. The pendulum clock which he made in 1658 for Dr Wilkins, afterwards Bishop of Chester, is mentioned by the inventor as peculiarly suited to the moderate swing of a pendulum; and he opposes this circumstance to a general practice

practice of wide vibrations and trifling pendulums. The French are not in the practice of ascribing to us any thing that they can claim as their own; yet Lepaute says that the Échappement à l'Ancre came from England about the year 1665. It is also admitted by him that clock-making flourished in England at that time, and that the French artists went to London to improve in it. Putting these and other circumstances together, we think it highly probable that we are indebted to Dr Hooke for the escapement now in use. The principle of this is altogether different from the simple pallets and direct impulse already described; and is so far from being obvious, that the manner of action has been misunderstood, even by men of science, and writers of systems of mechanics.

In this escapement we employ those teeth of the wheel which are moving in one direction; whereas in the former escapement, opposite teeth were employed moving in contrary directions. Yet even here we must communicate an alternate motion to the axis of the pallets. The contrivance, in general, was as follows: On the axis A (See fig. 3.) of the pendulum or balance is fixed a piece of metal BAC, called the CRUTCH by our artists, and the ANCHOR by the French. It terminates in two faces B b C c of tempered steel, or of some hard stone. These are called the PALLETS, and it is on them that the teeth of the wheel act. The faces B b C c are set in such positions that the teeth push them out of the way. Thus B pushes the pallet to the left, and C pushes its pallet to the right. Both push their pallets sidewise outward from the centre of the wheel. The pallet B is usually called the leading, and C the driving pallet by the artists, although it appears to us that these names should be reversed, because B drives the pallet out of the way, and C pulls or leads it out of the way. They might be called the first and second pallet, in the order in which they are acted on by the wheel. We shall use either denomination. The figure is accommodated to the inactive or resting position of the pendulum. Suppose the pendulum drawn aside to the right at Q, and then let go. It is plain that the tooth B, pressing on the face of the pallet B b all the way from s to b, thrusts it aside outwards, and thus, by the connection of the crutch with the pendulum rod, aids the pendulum's motion along the arch QPR. When the pendulum reaches R, the point of the tooth B has reached the angle b of the pallet, and escapes from it. The wheel pressing forward, another tooth C drops on the pallet face C c, and, by pressing this pallet outward, evidently aids the pendulum in its motion from R to P. The tooth C escapes from this pallet at the angle c, and now a tooth B' drops on the first pallet, and again aids the pendulum; and this operation is repeated continually.

The mechanism of this communication of motion is thus explained by several writers of elements. The tooth B (fig. 2.) is urged forward in the direction BD, perpendicular to the radius MB of the SWING WHEEL. It therefore presses on the pallet, which is moveable only in the direction BE, perpendicular to BA the radius of the pallet. Therefore the force BD must be resolved into two, viz. BE, in the direction in which alone the pallet can move, and ED, or BF, perpendicular to that direction. The last of these only presses the pallet and crutch against the pivot hole A.

BE is the only useful force, or the force communicated to the pallet, enabling it to maintain the pendulum's motion, by restoring the momentum lost by friction and other causes.

But this is a very erroneous account of the modus operandi, as may be seen at once, by supposing the radius of the pallets to be a tangent to the wheel. This is a position most frequently given to them, and is the very position in fig. 3. In this case MB is perpendicular to BA, and therefore BD will coincide with BA, and there will be no such force as BE to move the pendulum. It is a truth, deducible from what we know of the mechanical constitution of solid bodies, and confirmed by numberless observations, that when two solid bodies press on each other, either in impulse or in dead pressure, the direction in which the mutual pressure is exerted is always perpendicular to the touching surfaces, whatever has been the direction of the impelling body (See IMPULSION, SUPPL. n° 66. MACHINERY, SUPPL. n° 35. and several other parts of this Work.) Moreover this pressure is mutual, equal, and opposite. Whatever be the shapes of the faces of the tooth and pallet, we can draw a plane BN, which is the common tangent to both surfaces, and a line HBI through the point of contact perpendicular to BN. It is farther demonstrated in the article MACHINERY, SUPPL. n° 35, &c. that the action of the wheel on the pendulum is the same as if the whole crutch were annihilated, and in its stead there were two rigid lines AH, MI, from the centres of the crutch and wheel, perpendicular to HI, and connected by a third rigid line or rod HI, touching the two in H and I.

For if a weight V be hung at v, the extremity of the horizontal radius M v of the wheel, it will act on the lever v MI, pressing its point I upwards in the direction IH perpendicular to MI; the upper end of this rod IH will, in like manner, press the extremity H of the rod HA, and this will urge the pendulum from P toward R. To withstand this, the pendulum rod AP may be withheld by a weight z, hanging by a thread on the extremity of the horizontal lever A z, equal to M v, and connected with the crutch and pendulum. The weights V and z may be so proportioned to each other that, by acting perpendicularly on the crooked levers v MI, and z AH, the pressures at H and I shall be equal, and just balance each other by the intervention of the rod HI. When this is the case, we have put things into the same mechanical state, in respect of mutual action, as is effected by the crutch, pallets, and wheel, which, in like manner, produce equal pressures at B the point of contact, in the direction BI and BF. The weight V may be such as produces the very same effect at B that is produced by the previous train of wheel-work. The weight z therefore must be just equal to the force produced by the wheel-work on the point z of the pendulum rod, because by acting in the opposite direction it just balances it. Let us see therefore what force is communicated to the pendulum by the wheels.

Let x be the upward pressure excited at I, and y the equal opposite pressure excited at H. Then, by the property of the lever, we have MI : M v = V : x, and x x MI = V x M v. In like manner y x AH = Z x A z. Therefore, because x = y, and A z = M v, we have V : Z = MI : AH. That is, the force exerted

ed by the tooth of the wheel in the direction of its motion is to the force impressed on the pendulum rod at a distance equal to the radius of the wheel as $MI$ to $AH$. The force impressed on the ball of the pendulum is less than this in the proportion of $AP$ to $Az$, or $Mv$.

Cor. 1. If the perpendiculars $MN$, $AV$, be drawn on the tangent plane, the forces at $B$ and $z$ will be as $BN$ to $BO$. For these lines are respectively equal to $MI$ and $AH$.

Cor. 2. If $HI$ meet the line of the centres $AC$ in $S$, the forces will be as $SM$ to $SA$; that is, $V\; :\; Z\; =\; SM\; :\; SA$.

Cor. 3. If the face $Bb$ of the pallet be the involutrix of a circle described with the radius $AH$, and the face of the tooth be the involutrix of a circle described with the radius $MI$, the force impressed on the pendulum by the wheels will be constant during the whole vibration (MACHINERY, n^o 26). But these are not the only forms which produce this constancy. The forms of teeth described by different authors, such as De la Hire, Camus, &c. for producing a constant force in trains of wheel-work, will have the same effect here. It is also easy to see that the force impressed on the pendulum may be varied according to any law, by making these faces of a proper form. Therefore the face, from $B$ outwards, may be so formed that the force communicated to the pendulum by the wheels, during its descent from $Q$ to $P$, may be in one constant proportion to the acceleration of gravity, and then the sum of the forces will be such as produce isochronous vibrations. If the inner part $Bb$ of the face be formed on the same principle, the difference of the forces will have the same law of variation. If the face $Bb$ be the involutrix of a circle, and the tooth $B$ terminate in a point gently rounded, or quite angular, the force on the pendulum will continually increase as the tooth slides from $a$ to $b$. For the line $AH$ continues of the same magnitude, and $MI$ diminishes. The contrary will happen, if the pallet be a point, either sharp or rounded, and if the face of the tooth be the involutrix now mentioned; for $MI$ will remain the same, while $AH$ diminishes. If the tooth be pointed, and $Bb$ be a straight line, the force communicated to the pendulum will diminish, while the tooth slides from $a$ to $b$. For in this case $AH$ diminishes and $MI$ increases.

Cor. 4. In general, the force on the pendulum is greater as the angle $MBb$ increases, and as $ABb$ diminishes.

Cor. 5. The angular velocity of the wheel is to that of the pendulum, in any part of its vibration, as $AH$ to $MI$. This is evident, because the rod $HI$ moving (in the moment under consideration) in its own direction, the points $H$ and $I$ move through equal spaces, and therefore the angles at $A$ and $M$ must be inversely as the radii.

All that has now been said of the first pallet $AB$ may be applied to the second pallet $AC$.

If the perpendiculars $Cz$ be drawn to the touching plane $oCz$, cutting $AM$ in $x$, we shall have $V\; :\; x\; =\; sM\; :\; sA$, as in Cor. 2. And if the perpendiculars $Mi$, $Ah$, be drawn on $Cz$, we have $V\; :\; Z\; =\; Mi\; :\; Ah$, as in the general theorem. The only difference between the action on the two pallets is, that if the faces of both are plain, the force on the pendulum increases during the whole of the action on the pallet $C$, whereas it dimi-

nishes during the progress of the tooth along the other pallet.

The reader will doubtless remark that each tooth of the wheel acts on both pallets in succession; and that, during its action on either of them, the pendulum makes one vibration. Therefore the number of vibrations during one turn of the wheel is double the number of the teeth: consequently, while the tooth slides along one of the pallets, it advances half the space between two successive teeth; and when it escapes from the pallet, the other tooth may be just in contact with the other pallet. We say it may be so; in which case there will be no dropping of the teeth from pallet to pallet. This, however, requires very nice workmanship, and that every tooth be at precisely the same distance from its neighbour. Should the tooth which is just going to apply to a pallet chance to be a little too far advanced on the wheel, it would touch the pallet before the other had escaped. Thus, suppose that before $B$ escapes from the point $b$ of the pallet, the tooth $C$ is in contact with the pallet $CG$, $B$ cannot escape. Therefore when the pendulum returns from $R$ towards $Q$, the pallet $ab$, returning along with it, will push back the tooth $B$ of the wheel. It does this in opposition to the force of the wheel. Therefore, whatever motion the wheel had communicated to the pendulum, during its swing from $P$ to $Q$, will now be taken from it again. The pendulum will not reach $Q$, because it had been aided in its motion from $Q$, and had proceeded further than it would have done without this help. Its motion toward $Q$ is further diminished by the friction of the pallet. Therefore it will now return again from some nearer point $q$, and will not go so far as in the last vibration, but will return through a still shorter arch: And this will be still more contracted in the next vibration, &c. &c. Thus it appears that if a tooth chances to touch the pallet before the escape of the other, the wheel will advance no farther, and soon after the pendulum will be brought to rest.

For such reasons it is necessary to allow one tooth to escape a little before the other reaches the pallet on which it is to act, and to allow a small drop of the teeth from pallet to pallet. But it is accounted bad workmanship to let the drop be considerable, and close scapement is accounted a mark of care and of good workmanship. It is evidently an advantage, because it gives a longer time of action on each pallet. This freeing the scapement cannot be accomplished by filing something from the face of the tooth; because this being done to all, the distance between them is diminished rather than augmented. The pallets must be first scaped as close as possible. This obliges the workman to be careful in making the teeth equidistant. Then a small matter is taken from the point of each pallet, by filing off the back $br$ of the pallet. The tooth will now escape before it has moved through half a space.

From all that has been said on this particular, it appears that the interval between the pallets must comprehend a certain number of teeth, and half a space more.

The first circumstance to be considered in contriving a scapement is the angular motion that is intended to be given to the pendulum during the action of the wheel. This is usually called the angle of scapement, or the angle of action. Having fixed on an angle $a$ that we think proper,

proper, we must secure it by the position and form of the face of the pallets. Knowing the number of teeth in the swing-wheel, divide $180^\backslash circ$ by this number, and the quotient is the angle $b$ of the wheel's motion during one vibration of the pendulum. In the line AM, joining the centres of the crutch and wheel, make SM to SA, and $rM$ to $rA$, as the angle $a$ to the angle $b$; and then, having determined how many teeth shall be comorehended between the pallets, call this number $n$. Multiply the angle $b$ by $n\; +\; r$, and take the half of the product. Set off this half in the circumference of the wheel (at the points of the teeth) on each side of the line joining the centres of the crutch and wheel, as at TB and TC. Through S and $s$ draw SB and $sC$, and through B draw $sBb$ perpendicular to SB, for the medium position of the face of the first pallet; that is, for its position when the pendulum hangs perpendicular. In like manner, drawing $sCn$ perpendicular to $sC$, we have the medium position of the second pallet. The demonstration of this construction is very evident from what has been said.

We have hitherto supposed that the pendulum furnishes its vibration at the instant that a tooth of the wheel escapes from a pallet, and another tooth drops on the other pallet. But this is never, or should never be, the case. The pendulum is made to swing somewhat beyond the angle of escapement: for if it do not when the clock is clean and in good order, but stop precisely at the drop of a tooth, then, when it grows foul, and the vibration diminishes, the teeth will not escape at all, and the clock will immediately stop. Therefore the force communicated by the wheels during the vibration within the limits of escapement, must be increased so as to make the pendulum throw (as the artists term it) farther out; and a clock is more valued when it throws out considerably beyond the angle of escapement. There are good reasons for this. The momentum of the pendulum, and its power to regulate the clock (which Mr Harrison significantly called its dominion), is proportional to the width of its vibrations very nearly.

This circumstance of exceeding the angle of escapement has a very great influence on the performance of the clock, or greatly affects the dominion of the pendulum. It is easy to see that, when the face $sBb$ of the leading pallet is a plane, if the pendulum continue its motion to the right, from P toward Q, after the tooth B has dropped on it, the pallet will push the wheel back again, while the tooth slides outward on the pallet toward $s$. Such pallets therefore will make a recoiling escapement, resembling, in this circumstance, the old pallet employed with the crown wheel, and will have the properties attached to this circumstance. One consequence of this is, that it is much affected by any inequalities of the maintaining power. It is a matter of the most familiar observation, that a common watch goes slower when within a quarter of an hour of being down, when the action of the spring is very weak, in consequence of its not pulling by a radius of the fusée. We observe the same thing in the beating of an alarm clock. Also if we at any time press forward the wheelwork of a common watch with the key, we observe its beats accelerate immediately. The reason of this is pretty plain. The balance, in consequence of the acceleration in the angle of escapement, would have gone much farther, employing a considerable time in

the excursion. This is checked abruptly, which both shortens the vibration and the time employed in it. In the return of the pendulum, the motion is accelerated the whole way, along an arch which is shorter than what corresponds to its velocity in the middle point; for it is again checked on the other side, and does not make its full excursion. Moreover, all this irregularity of force, or the great deviation from a resistance to the excursion proportional to the distance from the middle point, is exerted on the pendulum when it is near the end of the excursion, where the velocity being small, this irregular force acts long upon it, at the very time that it has little force wherewith to resist it. All temporary inequalities of force, therefore, will be more felt in this situation of the balance than if they had been exerted in the middle of its motion. And although the regulating power of a pendulum greatly exceeds that of the light balances used in pocket watches, something of the same kind may be expected even in pendulum clocks. Accordingly this appears by a series of experiments made by Mr Berthoud, a celebrated watchmaker of Paris. A clock, with a half second pendulum weighing five drams, was furnished with a recoiling escapement, whose pallets were planes. The angle of escapement was $5\backslash frac\{1\}\{2\}$ degrees. When actuated with a weight of two pounds, it swung $8^\backslash circ$, and lost $15\text{'}\text{'}$ per hour; with four pounds, it swung $10^\backslash circ$, and lost $6\text{'}\text{'}$. Thus it appears that by doubling the maintaining power, although the vibration was increased in consequence of the greater impulse, the time was lessened $9\text{'}\text{'}$ per hour, viz. about $\backslash frac\{1\}\{240\}$. It is plain, from what was said when we described the first escapement, that an increase of maintaining power must render the vibration more frequent. We saw, on that occasion, that, even when the gravity of the pendulum is balanced by a weight on the other end of the rod, the force of the wheels will produce a vibratory motion, and that an augmentation of this force will increase it, or make the vibrations more rapid. The precise effect of any particular form of teeth can be learned only by computing the force on the pendulum in every position, and then constructing the curve $sBb$ of fig. 1. The rapid increase of the ordinates beyond those of the triangle ADC, forms a considerable area $DA\; =\; o$, to compensate the area $sC$, and thus makes a considerable contraction $A^o$ of the vibration, and a sensible contraction $\backslash frac\{A^o\}\{2\}$ of the time.

Mr George Graham, the celebrated watchmaker in London, was also a good mathematician, and well qualified to consider this subject scientifically. He contrived a escapement, which he hoped would leave the pendulum almost in its natural state. The acting face of the pallet $aBc$ (fig. 4.) is a plane. The tooth drops on $a$, and escapes from $c$, and is on the middle point $b$ when the pendulum is perpendicular. Beyond $a$, the face of the pallet is an arch $ad$, whose centre is $A$, the centre of the crutch. The maintaining power is made so great as to produce a much greater vibration than the angle of active escapement $aAe$. The consequence of this is that, when the tooth drops on the angle $ae$, the pendulum, continuing its motion, carries the crutch along with it, and the tooth passes on the arch $ad$, in a direction passing through the centre of the crutch. This pressure can neither accelerate nor retard the motion

tion of the crutch and pendulum. As the pendulum was accelerated after it passed the perpendicular, by the other pallet, it will (if quite unobstructed) throw out farther than what corresponds to the velocity which it had in the middle point of its vibration; perhaps till the tooth passes from $a$ to $c$ on the circular arch of the pallet. But although it sustains no contrary action from the wheels during this excursion beyond the angle of scapement, it will not proceed so far, but will stop when the tooth reaches $d$; because there must be some resistance arising from the friction of the tooth along the arch $ad$, and from the clamminess of the oil employed to lubricate it: but this resistance is exceedingly minute, not amounting to $\backslash frac\{1\}\{4\}$th of the pressure on the arch. Nay, we think that it appears from the experiments of Mr Coulomb that, in the case of such minute pressures on a surface covered with oil, there is no sensible retardation analogous to that produced by friction, and that what retardation we observe arises entirely from the clamminess of the oil. We are so imperfectly acquainted with the manner in which friction and viscosity obstruct the motions of bodies, that we cannot pronounce decisively what will be their effect in the present case. Friction does not increase much, if at all, by an increase of velocity, and appears like a fixed quantity when the pressure is given. This makes all motions which are obstructed by friction terminate abruptly. This will shorten both the length and the time of the outward excursion of the pendulum. The viscosity of the oil resists differently, and more nearly in the proportion of the velocities. The diminution of motion will not be in this proportion, because in the greater velocities it acts for a shorter time. Were this accurately the case, the resistance of viscosity would also be nearly constant, and it would operate as friction does. But it does not stop a motion abruptly, and the motions are extinguished gradually. Therefore, although viscosity must always diminish the extent of the excursion, it may so vary as not to diminish the time. We apprehend, however, that it generally does. But whatever happens in the excursion, the return will certainly be slower, and employ more time than if it had not been obstructed, because the velocity in every point is less than if perfectly free. The whole arch, consisting of a returning arch and an excursion on the other side, may be either slower or quicker, according as the compensation is complete or not, or is even overdone.

All these reflections occurred to Mr Graham; and he was persuaded that the time of the tooth's remaining on the arch $ad$, both ascending and descending, would differ very little from that of the description of the same arch by a free pendulum. The great causes of irregularity seemed to be removed, viz. the inequalities in the action of the wheels in the vicinity of the extremity of the vibration, where the pendulum having little momentum is, long in the same little space, exposed to their action. The derangement produced by any force depends on the time of its action, and therefore must be greatest when the motion is slowest. The pendulum gets its impulse in the very middle of its vibration, where its velocity is the greatest; and therefore the inequalities of the maintaining power act on it only for a short time, and make a very trifling alteration in the time of its describing the arch of scapement. Beyond this, it is nearly in the state of a free pendu-

lum; nay, even though it be affected by an inequality of the maintaining power, and it be accelerated beyond its usual rate in that arch, the chief effect of this will be to cause it to describe a larger arch of excursion. The shortening of the time of this description by the friction will be the same as before, happening at the very end of the excursion; but the return will be more retarded by the friction on a longer arch. And, by this, a compensation may be made for the trifling contraction of the time of describing the arch of scapement.

This circumstance of giving the impulse in the middle of the vibration, where its time of action is the smallest possible, and whereby the pendulum is so long left free from the action of the wheels, is of the very first importance in all scapements, and should ever be in the mind of the mechanician. When this is adhered to, the form of the face $abc$ is scarcely of any moment. Much has been written on this form, and many attempts have been made to make it such that the action of the wheels shall be proportional to the action of gravity. To do this is absolutely impossible. Mr Graham made them planes, not only because of easiest execution, but because a plane really conspires pretty well with the change of gravity. While the pendulum moves from $Q$ to $P$ (fig. 3.), the force of gravity, acting in the direction $QP$, is continually diminishing. So is the accelerating power of the pallet from $a$ to $b$. When the pendulum rises from $P$ to $R$, a force in the opposite direction $RP$ continually increases. This is analogous to the continual diminution of a force in the direction $PR$. Now we have such a diminution of such a force, in the action of the pallet from $b$ to $c$, and such an augmentation in the action of the other pallet.

For all these reasons, this construction of a scapement appeared very promising. Mr Graham put it in practice, and it answered his most sanguine expectation, and is now universally adopted in all nice clocks. Mr Graham, however, did not think it prudent to cause a tooth to drop on the very angle $a$ of the pallet. He made it drop on a point $f$ of the arch of excursion. This has also the advantage of diminishing the angle of action, which we have proved to be of service. It requires, indeed, a greater maintaining power; but this can easily be procured, and is less affected by the changes to which it is liable by the effect of heat and cold on the oil. Our observations on the effects of friction and viscosity in the arch $ad$ seem to be confirmed by the observations of several artists, who agree in saying that a great increase of maintaining power increases the vibrations, but makes them perceptibly slower. When they wrote, much oil was applied to diminish the friction on the arch of repose; but, since that time, the rubbing parts were made such as required no oil, and this retardation disappeared. In the clock of the transit room of the Royal Observatory, the angle of action seldom exceeds one-third of the swing of the pendulum. The pallets are of oriental ruby, and the wheel is of steel tempered to the utmost degree of hardness. This clock never varies a whole second from equable motion in the course of five days.

This contrivance is known by the name of the DEAD BEAT, the DEAD SCAPMENT; because the seconds index stands still after each drop, whereas the index of a clock with a recoiling scapement is always in motion, hobbling backward and forward.

These escapements, both recoiling and dead beat, have been made in a thousand forms; but any person tolerably acquainted with mechanics, will see that they are all on the same principles, and differ only in shape or some equally unimportant circumstance. Perhaps the most convenient of any is that represented in fig. 5, where the shaded part is the crutch, made of brass or iron, and A and B are two pieces of agate, flint, or other hard stone, cut into the proper shape for a pallet of either kind, and firmly fixed in proper sockets. They project half an inch, or thereabouts, in front of the crutch, so that the swing wheel is also before the crutch, distant about $\backslash frac\{1\}\{4\}$th of an inch or so. Pallets of ruby, driven by a hard steel swing wheel, need no oil, but merely to be once rubbed clean with an oily cloth.

Sometimes the wheel has pins instead of teeth. They are ranged round the rim of the wheel, perpendicular to its plane, and both pallets are on one side of the wheel, standing perpendicular to its plane. One of these pins drops from the first to the second pallet at once. The pallets are placed on two arms, as in fig. 6, in which case the pins are alternately on different sides of the wheel; or on one, as in fig. 7. By the motion of the pendulum to the right, the pin (in fig. 7), after resting on the concave arch da, acts on the face ae, and drops from e on the other concave arch ig, which continues to move a little way to the right. It then returns, and the pin slides and acts on the pallet ib, and escapes at b; and the next pin is then on the arch of repose da.

It being evident that the recoiling escapement accelerates the vibrations beyond the rate of a free pendulum, and it also appearing to many of the first artists that the dead escapement retards them, they have attempted to form a escapement which shall avoid both of these defects, by forming the arches ad, ig, so as to produce a very small recoil. Mr Berthoud does this in a very simple manner, by placing the centre of ad at a small distance from that of the crutch, so as to make the rise of the pallet above the concentric arch about one-third of the arch itself. Applying such a crutch to the light pendulum mentioned in a former paragraph, he found that doubling, and even trebling the maintaining power, produced no change in the time of vibration, though it increased the width from $8^\{\backslash circ\}$ to $12^\{\backslash circ\}$ and $14^\{\backslash circ\}$. We have no doubt of the efficacy of this contrivance, and think it very proper for all clocks which require much oil, such as turret clocks, &c. But we apprehend that no rule can be given for the angle that the recoiling arch should make with the concentric one. We imagine that this depends entirely on the share which friction and oil have in producing the retardation of the dead beat.

Other artists have endeavoured to avoid the inconveniences of friction and oil on the arch of repose in another way. Instead of allowing the tooth of the wheel to drop on the back of the pallet, which we called the arch of excursion, and others call the arch of repose, it drops on a detent ota (fig. 8.), of which the part ta is part of an arch whose centre is A, the centre of the crutch, and the part to is in the direction of the radius. This piece does not adhere to the pallet, but is on the end of an arm oA, which turns round the axis A of the crutch on fine pivots: it is made to apply itself to the back of the pallet by means of a slender spring Ap, attached to the pallet, and pressing inward on a pin p,

fixed in the arm of the detent. When so applied, its arch ta makes the repose, and its point a makes a small portion of the face ae of the pallet.

The action of this apparatus is very easily understood. When a tooth escapes from the second pallet, by the motion of the pendulum from the left to the right, another tooth drops on this pallet (which the figure shews to be the first or leading pallet) at the angle i, and rests on the small portion ta of an arch of repose. But the crutch, continuing its motion to the right, immediately quits the arm oA, carrying the pallet acr along with it, and leaving the wheel locked on the detent ota. By and bye the pendulum finishes its excursion to the right, and returns. When it enters the arch of action, the pallet has applied itself to the detent ota, and withdraws it from the tooth. The tooth immediately acts on the face ae of the pallet, and restores the motion lost during the last vibration. The use of the spring is merely to keep the detent applied to the pallet without shaking. It is a little bent during their separation, and adds something of an opposing force to the ascent of the pendulum on the other side of the wheel, and accelerates its return. A similar detent on the back of the second pallet performs a similar office, supporting the wheel while the pendulum is beyond the arch of escapement, and quitting it when the pendulum enters that arch.

We do not know who first practised this very ingenious and promising invention. Mr Hudge certainly did so early as 1753 or 1754. Mr Berthoud speaks obscurely of contrivances of the same nature. So does Le Roy, and (we think) Le Paute. We say that it is very promising. Friction is almost annihilated by transferring it to the pivots at A; so that, in the excursion beyond the angle of escapement, the pendulum seems almost free. Indeed some artists of our acquaintance have even avoided the friction of the pivots at A, by making the arm of the detent a spring of considerable thickness, except very near to A, where it is made very thin and broad. But we do not find that this construction, though easily executed, and susceptible of great precision and steadiness of action, is much practised. We presume that the performance has not answered expectations. It has not been superior to the incomparably more simple dead escapement of Graham. Indeed we think that it cannot. A part of the friction still remains, which cannot be removed; namely, while the arch ta is drawn from between the tooth and pallet. Nay, we apprehend that something more than friction must be overcome here. The tooth is apt to force the detent outward, unless the part ta be a little elevated at its point a like a claw, above the concentric arch, and the face of the tooth be made to incline forward, so as to fit this shape of the detent. This will consume some force, when the momentum of the pendulum is by no means at its maximum. Should the clock be foul, and the excursions beyond escapement be very small, this disturbance must be exceedingly pernicious. But we have a much greater objection. During the whole excursion beyond escapement, there is a new force of a spring acting on the pendulum, which deviates considerably from the proportions of the accelerating power of gravity. It does not commence its action till the detent separates from the arm of the crutch. Then the spring of the detent acts as a retarding force against

against the excursion of the pendulum, now on the other side, bringing it sooner to rest, and then accelerating it in its way back to the beginning of the arch of scapement. In short, this construction should have the properties of a recoiling scapement. We got a clock-maker to make some experiments on one which he had made for an amateur, which fully confirmed our conjecture. When the detent spring was strong, an increase of maintaining power made the vibrations both wider and more rapid. The artist reduced the strength of the spring till this effect was rendered very small. It might perhaps be quite removed by means of a still weaker spring: But the spring was already so weak that a hard step on the floor of the room did sometimes disengage the detent from the wheel. It appears, therefore, that nothing can be reasonably expected from this construction that is not as well performed by the dead scapement of Mr Graham, of much easier execution, and more certain performance.

Very similar to this construction (at least in the excursion beyond the angle of scapement) is the construction of Mr Cumming, and it has the same defects. His pallets are carried, as in the one described, by the crutch. The detents press on them behind by their weight only: therefore, when the tooth is locked on the detent of one pallet, its weight is taken off from the pendulum on that side, and the weight of the detent on the other side opposes the ascent, and accelerates the descent of the pendulum.

Mr Cumming executed another scapement, consisting, like those, of a pallet and detent. But the manner of applying the maintaining power is extremely different in principle from any yet described. It is exceedingly ingenious, and seems to do all that is possible for removing every source of irregularity in the maintaining power, and every obstruction to free motion arising from friction and oil in the scapement. For this reason we shall give such an account of its essential circumstances as may suffice to give a clear conception of its manner of acting, and its good properties and defects; but referring the inquisitive reader to Mr Cumming's Elements of Clock and Watch Work, published in 1766, for a more full account.

In the scapements last described, the pallets were fixed to the crutch and pendulum, and the maintaining power, during its action, was applied to the pendulum by means of the pallets, in the same way as in ordinary scapements. The detents were unconnected with the pendulum, and it was free during the whole excursion. In the present scapement both the pallets and detents are detached from the pendulum, except in the moment of unlocking the wheel; so that the pendulum may be said to be free during its whole vibration, except during this short moment.