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 SCAPMENT, 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 justness 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.

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 constancy, 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 transit of the stars, and renewing them by a little push with the finger when they became too small. Galileo, Riccioli, and others, in more recent times, followed this example. The celebrated physician Sanctarius 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 Albans', in 1325 (a). It appears to have been regulated by a fly like a kitchen jack*. Not long after this Giacomo Dondi made one at Padua, which had a motus faciens figuram, 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 measures. 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 measures of time. Mathematicians saw that their vibrations had some regularity.

(a) Professor Beekman, in the first volume of his History of Inventions, expresses a belief that clocks of this kind were used in some monasteries as early as the 14th century, and that they were derived to the monks from the Saracens. His authorities, however, are discordant, and seem not completely satisfactory even to himself. 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 so 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° 103. Cor. 7. Suppl.) and therefore, since pendulums swinging in small circular arches are fennyly 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 AFHd.

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(b) We are provoked to make this observation, by observing at this moment, in a literary journal, a pert and petulant upstart speaking of Newton's optical discoveries in terms of ridicule and abuse, employing these very discoveries to diminish his authority. Is it not thus that Christianity is now slighted by those who enjoy the fruits of the pure morality which it introduced? Through any points B, K, k, b, &c. of 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, &c., kL, &c., to the straight line DC.

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; 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, and another describe the arch KC, 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, the forces AD, BE, KL accelerate the pendulum, and the similar forces a, b, 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 wheel-work. Suppose it is an addition of pressure equal to a certain number of grams. We can make AD to DJ 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 of forces, 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 perpendicular to AC, making the space C equal to CAD, the point will be the limit of the oscillation outward from C, where the initial velocity HC is extinguished. If the line cut the same circle in s, 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 half circumference 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 be drawn only by the judgment 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 escapement among our artists, from the French word échappement. We proceed, therefore, to consider this subject more particularly, first considering the escapements 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 escapements for pendulums and balances very different.

I. Of the Action of a Wheel and Pallet.

The escapement 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 AFEB. 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 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 registering 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 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, Escap.). 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 affording us any thing that they can claim as their own; yet Lepaute says that the Echappement 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 misundertood, 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.

Suppl. Vol. II. Part II. 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 evolute of a circle, described with the radius AH, and the face of the tooth be the evolute 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, § 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, Casus, &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 evolute 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 b 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 evolute 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 b 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 IH 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 Cz, cutting AM in z, we shall have V : z = SM : A, as in Cor. 2. And if the perpendiculars Mi, Ab, be drawn on Cz, we have V : Z = Mi : Ab, 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 diminishes 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 sA, 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 clearance 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 clearance 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 scraped 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 b 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 clearance 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 clearance, 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 $180^\circ$ 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 $SM$ to $SA$, as the angle $a$ to the angle $b$; and then, having determined how many teeth shall be comprehended between the pallets, call this number $n$. Multiply the angle $b$ by $n + 1$, 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 $A$, draw $SB$ and $AC$, and through $B$ draw $BC$ 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 $OC$ perpendicular to $AC$, 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 finishes 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 flop 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 flop. 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 $b$ 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 $b$. Such pallets therefore will make a receding 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 fusee. 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 reliance 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 $54^\circ$ degrees. When actuated with a weight of two pounds, it swung $8^\circ$, and lost $15^\circ$ per hour; with four pounds, it swung $10^\circ$, and lost $6^\circ$. Thus it appears that by doubling the maintaining power, although the vibration was increased in consequence of the greater impulse, the time was lessened $9^\circ$ per hour, viz. about $\frac{1}{2}$ of $1^\circ$. 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 $C$ of fig. 1. The rapid increase of the ordinates beyond those of the triangle $ADC$, forms a considerable area $DA \cdot e$, to compensate the area $OC$, and thus makes a considerable contraction $A$ of the vibration, and a sensible contraction $\frac{A}{2}$ 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 $a$ & $c$ (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 $aA \cdot c$. The consequence of this is that, when the tooth drops on the angle $a$, 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 prelude can neither accelerate nor retard the mo- Watchwork.

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 \(e\) on the circular arch of the pallet. But although it sustains no contrary action from the wheels during this excursion beyond the angle of escapement, 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 chumminess of the oil employed to lubricate it; but this resistance is exceedingly minute, not amounting to \(\frac{1}{4}\)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 chumminess 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 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 escapement. 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 escapement.

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 escapements, and should ever be in the mind of the mechanician. When this is adhered to, the form of the face \(a\) \(b\) \(c\) 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 compiles 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 escapement 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 escapement; because the seconds index flanks still after each drop, whereas the index of a clock with a recoiling escapement is always in motion, hobbling backward and forward.

These 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 1/3rd 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 d, sets on the face e, and drops from e on the other concave arch g, which continues to move a little way to the right. It then returns, and the pin slides and acts on the pallet h, and escapes at b; and the next pin is then on the arch of repose d.

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 an escapement which shall avoid both of these defects, by forming the arches d, g, so as to produce a very small recoil. Mr Berthoud does this in a very simple manner, by placing the centre of d 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° to 12° and 14°. 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 o t a (fig. 8.), of which the part t a is part of an arch whose centre is A, the centre of the crutch, and the part t o is in the direction of the radius. This piece does not adhere to the pallet, but is on the end of an arm o A, 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 A P attached to the pallet, and pressing inward on a pin p, fixed in the arm of the detent. When so applied, its Watch-arch t a makes the repose, and its point o makes a small portion of the face a c 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 shows to be the first or leading pallet) at the angle r, and rests on the small portion t a of an arch of repose. But the crutch, continuing its motion to the right, immediately quits the arm o A, carrying the pallet a c r along with it, and leaving the wheel locked on the detent o t a. 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 o t a, and withdraws it from the tooth. The tooth immediately acts on the face a c 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 flanking. 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 Mudge 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 t a 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 t a be a little elevated at its point o 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 escapement. In short, this construction should have the properties of a recoiling escapement. We got a clockmaker 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 artificial 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 escapement 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 escapement) 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 escapement, 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 escapement. 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 escapements 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 escapements. The detents were unconnected with the pendulum, and it was free during the whole excursion. In the present escapement 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.

ABC (fig. 9.) represents a portion of the swing wheel, of which O is the centre, and A one of the teeth; Z is the centre of the crutch, pallets, and pendulum. The crutch or detents is represented of a form resembling the letter A, having in the circular cross piece a slit i k, also circular, Z being the centre. This form is very different from Mr Cumming's, and inferior to his, but was adopted here in order to avoid a long description. The arm ZF forms the first detent, and the tooth A is represented as locked on it at F. D is the first pallet on the end of the arm Z, movable round the same centre with the detents, but moveable independently of them. The arm d e, to which the pallet D is attached, lies altogether behind the arm ZF of the detent, being fixed to a round piece of brass e f g, which has pivots turning concentric with the verge or axis of the pendulum. To the same round piece of brass is fixed the horizontal arm e H, carrying at its extremity the ball H, of such size, that the action of the tooth A on the pallet D is just able (but without any risk of failing) to raise it up to the position here drawn. ZP p represents the fork, or the pendulum rod, behind both detent and pallet. A pin p projects forward, coming through the slit i k, without touching the upper or under margin of it. There is also attached to the fork the arm m n (and a similar one on the other side), of such length that, when the pendulum rod is perpendicular, as is represented here, the angular distance of n q from the rod e g H is precisely equal to the angular distance of the left side of the pin p from the left end i of the slit i k.

The mode of action on this apparatus is abundantly simple. The natural position of the pallet D is at i, represented by the dotted lines, resting on the back of the detent F. It is naturally brought into this position by its own weight, and still more by the weight of the ball H. The pallet D, being set on the fore side of the arm at Z, comes into the same plane with the detent F and the swing-wheel. It is drawn, however, in the figure in another position. The tooth C of the wheel is supposed to have escaped from the second pallet, on which the tooth A immediately engages with the pallet D, situated at i, forces it out, and then rests on the detent F, the pallet D leaning on the tip of the tooth. F is brought into this situation in a way that will appear presently. After the escape of C, the pendulum, moving down the arch of semivibration, is represented as having attained the vertical position. Proceeding still to the left, the pin p reaches the extremity i of the slit i k; and, at the same instant, the arm n touches the rod e H in q. The pendulum proceeding a hair's breadth further, withdraws the detent F from the tooth, which now even pushes off the detent, by acting on the flint face of it. The wheel being now unlocked, the tooth following C on the other side acts on its pallet, pushes it off, and rests on its detent, which has been rapidly brought into a proper position by the action of A on the flint face of F. It was a similar action of C on its detent, in the moment of escape, which brought F into a fit position for locking the wheel by the tooth A. The pendulum still going on, the arm m n carries the weight of the ball H, and the pallet connected with it, and it comes to rest before the pin p again reaches the end of the slit, which had been suddenly withdrawn from it by the action of A on the flint face of F. The pendulum now returns towards the right, loaded on the left with the ball H, which restores the motion which it had lost during the last vibration. When, by its motion to the right, the pin p reaches the end k of the slit i k, it unlocks the wheel on the right side. At the same instant the weight H ceases to act on the pendulum, being now raised up from it by the action of a tooth like B on the pallet D.

Let us now consider the mechanism of these motions. The prominent feature of the contrivance is the almost complete disengagement of the regulator from the wheels. The wheels, indeed, act on the pallets; but the pallets are then detached from the pendulum. The sole use of the wheel is to raise the little weights while the pendulum is on the other side, in order to have them in readiness at the arrival of the pendulum. They are then laid on the pendulum, and supply an accelerating force, which returns to the pendulum the momentum lost during the preceding vibration. Therefore no inequalities in the action of the wheel on the pallets, whether arising from friction or oil, has any effect on the maintaining power. It remains always the same, namely, the rotative momentum of the two weights. The only circumstance, in which the irregularity of the action of the wheels can affect the pendulum is at the moment of unlocking. Here indeed the regulator may be affected; but this moment is so short, in comparison with other escapements, that it must be considered as a real improvement.

It is very unkind to refuse the author a claim to the character of an ingenious artist on account of this contrivance, as has been done by a very ingenious university Professor, who taxes Mr. Cumming with ignorance of the first elements of mechanics, and says that the best thing in his book is his advice to suspend the pendulum from a great block of marble, firmly fixed in the wall*. This is certainly a good advice, and we doubt not but that the Professor's clock would have performed still better if he had condescended to follow it. It is still less candid to question the originality of the invention. We know for certain that it was invented at a time and place where the author could not know what had been done by others. It would have been more like the humanity of a well-educated man to have acknowledged the genius, which, without similar advantages, had done so much.

But, while we thus pay the tribute of justice to Mr. Cumming, we do not adopt all his opinions. The clock has the same defects of the former in respect of the laws of the force which accelerates the pendulum. The sudden addition of the small weight, and this almost at the extremity of the vibration, would derange it very much, if the addition were susceptible of any sensible variation. The irregularity of the action of the wheels may sensibly affect the motion during the unlocking, when the clock is foul, and the pendulum fails to unlock; for any disturbance at the extremity of the vibration greatly affects the time. We acknowledge that the parts which we here suppose to be foul may not be so in the course of twenty years; these parts being only the pivots of the escapement. The great defect of the escapement is its liability to unlock by any jolt. It is more subject to this than the others already mentioned. This risk is much increased by the slender make of the parts, in Mr. Cumming's drawings, and in the only clock of the kind we have seen; but this is not necessary; and it should be avoided for another reason; the interposing too many slender and crooked parts between the moving power and the pendulum weakens the communication of power, and requires a much more powerful wheelwork.

All these, however, are slight defects, and only the last can be called a fault. The clocks made on this principle have gone remarkably well, as may be seen by the registers of his majesty's private observatory. But the greatest objection is, that they do not perform better than a well-made dead escapement; and they are vastly more troublesome to make and to manage. This is strictly true, and is a serious objection. The fact is, that the division of a heavy pendulum is so great, that if any one of the escapements now described be well executed with pallets of agate, and a wheel of hard steel, and if the pendulum be suspended agreeably to Mr. Cumming's advice, there is hardly any difference to be observed in their performance. We shall content ourselves with a single proof of this from fact. The clock invented by the celebrated Harrison is at least equal in its performance to any other. Friction is almost annihilated, and no oil is required. It went fourteen years without being touched, and during that time did not vary one complete second from one day to another; nor ever deviated half a minute by accumulation from equable motion: Yet the escapement, in so far as it reflects the law of the accelerating force, deviates more from the proportion of the spaces than the most rounding escapement that ever was put to a good clock. It is so different from all hitherto described, both in form and principle, that we must not omit some account of it, and with it we shall conclude our escapements for clocks.

Let GDO represent the swing wheel, of which M is the centre. A is the verge or axis of the pendulum. It has two very short arms AB, AE. A flender rod BC turns on fine pivots in the joint B, and has at its extremity C a hook or claw, which takes hold of a tooth D of the swing-wheel when the pendulum moves from the right side to the left. This claw, when at liberty, stands at right angles, or, at least, in a certain determinate angle, with regard to the arm AB; and when drawn a little from that position, it is brought back to it again by a very flender spring. The arm AE is furnished with a detent EF, which also, when at liberty, maintains its position on the arm by means of a very flender spring.

Let us now suppose that the tooth D is pressing on the claw C, while the pendulum is moving to the right. The joint B yields, by its motion round A, to the pressure of the tooth on the claw. By this yielding, the angle ABC opens a little. In the mean time, the same motion round A causes the point F of the detent on the other side to approach the circumference of the wheel in the arch of a circle, and the tooth G at the same time advances. They meet, and the point of G is lodged in the notch under the projecting heel f. When this takes place, it is evident that any farther motion of the point E round A must pull the tooth G a little backward, by means of the detent EF. It cannot come any nearer to the wheel, because the point of the tooth stops the heel f. The instant that F pushes G back, the tooth D is withdrawn from the claw C, and C flies out, by the action of its spring, and resumes its position at right angles to BA; and the wheel is now free from the claw, but is pushing at the detent F (c). The pendulum, having

(c) The reader may here remark the manner in which the pressure of the tooth G on the detent is transferred to the joint E by the intervention of the shank FE, and from the joint E to the pendulum rod, by the intervention... vies finished its excursion to the right (in which it causes the wheel to recoil by means of the detent F), returns toward the left. The wheel now advances again, and, by pressing on F, aids the pendulum through the whole angle of escapement. By this motion the claw C describes an arch of a circle round A, and approaches the wheel, till it take hold of another tooth, namely, the one following D, and pulls it back a little. This immediately frees the detent F from the pressure of the tooth G, and it flies out a little from the wheel, refusing its natural position by means of its spring. Soon after, the motion of the pendulum to the left ceases, and the pendulum returns; D pulling forward the hook C to aid the pendulum, and the former operation is repeated, &c. &c.

Such is the operation of the pallets of Harrison and Hindley. Friction is almost totally avoided, and oil entirely (a). The motion is given to the pendulum by a fair pull or push, and the teeth of the wheel only apply themselves to the detents without rubbing. There is no drop, and the escapement makes no noise, and is what the artists call a silent escapement. The mechanic will readily perceive, that by properly disposing the arms AB, AE, and disposing the pallets on the circumference of the wheel, the law, by which the action of the wheel on the pendulum is regulated, may be greatly varied, so as to harmonize, as far as the nature of escapement, alternately pushing and pulling, will admit, with the action of gravity.

But this is evidently a receding escapement, and one of the worst kind; for the recoil is made at the very confines of the vibration, where every disturbance of the regular cycloidal vibration occasions the greatest disturbance to the motion. Yet this clock kept time with most unexampled precision, far exceeding all that had been made before, and equal to any that have been made since. This is entirely owing to the immense superiority of the momentum of the pendulum over the maintaining power.

II. Of Escapements for a Watch.

The execution of a proper escapement for watches is a far more delicate and difficult problem than the foregoing, on account of the small size, which requires much more accurate workmanship, because the error of the hundredth part of an inch has as great a proportion to the dimensions of the regulator as an inch in a common house clock. It is much more difficult on another account. We have no such means of accumulating such a dominion (to use Mr Harrison's expressive term) over the wheel-work in the regulator of a watch as in that of a clock. The heaviest balance that we can employ, without the certainty of snapping its pivots by every flight jolt, is a mere trifle, in comparison with the pendulum of the most ordinary clock. A dozen or twenty grains is the utmost weight of the balance, even of a very large pocket watch. The only way that we can accumulate any notable quantity of regulating power in such a small pittance of matter is by giving it a very great velocity. This we do by accumulating all its weight in the rim, by giving it very wide vibrations, and by making them extremely frequent. The balance-rim of a middling good watch should pass through at least ten inches in every second. Now, when we reflect on the small momentum of this regulator, the inevitable inequalities of the maintaining power, and the great arch of vibration on which these inequalities will operate, and the comparative magnitude even of an almost invisible friction or clammings, it appears almost chimerical to expect anything near to equilibria in the vibrations, and incredible that a watch can be made which will not vary more than one beat in 86400. Yet such have been made. They must be considered as the most matterly exertions of human art. The performance of a reflecting telescope is a great wonder; the world that can find a market must have its mirrors executed without an error of the ten thousandth part of an inch; but we now know that this accuracy is attained almost in spite of us, and that we scarcely can make them of a worse figure. But the case is far otherwise in watch work. Here all those wonderful approaches to perfection are the results of rational diffusion, by means of sound principles of science; and, unless the artist who puts these principles into practice be more than a mere copyist, unless the principles themselves are perceived by him, and actually direct his hand, the watch may still be good for nothing. Surely, then, this is a liberal art, and far above a manual knack. The study of the means by which such wonders are steadily effected, is therefore the study of a gentleman.

In the account given above of the escapements for pendulums, we assumed as one leading principle that the natural vibrations of a pendulum are performed in equal times, whether wide or narrow. This is so nearly true, when the arches on each side of the perpendicular do not exceed four degrees, that the retardation of the wider arches within that limit will not become sensible, though accumulated for a long time. The common escapement with a plane face of the pallet, helps to correct even this small inequality much better than the nicest form of the cycloidal cheeks propounded by Huyghens.

In watch-work we assume a similar principle, namely, that the oscillations of a balance, urged by its spring, and undisturbed... undisturbed by all foreign forces, are performed in equal times, whether they be wide or narrow. This principle was affirmed by the celebrated mechanician Dr Robert Hooke, on the authority of many experiments which he had made on the bending and unbending of springs. He found that the force necessary for retaining a spring in any constrained position was proportional to its tension, or deflection from its natural form. He expressed this in an anagram, which he published about the year 1660, in order to establish his claim to the discovery, and yet conceal it, till he had made some important application of it. When the anagram was explained some years afterwards, it was, "Uttermost, sic vis." Dr Hooke thought of applying this discovery to the regulation of watch movements. For, if a slender spring be properly applied to the axis of a watch balance, it will put that balance in a certain determinate position. If the balance be turned aside from this position, it seems to follow that it will be urged back toward it by a force proportional to its distance from it. He immediately made the application to an old watch, which he afterward gave to Dr Wilkins, Bishop of Chester. This was in 1658. Its motion was so amazingly improved, that Hooke was persuaded of the perfection of his principle, and thought that nothing was now wanting for making a watch of this kind a perfect chronometer but the hand of a good workman. For his watch seemed almost perfect, though made in a small country town, in a very coarse manner. Mr Huyghens also claims this discovery. He published his claim about the year 1675, and proposed to make watches for discovering the longitude of a ship at sea. But there is the most unquestionable evidence of Dr Hooke's priority by fifteen years, and of his having made several watches of this kind. One of them was in the possession of his majesty king Charles II. Dr Hooke's first balance spring was straight, and acted on the balance in a very imperfect manner. But he soon saw the imperfections, and made several successive alterations; and, among others, he employed the cylindrical spiral now employed by Mr Arnold; but he gave it up for the flat spiral; and the king's watch had one of this kind before Mr Huyghens published his invention. His project of longitude watches had been carried on along with Lord Brouncker and Sir Robert Moray, and they had quarrelled some years before that publication. See Watch, Encycl.

But both Dr Hooke and Mr Huyghens were too fainthearted in their expectations. We, by no means, have the evidence for the truth of this principle that we have for the accelerating action of gravity on a pendulum. It rests on the nicety and the propriety of the experiments; and long experience has shown that it is feebly true only within certain limits. The demonstrations by which Bernoulli supports the unqualified principle of Mr Huyghens, proceed on hypothetical doctrines concerning the nature of elasticity. And even these show that the law of elasticity which he assumed was selected, not because founded on simpler principles than any other, but because it was consistent with the experiments of Hooke and Huyghens. Besides, although this should be the true law of a spring, it does not follow that this spring, applied in any way to the axis of a balance, will urge that balance agreeably to the same law; and if it did, it still does not follow that the oscillations of the balance will be isochronous; for the force has to move not only the balance but also the spring. Part of the restoring force of the spring is employed in restoring it rapidly to its quiescent shape, and thus enabling it to follow and fill up the yielding balance. It is therefore only the surplus which is employed in actually moving the balance, and it is uncertain whether this surplus varies according to the same law, being always the same proportion of the whole force of the spring. We find it an extremely difficult problem to determine the law of variation of this surplus, even in the simplest form of the spring; nay, it is by no means an easy problem to determine the law of oscillation of a spring, unloaded with any balance; and we can easily show that there are such forms of a spring, that although the velocity with which the different parts approach to their quiescent position be exactly as their excursion from it, this is by no means the law of velocity which this spring will produce in a balance. The matter of fact is, that when the spring is a simple straight steel wire, suspending the balance in the direction of its axis, the motions of it, if not immoderate, are precisely agreeable to Huyghens's and Hooke's rule; and that the motion of a balance urged by a spring wound up into a flat, or a cylindrical spiral, as in common watches, and those of Arnold, deviates sensibly from it, unless a certain analogy be preserved between the length and the elasticity of the spring. If the spring be immoderately long, the wide vibrations are slower than the narrow ones; and the contrary is observed when the spring is immoderately short. A certain taper, or gradual diminution of the spring, is also found to have an effect in equalizing the wide and narrow vibrations. There is also a great difference between the force with which a part of the spring unbends itself, and the action of that force in urging the balance round its axis; and the performance of many watches, good in other respects, is often faulty from the manner in which this unbending force is employed.

But, since these corrections are in our power in a considerable degree, we may suppose them applied, and the true motion (which we shall call the cycloidal) attained; and we may then adapt the construction of the escapement to the preserving this motion undisturbed. And here we must see at once that the problem is incomparably more delicate than in the case of pendulums. The vibrations must be very wide, and the angular motion rapid, that it may be little affected by external motions. The smallest inequalities of maintaining power acting through so great a space, must bear a considerable proportion to the very minute momentum of a watch balance. Oil is as clammy on the pallets of a watch as on those of a clock; a viscosity which would never be felt by a pendulum of 20 pounds weight will stop a balance of 20 grams altogether. For the same reason, it is evident that any impropriety in the form of the pallet must be incomparably more pernicious than in the case of a pendulum; the deviation which this may occasion from a force proportional to the angular distance from the middle point, must bear a great proportion to the whole force.

The common recoiling escapement of the old clocks still holds its place in the ordinary pocket watches, and answers all the common purposes of a watch very well. A well finished watch, with a recoiling escapement, will will keep time within a minute in the day. This is enough for the ordinary affairs of life. But such watches are subject to great variation in their rate of going, by any change in the power of the wheels. This is evident; for if the watch be held back, or pressed forward, by the key applied to the fusee square, we hear the beating greatly retarded or accelerated. The maintaining power, in the best of such watches, is never less than one-fifth of the regulating power of the spring. For, if we take off the balance spring, and allow the balance to vibrate by the impulse of the wheels alone, we shall find the minute hand to go forward from 25 to 30 minutes per hour. Suppose it 30. Then, since the wheels act through equal spaces with or without a spring, the forces are as the squares of the acquired velocities. (Dynamics, Suppl. p. 95.) The velocity in this case is double; therefore the accelerating force is quadruple, and the force of the spring is three times that of the wheels. If the hand goes forward 25 minutes, the force of the wheels is about one-fifth of that of the spring. This great proportion is necessary, as already observed, that the watch may go as soon as unflipped.

We have but little to say on this escapement; its principle and manner of action, and its good and bad qualities, being the same with those of the similar escapement for pendulums. It is evident that the maintaining power being applied in the most direct manner, and during the whole of the vibration, it will have the greatest possible influence to move the balance. A given mainspring and train will keep in motion a heavier balance by means of this escapement than by any other. But, on the other hand, and for the same reason, the balance has lost dominion over the wheel-work, and its vibrations are more affected by any irregularities of the wheel-work. Moreover, the chief action of the wheel being at the very extremities of the vibrations, and being very abrupt, the variations in its force are most hurtful to the isochronism of the vibrations.

Although this escapement is extremely simple, it is susceptible of more degrees of goodness or imperfection than almost any other, by the variation of the few particulars of its construction. We shall therefore briefly describe that construction which long experience has sanctioned as approaching near to the best performance that can be obtained from the common escapement. Fig. 11 represents it in what are thought its best proportions, as it appears when looking straight down on the end of the balance arbor. C is the centre of the balance and verge. CA and CB are the two pallets; CA being the upper pallet, or the one next to the balance, and CB being the lower one. F and D are two teeth of the crown wheel, moving from left to right; and E, G, are two teeth on the lower part of the circumference, moving from right to left. The tooth D is represented as just escaped from the point of CA, and the lower tooth E as just come into contact with the lower pallet. The escapement should not, however, be quite so close, because an inequality on the teeth might prevent D from escaping at all. For if E touch the pallet CB before D has quitted CA, all will stand still. This fault will be corrected by withdrawing the wheel a little from the verge, or by shortening the pallets.

The proportions are as follow: The distance between the front of the teeth (that is, of G, F, E, D) and the axis C of the balance is one-fifth of FA, the distance between the points of the teeth. The length CA, CB of the pallets is three-fifths of the same distance. The pallets make an angle ACB of 95 degrees, and the front DH or FK of the teeth make an angle of 25° with the axis of the crown-wheel. The sloping side of the tooth must be of an epicycloidal form, suited to the relative motion of the tooth and pallet.

From these proportions it appears that the pallet A can throw out, by the action of the tooth D, till it reaches a 120 degrees from CL, the line of the crown-wheel axis. For it can throw out till the pallet B strike against the front of E, which is inclined 25° to CL. To this add BCA = 95°, and we have LC a = 120°. In like manner B will throw out as far on the other side. From 25°, the sum of these angles, take the angle of the pallets 95°, and there remains 145° for the greatest vibration which the balance can make without striking the front of the teeth. This extent of vibration supposes the teeth to terminate in points, and the acting surfaces of the pallets to be planes directed to the very axis of the verge. But the points of the teeth must be rounded off a little for strength, and to diminish friction on the face of the pallets. This diminishes the angle of escapement very considerably, by shortening the teeth. Moreover, we must by no means allow the point of the pallet to bank or strike on the foreside of a tooth. This would greatly derange the vibration by the violence and abruptness of the check which the wheel would give to the pallet. This circumstance makes it improper to continue the vibrations much beyond the angle of escapement. One-third of a circle, or 120°, is therefore reckoned a very proper vibration for a escapement made in these proportions. The impulse of the wheels, or the angle of escapement, may be increased by making the face of the pallets a little concave (preserving the same angle at the centre). The vibration may also be widened by pushing the wheel nearer to the verge. This would also diminish the recoil. Indeed this may be entirely removed by bringing the front of the wheel up to C, and making the face of the pallet not a radius, but parallel to a radius and behind it, i.e., by placing the pallet CA so that its acting face may be where its back is just now. In this case, the tooth D would drop on it at the centre, and lie there at rest, while the balance completes its vibration. But this would make the banking (as it is called) on the teeth almost unavoidable. In short, after varying every circumstance in every possible manner, the best makers have settled on a escapement very nearly such as we have described. Precise rules can scarcely be given; because the law by which the force acting on the pallets varies in its intensity, deviates so widely from the action of the balance spring, especially near the limits of the excursion.

The discoveries of Huyghens and Newton in rational mechanics engaged all the mathematical philosophers of Europe in the solution of mechanical problems, about the end of the last century. The vibrations of elastic plates or wires, and their influence on watch balances, became familiar to every body. The great requisites for producing isochronous vibrations were well understood, and the artists were prompted by the speculators to attempt constructions of escapements proper for this purpose. It appeared clearly, that the most effectual means for this purpose was to leave the balance unconnected with the wheels, especially near the extremities of the vibration, where the motion is languid, and where every inequality of maintaining power must act for a longer time, and therefore have a great effect on the whole duration of the vibrations. The maxim of construction that naturally arises from these reflections is to confine, if possible, the action of the wheels to the middle of the vibration, where the motion is rapid, and where the chief effect of an increase or diminution of the maintaining power will be to enlarge or contract the angular motions, but will make little change on their duration; because the greatest part of the motion will be effected by the balance spring alone. This maxim was inculcated in express terms by John Bernoulli, in his *Recherches Mécaniques et Physiques*; but it had been fugged by common sense to several unlettered artists before that time. About the beginning of this century watches were made in London, where the verge had a portion $e d b$ (fig. 12.) of a small cylinder, having its centre $c$ in the axis, and a radial pallet $b a$ proceeding from it. Suppose a tooth just escaped from the point of the pallet, moving in the direction $b d e$, the cylindrical part was so situated that the next tooth dropped on it at a small distance from its termination. While the verge continues turning in the direction $b d e$, the tooth continues resting on the cylinder, and the balance sustains no action from the wheels, and has only to overcome the minute frictions on the polished surface of a hard steel cylinder. This motion may perhaps continue till the pallet acquires the position $f$, almost touching the tooth. It then stops, its motion being extinguished by the increasing force of the spring. It now returns, moving in the direction $e d b$; and when the pallet has acquired the position $e i$, the tooth $g$ quits the circumference of the cylinder, and drops on the pallet at the very centre. The crooked form of the tooth allows the pallet to proceed still farther, before there is any danger of hanging on the tooth. This vibration being also ended, the balance resumes its first direction, and the tooth now acts on the face of the pallet, and restores to the balance all the motion which it had lost by friction, &c., during the two preceding vibrations.

It is evident that this construction obviates all the objections to the former recoiling escapement, and that, by sufficiently diminishing the diameter of the cylindrical part, the friction may be reduced to a very small quantity, and the balance be made to move by the action of the spring during the whole of the excursion, and of the returning vibration. Yet this construction does not seem to have come much into use, owing, in all probability, to the great difficulty of making the drop to accurate in all the teeth. The smallest inequality in the length of a tooth would occasion it to drop sooner or later; and if the cylinder was made very small, to diminish friction, the formation of the notch was almost a microscopical operation, and the smallest flake in the axis of the verge or the balance-wheel would make the tooth slip past the cylinder, and the watch run down again.

About the same time, a French artist in London (then the school of this art) formed another escapement, with the same views. We have not any distinct account of it, but are only informed (in the 7th volume of the *Mémoires approvées par l'Acad. des Sciences*) that the tooth rested on the surface of a hollow cylinder, and then escaped by acting on the inclined edge of it. But we may presume that it had merit, being there told that Sir Isaac Newton wore a watch of this kind.

A much superior escapement, on the same principle, was invented by Mr Geo. Graham, at the same time that he changed the recoiling escapement for pendulums into the dead beat. Indeed it is the same escapement, accommodated to the large vibrations of a balance. In fig. 13. DE represents part of the rim of the balance-wheel. A and C are two of its teeth, having their faces $b e$ formed into planes, inclined to the circumference of the wheel, in an angle of about 15 degrees; so that the length $b e$ of the face is nearly quadruple of its height $e m$. Suppose a circular arch ABC described round the centre of the wheel, and through the middle of the faces of the teeth. The axis of the balance passes thro' some point B of this arch, and we may say that the mean circumference of the teeth passes through the centre of the verge. On this axis is fixed a portion of a thin hollow cylinder $b c d$, made of hard tempered steel, or of some hard and tough stone, such as ruby or sapphire. Agates, though very hard, are brittle. Chalcedony and cornelian are tough, but inferior in hardness. This cylinder is so placed on the verge, that when the balance is in its quietest position, the two edges $b$ and $d$ are in the circumference which passes through the points of the teeth. By this construction the portion of the cylinder will occupy 210° of the circumference, or 30° more than a semicircle. The edge $b$, to which the tooth approaches from without, is rounded off on both angles. The other edge $d$ is formed into a plane, inclined to the radius about 30°.

Now, suppose the wheel pressed forward in the direction AC. The point $b$ of the tooth, touching the rounded edge, will push it outwards, turning the balance round in the direction $b c d$. The heel $e$ of the tooth will escape from this edge when it is in the position $b$, and $e$ is in the position $f$. The point $b$ of the tooth is now at $d$, but the edge of the cylinder has now got to $i$. The tooth, therefore, rests on the inside of the cylinder, while the balance continues its vibration a little way, in consequence of the shove which it has received from the action of the inclined plane pushing it out of the way, as the mould board of a plough throws a stone aside. When this vibration is ended, by the opposition of the balance-spring, the balance returns, the tooth (now in the position $b$) rubbing all the while on the inside of the cylinder. The balance comes back into its natural position $b c d$, with an accelerated motion, by the action of its spring, and would, of itself, vibrate as far, at least, on the other side. But it is aided again by the tooth, which, pressing on the edge $d$, pushes it aside, till it come into the position $k$, when the tooth escapes from the cylinder altogether. At this moment the other edge of the cylinder is in the position $l$, and therefore is in the way of the next tooth, now in the position $A$. The balance continues its vibration, the tooth all the while resting, and rubbing on the outside of the cylinder. When this vibration, in the direction $d c e$, is finished, the balance resumes its first motion $b c d$, by the action of the spring, and the tooth begins to act on the first edge $b$, as soon as the balance gets into its natural position, throws it aside, escapes from it, and drops on the inside of the cylinder. In this manner are the vibrations produced, gradually increased to their maximum, and maintained in that state. Every succeeding tooth of the wheel acts first on the edge \( b \), and then on the edge \( d \); resting first on the outside, and then on the inside of the cylinder. The balance is under the influence of the wheels while the edge \( b \) passes to \( b \), and while \( d \) passes to \( d \); and the rest of the vibration is performed without any action on the part of the wheels, but is a little obstructed by friction, and by the clamminess of the oil. In the construction now described, the arch of action or escapement is evidently \( 32^\circ \), being twice the angle which the face of a tooth makes with the circumference.

The reader will perceive, that when this escapement is executed in such a manner that the succeeding tooth is in contact with the cylinder at the instant that the preceding one escapes from it, the face of the tooth must be equal to the inside diameter of the cylinder, and that the distance between the heel of one tooth and the point of the following one must be equal to the outside diameter. When the escapement is so close there is no drop. A good artist approaches as near to this adjustment as possible; because, while a tooth is dropping, but not yet in contact, it is not acting on the balance, and some force is lost. The execution is accounted very good, if the distance between the centres of two teeth is twice the external diameter of the cylinder. This allows a drop equal to the thickness of the cylinder, which is about \( \frac{1}{4} \) th of its diameter.

We must also explain how this cylinder is connected with the verge as to make such a great revolution round the tooth of the wheel. The triangular tooth \( s \) is placed on the top of a little pillar or pin fixed into the extremity of the piece of brass \( M \) formed on the rim of the wheel. Thus the wedge-tooth has its plane parallel to the plane of the wheel, but at a small distance above it. Fig. B represents the verge, a long hollow cylinder of hard steel. A great portion of the metal is cut out. If it were spread out flat, it would have the shape of fig. C. Suppose this rolled up till the edges \( G H \) and \( G'H' \) are joined, and we have the exact form. The part acted on by the point of the tooth is the dotted line \( b \). The part \( DIF'E' \) serves to connect the two ends. Thus it appears to be a very slender and delicate piece; but being of tempered steel, it is strong enough to resist moderate jolts. The ruby cylinders are much more delicate.

Such is the cylinder escapement of Mr Graham, called also the horizontal escapement, because the balance wheel is parallel to the others. Let us see how far it may be expected to answer the intended purposes. If the excursions of the balance beyond the angle of impulsion were made altogether unconnected with the wheels, the whole vibration would be quicker than one of the same extent, made by the action of the balance-spring alone, because the middle part of it is accelerated by the wheels. But the excursions are obstructed by friction and the clamminess of oil. The effect of this in obliterating the motion is very considerable. Mr Le Roy placed the balance so, that it rested when the point of the tooth was on the middle of the cylindrical surface. When the wheel was allowed to press on it, and it was drawn \( 8^\circ \) from this position, it vibrated only during \( 4\frac{1}{2} \) seconds. When the wheel was not allowed to touch the cylinder, it vibrated \( 90 \) seconds, or \( 20 \) times as long; so much did the friction on the cylinder exceed that of the pivots. We are not sufficiently acquainted with the laws of either of these obstructions to pronounce decidedly whether they will increase or diminish the time of the whole vibrations. We observe distinctly, in motions with considerable friction, that it does not increase nearly so fast as the velocity of the motion; nay, it is often less when the velocity is very great. In all cases it is observed to terminate motions abruptly. The friction requires a certain force to overcome it, and if the body has any less it will stop. Now this will not only contract the excursion of the balance, but will shorten the time. But the return to the angle of impulsion will undoubtedly be of longer duration than the excursion; for the arch of return, from the extremity of the excursion to its beginning, where the angle of impulsion ends, is the same with the arch of excursion. The velocity which the balance has in any point of the return is less than what it had in the same point of the excursion; because, in the excursion, it had velocity enough to carry it to the extremity, and also to overcome the friction. In the return, it could, even without friction, only have the velocity which would have carried it to the extremity; and this smaller velocity is diminished by friction during the return. The velocity being less through the whole return than during the excursion, the time must be greater. It may therefore happen that this retardation of the return may compensate the contraction of the excursion and the diminution of its duration. In this case the vibration will occupy the same time as if the balance had been free from the wheels. But it may more than compensate, and the vibrations will then be slower; or it may not fully compensate, and they will be quicker. We cannot therefore say, a priori, which of the two will happen; but we may venture to say that an increase of the force of the wheels will make the watch go slower; for this will exert a greater prelude, give a greater impulsion, produce a wider excursion, and increase the friction during that greater excursion, making the wide vibrations slower than the narrow ones; because the angle of impulsion remaining the same, the pressures exerted must be quadrupled, in order to double the excursion (see Dynamics, p. 95, Supp.), and therefore the friction will be increased in a greater proportion than the momentum which is to overcome it. But, with respect to the obstruction arising from the viscosity of the oil, we know that it follows a very different law. It bears a manifest relation to the velocity, and is nearly proportional to it. But still it is difficult to say how this will affect the whole vibration. The duration of the excursion will not be so much contracted as by an equal obstruction from friction, because it will not terminate the motion abruptly. There are therefore more chances of the increased duration of the return exceeding the diminution of it in the excursion. All that we can say, therefore, is, that there will be a compensation in both cases. The time of excursion will be contracted, and that of return augmented.

Now, as the friction may be greatly diminished by fine polish, fine oil, and a small diameter of the cylinder, we may reasonably expect that the vibrations of such a balance will not vary nearly so much from isochronism as with a recoiling escapement, and will be little affected by changes in the force of the wheels. Accord- Accordingly, Graham's cylindrical escapement supplanted all others as soon as it was generally known. We cannot compare the vibrations with those of a free balance, because we have no way of making a free balance vibrate for some hours. But we find that doubling or trebling the force of the wheels makes very little alteration in the rate of the watch, though it greatly enlarges the angular motion. Any one may perceive the immense superiority of this escapement over the common recording escapement, by pressing forward the movement of a horizontal watch with the key, or by keeping it back. No great change can be observed in the frequency of the beats, however hard we press. But a more careful examination shows that an increase of the power of the wheels generally causes the watch to go slower; and that this is more remarkable as the watch has been long going without being cleaned. This shows that the cause is to be ascribed to the friction and oil operating on the wide arches of excursion. But when this escapement is well executed, in the best proportions of the parts, the performance is extremely good. We know such watches, which have continued for several weeks without ever varying more than 5° in one day from equable motion. We have seen one whose cylinder was not concentric with the balance, but so placed on the verge that the axis of the verge was at o (fig. 13.), between the centre B of the cylinder and the entering edge b, and Be was equal to the thickness of the cylinder. The watch was made by Emery of London, and was said to go with astonishing regularity, so as to equal any time piece while the temperature of the air did not vary; and when clean, was said to be less affected by the temperature than a watch with a free escapement, but unprovided with a compensation piece. It is evident that this watch must have a minute recoil. This was said to be the aim of the artist, in order to compensate for the obstruction caused by friction during the return of the balance from its excursions. It indeed promises to have this effect; but we should fear that it subjects the excursions to the influence of the wheels. We suspect that the indifferent performance of cylinder watches may often arise from the cylinder being off the centre in some disadvantageous manner.

The watch from which the proportions here stated were taken, is a very fine one made by Graham for Archibald Duke of Argyle, which has kept time with the regularity now mentioned. We believe that there are but few watches which have to large a portion of the cylinder; few indeed have more than one half, or 180° of the circumference. But this is too little. The tooth of the wheel does not begin to act on the retting cylinder till its middle point A or B touch one of the edges. To obtain the same angle of escapement, the inclination of the face of the tooth must be increased (it must be doubled); and this requires the maintaining power to be increased in the same proportion. Besides, in such a escapement it may happen that the tooth will never rest on the cylinder; because the instant that it quits one edge it falls on the other, and pushes it aside, so that the balance acquires no wider vibration than the angle of escapement, and is continually under the influence of the wheels. The escapement is in its best state when the portion of the cylinder exceeds 180° by twice the inclination of the teeth to the circumference of the wheel.

It would employ volumes to describe all the escapements which have been contrived by different artists, aiming at the same points which Graham had in view. We shall only take notice of such as have some essential difference in principle.

Fig. 14. represents a escapement invented in France, and called the Escapement à Viroque, because the pallet resembles a comma. The teeth A, B, C, of the balance wheel are set very oblique to the radius, and there is formed on the point of each a pin, standing up perpendicular to the plane of the wheel. This greatly resembles the wheel of Graham's escapement, when the triangular wedge is cut off from the top of the pin on which it stands. The axis c of the verge is placed in the circumference passing through the pins. The pallet is a plate of hard steel a c f d b, having its plane parallel to the plane of the wheel. The inner edge of this plate is formed into a concave cylindrical surface between a and b, whose axis c coincides with the axis of the verge. Adjoining to this is the acting face b d of the pallet. This is either a straight line b d, making an angle of nearly 30° with a line e b g drawn from the centre, or it is more generally curved, according to the nostrum of the artist. The back of the pallet a e f is also a cylindrical surface (convex) concentric with the other. This extends about 100° from a to f. The part between f and d may have any shape. The interval a o is formed into a convex surface, in such a manner as to be everywhere intersected by the radius in an angle of 30° nearly; i.e., it is a portion of an equiangular spiral. The whole of this is connected with the verge by a crank, which passes perpendicularly through it between f and c; and the plate is set at such height on the crank or verge, that it can turn round clear of the wheel, but not clear of the pins. The teeth of the wheel are set so obliquely, and made so slender, that the verge may turn almost quite round, without the crank's banking on the teeth. The part f d b, called the horn, is of such a length, that when one pin B rests on the outside cylinder at a, the point d is just clear of the next pin A.

When the wheel is not acting, and the balance spring is in equilibrium, the position of the balance is such that the point d of the horn is near i, about 30° from d. The figure represents it in the position which it has when the tooth A has just escaped from the point d of the horn. In this position the next tooth B is applied to the convex cylinder, a very little way (about 5°) from its extremity a. This description will enable the reader to understand the operation of the virgule escapement.

Now suppose the pin A just escaped from the horn. The succeeding pin B is now in contact with the back of the cylinder; and the balance, having got an impulse by the action of A along the concave pallet b d, continues its motion in the direction d g b, till its force is spent, the point of the horn arriving perhaps at b, more than 90° from d. All this while the following tooth B is resting on the back e f of the cylinder. The balance now returns, by the action of its spring; and when the horn is at i, the pin gets over the edge a o, and drops on the opposite side of the concave cylinder, where it rests, while the horn moves from i to k, where it flops, the force of the balance being again spent. The balance then returns; and when the horn comes within The pin gets out of the hollow cylinder, shoves the horn out of its way, and escapes at d. Besides the impulse which the balance receives by the action of the wheel on the horn b d, there is another, though smaller, action in the contrary direction, while the point of B passes over the surface a o; for this surface being inclined to the radius, the pressure on it urges the balance round in the direction b d.

The chief difference of this escapement from the former is that the inclined plane is taken from the teeth of the wheel, and placed on the verge. This alone is a considerable improvement; for it is difficult to shape all the teeth alike; whereas the horn b d is invariable. Moreover, the rearing parts, although they be drawn large in this figure for the sake of distinctness, may be made vastly smaller than Graham's cylinder, which must be big enough to hold a tooth within it. By this change, the friction, during the repose of the wheel, that is, during the excursions of the balance, may be vastly diminished. The inside cylinder need be no bigger than to receive the pin. But although the performance of these escapements is excellent, they have not come into general use in this country. The cause seems to be the great nicety requisite in making the pins of the wheel pass exactly through the axis of the verge. The least flake in the pivots of the balance and balance-wheel must greatly change the action. A very minute increase of distance between the pivots will cause the pin B to slide from the edge a to the horn, without rearing at all on the inside cylinder; and when it does so, it will flop the balance at once, and, immediately after, the watch will run down. The same irregularities will happen if all the pins be not at precisely the same distance from the axis of the wheel.

This escapement was greatly improved, and, in appearance, totally changed, by Mr Lepaute of Paris in 1753. By placing the pins alternately on the two sides of the rim of the balance-wheel, he avoided the use of the outside cylinder altogether. The escapement is of such a singular form, that it is not easy to represent it by any drawing. We shall endeavour, however, to describe it in such a manner as that our readers, who are not artists, will understand its manner of acting. Artists by profession will easily comprehend how the parts may be united which we represent as separate.

Let ABC (fig. 15.) represent part of the rim of the balance-wheel, having the pins 1, 2, 3, 4, 5, &c., projecting from its faces; the pins 1, 3, 5, being on the side next the eye, but the pins 2 and 4 on the farther side. D is the centre of the balance and verge, and the small circle round D represents its thickness. But the verge in this place is crooked, like a crank, that the rim of the wheel may not be interrupted by it. This will be more particularly described by and bye. There is attached to it a piece of hard tempered steel a b c d, of which the part a b c is a concave arch of a circle, having D for its centre. It wants about 30° of a semicircle. The rest of it c d is also an arch of a circle, having the same radius with the balance-wheel. The natural position of the balance is such, that a line drawn from D, through the middle of the face c d, is a tangent to the circumference of the wheel. But, suppose the balance turned round till the point d of the horn comes to a, and the point c comes to 2, in the circumference in which the pins are placed. Then the pin, pressing on the beginning of the horn or pallet, pushes it aside, slides along it, and escapes at d, after having generated a certain velocity in the balance. So far this escapement is like the virgule escapement described already. But now let another pallet, similar to the one now described, be placed on the other side of the wheel, but in a contrary position, with the acting face of the pallet turned away from the centre of the wheel. Let it be so placed at E, that the moment that the pin 1, on the upper side of the wheel, escapes from the pallet c d, the pin 4, on the under side of the wheel, falls on the end of the circular arch e f g of the other pallet. Let the two pallets be connected by means of equal pulleys G and F on the axis of each, and a thread round both, so that they shall turn one way. The balance on the axis D, having gotten an impulse from the action of the pin 1, will continue its motion from A towards i, and will carry the other pallet with a similar motion round the centre E from b towards k. The pin 4 will therefore rest on the concave arch g f e as the pallet turns round. When the force of the balance is spent, the pallet c d returns towards its first position. The pallet g b turns along with it; and when the point of the first has arrived at d, the beginning g of the other arrives at the pin 4; and, proceeding a little farther, this pin escapes from the concave arch e f g, and slides along the pallet g b, pushing it aside, and therefore urging the pallet round the centre E, and consequently (by means of the connection of the pulleys) urging the balance on the axis D round at the same time, and in the same direction. The pin 4 escapes from the pallet g b, when b arrives at 3; but in the time that the pin 4 was sliding along the yielding pallet g b, the pin 3 is moving in the circumference B D A; and the instant that the pin 4 escapes from g b at 3, the pin 3 arrives at 2, and finds the beginning e of the concave arch e b a ready to receive it. It therefore rests on this arch, while the balance continues its motion. This perhaps continues till the point b of the arch comes to 2. The balance now flops, its force being spent, and then returns; and the pin 3 escapes from the circle at e, slides along the yielding pallet c d, and when it escapes at 1, another pin on the under side of the wheel arrives at 4, and finds the arch g f e ready to receive it. And in this manner will the vibration of the balance be continued.

This description of the mode of action at the same time points out the dimensions which must be given to the parts of the pallet. The length of the pallet c d or g b must be equal to the interval between two succeeding pins, and the distance of the centres D and E must be double of this. The radius D e or E g may be as small as we please. The concave arches e b a and g f e must be continued far enough to keep a pin resting on them during the whole excursion of the balance. The angle of escapement, in which the balance is under the influence of the wheels, is had by drawing D c and D d. This angle c D d is about 30°, but may be made greater or less.

Fig. B will give some notion how the two pallets may be combined on one verge. K L represents the verge with a pivot at each end. It is bent into a crank M N O, to admit the balance wheel between its branches. BC represents this wheel, seen edgewise, with its pins, alternately on different sides. The pallets are also represented. presented edgewise by b c d and b g f fixed to the inside of the branches of the crank, facing each other. The position of their acting faces may be seen in the preceding figure; on the verge D, where the pallet g b is represented by the dotted line z i, as being situated behind the pallet c d. The remote pallet z i is placed so that when the point d of the near pallet is just quitted by a pin t on the upper side of the wheel, the angle formed by the face and the arch of rest of the other pallet is just ready to receive the next pin z, which lies on the under side of the rim. A little attention will make it plain, that the action will be precisely the same as when the pallets were on separate axes. The pin t escapes from d, and the pin z is received on the arch of rest, and locks the wheel while the balance is continuing its motion. When it returns, z gets off the arch of rest, pushes aside the pallet z i, escapes from it when i gets to t, and then the pin z finds the point c ready to receive it, &c. The vibrations may be increased by giving a sufficient impulse through the angle of escapement. But they cannot be more than a certain quantity, otherwise the top N of the crank will strike the rim of the wheel. By placing the pins at the very edge of the wheel, the vibrations may easily be increased to a semicircle. By placing them at the points of long teeth, the crank may get in between them, and the vibrations extended still farther, perhaps to 240°.

This escapement is unquestionably a very good one; and when equally well executed, should excel Graham's, both by having but two acting faces to form (and these of hard steel or of stone), and by allowing us to make the circle of rest exceedingly small without diminishing the acting face of the pallet. This will greatly diminish the friction and the influence of oil. But, on the other hand, we apprehend that it is of very difficult execution. The figure of the pallets, in a manner that shall be susceptible of adjustment and removal for repair, and yet sufficiently accurate and steady, seems to us a very delicate job.

Mr. Cumming, in his Elements of Clock and Watchwork, describes (slightly) pallets of the very same construction, making what he conceives to be considerable improvements in the form of the acting faces and the curves of rest. He has also made some watches with this escapement; but they were so difficult, that few workmen can be found fit for the task; and they are exceedingly delicate, and apt to be put out of order. The connection of the pallets with each other, and with the verge, makes the whole such a contorted figure, that it is easily bent and twisted by any jolt or unskillful handling.

There remains another escapement of this kind, having the tooth of the balance-wheel resting on a cylindrical surface on the axis of the verge during the excursions of the balance beyond the angle of escapement, and which differs somewhat in the application of the maintaining power from all those already described.

This is known by the name of Duplex's escapement, and is as follows: Fig. 16 represents the essential parts greatly magnified. AD is a portion of the balance-wheel, having teeth f, b, g, at the circumference. These teeth are entirely for producing the roll of the wheel, while the balance is making excursions beyond the escapement. This is effected by means of an agate cylinder o p q s on the verge. This cylinder has a notch e. When the cylinder turns round in the direction o p q, the notch easily passes the tooth B which is resting on the cylindrical surface; but when it returns in the direction q p o, the tooth B gets into the notch, and follows it, pressing on one side of it till the notch comes into the position o. The tooth, being then in the position b, escapes from the notch, and another tooth drops on the convex surface of the cylinder at B.

The balance-wheel is also furnished with a set of stout flat-sided pins, standing upright on its rim, as represented by a, D. There is also fixed on the verge a larger cylinder G F C above the smaller one o p q, with its under surface clear of the wheel, and having a pallet C, of ruby or sapphire, firmly indented into it, and projecting so far as just to keep clear of the pins on the wheel. The position of this cylinder, with respect to the smaller one below it, is such that, when the tooth b is escaped from the notch, the pallet C has just passed the pin a, which was at A while B rested on the small cylinder; but it moved from A to a, while B moved to b. The wheel being now at liberty, the pin a exerts its pressure on the pallet C in the most direct and advantageous manner, and gives it a strong impulsion, following and accelerating it till another tooth drops on the little cylinder. The angle of escapement depends partly on the projection of the pallet, and partly on the diameter of the small cylinder and the advance of the tooth B into the notch. Independent of the action on the small cylinder, the angle of escapement would be the whole arch of the large cylinder between C and a. But a stops before it is clear of the pallet, and the arch of impulsion is shortened by all the space that is described by the pin while a tooth moves from B to b. It stops at a'.

We are informed by the best artists, that this escapement gives great satisfaction, and equals, if it do not exceed, Graham's cylindrical escapement. It is easier made, and requires very little oil on the small cylinder, and none at all on the pallet. They say that it is the best for pocket watches, and is coming every day more into repute. Theory seems to accord with this character. The resting cylinder may be made very small, and the direct impulse on the pallet gives it a great superiority over all those already described, where the action on the pallet is oblique, and therefore much force is lost by the influence of oil. But we fear that much force is lost by the tooth B shifting its place, and thus shortening the arch of impulsion; for we cannot reckon much on the action of B on the side of the notch, because the lever is so extremely short. Accordingly, all the watches which we have seen of this kind have a very strong main spring in proportion to the size and vibration of the balance. If we lessen this diminution of the angle of impulsion, by lessening the cylinder o p q, and by not allowing B to penetrate far into the notch, the smallest inequality of the teeth, or shake in the pivots of the balance or wheel, will cause irregularity, and even uncertainties in the locking and unlocking the wheel by this cylinder.

A escapement exceedingly like this was applied long ago by Dutertre, a French artist, to a pendulum. The only difference is, that in the pendulum escapement the small cylinder is cut through to the centre, half of it only being left; but the pendulum escapement gives a more effective employment of the maintaining power, because... the wheel acts on the pallet during the whole of the assisted vibration. In a balance escapement, if we attempt to diminish the insufficient motion of the pin from A to a, by lessening the diameter of the small cylinder, the hold given to the tooth in the notch will be so trifling, that the tooth will be thrown out by the smallest play in the pivot holes, or inequality in the length of the teeth.

With this we conclude our account of escapements, where the action of the maintaining power on the balance is suspended during the excursion beyond the angle of impulsion, by making a tooth rest on the surface of a small concentric cylinder. In such escapements, the balance, during its excursions, is almost free from any connection with the wheels, and its isochronism is disturbed by nothing but the friction on this surface.

We come now to escapements of more artful construction, in which the balance is really and completely free during the whole of its excursion, being altogether disengaged from the wheelwork. These are called detached escapements. They are of more recent date. We believe that Mr Le Roy was the first inventor of them, about the year 1748. In the Memoirs of the Academy of Paris for that year, and in the Collection of approved Machines and Inventions, we have descriptions of the contrivance. The balance-wheel rests on a detent, while the balance is vibrating in perfect freedom. It has a pallet standing out from the centre, which, in the course of vibration, passes close by the point of a tooth of the wheel. At that instant a pin, connected with this pallet, withdraws the detent from the wheel, and the tooth just now mentioned follows the pallet with rapidity, and gives it a smart push forward. Immediately after, another tooth of the wheel meets the other claw of the detent, and the wheel is again locked. When the balance returns, the pin pushes the detent back into its former place, where it again locks the wheel. Then the balance, returning its first direction, unlocks the wheel, and receives another impulsion from it. Thus the balance is unconnected with the wheels, except while it gets the impulsion, and at the moments of unlocking the wheels.

This contrivance has been reduced to the greatest possible simplicity by the British artists, and seems scarcely capable of farther improvement. The following is one of the most approved constructions. In fig. 17, abe represents the pallet, which is a cylinder of hard steel or stone, having a notch a. A portion of the balance-wheel is represented by A.B. It is placed so near to the cylinder that the cylinder is no more than clear of two adjoining teeth. DE is a long spring, fixed to the watch plate at E, so as to press very gently on the flop pin G. A small fluid F is fixed to that side of the spring that is next to the wheel. The tooth of the wheel rests on this fluid, in such a manner that the tooth a is just about to touch the cylinder, and the tooth f is just clear of it. Another spring, extremely slender, is attached to the spring DE, on the side next the balance-wheel, and claps close to it, but keeping clear of the fluid F, and having its point e projecting about 1/36th of an inch beyond its extremity. When the point e is pressed towards the wheel, it yields most readily; but, when pressed in the opposite direction, it carries the spring DE along with it. The cylinder being so placed on the verge that the edge a of the notch is close by the tooth a, a hole is drilled at i, close by the projecting point of the slender spring, and a small pin is driven into this hole. This is the whole apparatus; and this situation of the parts corresponds to the quiescent position of the balance.

Now, let the balance be turned out of this position 80 or 90 degrees, in the direction a.b.c. When it is let go, it returns to this position with an accelerated motion. The pin i strikes on the projecting point of the slender spring, and pressing the strong spring DE outward from the wheel, withdraws the fluid F from the tooth; and thus unlocks the wheel. The tooth a engages in the notch, and urges round the balance. The pin e quits the slender spring before the tooth quits the notch; so that when it is clear of the pallet, the wheel is locked again on the fluid F, and another tooth g is now in the place of a, ready to act in the same manner. When the force of the balance is spent, it flows, and then returns toward its quiescent position with a motion continually accelerated. The pin i arrives at the point e of the slender spring, raises it from the strong spring without disturbing the latter, and almost without being disturbed by this trifling obstacle; and it goes on, turning in the direction a.b.c., till its force is again spent; it stops, returns, again unlocks the wheel, and gets a new impulsion. And in this manner the vibrations are continued. Thus we see a vibration, almost free, maintained in a manner even more simple than the common crutch escapement. The impulse is given direct, without any decomposition by oblique action, and it is continued through the whole motion of the wheel. No part of this motion is lost, as in Dupleix's escapement, by the gradual approach of the tooth to its active position. Very little force is required for unlocking the wheel, because the spring DE is made slender at the remote end E, so that it turns round E almost like a lever turning on pivots. A sudden twitch of the watch, in the direction a.b.c., might chance to unlock the wheel. But this will only derange one vibration, and even that not considerably, because the teeth are so close to the cylinder that the wheel cannot advance till the notch comes round to the place of escape- ment. A tooth will continue pressing on the cylinder, and by its friction will change a little the extent and duration of a single vibration. The greatest derangement will happen if the wheel should thus unlock by a jolt, while the notch passes through the arch of escapement in the returning vibration. Even this will not greatly derange it, when the watch is clean and vibrating wide; because, in this position, the balance has its greatest momentum, and the direction of the only jolt that can unlock the wheel tends to increase this momentum relatively. In short, considering it theoretically, it seems an almost perfect escapement; and the performance of many of these watches abundantly confirms that opinion. They are known to keep time for many days together, without varying one second from day to day; and this even under considerable variations of the maintaining power. Other detached escapements may equal this, but we scarcely expect any to exceed it; and its simplicity is so much superior to any that we have seen, that, on this account, we are disposed to give it the preference. We do not mean to say that it is the best for a pocket watch. Perhaps the escapement of Dupleix or Graham may be preferable, as being sus-ceptible centible of greater strength, and more able to withstand jolts. Yet it is a fact that some of the watches made in this form by Arnold and others have kept time in the wonderful manner abovementioned while carried about in the pocket.

Mr Mudge of London invented, about the year 1765, another detached escapement, of a still more ingenious construction. It is a counterpart of Mr Cumming's escapement for pendulums. The contrivance is to this effect. In fig. 18, \(a b c\) represents the balance. Its axis is bent into a large crank \(EFGHIK\), sufficiently roomy to admit within it two other axes \(M\) and \(L\), with the proper cocks for receiving their pivots. The three axes form one straight line. About these smaller axes are coiled two auxiliary springs, in opposite directions, having their outer extremities fixed in the fluids \(A\) and \(B\). The balance has its spring also, as usual, and the three springs are so disposed that each of them alone would keep the balance at rest in the same position, which we may suppose to be that represented in the figure. The auxiliary springs \(A\) and \(B\) are connected with the balance only occasionally, by means of the arms \(m\) and \(n\) projecting from their respective axes. These arms are attached on opposite sides by the pins \(o\), \(p\), in the branches of the crank; so that when the balance turns round, it carries one or other of those arms round with it; and during this motion, it is affected by the auxiliary spring connected with the arm so carried round by it.

Let us suppose that the balance vibrates \(120^\circ\) on each side of its quiescent position \(abc\), so that the radius \(Ea\) acquires, alternately, the positions \(Eb\) and \(Ec\). The auxiliary springs are connected with the wheels by a common dead-beat pendulum escapement, so that each can be separately wound up about \(30^\circ\), and retained in that position. Let us also suppose that the spring \(A\) has been wound up \(30^\circ\) in the direction \(ab\), by the wheel-work, and that the point \(a\) of the rim of the balance, having come from \(c\), is passing through \(a\) with its greatest velocity. When the radius \(Ea\) has passed \(a 30^\circ\) in its course toward \(b\), the pin \(o\) finds the arm \(m\) in its way, and carries it along with it till \(a\) gets to \(b\). But, by carrying away the arm \(m\), it has unlocked the wheel-work, and the spring \(B\) is now wound up \(30^\circ\) in the other direction, but has no connection with the balance during this operation. Thus the balance finishes its semivibration \(ab\) of \(120^\circ\), opposed by its own spring the whole way, and by the auxiliary spring \(A\) through an angle of \(90^\circ\). It returns to the position \(Ec\), aided by \(A\) and by the balance spring, through an angle of \(120^\circ\). In like manner, when \(Ea\) has moved \(30^\circ\) toward the position \(Ec\), the pin \(p\) meets with the arm \(n\), and carries it along with it through an angle of \(90^\circ\), opposed by the spring \(B\), and then returns to the position \(Ec\), assisted by the same spring through an arch of \(120^\circ\).

Thus it appears that the balance is opposed by each auxiliary spring through an angle of \(90^\circ\), and assisted through an angle of \(120^\circ\). This difference of action maintains the vibrations, and the necessary winding up of the auxiliary springs is performed by the wheel-work, at a time when they are totally disengaged from the balance. No irregularity of the wheel-work can have any influence on the force of the auxiliary springs, and therefore the balance is completely disengaged from all these irregularities, except in the short moment of unlocking the wheel that winds up the springs.

This is a most ingenious construction, and the nearest approach to a free vibration that has yet been thought of. It deserves particular remark that, during the whole of the returning or accelerated semivibration, the united force of the springs is proportional to the distance from the quiescent position. The same may be said of the retarded excursion beyond the angle of impulse: therefore the only deviation of the forces from the law of cycloidal vibration is during the motion from the quiescent position to the meeting with the auxiliary spring. Therefore, as the forces, on both sides, beyond this angle, are in their due proportion, and the balance always makes such excursions, there seems nothing to disturb the isochronism, whether the vibrations are wide or narrow. Accordingly, the performance of this escapement, under the severest trials, equalled any that were compared with it, in so far as it depended on escapement alone. But it is evident that the execution of this escapement, though most simple in principle, must always be vastly more difficult than the one described before. There is so little room, that the parts must be exceedingly small, requiring the most accurate workmanship. We think that it may be greatly simplified, preserving all its advantages, and that the parts may be made of more than twice their present size, with even less load on the balance from the inertia of matter. This improvement is now carrying into effect by a friend.

Still, however, we do not see that this escapement is, theoretically, superior to the last. The irregularities of maintaining power affect that escapement only in the arch of impulsion, where the velocity is great, and the time of action very small. Moreover, the chief effect of the irregularities is only to enlarge the excursions; and in these the wheels have no concern.

Mr Mudge has also given another detached escapement, which he recommends for pocket watches, and executed entirely to his satisfaction in one made for the Queen. A dead beat pendulum escapement is introduced, as in the last, between the wheels and the balance. The crutch \(EDF\) (fig. 19.) has a third arm \(DG\), standing outwards from the meeting of the other two, and of twice their length. This arm terminates in a fork \(AGB\). The verge \(V\) has a pallet \(C\), which, when all is at rest, would stand between the points \(A, B\) of the fork. But the wheel, by its action on the pallet \(E\), forces the fork into the position \(BG\); the point \(A\) of the fork being now where \(B\) was before, just touching the cylindrical surface of the verge. The escapement of the crutch \(EDF\) is not accurately a dead beat escapement, but has a very small recoil beyond the angle of impulsion. By this circumstance the branch \(A\) (now at \(B\)) is made to press most gently on the cylinder, and keeps the wheel locked, while the balance is going round in the direction \(BHA\). The point \(A\) gets moving from \(A\) to \(B\) by means of a notch in the cylinder, which turns round at the same time by the action of the branch \(AG\) on the pallet \(C\); but \(A\) does not touch the cylinder during this motion, the notch leaving free room for its passage. When the balance returns from its excursion, the pallet \(C\) strikes on the branch \(A\) (still at \(B\)), and unlocks the wheel. This now acting on the crutch pallet F, causes the branch & of the fork to follow the pallet C, and give it a strong impulse in the direction in which it is then moving, causing the balance to make a semivibration in the direction AHB. The fork is now in the situation A & a, similar to B & b, and the wheel is again locked on the crutch pallet E.

The intelligent reader will admit this to be a very steady and effective escapement. The lockage of the wheel is procured in a very ingenious manner; and the friction on the cylinder, necessary for effecting this, may be made as small as we please, notwithstanding a very strong action of the wheel: For the pressure of the fork on the cylinder depends entirely on the degree of recoil that is formed on the pallets E and F. Pressure on the cylinder is not indispensably necessary, and the crutch escapement might be a real dead beat. But a small recoil, by keeping the fork in contact with the cylinder, gives the most perfect pendulums to the motion. The ingenious inventor, a man of approved integrity and judgment, declares that her Majesty's watch was the best pocket watch he had ever seen. We are not disposed to question its excellency. We saw an experiment watch of this construction, made by a country artist, having a balance so heavy as to vibrate only twice in a second. Every vibration was sensibly beyond a turn and a half, or $540^\circ$. The artist assured us, that when its proper balance was in, vibrating somewhat more than five times in a second, the vibrations even exceeded this. He had procured it this great mobility by substituting a roller with fine pivots in place of the simple pallet of Mudge. This great extent of detached vibration is an unquestionable excellence, and is peculiar to those two escapements of this ingenious artist.

Very ingenious escapements have been made by Earnshaw, Howel, Hayley, and other British artists; and many by the artists of Paris and Geneva. But we must conclude the article, having described all that have any difference in principle.

The escapement having been brought to this degree of perfection, we have an opportunity of making experiments on the law of action of springs, which has been too readily assumed. We think it only to demonstrate, that the figure of a spring, which must have a great extent of rapid motion, will have a considerable influence on the force which it imparts on a balance in actual motion. The accurate determination of this influence is not very difficult in some simple cases. It is the greatest of all in the plane spiral, and the least in the cylindrical; and, in this last form, it is so much less as the diameter is less, the length of the spring being the same. By employing many turns, in order to have the same ultimate force at the extremity of the excursion, this influence is increased. A particular length of spring, therefore, will make it equal to a given quantity; and it may thus compensate for a particular magnitude of friction, and other obstructions. This accounts for the observation of Le Roy, who found that every spring, when applied to a movement, had a certain length, which made the wide and narrow vibrations isochronous. His method of trial was so judicious, that there can be no doubt of the justness of his conclusion. His time-keeper had no fuzze; and when the last revolution of the main wheel was going on, the vibrations were but of half the extent of those made during the first revolution. Without minding the real rate of going, he only compared the duration of the first and last revolution of the minute hand. An artist of our acquaintance repeated these experiments, and with the same result: But, unfortunately, could derive little benefit from them; because in one state of the oil, or with one balance, he found the lengths of the same spring, which produced isochronous vibrations, were different from those which had this effect in another state of the oil, or with another balance. He also observed another difference in the rate, arising from a difference of position, according as XII, VI, III, or IX, was uppermost; which difference plainly arises from the swaying of the spring by its weight, and, in that state, acting as a pendulum. This unluckily put a stop to his attempts to lessen this hurtful influence by employing a cylindrical spiral of small diameter and great length.