by the whole power of the remontoire-spring; so that the balance has only to remove the small remaining pressure, which does away that objection, and also that of the disadvantage of detents, as this locking may be compared to a light balance turning on fine pivots, without a pendulum-spring; and has only the advantage of banking safe at two turns of the balance, and of being firmer, and less liable to be out of repair than any locking where spring-work is used, but likewise of unlocking with much less. power.-It was then observed, it required more power to make it go than usual. Permit me to say, it requires no more power than any other remontoire-scapement, as the power is applied in the most mechanical manner possible.-And, lastly, it was said, that it set or required the balance to vibrate an unusually large arch before the piece would go. This depends on the accuracy of the execution, the proportionate diameter and weight of the balance, the strength of the remontoire-spring, and the length of the pallets. If these circumstances are well attended to, it will set but little more than the most generally detached scapements."

A, shews the scape-wheel, pl. XXIX.

B, the lever-pallet, on an arbor with fine pivots, having at the lower end

c, the remontoire or spiral-spring fixed with a collar and stud, as pendulum-springs are.

D, the pallet of the verge, having a roller turning in small pivots for the lever-pallet to act against.

E, Pallets to discharge the locking, with a roller between, as in fig. 10.

F, the arm of the locking-pallet continued at the other end to - make it poise, having studs and screws to adjust and bank the quantity of motion.

a and b, the locking-pallets, being portions of circles, fastened on an arbor turning on fine pivots.

G, the triple fork, at the end of the arm of the locking pallets.

The centre of the lever-pallet in the draft, is in a right line between the centre of the scape-wheel and the centre of the verge, though in the model it is not: but may be made so or not, as best suits the calliper, &c.

"The scape-wheel A, with the tooth 1, is acting on the lever-pallet B, and has wound up the spring c: the verge-pallet D (turning the way represented by the arrow) the moment it comes within the reach of the lever pallet, the discharging pallet E, taking hold of one prong of the fork, removes the arm F, and relieves the tooth 3 from the convex part of the lock a. The wheel goes forward a little, just sufficient to per

mit the lever-pallet to pass, while the other end gives the impulse to the balance: the tooth 4 of the wheel is then locked on the concave side of the lock b, and the lever-pallet is stopped against the tooth 5, as in fig. 11. So far the operation of giving the impulse, in order again to wind the remontoire-spring (the other pallet at E, in the return, removing the arm F the contrary direction), relieves the tooth 3 from the lock b. The wheel again goes forward, almost the whole space, from tooth to tooth, winds the spiral spring again, and comes into the situation of fig. 1, and thus the whole performance is completed. The end of the lower pallet в resting on the point of the tooth 1, prevents the wheel exerting its full force on the lock a, as in fig. 1. The same effect is produced by the pallet lying on the tooth 5, by preventing the wheel from pressing on 6; and thus the locking becomes the tightest possible. This scapement may be much simplified by putting a spring with a pallet made in it, as in fig. 12. instead of the lever-pallet, and spiral-spring. The operation will be in other respects exactly the same, avoiding the friction of the pivots of the lever-pallet. This method I prefer for a piece to be in a state of rest, as a clock; but the disadvantage, from the weight of the spring in different positions, is obvious. The locking may be on any two teeth of the wheel, as may be found most convenient."

Many other ingenious scapements have been contrived by Harrison, Hindley, Ellicott, Lepaute, Le Roy, Berthoud, Arnold, Whitehurst, Earnshaw, Nicholson, &c. But descriptions of them would extend this article to much too great a length. What is here collected will, we trust, furnish some insight into the nature of a few of the most approved scapements.

ARCHIMEDES's SCREW, or the Watersnail, is a machine for raising water, which consists either of a pipe wound spirally round a cylinder, or of one or more spiral excavations formed by means of spiral projections from an internal cylinder, covered by an external coating, so as to be watertight. This screw is one of the most ancient, and at the same time ingenious, machines we know, being truly worthy of the name it bears, supposing Archimedes to be the real inventor. Though simple in its general manner of operation, its theory is attended with soine difficulties which could only be conquered by the modern analysis: it was first stated correctly, as far as we have been able to ascertain, by M. Pitot, in the Mémoires de l'Academie Royale des Sciences, and afterwards more elaborately by Euler in Nov. Comment. Petropol. tom. 5. Later attempts by Langsdorf in his Handbuch der Maschinenlehre, and some other authors, are not to be relied on. That the nature of this curious machine may be the better understood, we shall first state generally its manner of operation; and then present a more particular view

of the calculus necessary to shew the work it will really perform, and the force required as a first mover.

1. If we conceive that a flexible tube is rolled regularly about a cylinder from one end to another; this tube or canal will be a screw or spiral, of which we suppose the intervals of the spires or threads to be equal. The cylinder being placed with its axis in a vertical position, if we put in at the upper end of the spiral tube a small ball of heavy matter, which may move freely, it is certain that it will follow all the turnings of the screw from the top to the bottom of the cylinder, descending always as it would have done had it fallen in a right line along the axis of the cylinder, only it would occupy more time in running through the spiral. If the cylinder were placed with its axis horizontally, and we again put the ball into one opening of the canal, it will descend, following the direction of the first demi-spire; but when it arrives at the lowest point in this portion of the tube it will stop. It must be remarked that, though its heaviness has no other tendency than to make it descend in the demispire, the oblique position of the tube, with respect to the horizon, is the cause that the ball, by always descending, is always advancing from the extremity of the cylinder whence it commenced its motion, to the other extremity. It is impossible that the ball can ever advance more towards the further, or as we shall call it, the second extremity of the cylinder, if the cylinder placed horizontally remains always immoveable: but if, when the ball is arrived at the bottom of the first demi-spire, we cause the cylinder to turn on its axis without changing the position of that axis, and in such manner that the lowest point of the demi-spire on which the ball presses becomes elevated, then the ball falls necessarily from this point upon that which succeeds, and which becomes lowest; and since this second point is more advanced towards the second extremity of the cylinder than the former was, therefore by this new descent the ball will be advanced towards that extremity, and so on throughout, in such a manner that it will at length arrive at the second extremity by always descending, the cylinder having its rotatory motion continued. Moreover, the ball, by constantly following its tendency to descend, has advanced through a right line equal to the axis of the cylinder, and this distance is horizontal, because the sides of the cylinder were placed horizontally. But if the cylinder had been placed oblique to the horizon, and we suppose it to be turned on its axis always in the same direction, it is easy to see that if the first quarter of a spire actually descends, the ball will move from the lower end of the spiral tube, and be carried solely by gravity to the lowest point of the first demi-spire, where, as in the preceding case, it will be abandoned

by this point as it is elevated by the rotation, and thrown by its heaviness upon that which has taken its place: whence, as this succeeding point is further advanced towards the second extremity of the cylinder, than that which the ball occupied just before, and consequently more elevated; therefore the ball while following its tendency to descend by its heaviness, will be always more and more elevated by virtue of the rotation of the cylinder. Thus it will, after a certain number of turns, be advanced from one extremity of the tube to the other, or through the whole length of the cylinder; but it will only be raised through the vertical height determined by the obliquity of the position of the cylinder.

Instead of the ball let us now consider water as entering by the lower extremity of the spiral canal, when immersed in a reservoir: this water descends at first in the canal solely by its gravity; but the cylinder being turned, the water moves on in the canal to occupy the lowest place; and thus by the continual rotation is made to advance further and further in the spiral, till at length it is raised to the upper extremity of the canal where it is expelled. There is, however, an essential difference between the water and the ball: for the water, by reason of its fluidity, after having descended by its heaviness to the lowest point of the demi-spire, rises up on the contrary side to the original level; on which account more than half one of the spires may soon be filled with the fluid. This is an important particular, which, though it need not be regarded in a popular illustration, must be attended to in the more particular exhibition of the theory to which we now proceed.

2. The most simple method of tracing a screw or a helix upon a cylinder is well known to be this: take the height or length of a cylinder for one leg of a right-angled triangle, and make the other leg equal to as many times the circumference of the base of the cylinder, as the screw is to make convolutions about the cylinder itself; then if this triangle be enveloped about the surface of the solid, the two legs being made, the one to lie parallel to the axis of the cylinder, the other to fold upon the circumference of its base, the hypothenuse will form the contour of the screw. Suppose therefore here, that upon the cylinder ABCD (fig. 6. pl. XXIV.) we have rolled the rightangled triangle BDE, and that its hypothenuse DE traces upon the cylinder the contour of the helix or the spires BF, GH, &c. Then if a tube be formed according to the direction of this spiral, and a small ball put into it, if the cylinder were placed upright, the ball would roll to the bottom with the same velocity and the same force, as it would have descended upon the plane DE, if BE were horizontal and BD vertical. But if the

cylinder be inclined until it makes with the vertical CL an angle ACL equal to the angle RED, or the angle which the threads of the screw make constantly with the base of the cylinder, in that case DE will be parallel to the horizon; and whether the spires be few or many, they will all be parallel to the horizon: so that there being nothing to occasion the ball P to move toward either Gor H, it will remain immoveable, supposing the cylinder to be at rest: but if the cylinder be turned on its axis in one direction, the ball (abstracting from friction) will move the contrary way, in conformity with the first law of motion.

3. The inclination ACL BED which we have just assigned, is the least we can give, so that the ball shall not descend of itself: but if we augment this inclination, or make the angle LAC smaller, then by turning the cylinder in the direction CMD, the ball will always have a descent on the side towards н, and will mount, so to speak, by descending. The reason is very simple: the plane which carries it makes it rise more in consequence of the rotatory motion, than it descends by virtue of the force of gravity.

4. There are several methods of determining the ratio of the weight of the ball P to the force F, necessary to make it rise by turning the screw. The following is perhaps the most simple: the force or power is to the weight elevated, as the vertical space passed over by the weight, is to the space passed through by the power in moving it. Here the vertical space is c1, and if the moving force act at the circumference of the cylinder, the space passed over by that force will be equal to as many times the circumference of the cylinder's base, as there are convolutions of the helix: thus we shall have BE: CL:: P: F.

Example. Let the diameter AB of the cylinder be 14 inches, the vertical altitude CL = 12 feet or 144 inches, and 12 the convolutions of the spiral, the cylinder being so placed that the angle LAC is less than BED; the weight to be raised being a 48 lb. ball. Then the circumference of the cylinder will be nearly 44 inches, and the 12 turns equal to 12 × 44 = 528 = BE. Hence we have 528 144: P: F:: 48: 134 lbs. the measure of the requisite force at the surface of the cylinder. If the moving force describe a circle whose diameter is 3 times that of the cylinder, or act at a winch, whose distance from the axis of motion is 21 inches, that force will then be reduced to of 13 or 44 lbs. which is less than of the weight of the ball. The friction upon the pivots, &c. is not here considered.

Thus it appears that Archimedes's screw may be used for other purposes than raising of water. It might be adapted with advantage in raising cannon balls from a ship to a wharf: and