an hydraulic machine for raising water by means of the pressure of the atmosphere.

It would be an entertaining and not an uninteresting piece of information to learn the progressive steps invention by which the ingenuity of man has invented the various pumps, methods of raising water. A pump must be considered as the last step of this progress. Common as it is, and overlooked even by the curious, it is a very admirable and refined invention. Nothing like it has been found in any of the rude nations whom the restless spirit of the Europeans has discovered, either in the new continent of America, or the islands of the Pacific ocean. Nay, it was unknown in the cultivated empire of China at the time of our arrival there by sea; and it is still a rarity everywhere in Asia, in places unfrequented by the Europeans. It does not appear to have been known to the Greeks and Romans in early times; and perhaps it came from Alexandria, where physical and mathematical science was much cultivated by the Greek school under the protection of the Ptolemies. The performances of Ctesibius and Hero are spoken of by Pliny and Vitruvius as curious novelties (A). It is perhaps not difficult to trace the steps by which these mechanicians were led

(A) In the early Greek writings, it does not appear that the words ἀνθρώποι, ἄνθρωποι, ἄνθρωποι, &c. were used to express anything like what we call a pump. In all these passages the words either express generally the drawing of water, The Egyptian wheel was a common machine all over Asia, and is still in use in the remotest corners, and was brought by the Saracens into Spain, where it is still very common under its ancient name noria. The Danish missionaries found in a remote village in the kingdom of Siam the immediate offspring of the noria (Lettres Edifiantes et Curieuses). It was a wheel turned by an ass, and carrying round, not a string of earthen pots, but a string of wigs of hay, which it drew through a wooden trunk. This rude chain-pump was in frequent use for watering the rice fields. It is highly probable that it is of great antiquity, although we do not recollect its being mentioned by any of the Greek or Roman writers. The Arabs and Indians were nothing less than innovators; and we may suppose with great safety, that what arts we now find among them they possessed in very remote periods. Now the step from this to the pump is but short, though it is nice and refined; and the forcing pump of Ctesibius is the easiest and most natural.

Let AB (fig. 1.) be the surface of the water in the well, and D the height where it is to be delivered. Let DC be a long wooden trunk, reaching as deep under water as possible. Let the rope EF be fitted with its knot of hay F. When it is drawn up through the trunk, it will bring up along with it all the water lying between C and A, which will begin to run out by the spout D as soon as the knot gets to G, as far below D as C is below A. All this is very obvious; and it required but little reflection to be assured, that if F was let down again, or pushed down, by a rod instead of a rope, it would again perform the same office. Here is a very simple pump. And if it was ever put in practice, it behoved to show the supporting power of the atmosphere, because the water would not only be lifted by the knot, but would even follow it. The imperfection of this pump behoved to appear at first sight, and to suggest its remedy. By pushing down the knot F, which we shall henceforth call the piston, all the force expended in lifting up the water between A and G is thrown away, because it is again let down. A valve G, at the bottom would prevent this. But then there must be a passage made for the water by a lateral tube KBD (fig. 2.). And if this be also furnished with a valve H, to prevent its losing the water, we have the pump of Ctesibius, as sketched in fig. 2. The valve is the great refinement; but perhaps even this had made its appearance before in the noria. For, in the more perfect kinds of these machines, the pots have a stop or valve in their bottom, which hangs open while the pot descends with its mouth downwards, and then allows it to fill readily in the cistern; whereas, without the valve, it would occasion a double load to the wheel. If we suppose that the valve had made its appearance so early, it is not improbable that the common pump sketched in fig. 3. was as old as that of Ctesibius. In this place we shall first give a short description of the chief varieties of these engines, considering them in their simplest form, and we shall explain in very general terms their mode of operation. We shall then give a concise and popular theory of their operation, furnishing principles to direct us in their construction; and we shall conclude with the description of a few peculiarities which may contribute to their improvement or perfection.

There are but two sorts of pumps which essentially differ; and all the varieties that we see are only modifications of these. One of these original pumps has a solid piston; the other has a piston with a perforation and a valve. We usually call the first a FORCING PUMP, and the second a LIFTING or SUCKING PUMP.

Fig. 2. is a sketch of the forcing pump in its most Forcing simple form and situation. It consists of a hollow cylinder AC, called the WORKING BARREL, open at both ends, and having a valve G at the bottom, opening upwards. This cylinder is filled by a solid piston EF, covered externally with leather or tow, by which means it fits the box of the cylinder exactly, and allows no water to escape by its sides. There is a pipe KHD, which communicates laterally with this cylinder, and has a valve at some convenient place H, as near as possible to its junction with the cylinder. This valve also opens upwards. This pipe, usually called the RISING PIPE, or MAIN, terminates at the place D, where the water must be delivered.

Now suppose this apparatus set into the water, so that the upper end of the cylinder may be under or even operation, with the surface of the water AB; the water will open the valve G, and after filling the barrel and lateral pipe, will also open the valve H, and at last stand at an equal height within and without. Now let the piston be put in at the top of the working barrel, and thrust down to K. It will push the water before it. This will shut the valve G, and the water will make its way through the valve H, and fill a part Bb of the rising pipe, equal to the internal capacity of the working barrel. When this downward motion of the piston ceases, the valve H will fall down by its own weight and shut this passage. Now let the piston be drawn up again: The valve H hinders the water in the rising pipe from returning into the working barrel. But now the valve G is opened by the pressure of the external water, and the water enters and fills the cylinder as the piston rises. When the piston has got to the top, let it be thrust down again: The valve G will again be shut, and the water will be forced through the passage at H, and rise along the main, pushing before it the water already there, and will now have its surface at I. Repeating this operation, the water must at last arrive at D, however... ever remote, and the next stroke would raise it to e; so that during the next rise of the piston the water in eD will be running off by the spout.

The effect is the same whatever be the position of the working barrel, provided only that it be under water. It may lie horizontally or sloping, or it may be with its mouth and piston rod undermost. It is still the same forcing pump, and operates in the same manner and by the same means, viz. the pressure of the surrounding water.

The external force which must be applied to produce this effect is opposed by the pressure exerted by the water on the opposite face of the piston. It is evident, from the common laws of hydrostatics, that this opposing pressure is equal to the weight of a pillar of water, having the face of the piston for its base, and the perpendicular height dA of the place of delivery above the surface of the water AB in the cistern for its height. The form and dimensions of the rising pipe are indifferent in this respect, because heavy fluids press only in the proportion of their perpendicular height. Observe that it is not dF, but dA, which measures this pressure, which the moving force must balance and surmount. The whole pressure on the under surface Ff of the piston is indeed equal to the weight of the pillar dFf; but part of this is balanced by the water AFfa. Indeed the water does not get into the upper part of the working barrel, this compensation does not obtain. While we draw up the piston, this pressure is removed, because all communication is cut off by the valve H, which now bears the whole pressure of the water in the main. Nay, the ascent of the piston is even assisted by the pressure of the surrounding water. It is only during the descent of the piston therefore that the external force is necessary.

Observe that the measure now given of the external force is only what is necessary for balancing the pressure of the water in the rising pipe. But in order that the pump may perform work, it must surmount this pressure, and cause the water to issue at D with such a velocity that the required quantity of water may be delivered in a given time. This requires force, even although there were no opposing pressure; which would be the case if the main were horizontal. The water fills it, but it is at rest. In order that a gallon, for instance, may be delivered in a second, the whole water in the horizontal main must be put in motion with a certain velocity. This requires force. We must therefore always distinguish between the state of equilibrium and the state of actual working. It is the equilibrium only that we consider at present; and no more is necessary for understanding the operation of the different species of pumps. The other force is of much more intricate investigation, and will be considered by itself.

The simplest form and situation of the lifting pump is represented by the sketch fig. 3. The pump is immersed in the cistern till both the valve G and piston F are under the surface AB of the surrounding water. By this means the water enters the pump, opening both valves, and finally stands on a level within and without.

Now draw up the piston to the surface A. It must lift up the water which is above it (because the valve in the piston remains shut by its own weight); so that its surface will now be at a, Aa being made equal to AF. In the mean time, the pressure of the surrounding water forces it into the working barrel, through the valve G; and the barrel is now filled with water. Now, let the piston be pushed down again; the valve G immediately shuts by its own weight, and in opposition to the endeavours which the water in the barrel makes to escape this way. This attempt to compress the water in the barrel causes it to open the valve F in the piston; or rather, this valve yields to our endeavour to push the piston down through the water in the working barrel. By this means we get the piston to the bottom of the barrel; and it has now above it the whole pillar of water reaching to the height a. Drawing up the piston to the surface A a second time, must lift this double column along with it, and its surface now will be at b. The piston may again be thrust down through the water in the barrel, and again drawn up to the surface; which will raise the water to c. Another repetition will raise it to d; and it will now show itself at the intended place of delivery. Another repetition will raise it to e; and while the piston is now descending to make another stroke, the water in ed will be running off through the spout D; and thus a stream will be produced, in some degree continual, but very unequal. This is inconvenient in many cases: thus, in a pump for domestic use, such a hobbling stream would make it very troublesome to fill a bucket. It is therefore usual to terminate the main by a cistern LMNO, and to make the spout small. By this means the water brought up by the successive strokes of the piston rises to such a height in this cistern, as to produce an efflux by the spout nearly equable. The smaller we make the spout D the more equable will be the stream; for when the piston brings up more water than can be discharged during its descent, some of it remains in the cistern. This, added to the supply of next stroke, makes the water rise higher in the cistern than it did by the preceding stroke. This will cause the efflux to be quicker during the descent of the piston, but perhaps not yet sufficiently quick to discharge the whole supply. It therefore rises higher next stroke; and at last it rises so high, that the increased velocity of efflux makes the discharge precisely balance the supply. Now, the quantity supplied in each stroke is the same, and occupies the same room in the cistern at top; and the surface will sink the same number of inches during the descent of the piston, whether that surface has been high or low at the beginning. But because the velocities of the efflux are as the square roots of the heights of the water above the spout, it is evident that a fall of two or three inches will make a smaller change in the velocity of efflux when this height and velocity are great. This seems but a trifling observation; but it serves to illustrate a thing to be considered afterwards, which is important and abstruse, but perfectly similar to this.

It is evident, that the force necessary for this operation must be equal to the weight of the pillar of water dAaD, if the pipe be perpendicular. If the pump be standing sloped, the pressure which is to be balanced is still equal to the weight of a pillar of water of this perpendicular height, and having the surface of the piston for its base.

Such is the simplest, and, we may add, by far the best, form of the forcing and lifting pumps; but it is not the most usual. Circumstances of convenience, economy, and more frequently of fancy and habit, have caused the pump-makers to deviate greatly from this form. It is not usual to have the working barrel in the water; this, especially in deep wells, makes it of difficult access for repairs, and requires long piston rods. This would not do in a forcing pump, because they would bend.

We have supposed, in our account of the lifting pump, that the rise of the piston always terminated at the surface of the water in the cistern. This we did in order that the barrel might always be filled by the pressure of the surrounding water. But let us suppose that the rise of the piston does not end here, and that it is gradually drawn up to the very top: it is plain that the pressure of the atmosphere is by this means taken off from the water in the pipe (see Pneumatics), while it remains pressing on the water of the cistern. It will therefore cause the water to follow the piston as it rises through the pipe, and it will raise it in this way 33 feet at a medium. If, therefore, the spout D is not more than 33 feet above the surface of the water in the cistern, the pipe will be full of water when the piston is at D. Let it be pushed down to the bottom; the water will remain in the pipe, because the valve G will shut; and thus we may give the piston a stroke of any length not exceeding 33 feet. If we raise it higher than this, the water will not follow; but it will remain in the pipe, to be lifted by the piston, after it has been pushed down through it to the bottom.

But it is not necessary, and would be very inconvenient, to give the piston so long a stroke. The great use of a pump is to render effectual the reciprocation of a short stroke which we can command, while such a long stroke is generally out of our power. Suppose that the piston is pushed down only to b; it will then have a column bf incumbent on it, and it will lift this column when again drawn up. And this operation may be repeated like the former, when the piston was always under water; for the pressure of the atmosphere will always cause the water to follow the piston to the height of 33 feet.

Nor is it necessary that the fixed valve G be placed at the lower orifice of the pipe, nor even under water. For, while things are in the state now described, the piston drawn up to f, and the whole pipe full of water; if we suppose another valve placed at b above the surface of the cistern, this valve can do no harm. Now let the piston descend, both valves G and b will shut. G may now be removed, and the water will remain supported in the space bg by the air; and now the alternate motions of the piston will produce the same effect as before.

We found in the former case that the piston was carrying a load equal to the weight of a pillar of water of the height AD, because the surrounding water could only support it at its own level. Let us see what change is produced by the affluence of the pressure of the atmosphere. Let the under surface of the piston be at b; when the piston was at f, 33 feet above the surface of the cistern, the water was raised to that height by the pressure of the atmosphere. Suppose a partition made at b by a thin plate, and all the water above it taken away. Now pierce a hole in this plate. The pressure of the atmosphere was able to carry the whole column fa. Part of this column is now removed, and the remainder is not a balance for the air's pressure. This will therefore cause the water to spout up through this hole and rise to f. Therefore the under surface of this plate is pressed up by the contiguous water with a force equal to the weight of that pillar of water which it formerly supported; that is, with a force equal to the weight of the pillar fb. Now the under surface of the piston, when at b, is in the same situation. It is pressed upwards by the water below it, with a force equal to the weight of the column fb. But it is pressed downwards by the whole pressure of the atmosphere, which presses on all bodies; that is, with the weight of the pillar fa. On the whole, therefore, it is pressed downwards by a force equal to the difference of the weights of the pillars fa and fb; that is, by a force equal to the weight of the pillar ba.

It may be conceived better perhaps in this way. When the piston was under the surface of the water in the cistern, it was equally pressed on both sides, both by the water and atmosphere. The atmosphere exerted its pressure on it by the intervention of the water; which being, to all senses, a perfect fluid, propagates every external pressure undiminished. When the piston is drawn up above the surface of the pit-water, the atmosphere continues to press on its upper surface with its whole weight, through the intervention of the water which lies above it; and its pressure must therefore be added to that of the incumbent water. It also continues to press on the under surface of the piston by the intervention of the water; that is, it presses this water to the piston. But, in doing this, it carries the weight of this water which it is pressing on the piston. The pressure on the piston therefore is only the excess of the whole pressure of the atmosphere above the weight of the column of water which it is supporting. Therefore the difference of atmospheric pressure on the upper and under surfaces of the piston is precisely equal to the weight of the column of water supported in the pipe by the air. It is not, however, the individual weight of this column that loads the piston; it is the part of the pressure of the atmosphere on its upper surface, which is not balanced by its pressure on the under surface.

In attempting, therefore, to draw up the piston, we have to surmount this unbalanced part of the pressure of the atmosphere, and also the weight of the water which lies above the piston, and must be lifted by it: and thus the whole opposing pressure is the same as before, namely, the weight of the whole vertical pillar reaching from the surface of the water in the cistern to the place of delivery. Part of this weight is immediately carried by the pressure of the atmosphere; but, in lieu of it, there is an equal part of this pressure of the atmosphere abstracted from the under surface of the piston, while its upper surface sustains its whole pressure.

So far, then, these two states of the pump agree. But they differ exceedingly in their mode of operation; and there are some circumstances not very obvious which must be attended to, in order that the pump may deliver any water at the spout D. This requires, therefore, a serious examination.

Let the fixed valve G (fig. 4.) be supposed at the surface of the cistern water. Let Mm be the lowest, and Nn the highest, positions of the piston, and let HA=h be the height of a column of water equiponderant with the atmosphere.

When the pump is filled, not with water, but with air, and the piston is in its lowest position, and all in equilibrium, the internal air has the same density and elasticity with the external. The space MA am, therefore, contains air of the common density and elasticity. These may be measured by h, or the weight of a column of water whose height is h. Now, let the piston be drawn up to N n. The air which occupied the space MA am now occupies the space NA an, and its density is now \(\frac{MA am}{NA an}\). Its elasticity is now diminished, being proportional to its density (see Pneumatics), and no longer balances the pressure of the atmosphere. The valve G will therefore be forced up by the water, which will rise to some height SA. Now let the piston again descend to M m. It cannot do this with its valve shut; for when it comes down so far as to reduce the air again to its common density, it is not yet at M, because the space below it has been diminished by the water which got into the pipe, and is retained there by the valve G. The piston valve, therefore, opens by the air which we thus attempt to compress, and the superfluous air escapes. When the piston has got to M, the air is again of the common density, and occupies the space MS sm. Now draw the piston up to N. The air will expand into the space NS sn, and its density will be reduced to \(\frac{MS sm}{NS sn}\), and its elasticity will no longer balance the pressure of the atmosphere, and more water will enter, and it will rise higher. This will go on continually. But it may happen that the water will never rise so high as to reach the piston, even though not 33 feet above the water in the cistern: For the successive diminutions of density and elasticity are a series of quantities that decrease geometrically, and therefore will have a limit. Let us see what determines this limit.

At whatever height the water stands in the lower part of the pipe, the weight of the column of water SA as, together with the remaining elasticity of the air above it, exactly balances the pressure of the atmosphere (see Pneumatics, No 108.). Now the elasticity of the air in the space NS sn is equal to \(h \times \frac{MS sm}{NS sn}\). Therefore, in the case where the limit obtains, and the water rises no farther, we must have \(h = AS + h \times \frac{MS sm}{NS sn}\), or, because the column is of the same diameter throughout, \(h = AS + h \times \frac{MS}{NS}\) and \(\frac{MS}{NS} = h - AS = HS\), and NS : MS = HA : HS, and NS - MS : NS = HA - HS : HA, or NS : NS = AS : AH, and NM \(\times\) AH = NS \(\times\) AS. Therefore, if AN, the distance of the piston in its highest position from the water in the cistern, and NM the length of its stroke, be given, there is a certain determined height AS to which the water can be raised by the pressure of the air: For AH is a constant quantity; and therefore when MN is given, the rectangle AS \(\times\) SN is given. If this height AS be less than that of the piston in its lowest position, the pump will raise no water, although AN may be less than AH. Yet the same pump will raise water very effectually, if it be first of all filled with water; and we have seen professional engineers much puzzled by this capricious failure of their pumps. A little knowledge of the principles would have prevented their disappointment.

To insure the delivery of water by the pump, the stroke must be such that the rectangle MN \(\times\) AH may be greater than any rectangle that can be made of the parts of AN, that is, greater than the square of half AN. Or, if the length of the stroke be already fixed during the other circumstances, which is a common case, we deliver must make AN so short that the square of its half, measured in feet, shall be less than 33 times the stroke of the piston.

Suppose that the fixed valve, instead of being at the surface of the water in the cistern, is at S, or anywhere between S and A, the performance of the pump will be the same as before: But if it be placed anywhere above S, it will be very different. Let it be at T. It is plain that when the piston is pushed down from N to M, the valve at T prevents any air from getting down; and therefore, when the piston is drawn up again, the air contained in the space MT tm will expand into the space NT tn, and its density will be \(\frac{MT}{NT}\). This is less than \(\frac{MS}{NS}\), which expresses the density of the air which was left in the space TS st by the former operations.—The air, therefore, in TS st will also expand, will open the valve, and now the water will rise above S. The proportion of NS to NT may be evidently such that the water will even get above the valve T. This diminishes the space NT tn; and therefore, when the piston has been pushed down to M, and again drawn up to N, the air will be still more rarefied, and the water will rise still higher. The foregoing reasoning, however, is sufficient to show that there may still be a height which the water will not pass, and that this height depends on the proportion between the stroke of the piston and its distance from the water in the cistern. We need not give the determination, because it will come in afterwards in combination with other circumstances. It is enough that the reader sees the physical causes of this limitation: And, lastly, we see plainly that the utmost security will be given for the performance of the pump, when the fixed valve is so placed that the piston, when in its lowest position, shall come into contact with it. In this case, the rarefaction of the air will be the complete easily kept possible; and, if there were no space left between the air-tight piston and valve, and all were perfectly air-tight, the rarefaction would be complete, and the valve might be anything less than 33 feet from the surface of the water in the cistern.

But this perfect contact and tightness is unattainable; and though the pump may be full of water, its continual downward pressure causes it to filtrate slowly through every crevice, and the air enters through every pore, and even disengages itself from the water, with which a considerable portion had been chemically combined. The pump by this means loses water, and it requires several strokes of brisk working to fill it again: and if the leathers have become dry, so much admission may be given to the air, that the pump will not fill itself with water by any working. It is then necessary to pour water into it, which flushes up these passages, and soon sets all to rights again. For these reasons, it is always prudent to place the fixed valve as low as other circumstances will permit, and to make the piston rod of such a length, that when it is at the bottom of its stroke it shall be almost in contact with the valve. When we are not limited by other circumstances, it is evident that the best possible form is to have both the piston and the fixed valve under the surface of the water of the cistern. In this situation they are always wet and air-tight. The chief objection is, that by this disposition they are not easily come at when needing repair. This is a material objection in deep mines. In such situations, therefore, we must make the best compensation of different circumstances that we can. It is usual to place the fixed valve at a moderate distance from the surface of the water, and to have a hole in the side of the pipe, by which it may be got out. This is carefully fitted up by a plate firmly screwed on, with leather or cement between the parts. This is called the clack door. It would, in every case, be very proper to have a fixed valve in the lower end of the pipe. This would combine all advantages. Being always tight, the pipe would retain the water, and it would leave to the valve above it its full effect of increasing the rarefaction. A similar hole is made in the working barrel, a little above the highest position of the piston. When this needs repair, it can be got at through this hole, without the immense trouble of drawing up the whole rods.

Thus we have conducted the reader step by step, from the simplest form of the pump to that which long experience has at last selected as the most generally convenient. This we shall now describe in some detail.

The Sucking Pump consists of two pipes DCCD, BAAB (fig. 5); of which the former is called the Barrel, or the Working Barrel, and the other is called the Suction-pipe, and is commonly of a smaller diameter. These are joined by means of flanges E, F, pierced with holes to receive forewed bolts. A ring of leather, or of lead, covered with a proper cement, is put between them; which, being strongly compressed by the screw-bolts, renders the joint perfectly air-tight. The lower end A of the suction-pipe is commonly spread out a little to facilitate the entry of the water, and frequently has a grating across at AA to keep out filth or gravel. This is immersed in the standing water YZ. The working barrel is cylindrical, as evenly and smoothly bored as possible, that the piston may fill it exactly through its whole length, and move along it with as little friction as may be consistent with air-tightness.

The piston is a sort of truncated cone OPKL, generally made of wood not made to slip, such as elm or beech. The small end of it is cut off at the sides, so as to form a sort of arch OQP, by which it is fastened to the iron rod or spear. It is exhibited in different positions in figures 6, 7, which will give a more distinct notion of it than any description. The two ends of the conical part may be hooped with brass. This cone has its larger end surrounded with a ring or band of strong leather fastened with nails, or by a copper hoop, which is driven on it at the smaller end. This band should reach to some distance beyond the base of the cone; the farther the better; and the whole must be of uniform thickness all round, so as to suffer equal compression between the cone and the working barrel.

This would occasion bumps or inequalities, which would spoil its tightness; and no harm can result from the want of it, because the two edges will be squeezed close together by the compression in the barrel. It is by no means necessary that this compression be great. This is a very detrimental error of the pump-makers. It occasions enormous friction, and destroys the very purpose which they have in view, viz., rendering the piston air-tight; for it causes the leather to wear through very soon at the edge of the cone, and it also wears the working barrel. This very soon becomes wide in that part which is continually passed over by the piston, while the mouth remains of its original diameter, and it becomes impossible to thrust in a piston which shall completely fill the worn part. Now, a very moderate pressure is sufficient for rendering the pump perfectly tight, made of a piece of glove leather would be sufficient for this purpose, if loose or detached from the solid cone; for tight. Suppose such a loose and flexible, but impervious, band of leather put round the piston, and put into the barrel; and let it even be supposed that the cone does not compress it in the smallest degree to its internal surface. Pour a little water carefully into the inside of this sort of cup or dish; it will cause it to swell out a little, and apply itself close to the barrel all round, and even adjust itself to all its inequalities. Let us suppose it to touch the barrel in a ring of an inch broad all round. We can easily compute the force with which it is pressed. It is half the weight of a ring of water an inch deep and an inch broad. This is a trifle, and the friction occasioned by it not worth regarding; yet this trifling pressure is sufficient to make the passage perfectly impervious, even by the most enormous pressure of a high column of incumbent water: for let this pressure be ever so great, the pressure by which the leather adheres to the barrel always exceeds it, because the incumbent fluid has no preponderating power by which it can force its way between them, and it must insinuate itself precisely so far, that its pressure on the inside of the leather shall still exceed, and only exceed, the pressure by which it endeavours to infill itself; and thus the piston becomes perfectly tight with the smallest possible friction. This reasoning is perhaps too refined for the uninitiated artist, and probably will not persuade him. To such we would recommend an examination of the pistons and valves contrived and executed by that best practising artist, whose skill far surpasses our highest conceptions, the human all-wise Creator of this world. The valves which shut up the passages of the veins, and this in places where an extravasation would be followed by instant death, are cups of thin membrane which adhere to the sides of the channel about half way round, and are detached in the rest of their circumference. When the blood comes in the opposite direction, it pushes the membrane aside, and has a passage perfectly free. But a stagnation of motion allows the tone of the muscular (perhaps) membrane, to restore it to its natural shape, and the least motion in the opposite direction causes it instantly to clap close to the sides of the vein, and then no pressure whatever can force a passage. We shall recur to this again, when describing the various contrivances of valves, &c. What we have said is enough for supporting our directions for constructing a tight piston of a piston; but we recommended thick and strong leather, while recommending seems to render thin leather preferable. If the leather be thin, and the solid piston in any part does not press it gently to the barrel, there will be in this part an unbalanced pressure of the incumbent column of water, which would instantly burst even Pump.

A strong leather bag; but when the solid piston, covered with leather, exactly fills the barrel, and is even pressed a little to it, there is no such risk; and now that part of the leather band which reaches beyond the solid piston performs its office in the completest manner. We do not hesitate, therefore, to recommend this form of a piston, which is the most common and simple of all, as preferable, when well executed, to any of those more artificial, and frequently very ingenious, contrivances, which we have met with in the works of the first engineers. To proceed, then, with our description of the fucking-pump.

At the joining of the working barrel with the suction-pipe there is a hole H, covered with a valve opening upwards. This hole H is either made in a plate which makes a part of the suction-pipe, being cast along with it, or it is made in a separate plate. This last is the most convenient, being easily removed and replaced.

Different views are given of this valve in figs. 8, 9, 10. The diameter EF (fig. 10.) of this plate is the same with that of the flanges, and it has holes corresponding to them, through which their bolts pass which keep all together. A ring of thick leather NKL is applied to this plate, having a part cut out between N and L, to make room for another piece of strong leather NR (fig. 9.) which composes the valve. The circular part of this valve is broader than the hole in the middle of fig. 10. but not quite so broad as to fill up the inside of the ring of leather OQP of this fig. which is the same with GKI of fig. 10. The middle of this leather valve is strengthened by two brafs (not iron) plates, the uppermost of which is seen at R of fig. 9.: the one on its underside is a little smaller than the hole in the valve-plate, that it may go freely in; and the upper plate R is larger than this hole, that it may compress the leather to its brim all round. It is evident, that when this plate with its leathers is put between the joint flanges, and all is screwed together, the tail of leather N of fig. 9. will be compressed between the plates, and form a hinge, on which the valve can turn, rising and falling. There is a similar valve fastened to the upper side, or broadest base of the piston. This description serves for both valves, and in general for most valves which are to be found in any parts of a pump.

The reader will now understand, without any repetition, the process of the whole operation of a fucking-pump. The piston rarifies the air in the working barrel, and that in the suction-pipe expands through the valve into the barrel; and, being no longer a balance for the atmospheric pressure, the water rises into the suction-pipe; another stroke of the piston produces a similar effect, and the water rises farther, but by a smaller step than by the preceding stroke: by repeating the strokes of the piston, the water gets into the barrel; and when the piston is now pushed down through it, it gets above the piston, and must now be lifted up to any height. The suction-pipe is commonly of smaller size than the working barrel, for the sake of economy. It is not necessary that it be so wide; but it may be, and often is, made too small. It should be of such a size, that the pressure of the atmosphere may be able to fill the barrel with water as fast as the piston rises. If a void is left below the piston, it is evident that the piston must be carrying the whole weight of the atmosphere, besides the water which is lying above it. Nay, if the pipe be only so wide, that the barrels shall fill precisely as fast as the piston rises, it must sustain all this pressure. The suction-pipe should be wider than this, that all the pressure of the atmosphere which exceeds the weight of the pillar in the suction-pipe may be employed in pressing it on the under surface of the piston, and thus diminish the load. It cannot be made too wide; and too strict an economy in this respect may very sensibly diminish the performance of the pump, and more than defeat its own purpose. This is most likely when the suction-pipe is long, because there the length of the pillar of water nearly balances the air's pressure, and leaves very little accelerating force; so that water will rise but slowly even in the widest pipe. All these things will be made the subjects of computation afterwards.

It is plain that there will be limitations to the rise of the water in the suction-pipe, similar to what we found when the whole pump was an uniform cylinder. Let a be the height of the fixed valve above the water in the cistern: let B and b be the spaces in cubic measure between this valve and the piston in its highest and lowest positions, and therefore express the bulk of the air which may occupy these spaces: let y be the distance between the fixed valve and the water in the suction-pipe, when it has attained its greatest height by the rarefaction of the air above it: let h be the height of a column of water in equilibrium, with the whole pressure of the atmosphere, and therefore having its weight in equilibrium with the elasticity of common air; and let x be the height of the column whose weight balances the elasticity of the air in the suction-pipe, when rarified as much as it can be by the action of the piston, the water standing at the height a—y.

Then, because this elasticity, together with the column a—y in the suction-pipe, must balance the whole pressure of the atmosphere, (see Pneumatics, No. 108.), we must have \( h = x + a - y \), and \( y = a + x - h \).

When the piston was in its lowest position, the bulk of the air between it and the fixed valve was b. Suppose the valve kept shut, and the piston raised to its highest position, the bulk will be B, and its density \( \frac{b}{B} \), and its elasticity, or the height of the column whose weight will balance it, will be \( \frac{h}{B} \). If the air in the suction-pipe be denser than this, and consequently more elastic, it will lift the valve, and some will come in; therefore, when the pump has rarified the air as much as it can, so that none does, in fact, come in, the elasticity of the air in the suction-pipe must be the same.

Therefore \( x = \frac{h}{B} \).

We had \( y = a + x - h \). Therefore \( y = a + \frac{h}{B} \).

\( h = a + \frac{b}{B} \).

Therefore when \( \frac{B - b}{B} \) is less than \( a \), the water will stop before it reaches the fixed valve. But when \( a \) is less than \( \frac{B - b}{B} \), the water will get above the fixed valve, \( y \) becoming negative. But it does not follow that the water will reach the piston, that is, will rise so high that the piston will pass through it in its descent. Things now come into the condition of a pump of uniform dimensions from top to bottom; and this point will be determined by what was said when treating of such a pump.

There is another form of the fucking-pump which is much used in great water works, and is of equal efficiency with the one now described. It is indeed the same pump in an inverted position. It is represented in fig. 11, where ABCD is the working barrel, immersed, with its mouth downwards, in the water of the cistern. It is joined by means of flanges to the rising pipe or main.

This usually consists of two parts. The first, BEFC, is bent to one side, that it may give room for the iron frame TXYV, which carries the rod NO of the piston M, attached to the traverses RS, TOV of this frame. The other part, EGHF, is usually of a less diameter, and is continued to the place of delivery. The piston frame XIVY hangs by the rod Z, at the arm of a lever or working beam, not brought into the figure. The piston is perforated like the former, and is surrounded like it with a band of leather in form of a taper-dish. It has a valve K on its broad or upper base, opening when pressed from below. The upper end of the working barrel is pierced with a hole, covered with a valve I, also opening upwards.

Now suppose this apparatus immersed into the cistern till the water is above it, as marked by the line 2, 3, and the piston drawn up till it touch the end of the barrel. When the piston is allowed to descend by its own weight, the water rises up through its valve K, and fills the barrel. If the piston be now drawn up by the moving power of the machinery with which it is connected, the valve K shuts, and the piston pushes the water before it through the valve I into the main pipe EFGH. When the piston is again let down, the valve I shuts by its own weight and the pressure of the water incumbent on it, and the barrel is again filled by the water of the cistern. Drawing up the piston pushes this water into the main pipe, &c., and then the water is at length delivered at the place required.

This pump is usually called the lifting pump; perhaps the simplest of all in its principle and operation.—It needs no farther explanation: and we proceed to describe

The Forcing Pump, represented in fig. 12. It consists of a working barrel ABCD, a suction-pipe CDEF, and a main or rising pipe. This last is usually in three joints. The first GHKI may be considered as making part of the working barrel, and is commonly cast in one piece with it. The second IKLM is joined to it by flanges, and forms the elbow which this pipe must generally have. The third LNOM is properly the beginning of the main, and is continued to the place of delivery. At the joint IK there is a hanging valve or clack S; and there is a valve R on the top of the suction-pipe.

The piston PQTV is solid, and is fastened to a stout iron rod which goes through it, and is fixed by a key drawn through its end. The body of the piston is a sort of double cone, widening from the middle to each end, and is covered with two bands of very strong leather, fitted to it in the manner already described.

The operation of this pump is abundantly simple. When the piston is thrust into the pump, it pushes the air before it through the valve S, for the valve R remains shut by its own weight. When it has reached near the bottom, and is drawn up again, the air which filled the small space between the piston and the valve S now expands into the barrel; for as soon as the air begins to expand, it ceases to balance the pressure of the atmosphere, which therefore shuts the valve S. By the expansion of the air in the barrel the equilibrium at the valve R is destroyed, and the air in the suction-pipe lifts the valve, and expands into the barrel; consequently it ceases to be a balance for the pressure of the atmosphere, and the water is forced into the suction-pipe. Pushing the piston down again forces the air in the barrel through the valve S, the valve R in the mean time shutting. When the piston is again drawn up, S shuts, R opens, the air in the suction-pipe dilates anew, and the water rises higher in it. Repeating these operations, the water gets at last into the working barrel, and is forced into the main by pushing down the piston, and is pushed along to the place of delivery.

The operation of this pump is therefore two-fold, forcing and sucking. In the first operation, the same force must be employed as in the fucking-pump, namely, a force equal to the weight of a column of water having the section of the piston for its base, and the height of the piston above the water in the cistern for its height. It is for the sake of this part of the operation that the upper cone is added to the piston. The air and water would pass by the sides of the lower cone while the piston is drawn up; but the leather of the upper cone applies to the surface of the barrel, and prevents this. The space contained between the barrel and the valve S is a great obstruction to this part of the operation, because this air cannot be rarefied to a very great degree. For this reason, the suction-pipe of a forcing-pump must not be made long. It is not indeed necessary; for by placing the pump a few feet lower, the water will rise into it without difficulty, and the labour of suction is as much diminished as that of impulsion is increased. However, an intelligent artist will always endeavour to make this space between the valve S and the lowest place of the piston as small as possible.

The power employed in forcing must evidently surmount the pressure of the whole water in the rising pipe, and (independent of what is necessary for giving the water the required velocity, so that the proper quantity per hour may be delivered), the piston has to withstand a force equal to the weight of a column of water having the section of the piston for its base, and the perpendicular altitude of the place of delivery above the lower surface of the piston for its height. It is quite indifferent in this respect what is the diameter of the rising pipe; because the pressure on the piston depends on the altitude of the water only, independent of its quantity. We shall even see that a small rising pipe will require a greater force to convey the water along it to any given height or distance.

When we would employ a pump to raise water in a crooked pipe, or in any pipe of moderate dimensions, this form of pump, or something equivalent, must be used. In bringing up great quantities of water from mines, the common fucking-pump is generally employed. ed, as really the best of them all; but it is the most expensive, because it requires the pipe to be perpendicular, straight, and of great dimensions, that it may contain the piston rods. But this is impracticable when the pipe is crooked.

If the forcing pump, constructed in the manner now described, be employed, we cannot use forces with long rods. These would bend when pushed down by their further extremity. In this case, it is usual to employ only a short and stiff rod, and to hang it by a chain, and load it with a weight superior to the weight of water to be raised by it. The machinery therefore is employed, not in forcing the water along the rising-pipe, but in raising the weight which is to produce this effect by its subsequent descent.

In this case, it would be much better to employ the lifting-pump of fig. 11. For as the load on the forces must be greater than the resistances which it must surmount, the force exerted by the machine must in like manner be greater than this load. This double excess would be avoided by using the lifting-pump.

It will readily occur to the reader that the quantity of water delivered by any pump will be in the joint proportion of the surface or base of the piston and its velocity: for this measures the capacity of that part of the working barrel which the piston passes over. The velocity of the water in the conduit pipe, and in its passage through every valve, will be greater or less than the velocity of the piston, in the same proportion that the area of the piston or working barrel is greater or less than the area of the conduit or valve. For whatever quantity of water passes through any section of the working barrel in a second, the same quantity must go through any one of these passages. This enables us to modify the velocity of the water as we please: we can increase it to any degree at the place of delivery by diminishing the aperture through which it passes, provided we apply sufficient force to the piston.

It is evident that the operation of a pump is by starts, and that the water in the main remains at rest, pressing on the valve during the time that the piston is withdrawn from the bottom of the working barrel. It is in most cases desirable to have this motion equable, and in some cases it is absolutely necessary. Thus, in the engine for extinguishing fires, the spout of water going by jerks could never be directed with a certain aim, and half of the water would be lost by the way; because a body at rest cannot in an instant be put in rapid motion, and the first portion of every jerk of water would have but a small velocity. A very ingenious contrivance has been fallen upon for obviating this inconvenience, and procuring a stream nearly equable. We have not been able to discover the author. At any convenient part of the rising-pipe beyond the valve S there is annexed a spacious vessel VZ (fig. 13. No. 1. and 2.) close a-top, and of great strength. When the water is forced along this pipe, part of it gets into this vessel, keeping the air confined above it, and it fills it to such a height V, that the elasticity of the confined air balances a column reaching to T, we shall suppose, in the rising-pipe. The next stroke of the piston sends forward more water, which would fill the rising-pipe to some height above T. But the pressure of this additional column causes some more of it to go into the air vessel, and compresses its air so much more that its elasticity now balances a longer column. Every succeeding stroke of the piston produces a like effect. The water rises higher in the main pipe, but some more of it goes into the air vessel. At last the water appears at the place of delivery; and the air in the air vessel is now so much compressed that its elasticity balances the pressure of the whole column. The next stroke of the piston sends forward some more water. If the diameter of the orifice of the main be sufficient to let the water flow out with a velocity equal to that of the piston, it will so flow out, rising no higher, and producing no sensible addition to the compression in the air vessel. But if the orifice of the main be contracted to half its dimensions, the water sent forward by the piston cannot flow out in the time of the stroke without a greater velocity, and therefore a greater force. Part of it, therefore, goes into the air vessel, and increases the compression. When the piston has ended its stroke, and no more water comes forward, the compression of the air in the air vessel being greater than what was sufficient to balance the pressure of the water in the main pipe, now forces out some of the water which is lying below it. This cannot return towards the pump, because the valve S is now shut. It therefore goes forward along the main, and produces an efflux during the time of the piston's rising in order to make another stroke. In order that this efflux may be very equable, the air vessel must be very large. If it be small, the quantity of water that is discharged by it during the return of the piston makes so great a portion of its capacity, that the elasticity of the confined air is too much diminished by this enlargement of its bulk, and the rate of efflux must diminish accordingly. The capacity of the air vessel should be so great that the change of bulk of the compressed air during the action of the piston may be inconsiderable. It must therefore be very strong.

It is pretty indifferent in what way this air vessel is connected with the rising-pipe. It may join it laterally, as in fig. 13. No. 1. and the main pipe go on without interruption; or it may be made to surround an interruption of the main pipe, as in fig. 13. No. 2. It may also be in any part of the main pipe. If the sole effect intended by it is to produce an equable jet, as in ornamental water-works, it may be near the end of the main. This will require much less strength, because there remains but a short column of water to compress the air in it. But it is, on the whole, more advantageous to place it as near the pump as possible, that it may produce an equable motion in the whole main pipe. This is of considerable advantage: when a column of water several hundred feet long is at rest in the main pipe, and the piston at one end of it put at once into motion, even with a moderate velocity, the strain on the pipe would be very great. Indeed if it were possible to put the piston instantaneously into motion with a finite velocity, the strain on the pipe, tending to burst it, would be next to infinite. But this seems impossible in the nature; all changes of motion which we observe are gradual, because all impelling bodies have some elasticity of or softness by which they yield to compression. And, in the way in which pistons are commonly moved, viz. by cranks, or something analogous to them, the motion is very sensibly gradual. But still the air vessel tends to make the motion along the main pipe less deftly, and therefore diminishes those strains which would really take place. place in the main-pipe. It acts like the springs of a travelling-carriage, whose jolts are incomparably less than those of a cart; and by this means really enables a given force to propel a greater quantity of water in the same time.

We may here by the way observe, that the attempts of mechanicians to correct this unequal motion of the piston-rod are misplaced, and if it could be done, would greatly hurt a pump. One of the best methods of producing this effect is to make the piston-rod consist of two parallel bars, having teeth in the sides which front each other. Let a toothed wheel be placed between them, having only the half of its circumference furnished with teeth. It is evident, without any farther description, that if this wheel be turned uniformly round its axis, the piston-rod will be moved uniformly up and down without intermission. This has often been put in practice; but the machine always went by jolts, and seldom lasted a few days. Unskilled mechanicians attributed this defect in the execution: but the fault is essential, and lies in the principle.

The machine could not perform one stroke, if the first mover did not slacken a little, or the different parts of the machine did not yield by bending or by compression; and no strength of materials could withstand the violence of the strains at every reciprocation of the motion. This is chiefly experienced in great works which are put in motion by a water-wheel, or some other equal power exerted on the mass of matter of which the machine consists. The water-wheel being of great weight, moves with considerable steadiness or uniformity; and when an additional resistance is opposed to it by the beginning of a new stroke of the piston, its great quantity of motion is but little affected by this addition, and it proceeds very little retarded; and the machine must either yield a little by bending and compression, or go to pieces, which is the common event. Cranks are free from this inconvenience, because they accelerate the piston gradually, and bring it gradually to rest, while the water-wheel moves round with almost perfect uniformity. The only inconvenience (and it may be considerable) attending this slow motion of the piston at the beginning of its stroke is, that the valves do not shut with rapidity, so that some water gets back through them. But when they are properly formed and loaded, this is but trifling.

We must not imagine, that because the stream produced by the assistance of an air-vesSEL is almost perfectly equable, and because as much water runs out during the returning of the piston as during its active stroke, it therefore doubles the quantity of water. No more water can run out than what is sent forward by the piston during its effective stroke. The continued stream is produced only by preventing the whole of this water from being discharged during this time, and by providing a propelling force to act during the piston's return. Nor does it enable the moving force of the piston to produce a double effect: for the compression which is produced in the air-vesSEL, more than what is necessary for merely balancing the quiescent column of water, reacts on the piston, resisting its compression just as much as the column of water would do which produces a velocity equal to that of the efflux. Thus if the water is made to spout with the velocity of eight feet per second, this would require an additional column of one foot high, and this would just balance the compression in the air-vesSEL, which maintains this velocity during the non-action of the piston. It is, however, a matter of fact, that a pump furnished with an air-vesSEL delivers a little more water than it would do without it. But the difference depends on the combination of many very dissimilar circumstances, which it is extremely difficult to bring into calculation. Some of these will be mentioned afterwards.

To describe, or even to enumerate, the immense variety of combinations of these three simple pumps would fill a volume. We shall select a few, which are more deserving of notice.

I. The common sucking-pump may, by a small addition, be converted into a lifting-pump, fitted for propelling the water to any distance, and with any velocity.

Fig. 14. is a sucking pump, whose working-barrel ACDB has a lateral pipe AEGHF connected with it close to the top. This terminates in a main or rising pipe IK, furnished or not with a valve L. The top of the barrel is shut up by a strong plate MN, having a hollow neck terminating in a small flanch. The piston rod QR passes through this neck, and is nicely turned and polished. A number of rings of leather are put over the rod, and strongly compressed round it by another flanch and several screwed bolts, as is represented at OP. By this contrivance the rod is closely grasped by the leathers, but may be easily drawn up and down, while all passage of air or water is effectually prevented.

The piston S is perforated, and furnished with a valve opening upwards. There is also a valve T on the top of the suction-pipe YX; and it will be of advantage, though not absolutely necessary, to put a valve L at the bottom of the rising pipe. Now suppose the piston at the bottom of the working-barrel. When it is drawn up, it tends to compress the air above it, because the valve in the piston remains shut by its own weight. The air therefore is driven through the valve L into the rising pipe, and escapes. In the mean time, the air which occupied the small space between the piston and the valve T expands into the upper part of the working-barrel; and its elasticity is so much diminished thereby, that the atmosphere presses the water of the cistern into the suction-pipe, where it will rise till an equilibrium is again produced. The next downward stroke of the piston allows the air, which had come from the suction-pipe into the barrel during the ascent of the piston, to get through its valve. Upon drawing up the piston, this air is also drawn off through the rising pipe. Repeating this process brings the water at last into the working-barrel, and it is then driven along the rising-pipe by the piston.

This is one of the best forms of a pump. The reaction may be very perfect, because the piston can be brought so near to the bottom of the working-barrel; and, for forcing water in opposition to great pressures, it appears preferable to the common forcing-pump; because in that the piston rods are compressed and exposed to bending, which greatly hurts the pump by wearing the piston and barrel on one side. This soon renders it less tight, and much water squirts out by the sides of the piston. But in this pump the piston rod is always drawn or pulled, which keeps it straight. and rods exert a much greater force in opposition to a pull than in opposition to compression. The collar of leather round the piston-rods is found by experience to need very little repairs, and is very impervious to water. The whole is very accessible for repairs; and in this respect much preferable to the common pump in deep mines, where every fault of the piston obliges us to draw up some hundred feet of piston-rods. By this addition, too, any common pump for the service of a house is converted into an engine for extinguishing fire, or may be made to convey the water to every part of the house; and this without hurting or obstructing its common uses. All that is necessary is to have a large cock on the upper part of the working barrel opposite to the lateral pipe in this figure. This cock serves for a spout when the pump is used for common purposes; and the merely shutting this cock converts the whole into an engine for extinguishing fire or for supplying distant places with water. It is scarcely necessary to add, that for these services it will be proper to connect an air-vessel with some convenient part of the rising pipe, in order that the current of the water may be continual.

We have frequently spoken of the advantages of a continued current in the main pipe. In all great works a considerable degree of uniformity is produced by the manner of disposing the actions of the different pumps; for it is very rarely that a machine works but one pump. In order to maintain some uniformity in the resistance, that it may not all be opposed at once to the moving power, with intervals of total inaction, which would produce a very hobbling motion, it is usual to distribute the work into portions, which succeed alternately; and thus both diminish the strain, and give greater uniformity of action, and frequently enable a natural power which we can command, to perform a piece of work, which would be impossible if the whole resistance were opposed at once. In all pump machines therefore, we are obviously directed to construct them so that they may give motion to at least two pumps, which work alternately. By this means a much greater uniformity of current is produced in the main pipe. It will be rendered still more uniform if four are employed, succeeding each other at the interval of one quarter of the time of a complete stroke.

But ingenious men have attempted the same thing with a single pump, and many different constructions for this purpose have been proposed and executed. The thing is not of much importance, or of great research. We shall content ourselves therefore with the description of one that appears to us the most perfect, both in respect of simplicity and effect.

II. It consists of a working-barrel AB (fig. 15.) close at both ends. The piston C is solid, and the rod OP passes through a collar of leathers in the plate, which closes the upper end of the working-barrel. This barrel communicates laterally with two pipes H, K; the communications m and n being as near to the top and bottom of the barrel as possible. Adjoining to the passage m are two valves F and G opening upwards. Similar valves accompany the passage n. The two pipes H and K unite in a larger rising pipe L. They are all represented as in the same plane; but the upper ends must be bent backwards, to give room for the motion of the piston-rod OP.

Suppose the piston close to the entry of the lateral pipe n, and that it is drawn up: it compresses the air above it, and drives it through the valve G, where it escapes along the rising pipe; at the same time it rarefies the air in the space below it. Therefore the weight of the atmosphere shuts the valve E, and causes the water of the cistern to rise through the valve D, and fill the lower part of the pump. When the piston is pushed down again, this water is first driven through the valve E, because D immediately shuts; and then most of the air which was in this part of the pump at the beginning goes up through it, some of the water coming back in its stead. In the mean time, the air which remained in the upper part of the pump after the ascent of the piston is rarefied by its descent; because the valve G shuts as soon as the piston begins to descend, the valve F opens, the air in this suction pipe F expands into the barrel, and the water rises into the pipes by the pressure of the atmosphere. The next rise of the piston must bring more water into the lower part of the barrel, and must drive a little more air through the valve G, namely, part of that which had come out of the suction-pipe F; and the next descent of the piston must drive more water into the rising pipe H, and along with it most if not all of the air which remained below the piston, and must rarely fill more the air remaining above the piston; and more water will come in through the pipe F, and get into the barrel. It is evident, that a few repetitions will at last fill the barrel on both sides of the piston with water. When this is accomplished, there is no difficulty in perceiving how, at every rise of the piston, the water of the cistern will come in by the valve D, and the water in the upper part of the barrel will be driven through the valve G; and, in every descent of the piston, the water of the cistern will come into the barrel by the valve F, and the water below the piston will be driven through the valve E; and thus there will be a continual influx into the barrel through the valves D and F, and a continual discharge along the rising pipe L through the valves E and G.

This machine is, to be sure, equivalent to two forcing pumps, although it has but one barrel and one piston; lent to two but it has no sort of superiority. It is not even more forcing-economical in most cases; because we apprehend that the additional workmanship will fully compensate for the barrel and piston that is saved. There is indeed a saving in the rest of the machinery, because one lever produces both motions. We cannot therefore say that it is inferior to two pumps; and we acknowledge that there is some ingenuity in the contrivance.

We recommend to our readers the perusal of Belidor's Architecture Hydraulique, where is to be found a commendable variety of combinations and forms of the simple pumps; but we must caution them with respect to his theories, which in this article are extremely defective. Also in Leupold's Theatrum Machinarum Hydraulicae, there is a prodigious variety of all kinds of pumps, many of them very singular and ingenious, and many which have particular advantages, which may suit local circumstances, and give them a preference. But it would be improper to dwell a work of this kind with too many peculiarities; and a person who makes himself master of the principles delivered here in sufficient detail, can be at no loss to suit a pump to his particular views. views, or to judge of the merit of such as may be proposed to him.

We must now take notice of some very considerable and important varieties in the form and contrivance of the essential parts of a pump.

III. The forcing pump is sometimes of a very different form from that already described. Instead of a piston, which applies itself to the inside of the barrel, and slides up and down in it, there is a long cylinder POQ (fig. 16.) nicely turned and polished on the outside, and of a diameter somewhat less than the inside of the barrel. This cylinder (called a plunger) slides through a collar of leathers on the top of the working-barrel, and is constructed as follows. The top of the barrel terminates in a flanch ab, pierced with four holes for receiving screw-bolts. There are two rings of metal, cd ef, of the same diameter, and having holes corresponding to those in the flanch. Four rings of soft leather, of the same size, and similarly pierced with holes, are well soaked in a mixture of oil, tallow, and a little rosin. Two of these leather rings are laid on the pump flanch, and one of the metal rings above them. The plunger is then thrust down through them, by which it turns their inner edges downwards. The other two rings are then slipped on at the top of the plunger, and the second metal ring is put over them, and then the whole are slid down to the metal ring. By this the inner edges of the last leather rings are turned upwards. The three metal rings are now forced together by the screwed bolts; and thus the leathern rings are strongly comprimbed between them, and made to grasp the plunger so closely that no pressure can force the water through between. The upper metal ring just allows the plunger to pass through it, but without any play; so that the turned-up edges of the leathern rings do not come up between the plunger and the upper metal ring, but are lodged in a little conical taper, which is given to the inner edge of the upper plate, its hole being wider below than above. It is on this trifling circumstance that the great tightness of the collar depends. To prevent the leathers from shrinking by drought, there is usually a little cistern formed round the head of the pump, and kept full of water. The plunger is either forced down by a rod from a working beam, or by a set of metal-weights laid on it, as is represented in the figure.

It is hardly necessary to be particular in explaining the operation of this pump. When the plunger is at the bottom of the barrel, touching the fixed valve M with its lower extremity, it almost completely fills it. That it may do it completely, there is sometimes a small pipe RSZ branching out from the top of the barrel, and fitted with a cock at S. Water is admitted till the barrel is completely filled, and the cock is then shut. Now when the plunger is drawn up, the valve N in the rising pipe must remain shut by the pressure of the atmosphere, and a void must be made in the barrel. Therefore the valve M on the top of the function-pipe must be opened by the elasticity of the air in this pipe, and the air must expand into the barrel; and being no longer a balance for the atmosphere, the water in the cistern must be forced into the function-pipe, and rise in it to a certain height. When the plunger descends, it must drive the water through the valve N (for the valve M will immediately shut), and along with it most of the air which had come into the barrel. And as this air occupied the upper part of the barrel, part of it will remain when the plunger has reached the bottom; but a stroke or two will expel it all, and then every succeeding stroke of the descending piston will drive the water along the rising pipe, and every ascent of the plunger will be followed by the water from the cistern.

The advantage proposed by this form of piston is that it may be more accurately made and polished than the inside of a working barrel, and it is of much easier repair. Yet we do not find that it is much used, although an invention of the 17th century (we think by Sir Samuel Morland), and much praised by the writers on these subjects.

It is easy to see that the sucking-pump may be varied in the same way. Suppose this plunger to be open pump fitted both at top and bottom, but the bottom filled with a large valve opening upward. When this is pushed to the right bottom of the barrel, the air which it tends to compress lifts the valve (the lateral pipe FIK being taken away and the passage shut up), and escapes through the plunger. When it is drawn up, it makes the same rarefaction as the solid plunger, because the valve at O shuts, and the water will come up from the cistern as in the former case. If the plunger be now thrust down again, the valve M shuts, the valve O is forced open, and the plunger is filled with water. This will be lifted by it during its next ascent; and when it is pushed down again, the water which filled it must now be pushed out, and will flow over its sides into the cistern at the head of the barrel. Instead of making the valve at the bottom of the piston, it may be made at the top; but this disposition is much inferior, because it cannot rarify the air in the barrel one half. This is evident; for the capacity of the barrel and plunger together cannot be twice the capacity of the barrel.

IV. It may be made after a still different form, as another represented in fig. 17. Here the suction-pipe CO form of the comes up through a cistern KMNL deeper or longer than the intended stroke of the piston, and has a valve C at top. The piston, or what acts in lieu of it, is a tube AHGB, open at both ends, and of a diameter somewhat larger than that of the suction-pipe. The interval between them is filled up at HG by a ring or belt of soft leather, which is fastened to the outer tube, and moves up and down with it, sliding along the smoothly polished surface of the suction-pipe with very little friction. There is a valve I on the top of this piston, opening upwards. Water is poured into the outer cistern.

The outer cylinder or piston being drawn up from its the bottom, there is a great rarefaction of the air which mode of was between them, and the atmosphere presses the water up through the suction-pipe to a certain height; for the valve I keeps shut by the pressure of the atmosphere and its own weight. Pushing down the piston causes the air, which had expanded from the suction-pipe into the piston, to escape through the valve I; drawing it up a second time, allows the atmosphere to press more water into the suction-pipe, to fill it, and also part of the piston. When this is pushed down again, the water which had come through the valve C is now forced out through the valve I into the cistern KMNL, and now the whole is full of water. When, therefore, the piston is drawn up, the water follows, and fills it, if not 3 feet above the water in the cistern; and when it is pulled down again, the water which filled the piston is all thrown out into the cistern; and after this it delivers its full contents of water every stroke. The water in the cistern KMNL effectually prevents the entry of any air between the two pipes; so that a very moderate compression of the belt of soft leather at the mouth of the piston cylinder is sufficient to make all perfectly tight.

It might be made differently. The ring of leather might be fastened round the top of the inner cylinder at DE, and slide on the inside of the piston cylinder; but the first form is most easily executed. Mutchenerbroek has given a figure of this pump in his large system of natural philosophy, and speaks very highly of its performance. But we do not see any advantage which it possesses over the common fucking-pump. He indeed says that it is without friction, and makes no mention of the ring of leather between the two cylinders. Such a pump will raise water extremely well to a small height, and it seems to have been a model only which he had examined: But if the suction-pipe is long, it will by no means do without the leather; for on drawing up the piston, the water of the upper cistern will rise between the pipes, and fill the piston, and none will come up through the suction-pipe.

We may take this opportunity of observing, that the many ingenious contrivances of pumps without friction are of little importance in great works; because the friction which is completely sufficient to prevent all escape of water in a well-constructed pump is but a very trifling part of the whole force. In the great pumps which are used in mines, and are worked by a steam-engine, it is very usual to make the pistons and valves without any leather whatever. The working barrel is bored truly cylindrical, and the piston is made of metal of a size that will just pass along it without sticking. When this is drawn up with the velocity competent to a properly loaded machine, the quantity of water which escapes round the piston is insignificant. The piston is made without leathers, not to avoid friction, which is also insignificant in such works; but to avoid the necessity of frequently drawing it up for repairs through such a length of pipes.

VI. If a pump absolutely without friction be wanted, the following seems preferable for simplicity and performance to any we have seen, when made use of in proper situations. Let NO (fig. 18.) be the surface of the water in the pit, and K the place of delivery. The pit must be as deep in water as from K to NO. ABCD is a wooden trunk, round or square, open at both ends, and having a valve P at the bottom. The top of this trunk must be on a level with K, and has a small cistern EADF. It also communicates laterally with a rising pipe GHK, furnished with a valve at H opening upwards. LM is a beam of timber so fitted to the trunk as to fill it without sticking, and is of at least equal length. It hangs by a chain from a working beam, and is loaded on the top with weights exceeding that of the column of water which it displaces. Now suppose this beam allowed to descend from the position in which it is drawn in the figure; the water must rise all around it, in the crevice which is between it and the trunk, and also in the rising pipe; because the valve P shuts, and H opens; so that when the plunger has got to the bottom, the water will stand at the level of K. When the plunger is again drawn up to the top by the action of the moving power, the water sinks again in the trunk, but not in the rising pipe, because it is stopped by the valve H. Then allowing the plunger to descend again, the water must again rise in the trunk to the level of K, and it must now flow out at K; and the quantity discharged will be equal to the part of the beam below the surface of the pit-water, deducting the quantity which fills the small space between the beam and the trunk. This quantity may be reduced almost to nothing; for if the inside of the trunk and the outside of the beam be made tapering, the beam may be let down till they exactly fit; and as this may be done in square work, a good workman can make it exceedingly accurate. But in this case, the lower half of the beam and trunk must not taper; and this part of the trunk must be of sufficient width round the beam to allow free passage into the rising pipe. Or, which is better, the rising pipe must branch off from the bottom of the trunk. A discharge may be made from the cistern EADF, so that as little water as possible may descend along the trunk when the piston is raised.

One great excellence of this pump is, that it is perfectly free from all the deficiencies which in common pumps result from want of being air-tight. Another is, that the quantity of water raised is precisely equal to the power expended; for any want of accuracy in the work, while it occasions a diminution of the quantity of water discharged, makes an equal diminution in the weight which is necessary for pushing down the plunger. We have seen a machine consisting of two such pumps suspended from the arms of a long beam, the upper side of which was formed into a walk with a rail on each side. A man stood on one end till it got to the bottom, and then walked soberly up to the other end, the inclination being about twenty-five degrees at first, but gradually diminished as he went along, and changed the load of the beam. By this means he made the other end go to the bottom, and so on alternately, with the easiest of all exertions, and what we are most fitted for by our structure. With this machine, a very feeble old man, weighing 110 pounds, raised 7 cubic feet of water 11½ feet high in a minute, and continued working 8 or 12 hours every day. A stout young man, weighing nearly 135 pounds, raised 8½ to the same height, and when he carried 30 pounds, conveniently flung about him, he raised 9½ feet to this height, working 10 hours a-day without fatiguing himself. This exceeds Defagulier's maximum of a hoghead of water 10 feet high in a minute, in the proportion of 9 to 7 nearly. It is limited to very moderate heights; but in such situations it is very effectual. It was the contrivance of an untaught labouring man, possessed of uncommon mechanical genius. We shall have occasion to mention, with respect, some other contrivances of the same person, in the article WATER-WORKS.

VI. The most ingenious contrivance of a pump without friction is that of Mr Hafkin's, described by Defagulier, and called by him the Quicksilver Pump. Its construction and mode of operation are pretty complicated; but the following preliminary observations will, we hope, render it abundantly plain. Let \( ilmk \) (fig. 19.) be a cylindrical iron pipe, about six feet long, open at top. Let \( eghf \) be another cylinder, connected with it at the bottom, and of smaller diameter. It may either be solid, or, if hollow, it must be close at top. Let \( acdb \) be a third iron cylinder, of an intermediate diameter, so that it may move up and down between the other two without touching either, but with as little interval as possible. Let this middle cylinder communicate, by means of the pipe \( AB \), with the upright pipe \( FE \), having valves \( C \) and \( D \) (both opening upwards) adjoining to the pipe of communication. Suppose the outer cylinder suspended by chains from the end of a working beam, and let mercury be poured into the interval between the three cylinders till it fills the space to \( op \), about \( \frac{1}{4} \) of their height. Also suppose that the lower end of the pipe \( FE \) is immersed into a cistern of water, and that the valve \( D \) is less than 33 feet above the surface of this water.

Now suppose a perforation made somewhere in the pipe \( AB \), and a communication made with an air-pump. When the air-pump is worked, the air contained in \( CE \), in \( AB \), and in the space between the inner and middle cylinders, is rarefied, and is abstracted by the air-pump; for the valve \( D \) immediately shuts. The pressure of the atmosphere will cause the water to rise in the pipe \( CE \), and will cause the mercury to rise between the inner and middle cylinders, and sink between the outer and middle cylinders. Let us suppose mercury 12 times heavier than water: Then for every foot that the water rises in \( EC \), the level between the outside and inside mercury will vary an inch; and if we suppose \( DE \) to be 30 feet, then if we can rarefy the air so as to raise the water to \( D \), the outside mercury will be depressed to \( q \), \( r \), and the inside mercury will have risen to \( s \), \( t \), \( s \) and \( t \), being about 30 inches. In this state of things, the water will run over by the pipe \( BA \), and everything will remain nearly in this position. The columns of water and mercury balance each other, and balance the pressure of the atmosphere.

While things are in this state of equilibrium, if we allow the cylinders to descend a little, the water will rise in the pipe \( FE \), which we may now consider as a suction-pipe; for by this motion the capacity of the whole is enlarged, and therefore the pressure of the atmosphere will still keep it full, and the situation of the mercury will again be such that all shall be in equilibrium. It will be a little lower in the inside space and higher in the outside.

Taking this view of things, we see clearly how the water is supported by the atmosphere at a very considerable height. The apparatus is analogous to a syphon which has one leg filled with water and the other with mercury. But it was not necessary to employ an air-pump to fill it. Suppose it again empty, and all the valves shut by their own weight. Let the cylinders descend a little. The capacity of the spaces below the valve \( D \) is enlarged, and therefore the included air is rarefied, and some of the air in the pipe \( CE \) must diffuse itself into the space quitted by the inner cylinder. Therefore the atmosphere will press some water up the pipe \( FE \), and some mercury into the inner space between the cylinders. When the cylinders are raised again, the air which came from the pipe \( CE \) would return into it again, but is prevented by the valve \( C \).

Raising the cylinders to their former height would compress this air; it therefore lifts the valve \( D \), and escapes. Another depression of the cylinders will have a similar effect. The water will rise higher in \( FC \), and the mercury in the inner space; and then, after repeated strokes, the water will pass the valve \( C \), and fill the whole apparatus, as the air-pump had caused it to do before.

The position of the cylinders, when things are in this situation, is represented in fig. 20. The outer and inner cylinders in their lowest position having descended about 30 inches. The mercury in the outer space stands at \( q \), \( r \), a little above the middle of the cylinders, and the mercury in the inner space is near the top \( t \) of the inner cylinder. Now let the cylinders be drawn up. The water above the mercury cannot get back again through the valve \( C \), which shuts by its own weight. We therefore attempt to compress it; but the mercury yields, and descends in the inner space, and rises in the outer, till both are quickly on a level, about the height \( u \). If we continue to raise the cylinders, the compression forces out more mercury, and it now stands lower in the inner than in the outer space. But that there may be something to balance this inequality of the mercurial columns, the water goes through the valve \( D \), and the equilibrium is restored when the height of the water in the pipe \( ED \) above the surface of the internal mercury is 12 times the difference of the mercurial columns (on the former supposition of specific gravity). If the quantity of water is such as to rise two feet in the pipe \( ED \), the mercury in the outer space will be two inches higher than that in the inner space. Another depression of the cylinders will again enlarge the space within the apparatus, the mercury will take the position of fig. 19, and more water will come in. Raising the cylinders will send this water four feet up the pipe \( ED \), and the mercury will be four inches higher in the inner than in the outer space. Repeating this operation, the water will be raised still higher in \( DE \); and this will go on till the mercury in the outer space reaches the top of the cylinder; and this is the limit of the performance. The dimensions with which we set out will enable the machine to raise the water about 30 feet in the pipe \( ED \); which, added to the 30 feet of \( CF \), makes the whole height above the pit-water 60 feet. By making the cylinders longer, we increase the height of \( FD \). This machine must be worked with great attention, and but slowly; for at the beginning of the forcing stroke the mercury very rapidly sinks in the inner space and rises in the water, and will dash out and be lost. To prevent this as much as possible, the outer cylinder terminates in a sort of cup or dish, and the inner cylinder should be tapered at top.

The machine is exceedingly ingenious and refined; ingenuity and there is no doubt but that its performance will exceed that of any other pump which raises the water to great height, because friction is completely avoided, and there can be no want of tightness of the piston. But this is all its advantage; and from what has been observed, it is but trifling. The expense would be enormous; for whatever care the cylinders are made, the interval between the inner and outer cylinders must contain a very great quantity of mercury. The middle cylinder must be made of iron plate, and must be without a seam, for the mercury would dissolve every folder. For such such reasons, it has never come into general use. But it would have been unpardonable to have omitted the description of an invention which is so original and ingenious; and there are some occasions where it may be of great use, as in nice experiments for illustrating the theory of hydraulics, it would give the finest pistons for measuring the pressures of water in pipes, &c. It is on precisely the same principle that the cylinder bellows, described in the article Pneumatics, are constructed.

We beg leave to conclude this part of the subject with the description of a pump without friction, which may be constructed in a variety of ways by any common carpenter, without the assistance of the pump-maker or plumber, and will be very effective for raising a great quantity of water to small heights, as in draining marshes, marl-pits, quarries, &c. or even for the service of a house.

VII. ABCD (fig. 21.) is a square trunk of carpenter's work, open at both ends, and having a little cistern and spout at top. Near the bottom there is a partition made of board, perforated with a hole E, and covered with a clack. ffff represents a long cylindrical bag or pudding, made of leather or of double canvas, with a fold of thin leather such as sheepskin between the canvas bags. This is firmly nailed to the board E with soft leather between. The upper end of this bag is fixed on a round board, having a hole and valve F. This board may be turned in the lathe with a groove round its edge, and the bag fastened to it by a cord bound tight round it. The fork of the piston-rod FG is firmly fixed into this board; the bag is kept distended by a number of wooden hoops or rings of strong wire ffff, &c., put into it at a few inches distance from each other. It will be proper to connect these hoops before putting them in, by three or four cords from top to bottom, which will keep them at their proper distances. Thus will the bag have the form of a barber's bellows powder-puff. The distance between the hoops should be about twice the breadth of the rim of the wooden ring to which the upper valve and piston-rod are fixed.

Now let this trunk be immersed in the water. It is evident that if the bag be stretched from the compressed form which its own weight will give it by drawing up the piston-rod, its capacity will be enlarged, the valve F will be shut by its own weight, the air in the bag will be rarefied, and the atmosphere will press the water into the bag. When the rod is thrust down again, this water will come out by the valve F, and fill part of the trunk. A repetition of the operation will have a similar effect; the trunk will be filled, and the water will at last be discharged by the spout.

Here is a pump without friction, and perfectly tight. For the leather between the folds of canvas renders the bag impervious both to air and water. And the canvas has very considerable strength. We know from experience that a bag of six inches diameter, made of sail-cloth No. 3, with a sheepskin between, will bear a column of 15 feet of water, and stand six hours work per day for a month without failure, and that the pump is considerably superior in effect to a common pump of the same dimensions. We must only observe, that the length of the bag must be three times the intended length of the stroke; so that when the piston-rod is in its highest position, the angles or ridges of the bag may be pretty acute. If the bag be more stretched than this, the force which must be exerted by the labourer becomes much greater than the weight of the column of water which he is raising. If the pump be laid alope, which is very usual in these occasional and hasty drawings, it is necessary to make a guide for the piston-rod within the trunk, that the bag may play up and down without rubbing on the sides, which would quickly wear it out.

The experienced reader will see that this pump is very like that of Goffet and De la Deuille, described by Belidor, vol. ii. p. 120, and most writers on hydraulics. It would be still more like it, if the bag were on the under side of the partition E, and a valve placed farther down the trunk. But we think that our form is greatly preferable in point of strength. When in the other situation, the column of water lifted by the piston tends to burst the bag, and this with a great force, as the intelligent reader well knows. But in the form recommended here, the bag is compressed, and the strain on each part may be made much less than that which tends to burst a bag of six inches diameter. The nearer the rings are placed to each other the smaller will the strain be.

The same bag-piston may be employed for a forcing pump, by placing it below the partition, and inverting the valve; and it will then be equally strong, because the resistance in this case too will act by compression.

We now come naturally to the consideration of the different forms which may be given to the pistons and valves of a pump. A good deal of what we have been describing already is reducible to this head; but, having a more general appearance, changing as it were the whole form and structure of the pump, it was not improper to keep these things together.

The great desideratum in a piston is, that it be as piston tight as possible, and have as little friction as is consistent should have with this indispensable quality. We have already said, little friction is the common form, when carefully executed, has these properties in an eminent degree. And accordingly this form has kept its ground amidst all the improvement which ingenious artists have made. Mr Belidor, an author of the first reputation, has given the description of a piston which he highly extols, and is undoubtedly a very good one, constructed from principle, and extremely well composed.

It consists of a hollow cylinder of metal gh (fig. 22.) pierced with a number of holes, and having at top aved one by flanch AB, whose diameter is nearly equal to that of Belidor, the working-barrel of the pump. This flanch has a groove round it. There is another flanch IK below, by which this hollow cylinder is fastened with bolts to the lower end of the piston, represented in fig. 23. This consists of a plate CD, with a grooved edge similar to AB, and an intermediate plate which forms the seat of the valve. The composition of this part is better understood by inspecting the figure than by any description. The piston-rod HL is fixed to the upper plate by bolts through its different branches at G, G. This metal body is then covered with a cylindrical bag of leather fastened on it by cords bound round it, filling up the grooves in the upper and lower plates. The operation of the piston is as follows.

A little water is poured into the pump, which gets past past the sides of the piston, and lodges below in the fixed valve. The piston being pushed down dips into this water, and it gets into it by the valve. But as the piston in descending compresses the air below it, this compressed air also gets into the inside of the piston, swells out the bag which surrounds it, and compresses it to the sides of the working-barrel. When the piston is drawn up again, it must remain tight, because the valve will shut, and keep in the air in its most compressed state; therefore the piston must perform well during the suction. It must act equally well when pushed down again, and acting as a force; for however great the resistance may be, it will affect the air within the piston to the same degree, and keep the leather close applied to the barrel. There can be no doubt therefore of the piston's performing both its offices completely; but we imagine that the adhesion to the barrel will be greater than is necessary: it will extend over the whole surface of the piston, and be equally great in every part of its surface; and we suspect that the friction will therefore be very great. We have very high authority for supposing that the adhesion of a piston of the common form, carefully made, will be such as will make it perfectly tight; and it is evident that the adhesion of Belidor's piston will be much greater, and it will be productive of worse consequences. If the leather bag be worn through in any one place, the air escapes, and the piston ceases to be compressed altogether; whereas in the common piston there will very little harm result from the leather being worn through in one place, especially if it project a good way beyond the base of the cone. We still think the common piston preferable. Belidor's piston would do much better inverted as the piston of a sucking pump; and in this situation it would be equal, but not superior, to the common.

Belidor describes another forcing piston, which he had executed with success, and prefers to the common wooden force. It consists of a metal cylinder or cone, having a broad flanch united to it at one end, and a similar flanch which is screwed on the other end. Between these two plates are a number of rings of leather strongly compressed by the two flanches, and then turned in a lathe like a block of wood, till the whole fits tight, when dry, into the barrel. It will swell, says he, and soften with the water, and withstand the greatest pressures. We cannot help thinking this but an indifferent piston. When it wears, there is nothing to squeeze it to the barrel. It may indeed be taken out and another ring or two of leather put in, or the flanches may be more strongly screwed together; but all this may be done with any kind of piston; and this has therefore no peculiar merit.

The following will, we presume, appear vastly preferable. ABCD (fig. 24.) is the solid wooden or metal block of the piston; EF is a metal plate, which is turned hollow or dish-like below, so as to receive within it the solid block. The piston rod goes through the whole, and has a shoulder above the plate EF, and a nut H below. Four screw-bolts, such as i k, l m, also go through the whole, have their heads k, m sunk into the block, and nuts above at i, l. The packing or stuffing, as it is termed by the workmen, is represented at NO. This is made as solid as possible, and generally consists of soft hempen twine well soaked in a mixture of oil, tallow, and rosin. The plate EF is gently screwed down, and the whole is then put into the barrel, fitting it as tight as may be thought proper. When it wears loose, it may be tightened at any time by screwing down the nuts i l, which cause the edges of the dish to squeeze out the packing, and compress it against the barrel to any degree.

The greatest difficulty in the construction of a piston difficulties is to give a sufficient passage through it for the water, and yet allow a firm support for the valve, and fixture for the piston rod. We shall see presently that it occasions a considerable expense of the moving power to force a piston with a narrow perforation through the water lodged in the working barrel. When we are raising water to a small height, such as 10 or 20 feet, the power expended amounts to a fourth part of the whole, if the water-way in the piston is less than one-half of the section of the barrel, and the velocity of the piston two feet per second, which is very moderate. There can be no doubt, therefore, that metal pistons are preferable, because their greater strength allows much wider apertures.

The following piston, described and recommended by Belidor, seems as perfect in these respects as the nature of things will allow. We shall therefore describe it in detail, the author's own words as a model, which may be adopted with confidence in the greatest works.

"The body of the piston is a truncated metal cone (fig. 25.), having a small fillet at the greater end. Fig. 26. shows the profile, and fig. 27. the plan of its upper base; where appears a cross bar DD, pierced with an oblong mortise E for receiving the tail of the piston-rod. A band of thick and uniform leather AA (fig. 26. and 28.) is put round this cone, and secured by a brass hoop BB firmly driven on its smaller end, where it is previously made thinner to give room for the hoop.

"This piston is covered with a leather valve fortified with metal plates GG (fig. 29.). These plates are wider than the hole of the piston, so as to rest on its rim. There are similar plates below the leather of a smaller size, that they may go into the hollow of the piston; and the leather is firmly held between the metal plates by screws H, H, which go through all. This is represented by the dotted circle IK. Thus the pressure of the incumbent column of water is supported by the plates GG, whose circular edges rest on the brim of the water-way, and their straight edges rest on the cross bar DD of fig. 26. and 27. This valve is laid on the top of the conical box in such a manner that its middle FF rests on the cross bar. To bind all together, the end of the piston-rod is formed like a cross, and the arms MN (fig. 30.) are made to rest on the diameter FF of the valve, the tail EP going through the hole E in the middle of the leather, and through the mortise E of the cross bar of the box; and also through another bar QR (fig. 28. and 29.) which is notched into the lower brim of the box. A key V is then driven into the hole T in the piston-rod; and this wedges all fast. The bar QR is made strong; and its extremities project a little, so as to support the brass hoop BB which binds the leather band to the piston-box. The adjoining scale gives the dimensions of all the parts, as they were executed for a steam-engine near Condé, where the piston gave complete satisfaction."

This piston has every advantage of strength, tightness, and... and large water-way. The form of the valve (which has given it the name of the butterfly-valve) is extremely favourable to the passage of the water; and as it has but half the motion of a complete circular valve, less water goes back while it is shutting.

The following piston is also ingenious, and has a good deal of merit. OPPO (fig. 31.) is the box of the piston, having a perforation Q, covered above with a flat valve K, which rests in a metal plate that forms the top of the box. ABCBA is a flirrup of iron to which the box is fixed by screws a, a, a, whose heads are sunk in the wood. This flirrup is perforated at C, to receive the end of the piston-rod, and a nut H is screwed on below to keep it fast. DEFED is another flirrup, whose lower part at DD forms a hoop like the sole of a flirrup, which embraces a small part of the top of the wooden box. The lower end of the piston-rod is screwed; and before it is put into the holes of the two flirrups (through which holes it slides freely) a broad nut G is screwed on it. It is then put into the holes, and the nut H firmly screwed up. The packing RR is then wound about the piston as tight as possible till it completely fills the working barrel of the pump. When long use has rendered it in any degree loose, it may be tightened again by screwing down the nut G. This causes the ring DD to compress the packing between it and the projecting shoulder of the box at PP; and thus causes it to swell out, and apply itself closely to the barrel.

We shall add only another form of a perforated piston; which being on a principle different from all the preceding, will suggest many others; each of which will have its peculiar advantages. OO in fig. 32. represents the box of this piston, fitted to the working barrel in any of the preceding ways as may be thought best. AB is a cross bar of four arms, which is fixed to the top of the box. CF is the piston-rod going through a hole in the middle of AB, and reaching a little way beyond the bottom of the box. It has a shoulder D, which prevents its going too far through. On the lower end there is a thick metal plate, turned conical on its upper side, so as to fit a conical seat PP in the bottom of the piston-box.

When the piston-rod is pushed down, the friction on the barrel prevents the box from immediately yielding. The rod therefore slips through the hole of the cross bar AB. The plate E, therefore, detaches itself from the box. When the shoulder D presses on the bar AB, the box must yield, and be pushed down the barrels, and the water gets up through the perforation. When the piston-rod is drawn up again, the box does not move till the plate E lodges in the seat PP, and thus shuts the water-way; and then the piston lifts the water which is above it, and acts as the piston of a fucking pump.

This is a very simple and effective construction, and makes a very tight valve. It has been much recommended by engineers of the first reputation, and is frequently used; and from its simplicity, and the great solidity of which it is capable, it seems very fit for great works. But it is evident that the water-way is limited to less than one-half of the area of the working-barrel. For if the perforation of the piston be one-half of the area, the diameter of the plate or ball EF must be greater; and therefore less than half the area will be left for the passage of the water by its sides.

We come now to consider the forms which may be given to the valves of a hydraulic engine.

The requisites of a valve are, that it shall be tight, valves of sufficient strength to resist the great pressures to which it is exposed, that it afford a sufficient passage for the water, and that it do not allow much to go back while it is shutting.

We have not much to add to what has been said already on this subject. The valves which accompany valves, the pump of fig. 5., are called clack valves, and are of all the most obvious and common; and the construction described on that occasion is as perfect as any. We only add, that as the leather is at last destroyed at the hinge by such incessant motion, and it is troublesome, especially in deep mines, and under water, to undo the joint of the pump in order to put in a new valve, it is frequently annexed to a box like that of a piston, made a little conical on the outside, so as to fit a conical seat made for it in the pipe, as represented in fig. 33., and it Fig. 33. has an iron handle like that of a basket, by which it can be laid hold of by means of a long grappling-hook let down from above. Thus it is drawn up; and being very gently tapered on the sides, it sticks very fast in its place.

The only defect of this valve is, that by opening very wide when pushed up by the stream of water, it allows a good deal to go back during its shutting again.

In some great machines which are worked by a slow turning crank, the return of the piston is so very slow, that a sensible loss is incurred by this; but it is nothing like what Dr Defaguliers says, one-half of a cylinder whose height is equal to the diameter of the valve.—For in such machines, the last part of the upward stroke is equally slow, and the velocity of the water through the valve exceedingly small, so that the valve is at this time almost shut.

The butterfly-valve represented in figures 29, &c. is utility of free-from most of those inconveniences, and seems the butterfly most perfect of the clack valves. Some engineers make fly-valve. their great valves of a pyramidal form, consisting of four clacks, whose hinges are in the circumference of the water-way, and which meet with their points in the middle, and are supported by four ribs which rise up from the sides, and unite in the middle. This is an excellent form, affording the most spacious water-way, and shutting very readily. It seems to be the best possible for a piston. The rod of the piston is branched out on four sides, and the branches go through the piston-box, and are fastened below with screws. These branches form the support for the four clacks. We have seen a valve of this form in a pump of six feet diameter, which discharged 20 hogsheads of water every stroke, and made 12 strokes in a minute, raising the water above 22 feet.

There is another form of valve, called the button or button tail valve. It consists of a plate of metal AB (fig. 34.) valves, turned conical, so as exactly to fit the conical cavity a b Fig. 34. of its box. A tail CD projects from the under side, which passes through a cross bar EF in the bottom of the box, and has a little knob at the end, to hinder the valve from rising too high.

This valve, when nicely made, is unexceptionable. It has great strength, and is therefore proper for all severe strains, and it may be made perfectly tight by grinding. Accordingly it is used in all cases where this is of indispensible consequence. It is most durable, and the only kind that will do for passages where steam or hot water is to go through. Its only imperfection is a small water-way; which, from what has been said, cannot exceed, or indeed equal, one-half of the area of the pipe.

If we endeavour to enlarge the water-way, by giving the cone very little taper, the valve frequently sticks so fast in the seat that no force can detach them.—And this sometimes happens during the working of the machine; and the jolts and blows given to the machine in taking it to pieces, in order to discover what has been the reason that it has discharged no water, frequently detach the valve, and we find it quite loose, and cannot tell what has deranged the pump. When this is guarded against, and the diminution of the water-way is not of very great consequence, this is the best form of a valve.

Analogous to this is the simplest of all valves, represented in fig. 35. It is nothing more than a sphere of metal A, to which is fitted a seat with a small portion BC of a spherical cavity. Nothing can be more effectual than this valve; it always falls into its proper place, and in every position fits it exactly. Its only imperfection is the great diminution of the water-way. If the diameter of the sphere does not considerably exceed that of the hole, the touching parts have very little taper, and it is very apt to stick fast. It opposes much less resistance to the passage of the water than the flat under-surface of the button-valve. N.B. It would be an improvement of that valve to give it a taper-shape below like a boy’s top. The spherical valve must not be made too light, otherwise it will be hurried up by the water, and much may go back while it is returning to its place.

Belidor describes with great minuteness (vol. ii. p. 221, &c.) a valve which unites every requisite. But it is of such nice and delicate construction, and its defects are so great when this exactness is not attained, or is impaired by use, that we think it hazardous to introduce it into a machine in a situation where an intelligent and accurate artist is not at hand. For this reason we have omitted the description, which cannot be given in few words, nor without many figures; and desire our curious readers to consult that author, or peruse Dr Desagulier’s translation of this passage. Its principle is precisely the same with the following rude contrivance, with which we shall conclude the descriptive part of this article.

Suppose ABCD (fig. 36.) to be a square wooden trunk. EF is a piece of oak board, exactly fitted to the trunk in an oblique position, and supported by an iron pin which goes through it at I, one-third of its length from its lower extremity E. The two ends of this board are bevelled, so as to apply exactly to the sides of the trunk. It is evident, that if a stream of water come in the direction BA, its pressure on the part IF of this board will be greater than that upon EI. It will therefore force it up and rush through, making it stand almost parallel to the sides of the trunk. To prevent its rising so far, a pin must be put in its way. When this current of water changes its direction, the pressure on the upper side of the board being again greatest on the portion IF, it is forced back again to its former situation; and its two extremities rising on the opposite sides of the trunk, the passage is completely stopped. This board therefore performs the office of a valve; and this valve is the most perfect that can be, because it offers the freest passage to the water, and it allows very little to get back while it is shutting; for the part IE brings up half as much water as IF allows to go down. It may be made extremely tight, by fixing two thin fillets H and G to the sides of the trunk, and covering those parts of the board with leather which applies to them; and in this state it perfectly resembles Belidor’s fine valve.

And this construction of the valve suggests, by the way, a form of an occasional pump, which may be of an occasional very effectual in small heights. Let a b c d e (fig. 36.) be a square box made to slide along this wooden trunk without shake, having two of its sides projecting upwards, terminating like the gable-ends of a house. A piece of wood e is mortised into these two sides, and to this the piston-rod is fixed. This box being furnished with a valve similar to the one below, will perform the office of a piston. If this pump be immersed so deep in the water that the piston shall also be under water, we scruple not to say that its performance will be equal to any. The piston may be made abundantly tight by covering its outside neatly with soft leather. And as no pipe can be bored with greater accuracy than a very ordinary workman can make a square trunk, we presume that this pump will not be very deficient even for a considerable suction.

We now proceed to the last part of the subject, to consider the motion of water in pumps, in reference to the force which must be employed. What we have hitherto said with respect to the force which must be applied to a piston, related only to the sustaining the water at a certain height: but in actual service we must not only do this, but we must discharge it at the place of delivery in a certain quantity; and this must require a force superadded to what is necessary for its mere support at this height.

This is an extremely intricate and difficult subject, an intrinsically imperfectly understood even by professed engineers. The principles on which this knowledge must be founded are of a much more abstruse nature than the ordinary laws of hydrostatics; and all the genius of Newton was employed in laying the foundation of this part of physical science. It has been much cultivated in the course of this century by the first mathematicians of Europe. Daniel and John Bernoulli have written very elaborate treatises on the subject, under the very apposite name of Hydrodynamics; in which, although they have added little or nothing to the fundamental propositions established in some sort by Newton, and acquired in by them, yet they have greatly contributed to our progress in it by the methods which they have pursued in making application of those fundamental propositions to the most important cases. It must be acknowledged, however, that both these propositions, and the extensions given them by these authors, are supported by a train of argument that is by no means unexceptionable; and that they proceed on assumptions or postulates which are but nearly true in any Pump any case, and in many are inadmissible; and it remains to this hour a wonder or puzzle how these propositions and their results correspond with the phenomena which we observe.

But fortunately this correspondence does obtain to a certain extent. And it seems to be this correspondence chiefly which has given these authors, with Newton at their head, the confidence which they place in their respective principles and methods: for there are considerable differences among them in those respects; and each seems convinced that the others are in a mistake. Messieurs d'Alembert and De la Grange have greatly corrected the theories of their predecessors, and have proceeded on postulates which come much nearer to the real state of the case. But their investigations involve us in such an inextricable maze of analytical investigation, that even when we are again conducted to the light of day by the clue which they have given us, we can make no use of what we there discovered.

But this theory, imperfect as it is, is of great service. It generalizes our observations and experiments, and enables us to compose a practical doctrine from a heap of facts which otherwise must have remained solitary and unconnected, and as cumbersome in their application as the characters of the Chinese writing.

The fundamental proposition of this practical hydrodynamics is, that water or any fluid contained in an open vessel of indefinite magnitude, and impelled by its weight only, will flow through a small orifice with the velocity which a heavy body would acquire by falling from the horizontal surface of the fluid. Thus, if the orifice is 16 feet under the surface of the water, it will issue with the velocity of 32 feet in a second.

Its velocity corresponding to any other depth \( h \) of the orifice under the surface, will be had by this easy proportion: "As the square root of 16 is to the square root of \( h \); so is 32 feet to the velocity required: or,

\[ \sqrt{16} : 32 = \sqrt{h} : v, \quad \text{and} \quad v = \frac{32 \sqrt{h}}{\sqrt{16}}, \]

alternately, \( \sqrt{16} : 32 = \sqrt{h} : v \), and \( v = \frac{32 \sqrt{h}}{\sqrt{16}}, \)

\( \frac{32}{4} \sqrt{h}, = 8 \sqrt{h} \): that is, multiply the square root of the height in feet by eight, and the product is the required velocity.

On the other hand, it frequently occurs, that we want to discover the depth under the surface which will produce a known velocity \( v \). Therefore, \( \sqrt{h} = \frac{v}{8} \),

and \( h = \frac{v^2}{64} \): that is, divide the square of the velocity by 64, and the quotient is the depth wanted in feet.

This proposition is sufficient for all our purposes. For since water is nearly a perfect fluid, and propagates all impressions undiminished, we can, in place of any pressure of a piston or other case, substitute a perpendicular column of water whose weight is equal to this pressure, and will therefore produce the same efflux.—Thus, if the surface of a piston is half a square foot, and it be pressed down with the weight of 500 pounds, and we would wish to know with what velocity it would cause the water to flow through a small hole, we know that a column of water of this weight, and of half a foot base, would be 16 feet high. And this proposition teaches us, that a vessel of this depth will have a velocity of efflux equal to 32 feet in a second.

If therefore our pressing power be of such a kind that it can continue to press forward the piston with the force of 500 pounds, the water will flow with this velocity, whatever be the size of the hole. All that remains is, to determine what change of actual pressure on the piston results from the motion of the piston itself, and to change the velocity of efflux in the subduplicate ratio of the change of actual pressure.

But before we can apply this knowledge to the circumstances which take place in the motion of water in previous pumps, we must take notice of an important modification of the fundamental proposition, which is but very obscurely pointed out by any good theory, but is established on the most regular and unexceptionable observation.

If the efflux is made through a hole in a thin plate, and the velocity is computed as above, we shall discover the quantity of water which issues in a second by observing, that it is a prism or cylinder of the length indicated by the velocity, and having its transverse section equal to that of the orifice. Thus, in the example already given, supposing the hole to be a square inch, the solid contents of this prism, or the quantity of water issuing in a second, is \( 1 \times 32 \times 12 \) cubic inches, or 384 cubic inches. This we can easily measure by receiving it in a vessel of known dimensions. Taking this method, we uniformly find a deficiency of nearly 38 parts in 100; that is, if we should obtain 100 gallons in any number of seconds, we shall in fact get only 62. This is a most regular fact, whether the velocities are great or small, and whatever be the size and form of the orifice. The deficiency increases indeed in a very minute degree with the velocities. If, for instance, the depth of the orifice be one foot, the discharge is \( \frac{67}{100} \); if it be 15 feet, the discharge is \( \frac{67}{100} \).

This deficiency is not owing to a diminution of velocity; for the velocity may be easily and accurately measured by the distance to which the jet will go, if directed horizontally. This is found to correspond very nearly with the proposition, making a very small allowance for friction at the border of the hole, and for the resistance of the air. Sir Isaac Newton ascribed the deficiency with great justice to this, that the lateral columns of water, surrounding the column which is incumbent on the orifice, press towards the orifice, and contribute to the expense equally with that column. These lateral filaments, therefore, issue obliquely, crossing the motion of the central stream, and produce a contraction of the jet; and the whole stream does not acquire a parallel motion and its ultimate velocity till it has got to some distance from the orifice. Careful observation showed him that this was really the case. But even his genius could not enable him to ascertain the motion of the lateral filaments by theory, and he was obliged to measure every thing as he saw it. He found the diameter of the jet at the place of the greatest contraction to be precisely such as accounted for the deficiency. His exposition has been unanimously acquiesced in; and experiments have been multiplied to ascertain all those circumstances which our theory cannot determine a priori. The most complete set of experiments are those of Michelotti, made at Turin at the expense of the prince of Piedmont. Piedmont. Here jets were made of 1, 2, 3, and 4 inches diameter; and the water received into cisterns most accurately formed of brick, and lined with stucco. It is the result of these experiments which we have taken for a measure of the deficiency.

We may therefore consider the water as flowing through a hole of this contracted dimension, or substitute this for the real orifice in all calculations. For it is evident that if a mouth-piece (so to call it) were made, whose internal shape precisely tallied with the form which the jet assumes, and if this mouth-piece be applied to the orifice, the water will flow out without any obstruction. The vessel may therefore be considered as really having this mouth-piece.

Nay, from this we derive a very important observation, "that if, instead of allowing the water to flow through a hole of an inch area made in a thin plate, we make it flow through a hole in a thick plank, so formed that the external orifice shall have an inch area, but be widened internally agreeably to the shape which nature forms, both the velocity and quantity will be that which the fundamental proposition determines. Michelotti measured with great care the form of the great jets of three and four inches diameter, and found that the bounding curve was an elongated trochoid. He then made a mouth-piece of this form for his jet of one inch, and another for his jet of two inches; and he found the discharges to be $\frac{7}{9}$ and $\frac{8}{9}$; and he, with justice, ascribed the trifling deficiency which still remained, partly to friction and partly to his not having exactly suited his mouth-piece to the natural form. We imagine that this last circumstance was the sole cause: For, in the first place, the water in his experiments, before getting at his jet-holes, had to pass along a tube of eight inches diameter. Now a jet of four inches bears too great a proportion to this pipe; and its narrowness undoubtedly hindered the lateral columns from contributing to the efflux in their due proportion, and therefore rendered the jet less convergent. And, in the next place, there can be no doubt (and the observations of Daniel Bernoulli confirm it) but that this convergency begins within the vessel, and perhaps at a very considerable distance from the orifice. And we imagine, that if accurate observations could be made on the motion of the remote lateral particles within the vessel, and an internal mouth-piece were shaped according to the curve which is described by the remotest particle that we can observe, the efflux of water would almost perfectly tally with the theory. But indeed the coincidence is already sufficiently near for giving us very valuable information. We learn that the quantity of water which flows through a hole, in consequence of its own weight, or by the action of any force, may be increased one half by properly shaping the passage to this hole; for we see that it may be increased from 62 to near 99.

But there is another modification of the efflux, which we confess our total incapacity to explain. If the water issues through a hole made in a plate whose thickness is about twice the diameter of the hole, or, to express it better, if it issues through a pipe whose length is about twice its diameter, the quantity discharged is nearly $\frac{8}{9}$ of what results from the proposition. If the pipe be longer than this, the quantity is diminished by friction, which increases as the length of the pipe increases. If the pipe be shorter, the water will not fill it, but detaches itself at the very entry of the pipe, and flows with a contracted jet. When the pipe is of this length, and the extremity is stopped with the finger, so that it begins to flow with a full mouth, no subsequent contraction is observed; but merely striking on the pipe with a key or the knuckle is generally sufficient to detach the water in an instant from the sides of the pipe, and reduce the efflux to $\frac{1}{3}$. This effect is most unaccountable. It certainly arises from the mutual adhesion or attraction between the water and the sides of the pipe; but how this, acting at right angles to the motion, should produce an increase from 62 to 82, nearly $\frac{1}{3}$, we cannot explain. It shows, however, the prodigious force of this attraction, which in the space of two or three inches is able to communicate a great velocity to a very great body of water. Indeed the experiments on capillary tubes show that the mutual attraction of the parts of water is some thousands of times greater than their weight.

We have only further to add, that every increase of pipe beyond two diameters is accompanied with a diminution of the discharge; but in what ratio this is diminished it is very difficult to determine. We shall only observe at present that the diminution is very great. A pipe of 2 inches diameter and 30 feet long has its discharge only $\frac{1}{5}$ of what it would be if only 4 inches long. If its length be 60 feet, its discharge will be no more than $\frac{1}{10}$. A pipe of 1 inch diameter would have a discharge of $\frac{4}{5}$, and $\frac{1}{10}$, in the same situation. Hence we may conclude that the discharge of a 4 inch pipe of 30 feet long will not exceed $\frac{1}{5}$ of what it would be if only 8 inches long. This will suffice for our present purposes; and the determination of the velocities and discharges in long conduits from pump machines must be referred to the article WATER-WORKS. At present we shall confine our attention to the pump itself, and to what will contribute to its improvement.

Before we can proceed to apply this fundamental proposition to our purpose, we must anticipate in a loose way a proposition of continual use in the construction of water-works.

Let water be supposed stagnant in a vessel EFGH (fig. 37), and let it be allowed to flow out by a cylindrical pipe HIKL, divided by any number of partitions B, C, D, &c. Whatever be the areas B, C, D, of these orifices, the velocity in the intermediate parts of the pipe will be the same; for as much passes through any one orifice in a second as passes through any other in the same time, or through any section of the intervening pipe. Let this velocity in the pipe be V, and let the area of the pipe be A. The velocity in the orifices B, C, D, must be $\frac{VA}{B}$, $\frac{VA}{C}$, $\frac{VA}{D}$, &c. Let $g$ be the velocity acquired in a second by a heavy body. Then, by the general proposition, the height of water in the vessel which will produce the velocity $\frac{VA}{B}$ in the first orifice alone, is $\frac{V^2A^2}{2gB^2}$. After this passage the velocity is again reduced to V in the middle of the space between the first and second orifices. In the second orifice this velocity is changed to $\frac{VA}{C}$. This alone would have required a height of water \( \frac{V^2 A^2}{2g C^2} \).

But the water is already moving with the velocity \( V \), which would have resulted from a height of water in the vessel (which we shall, in the language of the art, call the head of water) equal to \( \frac{V^2}{2g} \). Therefore there is only required a head of water \( \frac{V^2 A^2}{2g C^2} - \frac{V^2}{2g} \), or

\[ \frac{V^2 A^2}{2g C^2} = 1. \]

Therefore the whole height necessary for producing the efflux through both orifices, so as fill to preserve the velocity \( V \) in the intervening pipe, is

\[ \frac{V^2 A^2}{2g B^2} + \frac{A^2}{C^2} = 1. \]

In like manner the third orifice \( D \) would alone require a head of water \( \frac{V^2 A^2}{2g D^2} - 1; \)

and all the three would require a head \( \frac{V^2 A^2}{2g B^2} + \frac{A^2}{C^2} + \frac{A^2}{D^2} = 2. \)

By this induction may easily be seen what head is necessary for producing the efflux through any number of orifices.

Let the expense or quantity of water discharged in an unit of time (suppose a second) be expressed by the symbol \( Q \). This is measured by the product of the velocity by the area of the orifice, and is therefore \( V A \),

or \( \frac{VA}{B} \times B \), or \( \frac{VA}{C} \times C \), &c., and \( V^2 = \frac{Q^2}{A^2} \). Therefore we may compute the head of water (which we shall express by \( H \)) in reference to the quantity of water discharged, because this is generally the interesting circumstance. In this view we have \( H = \frac{Q^2}{2g A} \times \frac{A^2}{B^2} + \frac{A^2}{C^2} + \frac{A^2}{D^2} = 2 \); which shows that the head of water necessary for producing the discharge increases in the proportion of the square of the quantity of water which is discharged.

These things being premised, it is an easy matter to determine the motion of water in a pump, and the quantity discharged, resulting from the action of any force on the piston, or the force which must be applied to the piston in order to produce any required motion or quantity discharged. We have only to suppose that the force employed is the pressure of a column of water of the diameter of the working barrel; and this is over and above the force which is necessary for merely supporting the water at the height of the place of delivery. The motion of the water will be the same in both cases.

Let us, first of all, consider a sucking-pump. The motion here depends on the pressure of the air, and will be the same as if the pump were lying horizontally, and communicated with a reservoir, in which is a head of water sufficient to overcome all the obstructions to the motion, and produce a velocity of efflux such as we desire. And here it must be noted that there is a limit. No velocity of the piston can make the water rise in the suction-pipe with a greater velocity than what would be produced by the pressure of a column of water 33 feet high; that is, about 46 feet per second.

Let the velocity of the piston be \( V \), and the area of the working barrel be \( A \). Then, if the water fills the barrel as fast as the piston is drawn up, the discharge during the rise of the piston, or the number of cubic feet of water per second, must be \( V \times A \). This is always supposed, and we have already ascertained the circumstances which ensure this to happen. If, therefore, the water arrived with perfect freedom to the piston, the force necessary for giving it this velocity, or for discharging the quantity \( V \times A \) in a second, would be equal to the weight of the pillar of water whose height is \( \frac{V^2}{2g} \) and base \( A \).

It does not appear at first sight that the force necessary for producing this discharge has anything to do with the obstructions to the efflux of the water into the pump, because this is produced by the pressure of the atmosphere, and it is the action of this pressure which is measured by the head of water necessary for producing the internal motion in the pump. But we must always recollect that the piston, before bringing up any water, and supporting it at a certain height, was pressed on both sides by the atmosphere. While the air supports the column below the piston, all the pressure expended in this support is abstracted from its pressure on the under part of the piston, while its upper part still supports the whole pressure. The atmosphere continues to press on the under surface of the piston, through the intermediate of the water in the suction-pipe, with the difference of these two forces. Now, while the piston is drawn up with the velocity \( V \), more of the atmospheric pressure must be expended in raising the water to follow the piston; and it is only with the remainder of its whole pressure that it continues to press on the under surface of the piston. Therefore, in order that the piston may be raised with the velocity \( V \), a force must be applied to it, over and above the force necessary for merely supporting the column of water, equal to that part of the atmospheric pressure thus employed; that is, equal to the weight of the head of water necessary for forcing the water up through the suction-pipe, and producing the velocity \( V \) in the working barrel.

Therefore let \( B \) be the area of the mouth of the suction-pipe, and \( C \) the area of the fixed valve, and let the suction-pipe be of equal diameter with the working barrel. The head necessary for producing the velocity \( V \) on the working barrel is \( \frac{V^2}{2g} \left( \frac{A^2}{B^2} + \frac{A^2}{C^2} - 1 \right) \). If \( d \) expresses the density of water; that is, if \( d \) be the number of pounds in a cubic foot of water, then \( d \frac{V^2}{2g} \) will express the weight of a column whose base is \( A \), and height \( \frac{V^2}{2g} \), all being reckoned in feet. Therefore the force which must be applied, when estimated in pounds, will be \( p = d \frac{AV^2}{2g} \left( \frac{A^2}{B^2} + \frac{A^2}{C^2} - 1 \right) \).

The first general observation to be made on what has been said is, that the power which must be employed to produce the necessary motion, in opposition to all the obstacles, is in the proportion of the square of the velocity. city which we would produce, or the square of the quantity of water we would discharge.

We have hitherto proceeded on the supposition, that there is no contraction of the jet in passing through these two orifices. This we know would be very far from the truth. We must therefore accommodate things to these circumstances, by diminishing B and C in the ratio of the contraction, and calling the diminished areas b and c; then we have \( p = \frac{AdV^2}{2g} \left( \frac{A^2}{b^2} + \frac{A^2}{c^2} - 1 \right) \).

What this diminution may be, depends on the form of the parts. If the fixed valve, and the entry into the pump, are simply holes in thin plates, then \( b = \frac{a}{\sqrt{3}} B \) and \( c = \frac{a}{\sqrt{3}} C \). The entry is commonly widened or trumpet-shaped, which diminishes greatly the contraction: but there are other obstructions in the way, arising from the strainer usually put round it to keep out filth. The valve may have its contraction greatly diminished also by its box being made bell-shaped internally; nay, even giving it a cylindrical box, in the manner of fig. 33., is better than no box at all, as in fig. 5.; for such a cylindrical box will have the unaccountable effect of the short tube, and make \( b = \frac{a}{\sqrt{3}} B \), instead of \( \frac{a}{\sqrt{3}} B \). Thus we see that circumstances seemingly very trifling may produce great effects in the performance of a pump. We should have observed that the valve itself presents an obstacle which diminishes the motion, and requires an increase of power; and it would seem that in this respect the clock or butterfly valve is preferable to the button valve.

Example. Suppose the velocity of the piston to be 2 feet or 24 inches per second, and that the two contracted areas are each \( \frac{1}{3} \) of the area of the pump, which is not much less than what obtains in ordinary pumps. We have \( \frac{V^2}{2g} \left( \frac{A^2}{b^2} + \frac{A^2}{c^2} - 1 \right) = \frac{576}{75} (25 + 25 - 1) = 36.75 \) inches, and the force which we must add to what will merely support the column is the weight of a pillar of water incumbent on the piston, and something more than three feet high. This would be a sensible portion of the whole force in raising water to small heights.

We have supposed the suction-pipe to be of the same diameter with the working barrel; but it is usual to make it of smaller diameter, generally equal to the water way of the fixed valve. This makes a considerable change in the force necessary to be applied to the piston. Let \( a \) be the area of the suction-pipe, the area of the entry being still B; and the equivalent entry without contraction being still b, we have the velocity at the entrance \( = \frac{AV}{b} \), and the producing head of water \( = \frac{A^2V^2}{2g} \). After this the velocity is changed, to \( \frac{AV}{a} \) in the suction-pipe, with which the water arrives at the valve, where it is again changed to \( \frac{AV}{c} \), and requires for this change a head of water equal to \( \frac{A^2V^2}{2g} \). But the velocity retained in the suction-pipe is equivalent to the effect of a head of water \( \frac{A^2V^2}{2g} \). Therefore the head necessary for producing such a current through the fixed valve, that the water may follow the piston with the velocity V, is \( \frac{A^2V^2}{2g} \left( \frac{A^2}{b^2} + \frac{A^2}{c^2} - \frac{A^2}{a^2} \right) \), or \( = \frac{V^2}{2g} \left( \frac{A^2}{b^2} + \frac{A^2}{c^2} - \frac{A^2}{a^2} \right) \). This is evidently less than before, because \( a \) is less than A, and therefore \( \frac{A^2}{a^2} \) is greater than unity, which was the last term of the former formula. There is some advantage, therefore, derived from making the diameter of the suction-pipe less than that of the working barrel: but this is only because the passage of the fixed valve is smaller, and the inspection of the formula plainly points out that the area of the suction-pipe should be equal to that of the fixed valve. When it is larger, the water must be accelerated in its passage through the valve; which is an useless expense of force, because this velocity is to be immediately reduced to V in the working-barrel. If the foregoing example be computed with \( a \) equal to \( \frac{1}{4} \) of A, we shall find the head H equal to 29 inches instead of 37.

But this advantage of a smaller suction-pipe is in all cases very moderate; and the pump is always inferior to one of uniform dimensions throughout, having the orifice at the fixed valve of the same area. And if these orifices are considerably diminished in any proportion, the head necessary for overcoming the obstructions, so that the required velocity V may still be produced in the working barrel, is greatly increased. If we suppose the area \( \frac{1}{2} \) of A, which is frequently done in house pumps, where the diameter of the suction-pipe does seldom exceed \( \frac{1}{4} \) of that of the working-barrel; and suppose everything made in proportion to this, which is also usual, because the unskilled pump-makers study a symmetry which satisfies the eye; we shall find that the pump taken as an example will require a head of water \( = 13 \) feet and upwards. Besides, it must be observed that the friction of the suction-pipe itself has not been taken into the account. This alone is greater, in most cases, than all the obstructions we have been speaking of; for if this pipe is three inches diameter, and that of the working-barrel is six, which is reckoned a liberal allowance for a suction-pipe, and if the fixed valve is 25 feet above the surface of the pit-water; the friction of this pipe will amount to one-third of the whole propelling force.

Thus we have enabled the reader to ascertain the force necessary for producing any required discharge of water from a pump of known dimensions; and the converse of this determination gives us the discharge which will be produced by any given force. For making \( \frac{A^2}{b^2} + \frac{A^2}{c^2} - \frac{A^2}{a^2} \) (which is a known quantity, resulting from the dimensions of the pump) \( = M \), we have \( H = \frac{V^2}{2g} M \), and \( V^2 = \frac{2gH}{M} \), and \( V = \sqrt{\frac{2gH}{M}} \). Now H is that part of the natural power which we have at command which exceeds what is necessary for merely supporting the column of water. Thus, if we have a pump whose piston has an area of \( \frac{1}{4} \) of a square foot, its diameter being 6\( \frac{1}{2} \) inches; and we have to raise the water 32 feet, and can apply a power of 525 pounds to the piston; we wish to know at what rate the piston will be moved, and the quantity of water discharged? Pump. Merely to support the column of water of this height and diameter, requires 500 pounds. Therefore the remaining power, which is to produce the motion, is 25 pounds. This is the weight of a column 1 foot 4 inches high, and $H = 1,333$ feet. Let us suppose the diameter of the suction-pipe $\frac{1}{2}$ of that of the working-barrel, so that $\frac{A}{B} = 4$. We may suppose it executed in the best manner, having its lower extremity trumpet-shaped, formed by the revolution of the proper trochoid. The contraction at the entry may therefore be considered as nothing, and $\frac{A}{b} = 4$, and $\frac{A^2}{b^2} = 16$. We may also suppose the orifice of the fixed valve equal to the area of the suction-pipe, so that $\frac{A^2}{C^2}$ is also $= 16$, and there is no contraction here; and therefore $\frac{A^2}{c^2}$ is also $16$. And lastly, $\frac{A^2}{a^2}$ is also $16$. Therefore

$$\frac{A^2}{b^2} + \frac{A^2}{c^2} - \frac{A^2}{a^2} \text{ or } M, = 16 + 16 - 16, = 16.$$

We have also $2g = 64$. Now $N = \sqrt{\frac{2gH}{M}} = \sqrt{\frac{64 \times 1,333}{16}}, = 2,309$ feet, and the piston will move with the velocity of 2 feet 4 inches nearly. Its velocity will be less than this, on account both of the friction of the piston and the friction of the water in the suction-pipe. These two circumstances will probably reduce it to one foot eight inches; and it can hardly be less than this.

We have taken no notice of the friction of the water in the working-barrel, or in the space above the piston; because it is in all cases quite insignificant. The longest pipes employed in our deep mines do not require more than a few inches of head to overcome it.

But there is another circumstance which must not be omitted. This is the resistance given to the piston in its descent. The pistons of an engine for drawing water from deep mines must descend again by their own weight in order to repeat their stroke. This must require a preponderance on that end of the working-beam to which they are attached, and this must be overcome by the moving power during the effective stroke. It makes, therefore, part of the whole work to be done, and must be added to the weight of the column of water which must be raised.

This is very easily ascertained. Let the velocity of the piston in its descent be $V$, the area of the pump-barrel $A$, and the area of the piston-valve $a$. It is evident, that while the piston descends with the velocity $V$, the water which is displaced by the piston in a second is $(A-a)V$. This must pass through the hole of the piston, in order to occupy the space above, which is left by the piston. If there were no contraction, the water would go through with the velocity $\frac{A-a}{a}V$; but as there will always be some contraction, let the diminished area of the hole (to be discovered by experiment) be $b$, the velocity therefore will be $V\frac{A-a}{b}$. This requires for its production a head of water $\frac{V^2(A-a)^3}{2g}$. This is the height of a column of water whose base is not $A$ but $A-a$. Calling the density of water $d$, we have for the weight of this column, and the force $p$ in $d \times A-a + \left(\frac{A-a}{b}\right) \times \frac{V^2}{2g} = \frac{dV^2(A-a)^3}{2gb^2}$. This, we see again, is proportional to the square of the velocity of the piston in its descent, and has no relation to the height to which the water is raised.

If the piston has a button valve, its surface is at least equal to $a$; and therefore the pressure is exerted on the water by the whole surface of the piston. In this case we shall have $p = \frac{dV^2A^3}{2gb^2}$ considerably greater than before. We cannot ascertain this value with great precision, because it is extremely difficult, if possible, to determine the resistance in so complicated a case. But the formula is exact, if $b$ can be given exactly; and we know within very moderate limits what it may amount to. In a pump of the very best construction, with a button-valve, $b$ cannot exceed one-half of $A$; and therefore $\frac{A^3}{b^2}$ cannot be less than 8. In this case, $\frac{V^2A^3}{2gb^2}$ will be $\frac{V^2}{8}$. In a good steam-engine pump $V$ is about three feet per second, and $\frac{V^2}{8}$ is about 1$\frac{1}{2}$ feet, which is but a small matter.

We have hitherto been considering the sucking-pump and the alone: but the forcing pump is of more importance, forcing and apparently more difficult of investigation.—Here pump, we have to overcome the obstructions in long pipes, with many bends, contractions, and other obstructions. But the consideration of what relates merely to the pump is abundantly simple. In most cases we have only to force the water into an air-vessel, in opposition to the elasticity of the air compressed in it, and to send it thither with a certain velocity, regulated by the quantity of water discharged in a given time. The elasticity of the air in the air-vessel propels it along the Main. We are not now speaking of the force necessary for counterbalancing this pressure of the air in the air-vessel, which is equivalent to all the subsequent obstructions, but only of the force necessary for propelling the water out of the pump with the proper velocity.

We have in a manner determined this already. The piston is solid, and the water which it forces has to pass through a valve in the lateral pipe, and then to move in the direction of the main. The change of direction requires an addition of force to what is necessary for merely impelling the water through the valve. Its quantity is not easily determined by any theory, and it varies according to the abruptness of the turn. It appears from experiment, that when a pipe is bent to a right angle, without any curvature or rounding, the velocity is diminished about $\frac{1}{8}$. This would augment the head of water about $\frac{1}{8}$. This may be added to the contraction of the valve hole. Let $c$ be its natural area, and whatever is the contraction competent to its form, increase it $\frac{1}{8}$, and call the contracted area $c$. Then this will require a head of water $=\frac{V^2A^2}{2gc^2}$. This must be added to the head $\frac{V^4}{2g}$ necessary for merely giving the velocity $V$ to the water. Therefore the whole is $\frac{V^4}{2g} \left( \frac{A^2}{c^2} + 1 \right)$; and the power $p$ necessary for this purpose is $\frac{dAV^3}{2g} \left( \frac{A^2}{c^2} + 1 \right)$.

It cannot escape the observation of the reader, that in all these formulae, expressing the height of the column of water which would produce the velocity $V$ in the working barrel of the pump, the quantity which multiplies the constant factor $\frac{dAV^3}{2g}$ depends on the contracted passages which are in different parts of the pump, and increases in the duplicate proportion of the sum of those contractions. It is therefore of the utmost consequence to avoid all such, and to make the main which leads from the forcing-pump equal to the working barrel. If it be only of half the diameter, it has but one-fourth of the area, the velocity in the main is four times greater than that of the piston, and the force necessary for discharging the same quantity of water is 16 times greater.

It is not, however, possible to avoid these contractions altogether, without making the main pipe wider than the barrel. For if only so wide, with an entry of the same size, the valve makes a considerable obstruction. Unskilful engineers endeavour to obviate this by making an enlargement in that part of the main which contains the valve. This is seen in fig. 14, at the valve L. If this be not done with great judgment, it will increase the obstructions. For if this enlargement is full of water, the water must move in the direction of its axis with a diminished velocity; and when it comes into the main, it must again be accelerated. In short, any abrupt enlargement which is to be afterwards contracted, does as much harm as a contraction, unless it be so short that the water in the axis keeps its velocity till it reaches the contraction. Nothing would do more service to an artist, who is not well founded in the theory of hydrodynamics, than to make a few simple and cheap experiments with a vessel like that of fig. 37. Let the horizontal pipe be about three inches diameter, and made in joints which can be added to each other. Let the joints be about six inches long, and the holes from one-fourth to a whole inch in diameter. Fill the vessel with water, and observe the time of its sinking three or four inches. Each joint should have a small hole in its upper side to let out the air; and when the water runs out by it, let it be stopped by a peg. He will see that the larger the pipe is in proportion to the orifices made in the partitions, the efflux is more diminished. We believe that no person would suspect this who has not considered the subject minutely.

All angular enlargements, all boxes, into which the pipes from different working barrels, unite their water before it goes into a main, must therefore be avoided by an artist who would execute a good machine; and the different contractions which are unavoidable at the seats of valves and the perforations of pistons, &c., should be diminished by giving the parts a trumpet-shape.

In the air-vessels represented in fig. 13, this is of very great consequence. The throat O, through which the water is forced by the expansion of the confined air, should always be formed in this manner. For it is this which produces the motion during the returning part of the stroke in the pump constructed like fig. 13, No. 1, and during the whole stroke in No. 2. Neglecting this seemingly trifling circumstance will diminish the performance at least one-fifth. The construction of No. 1 is the best, for it is hardly possible to make the passage of the other so free from the effects of contraction. The motion of the water during the returning stroke is very much contorted.

There is one circumstance that we have not taken any notice of, viz., the gradual acceleration of the motion of the water in pumps. When a force is applied to the piston, it does not in an instant communicate all the velocity which it acquires. It acts as gravity acts on heavy bodies; and if the resistances remained the same, it would produce, like gravity, an uniformly accelerated motion. But we have seen that the resistances (which are always measured by the force which just overcomes them) increase as the square of the velocity increases. They therefore quickly balance the action of the moving power, and the motion becomes uniform, in a time so short that we commit no error of any consequence by supposing it uniform from the beginning. It would have prodigiously embarrassed our investigations to have introduced this circumstance; and it is a matter of mere speculative curiosity: for most of our moving powers are unequal in their exertions, and these exertions are regulated by other laws. The pressure on a piston moved by a crank is as variable as its velocity, and in most cases is nearly in the inverse proportion of its velocity, as any mechanician will readily discover. The only case in which we could consider this matter with any degree of comprehensibility is that of a steam-engine, or of a piston which forces by means of a weight lying on it. In both, the velocity becomes uniform in a very small fraction of a second.

We have been very minute on this subject. For although it is the only view of a pump which is of any of elemental importance, it is hardly ever understood even by professional books on hydraulics. And this is not peculiar to hydraulics, but is seen in all the branches of practical mechanics. The elementary knowledge to be met with in such books as are generally perused by them, goes no farther than to state the forces which are in equilibrium by the intervention of a machine, or the proportion of the parts of a machine which will set two known forces in equilibrium. But when this equilibrium is destroyed by the superiority of one of the forces, the machine must move; and the only interesting question is, what will be the motion? Till this is answered with some precision, we have learned nothing of any importance. Few engineers are able to answer this question even in the simplest cases; and they cannot, from any confident science, say what will be the performance of an untried machine. They guess at it with a success proportioned to the multiplicity of their experience and their own sagacity. Yet this part of mechanics is as susceptible of accurate computation as the cases of equilibrium.—We therefore thought it our duty to point out the manner of proceeding so circumstantially, that every step should be plain and easy, and that conviction should always accompany our progress. This we think it has been in our power to do, by the very simple method of substituting a column lumn of water acting by its weight in lieu of any natural power which we may chance to employ.

To such as wish to prosecute the study of this important part of hydraulics in its most abstruse parts, we recommend the perusal of the dissertations of Mr Pitot and Mr Boffut, in the Memoirs of the Academy of Paris; also the dissertations of the Chevalier de la Borda, 1766 and 1767; also the *Hydraulique* of the Chevalier De Buat. We shall have occasion to consider the motion of the water in the mains of forcing or lifting pumps which send the water to a distance, in the article *Water-Works*; where the reader will see how small is the performance of all hydraulic machines, in comparison of what the usual theories, founded on equilibrium only, would make him expect.