Art. 1. OF THE FIRST PRINCIPLES OF MECHANICS.

MOTION may be said to be the Beginning or Foundation of all Mechanics, because no Mechanical Operation can be performed without Motion.

Art. 2. OF THE PRINCIPLES OF MECHANICS.

THERE are two principles, which are the foundation of all mechanical motion and mechani­cal powers, viz. Gravity and Elasticity; or Weight and Spring.

By one or the other of these principles or pow­ers every mechanical operation is performed.

[Page 3] Gravity, Art. 2. in the extent of the word, means every species of attraction; but more especially that species which is common to, and mutual be­tween, all bodies; and is evident between the sun and its planetary attendants, as also the earth and moon. ^* But we will only consider it, as it relates to that tendency which all bodies on this earth has to fall towards its centre; thus far it concerns the mechanical arts, and its laws are as follows, viz.

Art. 3. ELASTICITY.

Elasticity is that strength or power, which any body or quantity of matter, being confined or compressed, has to expand itself; such as a spring that is bent or wound up, heated air or steam con­fined in a vessel, &c. and by it many mechanical operations are performed.

Elasticity, in the full sense of the word, here means every species of repulsion.

The limits of the prodigious power of repulsion, which takes place between the particles of heated air and steam, are not yet known. Their effects are seen in the explosion of gunpowder, the burst­ing and cracking of wood in the fire, &c. In short, in every instance, where steam could not find room to expand itself, it has burst the vessel that confined it, endangering the lives of those who were near it. ^†

[Page 5] Having premised what was necessary to the right understanding of the science of mechanics, Art. 3. which mostly depends upon the principles of gra­vitation,

We come to consider the Objects thereof, viz. the Nature, Kinds, and various Effects of Motion and moving Bodies, and the Structure and Me­chanism of all Kinds of Machines, called Mecha­nical Powers, whether Simple or Compound.

Art. 4. OF MOTION AND ITS GENERAL LAWS.

MOTION is the continual and successive change of space or place, Motion, and is either ab­solute or relative.

Absolute motion is the change of space or place of bodies, Absolute. such as the flight of a bird, or the mo­tion of a ball projected in the air.

Relative motion is the motion one body has with respect to another, Relative. such as the difference of mo­tion of the flight of two birds, or of two ships sailing. ^*

Art. 5.

MOTION is either Equable, Accelerated, Motion, [...] Retarded.

Equable motion is when a body passes over equal distances in equal times. Equable.

Accelerated motion, Accelerated. is that which is continually increased; such is the motion of falling bodies. ^*

Retarded motion, Retarded. is that which continually de­creases; such is the motion of a cannon-ball throw perpendicularly upwards. ^†

Art. 6.

THE Momentum or quantity of motion, is [...] the power or force which a moving body has [...] strike an obstacle to produce effects, and is equal to that impressed force by which a body is com­pelled to change its place, by axiom 3, art. 1 [...] [Page 7] which, Art. 6. I think, ought to be distinguished by two names, viz. Instant and Effective Momentums.

1. The Instant Momentum, or force of moving bodies, Momentum of bodies in mo­tion. is in the compound ratio of their quantities of matter and simple velocities conjointly; that is, as the weight of the body A, multiplied into its velocity, is to the weight of the body B, mul­tiplied into its velocity, so is the instant force of A to the instant force of B. If A has 4lbs. of matter, and 1 degree of velocity, and B has 2lbs. of matter, and 4 degrees of velocity; then the momentum of their strokes will be as 4 is to 8; that is, supposing them to be instantaneously stop­ped by an obstacle.

2. The Effective Momentum, or force of moving bodies, is all the effect they will produce by im­pinging on any yielding obstacle, and is in the compound duplicate ratio of their quantities (or weights) multiplied into the squares of their ve­locities; that is, as the weight of the body A, multiplied into the square of its velocity, is to the weight of the body B, multiplied into the square of its velocity, so is the effective momen­tum of A to that of B. If A has 2lbs of matter and 2 degrees of velocity, and B 2lbs of matter and 4 degrees of velocity, then their effective mo­mentums are as 8 to 32; that is, a double velocity produces a quadruple effect.

Art. 7.

THE general Laws of Motion are the three following, viz.

Law 1. Every body will continue in its present [...]ate, Laws of mo­tion. whether it be at rest or moving uniformly [...] a right line, except it be compelled to change that state by some force impressed. ^*

[Page 8] Law 2. Art. 7. The change of motion or velocity is always proportional to the square root of the moving force impressed, New position. and in a right line with that force, and not as the force directly. ^*

Law 3. Action and re-action are always equal, and in contrary directions to each other. ^†

Art. 15.

HAVING now premised and considered all that is necessary for the better understand­ing those machines called Mechanical Powers, we come to treat of them, and they are six in number, viz.

The Lever, the Pulley, the Wheel and Axle, the Inclined Plain, the Wedge, and the Screw.

They are called Mechanical Powers, because they increase our power of raising or moving hea­vy bodies; and, But one prin­ciple in me­chanical pow­ers. although they are six in number, they seem to be reducible to one, viz. the Lever, and appear to be governed by one simple principle, which I shall call the First General Law of Me­chanical Powers; which is this, General laws of mechanical powers. viz. the Momen­tums [Page 28] of the power and weight are always equal when the engine is in equilibrio. Art. 15.

Momentum, her [...] means the product of [...] weight of the body multiplied into the distance [...] moves; that is, the power multiplied into its dis­tance moved, or into its distance from the centre of motion, or into its velocity, is equal to [...] weight multiplied into its distance moved, or into its distance from the centre of motion, or [...]nto [...] velocity; or, the power multiplied into its per­pendicular descent, is equal to the weight multi­plied into its perpendicular ascent.

The Second General Law of Mechanical Pow­ers, is,

The power of the engine, and velocity of the weight moved, are always in the inverse propor­tion to each other; that is, the greater the velo­city of the weight moved, the less it must be; and the less the velocity, the greater the weight may be; and that universally in all cases. There­fore,

The Third General Law is,

Part of the original power is always lost in over coming friction, inertia, &c. but no power can be gained by engines, when time is considered i [...] the calculation.

IN the theory of this science, we suppose al [...] plains to be perfectly smooth and even, levers [...] have no weight, cords to be perfectly pliable, and machines to have no friction; in short, all imper­fections are to be laid aside, until the theory [...] established, and then proper allowances are to be made.

Art. 16. Of the Lever.

A Lever is a bar of iron, wood, &c. one part of which is supported by a prop, and all other parts turn or move on that prop, as their centre of motion; and its length, on each side of the prop, is called its arms: the velocity or motion of every part of these arms, is directly as its dis­tance from its centre of motion, by 3d law of cir­cular motion.

The Lever—Observe the following laws:

1. The power and weight are to each other, Laws of the lever. as their distances from the centre of motion, or from the prop, respectively. ^*

2. The power is to the weight, as the distance the weight moves is to the distance the power moves, respectively. ^†

3. The power is to the weight, as the perpen­dicular ascent of the weight is to the perpendicu­lar descent of the power. ^‡

4. Their velocities are as their distances from their centre of motion, by 3d law of circular mo­tion.

These simple laws hold universally true in all mechanical powers or engines; Laws of the lever hold uni­versally true in all mecha­nical powers or engines. therefore it is easy (from these simple principles) to compute the power of any engine, either simple or com­pound; for it is only to find how much swifter the power moves than the weight, or how much farther it moves in the same time; and so much is the power, (and time of producing it) increased by the help of the engine.

Art. 17. GENERAL RULES FOR COMPUTING THE POWER OF ANY ENGINE.

1. DIVIDE either the distance of the power from its centre of motion, by the distance of the weight from its centre of motion. Or,

2. Divide the space passed through by the pow­er, by the space passed through by the weight. This space may be counted either on the arch de­scribed, or perpendiculars. And the quotient will shew how much the power is increased by the help of the engine.

Then multiply the power applied to the en­gine, by that quotient, and the product will be the power of the engine, whether simple or com­pound.

Art. 18. There are Four kinds of Levers.

1. THE common kind, Different kinds of le­vers. where the prop is placed between the weight and power, but gene­rally nearest the weight.

2. When the prop is at one end, the power at the other, and the weight between them.

3. When the prop is at one end, the weight at the other, and the power applied between them.

4. The bended lever, which differs only in form, but not in properties, from the others.

Those of the first and second kind have the same properties and powers, and are real mechanical powers, because they increase the power; but the third kind is a decrease of power, and only used to increase velocity, as in clocks, watches, and mills, where the first mover is too slow, and the velocity increased by the gearing of the wheels.

The machinery of the human frame is composed of the last kind of lever; Great power exerted by the muscles of the human frame. for when we lift a weight by the hand, resting the elbow on any thing, the muscle that exerts the force to raise the weight, is fastened at about one tenth of the distance from [Page 32] the elbow to the hand, Art. 18. and must exert a for [...] ten times as great as the weight raised; there­fore, he that can lift 56lbs with his arm at [...] right angle at the elbow, exerts a force equal [...] 560lbs. by the muscles of his arm. Wonder [...] is the power of the muscles in these cases. He [...] appears the reason, why men of low stature stronger than those of high, in proportion to the thickness, as is generally the case.

Art. 19. Compound Lever.

IF several levers are applied to act one upon another, Fig.9 Compound Lever. as 2 1 3, in fig. 9, where No. 1 is [...] the first kind, No. 2 of the second, and No. 3 [...] the third. The power of these levers, united act on the weight w, is thus found by the follow­ing rule, which will hold universally true in an [...] number of levers united, or wheels (which is si­milar thereto) acting upon one another.

Art. 20. Foundation of the rules for finding the motion of wheels, or number of cogs to pro­duce motions.

THE same rule holds good in calculating the powers of machines, consisting of wheels whether simple or compound, by counting the radius of the wheels as the levers; and because the diame­ters and circumferences of circles are proportional; we may take the circumference instead of the ra­dius, and it will be the same. Then again, be­cause the number of cogs in the wheels, constitute the circle, we may take the number of cogs and rounds instead of the circle or radius, and the re­sult will be the same.

Let Fig. 11 represent a water-mill (for grind­ing grain) double geared: Fig. 11.

• Number 8 The water-wheel,
• 4 The great cog-wheel,
• 2 The wallower,
• 3 The counter cog-wheel,
• 1 The trundle.
• 2 The mill-stones,

[Page 35] And let the above numbers also represent the radius of the wheels in feet. Art. 20.

Now suppose there be a power of 500lb. on the water-wheel, required what will be the force ex­erted on the mill-stone, 2 feet from the centre,

Then, The power on the water-wheel given, to find the power exerted on the mill-stone. by the rule, 500×8×2×1=8000, and 4×3×2=24, by which divide 8000, and it quotes 333,33 lb. the power or force required, exerted on the mill-stone two feet from its centre, which is the mean circle of a 6 feet stone.—And as the velocities are as the distance from the centre of mo­tion, by 3d law of circular motion, art. 13, there­fore, to find the velocity of the mean circle of the stone 2, deduce the following rule, viz.

1st. Multiply the velocity of the water-wheel into the radius or circumference of all the driving wheels, Rule to find the velocity of the mean cir­cle of a mill­stone. successively, and note the product.

2. Multiply the radius or circumference of all the leading wheels, successively, and note the pro­duct; divide the first by the last product, and the quotient will be the answer.

But observe here, that the driving wheels in this rule, are the leading levers in the last rule.

Art. 21. Power decreases as motion increases.

IT may be proper to observe here, that as the velocity of the stone is increased, the power to move it is decreased, and as its velocity is decreas­ed, [Page 36] the power [...] it to move it is increased, Art. 21. by 2d general law of mechanical powers. This [...] universally true in all engines that can possibly [...] contrived; which is evident from the 1st law of the lever, viz. the power multiplied into its ve­locity or distance moved, is equal to the weight multiplied into its velocity or distance moved.

Hence the general rule to compute the power of any engine, Rule to find the force ex­erted to move a mill-stone. simple or compo [...]nd, art. 17. [...] you have the moving power, and its velocity o [...] distance moved, given, and the velocity or dis­tance of the weight, then, to find the weight (which, in mills, is the force to move the stone, &c.) divide that product by the velocity of the weight or mill-stone, &c. and it quotes the weight or force exerted on the stone to move it: But a certain quantity or proportion of this force is lost, in order to obtain a velocity to the stone; which is shewn in art. 29. ^*

Art. 22. No Power gained by enlarging Undershot Water-Wheels.

THIS seems a proper time to shew the absurdi­ty of the idea of increasing the power of the mill, No power gained by in­creasing the diameter of undershot wa­ter-wheels, on the principle of lengthening the lever. by enlarging the diameter of the water-wheel, on the principle of lengthening the lever, or by dou­ble gearing mills where single gears will do; be­cause the power can neither be increased nor di­minished by the help of engines, while the velo­city of the body moved is to remain the same.

Art. 23. No Power gained by double gearing Mills, but some lost.

I might also go on to shew that no power or advantage is to be gained by double gearing Mills, No power gained by dou­ble gears, but some loft. upon any other principles than the following, viz.

1. The motion necessary for the stone, can sometimes be obtained without having the trun­dle too small, because we are obliged to have the pitch of the cogs and rounds, and the size of the spindle large enough, to bear the stress of the power. This pitch of gear and size of spindle may bear too great a proportion to the radius of the trundle (as does the size of the float to the ra­dius of the water-wheel, art 22) and may work hard. Therefore there may be a loss of power on that account; as there can be a loss but no gain, by 3. general law of mechanical powers, art. 15.

2. The mill may be made more convenient for two pair of stones to one water-wheel. ^*

Art. 24. Of the Pulley.

2. The pulley is a mechanical power well-known. One pulley, Fig. 10. if it be moveable by the [Page 39] weight doubles the power, Art. 24. because each rope sus­tains half the weight.

But if two or more pulleys be joined together in the common way, Of the pulley, an easy way to compute its power. then the easiest way of com­puting their power is, to count the number of ropes that join to the lower or moveable block, and so many times is the power increased; because all these ropes have to be shortened, and all run into one rope (called the fall) to which the mov­ing power is applied. If there be 4 ropes the power is increased fourfold. ^* See plate 1, fig. 10.

Art. 25. Of the Wheel and Axle.

3. THE wheel and axle, fig. 17, is a mechanical power, the same as the lever of the first kind; there fore the power is to the weight, as the diameter of the axle is to the diameter of the wheel; or the pow­er multiplied into the radius of the wheel, is equal to the weight multiplied into the radius of the axle, ^† in an equilibrium of this engine.

Art. 26. Of the inclined Plain.

4. The inclined plain is the fourth mechanical power: and in this the power is to the weight, Inclined plain its use. as the height of the plain is to its length. This [Page 40] is of use in rolling heavy bodies, Art. 26. such as barreled hogsheads, &c. into wheel-carriages, &c. and for letting them down again. See plate V, fig. 1. If the height of the plain be half its length, the [...] half the force will roll the body up the plain, that would lift it perpendicularly.

Art. 27. Of the Wedge.

5. The wedge is only a [...] inclined plain. Whence, The wedge equal to an in­clined plain. in the common form of it, the power applied will be to the resistance to be overcome as the thickness of the wedge is to the length thereof. This is a very great mechanical power, and may be said to excel all the rest; because wi [...] it we can effect, what we cannot with any [...] in the same time, and I think may be comput [...] in the following manner.

If the wedge be 12 inches long and 2 inches thick, Rule to com­pute the po­wer of the wedge. then the power to hold it in equilibrio [...] as 1 to balance 12 resistance; that is, 12 resistance pressing on each side of the wedge, ^* and whe [...] struck with a mallet, the whole force of the gra­vity of the mallet, added to the whole force [...] the agent exerted in the stroke, is communicated to the wedge in the time it continues to move [...] and this force to produce effect, is as the square of the velocity, with which the mallet strike [...] [Page 41] multiplied into its weight; Art. 27. therefore the mallet should not be too large, (see art. 44) because it may be too heavy for the workman's strength, and will meet too much resistance from the air, so that it will loose more by lessening the veloci­ty, than it will gain by its weight. Suppose a mallet of 10lb. strike with 5 velocity, its effect­ive momentum 250; but if it strike with 10 ve­locity, then its effective momentum is 1000. The effects produced by the strokes will be as 250 to 1000; and all the force of each stroke, ex­cept what may be destroyed by the friction of the wedge, is added in the wedge, until the sum of these forces amount to more than the resistance of the body to be split, therefore it must give way; but when the wedge does not move the whole force is destroyed by the friction. Therefore the less the inclination of the sides of the wedge, the greater resistance we can overcome by it, because it will be easier moved by the stroke.

Art. 28. Of the Screw.

6. THE Screw is the last mentioned mechani­cal power, Principles and power of the screw. and is a circular inclined plain (which will appear by wrapping a paper, cut in form of an inclined plain round a cylinder) and the lever of the first kind combined (the lever being applied to force the weight up in the inclined plain) and is a great mechanical power; its use is both for pres­sure and raising great weights. The power ap­plied is to the weight it will raise, as the distance through which the weight moves, is to the dis­tance through which the power moves; that is, as the distance of the threads of the screw, is to the cir­cle the power describes: so is the power to the weight it will raise. If the distance of the thread [Page 42] be half an inch, Art. 28. and the lever be 15 inches radius and the power applied be 10lb. then the power will describe a circle of 94 inches, while the weight raises half an inch; then, as half an inch is to 94 inches, so is 10lb to 1888lb the weight [...] the engine would raise with 10lb power. But this is supposing the screw to have no friction, [...] which it has a great deal.

Perhaps an improvement might be made on the screw, for some particular uses, by introducing rollers to take off the friction. See art. 33.

WE have hitherto considered the action [...] effect of these engines, Art. 29. as they would answer [...] the strictness of mathematical theory, were the [...] no such thing as friction or rubbing of parts upon each other; by which means, philosophers have allowed, that one third of the effect of the ma­chine is, at a medium, destroyed: which bring us to treat of it next in course. ^*

Art. 30. Of the Fly-wheel, and its Use.

BEFORE I dismiss the subject of mechanics powers, I shall take notice of the fly-wheel, [...] [Page 43] use of which is to regulate the motion of engines, Art. 30. Fly-wheel no increaser of power. and should be made of cast metal, of a circular form, that it may not meet with much resistance from the air.

Many have taken this wheel for an increaser of power, whereas it is, in reality, a considerable de­stroyer of it; which appears evident, when we consider that it has no motion of its own, but re­ceives all its motion from the first mover, and, as the friction of the gudgeons and resistance of the air are to be overcome, it cannot be done without some power; yet this wheel is of great use in many cases, viz.

1st. For regulating the power, Its use. where it is ir­regularly applied, such as the treadle or crank moved by foot or hand, as spinning-wheels, turn­ing lathes, flax-mills, or where steam is applied, by a crank, to produce a circular motion.

2d. Where the resistance is irregular, by jerks, &c. such as saw-mills, forges, flitting-mills, pow­der-mills, &c.

The fly-wheel, by its inertia, regulates the motion; because, if it be very heavy, it will re­quire a great many little shocks or impulses of power to give it a considerable velocity, and it will require as many equal shocks of resistance to destroy said velocity, by axiom 3, art. 1.

While a rolling or flitting mill is running emp­ty, the force of the water is employed in generating velocity to the fly-wheel [a heavy water-wheel will have the same effect] which force, summed up in the fly, will be sufficient to continue the motion, without much abatement, while the sheet is running between the rollers; whereas, had the force of the water been lost while the mill was empty, she would have slack­ [...]ned in motion too much before the sheet got through. This may be the case where water is [...]carce.

Art. 31.

FROM what I can gather from different authors, Friction. ^* and by my own experiments, [...] conclude that the doctrine of friction is as follows and we may say it is subject to the following laws [...] viz.

Art. 32. Of reducing Friction.

TO reduce friction, we must, by mechanical contrivances, To reduce friction. reduce the motion of the rubbing parts as much as possible; which is done, either by making the gudgeons small and the diameter of wheels large, or by fixing the gudgeons to run o [...] friction-wheels. By friction wheels. Fig. 14. Thus, let A, fig. 14, represent the gudgeon of a wheel set to run on the ver [...] of two wheels of cast metal passing each other [...] little, and the gudgeon laying between them. It is evident, that as A turns, it will turn both friction-wheels; and, if the diameter of gudgeon A is 2 inches, and that of the wheels 12, then the wheels will turn once while A turns 6 times, so that the velocity of the gudgeons C C of the wheels, is to the velocity of the gudgeon A, as [...] is to 6, supposing them to be equal in size; but [...] there are 4 of them to bear A, they may be bu [...] half the diameter, and then their velocity will be to that of A, as 1 is to 12; or A might be set on one wheel, as at B, with supporters to keep it on; and, if friction-wheels are added to friction wheels, the friction may be reduced to almost no­thing by that means.

Art. 33. Late Invention to reduce Friction.

WHEEL-CARRIAGES, Rollers appli­ed to reduce friction. pullies, and such wheels as have large axles in proportion to their diameters, have much friction. There has been a late discovery, in England, of applying the prin­ciple of the roller to them; which may be so done as almost totally to destroy the friction.

The easiest method possible, of moving heavy bodies horizontally, is the roller.

Let A B, fig. 15, Fig. 15. represent a body of 100 tons weight (with the underside perfectly smooth and even) set on two rollers, perfectly hard, smooth, and round, rolling on the horizontal plain C D, No friction in rollers. perfectly hard, smooth, and even; it is evident that this body is supported by two lines perfectly perpendicular, and, if globes were used instead of rollers, the least force would move it in any horizontal direction; even a spider's web would be sufficient, giving it time to overcome the visinertia of the body: But as perfect hard­ness, smoothness, &c. are not attainable, a little friction will still remain.

This principle is, or may be, applied to wheel-carriages, in the following manner:

Let the outside ring B C D, fig. 16, Fig.16. represent the box of a carriage-wheel, the inside circle A the axle, the circles a a a a a a the rollers round the axle between it and the box, and the inner [...]ring a thin plate for the pivots of the rollers to run in, to keep them at a proper distance from each other. When the wheel turns the rollers pass round on the axle, and on the inside of the box, and we may say without friction, because there is no rubbing of the parts past one another. ^*

Art. 34.

THE effect of a machine, Effect of a machine, what. is the distance which it moves or the velocity with which i [...] moves any body to which it is applied to give mo­tion, in a given time; and the weight of the body multiplied into its distance moved, or into its ve­locity, shews the effect.

The theory published by philosophers, Old theory of maximum, motion, and load of en­gines. and re­ceived and taught as true, for several centuries past, is, that any machine will work with its greatest perfection when it is charged with [...] [Page 49] 4-9 of the power that would hold it in equilibrio, Art. 34. and then its velocity will be just 1-3 of the great­est velocity of the moving power.

To explain this, they suppose the water-wheel, fig.17, Fig. 17. to be of the undershot kind, 16 feet dia­meter, turned by water issuing from under a 4 feet head, with a gate 1. foot wide, 1 foot high drawn; then the force will be 250lbs. because that is the weight of the column of water above the gate, and its velocity will be 16,2 feet per second, as shall be shewn under the head of Hydraulics; then the wheel will be moved by a power of 250lbs. and if let run empty, will move with a velocity of 16 feet per second; but if we hang the weight W to the axle (of 2 feet diameter) with a rope, and continue to add to it until it stops the wheel, and holds it in equilibrio, the weight will be found to be 2000lbs. by the rule, art. 19; and then the effect of the machine is nothing, because the velocity is nothing: But as we decrease the weight W, the wheel begins to move, and its velocity increases accordingly; and then the pro­duct of the weight multiplied into its velocity, will increase until the weight is decreased to 4-9 of 2000=888,7, which, multiplied into its dis­tance moved or velocity, will produce the great­est effect, and the velocity of the wheel will then be 1-3 of 16 feet, or 5,33 feet per second. So say those who have treated of it.

This will appear plainer to a young learner, Theory of maximums explained by application of an elevator. Fig. 17. if he will conceive this wheel to be applied to work an elevator, as E, fig. 17, to hoist wheat, and suppose that the buckets, when all full, contain 9 pecks, and will hold the wheel in equilibrio, it is evident it will then hoist none, because it has no motion; then, in order to obtain motion, we must lessen the quantity in the buckets, when the wheel will begin to move, and hoist faster and [Page 50] faster until the quantity is decreased to 4-9, Art. 34. or 4 pecks, and then, by the theory, the velocity of the machine will be 1-3 of the greatest velocity, when it will hoist the greatest quantity possible in a given time: for if we lessen the quantity in the buckets below 4 pecks, the quantity hoisted in any give [...] time will be lessened.

This is the theory established, for demonstra­tion of which, see Martin's Philosophy, vol. 1, page 185-187.

Art. 35. Old Theory investigated.

IN order to investigate this theory, Investigation of the old theory. and the better to understand what has been said, let us consider as follows, viz.

1. That the velocity of spouting water, under 4 feet head, is 16 feet per second, nearly.

2. The section or area of the gate drawn, in feet, multiplied by the height of the head in feet, gives the cubic feet in the whole column, which multiplied by 62,5 (the weight of a cubic foot of water) gives the weight or force of the whole column pressing on the wheel.

3. That the radius of the wheel, multiplied by the force, and that product divided by the radius of the axle, gives the weight that will hold the wheel in equilibrio.

4. That the absolute velocity of the wheel, subtracted from the absolute velocity of the wa­ter, leaves the relative velocity with which the water strikes the wheel in motion.

5. That as the radius of the wheel is to the ra­dius of the axle, so is the velocity of the wheel to the velocity of the weight hoisted on the axle.

[Page 51] 6. That the effects of spouting fluids are as the squares of their velocities (see art. 45, law 6) but the instant force of striking fluids, Art. 35. are as their velocities simply. See art. 8.

7. That the weight hoisted, multiplied into its perpendicular ascent, gives the effect.

8. That the weight of water expended, multi­plied into its perpendicular descent, gives the power used per second.

On these pinciples I have calculated the follow­ing scale; first supposing the force of striking fluids to be as the square of their striking or rela­tive velocity, which brings out the maximum a­greeably to the old theory, viz.

When the load, at equilibrio, is 2000, then the maximum load is 888,7=4/9; of 2000, when the effect is at its greatest, viz. 591,98, as appears in the 6th column, and then the velocity of the wheel is 5,333 feet per second, equal to 1-3 of 16, the velocity of the water, as appears in the 5th line of the scale: but as there is an evi­dent error in the first principle of this theory, Old theory doubted. by counting the instant force of the water on the wheel to be as the square of its striking velocity, therefore it cannot be true. See art. 41.

I then calculate upon this principle, viz. That the instant force of striking fluids is as their velo­city simply, then the load that the machine will carry, with its different velocities, will be as the velocity simply, as appears in the 7th column, and the load, at a maximum, is 1000lb=½ of 2000, the load at equilibrio, when the velocity of the wheel is 8 feet=½ of 16 the velocity of the water per second; and then the effect is at its greatest, as shewn in the 8th column, viz. 1000, as appears in the 4th line of the scale.

This I call the new theory, New theory. (because I found that William Waring had also, about the same time, established it, see art. 38) viz. That when [Page 52] any machine is charged with just 1-2 of the load that will hold it in equilibrio, Art. 35. its velocity will be just 1-2 of the natural velocity of the moving power, and then its effect will be at a maximum, or greatest possible.

This appears to be the way by which this great error has been so long overlooked by philosophers, and which has rendered the theory of no use in practice, but led many into expensive errors, thereby bringing great discredit upon philoso­phy.

For demonstrations of the old theory, see Mar­tin's Phil. vol. I, page 185—187.

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Art. 36. New Theory doubted.

BUT although I know that the velocity of the wheel, by this new theory is much nearer prac­tice than the old, (tho' rather slow) yet I am led to doubt the theory, for the following reasons, viz.

When I consider that there are 16 cubic feet of water, equal 1000lbs. expended in a second, which multiplied by its perpendicular descent, 4 feet, pro­duces the power 4000. The ratio of the power and effect by the old theory is as 10 to 1,47, and by the new as 4 to 1; as appears in the 9th co­lumn of the scale; which is a proof that the old theory is a great error, and sufficient cause of doubt that there is yet some error in the new. And as the subject is of the greatest consequence in prac­tical mechanics. Therefore I proceed, to endea­vour to discover a true theory, and will shew my work in order, that if I establish a theory it may be the easier understood, if right; or detected, if wrong.

Art. 37. Scale of Experiments.

THE following scale of actual experiments were made to prove whether the resistance occu­pies its proportion of the velocity, in order that I might judge whether the foregoing scale was foun­ded on true principles; the experiments were not very accurately performed, but often repeated, and proved always nearly the same. See plate II, fig. 18.

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[Page 59] By this scale it appears, Art. 37. that when the power P falls freely without any load, it descends 40 feet in five equal parts of time, but when charged with 3,5lbs.=1/2 P, which was 7lbs. it then took up 10 of those parts of time to descend the same distance; which seems to shew, that the charge occupies its proportional part of the whole velocity, which was wanted to be known, and the maximum ap­pear as in the last scale. ^* It also shews, that the effect is not as the weight multiplied into the square of its ascending velocity, this being the measure of the effect that would be produced by the stroke on a non-elastic body.

This experiment partly confirmed me in what I have called the New Theory; but still doubt­ing, and after I had formed the foregoing tables, I called on the late ingenius and worthy friend, William Waring, teacher in the Friends' Acade­my, Philadelphia, for his assistance. He told me he had discovered the error in the old theory, and corrected it in a paper which he had laid be­fore the Philosophical Society of Philadelphia, wherein he had shewn that the velocity of the undershot water-wheel, to produce a maximum effect, must be just one half the velocity of the water.

Art. 38. William Waring's Theory.

The following are extracts from the above mentioned paper, William war­ing's theory. published in the third volume of the Transactions of the American Philosophical Society, held at Philadelphia, p. 144.

After his learned and modest introduction, in which he shews the necessity of correcting so great an error as the old theory, he begins with these words, viz.

[Page 60] "But, Art. 38. to come to the point, I would just pro­mise these

Art. 39. Extract from a further paper, read in the phi­losophical society, April 5th, 1793.

"Since the philosophical society were pleased favour my crude observations on the theory mills, Further ex­tracts from W. Waring's pub­lication con­cerning his new theory. with a publication in their transaction [...] am apprehensive some part thereof may be [...] ­plied, it being therein demonstrated, that ' [...] force of an invariable stream, impinging against mill-wheel in motion, is in the simple direct ratio of the relative velocity.' Some may suppose that the effect produced, should be in the [...] proportion, and either fall into an error, or [...] ­ing by experiment, the effect to be as the sq [...] of the velocity, conclude the new theory to be [...] well founded; I therefore wish there had been little added, to prevent such misapplication, [...] ­fore the society had been troubled with the [...] ­ing of my paper on that subject: perhaps some­thing like the following.

The maximum effect of an undershot wh [...] produced by a given quantity of water, The effect of under shot wheels as the squares of the velocities of the water. in a [...] ­en time, is in the duplicate ratio, of the velocity of the water; for the effect must be as the ime­tus acting on the wheel, multiplied into the ve­locity thereof: but this impetus is demonstrate to be simply as the relative velocity, prop [Page 63] and the velocity of the wheel, Art. 39. producing a max­imum, being half of the water by prop. II. is likewise as the velocity of the water; hence the power acting on the wheel, multiplied into the velocity of the wheel, or the effect produc­ed, must be in the duplicate ratio of the velocity of the water. Q. E. D.

COROL. Hence the effect of a given quantity of water, in a given time, will be as the height of the head, because this height is as the square of the velocity. This also agrees with experi­ment.

If the force, acting on the wheel, were in duplicate ratio of the water's velocity, as usual­ly asserted, then the effect would be as the cube thereof, when the quantity of water and time are given, which is contrary to the result of experi­ment."

Art. 40. Waring's Theory doubted.

From the time I first called on William Waring, W. Waring's theory doubt­ed. until I read his publication on the subject (after his death) I had rested partly satisfied, with the new theory, as I have called it, with respect to the velocity of the wheel, at least; but finding that he had not determined the charge, as well as the velocity, by which we might have compared the ratio of the power and the effect produced, It not agree­ing fully with practice. and that he had assigned reasons somewhat different for the error; and having found the motion to be rather too slow to agree with practice, I began to suspect the whole, and resumed the search for a true theo­ry, thinking that perhaps no person had ever yet considered every thing that affects the calcula­tion, I therefore premised the following

Art: 41. Search for a true Theory, commenced on a new Plan.

IT appears, that we have applied wrong prin­ciples in our search after a true theory of the max­imum velocity and load of undershot water-wheels, or other engines moved by a constant power, that does not increase or decrease in quantity on the engine, as on an overshot water-wheel, as the ve­locity varies.

Let us suppose water to issue from under a head of 16 feet, on an undershot water-wheel; then, if the wheel moves freely with the water, its velo­city will be 32, 4 feet per second; but will bear no load.

Again, suppose we load it, so as to reduce its motion to be equal the velocity of water spouting from under 15 feet; it appears evident that the load will then be just equal to that 1 foot of the head, the velocity of which is checked; and this load multiplied into the velocity of the wheel, viz. 31, 34×1=31,34 for the effect.

This appears to be the true principle, from which we must seek the maximum velocity and load, for such engines as are moved by one constant power; and on this principle I have calculated the following scale.

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[Page 67] In this scale, Art. 41. let us suppose the aperture of the gate to be a square foot; then the greatest load that will balance the head, will be 16 cubic feet of water, and the different loads will be shewn in cubic feet of water.

And then it appears, by this scale, that when the wheel is loaded with 10,66 cubic feet of wa­ter, just 2-3 of the greatest load, its velocity will be 18,71 feet per second, just ,577 parts of the velo­city of the water, and the effect produced is at a maximum, or the greatest possible, viz. 199,44.

To make this more plain, Fig. 19. let us suppose A B, plate II, fig. 19, to be a fall of water 16 feet, which we wish to apply to produce the greatest effect possible, by hoisting water on its side oppo­site to the power applied; first, on the undershot principle, where the water acts by its impulse on­ly. Now let us suppose the water to strike the wheel at I, then, if we let the wheel move free­ly without any load, it will move with the velo­city of the water, viz. 32,4 feet per second, but will produce no effect, if the water issue at C; al­though there be 32,4 cubic feet of water expend­ed, under 16 feet perpendicular descent. Let the weight of a cubic foot of water be represented by unity or 1, for ease in counting; then 32,4×16 will shew the power expended, per second, viz. 518,4; and the water it hoists multiplied into its perpendicular ascent, or height hoisted, will shew the effect. Then, in order to obtain effect from the power, we load the wheel; the simplest way of doing which, is, to cause the tube of water C D to act on the back of the bucket at I; then, if CD be equal to AB, the wheel will be held in equili­brio; this is the greatest load, and the whole of the fall AB is balanced, and no part left to give the wheel velocity; therefore the effect=0. But if we make C D=12 feet of A B, then from 4 to A=4 feet, is left unbalanced, to give velocity to the wheel, which is now loaded with 12 feet, and [Page 68] exactly balanced by 12 on the other side, Art. 41. and per­fectly free to move either way by the least for [...] applied: Therefore it is evident, that the [...] pressure or force of 4 feet of A B will act to [...] velocity to the wheel, and, as there is no re [...] ­ance to oppose the pressure of these 4 feet, [...] velocity will be the same that water will [...] from under 4 feet head, viz. 16,2 feet per second which is shewn by the horizontal line 4=16 [...] and the perpendicular line 12=12 represents [...] load of the wheel; the rectangle or product of the [...] two lines, form a parallelogram, the area of [...] is a true representation of the effect, viz. the [...] 12 multiplied into 16,2 the distance it moves [...] second=194,4, the effect. In like manner we [...] try the effect of different loads; the less the [...] the greater will be the velocity. The horizon lines all shew the velocity of the wheel, produ [...] by the respective heads left unbalanced, and the per­pendicular lines shew the load on the wheel; [...] we find that when the load is 10,66=⅔ 16, [...] load at equilibrio, the velocity of the wheel [...] be 18,71 feet per second; which is 577/1000 parts, [...] a little less than 6 tenths, or 2/4 the velocity the water, Power and ef­fect as 3 to 2 on overshot wheels. and the effect is 199,44, the maximum or greatest possible; and if the aperture of [...] gate be 1 foot, the quantity will be 18,71 [...] feet per second. The power being 18,71 [...] feet expended per second, multiplied by 16 [...] the perpendicular descent, produces 299,36, [...] the ratio of the power and effect being 10 to 6 5/ [...] as 3: 2; but this is supposing none of the [...] lost by non-elasticity.

This may appear plainer, if we suppose the wa­ter to descend the tube A B, and, by its pressure to raise the water in the tube C D; now it is [...] ­dent, that if we raise the water to D, we [...] no velocity, therefore effect=o. Then again, we open the gate at C, we have 32,4 feet per se­cond velocity, but because we do not hoist [...] [Page 69] water any distance, Art. 41. effect=o. Therefore the maximum is somewhere between C and D. Then suppose we open gates of 1 foot area, at different heights, the velocity will shew the quantity of cubic feet raised; which multiplied by the per­pendicular height of the gate from C, or height raised, gives the effect as before, and the maxi­mum as before. But here we must consider, that in both these cases, the water acts as a perfect definite quantity, which will produce effects equal to elastic bodies, or equal to its gravity (see art.59) which is impracticable in practice: Whereas when it acts by percussion only, it communicates only half of its original force, on account of its non-elasticity, the other half being spent in splashing about (see art. 8); therefore the true effect will be 33/100,(a little more than 1-3) of the moving pow­er; because nearly 1-3 is lost to obtain velocity, and half of the remaining 2-3 is lost by non-elasti­city. These are the reasons, why the effects pro­duced by an undershot wheel is only half of that produced by an overshot wheel, the perpendicu­lar descent and quantity of water being equal. And this agrees with Smeaton's experiments (see art.68); but if we suppose the velocity of the wheel to be 1-3 that of the water-10,8, and the load to be 4-9 of 16, the greatest load at equilibrio; which is-7, 111, as by old theory, then the effect will be 10,8×4-9 of 16-76,79 for the effect, which is quite too little, the moving power be­ing 32,4 cubic feet of water, multiplied by 16 feet descent-518,4, the effect by this theory be­ing less than , 15/100 of the power, about half equal to the effect by experiment, which effect is set on the outside of the dotted circle in the fig.(19.) The dotted lines join the corner of the parallelo­grams, formed by the lines that represent the loads and velocities, in each experiment or sup­position, the areas of which truly represent the effect, and the dotted line A a d x, meeting the [Page 70] perpendicular line x E in the point x, Art. 41. formi [...] the parallelogram ABCx, truly represents the power=518,4.

Again, if we suppose the wheel to move w [...] half the velocity of the water, viz. 16,2 feet [...] second, and be loaded with half the greatest [...]=8, according to Waring's theory, then the ef­fect will be 16,2×8=129,6 for the effect, abo [...] 23/100 of the power, which is still less than by ex­periment. All this seems to confirm the maxi­mum brought out on the new principles.

But, if we suppose according to the new prin­ciple, that, when the wheel moves with the ve­locity of 16,2 feet per second, which is the velo­city of a 4 feet head, that it will then bear as load the remaining 12 feet, then the effect [...] be 16,2×12=194,4, which nearly agrees wi [...] practice: but as most mills in practice move [...] ­ter, rather than slower, than what I call the [...] maximum, shews it to be nearest the truth, [...] true maximum velocity being, 577 of the veloci­ty of the water, and the mills in practice movi [...] with 2-3, and generally quicker. ^*

[Page 71] This scale also establishes a true maximum charge for an overshot wheel, Art. 41. when the case is such, Maximum charge of overshot wheels, sup­posing the same quantity to be always in the buck­ets. that the power or quantity of water on the wheel at once, is always the same, even although the velocity vary, which would be the case, if the buckets were kept always full: for, suppose the water to be shot into the wheel at a, and by its gravity to raise the whole water again on the opposite side; then, as soon as the water rises in the wheel to d, it is evident that the wheel will stop, and effect=o; therefore we must let the water out of the wheel, before it rises to [...], which will be in effect to loose part of the power to obtain velocity. If the buckets both descending and ascending, carry a column of wa­ter 1 foot square, then the velocity of the wheel will shew the quantity hoisted as before, which, multiplied by the perpendicular ascent, shews the effect, and the quantity expended, multiplied by the perpendicular descent shews the power; and we find, that when the wheel is loaded with 2-3 of the power, the effect will be at a maximum, i. e. the whole of the water is hoisted, 2-3 of its whole descent, or 2-3 of the water the whole of the descent, therefore the ratio of the power to the effect is as 3 to 2, double to the effect of an undershot wheel: but this is, supposing the quan­tity in the buckets to be always the same; where­ [...], in overshot wheels, the quantity in the buck­ets is universally as the velocity of the wheel, i.e. the slower the motion of the wheel, the greater the quantity in the buckets, and the greater the velocity the less the quantity: but, [...]gain, as we are obliged to let the overshot wheel love with a considerable velocity, in order to obtain a steady, regular motion to the mill, we will find this charge to be always nearly right; hence I deduce the following theory.

Art. 42.

Art. 43. Of the Maximum velocity for Overshot Wheels, or those that are moved by the weight of the Wa­ter.

BEFORE I dismiss the subject of maximums, I think it best to consider, whether this doctrine will apply to the motion of the overshot wheels. It seems to be the general opinion of those, who consider the matter, that it will not; but, that the slower the wheel moves, provided it be ca­pacious enough to hold all the water, without losing any until it be delivered at the bottom of the wheel, the greater will be the effect, which appears to be the case in theory (see art. 36); but how far this theory will hold good in practice, is to be considered. Having met with the ingeni­ous James Smeaton's experiments, where he shews, that, when the circumference of his lit­tle wheel, of 24 inches diameter, (head 6 inches) moved with about 3,1 feet per second (although the greatest effect was diminished about 1/20 of the whole) he obtained the best effect, with a steady, regular motion. Hence he concludes about 3 feet to be the best velocity for the circumference of overshot mills. See art. 68. I undertook to compare this theory of his, with the best mills [Page 76] in practice, Art. 43. and, finding that those of about 17 feet diameter, generally moved about 9 feet per second, Smeaton's opinion of the proper veloci­ty for the cir­cumference of overshot wheels, does not agree with practice. being treble the velocity assigned by Smea­ton, I began to doubt the theory, which led [...] to inquire into the principle, that moves an over­shot wheel, and this I found to be a body descend­ing by its gravity, and subject to all the laws of falling bodies, (art. 9.) or of bodies descending in­clined plains, The principle of the power that moves overshot wheels is that of a fall­ing body. and curved surfaces (art. 10, 11,) the motion being equably accelerated in the who [...] of its descent, its velocity being as the square ro [...] of the distance descended through, and that the diameter of the wheel was distance the water descended through. Their veloci­ties vary, and will be as the square root of their diame­ters. From thence I concluded that the velocity of the circumference of the overshot wheels, was, as the square root of their diameters, and of the distance the water has [...] descend, if it be a breast of pitch-back wheel [...] then, taking Smeaton's experiments, with [...] wheel of 2 feet diameter, for a foundation, [...] say, As the square root of the diameter of Smea­ton's wheel, is to its maximum velocity, so is the square root of the diameter of any other wheel to its maximum velocity. Upon these principles I have calculated the following table; This rule found to agree with practice. and, hav­ing compared it with at least 50 mills in practice found it to agree so nearly with all the best con­structed ones, that I have reason to believe it is founded on true principles.

If an overshot wheel moves freely without re­sistance, Their veloci­ties will be a mean be­tween the least and greatest of a body falling through its diameter. it will acquire a mean velocity, between that of the water coming on the wheel, and the greatest velocity it would acquire, by falling free­ly through its whole descent: therefore this mean velocity will be greater, than the velocity of the water coming on the wheel; consequent­ly the backs of the buckets will overtake the wa­ter, and drive a great part of it out of the wheel. But, the velocity of the water being accelerated by its gravity, overtakes the wheel, perhaps ha [...] [Page 77] way down, Art. 43. and presses on the buckets, until it leaves the wheel: Water presses harder on the lower than upper quarter of the whele. therefore the water presses harder upon the buckets in the lower, than in the upper quarter of the wheel. Hence appears the reason why some wheels cast their water, which is always the case, when the head is not sufficient to give it velocity enough to enter the buckets. But this depends also much on the po­sition of the buckets, and direction of the shute into them. It, however, appears evident that the head of water above the wheel, should be nicely adjusted, to suit the velocity of the wheel. Here we may consider, that the head above the wheel acts by percussion, or on the same principles with the undershot wheel, and, as we have shewn (art. 41) that the undershot wheel should move with nearly 2-3 of the velocity of the water, it appears, that we should allow a head over the wheel, that will give such velocity to the water, as will be to that of the wheel as 3 to 2. Thus the whole descent of the water of a mill-seat should be nicely divided, The whole descent must be nicely di­vided be­tween head and fall. between head and fall, to suit each other, in order to obtain the best effect, and a steady-moving mill. First find the velocity that the wheel will move with, by the weight of the water, for any diameter you may suppose you will take for the wheel, and divide said velocity into two parts; then try if your head is such, as will cause the water to come on with a velocity of 3 such parts, mak­ing due allowances for the friction of the water, according to the aperture. See art. 55. Then, if the buckets and direction of the shute be right, the wheel will receive the water well, and move to the best advantage, keeping a steady, regular motion when at work, loaded or charged with a resistance equal to 2-3 of its power, (art. 41. 42.)

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Art. 44. Application of the doctrine of maxi­mums.

THIS doctrine of maximums is very interest­ing, and is to be met with in many occurrences through life.

1. It has been shewn, that there is a maximum load and velocity for all engines, to suit the pow­er and velocity of the moving power.

2. There is also a maximum size, velocity and feed for mill-stones, to suit the power; and velo­city for rolling-screens, and bolting-reels, by which the greatest work can be done in the best manner, in a given time.

3. A maximum degree of perfection and close­ness, with which grain is to be manufactured in­to flour, so as to yield the greatest profit by the mill in a day or week, and this maximum is con­tinually changing with the prices in the market, so that what would be the greatest profit at one time, will sink money at another. See art. 113.

4. A maximum weight for mallets, axes, sledges, &c. according to the strength of those that use them.

A true attention to the principles of maxi­mums, will prevent us from running into many errors.

Art. 47. Application of the Laws of Motion to Undershot Wheels.

To give a short and comprehensive detail of the ideas, I have collected from the different authors, and from the result of my own reasoning on the laws of motion, and of spouting fluids, as they ap­ply. to move undershot mills, I constructed fig. 44 plate V.

Let us suppose two large wheels, one of 1 [...] feet, and the other of 24 feet radius, then the cir­cumference of the largest, will be double that [...] the smallest: and let A 16, and C 16, be two [...] stocks of water, of 16 feet head, each.

1. Then, if we open a gate of 1 square foot at, [...] to issue from the penstock A 16, Laws of moti­on and spout­ing fluids app­lied in prac­tice. and impinge [...] the small wheel at I, the water being pressed by 4 feet head, will move 16 feet per second.(we omit fractions) The instant pressure or force on that gate, being 4 cubic feet of water, it will require a resistance of 4 cubic feet of water, from the head C 16 to stop it, and hold it in equilibrio (but we suppose the water cannot escape unless the wheel moves, so that no force be lost by non-elasticity) Here equal quantities of matter, with equal velocities, have their momentums equal.

2. Again, suppose we open a gate of 1 square foot at A 16 under 16 feet head, it will strike the large wheel at k, with velocity 32, its instant force or pressure being 16 cubic feet of water, it will require 16 cubic feet resistance, from the head C 16, to stop or balance it. In this case the [Page 85] pressure or instant force is quadruple, Art. 47. and so is the resistance, but the velocity only double, to the first case. In these two cases the forces and resistances being equal quantities, with equal ve­locities, their momentums are equal.

3. Again, suppose the head C 16 to be raised to E, 16 feet above 4, and a gate drawn ¼ of a square foot, then the instant pressure on the float I of the small wheel, will be 4 cubic feet, pressing on ¼ of a square foot, and will exactly balance 4 cu­bic feet, pressing on 1 square foot, from the head A 16; and the wheel will be in equilibrio, (sup­posing the water cannot escape until the wheel moves as before) although the one has power of velocity 32, and the other only 16 feet per se­cond. Their loads at equilibrio are equal, conse­quently their loads at a maximum velocity and charge, will be equal, but their velocities differ­ent.

Then, to try their effects, suppose, first, the wheel to move by the 4 feet head, its maximum velocity to be half the velocity of the water, which is 16, and its maximum load to be half its greatest load, which is 4, by Waring's theory; then the velocity 16|2 × by the load 4|2=16, the effect of the 4 feet head, with 16 cubic feet expended; because the velocity of the water is 16, and the gate 1 foot.

Again, suppose it to move by the 16 feet head and gate of ¼ of a foot; then the velocity 32|2X by the load 4|2=32, the effect, with but 8 cu­bic feet expended, because the velocity of the water is 32, and the gate but ¼ of a foot.

In this case the instant forces are equal, each being 4; but the one moving a body only [...] as heavy as the other, moves with velocity 32, and produces effect 32, while the other, moving with velocity 16, produces effect 16. A double velo­city, with equal instant pressure, produces a dou­ble effect, which seems to be according to the Newtonian theory. And in this sense the [Page 86] momentums of bodies in motion, Art. 47. are as th [...] quantities, multiplied into their simple velocities, and this I call the instant momentums.

But when we consider, that in the above case, it was the quantity of matter put in motion, [...] water expended, that produced the effect, we find that the quantity 16, with velocity 16, pro­duced effect 16; while qu. 8, with velocity 32 produced effect 32. Here the effects are as the [...] quantities, multiplied into the squares of their velocities; and this I call the effective momen­tums.

Again, if the quantity expended under each head, had been equal, their effects would have been 16 and 64, which is as the squares of their velocities, 16 and 32.

4. Again, suppose both wheels to be on one shaft, and let a gate of ½ of a square foot be drawn at 16 C, to strike the wheel at k, the head being 16 feet, the instant pressure on the gate will be 2 cu­bic feet of water, which is half of the 4 feet head with 1 foot gate, from A 4 striking at I; but the 16 feet head, with instant pressure 2, acting [...] the great wheel, will balance 4 feet on the small one, because the lever is of double length, and the wheels will be in equilibrio. Then, by Wa­ring's theory, the greatest load of the 16 feet head being 2, its load at a maximum will be 4 and the velocity of the water being 32, the max­imum velocity of the wheel will be 16. Now the velocity 16×1=16, the effect of the 16 feet head, and gate of 1/8 of a foot. The greatest load of the 4 feet head being 4, its maximum load [...] the velocity of the water 16, and the velocity of the wheel 8, now 8×2=16, the effect. Here the effects are equal: and here again the effects are as the instant pressures, multiplied into their simple velocities; and the resistances that would instant­ly stop them, must be equal thereto, in the same ratio.

[Page 87] But when we consider, Art. 47. that in this case, the 4 feet head expended 16 cubic feet of water, with velocity 16, and produced effect 16; while the 16 feet head expended only 4 cubic feet of water, with velocity 32, and produced effect 16, we find, that the effects are as their quantities, multiplied into the squares of their velocities.

And when we consider, that the gate of ⅓ of a square foot, with velocity 32, produced effects equal to the gate of 1 square foot, with velocity 16, it is evident, that if we make the gates equal, the effects will be as 8 to 1; that is, the effects of spouting fluids, with equal apertures, are as the cubes of their velocities; because, their instant forces are as the squares of their velocities by 6th law.

Art. 48. The Hydrostatic Paradox.

THE pressure of fluids is as their perpendicu­lar heights, without any regard to their quanti­ty: and their pressure upwards is equal to their pressure downwards. In short, their pressure is every way equal, at any equal distance from their surface. ^*

[Page 88] In a vessel of a cubic form, Art. 48. whose sides and bot­tom are equal, the pressure on each side is just half the pressure on the bottom; therefore the pressure on the bottom and sides, is equal to [...] times its pressure on the bottom. ^*

And in this sense fluids may be said to act with three times the force of solids. Solids act by gravity only, but fluids by gravity and pressure jointly. Solids act with a force proportional to their quantity of matter; but fluids act with [...] pressure proportional to their altitude only.

Art. 50.

The weight of a cubic foot of water is found by experience, to be 1000 ounces avoirdupoise, or 62,5lb. On these principles is founded the following theorem.

Art. 51. Rule for finding the Velocity of spouting Water.

BY experiments it has been found, that water will spout from under a 4 feet head, with a velocity equal to 16,2 feet per second, and from under 16 feet head, with a velocity equal to 32,4 feet per second.

On these experiments, and the 2nd law of spout­ing fluids, is founded the following theorem, or [Page 90] general rule for finding the velocity of water un­der any given head. Art. 51.

Art. 52.

FROM the laws of spouting fluids, theorems I. and II. the theory for finding the maximum charge and velocity of undershot wheels, (art. 42) and the principle of non-elasticity, is deduc­ed the following theorem for finding the effect of any gate, drawn under any given head, upon an undershot water-wheel.

Art. 53. Water applied on Wheels to act by Gravity.

BUT when fluids are applied to act on wheels to produce effects by their gravity, they act on very different principles, producing double ef­fects, to what they do by percussion, and then [Page 93] their powers are directly as their quantity or weight, Art. 53. multiplied into their perpendicular de­scent.

Art. 54. Investigation of the Principles of Overshot Mills

SOME have asserted, and many believed, that water is applied to great disadvantage on [Page 95] the principle of an overshot mill; Art. 54 because, say they, there are never more than two buckets, at once Water acts with as much power on an overshot, as if it was all to act fairly on the breast bucket, in the whole of its perpendicular descent. that can be said to act fairly on the end of the lever, as the arms of the wheel are called in these arguments. But we must consider well the laws of bodies, descending inclined plains, and curved surfaces. See art. 10, 11. This matter will be cleared up, if we consider the circumference of the wheel to be the curved surface: for the fact is, that the water acts to the best advantage, and produces effects equal to what it would, in case the whole of it acted upon the very end of the lever, in the whole of its perpendicular descent. ^*

Art. 55. Of the Friction of the Apertures of Spouting Fluids.

THE doctrine of this species of friction ap­pears to be as follows: Of the fric­tion of the apertures of spouting fluids.

1. The ratio of the friction of round apertures are as their diameters, nearly, while their quanti­ties expended, are as the squares of their diameters.

2. The friction of an aperture, of any re­gular or irregular figure, is as the length of the sum of the circumscribing lines, nearly; the [Page 98] quantities being as the areas of the aperture. Art. 55. Therefore,

3. The less the head or pressure, and the large the aperture, the less the ratio of the friction therefore,

4. This friction need not be much regards in the large openings or apertures of undershot mills, Need not be regarded in undershot mills; But must be noted in overshots. where the gates are from 2 to 15 inches [...] their shortest sides; but it very sensibly affect the small apertures of high overshot or undershot mills, with great heads, where their shortest [...] are from five tenths of an inch to two inches. ^†

Art. 56. Of the Pressure of the Air on Fluids.

THE second cause of the motion or rise fluids, Pressure of the air the cause of the rise of fluids. is the pressure of the air on the surface them, in the fountain or reservoir; and this [...] sure is equal to a head of water 33; feet per­pendicular height, under which pressure or [...] of head, the velocity of spouting water is 46. [...] feet per second.

Therefore, Art. 56. if we could by any means take off the pressure of the atmosphere, from any one part of the surface of a fluid, that part would spout up with a velocity of 46,73 feet per second, and rise to the height of 331/3 feet, nearly. ^*

On this principle act all syphons or cranes, and all pumps for raising water by suction, as it is called.—Let fig. 23, Fig. 23. pl. III. pl. III. represent a cask of water, with a syphon therein, to extend 33⅓ feet above the surface of the water in the cask. Principles of syphons for decanting li­quors. Now if the bung be made perfectly air-tight, round the syphon, so that no air can get into the cask, and the cask be full, then, if all the air be drawn cut of the syphon, at the bended part A, the fluid will not rise in the syphon, because the air can­not get to it to press it up; but take out the plug P, and let the air into the cask, to press on the surface of the water, and it will spout up the short leg of the syphon B A, with the same force and velocity, as if it had been pressed with a head of water 33⅓ feet high, and will run into the long leg and will fill it. Then if we turn the cock c, and let the water run out, its weight in the long leg will overbalance the weight in the short one, drawing the water out of the cask, until the water sink so low, that the leg B A will be 33⅓ feet high, above the surface of the water in the cask; then it will stop, because the weight of water in the leg, in which it rises, will be equal to the weight of a column of the air of equal size, and of the whole height of the atmosphere. The water will not run out of the leg [Page 10] A c, but will stand full 33⅓ feet above its mouth because the air will press up the mouth c, Art. 56. with force that will balance 33⅓ feet of water [...] the leg c A. This will be the case, let the [...] ­per part of the leg be any size whatever— [...] there will be a small vacuum in the top of the long leg.

Art. 57. Of Pumps.

LET fig. 24, Fig. 24, pl. III. pl. III. represent a pump of the common kind, used for drawing water out [...] wells. The moveable valve or bucket A, principles of Pumps for raising water. is cas [...] with leather, which springs outwards, and [...] the tube so nicely, that neither air nor water [...] pass freely by it. When the lever L is worked the valve A opens as it descends, letting the [...] or water pass through it. As it ascends again the valve shuts; the water which is above the bucket A is raised, and there would be a vacuum between the valves, but the weight of the [...] presses on the surface of the water in the well at W, forcing it up through the valve B to [...] the space between the buckets; and as the valve. A descends, B shuts, and prevents the water [...] descending again: But if the upper valve A be [...] more than 33⅓ feet above the surface of the water in the well, the pump cannot be made to draw because the pressure of the atmosphere will not cause the water to rise more than 33⅓ feet.

[Page 101]

"All pumps should be so constructed as to work with equal ease, in rais­ing the water to any given height above the surface of the well: and this may be done by observing adue proportion between the diameter of that part of the pump bore in which the piston or bucket works, and the height to which the water must be raised.

"For this purpose I have calculated the above table, in which the han­dle of the pump is supposed to be a lever, increasing the power five times: that is, the distance or length of that part of the handle that lies between the pin on which it moves, and the top of the pump-rod to which it is fixed, to be only one fifth part of the length of the handle, from the said pin to the part where the man (who works the pump) applies his force or power.

"In the first column of the table, find the height at which the pump must discharge the water above the surface of the well; then in the second co­lumn, you have the diameter of that part of the bore in which the piston or bucket works, in inches and hundredth parts of an inch; in the third column is the quantity of water, (in wine measure) that a man of common strength can raise in a minute.—And by constructing according to this me­thod, pumps of all heights may be wrought by a man of ordinary strength so as to be able to hold out for an hour."

Art. 58. Of conveying Water under Valleys and over Hills.

WATER, by its pressure, and the pressure of the atmosphere, Fig. 20. may be conveyed under val­leys and over hills, to supply a family, a mill, or a town. To convey water under valleys and over hills. See fig. 20. pl. III. F H is a canal for conveying water to a mill-wheel. Now let us suppose F G 16 H to be a tight tube or trunk—the water being let in at F, it will descend from F to G, and its pressure at F will cause it to rise to H, passing along if permitted, and may be conveyed over a hill by a tube, acting on the principle of the syphon. (art. 56) But where some have had occasion thus to convey water under any obstacle for the convenience of a mill, which often occurs in practice they have gone into the following expensive error: They make the tube at G 16 smaller than if it had been on a level, because, say they, a greater quantity will pass through a tube, pressed by the head G F, than on a level. But they should consider that the head G F is balanced by the head H 16, and the ve­locity through the tube G 16 will only be such that a head equal to the difference between the perpendicular height of G F and H 16 would give it; (see art. 41, fig. 19) therefore it should be as large at G 16 as if on a level.

Art. 59. Of the Difference of the Force of indefinite and de­finite Quantities of Water Striking a Wheel. DEFINITIONS.

1. BY an indefinite quantity of water we here mean a river or large quantity, much larger than [Page 103] the float of the wheel, so that, Art. 59. when it strikes the float, it has liberty to move or escape from it in every lateral direction. Of the force of definite & indefinite quantities of water.

2. By a definite quantity of water we mean a quantity passing through a given aperture along a shute to strike, a wheel; but as it strikes the float, it has liberty to escape in every lateral direc­tion.

3. By a perfectly definite quantity, we mean a quantity passing along a close tube so confined, that when it strikes the float it has not liberty to escape in any lateral direction.

First, Indefinite quantity of water ap­plied when a float of a wheel is struck by an indefinite quantity, the float is struck by a co­lumn of water, the section of which is equal to the area of the float; and as this column is con­fined on every side by the surrounding water, which has equal motion, it cannot escape freely sideways; therefore more of its force is commu­nicated to the float than would be, in case it had free liberty to escape sideways in every direction.

Secondly, Definite quan­tity applied. The float being struck by definite quantity, with liberty to escape freely in every side direction, it acts as the most perfect non-elas­tic body; therefore (by art. 8) it communicates only a part of its force, the other part being spent in the lateral direction. Hence it appears, that in the application of water to act by impulse, we should draw the gate as near as possible to the float-board, and confine it as much as possible from escaping sideways as it strikes the float; but, taking care at the same time, that we do not bring the principle of the Hydrostatic Paradox into ac­tion, (art 48).

What proportion of the force of the water is spent in a lateral direction is not yet determined, but see Art. 8.

3. A perfectly definite quantity striking a plain, communicates its whole force; because no part can escape sideways, and is equal in power to an [Page 104] elastic body, Art. 59. or the weight of the water on [...] overshot wheel, in its whole perpendicular de­scent. A perfectly definite quan­tity applied, but cannot be applied to a wheel in prac­tice. But this application of water to [...] has been hither to impracticable; for whenever we attempt to confine the water totally from escap­ing sideways, we bring the paradoxical principle into action which defeats the scheme. ^*

To make this plain, let fig. 25, pl. III, be a wa­ter-wheel; and first, let us suppose the water is be brought to it by the penstock 4. 16 to act by im­pulse on the float b, and have free liberty to e­scape every way as it strikes; then by art. 8 [...] will communicate but half its force. But if it be confined both at sides and bottom and can escape only upwards, to which the gravity will make some opposition, it will communicate perhaps more than half its force, and will not [...] back against the float c. But if we put soali [...] to the wheel to prevent the water from escaping upwards, then the space between the floats [...] be filled, as soon as the wheel begins to be retard­ed, Paradoxical mill that will not move empty. and the paradoxical principle art. 48 is brought fully into action, viz. the pressure of water is [...] way equal, and presses backwards against the bottom of the float c, with a force equal to [...] pressure on the top of the float b, and the wheel will immediately stop and be held in equilibrio and will not start again although all resistance [...] removed. This we may call the paradoxical mill. There are many mills, where this princi­ple is, in part, brought into action, which very much lessens their power.

Art. 60. Of the Motion of Breast and Pitch-back wheels.

MANY have been of opinion, Fig. 25. that when wa­ter is put to act on the wheel as at a (called [...] [Page 105] low breast) with 12 feet head, Art. 60. that then the 4 feet fall below the point of impact a, is totally lost, Breast and pitch-back wheels does not run before the gravity of the water. because, say they, the impulse of the 12 feet head, will require the wheel to move with such velocity to suit the motion of the water as to move before the action of gravity, therefore the water cannot act after the stroke. But if they will consider well the principles of gravity act­ing on falling bodies (art. 9) they will find, that, if the velocity of a falling body be ever so great, the action of gravity is still the same to cause it to move faster, so that, although an overshot wheel may move before the power of the gravity, of the water thereon, yet no impulse downwards can give a wheel such velocity, as that the gravi­ty of the water acting thereon can be lessened thereby. ^*

Hence it appears, that when a greater head is used, Direct the wa­ter down­wards when there is too much head, on overshot, pitch-back, or breast wheels. than what is necessary to shoot the water fairly into the wheel, the impulse should be direct­ed downward a little as at D (which is called pitch-back) and have a circular sheeting to pre­vent the water from leaving the wheel, because if it be shot horizontally on the top of the wheel, the impulse in that case will not give the water any greater velocity downwards, then, in this case, the fall would be lost, if the head was very great, and the wheel moved to suit the velocity of the impulse, the water would be thrown out of the buckets by the centrifugal force; and if we attempt to retard the wheel, so as to retain the water the mill will be so ticklish and unsteady, that it will be almost impossible to attend it.

[Page 106] Hence may appear the reason why breast-wheels generally run quicker than overshots, al­though the fall after the water strikes be not so great.

1. There is generally more head allowed to breast-mills than overshots, and the wheel will incline to move with nearly 2/3 the velocity of the water, spouting from under the head, (art. 41).

2. If the water was permitted to fall freely after it issues from the gate, it would be accele­rated by the fall, so that its velocity at the low­est point, would be equal to its velocity, had it spouted from under a head equal to its whole per­pendicular descent. This accelerated velocity of the water, tends to accelerate the wheel; hence, to find the velocity of a breast-wheel, where the water is struck on in a tangent direc­tion as in fig. 31, 32, I deduce the following.

Art. 61. Rule for calculating the Power of any Mill-Seat.

THE only loss of power sustained by using too much head, in the application of water to turn a mill-wheel, is from the head producing only half its power. Therefore, in calculating the power of 16 cubic feet per second, on the different ap­plications of fig. 25, pl. III. we must add half the head to the whole fall, and count that sum [Page 108] the virtual perpendicular descent. Art. 61. Then by the­orem IV. (art. 53) multiply the weight of the water per second by its perpendicular descent, and you have the true measure of its power.

But to reduce the rule to a greater simplicity, Simple rule for calculat­ing the power of any mili­seat. let us call each cubic foot 1, and the rule will be simply this—Multiply the cubic feet expended per second, by its virtual perpendicular descent in feet, and the product will be a true measure of the power per second. This measure [...] have a name, which I call Cuboch; that is, [...] cubic foot of water, multiplied by 1 foot descent, is one cuboch, or the unit of power.

Art. 62. Theory and Practice compared.

I WILL here give a table of 18 mills in actual practice out of about 50 that I have taken an ac­count of, in order to compare theory with prac­tice, and in order to ascertain the power requir­ed [Page 111] on each superficial foot of the acting parts of the stone: Art. 62. But I must premise the following

Art. 63. Further observations on the following Table.

1. The mean power used to turn the 5 feet stones in the Experiments (No. 1. 7. 14. 17.) is 87,5 cubochs of the measure established art. Experiments. 61, and the mean velocity is 104 revolutions of the stones in a minute, the velocity of the mean cir­cle being 18,37 feet per second, and their mean quantity ground is 3,8lb. per minute, which is 3,8 bushels per hour, and the mean power used to each foot of the area of the stone is 4,69 of the measure aforesaid, done by 36582 superficial feet passing each other in a minute. Hence we may conclude, until better informed,

1. That 87,5 cubochs of power per second will turn a 5 feet stone 104 revolutions in a minute, and grind 3,8 bushels an hour.

2. That 4,69 cubochs of power is required to every superficial foot of a mill-stone, when their mean circles move with a velocity of 18,37 feet per second. Or,

3. That for every 36582 feet of the face of stones that pass each other we may expect 3,8lb. will be ground, when the stones, grain, &c. are in the state and condition, as were the above stones in the experiments.

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Note, the above table being large, obliged us to put it on two pages folding of which may set it a little out of place. In the 3rd, 4th, [...] 18th experiments, there is two pair of stones to one water-wheel, the [...] &c. of which, are shewn by the braces. If the reader will by a rule [...] small lines between the experiments, the table will be easier read.

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Art. 64. Of Canals for conveying Water to Mills.

IN digging canals we must consider that water will come to a level on its surface, be the form of the bottom as it may. If we have once de­termined [Page 123] on the area of the section of the canal necessary to convey a sufficient quantity of water to the mill, Art. 64. Directions for digging canals through rocky ground. we need only mind to keep to that area in the whole distance, and need not pay much regard to the depth or width, if there be rocks in the way. Much expence may be often-times saved, by making the canal deep where it cannot easily be got wide enough, and wide where it cannot easily be got deep enough. Thus, sup­pose we have determined it to be 4 feet deep and 6 feet wide, then the area of its section will be 24.—Let fig. 36, pl. IV. represent a canal, Fig. 36. the line A B the level or surface of the water, C D the side, E F the bottom, A C the width 6 feet, They need not be of equal width nor depth in all places; A E the depth 4 feet. Then, if there be rocks at G, so that we cannot without great expence obtain more than 3 feet width, but 8 feet depth at a small expence: then 8×3=24, the section required. Again, suppose a flat rock to be at H, so that we cannot, But the area of their section must be the same or not less than the given size. without great expence obtain more than 2 feet depth, but can, with small ex­pence obtain 12 feet width: then 2×12=24, the section required; and the water will come on equally well, even if it were not more than, 5 of a foot deep, provided it be proportionably wide. One disadvantage however arises in having canals too shallow in places, because the water in very dry seasons, may be too low to rise over them; but if the water was always to be of one height, the disadvantage would be but little. The cur­rent will keep the deep places open; light sand or mud will not settle in them. This will seem paradoxical to some, but seeing the experiment may be a saving of expence, it may be worth try­ing.

Art. 65. Of the Size and Fall of Canals.

AS to the size and fall necessary to convey any quantity of water required, to a mill, I do not find any rule laid down for either. But in order to establish one let us consider, that the size de­pends entirely upon the quantity of water and the velocity with which it is to pass: therefore, [...] we can determine on the velocity, which I [...] suppose to be from 1 to 2 feet per second— [...] the slower the better, as there will be the less fall lost. We can find the size of the canal by the following theorem.

Art. 66. Of Air-pipes to prevent tight Trunks from [...] when filled with water.

WHEN water is to be conveyed under ground or in a tight trunk below the surface of the wa­ter in the reservoir, Air-pipes ne­cessary to pre­vent tight trunks from bursting. to any considerable length, there must be air-pipes (as they have been called to prevent the trunk from bursting. To under­stand their use let us suppose a trunk 100 feet long, 16 feet below the surface of the water, [...] fill which we draw a gate at one end of equal size with the trunk. Then the water, in passing to the other end acquires great velocity if it meets [...] resistance, which velocity is suddenly to be stop­ped when the trunk is full. This great column of water in motion, in this case, would strike with a force equal to a solid body of equal weight and velocity, the shock of which would be suffi­cient to burst any trunk that ever was made of wood. If they are too small they are worse than none. Many having thought the use of these pipes to be to let out the air, have made then too small, so that they would vent the air fa [...] enough to let the water in with considerable ve­locity, but would not vent the water fast enough when full, to cheek its motion easily, in which case they are worse than none at all, for if the [Page 127] air cannot escape freely, Art. 66. the water cannot enter freely.

Whenever the air has been compressed in the trunk by the water coming in, Air prevents trunks from bursting. it has made a great blowing noise in escaping through the crevices, and therefore has been blamed as the cause of the bursting of the trunk; whereas it acted by its elastic principle as a great preventative against it. For I do suppose, that if we were to pump the air all out of a trunk, 100 feet long, and 3 by 3 feet wide, and let the water in with full force, that it would burst, if as thick as a cannon of cast me­tal: because in that case there would be 900 cu­bic feet of water, equal to 56250lbs. pressed on by the weight of the atmosphere, with a velocity of 47 feet per second, to be suddenly stopped, the shock would be inconceivable. ^*

Therefore I do conclude it best, to make an air-pipe for every 20 or 30 feet, of the full size of the trunk; but this will depend much on the depth of the trunk below the surface of the re­servoir, and many other circumstances.

Having now said what was necessary in order the better to understand the theory of the power and principles of mechanical engines, and water acting on the different principles on water-wheels, and for the establishing new and true theories of the motion of the different kinds of water-wheels, I here quote many of the ingenious Smeaton's ex­periments, that the reader may compare them with the theories established, and judge for him­self.

Art. 67. An experimental Enquiry, read in the Philosophical Society in London, May 3rd and 10, 1759, con­cerning the Natural Powers of Water to [...] Mills and other Machines, depending on a circu­lar Motion, by James Smeaton, F.R.S.

WHAT I have to communicate on this sub­ject was originally deduced from experiment made on working models, which I look upon [...] the best means of obtaining the outlines in mecha­nical enquiries. But in this case it is necessary to distinguish the circumstances in which a [...] differ from a machine in large: otherwise a [...] ­del is more apt to lead us from the truth than [...] ­wards it. Hence the common observation, [...] a thing may do very well in a model that [...] not do in large. And indeed though the [...] circumspection be used in this way the best struc­ture of machines cannot be fully ascertained, [...] by making trials with them of their proper [...] It is for this purpose that though the model [...] ­ferred to, and the greatest part of the following experiments, were made in the year 1752 [...] 1753 yet deferred offering them to the [...] till I had an opportunity of putting the deduction made therefrom in real practice in a variety [...] cases and for various purposes, so as to be able assure the society, that I have found them to [...] ­swer.

PART II. Art. 68. Concerning Overshot Wheels.

IN the former part of this essay, we have consi­dered the impulse of a confined stream, acting on undershot wheels; we now proceed to examine the power and application of water, when acting by its gravity on overshot wheels.

It will appear in the course of the following de­ductions, that the effect of the gravity of de­scending bodies, is very different from the effect of the stroke of such as are non-elastic, though ge­nerated by an equal mechanical power.

[Page 146] The alterations of the machinery already de­scribed, Art. 68. to accommodate the same for experiments on overshot wheels, were principally as follows.

Plate XII. The sluice I b being shut down, the rod H I was taken off. The undershot water wheel was taken off the axis, and instead thereof an overshot wheel of the same size and diameter was put in its place. Note, this wheel was 2 inches deep in the shroud or depth of the bucket, the number of buckets was 36.

A trunk for bringing the water upon the wheel was fixed according to the dotted lines fg, the aperture was adjusted by a shuttle, which also closed up the outer end of the trunk, when the water was to be stopped.

PART III. Art. 69. On the Construction and Effects of Wind-mill Sails. ^†

IN trying experiments on wind-mill sails, the wind itself is too uncertain to answer the pur­pose: we must therefore have recourse to artifi­cial wind.

This may be done two ways; either by caus­ing the air to move against the machine, or the machine to move against the air. To cause the air to move against the machine in a sufficient co­lumn with steadiness and the requisite velocity, is not easily put in practice: To carry the ma­chine forward in a right line against the air, [Page 155] would require a larger room than I could conve­niently meet with. Art.69. What I found most practi­cable therefore was, to carry the axis whereon the sails were to be fixed progressively round in the circumference of a large circle. Upon this idea the machine was constructed. ^*

The product is found by simply multiplying the weight in the scale by the number of turns.

By this set of experiments it appears that the maximum velocity is ⅔ of the greatest velocity, [Page 156] and that the ratio of the greatest load to that of a maximum is, Art. 69. as 9 to 7,5, but by adding the weight of the scale and friction to the load the ratio turns out to be as 10:8,4, or as 5 to 4 nearly. The following table is the result of 19 similar sets of experiments.

By the following table it appears, that the most general ratio between the velocity of the sails un­loaded and when loaded to a maximum, is 3 to 2 nearly.

And the ratio between the greatest load and the load at a maximum (taking such experiments where the sails answered best) is at a medium a­bout as 6 to 5 nearly.

And that the kind of sails used in the 15th and 16th experiments is best of all, because they pro­duce the greatest effect or product, in proportion to their quantity of surface, as appears in column 12.

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N. B. The sails were of the same kind as those of Nos. 10,11 and 12, table IV. Continuance of the experiment one mi­nute.

Of Undershot Mills.

UNDERSHOT wheels move by the per­cussion or stroke of the water, and are only half as powerful as other wheels that are moved by the gravity of the water. See art. 8. Therefore this construction ought not to be used, except where there is but little fall or great plenty of water. The undershot wheel, and all others that move by percussion, should move with a velocity nearly equal to two thirds of the velocity of the water. See art. 42. Fig. 28. plate iv. represents this con­struction.

For a rule for finding the velocity of the water, under any given head, see art. 51.

Upon which principles, Principles on which is founded the undershot table. and by said rule, is formed the following table of the velocity of spouting water, under different heads, from one to twenty-five feet high above the centre of the issue; to which is added the velocity of [Page 4] the wheel suitable thereto, Principles on which is founded the undershot table. and the number of revolutions a wheel of fifteen feet diameter (which I take to be a good size) will revolve in a minute; also, the number of cogs and rounds in the wheels, both for double and sin­gle gears, so as to produce about ninety-seven or one hundred revolutions for a five feet stone per minute, which I take to be a good mo [...] and size for a mill-stone, grinding for mer­chantable flour.

That the reader may fully understand how the following table is calculated, let him ob­serve,

1. That by art. 42, the velocity of the wheel must be just 577 thousandth parts of the velocity of the water; therefore, if the velo­city of the water, per second, be multiplied by, 577 the product will be the maximum ve­locity of the wheel, or velocity that will pro­duce the greatest effect, which is the third co­lumn in the table.

2. The velocity of the wheel per second, multiplied by 60, produces the distance the circumference moves per minute, which, di­vided by 47,1 feet, the circumference of a 15 feet wheel, quotes the number of revolutions of the wheel per minute, which is the fourth column.

3. That by art. 20 and 74, the number of revolutions of the wheel per minute, multi­plied by the number of cogs in all the driving wheels, successively, and that product divided by the product of the number of cogs in all the leading wheels, multiplied successively, the quotient is the revolutions of the stones per minute, which is the ninth and twelfth co­lumns.

[Page 5] 4. The cubochs of power required to drive the stone, Principles on which is founded the undershot table. being, by art. 61, equal to 111,78 cubochs per second, which, divided by half the head of water, added to all the fall (if any,) being the virtual or effective head by art. 61 quotes the quantity of water, in cubic feet, required per second, which is the thirteenth column.

5. The quantity required, divided by the velocity with which it is to issue, quotes the area of the aperture of the gate—fourteenth column.

6. The quantity required, divided by the velocity of the water proper for it to move a­long the canal, quotes the area of a section of the canal—fifteenth column.

7. Having obtained their areas, it is easy, by art. 65, to determine the width and depth, as may suit other circumstances.

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[Page 7] Note, that five feet fall is the least that a single gear can be built on, CHAP.1. to keep the cog-wheel clear of the water, and give the stone sufficient motion.

Although double gear is calculated to fifteen feet fall, yet I do not recommend them above ten feet, unless for some particular conveni­ence, such as two pair of stones to one wheel, &c. &c. The number of cogs in the wheels are even, and chosen to suit eight, fix, or four arms, so as not to pass through any of them, this being the common practice. But when the motion cannot be obtained without a trun­dle that will cause the same cogs and rounds to meet too often, such as 16 into 96, which will meet every revolution of the cog-wheel, or 18 into 96, which will meet every third re­volution—I advise rather to put in one more or less, as may best suit the motion, which will cause them to change oftener. See art. 82.

Note, that the friction at the aperture of the gate will greatly diminish both the veloci­ty and power of the water in this application, where the head is great, if the gate be made of the usual form, wide and shallow. Where the head is great, the friction will be great. See art. 55. Therefore the wheel must be narrow, and the aperture of the gate of a square form, to evade the friction and loss that may be under a wide wheel, if it does not run close to the sheeting.

Art. 71. Of Tub Mills.

A TUB MILL has a horizontal water-wheel, Tub Mills described. that is acted on by the percussion of the water altogether; the shaft is verticle, carrying the stone on the top of it, and serves in place of a spindle; the lower end of this shaft is set in a step fixed in a bridge-tree, by which the stone is raised and lowered, as by the bridge-tree of other mills; the water is shot on the upper side of the wheel, in a tan­gent direction with its circumference. See fig. 29, plate IV, which is a top view of the tub-wheel, and fig. 30 is a side view of it, with the stone on the top of the shaft, bridge-tree, &c. The wheel runs in a hoop, like a mill-stone hoop, projecting so far above the wheel as to prevent the water from shooting over the wheel, and whirls it about until it strikes the buckets, because the water is shot on in a deep narrow column, 9 inches wide and 18 inches deep, to drive a 5 feet stone, with 8 feet head—so that all this column can­not enter the buckets until part has passed half way round the wheel, so [...] there are always nearly half the buckets struck at once; the buckets are set obliquely, so that the water [Page 12] may strike them at right angles. CHAP.1. See fig. 30. As soon as it strikes it escapes under the wheel in every direction, as in fig. 29. ^*

[Page 13] The disadvantages of these wheels are, CHAP.1.

1. The water does not act to advantage on them, we being obliged to make them so small to obtain velocity to the stone (in most cases) that the buckets take up a third part of their diameter.

2. The water acts with less power than on undershot wheels, Tub mills act with less power than undershots. as it is less confined at the time of striking the wheel, and its non elastic principle takes place more fully. See art. 8.

3. It is with difficulty we can put a sufficient quantity of water to act on them to drive them with sufficient power, if the head be low; therefore I advise to strike the water on in two places, as in fig. 29; then the apertures need only be about 6 by 13 inches each, in­stead of 9 by 18, and will act to more advan­tage; and then, in this case, nearly all the buckets will be acted on at once.

Their advantages are,

Their exceeding simplicity and cheapness, They are preferable to others, with plenty of water. having no cogs nor rounds to be kept in re­pair; their wearing parts are few, and have but little friction; the step-gudgeon runs un­der water, therefore, if well fixed, will not get out of order in a long time; and they will move with sufficient velocity and power with 9 or 10 feet total fall, and plenty of water; and, if they be well fixed, they will not re­quire [Page 14] much more water than undershot wheels; therefore they are vastly preferable in all seats with plenty of water, and above 8 feet fall.

In order that the reader may fully under­stand how the following table is calculated, let him consider,

1. That as the tub-wheel moves altogether by percussion, Should move 2-3 the velocity of the water the water flying clear of the wheel the instant it strikes, and it being better, by art. 70, for such wheels to move faster in­stead of slower than the maximum velocity; therefore, instead of ,577 we will allow them to move ,66 velocity of the water; Rules To find the velocity of the water. then mul­tiplying the velocity of the water by ,66 gives the velocity of the wheel, at the centre of the buckets; which is the 3d column in the table.

2. And the velocity of the wheel per second, Its diameter multiplied by 60, and divided by the number of revolutions the stone is to make in a minute, gives the circumference of the wheel at the centre of the buckets; which circumference, multiplied by 7, and divided by 22, gives the diameter from the centre of the buckets, to produce the number of revolutions required; which are the 4th, 5th, 6th and 7th columns.

3. The cubochs of power required, The quanti­ty of water required. by art. 63, to drive the stone, divided by half the head, gives the cubic feet of water required to produce said power; which are the 8th and 10th columns.

4. The cubic feet of water, The size of the aper­ture. divided by the velocity, will give the sum of the apertures of the gates; which are the 9th and 11th co­lumns.

[Page 15] 5. The cubic feet of water, divided by 1,5 feet, And size of the canal. the velocity of the water in the canal, gives the area of a section of the canal; which are the 12th and 13th columns.

6. For the quantity of water, aperture of gate, and size of canal, for 5 feet stones, see table for undershot Mills, in art. 70.

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Art.72. Of Breast Mills.

BREAST WHEELS, which have the water shot on them in a tangent direction, On Breast Mills the water acts both by per­cussion and gravity. are acted on by the principles of both percussion and gravity; all that part above the point of impact, called head, acts by percussion, and all that part below said point, called fall, acts [...] gravity.

[Page 18] We are obliged, CHAP. I. in this structure of breast mills, Plate IV. Fig. 31. to use more head than will act to advan­tage; because we cannot strike the water [...] the wheel, in a true tangent direction, higher than I, the point of impact in fig. 31, which is a breast-wheel, with 12 feet perpendicular descent, 6,5 feet of which is above the point I, as head, and 5,5 feet below, as fall. The upper end of the shute, that carries the water down to the wheel, must project some inches above the point of the gate when full dra [...] else the water will strike towards the centre of the wheel; and it must not project too high else the water in the penstock will not c [...] fast enough into the shute when the head [...] a little. The bottom of the penstock is a little below the top end of the shute, to leave room for stones and gravel to settle, and prevent them from getting into the gate.

We might lay the water on higher, by set­ting the top of the penstock close to the whe [...] and using a sliding gate at bottom, as [...] by the dotted lines; but this is not approved of in practice. See Ellicott's mode, part [...] plate III, fig. 1.

But if the water in the penstock be nearly as high as the wheel, Pitchback wheels have their mo­tion equal to overshots it may be carried over as by the upper dotted lines, and shot on [...] ­wards, making that part next the wheel [...] shute to guide the water into the wheel, [...] the gate very narrow or shallow, allowing [...] water to run over the top of it when drawn by this method (called Pitchback) the [...] may be reduced to the same as it is for [...] overshot wheel; and then the motion of the circumference of the wheel will be equal [Page 19] the motion of an overshot wheel, CHAP. I. whose dia­meter is equal to the fall below the point of impact, and their power will be equal.

This structure of a wheel, Fig. 31, fig. 31, I take to be a good one, for the following reasons, viz.

1. The buckets, Wheel of new con­struction. or floats, receive the per­cussion of the water at right angles, which is the best direction possible.

2. It prevents the water from flying to­wards the centre of the wheel, without re­acting against the bottom of the buckets, and retains it in the wheel, to act by its gravity in its descent, after the stroke.

3. It admits air, and discharges the water freely, without lifting it at bottom; and this is an important advantage, because, if the buckets of a wheel be tight, and the wheel wades a little in back-water, they will lift the water a considerable distance as they empty; the pressure of the atmosphere prevents the water from leaving the buckets freely, and it requires a great force to lift them out of the water with the velocity of wheel; which may be proved by dipping a common water-bucket into water, and lifting it out, bot­tom up, with a quick motion, you have to [...]ft not only the water in the bucket, but it appears to suck a deal more up after it; which is the effect of the pressure of the atmosphere, see art. 56. This shews the necessity of air-holes to let air into the buckets, that the wa­ter may have liberty to get out freely.

Its disadvantages are,

1. It loses the water much, if it is not kept close to the sheeting. And,

[Page 20] 2. It requires too great a part of the total fall to be used as head, which is a loss of pow­er, one foot fall being equal in power to two feet head, by art. 8.

Fig. 32 is a draught, Fig. 32. shewing the position of the shute for striking the water on a wheel in a tangent direction, for all the total perpen­dicular descents from 6 to 15 feet; the points of impact are numbered inside the fig. with the number of the total fall, that each is for respectively. The top of the shute is only about 15 inches from the wheel, in order to set the point of impact as high as possible, al­lowing 3 feet above the upper end of the [...] to the top of the water in the penstock, which is little enough, when the head is often to be run down any considerable distance; but when the stream is steady, being always nearly the same height in the penstock, 2 feet would [...] sufficient, especially in the greatest total [...] where the quantity is less, raising the shute foot would raise the point of impact nearly the same, and increase the power, because I [...] fall is equal in power to 2 feet head, by art 61.

On these principles, to suit the application of water, as represented by fig. 32, I have calculated the following table for breast mill. And, in order that the reader may fully un­derstand the principles on which it is calcu­lated, let him consider as follows:

1. That all the water above the point of im­pact, Principles on which the breast mill table is founded. called head, acts wholly by percussion and all below said point, called fall, acts wh [...] ­ly by gravity (see art. 60) and form the [...] and 3d columns.

[Page 21] 2. That half the head, added to the whole fall, CHAP. I. constitutes the virtual or effective descent, by art. 61; which is the 4th column.

3. That if the water was permitted to de­scend freely down the circular sheeting, Rules after it passes the point of impact, To find the velocity of a breast wheel its velocity would be accelerated, by art. 60, to be, at the low­est point, equal to the velocity of water spout­ing from under a head equal to the whole de­scent: therefore the maximum velocity of this wheel will be a compound of the velocity to suit the head and the acceleration after it passes the point of impact. Therefore, to find the velocity of this wheel, I first multiply the ve­locity of the head, in column 5, by ,577 (as for the undershot mills) which gives the velo­city suitable to the head; I then (by the rule for determining the velocity of overshots) say, as the velocity of water descending 21 feet, equal to 37,11 feet per second, is to the velo­city of the wheel 10 feet per second, so is the acceleration of velocity, after it passes the point of impact, to the accelerated velocity of the wheel; and these two velocities added, gives the velocity of the wheel; which is the 6th column.

4. The velocity of the wheel per second, Its number of revolu­tions. multiplied by 60, and divided by the circum­ference of the wheel, gives the revolutions per minute: 7th column.

5. The number of cogs in the cog-wheel, Revolutions of the stone. multiplied by the number of revolutions of the wheel per minute, and divided by the rounds in the trundle-head, will give the number of revolutions of the stone per minute; and if we divide by the number of revolutions the stone [Page 22] is to have, CHAP. I. it gives the rounds in the trundle, and, when fractions arise, take the nearest whole number; columns 8, 9, and 10.

6. The cubochs of power required to turn the stone, Power re­quired. by art. 63, divided by the virtual descent, gives the cubic feet of water required per second; column II.

7. The cubic feet, Area of the canal. divided by the velocity of water allowed in the canal, suppose 1,5 feet per second, gives the area of the canal; column 12.

8. If the mill is to be double geared, take the revolutions of the wheel from column 7 of this table, and look in column 4 of the under­shot table, art. 70, for the number of revo­lutions nearest to it, and against that number you have the gears that will give a 5 feet stone the right motion.

[Page]

Art. 73. Of Overshot Mills.

FIG. 33, Fig. 33. Overshot Wheel de­scribed. plate IV, is an Overshot Wheel; the water is laid on at the top, so that the up­per part of the column will be in a true tan­gent direction with the circumference of the wheel, but so that all the water may strike with­in the circle of the wheel.

The gate is drawn about 12 inches behind the perpendicular line from the centre of the wheel, Of shooting on the water and the point of the shute ends at said perpendicular, with a direction a little down­wards, which gives the water a little velocity downwards to follow the wheel; for if it be directed horizontally, the head will give it no velocity downwards, and, if the head be great, the parabolic curve, which the spouting water forms, will extend beyond the outside of the [Page 26] circle of the wheel, and it will incline to [...] over. See art. 44 and 60.

The head above the wheel acts by per­cussion, Proper ve­locity of the water spouting on the wheel. as on an undershot wheel, and we have shewn, art. 43, that the head should be such as to give the water velocity 3 for 2 of the wheel. After the water strikes the wheel it acts by gravity; therefore, to calculate the power, we must take half the head and add it to the fall, for the virtual descent, as in breast mills.

The velocity of overshot wheels are as the square roots of their diameters. See art. 43.

On these principles, I have calculated the following table for overshot wheels; and, in order that the reader may understand it fully, let him consider well the following premises:

1. That the velocity of the water spouting on the wheel must be one and a half the velo­city of the wheel, Rule for proportion­ing the head to the wheel by art. 43; then, to find the head that will give said velocity, say, [...] the square of 16,2 feet per second, is to 4 feet, the head that gives that velocity, so is the square of the velocity required, to the hea [...] that will give that velocity—But, to this head so found, we must add a little by conjecture to overcome the friction of the aperture. See art. 55.

In this table, I have added to the heads o [...] wheels from 9 to 12 feet diameter, 1 of a foot and from 12 to 20 I have added, 1 tenth more for every foot increase of diameter, and from 20 to 30 feet I have added ,05 more to every foot diameter's increase; which gives a 30 feet wheel 1,5 feet additional head, while a 9 feet [Page 27] wheel has only, 1 tenth of a foot, to overcome the friction. The reason of this great difference will appear when we consider that the friction increases as the aperture decreases, and as the velocity increases—But this much depends on the form of the gate, for if that be nearly square, there will be but little friction, but if very oblong, say 24 inches by half an inch, then it will be very great.

The heads, thus found, compose the 3d co­lumn.

2. The head, added to the diameter of the wheel, Rules makes the total descent, and is co­lumn 1.

3. The velocity of the wheel per second, For finding the number of revolu­tions of the wheel. taken from the table in art. 43, and multiplied by 60, and divided by the circumference of the wheel, quotes the number of revolutions of the wheel per minute, and is column 4.

4. The number of revolutions of the wheel per minute, Ditto of the stone. multiplied by the number of cogs in all the driving wheels successively, and that product divided by the product of all the lead­ing wheels, quotes the number of revolutions of the stone per minute, and is column 9, double gear, for 5 feet stones; and column 12, single gear, for 6 feet stones.

5. The cubochs of power required to drive the stone, To find the quantity of water re­quired. by table in art. 63, divided by the virtual or effective descent, which is half the head added to the (fall or) diameter of the wheel, quotes the cubic feet of water required per second to drive the stone, and is column 13.

[Page 28] 6. The cubic feet required, Rule to find the size of the canal. divided by the velocity you intend the water to have in the canal, quotes the area of a section of the ca­nal. The width multiplied by the depth, must always produce this area. See art. 64.

7. The number of cogs in the wheel, mul­tiplied by the quarter inches in the pitch, pro­duces the circumference of the pitch circle which, multiplied by 7, and divided by 22, quotes the diameter in quarter inches; which, reduced to feet and parts, is column 15. The reader may here at once observe how near the cog wheel, in the single gear, will be to the water; that is, how near it is, in size, equal to the water-wheel.