, properly denotes a machine for grinding corn, &c; but in a more general signisication, is applied to all machines whose action depends on a circular motion. Of these there are several kinds, according to the various methods of applying the moving power; as water-mills, wind-mills, horse-mills, handmills, &c, and even steam-mills, or such as are worked by the force of steam; as that noble structure that was erected near Blackfriars Bridge, called the Albion Mills, but lately destroyed by sire.

The water acts both by its impulse and weight in an overshot water-mill, but only by its impulse in an undershot one; but here the velocity is greater, because the water is suffered to descend to a greater depth before it strikes the wheel. Mr. Ferguson observes, that where there is but a small quantity of water, and a fall great enough for the wheel to lie under it, the bucket or overshot wheel is always used: but where there is a large body of water, with a little fall, the breast or float-board wheel must take place: and where there is a large supply of water, as a river, or large stream or brook, with very little fall, then the undershot wheel is the easiest, cheapest, and most simple struc- ture.|

Dr. Desaguliers, having had occasion to examine many undershot and overshot Mills, generally found that a well made overshot Mill ground as much corn, in the same time, as an undershot Mill does with ten times as much water; supposing the fall of water at the overshot to be 20 feet, and at the undershot about 6 or 7 feet: and he generally observed that the wheel of the overshot Mill was of 15 or 16 feet diameter, with a head of water of 4 or 5 feet, to drive the water into the buckets with some momentum.

In Water-mills, some few have given the preference to the undershot wheel, but most writers prefer the overshot one. M. Belidor greatly preferred the undershot to any other construction. He had even concluded, that water applied in this way will do more than six times the work of an overshot wheel; while Dr. Desaguliers, in overthrowing Belidor's position, determined that an overshot wheel would do ten times the work of an undershot wheel with an equal quantity of water. So that between these two celebrated authors, there is a difference of no less than 60 to <*>. In consequence of such monstrous disagreement, Mr. Smeaton began the course of experiments mentioned below.

In the Philos. Trans. vol. 51, for the year 1759, we have a large paper with experiments on Mills turned both by water and wind, by that ingenious and experienced engineer Mr. Smeaton. From those experiments it appears, pa. 129, that the effects obtained by the overshot wheel are generally 4 or 5 times as great as those with the undershot wheel, in the same time, with the same expence of water, descending from the same height above the bottom of the wheels; or that the former performs the same effect as the latter, in the same time, with an expence of only one-4th or one- 5th of the water, from the same head or height. And this advantage seems to arise from the water lodging in the buckets, and so carrying the wheel about by their weight. But, in pa. 130, Mr. Smeaton reckons the effect of overshot only double to that of the undershot wheel. And hence he infers, in general, “that the higher the wheel is in proportion to the whole descent, the greater will be the effect; because it depends less upon the impulse of the head, and more upon the gravity of the water in the buckets. However, as every thing has its limits, so has this; for thus much is desirable, that the water should have somewhat greater velocity, than the circumference of the wheel, in coming thereon; otherwise the wheel will not only be retarded, by the buckets striking the water, but thereby dashing a part of it over, so much of the power is lost.” He is farther of opinion, that the best velocity for an overshot wheel is when its circumference moves at the rate of about 3 feet in a second of time. See Wind Mill.

Considerable differences have also arisen as to the mathematical theory of the force of water striking the floats of a wheel in motion. M. Parent, Maclaurin, Desaguliers, &c, have determined, by calculation, that a wheel works to the greatest effect, when its velocity is equal to one-third of the velocity of the water which strikes it; or that the greatest velocity that the wheel acquires, is one-third of that of the water. And this determination, which has been followed by all mathematicians till very lately, necessarily results from a position which they assume, viz, that the force of the water against the wheel, is proportional to the square of its relative velocity, or of the difference between the absolute velocity of the water and that of the wheel. And this position is itself an inference which they make from the force of water striking a body at rest, being as the square of the velocity, because the force of each particle is as the velocity it strikes with, and the number of particles or the whole quantity that strikes is also as the same velocity. But when the water strikes a body in motion, the quantity of it that strikes is still as the absolute velocity of the water, though the force of each particle be only as the relative velocity, or that with which it strikes. Hence it follows, that the whole force or effect is in the compound ratio of the absolute and relative velocities of the water; and therefore is greater than the before mentioned effect or force, in the ratio of the absolute to the relative velocity. The effect of this correction is, that the maximum velocity of the wheel becomes one-half the velocity of the water, instead of one-third of it only: a determination which nearly agrees with the best experiments, as those of Mr. Smeaton.

This correction has been lately made by Mr. W. Waring, in the 3d volume of the Transactions of the American Philosophical Society, pa. 144. This ingenious writer says, ‘Being lately requested to make some calculations relative to Mills, particularly Dr. Barker's construction as improved by James Rumsey, I found more difficulty in the attempt than I at first expected. It appeared necessary to investigate new theorems for the purpose, as there are circumstances peculiar to this construction, which are not noticed, I believe, by any author; and the theory of Mills, as hitherto published, is very imperfect, which <*> take to be the reason it has been of so little use to practical mechanics.

‘The first step, then, toward calculating the power of any water-mill (or wind-mill) or proportioning their parts and velocities to the greatest advantage, seems to be,

The Correction of an Essential Mislake adopted by Writers on the Theory of Mills.

‘This is attempted with all the deference due to eminent authors, whose ingenious labours have justly raised their reputation and advanced the sciences; but when any wrong principles are successively published by a feries of such pens, they are the more implicitly received, and more particularly claim a public rectification; which must be pleasing, even to these candid writers themselves.’

A very ingenious writer in England, ‘in his masterly treatife on the rectilinear motion and rotation of bodies, published so lately as 1784, continues this oversight, with its pernicious consequences, through his propositions and corollaries (pa. 275 to 284), although he knew the theory was suspected: for he observes (pa. 382) “Mr. Smeaton in his paper on mechanic “power (published in the Philosophical Transactions “for the year 1776) allows, that the theory usually “given will not correspond with matter of fact, when “compared with the motion of machines; and seems “to attribute this d sagreement, rather to desiciency “in the theory, thanito the obstacles which have pre-| “vented the application of it to the complicated mo“tion of engines, &c. In order to satisfy himself con“cerning the reason of this disagreement, he construct“ed a set of experiments, which, from the known “abilities and ingenuity of the author, certainly de“serve great consideration and attention from every “one who is interested in these inquiries.” ‘And notwithstanding the same learned author says, “The evidence upon which the theory rests is scarcely less than mathematical;” I am sorry to find, in the present state of the sciences, one of his abilities concluding (pa. 380) “It is not probable that the theory of motion, however incontestible its principles may be, can afford much assistance to the practical mechanic,” although indeed his theory, compared with the above cited experiments, might suggest such an inference. But to come to the point, I would just premise these


‘If a stream of water impinge against a wheel in motion, there are three different velocities to be considered, appertaining thereto, viz,

First, the absolute velocity of the water;

Second, the absolute velocity of the wheel;

Third, the relative velocity of the water to that of the wheel, i. e. the difference of the absolute velocities, or the velocity with which the water overtakes or strikes the wheel.’

‘Now the mistake consists in supposing the momentum or force of the water against the wheel, to be in the duplicate ratio of the relative velocity: Whereas,

Prop. I.

‘The force of an Invariable Stream, impinging against a Mill-wheel in Motion, is in the Simple Direct Proportion of the Relative Velocity.

‘For, if the relative velocity of a fluid against a single plane be varied, either by the motion of the plane, or of the fluid from a given aperture, or both, then, the number of particles acting on the plane in a given time, and likewise the momentum of each particle, being respectively as the relative velocity, the force on both these accounts, must be in the duplicate ratio of the relative velocity, agreeably to the common theory, with respect to this single plane: but, the number of these planes, or parts of the wheel acted on in a given time, will be as the velocity of the wheel, or inversely as the relative velocity; therefore, the moving force of the wheel must be in the simple direct ratio of the relative velocity. Q. E. D.

‘Or the proposition is manifest from this consideration; that, while the stream is invariable, whatever be the velocity of the wheel, the same number of particles or quantity of the fluid, must strike it somewhere or other in a given time; consequently the variation of force is only on account of the varied impingent velocity of the same body, occasioned by a change of motion in the wheel; that is, the momentum is as the relative velocity.’

‘Now, this true principle substituted for the erroneous one in use, will bring the theory to agree remarkably with the notable experiments of the ingenious Smeaton, before mentioned, published in the Philosophical Transactions of the Royal Society of London for the year 1751, vol. 51, for which the honorary annual medal was adjudged by the society, and presented to the author by their president. An instance or two of the importance of this correction may be adduced as below.’

Prop. II.

‘The velocity of a wheel, moved by the impact of a stream, must be half the velocity of the fluid, to produce the greatest possible effect.—For let V = the velocity, m = the momentum of the fluid; v = the velocity, p = the power of the wheel. Then V - v = the relative velocity, by def. 3d; and as (prop. 1); this multiplied by v, gives maximum; hence a maximum, and its fluxion (v being the variable quantity) is ; therefore , that is, the velocity of the wheel = half that of the fluid, at the place of impact, when the effect is a maximum. Q. E. D.’

‘The usual theory gives ; where the error is not less than one third of the true velocity of the wheel.’

‘This proposition is applicable to undershot wheels, and corresponds with the accurate experiments before cited, as appears from the author's conclusion (Philos. Trans. for 1776, pa. 457), viz, “The velocity of the “wheel, which according to M. Parent's determina“tion, adopted by Desaguliers and Maclaurin, ought to “be no more than one third of that of the water, varies “at the maximum in the experiments of table 1, be“tween one third and one half; but in all the cases “there related, in which the most work is performed “in proportion to the water expended, and which ap“proach the nearest to the circumstances of great “works when properly executed, the maximum lies “much nearer one half than one third, one half seeming “to be the true maximum, if nothing were lost by the “resistance of the air, the scattering of the water car“ried up by the wheel, &c.” Thus he fully shews the common theory to have been very defective; but, I believe, none have since pointed out wherein the deficiency lay, nor how to correct it; and now we see the agreement of the true theory with the result of his experiments.’ For another problem,

Prob. III.

‘Given, the momentum (m) and velocity (V) of the fluid at I, the place of impact; the radius (R = IS) of the wheel ABC; the radius (r = DS) of the small wheel DEF on the same axle or shaft; the weight (w) or resistance to be overcome at D, and the friction (f) or force necessary to move the wheel without the weight; required the velocity (v) of the wheel &c.”

‘Here we have the acting force at I in the direction KI, as before (prop. 2). Nov the power| at I necessary to counterpoise the weight w; hence the whole resistance opposed to the action of the fluid at I; which deducted from the moving force, leaves the accelerating for<*>e of the machine; which, when the motion becomes uniform, will be evanescent or = 0; therefore , which gives the true velocity required; or, if we reject the friction, then is the theorem for the velocity of the wheel. This, by the common theory, would be , which is too little by . No wonder why we have hitherto derived so little advantage from the theory.’

Corol. 1. If the weight (w) or resistance be required, such as just to admit of that velocity which would produce the greatest effect; then, by substituting (1/2)V for its equivalent v (by prop. 2), we have ; hence ; or, if f = 0, ; but theorists make this , where the error is .’

Corol. 2. We have also ; or, rejecting friction, , when the greatest effect is produced, instead of , as has been supposed: this is an important theorem in the construction of mills.’

In the same volume of the American Transactions, pa. 185, is another ingenious paper, by the same au- thor, on the power and machinery of Dr. Barker's Mill, as improved by Mr. James Rumsey, with a description of it. This is a Mill turned by the resisting force of a stream of water that issues from an orifice, the rotatory part, in which that orifice is, being impelled the contrary way by its reaction against the stream that issues from it.

Mr. Ferguson has given the following directions for constructing water mills in the best manner; with a table of the several corresponding dimensions proper to a great variety of perpendicular falls of the water.

When the float-boards of the water-wheel move with a 3d part of the velocity of the water that acts upon them, the water has the greatest power to turn the Mill: and when the millstone makes about 60 turns in a minute, it is found to perform its work the best: for, when it makes but about 40 or 50, it grinds too slowly; and when it makes more than 70, it heats the meal too much, and cuts the bran so small that a great part of it mixes with the meal, and cannot be separated from it by sifting or boulting. Consequently the utmost perfection of mill-work lies in making the train so as that the millstone shall make about 60 turns in a minute when the water wheel moves with a 3d part of the velocity of the water. To have it so, observe the following rules:

1. Measure the perpendicular height of the fall of water, in feet, above the middle of the aperture, where it is let out to act by impulse against the floatboards on the lowest side of the undershot wheel.

2. Multiply that height of the fall in feet by the constant number 64 1/3, and extract the square root of the product, which will be the velocity of the water at the bottom of the fall, or the number of feet the water moves per second.

3. Divide the velocity of the water by 3; and the quotient will be the velocity of the floats of the wheel in feet per second.

4. Divide the circumference of the wheel in feet, by the velocity of its floats; and the quotient will be the number of seconds in one turn or revolution of the great water-wheel, on the axis of which is fixed the cogwheel that turns the trundle.

5. Divide 60 by the number of seconds in one turn of the water-wheel or cog-wheel; and the quotient will be the number of turns of either of these wheels in a minute.

6. Divide 60 (the number of turns the millstone ought to have in a minute) by the abovesaid number of turns; and the quotient will be the number of turns the millstone ought to have for one turn of the water or cog-wheel. Then,

7. As the required number of turns of the millstone in a minute is to the number of turns of the cogwheel in a minute, so must the number of cogs in the wheel be to the number of staves or rounds in the trundle on the axis of the millstone, in the nearest whole number that can be found.

By these rules the following table is calculated; in which, the diameter of the water-wheel is supposed 18 feet, and consequently its circumference 56 4/7 feet, and the diameter of the millstone is 5 feet.|

The Mill-Wright's Table.
Perpendicular height of the fall of water.Velocity of the water in feet per second.Velocity of the wheel in feet per second.Number of turns of the wheel in a minute.Required n°. of turns of the millstone for each turn of the wheel.Nearest number of cogs and staves for that purpose.Number of turns of the millstone for one turn of the wheel by these cogs and staves.Number of turns of the millstone in a minute by these cogs and staves.

For the theory and construction of Wind-mills, see Wind-mill.

previous entry · index · next entry


Entry taken from A Mathematical and Philosophical Dictionary, by Charles Hutton, 1796.

This text has been generated using commercial OCR software, and there are still many problems; it is slowly getting better over time. Please don't reuse the content (e.g. do not post to wikipedia) without asking liam at holoweb dot net first (mention the colour of your socks in the mail), because I am still working on fixing errors. Thanks!

previous entry · index · next entry