Encyclopaedia Britannica [12, 7 ed.]

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ENCYCLOPAEDIA BRITANNICA SEVENTH EDITION.

THE

ENCYCLOPAEDIA BRITANNICA OR

DICTIONARY 9

OF 0

ARTS, SCIENCES, AND GENERAL LITERATURE.

SEVENTH EDITION, *

WITH PRELIMINARY DISSERTATIONS ON THE HISTORY OF THE SCIENCES, AND

OTHER EXTENSIVE IMPROVEMENTS AND ADDITIONS; INCLUDING THE LATE SUPPLEMENT.

A GENERAL INDEX, AND NUMEROUS ENGRAVINGS.

VOLUME XII.

ADAM AND CHARLES BLACK, EDINBURGH; M.DCCC.XLII.

ENCYCLOPEDIA BRITANNIC A

HYDRODYNAMICS. History. 1. TTYDRODYNAMICS, from “ water,” and ^ -11“ power,” is properly that science which v Definition. treats of the power of water, whether it acts by pressure or by impulse. In its more enlarged acceptation, however, it treats of the pressure, equilibrium, cohesion, and motion of fluids, and of the machines by which water is raised, or in which that fluid is employed as the first mover. Hydrodynamics is divided into two biancnes, Hydrostatics and Hydraulics. Hydrostatics comprehends the pressure, equilibrium, and cohesion of fluids, and Hydraulics their motion, together with the machines in which they are chiefly concerned. HISTORY. Hydrody. 2. The science of hydrodynamics was cultivated with namics in less success among the ancients than any other bianch m some re- mechanical philosophy. When the human mind had made spects a considerable progress in the other departments of physical science, the doctrine of fluids had not begun to occupy the attention of philosophers ; and, if we except a few propositions on the pressure and equilibrium of water, hydrodynamics must be regarded as a modern science, which owes its existence and improvement to those great men who adorned the seventeenth and eighteenth centuries. Discoveries 3. Those general principles of hydrostatics which are to of Archi- this day employed as the foundation of that part of the medes. science, were first given by Archimedes in his work risg< A. C. -50. ^ h or De Insidentibus Humido, about 250 years before the birth of Christ, and were afterwards applied to experiments by Marinus Ghetaldus in his Archimedes Hi omotus. Archimedes maintained that each particle of a fluid mass, when in equilibrio, is equally pressed in every direction; and he inquired into the conditions, according to which a solid body floating in a fluid should assume and preserve a position of equilibrium. We are also indebted to the philosopher of Syracuse for that ingenious VOL. XII.

hydrostatic process by which the purity of the precious History, metals can be ascertained, and for the screw engine which-v-—' goes by his name, the theory of which has lately exercised the ingenuity of some of our greatest mathematicians. 4. In the Greek school at Alexandria which flourished under the auspices of the Ptolemies, the first attempts were made at the construction of hydraulic machinery. About 120 years after the birth of Christ, the fountain of Inventions compression, the syphon, and the forcing pump, were in-ofCtesibius vented by Ctesibius and Hero ; and though these machines operated by the elasticity and weight of the air, yet their inventors had no distinct notions of these pieliminary branches of pneumatical science. 1 he syphon is a simple instrument which is employed to empty vessels full of water or spirituous liquors, and is of great utility in the arts. The forcing pump, on the contrary, is a complicated Forcing and abstruse invention, which could scarcely have been 1 unexpected in the infancy of hydraulics. It was probably suggested to Ctesibius by the Egyptian wheel or Noria, Egyptian which was common at that time, and which was a kind of wheel, chain pump, consisting of a number of earthern pots carried round by a wheel. In some of these machine s the pots have a valve in their bottom which enables them to descend without much resistance, and diminishes greatly the load upon the wheel; and if we suppose that this valve was introduced so early as the time of Ctesibius, it is not difficult to perceive how such a machine might have led this philosopher to the invention of the forcing 5. Notwithstanding these inventions of the Alexandrian Labours of school, its attention does not seem to have been directed exus^uto the motion of fluids. The first attempt to investigate nus in ^ this subject was made by Sextus Julius Frontinus, u-jspec- draulies. tor of the public fountains at Rome in the reigns of Nerva A. D. 110. and Trajan ; and we may justly suppose that his work entitled De Aquceductibus urbis Romm Commentanus contains all the hydraulic knowledge of the ancients. After

HYDRODYNAMICS.

2

History, describing the nine1 great Roman aqueducts, to which he

——v

' himself added more, and mentioning the dates of their erection, he considers the methods which were at that time employed for ascertaining the quantity of water discharged from adjutages, and the mode of distributing the waters of an aqueduct or a fountain. He justly remarks that the expense of water from an orifice, depended not only on the magnitude of the orifice itself, but also on the height of the water in the reservoir ; and that a pipe employed to carry off a portion of water from an aqueduct, should, as circumstances required, have a position more or less inclined to the original direction of the current. But as he was unacquainted with the true law of the velocities of running water as depending upon the depth of the orifice, we can scarcely be surprised at the want of precision which appears in his results. It has generally been supposed that the Romans were The Romans ac- ignorant of the art of conducting and raising water by quainted means of pipes ; but it can scarcely be doubted, fi'om the with the statement of Pliny and other authors, that they not only art of conwere acquainted with the hydrostatical principle, but that ducting water in they actually used leaden pipes for the purpose. Pliny asserts that water will always rise to the height of its source, pipes. and he also adds that, in order to raise water up to an eminence, leaden pipes must be employed.2 6. The labours of the ancients in the science of hydrodynamics terminated with the life of Frontinus. The sciences had already begun to decline, and that night of ignorance and barbarism was advancing apace, which for more than a thousand years brooded over the nations of Europe. During this lengthened period of mental degeneracy, when less abstruse studies ceased to attract the notice, and rouse the energies of men, the human mind could not be supposed capable of that vigorous exertion, and patient industry, which are so indispensable in physical reLabours of searches. Poetry and the fine arts, accordingly, had made Galileo. considerable progress under the patronage of the family of Born 1504, Medici, before Galileo began to extend the boundaries of died 1641. scjence> This great man, who deserves to be called the father and restorer of physics, does not appear to have directed his attention to the doctrine of fluids : but his discovery of the uniform acceleration of gravity, laid the foundation of its future progress, and contributed in no small degree to aid the exertions of genius in several branches of science. OfCastelli. 7* Castelli and Torricelli, two of the disciples of Galileo, Born 1577, applied the discoveries of their master to the science of died 1644. hydrodynamics. In 1628 Castelli published a small work, Della Misura dell ’acque correnti, in which he gave a very satisfactory explanation of several phenomena in the motion of fluids, in rivers and canals. But he committed a great paralogism in supposing the velocity of the water proportional to the depth of the orifice below the surface of the Of Torri- vessel. Torricelli observing that in a, jet d'eau where the celli. water pushed through a small adjutage, it rose to nearly 8 tlie Same w diYiar' hh the reservoir from which it was supte 1 ’ ‘ plied, imagined that it ought to move with the same velocity as if it had fallen through that height by the force of gravity. And hence he deduced this beautiful and important proposition, that the velocities of fluids are as the square roots of the pressures, abstracting from the resistance of the air and the friction of the orifice. This theorem was published in 1643, at the end of his treatise De Motu Gravium naturaliter accelerato. It was afterwards confirmed by the experiments of Raphael Magiotti, on

the quantities of water discharged from different adjutages History, under different pressures; and though it is true only in small orifices, it gave a new turn to the science of hydraulics. 8. After the death of the celebrated Pascal, who dis-Of Pascal, covered the pressure of the atmosphere, a treatise on the Born 1623, equilibrium of fluids (Sur VEquilibre des Liqueurs), was 1662. found among his manuscripts, and was given to the public in 1663. In the hands of Pascal, hydrostatics assumed the dignity of a science. The laws of the equilibrium of fluids were demonstrated in the most perspicuous and simple manner, and amply confirmed by experiments. The discovery of Torricelli, it may be supposed, would have incited Pascal to the study of hydraulics. But as he has not treated this subject in the work which has been mentioned, it was probably composed before that discovery had been made public. 9. The theorem of Torricelli was employed by manyofMari-

succeeding writers, but particularly by the celebrated Ma-otte. riotte, whose labours in this department of physics deserve Died 1684. to be recorded. His Traite du Mouvement des Eaux, which was published after his death in the year 1686, is founded on a great variety of well-conducted experiments on the motion of fluids, performed at Versailles and Chantilly. In the discussion of some points he has committed considerable mistakes. Others he has treated very superficially, and in none of his experiments does he seem to have attended to the diminution of efflux arising from the contraction of the fluid vein, when the orifice is merely a perforation in a thin plate ; but he appears to have been the first who attempted to ascribe the discrepancy between theory and experiment to the retardation of the water’s velocity arising from friction. His cotemporary Guglielmini, who was inspector of the rivers and canals in the Milanese, had ascribed this diminution of velocity in rivers, to transverse motions arising from inequalities in their bottom. But as Mariotte observed similar obstructions even in glass pipes, where no transverse currents could exist, the cause assigned by Guglielmini seemed destitute of foundation. The French philosopher, therefore, regarded these obstructions as the effects of friction. He supposes that the filaments of water which graze along the sides of the pipe lose a portion of their velocity; that the contiguous filaments having on this account a greater velocity, rub upon the former, and suffer a diminution of their celerity; and that the other filaments are affected with similar retardations proportional to their distance from the axis of the pipe. In this way the medium velocity of the current may be diminished, and consequently the quantity of water discharged in a given time, must, from the effects of friction, be considerably less than that which is computed from theory. 10. That part of the science of hydrodynamics which The morelates to the motion of rivers seems to have originated intion of Italy. This fertile country receives from the Apennines Bvers first a great number of torrents, which traverse several princi- ^tended palities before they mingle their waters with those of theto 111 Po, into which the greater part of them fall. To defend themselves from the inundations with which they were threatened, it became necessary for the inhabitants to change the course of their rivers ; and while they thus drove them from their own territories, they let them loose on those of their neighbours. Hence arose the* continual quarrels which once raged between the Bolognese and the inhabitants of Modena and Ferrara. The attention of the Italian engineers was necessarily directed to this branch of science; and from this cause a greater number of works

1 These nine aqueducts delivered every day 14.000 quinaria, or about 50,000,000 cubic feet of water, or about 50 cubic feet for the dady consumption of each inhabitant, supposing the population of Rome to have been a million. According to Professor Leslie the supply in modern Rome is forty cubic feet per person, in London three cubic feet, and in Paris one-half a cubic foot See Elements of Nat. Phil, p, 419. 2

xxxv

i* 7.

See also Palladius De Be Rustica ix. 11, &c., and Horace Epist. I. x. 20, Ovid Met. iv. 120.

HYDRODYNAMICS. History. were written on the subject in Italy than in all the rest of Europe. 11. Guglielmini was the first who attended to the moTheory of Gugliel- tion of water in rivers and open canals.1 Embracing the mini. theorem of Torricelli, which had been confirmed by repeated experiments, Guglielmini concluded that each particle in the perpendicular section of a current has a tendency to move with the same velocity as if it issued from an orifice at the same depth from the surface. The consequences deducible from this theory of running waters are in every respect repugnant to experience, and it is really surprising that it should have been so hastily adopted by succeeding writers. Guglielmini himself was sufficiently sensible that his parabolic theory was contrary to fact, and endeavoured to reconcile them by supposing the motion of rivers to be obstructed by transverse currents arising from irregularities in their bed. The solution of this difficulty, as given by Mariotte, was more satisfactory, and was afterwards adopted by Guglielmini, who maintained also that the viscidity of water had a considerable share in retarding its motion. 12. The effects of friction and viscidity in diminishing Discoveries of Sir the velocity of running water were noticed in the Principia Isaac New-0f g[r Isaac Newton, who has thrown much light upon several branches of hydrodynamics. AtatimewhentheCardie™1727 5 tcsian system of vortices universally prevailed, this great man found it necessary to investigate that absurd hypothesis, and in the course of his investigations he has shewn that the velocity of any stratum of the vortex is an arithmetical mean between the velocities of the strata which enclosed it; and from this it evidently follows, that the velocity of a filament of water moving in a pipe is an arithmetical mean between the velocities of the filaments which surround it. Taking advantage of these results, it was afterwards shewn by M. Pitot, that the retardations arising from friction are inversely as the diameters of the pipes in which the fluid moves. The attention of Newton was also directed to the discharge of water from orifices in the bottom of vessels. He supposed a cylindrical vessel full of water to be perforated in its bottom with a small hole by which the water escaped, and the vessel to be supplied with water in such a manner that it always remained full at the same height. He then supposed this cylindrical column of water to lie divided into two parts ; the first, which he calls the cataract, being a hyperboloid generated by the revolution of a hyperbola of the fifth degree around the axis of the cylinder which should pass through the orifice; and the second the remainder of the water in the cylindrical vessel. He considered the horizontal strata of this hyperboloid as always in motion, while the remainder of the water was in a state of rest; and imagined that there was a kind of ca taract in the middle of the fluid. When the results of this theory were compared with the quantity of water actually discharged, Newton concluded that the velocity with which the water issued from the orifice was equal to that which a falling body would receive by descending through half the height of water in the reservoir. This conclusion, however, is absolutely irreconcilable with the known fact, that jets of water rise nearly to the same height as their reservoirs, and Newton seems to have been aware of this objec-tion. In the second edition of his Principia, accordingly, which appeared in 1714, Sir Isaac has reconsidered his theory. He had discovered a contraction in the vein of fluid {vena contractd), which issued from the orifice, and found that, at the distance of about a diameter of the aperture, the section of the vein was contracted in the subduplicate ratio of two to one. He regarded, therefore, the section of the contracted vein as the true orifice from which the discharge of water ought to be deduced, and the velo-

city of the effluent water as due to the whole height of History, water in the reservoir; and by this means his theory be^ came more conformable to the results of experience. This theory, however, is still liable to serious objections. The formation of a cataract is by no means agreeable to the laws of hydrostatics; for when a vessel is emptied by the efflux of water through an orifice in its bottom, all the particles of the fluid direct themselves toward this orifice, and therefore no part of it can be considered as in a state of repose. 13. The subject of the oscillation of waves, one of the The oscilmost difficult in the science of hydrodynamics, was first lation of investigated by Sir Isaac Newton. In the forty-fourth proposition of the second book of his Principia, he has furnished us with a method of ascertaining the velocity of the waves oft^n< the sea, by observing the time in which they rise and fall. If the two vertical branches of a syphon, which communicate by means of a horizontal branch, be filled with a fluid of known density, the two fluid columns, when in a state of rest, will be in equilibrio and their surfaces horizontal. But if the one column is raised above the level of the other, and left to itself, it will descend below that level, and raise the other column above it, and, after a few oscillations, they will return to a state of repose. Newton occupied himself in determining the duration of these oscillations, or the length of a pendulum isochronous to their duration ; and he found, by a simple process of reasoning, that, abstracting from the effects of friction, the length of a synchronous pendulum is equal to one-half of the length of the syphon, that is, of the two vertical branches and the horizontal one, and hence he deduced the isochronism of these oscillations. From this Newton concluded, that the velocity of waves formed on the surface of water, either by the wind or by means of a stone, was in the subduplicate ratio of their size. When their velocity, therefore, is measured, which can be easily done, the size of the waves will be determined by taking a pendulum which oscillates in the time that a wave takes to rise and fall. 14. In the year 1718, the Marquis Poleni published, Labours of at Padua, his work De Castellis per quae derivantur Flu- the Marviorum aqua:, &c. He found, from a great number of'l1”8 i>0_ experiments, that if A be the aperture of the orifice, and D its depth below the surface of the reservoir, thedied ’ quantity of water discharged in a given time will be as 2 AD X O QQI,

while it ought to be as 2 AD, if the velo-

city of the issuing fluid was equal to that acquired by falling through D. By adapting to a circular orifice through which the water escaped, a cylindrical tube of the same diameter, the Marquis found that the quantity discharged in a determinate time was considerably greater than when it issued from the circular orifice itself; and this happened whether the water descended perpendicularly or issued in a horizontal direction. 15. Such was the state of hydrodynamics in 1738, when Daniel Daniel Bernouilli published his Hydrodynamica, seu de vi- Bernouil1 116 ribus et motibus Fluidorum Commentarii. His theory oH ^ ^ the motion of fluids was founded on two suppositions, which °on of appeared to him conformable to experience. He supposed that the surface of a fluid, contained in a vessel which was Born 1700. emptying itself by an orifice, remains always horizontal; Died 1782. and if the fluid mass is conceived to be divided into an infinite number of horizontal strata of the same bulk, that these strata remain contiguous to each other, and that all their points descend vertically, with velocities inversely proportional to their breadth, or to the horizontal sections of the reservoir. In order to determine the motion of each stratum, he employed the principle of the conservatio viri-

See his principal work, entitled La Misura deli acque correnH.

HYDRODYNAMICS. History, um vivarum, and obtained very elegant solutions. In the structed of the finest masonry. Basins (one of which was History. v '—-v-'—'' opinion of the Abbe Bossut, his work was one of the finest 289 feet square) built of masonry, and lined with stucco, received the effluent water, which was conveyed in canals productions of mathematical genius.1 Objected 16. The uncertainty of the principle employed by Daniel of brickwork, lined with stucco, of various forms and declitoby Mac-Bemouiiiij which has never been demonstrated in a gene- vities. The whole of Michelotti’s experiments were conducted with the utmost accuracy; and his results, which EornAGgs ra^ manner> deprived his results of that confidence which die ■ 1746. they would otherwise have deserved ; and rendered it de- are in every respect entitled to our confidence, were puband John' sirable to have a theory more certain, and depending solely lished in 1774 in his Sperienze Idrauliche. Bernouilli, on the fundamental laws of mechanics. Maclaurin and 19. The experiments of the Abbe Bossut, whose labours Of the Born 1G67, John Bernouilli, who were of this opinion, resolved the in this department of science have been very assiduous and Abbe Bosdied 1748. problem by more direct methods, the one in his Fluxions, successful, have, in as far as they coincide, afforded the8111, published in 1742; and the other in his Hydraulica nunc same results as those of Michelotti. Though performed on primum detecta, et directe demonstrata ex principiis pure a smaller scale, they are equally entitled to our confidence, mechanicis, which forms the fourth volume of his works. and have the merit of being made in cases which are most The method employed by Maclaurin has been thought not likely to occur in practice. In order to determine what, sufficiently rigorous ; and that of John Bernouilli is, in the were the motions of the fluid particles in the interior of a opinion of La Grange, defective in perspicuity and precision. vessel emptying itself by an orifice, M. Bossut employed a 17. The theory of Daniel Bernouilli was opposed also glass cylinder, to the bottom of which different adjutages D’Alem- by the celebrated D’Alembert. When generalising James were fitted; and he found that all the particles descend at bert apBernouilli’s Theory of Pendulums, he discovered a prin- first vertically, but that at a certain distance from the oriplies his ciple of dynamics so simple and general, that it reduced the fice they turn from their first direction towards the aperprinciple laws of the motions of bodies to that of their equilibrium. ture. In consequence of these oblique motions, the fluid of dvnamics to the He applied this principle to the motion of fluids, and gave vein forms a kind of truncated conoid, whose greatest base motion of a specimen of its application at the end of his Dynamics is the orifice itself, having its altitude equal to the radius fluids, in 1743. It was more fully developed in his Traite des of the orifice, and its bases in the ratio of 3 to 2 It apBorn 1717. Fluides, which was published in 1744, where he has re- pears also, from the experiments of Bossut, that when wasolved, in the most simple and elegant manner, all the pro- ter issues through an orifice made in a thin plate, the exblems which relate to the equilibrium and motion of fluids. pense of water, as deduced from theory, is to the real exHe makes use of the very same suppositions as Daniel Ber- pense as 16 to 10, or as 8'to 5 ; and, when the fluid issues nouilli, though his calculus is established in a very different through an additional tube, two or three inches long, and manner. He considers, at every instant, the actual motion follows the sides of the tube, as 16 to 13—In analyzing of a stratum, as composed of a motion which it had in the the effects of friction, he found, 1. That small orifices gave preceding instant, and of a motion which it has lost. The less water in proportion than great ones, on account of fr iclaws of equilibrium between the motions lost, furnish him tion ; and, 2. That when the height of the reservoir was with equations which represent the motion of the fluid. Al- augmented, the contraction of the fluid vein was also inthough the science of hydrodynamics had then made con- creased, and the expense of water diminished; and by siderable progress, yet it was chiefly founded on hypothe- means of these two laws he was enabled to determine the sis. It remained a desideratum to express by equations quantity of water discharged, with all the precision he could the motion of a particle of the fluid in any assigned direc- wish. In his experiments on the motion of water in canals tion. These equations were found by D’Alembert, from and tubes, he found that there was a sensible difference betwo principles, that a rectangular canal, taken in a mass of tween the motion of water in the former and in the latter. fluid in equilibrio, is itself in equilibrio ; and that a portion Under the same height of reservoir, the same quantity of of the fluid, in passing from one place to another, preserves water always flows in a canal, whatever be its length and the same volume when the fluid is incompressible, or dilates declivity; whereas, in a tube, a difference in length and itself according to a given law when the fluid is elastic. His declivity has a very considerable influence on the quantity very ingenious method was published in 17 52, in his Essai of water discharged. According to the theory of the resur la resistance des Jluides. It was brought to perfection sistance of fluids, the impulse upon a plane surface is as in his Opuscules Mathematiques, and has been adopted by the product of its area multiplied by the square of the the celebrated Euler. fluid’s velocity, and the square of the sine of the angle of Before the time of D’Alembert, it was the great object incidence. The experiments of Bossut, made in conjuncof philosophers to submit the motion of fluids to general tion with D’Alembert and Condorcet, prove, that this is formulae, independent of all hypothesis. Their attempts, sensibly true when the impulse is perpendicular; but that however, were altogether fruitless ; for the method of flux- the aberrations from theory increase with the angle of imions, which produced such important changes in the phy- pulsion. They found, that when the angle of impulsion sical sciences, was but a feeble auxiliary in the science of was between 50° and 90°, the ordinary theory may be emhydraulics. For the resolution of the questions concerning ployed, that the resistances thus found will be a little less the motion of fluids, we are indebted to the method of par- than they ought to be, and the more so as the angles recede tial differences, a new calculus, with which Euler enriched from 90°. The attention of Bossut was directed to a vathe sciences. This great discovery was first applied to the riety of other interesting points, which we cannot stop to motion of water by the celebrated D’Alembert, and enabled notice, but for which we must refer the reader to the works both him and Euler to represent the theory of fluids in of that ingenious author. formulae restricted by no particular-hypothesis. 20. The oscillation of waves, which was first discussed Inquiries F.xperi18. An immense number of experiments on the motion by Sir Isaac Newton, and afterwards by D’Alembert, in of Flaugerments of of water in pipes and canals was made by Professor Michethe article Ondes in the French Encyclopaedia, was now krues.conMichelotti of Turin, at the expense of the sovereign. In these revived by M. Flaugergues, who attempted to overthrow th^oseik lotti, A. I). 1764.experiments tlie water issued from holes of different sizes, the opinions of these philosophers. He maintained, that aiation 0f" under pressures of from 5 to 22 feet, from a tower con- wave is not the effect of a motion in the particles of water, waves.

1 The germ of Daniel Bernouilli’s theory was first published in his memoir entitled Theoria Nova de Motu Aquarum per Canales quocunque Fluentes, which he had communicated to the Academy of St Petersburg as early as 1726.

HYDRODYNAMICS. History, by which they rise and fall alternately, in a serpentine line, when moving from the centre where they commenced; but that it is a kind of intumescence, formed by a depression at the place where the impulse is first made, which propagates itself in a circular manner when removing from the point of impulse. A portion of the water, thus elevated, he imagines, flows from all sides into the hollow formed at the centre of impulse, so that the water being, as it were, heaped up, produces another intumescence, which propagates itself as formerly. From this theory M. Flaugergues concludes, and he has confirmed the conclusion by experiment, that all waves, whether great or small, have the same velocity. And of M. 21. This difficult subject has also been discussed by M. de la de la Grange, in his Mecanique Analytique. He found, (1 range, that the velocity of waves, in a canal, is equal to that which a heavy body would acquire by falling through a height ie ' ‘ equal to half the depth of the water in the canal. If this depth, therefore, be one foot, the velocity of the waves will be 5.945 feet in a second; and if the depth is greater or less than this, their velocity will vary in the subduplicate ratio of the depth, provided it is not very considerable. If we suppose that, in the formation of waves, the water is agitated but to a very small depth, the theory of La Grange may be employed, whatever be the depth of the water and the figure of its bottom. This supposition, which is very plausible, when we consider the tenacity and adhesion of the particles of water, has also been confirmed by experience. Experi22. The most successful labourer in the science of hyments and drodynamics was the Chevalier Buat, engineer in ordinary theory of to the King of France. Following in the steps of the Abbe th ® ‘ Bossut, he prosecuted the inquiries of that philosopher with BuaT

6

uncommon ingenuity; and in the year 1786, he published,

A.D.* 1779. in two volumes, his Principes d'Hydraulique,1 which con-

tains a satisfactory theory of the motion of fluids, founded solely upon experiments. The Chevalier du Buat considered, that if water were a perfect fluid, and the channels in which it flowed infinitely smooth, its motion would be continually accelerated, like that of bodies descending in an inclined plane. But as the motion of rivers is not continually accelerated, and soon arrives at a state of uniformity, it is evident that the viscidity of the water, and the friction of the channel in which it descends, must equal the accelerating force. M. Buat, therefore, assumes it as a proposition of fundamental importance, that when water flows in any channel or bed, the accelerating force, which obliges it to move, is equal to the sum of all the resistances which it meets with, whether they arise from its own viscidity or from the friction of its bed. This principle was employed by M. Buat, in the first edition of his work, which appeared in 1779 ; but the theory contained in that edition was founded on the experiments of others. He soon saw, however, that a theory so new, and leading to results so different from the ordinary theory, should be founded on new experiments more direct than the former, and he was employed in the performance of these from 1780 to 1783. The experiments of Bossut having been made only on pipes of a moderate declivity, M. Buat found it necessary to supply this defect. He used declivities of every kind, from the smallest to the greatest; and made his experiments upon channels, from a line and a half in diameter, to seven or eight square toises. All these experiments he arranged under some circumstances of resemblance, and produced the following proposition, which agrees in a most wonderful manner with the immense number of facts which he has brought together, viz.

5 History.

307 X>Jd- -01 0.3 X d—0.1, \!s—L s!s-\-1.6 where d is the hydraulic mean depth, s the slope of the pipe, or of the surface of the current, and V the velocity with which the water issues. 23. M. Venturi, Professor of Natural Philosophy in the Researches University of Modena, succeeded in bringing to light some of M. Vencurious facts respecting the motion of water, in his workturion the “Lateral Communication of Motion in Fluids.”^’ He observed, that if a current of water is introduced with a certain velocity into a vessel filled with the same fluid at rest, and if this current passing through a portion of the fluid is received in a curvilineal channel, the bottom of which gradually rises till it passes over the rim of the vessel itself, it will carry along with it the fluid contained in the vessel; so that after a short time has elapsed, there remains only the portion of the fluid which was originally below the aperture at which the current entered. This phenomenon has been called by Venturi, the lateral communication of motion in fluids; and, by its assistance, he has explained many important facts in hydraulics. He has not attempted to explain this principle; but has shewn, that the mutual action of the fluid particles does not afford a satisfactory explanation of it. The work of Venturi contains many other interesting discussions, which are worthy of the attention of every reader. 24. Although the Chevalier Buat had shown much saga- Discove. city in classifying the different kinds of resistances which ries of are exhibited in the motion of fluids, yet it was reserved for Coulomb, Coulomb to express the sum of them by a rational function VP, 1800. of the velocity. By a series of interesting experiments on the successive diminution of the oscillation of discs, arising from the resistance of the water in which they oscillated, he was led to the conclusion, that the pressure sustained by the moving disc is represented by two terms, one of which varies with the simple velocity, and the other with its square. When the motions are very slow, the part of the resistance proportional to the square of the velocity is insensible, and hence the resistance is proportional to the simple velocity. M. Coulomb found also, that the resistance is not perceptibly increased by increasing the depth of the oscillating disc in the fluid; and by coating the disc successively with fine and coarse sand, he found that the resistance arises solely from the mutual cohesion of the fluid particles, and from their adhering to the surface of the moving body, 25. The law of resistance discovered by Coulomb, was Experi. first applied to the determination of the velocity of running meats of water by M. Girard, who considers the resistance as repre- M. Girard, sented by a constant quantity, multiplied by the sum of the first and second powers of the velocity. He regards the water which moves over the wetted sides of the channel as at first retarded by its viscidity, and he concludes that the water will, from this cause, suffer a retardation proportional to the simple velocity. A second retardation, analogous to that of friction in solids, he ascribes to the roughness of the channel, and he represents it by the second power of the velocity, as it must be in the compound ratio of the force and the number of impulsions which the asperities receive in a given time. He then expresses the resistance due to cohesion by a constant quantity, to be determined experimentally, multiplied into'the product of the velocity of the perimeter of the section of the fluid. 26. The influence of heat in promoting fluidity was known to the ancients ;2 but M. Du Buat was the first person who A.D. 1814. investigated the subject experimentally. His results, however, were far from being satisfactory; and it was left to v_

i A. third volume of this work was published in 1816, entitled Principes d'Hydrauliqne et Pyrodynamique, relating chiefly to the 2 subject of heat and elastic fluids. FAny, Quaint. Nat.

HYDRODYNAMICS.

6

History* M. Girard to ascertain the exact effect of temperature on the motion of water in capillary tubes. When the length of the capillary tube is great, the velocity is quadrupled by an increase of heat from 0° to 85° centig.; but when its length is small, a change of temperature exercises little or no influence on the velocity. He found also, that, in ordinary conduit pipes, a variation of temperature exercises scarcely any influence over the velocity. Investiga27. The theory of running water was greatly advanced tions ot M. by the researches of M. Prony. From a collection of the A D 1804

experiments by Couplet, Bossut, and Du Buat, he selected 82, of which 51 were made on the velocity of water in conduit pipes, and 31 on its velocity in open canals; and by discussing these on physical and mechanical principles, he succeeded in drawing up general formulae, which afford a simple expression of the velocity of running water. The following is the formula for English feet, which answers both for pipes and canals:— V = 0.1541131 + V (0.023751 + 32806.6 G) When we use this formula for canals, we must take —, a representing the area III, R being ~ — and I A £ of the section of the pipe or canal, ^ the perimeter of the section in contact with the water, £ the difference of level between the two extremities of the pipe, and A the length of the pipe or canal. When the formula is applied to pipes, we must take G = i DK, D being the diameter of the pipe, and K = G

H 4- B — c X «iB ; then, by transposition, aynb — cxmB = ax &B + cx6B=a + C + 6B. But wB = aA and »nB = cC, therefore, by substitution, «xaA + cxcC = a + cx6B. By supposing the two weights a and c united in their common centre of gravity, the same demonstration may be extended to any number of weights.

HYDRODYNAMICS. Pressure, similar pressure; and therefore it follows, that the pressure &c. of upon the bottom Q.R is as great as if it supported the whole Fluids. column MNQIl. gi. The same truth may be deduced from Prop. IV. For since the fluid in the two comFig. 8. municating vessels AB, CD, will rise to the same level, whatever be their A size, the fluid in AB evidently balances the fluid in CD ; and any surface mn is pressed with the same force in the direction B?» by the small column A B, as it is pressed in „ the direction Dm by the larger coB lumnCD. Corolla62. COR. i. From this proposition it follows, that the •los. whole pressure on the sides of a vessel which are perpendicular to its base, is equal to the weight of a rectangular prism of the fluid, whose altitude is that of the fluid, and whose base is a parallelogram, one side of which is equal to the altitude of the fluid, and the other to half the perimeter of the vessel. COR. II. The pressure on the surface of a hemispherical vessel full of fluid, is equal to the product of its surface multiplied by its radius. COR. in. In a cubical vessel the pressure against one side is equal to half the pressure against the bottom; and the pressure against the sides and bottom together, is to that against the bottom alone as three to one. Hence, as the pressure against the bottom is equal to the weight of the fluid in the vessel, the pressure against both the sides and bottom will be equal to three times that weight. COR. iv. The pressure sustained by different parts of the side of a vessel are as the squares of their depths below the surface ; and if these depths are made the abscissae of a parabola, its ordinates will indicate the corresponding pressures. Definition.

13

lines CM, DM, EM. But the pressures upon the point Pressure, C, D, E, are as the lines CN, DO, EP, and these lines are &c. of to one another as CM, DM, EM; therefore the percussive Fhdds. forces of the points C, D, E, are as the pressures upon these points. Consequently, the centre of pressure will always coincide with the centre of percussion.

SECT.

II. Instruments and Experiments for illustrating the Pressure of Fluids.

65.

upon the bottoms of vessels filled with fluids does not de- f°r ihuspend upon the quantity of fluid which they contain, but |,ra^n? t!‘e upon its particular altitude. This proposition has been called the Hydrostatical Paradox, and is excellently illus- dox, trated by the following machine. In fig. 10, AB is a box 10_ which contains about a pound of water, and abed a, glass tube fixed to the end C of the beam of the balance, and the other end to a moveable bottom which supports the water in the box, the bottom and wire being of an equal weight with an empty scale hanging at the other end of the balance. If one pound weight be put into the empty scale, it will make the bottom rise a little, and the water will appear at the bottom of the tube a, consequently it will press with a force of one pound upon the bottom. If another pound be put into the scale, the water will rise to b, twice as high as the point a, above the bottom of the vessel. If a third, a fourth, and a fifth pound be put successively into the scale, the water will rise at each time to c, d, and e, Fig. 10.

Fig 11.

DEFINITION. 63. The centre of pressure is that point of a surface exposed to the pressure of a fluid, to which, if the total pressure were applied, the effect upon the plane would be the same as when the pressure was distributed over the whole surface: Or, it is that point to which, if a force equal to the total pressure were applied in a contrary direction, the one would exactly balance the other, or, in other words, the force applied and the total pressure would be in equilibria. PROP.

64. percussion. To find the centre of pressure.

VIII.

The centre of pressure coincides with the centre of

Let AB be a vessel full of water, and CE the section of Fig. 9. a plane whose centre of pressure is required. Prolong CE A till it cuts the surface of the water in M. Take any point D, and draw DO, EP, CN, perpendicular to the surface MP. Then if M be made the axis of suspension of the plane CE, the centre of percussion of the plane CE revolving round M will also be the centre of pressure. If MCE moves round M as a centre, and strikes any object, the percussive force of any point C is as its velocity, that is, as its distance CM from the centre of motion; therefore the percussive force of the points C, D, E, are as the

the divisions ab, be, cd, de, being all equal. This will be the case, however small be the bore of the glass tube ; and since, when the water is at b, c, d, e, the pressures upon the bottom are successively twice, thrice, four times, and five times as great as when the water was contained within the box, we are entitled to conclude that the pressure upon the bottom of the vessel depends altogether on the altitude of the water in the glass tube, and not upon the quantity it

14

HYDRODYNAMICS.

Pressure, contains. If a long narrow tube full of water, therefore, be &c. of fixed in the top of a cask likewise full of water, then though Fluids. j-}ie tube be so small as not to hold a pound of the fluid, the ^

Y

Fig. 13.

Fig. 14.

Pressure, &c. of Fluids.

J

pressure of the water in the tube will be so great on the bottom of the cask, as to be in danger of bursting it; for the pressure is the same as if the cask was continued up in its full size to the height of the tube, and filled with water. The small- Upon this principle it has been affirmed that a certain est quanti-quantity of water, however small, may be rendered capable ty of water of exerting a force equal to any assignable one, by inaTonxf ^1 creasing the height of the column, and diminishing the base equal to on which it presses. This, however, has its limits; for any assign- when the tube becomes so small as to belong to the capilable one,, lary kind, the attraction of the glass will support a considerable quantity of the water it contains, and therefore diminish the pressure upon its base. Construe66. The preceding machine must be so constructed, that tion of the the moveable bottom may have no friction against the inpreceding gj^g 0p t]ie box, and that no water may get between it and machine.

the box. The method of effecting this will be manifest from fig. 11, where ABCD is a section of the box, and abed its lid, which is made very light. The moveable bottom E, with a groove round its edges, is put into a bladder/I?, which is tied close around it in the groove by a strong waxed thread. The upper part of the bladder is put over the top of the box at a and d all around, and is kept firm by the lid abed, so that if water be poured into the box through the aperture //in its lid, it vail be contained in the space/E