HISTORICAL INTRODUCTION.

The word Hydromechanics is derived from the Greek vhpo-jxrjxavLKo., meaning the mechanics of water and fluids in general. The science is divided into three branches :—Hydrostatics, which deals with the equilibrium of fluids; Hydrodynamics, which deals with the mathe-matical theory of the motion of fluids, neglecting the viscosity; and Hydraulics, in which the motion of water in pipes and canals is considered, and hydrodynamical questions of practical application are investigated.

The science of hydromechanics was cultivated with less success among the ancients than any other branch of mechanical philosophy. When the human mind had made 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 proposi-tions on the pressure and equilibrium of water, hydro-mechanics must be regarded as a modern science, which owes its existence and improvement to these great men who adorned the 17th and 18th centuries. Archi- Those general principles of hydrostatics which are to medes. this day employed as the foundation of that part of the science were first given by Archimedes in his work Hepi TU>V 6-xpvp.iviav, or Be Us quae vehuntur in humido, about 250 B.C., and were afterwards applied to experiments by Marinus Ghetaldus in his Promotus Archimedes (1603). Archimedes maintained that each particle of a fluid mass, when in equili-brium, is equally pressed in every direction; and he in-quired 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 him for that in-genious hydrostatic process by which the purity of the precious metals can be ascertained, and for the screw engine which goes by his name. Alex- In the Greek school at Alexandria, which flourished andrian under the auspices of the Ptolemies, the first attempts school. were made at the construction of hydraulic machinery. About 120 B.C. the fountain of compression, the siphon, and the forcing pump were invented by Ctesibius and Hero ; and, though these machines operated by the pressure of the air, yet their inventors had no distinct notions of the preliminary branches of pneumatical science. The siphon is a simple instrument; but the forcing pump is a com-plicated and abstruse invention, which could scarcely have been expected in the infancy of hydraulics. It was pro-bably suggested to Ctesibius by the Egyptian Wheel or Noria, which was common at that time, and which was a kind of chain pump, consisting of a number of earthen pots carried round by a wheel. In some of these machines the pots have a valve in the 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 that philosopher to the invention of the forcing pump.

Notwithstanding these inventions of the Alexandrian school, its attention does not seem to have been directed to the motion of fluids. The first attempt to investigate this Fron- subject was made by Sextus Julius Frontinus, inspector of tinus. the public fountains at Rome in the reigns of Nerva and Trajan; and we may justly suppose that his work, entitled De Aquceductibus Urbis Romce Commentarius, contains all the hydraulic knowledge of the ancients. After describing the nine great Roman aqueducts, to which he himself added five 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 ajutages, and the mode of distributing the waters of an aqueduct or a fountain. He justly remarks that the flow 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 ignorant of the art of conducting and raising water by means of pipes; but it can scarcely be doubted, from the statement of Pliny and other authors, not only that they were acquainted with the hydrostatical principle, but that they actually used leaden pipes for the purpose. Pliny asserts that water will always rise to the height of its source, and he also adds that, in order to raise water up to an eminence, leaden pipes must be employed.8

Castelli and Torricelli, two of the disciples of Galileo, Castelli applied the discoveries of their master to the science of hydrodynamics. In 1628 Castelli published a small work, Delia 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 propor-tional to the depth of the orifice below the surface of the vessel. Torricelli, observing that in a jet where the water Torri-rushed through a small ajutage it rose to nearly the same celli. height with the reservoir from which it was supplied, 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 pro-position, that the velocities of fluids are as the square root of the head, allowing for the resistance of the air and the friction of the orifice. This theorem was published in 164:3, at the end of his treatise De Molu gravium Projectorum. It was afterwards confirmed by the ex-periments of Raphael Magiotti on the quantities of water discharged from different ajutages under different pressures ; and, though it is true only in small orifices, it gave a new turn to the science of hydraulics.

2 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 daily consumption of each inhabitant, supposing the population of Rome to have been a million. From measurements of Frontinus at the close of the 1st century, the total length of channels of aqueducts was 285 Roman miles (Roman mile = 1618 English yards). The supply measured by Frontinus amounted to 13,470 quinaries,—outside Rome 3164, inside 10,306. Measured at the head the supply was 24,413 quinaries, the difference being due to waste, and to some of the channels having fallen into decay. Parker says :—"It has been com-puted by a French engineer that the supply to Rome was 332,306,624 gallons daily. If we assume the population at a million, the rate was 332 gallons daily per person. In our day we consider 40 gallons suffi-cient, and many think this excessive." Modem supply varies from 24 to 50 gallons per head per day.
3 Plin. xxxvi. See also Palladius, Be Be Bustica, ix., xi.; Horace, Mpist., i. 10, 20; Ovid, Met., iv. 122.





After the death of the celebrated Pascal, who discovered Pascal, the pressure of the atmosphere, a treatise on the equilibrium of fluids (Sur I'Equilibre des Liqueurs) was 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 mentioned, it was probably composed before that discovery had been made public. Mariotte. The theorem of Torricelli was employed by many succeed-ing writers, but particularly by the celebrated Mariotte, whose labours in this department of physics deserve 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 contemporary 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 dis-charged in a given time must, from the effects of friction, be considerably less than that which is computed from theory. Guglielmini was the first who attended to the motion of water in rivers and open canals (La Misura dell' acque correnti). Embracing the theorem of Torricelli, which had been confirmed by repeated experiments, Guglielmini concluded that each particle in the perpen-dicular 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 trans-verse 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 viscosity of water had a con-siderable share in retarding its motion. Newton. The effects of friction and viscosity in diminishing the velocity of running water were noticed in the Principia of Sir Isaac Newton, who threw much light upon several branches of hydromechanics. At a time when the Carte-sian system of vortices universally prevailed, this great man found it necessary to investigate that absurd hypothesis, and in the course of his investigations he showed that the velocity of any stratum of the vortex is an arithmetical mean between the velocities of the strata which enclose 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 shown by 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 be divided into two parts,—the first, which he calls the " cataract," being an hyperboloid generated by the revolu-tion of an 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 cataract 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 objection. In the second edition of his Principia accordingly, which appeared in 1714, he reconsidered his theory. He had discovered a contraction in the vein of fluid (vena contracta) 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, there-fore, the section of the contracted vein as the true orifice from which the discharge of water ought to be deduced, and the velocity of the effluent water as due to the whole height of water in the reservoir; and by this means his theory becamemore 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.

The subject of the oscillation of waves, one of the most difficult in the science of hydrodynamics, was first in-vestigated by 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 of the sea, by observing the time in which they rise and fall. If the two vertical branches of a siphon, which communi-cate by means of a horizontal branch, are filled with a fluid of known density, the two fluid columns, when in a state of rest, will be in equilibrium 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, allowing for the effects of friction, the length of a syn-chronous pendulum is equal to one-half of the length of the siphon, 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 a body thrown into it, was in the sub-duplicate ratio of their size. When their velocity, there-fore, is measured, which can be easily done, the size of the waves will be determined by means of a pendulum which oscillates in the time that a wave takes to rise and fall, nioulli. Such was the state of hydrodynamics in 1738, when Daniel Bernoulli published his Hydrodynamica, sive de Viribits et Motibus Fluidorum Commentarii. The germ of Daniel Bernoulli'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. His theory of the motion of fluids was founded on two suppositions, which appeared to him conformable to experience. He supposed that the surface of a fluid, contained in a ves-sel which is emptying itself by an orifice, remains always horizontal; 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 virium vivarum, and obtained very elegant solutions. In the opinion of the Abb^ Bossut, his work was one of the finest productions of mathematical genius. The uncer-tainty of the principle employed by Daniel Bernoulli, which has never been demonstrated in a general manner, deprived his results of that confidence which they would otherwise have deserved, and rendered it desirable to have a theory more certain, and depending solely on the funda-mental laws of mechanics. Maclaurin and John Bernoulli, who were of this opinion, resolved the problem by more direct methods, the one in his Fluxions, published in 1742, and the other in his Hydraulica nunc primum detecta, et demonstrata directe ex fundamentis pure mechanicis, which forms the fourth volume of his works. The method employed by Maclaurin has been thought not sufficiently rigorous; and that of John Bernoulli is, in the opinion of Lagrange, defective in perspicuity and precision, D'Alem- The theory of Daniel Bernoulli was opposed also by Dert- the celebrated D'Alembert. When generalizing James Bernoulli's theory of pendulums he discovered a prin-ciple of dynamics so simple and general that it reduced the laws of the motions of bodies to that of their equili-brium. He applied this principle to the motion of fluids, and gave a specimen of its application at the end of his Dynamics in 1743. It was more fully developed in his Traile des Fluides, which was published in 1744, where he has resolved, in the most simple and elegant manner, all the problems which relate to the equilibrium and motion of fluids. He makes use of the very same suppositions as Daniel Bernoulli, though his calculus is established in a very different manner. He considers, at every instant, the actual motion of a stratum as composed of a motion which it had in the preceding instant and of a motion which it has lost, The laws of equilibrium between the motions lost furnish him with equations which represent the motion of the fluid, Although the science of hydrodynamics had then made considerable progress, yet it was chiefly founded on hypothesis. It remained a desideratum to express by equations the motion of a particle of the fluid in any assigned direction. These equations were found by D'Alembert from two principles,—that a rectangular canal, taken in a mass of fluid in equilibrium, is itself in equilibrium; and that a portion of the fluid, in passing from one place to another, preserves the same volume when the fluid is incompressible, or dilates itself according to a given law when the fluid is elastic. His very ingenious method was published in 1752, in his Essai sur la Resistance des Fluides. It was brought to perfection in his Opuscules Mathématiques, and was adopted by Euler.





Before the time of D'Alembert, it was the great object of philosophers to submit the motion of fluids to general formulae, independent of all hypothesis. Their attempts, however, were altogether fruitless ; for the method of fluxions, which produced such important changes in the physical sciences, was but a feeble auxiliary in the science of hydraulics. For the resolution of the questions concern-ing the motion of fluids, we are indebted to the method of partial differences, a new calculus, with which Euler enriched the sciences. This great discovery was first applied to the motion of water by D'Alembert, and enabled both him and Euler to represent the theory of fluids in formulae restricted by no particular hypothesis.

The most successful labourer in the science of hydro- Dubuat. dynamics was the Chevalier Dubuat, engineer in ordinary to the king of France. Following in the steps of the Abbé Bossut (Nouvelles expériences sur la resistance des fluides, 1777), he prosecuted the inquiries of that philosopher with uncommon ingenuity; and in the year 1786 he published, in two volumes, his Principes d'Hydraulique, which contains a satisfactory theory of the motion of fluids, founded solely upon experiments. Dubuat 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 viscosity of the water, and the friction of the channel in which it descends, must equal the accelerating force. Dubuat 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 viscosity or from the friction of its bed. This principle was emploj'ed by Dubuat 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, Dubuat found it necessary to supply this defect. He used declivities of every kind, and made his experiments upon channels from a line and a half in diameter to seven or eight square toises.

The theory of running water was greatly advanced by Prony. the researches of Prony. From a collection of the best experiments by Couplet, Bossut, and Dubuat he selected eighty-two (fifty-one on the velocity of water in conduit pipes, and thirty-one on its velocity in open canals) ; and, discussing these on physical and mechanical principles, he succeeded in drawing up general formulae, which afford a simple expression for the velocity of running water.

Eytelwein of Berlin published, in 1801, a valuable com- Eytel-pendium of Hydraulics, entitled Handbuch der Mechanik wein-mid der Hydraulik, which contains an account of many new and valuable experiments made by himself. He in-vestigates the subject of the discharge of water by com-pound pipes, the motions of jets, and their impulses against plane and oblique surfaces; and he shows theo-retically that a water wheel will have its effect a maximum when its circumference moves with half the velocity of the stream.

Mallet A series of interesting hydraulic experiments was made and at Rome in 1809 by Mallet and Vici. They found that ^'oi- a pipe whose gauge was five ounces French measure (or 0'03059 French kilolitres) furnished one-seventh more wrater than five pipes of one ounce, an effect arising from the velocity being diminished by friction in the ratio of the perimeters of the orifices as compared with their areas. Hachette. Hachette, in the year 1816, presented to the National Institute a memoir containing the results of experiments which he had made on the spouting of fluids, and the discharge of vessels. The objects he had in view were to measure the contracted part of a fluid vein, to examine the phenomena attendant on additional tubes, and to investigate and describe the figure of the fluid vein, and the results which take place when different forms of orifices are employed. Hachette showed in the second part of his memoir that greater or lesser volumes of water will be discharged in the same time through tubes of different figures, the apertures in all having the same dimensions. He also gave several remarkable results respecting other fluids issuing out of orifices into air or a vacuum. Investi- Several very interesting experiments on the propagation gâtions 0f waves have been made by the brothers Weber and by waves Bidone. Mr John Scott Russell performed a number of experiments on waves, which are described in the Edin-burgh Transactions, vol. xiv., and in the British Association Report for 1837. The mathematical theory has been-worked out by Green, Stokes, Rankine, and other mathe-maticians, but still offers an interesting field for the inves-tigator. Stokes's Report of the British Association for 1846 on Recent Researches in Hydrodynamics gives an account of the subject as it existed at that date. Bidone. In 1826 Bidone, besides his experiments on waves, made a series on the velocity of running water at the hydraulic establishment of the university of Turin, and he published an account of them in 1829. After giving a description of his apparatus and method of experimenting, he gives the figures obtained from fluid veins, sections of which were taken at different distances from the orifice. Poncelet. In the year 1827 Poncelet published a Mémoire sui-tes Rones Hydrauliques à Aubes Courbes, containing his experiments on the undershot wheel with curved palettes, which he had invented in the year 1824. The best undershot previous to the introduction of the Poncelet wheel never developed more than 0'25 of the work of the water, whereas this utilized 0-60 of that work, which is nearly equivalent to the maximum effect of the breast wheel. The principle on which the Poncelet wheel acts, and that which makes it utilize so much of the work of the water, is that the water is received by the curved floats without any shock, and is discharged finally with a small velocity. This undershot wheel is much used in France. Four- Previous to the year 1827, the wheels required in the neyron. miHs and manufactories of Germany and France were generally those which worked with the axis horizontal, or the tub and spoon wheels with the axis vertical ; but in that year a young mechanician named Fourneyron introduced a wheel working with the axis vertical, yet wholly different from the latter kind. Fourneyron showed that in existing wheels with a vertical axis the water left the wheel with considerable velocity in the direction of the motion of the wheel, and thus carried away and wasted much of the energy of the fall. By the introduction of a series of fixed guide blades, which gave the water initially a backward velocity of rotation, the water left the wheel with a much smaller velocity of discharge. He thus invented the first complete turbine, a kind of water motor which has largely superseded the more cumbrous water wheels previously in use. Shortly after the invention was made public, Fourneyron was awarded the prize of 6000 francs which was offered by the Society for the Encouragement of National Industry.

The most extensive experiments on the discharge of water Recent from orifices are those made under the direction of the investi-French Government by Poncelet and Lesbros (Experiences S"* ™"* Hydrauliques, Paris, 1851). Boileau (Traite de la mesure des eaux courantes) has discussed these results and added experiments of his own. Bornemann has re-examined all these results with great care, and has been able to express in formulas the variation of the coefficients of discharge in different conditions (Civilingenieur, 1880). Very valuable experiments leading to a modification of the usual formula for the discharge over weirs were made by Mr J. B. Francis (Lowell Hydraulic Experiments, Boston, Mass., 1855). Wiesbach also has made many experimental investigations of the discharge of fluids.

The friction of water investigated originally by Coulomb at slow speeds has been measured for higher speeds by Mr W. Froude, whose researches have very great value in the theory of ship resistance (Report of Brit. Assoc., 1869).

The flow of air and steam from orifices has been measured by many experimenters from Young to Saint Venant. Mr Napier in some interesting experiments first pointed out that when the ratio of the pressures on the two sides of an orifice exceeded a certain limit the measured discharge was very different from that calculated by the accepted formulae. Since then numerous experiments have been made, and the theory of the flow of elastic fluids has been discussed in numerous memoirs. The valuable investigations of Fliegner (Civilingenieur, 1878) deserve special mention,

A most valuable investigation of the flow of wTater in pipes and channels has been carried out with exceptional accuracy and on a very large scale by the late M. Darcy, and continued by his successor the late M. Bazin, at the expense of the French Government (Recherches Hydrau-liques, Paris, 1866). The measurement of the flow in rivers has been extensively carried out, especially by German engineers. Harlachens Beiträge zur Hyclrographie de* Königreiches Böhmen contains exceedingly valuable measure ments of this kind, aud a comparison of the experimental results with all the formulae of flow which have been pro-posed. Messrs Humphreys and Abbott's gaugings of the Mississippi for the United States Government, Mr Gordon's gaugings of the Irrawaddi, and Captain Cunningham's experiments on the Ganges Canal may be referred to as having materially advanced hydraulic science.

The first adequate theory of turbines is that of Poncelet in the Comptes Rendus de VAcademie de Paris, 1838. Redtenbacher's Theorie und Bau der Turbinen und Ven-tilatoren (Mannheim, 1844) is the first complete treatise on the subject. Girard's turbine, which was of an entirely new type, was discussed in Le Genie industrielle, 1856-1857, and lately by Fink (Civilingenieur, 1880). Important experiments on turbines were made by Francis (Lowell Hydraulic Experiments, Boston, Mass., 1855).


Footnotes

This historical sketch of the subject is a revised abridgment of
that written by David Buchanan, and prefixed to the article HYDRODYNAMICS in the 8th edition of this work.

Wellenlehre auf Experimente gegründet, Leipsic, 1825.
Turin Memoirs, vol. xxv.

Wellenlehre auf Experimente gegründet, Leipsic, 1825.