2. The science of hydrodynamics 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 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.

3. Those general principles of hydrostatics which are to this day employed as the foundation of that part of the science, were first given by Archimedes in his work Ἐπεξεργασίαι, or De Insidentibus Humidis, about 250 years before the birth of Christ, and were afterwards applied to experiments by Marinus Ghetaelhus in his Archimedes Prodromus. Archimedes maintained that each particle of a fluid mass, when in equilibrium, 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 hydrostatic process by which the purity of the precious metals can be ascertained, and for the screw engine which 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 invented 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 preliminary branches of pneumatical science. The 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 and abstruse invention, which could scarcely have been expected 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 earthen pots carried round by a wheel. In some of these machines 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 pump.

5. Notwithstanding these inventions of the Alexandrian Labourers of school, its attention does not seem to have been directed to the motion of fluids. The first attempt to investigate this subject was made by Sextus Julius Frontinus, inspector of the public fountains at Rome in the reigns of Nerva and Trajan; and we may justly suppose that his work entitled De Aqueductibus urbis Romae 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 adjuvatures, 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 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, that they not only were acquainted with the hydrosynthetic 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.

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 researches. Poetry and the fine arts, accordingly, had made considerable progress under the patronage of the family of Medici, before Galileo began to extend the boundaries of science. 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.

7. Castelli and Torricelli, two of the disciples of Galileo, applied the discoveries of their master to the science of 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 vessel. Torricelli observing that in a jet d'eau where the water rushed through a small adjuvature, it rose to nearly the same 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 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 adjuvatures 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 discovered the pressure of the atmosphere, a treatise on the equilibrium of fluids (Sur l'Equilibre des Liqueurs), was found among his manuscripts, and was given to the public in 1663. In the hands of Pascal, hydrosynetics 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 many succeeding writers, but particularly by the celebrated Mariotte, whose labours in this department of physics deserve to be recorded. His Traité 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 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 relates to the motion of rivers seems to have originated in Italy. This fertile country receives from the Apennines a great number of torrents, which traverse several principalities before they mingle their waters with those of the 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... were written on the subject in Italy than in all the rest of Europe.

11. Guglielmini was the first who attended to the motion of water in rivers and open canals. Embracing the 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 viscosity of water had a considerable share in retarding its motion.

12. The effects of friction and viscosity in diminishing the velocity of running water were noticed in the Principia of Sir Isaac Newton, who has thrown much light upon several branches of hydrodynamics. At a time when the Cartesian 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 shown 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 be 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 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, Sir Isaac has 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, therefore, 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 became 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 most difficult in the science of hydrodynamics, was first 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 of the sea, by observing the time in which they rise and fall. If the two vertical branches of a siphon, 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 equilibrium and their surfaces horizontal. But if 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 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 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, at Padua, his work De Castellis per quae derivantur Fluxionum aquarum, &c. He found, from a great number of experiments, that if A be the aperture of the orifice, and D its depth below the surface of the reservoir, the quantity of water discharged in a given time will be as

\[ \frac{0.571}{2 AD} \times \frac{1}{1000} \]

while it ought to be as \( 2 AD \), if the velocity 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 Bernoulli published his Hydrodynamica, seu de vi fluidorum et motibus Fluidorum Commentarii. 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 vessel which was 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 viri-

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1 See his principal work, entitled La Misura dell' acqua corrente. History. *om vivarium*, and obtained very elegant solutions. In the opinion of the Abbé Bossut, his work was one of the finest productions of mathematical genius.

Objected 16. The uncertainty of the principle employed by Daniel Bernouilli, 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 fundamental laws of mechanics. Maclaurin and John Bernouilli, 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 directe demonstrata ex principiis purè mechanicis*, which forms the fourth volume of his works. The method employed by Maclaurin has been thought not sufficiently rigorous; and that of John Bernouilli is, in the opinion of La Grange, defective in perspicuity and precision.

17. The theory of Daniel Bernouilli was opposed also by the celebrated D'Alembert. When generalising James Bernouilli's Theory of Pendulums, he discovered a principle of dynamics so simple and general, that it reduced the laws of the motions of bodies to that of their equilibrium. 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 *Traité 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 Bernouilli, 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 has been adopted by the celebrated 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 concerning 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 the celebrated D'Alembert, and enabled both him and Euler to represent the theory of fluids in formulae restricted by no particular hypothesis.

18. An immense number of experiments on the motion of water in pipes and canals was made by Professor Michelotti of Turin, at the expense of the sovereign. In these experiments the water issued from holes of different sizes, under pressures of from 5 to 22 feet, from a tower constructed of the finest masonry. Basins (one of which was 289 feet square) built of masonry, and lined with stucco, received the effluent water, which was conveyed in canals of brickwork, lined with stucco, of various forms and declivities. The whole of Michelotti's experiments were conducted with the utmost accuracy; and his results, which are in every respect entitled to our confidence, were published in 1774 in his *Sperienze Idrauliche*.

19. The experiments of the Abbé Bossut, whose labours in this department of science have been very assiduous and successful, have, in as far as they coincide, afforded the same results as those of Michelotti. Though performed on a smaller scale, they are equally entitled to our confidence, and have the merit of being made in cases which are most likely to occur in practice. In order to determine what were the motions of the fluid particles in the interior of a vessel emptying itself by an orifice, M. Bossut employed a glass cylinder, to the bottom of which different adjutages were fitted; and he found that all the particles descend at first vertically, but that at a certain distance from the orifice they turn from their first direction towards the aperture. In consequence of these oblique motions, the fluid vein forms a kind of truncated conoid, whose greatest base is the orifice itself, having its altitude equal to the radius of the orifice, and its bases in the ratio of 3 to 2.—It appears also, from the experiments of Bossut, that when water issues through an orifice made in a thin plate, the expense of water, as deduced from theory, is to the real expense as 16 to 10, or as 8 to 5; and, when the fluid issues through an additional tube, two or three inches long, and follows the sides of the tube, as 15 to 13.—In analyzing the effects of friction, he found, 1. That small orifices gave less water in proportion than great ones, on account of friction; and, 2. That when the height of the reservoir was augmented, the contraction of the fluid vein was also increased, and the expense of water diminished; and by means of these two laws he was enabled to determine the quantity of water discharged, with all the precision he could wish. In his experiments on the motion of water in canals and tubes, he found that there was a sensible difference between the motion of water in the former and in the latter. Under the same height of reservoir, the same quantity of water always flows in a canal, whatever be its length and declivity; whereas, in a tube, a difference in length and declivity has a very considerable influence on the quantity of water discharged. According to the theory of the resistance of fluids, the impulse upon a plane surface is as the product of its area multiplied by the square of the fluid's velocity, and the square of the sine of the angle of incidence. The experiments of Bossut, made in conjunction with D'Alembert and Condorcet, prove, that this is sensibly true when the impulse is perpendicular; but that the aberrations from theory increase with the angle of impulsion. They found, that when the angle of impulsion was between 50° and 90°, the ordinary theory may be employed, that the resistances thus found will be a little less than they ought to be, and the more so as the angles recede from 90°. The attention of Bossut was directed to a variety of other interesting points, which we cannot stop to notice, but for which we must refer the reader to the works of that ingenious author.

20. The oscillation of waves, which was first discussed by Sir Isaac Newton, and afterwards by D'Alembert, in the article *Ondes* in the French Encyclopaedia, was now revived by M. Flaugergues, who attempted to overthrow the opinions of these philosophers. He maintained, that a vibration of wave is not the effect of a motion in the particles of water, waves. 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 intumescent, 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 intumescent, 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.

21. This difficult subject has also been discussed by M. de la Grange, in his Mécanique Analytique. He found, that the velocity of waves, in a canal, is equal to that which a heavy body would acquire by falling through a height 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 tensility and adhesion of the particles of water, has also been confirmed by experience.

22. The most successful labourer in the science of hydrodynamics was the Chevalier Buat, engineer in ordinary to the King of France. Following in the steps of the Abbé Bossut, 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. 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 viscosity 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 viscosity 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.

\[ V = \frac{307 \times \sqrt{d - 0.1}}{\sqrt{s - L} \times \sqrt{s + 1.6}} - 0.3 \times \frac{d - 0.1}{s}, \]

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 University of Modena, succeeded in bringing to light some of Venturi's curious facts respecting the motion of water, in his work on the "Lateral Communication of Motion in Fluids," A.D. 1798. 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 curvilinear 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 shown, 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 sagacity in classifying the different kinds of resistances which rise 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 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 first applied to the determination of the velocity of running streams of water by M. Girard, who considers the resistance as represented 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 viscosity, 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, but M. Du Buat was the first person who investigated the subject experimentally. His results, however, were far from being satisfactory; and it was left to

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1 A third volume of this work was published in 1816, entitled Principes d'Hydraulique et Pyrodymanique, relating chiefly to the subject of heat and elastic fluids.

2 Pliny, Quæst. Nat. 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° centigr.; 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.

27. The theory of running water was greatly advanced by the researches of M. Prony. From a collection of the best experiments by Couplet, Bossut, and Du Bunt, 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:

\[ V = 0.1541131 + \sqrt{(0.023751 + 32806.6G)} \]

When we use this formula for canals, we must take

\[ G = RI, \quad R \text{ being } \frac{\omega}{z} \quad \text{and} \quad I = \frac{2}{\lambda}, \]

representing the area of the section of the pipe or canal, \( z \) the perimeter of the section in contact with the water, \( \xi \) the difference of level between the two extremities of the pipe, and \( \lambda \) the length of the pipe or canal.

When the formula is applied to pipes, we must take

\[ G = \frac{1}{4}DK, \quad D \text{ being the diameter of the pipe, and} \quad K = \frac{H + \xi}{\lambda}, \]

\( H \) being the height of the head of water above the superior orifice of the pipe.

28. M. Eytelwein of Berlin published, in 1801, a valuable compendium of Hydraulics, entitled *Handbuch der Mechanik und der Hydraulik*, which contains an account of many new and valuable experiments made by himself. His work is divided into 24 chapters, the most important of which are the 7th, which treats of the motion of water in rivers, and the 9th, which treats of the motion of water in pipes. He has shewn that the mean velocity of water in a second in a river or canal flowing in an equable channel, is \( \frac{1}{4} \)ths of a mean proportional between the fall in two English miles, and the hydraulic mean depth; and that the superficial velocity of a river is nearly a mean proportional between the hydraulic mean depth and the fall in two English miles. In treating of the motion of water in pipes, he obtains the following simple formula, which agrees wonderfully with experiment,

\[ V = 50 \sqrt{\left(\frac{dh}{l + 50d}\right)} \]

in which \( l \) is the length of the pipe, \( d \) the hydraulic mean depth, and \( h \) the height of the reservoir. The following are some of the other important results which are given in his work. The contraction of the fluid area is 0.64, the coefficient for additional pipes 0.65, the coefficient for a conical tube similar to the curve of contraction 0.98. For the whole velocity due to the height, the coefficient by its square must be multiplied by 8.0458. For an orifice, the coefficient must be multiplied by 7.8; for wide openings in bridges, sluices, &c., by 6.9; for short pipes 6.6; and for openings in sluices without side walls 5.1. Our author investigates the subject of the discharge of water by compound pipes, the motions of jets, and their impulses against plane and oblique surfaces, and he shews theoretically, that a water-wheel will have its effect a maximum when its circumference moves with half the velocity of the stream.

29. A series of interesting hydraulic experiments was made at Rome in 1809, by MM. Mallet and Vici. They found that a pipe, whose gauge was five ounces French Mallet and measure (or 0.03059 French kilolitres), furnished one-seventh more water 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.

Notwithstanding the investigations of Newton, D'Alembert, and Lagrange, the problem of waves was still unsolved; and the Institute of France was induced to propose, as the subject of its annual prize for 1816, "The Theory of Waves on the surface of a heavy Fluid of indefinite depth." M. Poisson had previously studied this difficult subject, and he lodged his first memoir in the bureau of the Institute on the 2d October 1815, at the expiration of the period allowed for competition. M. Poisson supposes the waves to be produced in the following manner. A body of the form of an elliptic paraboloid is immersed a little in the fluid, with its axis vertical and its vertex downwards. After being left in this position till the equilibrium of the fluid is restored, the body is suddenly withdrawn, and waves are formed round the place which it occupied. This first memoir contains the general formula for waves propagated with an uniformly accelerated motion; but in a second memoir, read in December, he gives the theory of waves propagated with a constant velocity. This last class of waves are much more sensible than the first, and are those which are seen to spread in circles round any disturbance made at the surface of water. In determining the superficial as well as the internal propagation of these waves, he considers only the case when the disturbance of the water is so small, that the second and the higher powers of the velocity of the oscillating particles may be neglected; and he assumes, that a fluid particle which is at any instant at the surface, continues there during the whole of the motion, a supposition which the condition of the continuity of the fluid renders necessary. He supposes the depth of the water constant throughout its whole extent, the bottom being considered as a fixed horizontal plane at a given distance beneath its natural surface. He then treats, first, the case in which the motion takes place in a canal of uniform width, over which obstruction is made of the horizontal dimension of the fluid; and, secondly, the case in which the fluid is considered in its true dimensions.

30. The prize offered by the Institute was gained by M. Augustin Louis Cauchy, then a young mathematician of the highest promise. In his memoir, which was published in the 3d volume of the *Mémoires des Sciences*, he treats only of the first kind of waves above mentioned; and his investigation claims to be more complete than that in the first memoir of Poisson, in so far as it leaves entirely arbitrary the form of the function relative to the initial form of the fluid surface, and, therefore, allows the analysis to be applied when bodies of different forms are used to produce the initial disturbance. From his analysis, M. Cauchy concludes, "that the heights and velocities of the different waves produced by the immersion of a cylindrical or prismatic body, depend not only on the width and height of the part immersed, but also on the form of the surface which bounds this part." He is also of opinion, that the number of the waves produced may depend on the form of the immersed body, and the depth of immersion.

31. The following abstract of the principal results obtained by theory, respecting the nature of waves, has been given by Mr. Challis:

"With respect first to the canal of uniform width, the law of the velocity of propagation found by Lagrange, is..." confirmed by Poisson's theory when the depth is small, but not otherwise.

"When the canal is of unlimited depth, the following are the chief results.

1. An impulse given to any point of the surface, affects instantaneously the whole extent of the fluid mass.

The theory determines the magnitude and direction of the initial velocity of each particle resulting from a given impulse.

2. The summit of each wave moves with a uniformly accelerated motion.

This must be understood to refer to a series of very small waves, called by M. Poisson dents, which perform their movements, as it were, on the surface of the larger waves, which he calls 'les ondes dentelées.' Each wave of the series is found to have its proper velocity, independent of the primitive impulse.

3. At considerable distances from the place of disturbance, there are waves of much more sensible magnitude than the preceding.

Their summits are propagated with a uniform velocity, which varies as the square root of the breadth à fleur d'eau of the fluid originally disturbed. Yet the different waves which are formed in succession, are propagated with different velocities; the foremost travels swiftest. The amplitudes of oscillations of equal duration, are reciprocally proportional to the square root of the distances from the point of disturbance.

4. The vertical excursions of the particles situated directly below the primitive impulse, vary according to the inverse ratio of the depth below the surface. This law of decrease is not so rapid but that the motion will be very sensible at very considerable depths: it will not be the true law, as the theory proves, when the original disturbance extends over the whole surface of the water, for the decrease of motion in this case will be much more rapid.

The results of the theory, when the three dimensions of the fluid are considered, are analogous to the preceding 1, 2, 3, 4, and may be stated in the same terms, excepting that the amplitudes of the oscillations are inversely as the distances from the origin of disturbance, and the vertical excursions of the particles situated directly below the disturbance, vary inversely as the square of the depth."

32. Several very interesting experiments on the propagation of waves, have been made by M. Weber and by M. Bidone. Although Weber's experiments were not made in exact conformity with the condition which the theory required, yet they, generally speaking, harmonize with it; and they particularly establish the existence of the small accelerated waves near the place of disturbance, and of a perceptible motion of the particles of the fluid at considerable depths below the surface. When an elliptic paraboloid is used to produce the waves, with its axis vertical, and its vertex downwards, and when, of course, the section of the solid in the plane of the surface of the water is an ellipse, the velocity of propagation is, according to the theory, greater in the direction of the major axis than in that of the minor, in the ratio of the square root of the one to the square root of the other; but this result is not confirmed by Weber's experiments.

M. Bidone of Turin has made experiments on waves more conformable to the condition of the theory, and he has in a great measure removed the obstacle arising from the adhesion of the water to the immersed body; and the experiments which he has thus made, confirm the laws of motion, as well as the existence of accelerated waves.

33. In 1826, M. Bidone made a series of experiments 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, discharged from rectilineal and curvilineal orifices with salient angles pierced in vertical plates, and whose perimeters are formed of straight and curve lines, varying in more than fifty different ways, with variable and invariable changes from zero to 22 French feet, the area of the water being equal to one square inch. The sections of the veins were taken at different distances from the orifice, and the results, which are extremely curious, are illustrated by diagrams.

In a subsequent memoir, M. Bidone gives a theoretical view of his experiments. He considers the greatest contraction of the fluid vein to take place at a distance not more than the greatest diameter of the orifice, whatever be its shape; and hence it follows, that the discharge from the orifice is equal to the sum of the product of each superficial element, multiplied by the velocity of the fluid vein; and since the area of the vena contracta was found to be from 0.50 to 0.62, the expenditure will be represented by the product of this coefficient of contraction, and the velocity due to the charge.

In the case of a fluid vein issuing from a small orifice relative to the section of the vessel, and reduced to a state of permanence, he finds that the area of the section of the vena contracta depends solely on the direction, and not on the velocity, of its component filaments; a result which experiment confirms. M. Bidone next ascertains that, in a circular orifice, the absolute diameter of the vena contracta is exactly two-thirds of that of the orifice, the correction which is due to the contraction depending on the adhesion and friction of the fluid against the perimeter of the orifice, and the ratio of the area of the vein to the area of the orifice; the same being true for all orifices.

34. A series of useful experiments was made in 1827, under the sanction of the French government, by General Sabatier, Commandant of the Military School at Metz. The General apparatus which he used consisted of four basins. The first had an area of 25,000 square metres; the second had an area of only 1500 square metres, and a depth of 3.70 metres, and was so contrived by means of sluices, as to have a complete command of the level of the water during the experiment: the third basin, which communicated directly with the second basin, was 3.68 metres long and 3 wide, to receive the water discharged by the orifices: and the fourth basin was a gauge capable of containing 24,000 litres. In carrying on the experiments, the opening of the orifices, the height or charge of the fluid in the reservoir, as well as the level of the water in the gauge basin relative to each discharge of fluid, were measured to the tenth of a millimetre, so that the approximation was at least \( \frac{1}{10} \)th of the whole result.

The following are some of the principal results of these experiments.

1. For complete orifices of 20 centimetres square and high charges, the coefficient is 0.600. When the charge is 4 or 5 times the opening of the orifice, the coefficient becomes 0.605, but beyond that charge the coefficient diminishes to 0.593.

2. The same law takes place for 10 and 5 centimetres in height, the coefficients being for 10 centimetres 0.611, 0.618, 0.611, respectively, and for 5 centimetres in height 0.618, 0.631, 0.623.

3. With orifices of 3, 2, and 1, centimetres in height, the law changes rapidly, and the coefficients increase as the opening of the orifice diminishes, being for 1 centimetre, the smallest height of the orifice, 0.698 to 0.640 for 3 centimetres.

35. In the year 1830, Mr George Rennie undertook a series of experiments... Hydrostats. of experiments on the friction of water against revolving cylinders and discs, on the direct resistances to globes and discs revolving in air and water alternately, on the coefficients of contraction, and on the expenditure of water through orifices, additional tubes, and pipes of different lengths. The following are some of his principal results.

1. It appears from his first series of experiments, that, with slow velocities, the friction varies with the surfaces; that an increase of surface did not materially affect the friction with increased velocities; and that, with equal surfaces, the resistances approximated to the squares of the velocities.

2. In the experiments on the resistances experienced by globes and discs revolving in air and water alternately, it appears that the resistances in both cases were as the squares of the velocities, and that the mean resistances were as follows:

| Resists. in Air | Resists. in Water | |-----------------|------------------| | Circular discs | 25.189 | | Square plates | 22.010 | | Round globes | 16.627 |

3. In circular orifices in brass plates, \(\frac{1}{6}\)th of an inch thick, and with apertures of \(\frac{1}{4}, \frac{1}{2}, \frac{3}{4}\), and \(\frac{3}{2}\) of an inch respectively, under pressures from 1 to 4 feet, the coefficients were—

For altitudes of 1 foot, \(0.619\) For altitudes of 4 feet, \(0.621\)

For additional tubes of glass, they were—

For altitudes of 1 foot, \(0.617\) For altitudes of 4 feet, \(0.606\)

4. The expenditure through orifices, additional tubes, and pipes of different lengths, of equal areas, and the same kind of water, as compared with the expenditure through a pipe 30 feet long, are—

For orifices, \(3\) to \(1\) For additional tubes, \(4\) to \(1\) For a pipe 1 foot long, \(3.7\) to \(1\) For a pipe 2 feet long, \(1.4\) to \(1\) For a pipe 4 feet long, \(2.0\) to \(1\) For a pipe 8 feet long, \(2.6\) to \(1\)

5. With bent rectangular pipes, half an inch in diameter, and 15 feet long, the expenditure was diminished two-thirds; with 15 bends as compared with a straight pipe, and with 24 bends, one-third; but it did not seem to follow any law.

6. Many very accurate experiments on the discharge of water from long pipes, were made by James Jardine, Esq., civil-engineer. In the case of the Comiston Main, which conducts the water from the reservoir at Comiston to the Castle at Edinburgh, the length of the pipe is 14,930 feet, its diameter 4\(\frac{1}{2}\) inches, the altitude 51 feet, and the quantity of water actually delivered 189.4 Scots pints. This valuable result Mr Jardine compared with the different formulae as follows:

| Actual delivery of water at Comiston Main | 189.4 | |------------------------------------------|-------| | Calculated do. by Eytelwein's formula | 189.77| | Do. by Gerard's formula | 183.20| | Do. by Dubuat's formula | 183.13| | Do. by Prony's simple formula | 182.32| | Do. by Prony's tables | 180.7 |

An improvement on the common theory of fluids was lately suggested by Professor Airy, in his lectures at Cambridge. It had been usual to assume the law of equal pressure as a datum of observation. Professor Airy, however, has shewn that this property may be deduced from another more simple and equally given by observation; fluids by namely, that the division of a perfect fluid may be effected Mr Airy, without the application of sensible force; and hence, it immediately follows, that the state of equilibrium or motion of a mass of fluid, is not altered by a mere separation of its parts by an indefinitely thin partition. Professor Miller has given a definition of fluids founded on this principle, and a proof of the law of equal pressure, at the beginning of his Elements of Hydrostatics, &c. published at Cambridge in 1831. Dr Thomas Young had employed an equivalent principle to that of Mr Airy in determining the manner of the reflection of waves of water; and Mr Challis considers it as necessary in the solution of some hydrostical and hydrodynamical problems.

There are certain cases in the analytical theory of hydrodynamics which require a more simple analysis than of Prof others; such, for example, as those of steady motion, or of motion which has arrived at a permanent state, so that the velocity is constant in quantity and direction at the same point. Equations applicable to this kind of motion seem to have been first given by Professor Mosely, in his Elementary Treatise on Hydrostatics and Hydrodynamics. The following is the principle from which he has derived them.

When the motion is steady, each particle, in passing from one point to another, passes successively through the states of motion of all the particles which at any instant lie in its path. This general principle is applicable to all kinds of fluids, and is true, whether or not the effect of heat is taken into account, provided the condition of steadiness remains. As it enables us to consider the motion of a single particle in place of that of a number, it readily affords the equations of motion.

As it was desirable to know how the same equation might be obtained from the general equations of fluid motion, Mr of Mr Challis undertook this inquiry, and published the results of Challis, it in the Transactions of the Cambridge Philosophical Society. In this paper, he has given a method of doing this both for incompressible and elastic fluids, and he has shewn that a term in the general formulae which occasions the complexity in most hydrodynamical questions, disappears in cases of steady motion. Mr Challis is of opinion, that these equations may be employed in very interesting researches, and he mentions, as instances, the motion of the atmosphere as affected by the rotation of the earth, and a given distribution of temperature due to solar heat.

The science of hydrodynamics has of late years been experimented with by M. Eytelwein of Berlin, whose practical conclusions coincide nearly with those of Bossut; by Dr Eytelwein and others Matthew Young, late Bishop of Clonfert, who has explained the cause of the increased velocity of efflux through additional tubes; and by Mr Vince, Dr T. Young, Coulomb, and Don George Juan; but the limits of this work will not permit us to give any further account of their labours at present. We must now proceed to initiate the reader into the science itself, beginning with that branch of it which relates to the pressure, equilibrium, and cohesion of non-elastic fluids.

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1 Nat. Phil. vol. ii. p. 64. 2 Vol. iii. Part iv. PART I.—HYDROSTATICS.

41. Hydrostatics is that branch of the science of hydrodynamics which comprehends the pressure and equilibrium of non-elastic fluids, as water, oil, mercury, &c.; the method of determining the specific gravities of substances, the equilibrium of floating bodies, and the phenomena of capillary attraction.

DEFINITIONS AND PRELIMINARY OBSERVATIONS.

42. A fluid is a collection of very minute particles, cohering so little among themselves, that they yield to the smallest force, and are easily moved among one another.

43. Fluids have been divided into perfect and imperfect. In perfect fluids the constituent particles are supposed to be endowed with no cohesive force, and to be moved among one another by a pressure infinitely small. But, in imperfect or viscous fluids, the mutual cohesion of their particles is very sensible, as in oil, varnish, melted glass, &c.; and this tenacity prevents them from yielding to the smallest pressure. Although water, mercury, alcohol, &c. have been classed among perfect fluids, yet it is evident that neither these nor any other liquid is possessed of perfect fluidity. When a glass vessel is filled with water above the brim, it assumes a convex surface; and when a quantity of it is thrown on the floor, it is dispersed into a variety of little globules, which can scarcely be separated from one another. Even mercury, the most perfect of all the fluids, is endowed with such a cohesive force among its particles, that if a glass tube, with a small bore, is immersed in a vessel full of this fluid, the mercury will be lower in the tube than the surface of the surrounding fluid;—if a small quantity of it be put in a glass vessel, with a gentle rising in the middle of its bottom, the mercury will desert the middle, and form itself into a ring, considerably rounded at the edges; or if several drops of mercury be placed upon a piece of flat glass, they will assume a spherical form; and if brought within certain limits, they will conglobulate and form a single drop. Now, all these phenomena concur to prove, that the particles of water have a mutual attraction for each other; that the particles of mercury have a greater attraction for one another, than for the particles of glass; and, consequently, that these substances are not entitled to the appellation of perfect fluids.

44. It was universally believed, till within the last seventy years, that water, mercury, and other fluids of a similar kind, could not be made to occupy a smaller space, by the application of any external force. This opinion was founded on an experiment made by Lord Bacon, who inclosed a quantity of water in a leaden globe, and by applying a great force attempted to compress the water into a less space than it occupied at first: The water, however, made its way through the pores of the metal, and stood on its surface like dew. The same experiment was afterwards repeated at Florence by the Academy del Cimento, who filled a silver globe with water, and hammered it with such force as to alter its form, and drive the water through the pores of the metal. Though these experiments were generally reckoned decisive proofs of incompressibility, yet Bacon himself seems to have drawn from his experiment a very different conclusion; for after giving an account of it, he immediately adds, that he computed into how much less space the water was driven by this violent pressure. This passage from Lord Bacon does not seem to have been noticed by any writer on hydrostatics, and appears a complete proof that the compressibility of water was fairly deducible from the issue of his experiment. In consequence of the reliance which was universally placed on the result of the Florentine experiment, fluids have generally been divided into compressible and incompressible, or elastic and non-elastic fluids: water, oil, alcohol, and mercury, being regarded as incompressible and non-elastic; and air, steam, and other aeriform fluids, as compressible or elastic.

45. About the year 1761, the ingenious Mr Canton began to consider this subject with attention, and distrusting the result obtained by the Academy del Cimento, resolved to bring the question to a decisive issue. Having procured a small glass tube, about two feet long, with a hall at one end, an inch and a quarter in diameter, he filled the Canton ball and part of the tube with mercury, and brought it to the temperature of 50° of Fahrenheit. The mercury then stood six inches and a half above the ball; but after it had been raised to the top of the tube by heat, and the tube sealed hermetically, then, upon bringing the mercury to its former temperature of 50°, it stood 1/10th of an inch higher in the tube than it did before. By repeating the same experiment with water exhausted of air, instead of mercury, the water stood 1/10th of an inch higher in the tube than it did at first. Hence, it is evident, that when the weight of the atmosphere was removed, the water and mercury expanded, and that the water expanded 1/10th of an inch more than the mercury. By placing the apparatus in the receiver of a condensing engine, and condensing the air in the receiver, he increased the pressure upon the water, and found that it descended in the tube. Having thus ascertained the fact, that water and mercury are compressible, he subjected other fluids to similar experiments, and obtained the results in the following table:

| Fluid | Compression | Specific Gravity | |------------------------|-------------|------------------| | Compression of Mercury | 3 | 13.595 | | Sea-Water | 40 | 1.028 | | Rain-Water | 46 | 1.000 | | Oil of Olives | 48 | 0.913 | | Spirit of Wine | 66 | 0.846 |

Lest it should be imagined that this small degree of compressibility arose from air imprisoned in the water, Mr Canton made the experiment on some water which had imbibed a considerable quantity of air, and found that its compressibility was not in the least augmented. By inspecting the preceding table, it will be seen that the compressibility of the different fluids is nearly in the inverse ratio of their specific gravities.

46. The experiments of Mr Canton have been lately confirmed by Professor Zimmerman. He found that sea-water was compressed 1/10th part of its bulk, when inclosed in the cavity of a strong iron cylinder, and under the influence of a force equal to a column of sea-water 1000 feet in height. From these facts, it is obvious that fluids are susceptible of contraction and dilatation, and that there is no foundation in nature for their being divided into compressible and incompressible. If fluids are compressible, they will also be elastic; for when the compressing force is removed, they will recover their former magnitude; and

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1 Bacon's Works, by Shaw, vol. ii. p. 521; Novum Organum, part ii. sect. 2. alph. 45. § 222. 2 See the Philosophical Transactions for 1762 and 1764, vols. iii. and iv. hence their division into elastic and non-elastic is equally improper.

A series of very valuable experiments on the compressibility of water has recently been made by Professor Oersted of Copenhagen. At the temperature at which water has a maximum density, which, according to Professor Stampfer of Vienna, is at $38^\circ75$ Fahrenheit, Professor Oersted found the true compressibility of water by one atmosphere (or 336 French lines of mercury) to be 46.1 millionths of the volume, the difference between the true and the apparent compressibility arising from the effect of the heat developed by the compression, by which the liquid and the bottle are dilated. He found also that the differences of volume in the compressed water are proportional to the compressing power; and that this law holds as far as the pressure of 65 atmospheres, and probably much farther; but how far he was not able to determine, as his apparatus could not resist a greater pressure.

A series of interesting experiments at high pressures were made by Mr Perkins, who has described the instrument which he employed in the Philosophical Transactions for 1826, p. 541. The following table contains the results which he obtained from 10 to 2000 atmospheres upon a column of water 190 inches long:

| No. of Atmospheres | Compression, Inch | No. of Atmospheres | Compression, Inch | No. of Atmospheres | Compression, Inch | |--------------------|------------------|--------------------|------------------|--------------------|------------------| | 10 | 0.189 | 80 | 1.187 | 500 | 5.037 | | 20 | 0.372 | 90 | 1.238 | 600 | 5.907 | | 30 | 0.547 | 100 | 1.422 | 700 | 6.715 | | 40 | 0.691 | 150 | 1.914 | 800 | 7.492 | | 50 | 0.812 | 200 | 2.440 | 900 | 8.243 | | 60 | 0.756 | 300 | 3.339 | 1000 | 9.062 | | 70 | 1.056 | 400 | 4.193 | 2000 | 15.823 |

If the number of atmospheres are made the abscissa of a curve, and the compressions its ordinates, it will be seen that the curve approximates to a hyperbolic one.

47. The doctrines of hydrostatics have been deduced by different philosophers from different properties of fluids. Euler has founded his analysis on the following property, "that when fluids are subjected to any pressure, that pressure is so diffused throughout the mass, that when it remains in equilibrio all its parts are equally pressed in every direction." D'Alembert at first deduced the principles of hydrostatics from the property which fluids have of rising to the same altitude in any number of communicating vessels; but he afterwards adopted the same property as Euler from the foundation which it furnishes for an algebraical calculus. The same property has been employed by Bossut, Prony, and other writers, and will form the first proposition of the following chapter.

CHAPTER I.—ON THE PRESSURE AND EQUILIBRIUM OF FLUIDS.

PROPOSITION I.

48. When a mass of fluid, supposed without weight, is subjected to any pressure, that pressure is so diffused throughout the whole, that when it remains in equilibrio all its parts are equally pressed in every direction.

As it is the distinguishing property of fluids that their particles yield to the smallest pressure, and are easily moved among themselves (26.), it necessarily follows, that if any particle is more pressed towards one side than towards another, it will move to that side where the pressure is least; and the equilibrium of the fluid mass will be instantly destroyed. But by the hypothesis the fluid is in equilibrio, consequently the particle cannot move towards one side, and must therefore be equally pressed in every direction.

In order to illustrate this general law, let EF (fig. 1.) be a vessel full of any liquid, and let mn, op be two orifices at equal depths below its surface; then, in order to prevent the water from escaping, it will be necessary to apply two pistons, A and B, to the orifices mn, op with the same force, whether the orifice be horizontal or vertical, or in any degree inclined to the horizon; so that the pressure to which the fluid mass is subject, which in this case is its own gravity, must be distributed in every direction. But if the fluid has no weight, then the pressure exerted against the fluid at the orifice op, by means of the piston B, will propagate itself through every part of the circular vessel EF, so that if the orifices mn, tu are shut, and rs open, the fluid would rush through this aperture, in the same manner as it would rush through mn or tu, were all the other orifices shut. This proposition, however, is true only in the case of perfect fluids; for when there is a sensible cohesion between the particles, as in water, an equilibrium may exist even when a particle is less pressed in one direction than in another; but this inequality of pressure is so exceedingly trifling, that the proposition may be considered as true, even in cases of imperfect fluidity.

PROPOSITION II.

49. If to the equal orifices mn, tu, op, rs of a vessel, containing a fluid destitute of weight, be applied equal powers A, B, C, D, in a perpendicular direction, or if the orifices mn, &c. be unequal, and the powers A, B, &c. which are respectively applied to them be proportional to the orifices, these powers will be in equilibrio.

It is evident from the last proposition, that the pressure exerted by the power B is transmitted equally to the orifices mn, rs, tu, that the pressure of the power C is transmitted equally to the orifices mn, op, tu, and so on with all other powers. Every orifice then is influenced with the same pressure, and, consequently, none of the powers A, B, C, D, can yield to the action of the rest. The fluid mass, therefore, will neither change its form nor its situation, and the powers A, B, C, D will be in equilibrio. If the powers A, B, C, D are not equal to one another, nor the orifices mn, op, rs, tu; but if A = B = mn : op, and so on with the rest, the fluid will still be in equilibrio. Let A be greater than B, then mn will be greater than op; and whatever number of times B is contained in A, so many times will op be contained in mn. If A = 2B, then mn = 2op, and since the orifice mn is double of op, the pressure upon it must also be double; and, in order to resist that pressure, the power A must also be double of B;

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1 Nov. Comment. Petropol. tom. xlii. p. 305. 2 Milanges de Littérature, d'Histoire, et Philosophie. Pressure, but, by hypothesis, \( A = 2B \), consequently the pressures upon the orifices, or the powers \( A, B \), will be in equilibrio. If the power \( A \) is any other multiple of \( B \), it may be shewn in the same way that the fluid will be in equilibrio.

**Prop. III.**

50. The surface of a fluid, influenced by the force of gravity and in equilibrio in any vessel, is horizontal, or at right angles to the direction of gravity.

Let the surface of the fluid be supposed to assume the waving form APEB. Any particle \( P \) in the surface of the fluid is influenced by the force of gravity, which may be represented by PS, and which may be decomposed into two forces \( Pm, Pn \) in the direction of the two elementary portions of the surface \( Pm, Pn \) (see Article Dynamics). But since the particle \( P \) is in a state of equilibrium, the force of gravity acting in the direction \( Pm, Pn \) must be destroyed by equal and opposite forces, exerted by the neighbouring particles against \( P \) in the direction \( mP, nP \); therefore the forces \( Pm, Pn \) are equal to the forces \( mP, nP \).

Now the particle \( P \) being in equilibrium, must be equally pressed in every direction (48). Therefore the forces \( Pm, Pn \) are equal, and by the doctrine of the composition of forces (see Article Dynamics), the angle \( mPn \) formed by the two elementary portions \( Pm, Pn \) of the surface of the fluid, must be bisected by PS, the line which represents the direction of gravity. The same may be proved of every other point of the surface of the fluid; and therefore this surface must be horizontal or perpendicular to the direction of gravity.

51. This proposition may be otherwise demonstrated. From the principles of mechanics, it is obvious, that when the centre of gravity of any body is at rest, the body itself is at rest; and that when this centre is not supported, the body itself will descend, till it is prevented by some obstacle from getting farther. In the same manner the centre of gravity of a fluid mass will descend to the lowest point possible; and it can be shewn that this centre will be in its lowest position only when the surface of the fluid mass is horizontal. For let FGHI (fig. 2) be any surface, whether solid or fluid, and C its centre of gravity, the point C is nearer the line HI when FG is parallel to HI and rectilineal, than when it has any other form or position. When the surface FGHI is suspended by the point C, or balanced upon it, it will be in equilibrio; but if the line FG is made to assume any other form as Fstop, by removing the portion Gop of the surface to rest, the equilibrium, will be destroyed, and the side FG will preponderate. In order, therefore, to restore the equilibrium, the surface must be balanced on a point e farther from HI; that is, the centre of gravity of the surface FstopHI is e. In the same way it may be shewn, that whatever be the form of the bounding line FG, the quantity of surface remaining the same, its centre of gravity will be nearest HI, when FG is rectilineal and parallel to it. On the truth contained in this proposition depends the art of levelling, and the construction of the spirit level, for an account of which see Levelling.

52. As the direction of gravity is in lines which meet near the centre of the earth; and as it appears from this proposition, that the surface of fluids is perpendicular to that direction, their surface will be a portion of a spheroid similar to the earth. When the surface has no great extent, it may be safely considered as a plane; but when it is pretty large, the curvature of the earth must be taken into account.

**Prop. IV.**

53. The surface of a fluid influenced by the force of gravity, and contained in any number of communicating vessels, however different in form and position, will be horizontal.

Let AB, CD, EF, GH, IK, be a system of communicating vessels into which a quantity of fluid is conveyed: It will rise to the same height in each vessel, and have a fluid in any horizontal surface. Suppose ABEF, fig. 4, a vessel full of water. By the last proposition, its surface ST will be horizontal. Now, if any body be plunged into this vessel, the cylinder AD for instance, the surface of the fluid will still be horizontal; for no reason could be assigned for the water's rising on one side of this body any more than on another. Let us now immerse into the fluid, successively, the solid bodies AD, CE, GHF, &c., then after each immersion the surface will still be horizontal; and when all these solids are immersed, the large vessel ABFE will be converted into the system of communicating vessels represented in fig. 4, in which the surface of the fluid will, of consequence, be horizontal.

54. This proposition may be also demonstrated by supposing the parts AD, CE, GHF, &c., converted into ice without changing their former magnitude. When this happens, the equilibrium will not be disturbed; and the fluid mass AF, whose surface was proved to be horizontal by the last proposition, will continue in the same state after the congelation of some of its parts. That is, the surface of the fluid in the communicating vessels will be horizontal.

55. When the communicating vessels are so small that they may be regarded as capillary tubes, the surface of position is not true which all fluids have for glass, they rise to a greater height when the fluid substance is depressed in capillary tubes, and tubes in smaller tubes than in larger ones, and the quantity of elevation is in the inverse ratio of the diameters of the sols are bores. In the case of mercury, and probably of melted metals, the fluid substance is depressed in capillary tubes, and tubes the depression is subject to the same law. The subject of capillary attraction will be treated at length in a subsequent part of this article.

56. This proposition explains the reason why the surface of small pools in the vicinity of rivers is always on Pressure, a level with the surface or the rivers themselves; when there is any subterraneous communication between the river and the pool. The river and the pool may be considered as communicating vessels.

**Prop. V.**

57. If a mass of fluid contained in a vessel be in equilibrium, any particle whatever is equally pressed in every direction, with a force equal to the weight of a column of particles whose height is equal to the depth of the particle pressed below the surface of the fluid.

![Fig. 5]

Immerse the small glass tube \( mp \) into the vessel \( AB \) filled with any fluid; then if the tube is not of the capillary kind, the fluid will rise to \( n \) on the same level with the surface \( AB \) of the fluid in the vessel. Now it is evident, that the particle \( p \) at the bottom of the tube \( mp \) is pressed downwards by the superincumbent column of particles \( np \), which is equal to the depth of the particle \( p \) below the surface of the fluid. But since the mass of fluid is in equilibrium, the particle \( p \) is pressed equally in every direction: Therefore, the particle \( p \) is pressed equally in every direction by a force equal to the superincumbent column \( np \).

**Prop. VI.**

58. A very small portion of a vessel of any form, filled with a fluid, is pressed with a force which is in the compound ratio of the number of particles contained in that surface, its depth below the surface of the fluid, and the specific gravity of the fluid.

Let \( DpEB \) be the vessel, and \( rs \) a very small portion of its surface, the pressure upon \( rs \) is in the compound ratio of the number of particles in \( rs \), and \( np \) its depth below the horizontal surface \( DB \). Suppose the glass tube \( mp \) to be inserted in the infinitely small aperture \( p \), then, abstracting from the influence of capillary attraction, the fluid in the glass tube will ascend to \( n \) on a level with \( DB \), the surface of the fluid in the vessel, and the particle \( p \) will be pressed with a column of particles, whose height is \( np \). In the same way it may be shewn, that every other particle contained between \( r \) and \( s \) is pressed with a similar column. Then, since \( p \times np \) will represent the pressure of the column \( np \) on the particle \( p \); if \( N \) be the number of particles in the space \( rs \), \( N \times np \) will be the force of the column supported by the space \( rs \). And as the weight of this column must increase with the specific gravity of the fluid, \( S \times N \times np \) will represent its pressure, \( S \) being the specific gravity of the fluid.

**Prop. VII.**

59. The pressure upon a given portion of the bottom of a vessel, whether plane or curved, filled with any fluid, is in the compound ratio of the area of that portion, and Pressure, the mean altitude of the fluid, that is, the perpendicular distance of the centre of gravity of the given portion from the surface of the fluid; or, in other words, the pressure is equal to the weight of a column of fluid whose base is equal to the area of the given portion, and whose altitude is the mean altitude of the fluid.

Let \( AEGB \) be the vessel, and \( AFB \) the surface of the fluid which it contains. Let \( GH \) be a given portion of its bottom, and \( C \) the centre of gravity of that portion: Then shall \( CF \) be the mean altitude of the fluid. For, conceive the portion \( GH \) to be divided into an infinite number of small elements \( Hh, Gg, \&c. \) then (58.) the pressure sustained by the elements \( Hh, Gg, \&c. \) will be respectively \( S \times Hh \times Hw; S \times Gg \times Gt, \&c. \) the specific gravity of the fluid being called \( S \). But it follows from the nature of the centre of gravity, that the sum of all these products is equal to the product of the whole portion \( GH \) into \( CF \) the distance of its centre of gravity from the horizontal surface of the fluid. Therefore the pressure upon the portion \( GH \) is in the compound ratio of its surface converted into a plane, and the mean altitude of the fluid.

60. From this proposition we may deduce what is generally called the Hydrostatic paradox, viz. that the pressure upon the bottoms of vessels filled with fluid does not depend upon the quantity of fluid which they contain, but upon its altitude; or, in other words, that any quantity of fluid, however small, may be made to balance any quantity or any weight, however great. Let \( ACOQRPDB \) be a vessel filled with water, the bottom \( QR \) will sustain the same pressure as if it supported a quantity of water equal to \( MQRN \). It is evident (59.) that the part \( EF \) is pressed with the column of fluid \( ABEF \), and that the part \( DG \) equal to \( CD \) is pushed upwards with the weight of a column equal to \( ABCD \). Now, as action and reaction are equal and contrary, the part \( DG \) reacts upon \( FH \) with a force equal to the weight of the column \( ABCD \), and \( FH \) evidently sustains the smaller column \( DGFH \); therefore \( FH \) sustains a pressure equal to the weight of the two columns \( ABCD \) and \( DGFH \), that is, of the column \( BIHF \). In the same way it may be shewn, that any other equal portion of the bottom \( QR \) sustains a

---

1 This will be evident from the following proposition. If every indefinitely small part of a surface be multiplied by its perpendicular distance from a given plane, the sum of the products will be equal to the product of the whole surface, multiplied by the perpendicular distance of its centre of gravity from the same plane. In Fig. 7, let \( a, c \) represent two weights suspended at their centre of gravity by the lines \( aA, cC \), attached to the horizontal plane of which \( ABC \) is a section, and let \( b \) be the common centre of gravity of these weights, and \( OB \) the distance of this centre from the given plane, then \( a \times aA + c \times cC = a \times c \times bB \). Draw \( am, cm \) at right angles to \( AB \). Then since \( b \) is the common centre of gravity of the weights \( a \) and \( c \), we shall have by the similar triangles \( abn, cmb \), (Euclid, vi. 4.) \( nb : mb = (ba : bc) : ac \). (See Mechanics, Centre of Gravity.) Hence \( a \times nb = c \times mb \), or \( a \times nB - c \times bB = c \times bB - c \times mB \), or \( a \times nb - a \times bB = c \times bB - c \times mB \); then, by transposition, \( a \times nb - c \times mB = a \times bB + c \times bB = a + c + bB \). But \( aB = aA \) and \( mB = cC \), therefore, by substitution, \( a \times aA + c \times cC = a + c + bB \). 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. Pressure, similar pressure; and therefore it follows, that the pressure upon the bottom QR is as great as if it supported the whole column MNQR.

61. The same truth may be deduced from Prop. IV. For since the fluid in the two communicating vessels AB, CD, will rise to the same level, whatever be their size, the fluid in AB evidently balances the fluid in CD; and any surface mn is pressed with the same force in the direction Bm by the small column AB, as it is pressed in the direction Dm by the larger column CD.

62. Cor. I. From this proposition it follows, that the 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. III. 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.

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 equilibrio.

Prop. VIII.

64. The centre of pressure coincides with the centre of percussion.

Let AB be a vessel full of water, and CE the section of a plane whose centre of pressure is required. Prolong CE 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 lines CM, DM, EM. But the pressures upon the point C, D, E, are as the lines CN, DO, EP, and these lines are to one another as CM, DM, EM; therefore the percussive 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. We have already shewn in Art. 57, that the pressure upon the bottoms of vessels filled with fluids does not depend upon the quantity of fluid which they contain, but upon its particular altitude. This proposition has been called the Hydrostatical Paradox, and is excellently illustrated by the following machine. In fig. 10, AB is a box Fig. 10. which contains about a pound of water, and abcd 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. Pressure, contains. If a long narrow tube full of water, therefore, be fixed in the top of a cask likewise full of water, then though the tube be so small as not to hold a pound of the fluid, the 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 smallness of this principle it has been affirmed that a certain quantity of water, however small, may be rendered capable of exerting a force equal to any assignable one, by increasing the height of the column, and diminishing the base on which it presses. This, however, has its limits; for when the tube becomes so small as to belong to the capillary kind, the attraction of the glass will support a considerable quantity of the water it contains, and therefore diminish the pressure upon its base.

66. The preceding machine must be so constructed, that the moveable bottom may have no friction against the inside of the box, and that no water may get between it and the box. The method of effecting this will be manifest from fig. 11, where ABCD is a section of the box, and abcd its lid, which is made very light. The moveable bottom E, with a groove round its edges, is put into a bladder fg, 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 abcd, so that if water be poured into the box through the aperture H in its lid, it will be contained in the space f'g'h, and the bottom may be raised by pulling the wire i fixed to it at E.

67. The upward pressure of fluids is excellently illustrated by the hydrostatic bellows. The form given to this machine by the ingenious Mr Ferguson (Lectures, vol. ii. p. 111) is represented in fig. 12, where ABCD is an oblong square box, into one of whose sides is fixed the upright glass tube aI, which is bent into a right angle at the lower end as at i, fig. 13.: To this bent extremity is tied the neck of a large bladder K, which lies in the bottom of the box. Over this bladder is placed the moveable board L, fig. 14, in which the upright wire M is fixed. Leaden weights NN, with holes in their centre, to the amount of 16 pounds, are put upon this wire, and press with all their weight upon the board L. The cross bar P is then put on, in order to keep the glass tube in an upright position; and afterwards the piece EFG for keeping the weights N, N horizontal, and the wire M vertical. Four upright pins, about an inch long, are placed in the corners of the box, for the purpose of supporting the board L, and preventing it from pressing together the sides of the bladder. When the machine is thus fitted up, pour water into the tube I till the bladder is filled up to the board L. Continue pouring in more water, and the upward pressure which it will excite in the bladder will raise the board with all the weights NN, even though the bore of the tube should be so small as to contain no more than an ounce of water.

68. That the pressure of fluids arises from their gravity, and is propagated in every direction, may be proved by the following experiment. Insert into an empty vessel a number of glass tubes bent into various angles. Into their lower orifices introduce a quantity of mercury, which will rest in the longer legs on a level with these orifices. Let the vessel be afterwards filled with water; and it will be seen, while the vessel is filling, that the mercury is gradually pressed from the lower orifices towards the higher, where the water is prevented from entering. Now, in consequence of the various angles into which the glass tubes are bent, the lower orifices point to almost every direction; and therefore it follows, that the pressure of the superincumbent water is propagated in every direction. When a straight tube is employed to shew the upward pressure of fluids, the mercury which is introduced into its lower extremity must be kept in by the finger till the height of the water above the orifice is equal to fourteen times the length of the column of quicksilver: When the finger is removed, the mercury will ascend in the tube.

69. The pressure of the superior strata of fluids upon the inferior strata may be shewn in the following manner. Immerse two tubes of different bores, but not of the capillary kind, in a vessel of mercury. The mercury will rise in the rior press tube on a level with its surface in the vessel. Let water upon the then be poured upon the mercury so as not to enter the inferior upper orifices of the tubes, the pressure of the water upon strata of the inferior fluid will cause the mercury to ascend in the fluids tubes above the level of that in the vessel, but to the same height in both tubes. The columns of quicksilver in the two tubes are evidently supported by the pressure of the water on the inferior fluid. The same experiment may be made with oil and tinge water, the latter being made the inferior fluid.

70. The syphon is an instrument which shows the gravitation of fluids, and is frequently employed for decantation of common liquors. It is nothing more than a bent tube BAC, fig. 15, having one of its legs longer than the other. The shorter leg AC is immersed in the fluid contained in the vessel M; and if, by applying the mouth to the orifice B, the air be sucked out of the tube, the water in the vessel M will flow off till it be completely emptied. Now, it is obvious, that the atmosphere which has a tendency to raise the water in the shorter leg AC by its pressure on the surface pressure, of the water at M, has the same tendency to prevent the water from falling from the orifice B, by its pressure there, and therefore if the syphon had equal legs, no water could possibly issue from the orifice. But when the leg AB is longer than AC, the column of fluid which it contains being likewise longer, will, by its superior weight, cause the water to flow from the orifice B, and the velocity of the issuing fluid will increase as the difference between the two legs of the syphon is made greater.

The syphon is greatly improved by fixing a stop-cock D at the end of the longer branch, and placing on the same branch a small bent tube DE, communicating with the tube AB above D. When the aperture C is placed in the water to be drawn off, the mouth of the stop-cock D is closed, and the air is drawn out by suction at E from the longer branch. When there is no stop-cock at D, the finger may be applied there, till the air is sucked out at E.

An improved syphon by M. Bunten is shewn in fig. 16, where a bulb A is placed on the long branch AB. This syphon requires neither to be blown into nor sucked. When the long branch AB and bulb A are filled with fluid, and the other branch plunged in the fluid, the flow will be unremitting, as the bulb A in emptying itself draws off the liquid in contact with the short branch.

Another improvement on the syphon is shewn in fig. 17, as made by M. Hempel of Berlin. The short branch has fitted into it a vertical tube BA, terminating in a funnel A. A part of the liquid to be drawn off is then poured into the funnel A, and as soon as the flow commences the long branch DC, the tube AB is withdrawn, and the flow continues.

Another improvement on the syphon made by Mr Hunter of Thurston, is shewn in fig. 18, which has the peculiar advantage of retaining its charge. Two small cups or boxes A, B are fixed to the ends of the unequal branches by two screws C, C. When it is charged in the common way, and has been in use, it will stand vertically in the boxes A, B as a base, so that, when it is tilted by the ring D, it may be replaced, and will act as before. The same effect may be obtained by turning up the ends of the branches, and fixing to them a plate or piece of metal, upon which they may stand.

A moveable branch syphon, in which there is a joint at the top D, is a most valuable instrument, and the idea of it was first given by the late Mr Bryce, who employed it in place of a stomach-pump, in order to throw fluids into the stomach, or to extract them from it. The two branches may be made of metal, glass, or any other substance, the two parts being united by an air-tight joint. In cases of exigency two glass tubes, or pieces of any tube, might be joined into a syphon, by making the joint of a piece of bladder. One of the branches may be raised into any position for the purpose of charging it, and the instrument may be hung up charged, and ready for use, by a ring at the extremity of each branch.

71. In order to shew that the effect of the syphon depends upon the gravitation of fluids, M. Pascal devised the following experiment: In the large glass vessel AB, fasten by means of bees-wax two cylindrical cups a, b, containing tinged water, whose surface is about an inch higher in the one than in the other. Into the tinged water insert the legs of a glass the effect syphon cd, having an open tube e of the syphon fixed into the middle of it, and put a wooden cover on the vessel with a hole in its centre to receive the tube, and keep it in a vertical position. Then through the funnel f, fixed in another part of the cover, pour oil of turpentine into the larger vessel till it flows into the cups a, b, and rise above the arch of the syphon. The pressure of the oil upon the tinged water in the cups will cause the water to pass through the syphon from the higher cup to the lower, till the surfaces of the water in both the cups be reduced to a level. In order to explain this, suppose a horizontal plane eb to pass through the legs of the syphon, and the tinged water in the cups, the parts of this plane within the legs when the syphon is full, will be equally pressed by the columns of tinged water ce, db within the syphon; but the equal parts of this plane between the circumference of each leg of the syphon, and the circumference of each cylindrical cup, their diameters being equal, will sustain unequal pressures from their superincumbent columns, though the altitudes of these columns be equal. For since the pressure upon c is exerted by a column of oil ac, and a column of water ae, whereas the pressure upon b is exerted by a column of oil hd, and a column of water hb; the column ce, which contains the greatest quantity of water, will evidently exert the greatest force, and by its pressure will drive the tinged water from the cup a, through the syphon acd into the cup b, until a perfect equilibrium is obtained by an equality between the columns of water ae and hb.

Sect. III.—Application of the Principles of Hydrostatics to the Construction of Dikes, &c., for resisting the Pressure of Water.

Definition.

A dike is an obstacle either natural or artificial, which opposes itself to the constant effort of water to spread itself in every direction.

72. In discussing this important branch of hydraulic architecture, we must inquire into the thickness and form which must be given to the dike in order to resist the pressure of the water. In this inquiry the dike may be considered as a solid body, which the water tends to overthrow, by turning it round upon its posterior angle C; or it may sure of be regarded as a solid, whose foundation is immovable, water, but which does not resist the pressure of the water through the whole of its height, and which may be separated into horizontal sections by the efforts of the fluid. A dike may considered also as a solid body which can be neither broken nor overturned, but which may be pushed horizontally from its base, and can preserve its stability only by the friction of its base on the ground which supports it. On these conditions are founded the calculations in the following proposition which contain the most useful information that theory can suggest upon the construction of dikes.

Prop. I.

73. To find the dimensions of a dike which the water tends to overthrow by turning it round its posterior angle. Let ABCD, fig. 20, be the section of the dike, considered as a continuous solid, or a piece of firm masonry, H the level of the water which tends to overthrow it, by turning it round its posterior angle C, supposed to be fixed, and let AC, BD, be right lines or known curves. It is required to determine CD, the thickness which must be given to its base to prevent it from being overturned.

To the surface of the water HK draw the ordinates PM, PM infinitely near each other, and let fall from the points H and M the perpendiculars HT, MX. Draw the horizontal line ML and raise the perpendicular CL, and suppose

\[ \begin{align*}

= x \\ PM & = y \\ Pp \text{ or } MV & = z \\ Vm & = y \\ HT & = b \\ DT & = a \\ CD & = z \end{align*} \]

The momentum of the area ABCD, or the force with which it resists being turned round the fulcrum C

\[ = Z \]

The specific gravity of water

\[ = s \]

The specific gravity of the dike

\[ = \sigma \]

74. It is obvious, from art. 57, that every element sustains a perpendicular pressure proportional to the height PM. Let RM perpendicular to Mm represent the force exerted by the column of water MmpP, and let it be decomposed into two other forces, one of which RQ is horizontal, and has a tendency to turn the dike round the point C, and the other RY is vertical, and tends to press the dike upon its base. The force RQ is evidently \( s \times y \times Mm \times \frac{RQ}{RM} \), and therefore the horizontal part of it will be only \( sy \times Mm \times \frac{RQ}{RM} \). But the triangles RQM, MVm are evidently similar, consequently

\[ RQ : RM = VM : Mm; \quad \text{hence} \quad \frac{RQ}{RM} = \frac{VM}{Mm} = \frac{y}{Mm}. \]

Wherefore by substitution we have the force \( RQ = sy \times Mm \times \frac{y}{Mm} \), and dividing by Mm, we have \( RQ = syy \). The force RQ, therefore, will always be the same as the force against VM, whatever be the nature of the curve BD. Now the momentum of this force with relation to the fulcrum C, or its power to make the dike revolve round C, is measured by the perpendicular CL, let fall from the centre of motion to the direction in which the force is exerted (see Mechanics), consequently this momentum will be \( syy \times CL = syy \times a - y \) (since \( CL = HT - PM = a - y \)) \( = sayy - syyy \), whose fluent is \( \frac{syy}{2} - \frac{sy^3}{3} \), which by supposing \( y = a \)

becomes \( \frac{1}{6} sa^3 \) for the total momentum of the horizontal force, effort of the water to turn the dike round C. The vertical force RY or QM, which presses the dike upon its base, is evidently \( sy \times Mm \times \frac{MQ}{RM} \), but on account of the similar triangles, \( \frac{MQ}{RM} = \frac{x}{Mm} \), consequently by substitution we shall have the force \( RY = sy \times Mm \times \frac{x}{Mm} = syx \), after division by Mm. The momentum, therefore, of the vertical force RY with relation to C, or its power to prevent the dike from moving round the fulcrum C, will be \( syx \times CX \); CX being the arm of the lever by which it acts, or the perpendicular let fall from the fulcrum upon the direction of the force. Now \( CX = CD - DT + TX \) or HP, that is \( CX = z - b + x \), therefore the momentum of the force \( RY = syx \times z - b + x \), and the sum of the similar momenta from F to H will be the fluent \( \int (z - b + x) syx \),

the combined momentum of all the vertical forces which resist the efforts of the horizontal forces to turn the dike round C. But the efforts of the horizontal forces are also resisted by the weight of the dike, whose momentum we have called Z, therefore \( \varepsilon Z \), \( \varepsilon \) being the specific gravity of the dike, will be the momentum of the dike. We have now three forces acting at once, viz. the horizontal force of the water striving to overturn the dike, and the vertical force of the water combined with the momentum of the dike, striving to resist its overthrow, therefore we shall have an equilibrium between these three forces, when the momentum of the horizontal forces is made equal to the momentum of the vertical forces, added to that of the dike itself, consequently

\[ \frac{1}{6} sa^3 = \int (z - b + x) syx + \varepsilon Z. \]

75. As it is necessary, however, to give more stability to the dike than what is just requisite to preserve its equilibrium, we must make its dimensions such as to resist a force greater than the horizontal forces, a force, for example, \( n \) times the momentum of the horizontal forces. The equation will therefore become

\[ (L) \quad n \times \frac{1}{6} sa^3 = \int (z - b + x) syx + \varepsilon Z, \]

which comprehends every possible case of stability; for if we wish the stability of the dike to have double the stability of equilibrium, we have only to make \( n = 2 \). The preceding general equation is susceptible of a variety of applications according to the nature of the curves which form the sides of the dike. It is at present worthy of remark, that since the momentum of the horizontal forces is always the same, whatever be the curvature of the sides AC, BD, and since the momentum of the vertical forces increases as Equation the angle CDH diminishes, it follows that it will always be containing advantageous to diminish the angle CDH, and give as much, the condition slope as possible to the sides of the dike.

76. Let us now consider the conditions that may be necessary to prevent the dike ABCD from sliding on its base the supposition CD. Since the base of the dike is supposed horizontal, the situation that force which the dike opposes to the horizontal efforts of the dike water arises solely from the adhesion of the dike to its base, may slide and from the resistance of friction. These two forces, therefore, Pressure, fore, combined with the weight of the dike, form the force &c. of which resists the horizontal efforts of the water; an equilibrium will consequently obtain when the three first forces are made equal to the last. But the force of adhesion, and the resistance of friction, being unknown, may be made equal to the weight of the dike multiplied by the constant quantity \( m \), which must be determined by experience.

Now, calling \( A \) the area of the section \( ABCD \), we shall have \( sA \) for its weight, and \( m \times A \) for the resistance which is opposed to the horizontal efforts of the water. But we have already seen that the horizontal forces of the water upon \( M \) are equal to \( syy \), whose fluent \( \frac{1}{2} sa^2 \) (when \( a = y \)) is the sum of all the horizontal forces; consequently, when an equilibrium takes place between these opposing forces, we shall have,

\[ (II.) \quad m \times A = \frac{1}{2} sa^2, \quad \text{or} \quad A = \frac{s}{m} \times \frac{a^2}{2}. \]

We might have added to the weight of the dike the vertical pressure of the water, but it has been neglected for the purpose of having the dike sufficiently strong to resist an additional force.

77. We shall now proceed to inquire into the form which the general equation assumes when the sides of the dike are rectilineal. Let \( AC, BD \), fig. 21, be two lines inclined to the horizon under given angles \( ACD, BDC \), and let \( AB, CD \) be two horizontal lines. Retaining the construction and symbols in art. 73, let fall \( AQ, BZ \), perpendicular to \( CD \), and make \( AQ = BZ = d, CQ = r, \) and \( DZ = r' \).


On account of the similar triangles \( HPM, FTH \) we shall have \( a : b = y : x \), and therefore \( x = \frac{by}{a} \). Substituting this value of \( x \), instead of \( x \) in the general equation, art. 75, we have

\[ \int (z - b + x) syx = \int \frac{zb}{a} (z - b + \frac{by}{a}) y \dot{y} = \frac{sbzyy}{2a} - \frac{sbbyy}{2a} + \frac{sbb^2}{3a^2} = \left( \text{making } y = a \right) \frac{sba}{2} - \frac{sba}{6}; \]

now the momentum of the dike \( ABCD \) with relation to \( C \), is equal to the whole area of the dike \( ABCD \) collected in its centre of gravity, and placed at the end of a lever whose length is the horizontal distance of that centre of gravity from the fulcrum \( C \). But the area of \( ABQZ = QZ \times ZB = z - r' - r \times d \); the area of the triangle \( ACQ = \frac{CQ \times QA}{2} = \frac{dr}{2} \), and the area of the triangle \( BZD = \frac{DZ \times ZB}{2} = \frac{dr'}{2} \). Now the lever by which the area \( ABQZ \), collected in its centre of gravity \( F \), acts upon the fulcrum, is evidently \( Cf = CQ + Qf = CQ + \frac{1}{2} QZ = r + \frac{z - r' - r}{2} \), consequently the momentum by which the area \( ABCD \) resists the horizontal forces that conspire to give it a motion of rotation about \( C \) will be \( z - r' - r \times d \times r + \frac{z - r' - r}{2} \). The lever by which the triangle \( BZD \) acts, when collected in its centre of gravity \( I \), is evidently \( Ci \); but by the property of the centre of gravity \( Di = \frac{3}{2} DZ = \frac{2r'}{3} \), hence \( Ci = CD - Di = z - \frac{2r'}{3} \), consequently the energy of the triangle \( BZD \) to resist the efforts of the water acting horizontally will be \( \frac{d}{2} \times z - \frac{2r'}{3} \). The lever of the triangle \( ACQ \) is plainly \( Cs = \frac{3}{2} CQ = \frac{2r}{3} \), consequently the momentum of \( ACQ \), collected in its centre of gravity \( S \), will be \( \frac{dr}{2} \times \frac{2r}{3} \). Having thus found the momentum of the rectangle \( ABQZ \), and of the triangles \( BZD, ACQ \), the sum of these momenta will be the momentum \( Z \), with which the dike opposes the horizontal efforts of the water, therefore we shall have

\[ Z = z - r' - r \times d \times r + \frac{z - r' - r}{2} + \frac{dr'}{2} \times z - \frac{2r'}{3} + \frac{dr}{2} \times \frac{2r}{3}; \]

and by multiplication,

\[ Z = \frac{dz}{2} - \frac{dr'}{2} + \frac{dr'}{6} - \frac{dr}{6}. \]

By substituting this value of \( Z \) in the general equation in art. 75, we shall have

\[ (III.) \quad n \times \frac{1}{2} sa^2 = \frac{sba}{2} - \frac{sbb}{6} + \frac{sba}{2} - \frac{sba}{6} + \frac{sba}{6} + \frac{sba}{6}, \]

a quadratic equation, which will determine in general the base \( z \) of a dike, when its sides are rectilineal and inclined at any angle to the horizon.

78. When the angle \( ACQ \) is a right angle, or when the posterior side \( AC \) of the dike is perpendicular to the horizon, the quantity \( r \) becomes \( = 0 \), and the last term of the preceding equation in which \( r \) appears will vanish, consequently the equation will now become

\[ (IV.) \quad n \times \frac{1}{2} sa^2 = \frac{sba}{2} - \frac{sbb}{6} + \frac{sba}{2} - \frac{sba}{6} + \frac{sba}{6}, \]

Resulting equation when both its sides are rectilineal and inclined.

79. When the angles \( ACQ \) and \( BDZ \) are both right, the dike becomes rectangular, with its sides perpendicular to its base. In this case both \( r \) and \( r' \) become each \( = 0 \), when both and therefore all the terms in which they are found will vanish. In this case too \( DT = b \) becomes \( = 0 \), and therefore the terms in which it appears will likewise vanish. The general equation will now become

\[ (V.) \quad n \times \frac{1}{2} sa^2 = \frac{sba}{2} \text{ a pure quadratic.} \]

80. In order to show the application of the preceding formulae, and at the same time the advantage of inclining the sides of the dike, let us suppose the depth of the water, formulae, and also the height of the dike, to be 18 feet, so that \( B \) will coincide with \( H \). Let us also suppose, what is generally the case in practice, that the declivity of the sides is \( \frac{1}{2} \) of their altitude, that is \( DZ = CQ = \frac{1}{2} BZ \). Let the specific gravity of the dike be to that of water as 12 to 7; and suppose it is wished to make the stability of the dike twice as great as the stability of equilibrium, that is, to make it capable of resisting a force twice as great as that which it really sustains. Then, upon these conditions, we shall have \( BZ = HT \), or \( a = d = 18 \) feet; \( CQ = DZ = DT \), or \( r' = r = b = 3 \) feet; \( s = 7 \); \( e = 12 \); and \( n = 2 \). By substituting these numerical values in the general equation No. III, it becomes

\[ \frac{45}{36} z = \frac{4599}{39} \text{ feet}; \]

a quadratic equation, which, after reduction, will give \( z = 12 \) feet nearly. When \( z = 12 \), the area of the dike ABCD will be 162 square feet.

81. Let us now suppose the sides of the dike to be vertical, the equation No. V. will give us \( z = 11 \) feet 2 inches, which makes the area of the dike more than 201 square feet. The area of the dike with inclined sides is therefore to its area with vertical sides nearly as 4 to 5; and hence we may conclude that a dike with inclined sides has the same stability as a dike with vertical sides, while it requires less materials.

**Prop. II.**

82. To find the dimensions of a dike which can neither slide upon its base, nor turn round its posterior angle; but which is composed of horizontal sections, which may be separated from each other.

In solving this proposition, we must find the curvature of the side exposed to the pressure of the water, which will make all the different sections or horizontal laminae equally capable of resisting the different forces which tend to separate them. If the lamina NM does not resist the column PM, which partly presses it in the direction MN as powerfully as the lamina nm resists the horizontal pressure of the column pm, the lamina NM is in danger of being separated from the lamina nm. But if all the laminae NM, nm resist with equal force the horizontal effects of the water, and if the dike cannot be made to slide upon its base nor turn round its posterior angle T, it cannot possibly yield to the pressure of the water; for it is impossible to separate one lamina from another, unless the one opposes a less resistance than the other. To simplify the investigation as much as possible, let us suppose the posterior side of the dike to be vertical, and the depth of the water to be equal to the height of the dike.

83. Let ABC be the section of the dike, AK the surface of the water, AC the curvature required, AB its posterior side; MN nm a horizontal lamina infinitely small, in the direction of which the dike has a tendency to break, in consequence of the efforts of the water upon AM.

If the dike should break in the direction MN, the superior part AMN will detach itself from the inferior part MNBC, by moving from M towards N; and at the moment when the impulse takes place, it will have a small motion of rotation round the point N. We must therefore determine the forces which act upon the lamina MN nm, and form an equation expressing their equilibrium round the point N. The forces alluded to are evidently, 1. The horizontal efforts of the water; 2. The vertical efforts of the water; 3. The weight of the part AMN; and, 4. The adhesion of the two surfaces MN, mn. Of these four forces, the first is the only one which has a tendency to overthrow the portion AMN of the dike; and its efforts are resisted by the three other forces. In order to find the momenta of these forces with regard to the point N, let us suppose

\[ AP = NM = x \\ PM = y \\ The specific gravity of water = s \\ The specific gravity of the dike = r \]

Then we shall have,

1. The momentum of the horizontal forces of the water will be \( \frac{1}{2} sy^2 \), by the same reasoning that was employed in art. 74.

2. The momentum of the part AMN of the dike will be \( \sigma \int x \dot{y} \) the area of the surface AMN, multiplied by the distance of its centre of gravity from the fulcrum N,

\[ \frac{1}{2} \int x \dot{y} \]

which is equal to \( \frac{\sigma}{\int x \dot{y}} \). See Mechanics.

84. In order to simplify the calculus, and at the same time increase the stability of the dike, we shall neglect the vertical force of the water, and the adhesion of the two surfaces MN, mn. The only forces, therefore, which we have to consider, are the horizontal efforts of the water acting against the momentum of the superior part AMN. By making an equilibrium between these forces, we shall have the following equation:

\[ \frac{1}{2} sy^2 = \sigma \int x \dot{y} \times \frac{1}{2} \int x \dot{y} = \sigma \times \int x \dot{y}. \]

By taking the fluxion we have,

\[ \frac{1}{2} sy^2 \dot{y} = \frac{1}{2} \sigma \times x \dot{y}. \quad \text{Dividing by } \dot{y} \text{ we have} \]

\[ \frac{1}{2} sy^2 = \frac{1}{2} \sigma \times x^2, \text{ which by reduction becomes} \]

\[ y = \sqrt{\frac{s}{\sigma}} \times x. \]

The line AMC therefore is rectilineal, and the base BC is to the altitude BA as \( \sqrt{s} : \sqrt{\sigma} \); that is, as the square root of the specific gravity of the water is to the square root of the specific gravity of the dike.

85. In order to prevent the superior portion AMN from sliding on its base MN, we must procure an equilibrium containing between the adhesion of the surfaces MN, mn and the horizontal force exerted by the water. Now, the sum of all the horizontal forces exerted by the water is (by art. 76,) equilibrium on \( \frac{1}{2} sy^2 \), and the adhesion may be represented by some multiple \( m \) of its weight, the constant quantity \( m \) being determined by experience. The adhesion will therefore be the dike may slide upon its base.

\[ m \times \sigma \int x \dot{y}, \text{ and the equation of equilibrium will be} \]

\[ \frac{1}{2} sy^2 = m \times \sigma \int x \dot{y}, \text{ the fluxion of which is} \]

\[ sy^2 \dot{y} = m \times \sigma \times x \dot{y}. \quad \text{Dividing by } \dot{y} \text{ we have} \]

\[ sy = m \times x, \text{ and therefore} \]

\[ x : y = s : m \times n. \]

Hence the base BC of the dike is to its altitude BA as the specific gravity of water is to a multiple \( m \) of the specific CHAPTER II.—OF SPECIFIC GRAVITIES.

Definition.

86. The absolute weights of different bodies of the same bulk are called their specific gravities or densities; and one body is said to be specifically heavier or specifically lighter than another, when under the same bulk it contains a greater or less quantity of matter. Brass, for example, is said to have eight times the specific gravity of water, because one cubic inch of brass contains eight times the quantity of matter, or is eight times heavier than a cubic inch of water.

Prop. I.

87. Fluids pressing against each other in two or more communicating vessels, will be in equilibrium when the perpendicular altitudes above the level of their junction are in the inverse ratio of their specific gravities.

If a quantity of mercury be poured into the vessel FMN, it will be in equilibrium when it rises to the same level AHIB, in both tubes. Take away an inch of mercury ACDH, and substitute in its room 13½ inches of water FCDG. Then since mercury is 13½ times heavier than water, 13½ inches of water will have the same absolute weight as one inch of mercury, and the equilibrium will not be disturbed; for the column of water FD will exert the same pressure upon the surface CD of the mercury, as the smaller column of mercury did formerly. The surface of the mercury, therefore, will remain at IB: now, since AB, CE, are horizontal lines, AC will be equal to IK; but FC was made 13½ times AC, therefore FC = 13½ times IK, that is FC : IK = 13½ : 1, the ratio between the specific gravities of mercury and water.

88. On this proposition depends the theory of the barometer. Let a quantity of mercury be introduced into the tube FMN, and let the pressure of the atmosphere be removed from the surface IB; the pressure of the air upon the other surface CD will be the same as if the tube FD were continued to the top of the atmosphere, and therefore, instead of the column of water FD we have a column of air equal to the height of the atmosphere acting against the mercury CDMIB; the mercury consequently will rise towards N, so that its height will be to the height of the atmosphere as the specific gravity of air is to the specific gravity of mercury; but as the density of the air diminishes as it recedes from the earth, we must take the specific gravity of the air at a mean height in the atmosphere. It is obvious from the proposition, that the altitude of the column of mercury which balances the column of air, must be reckoned from CD, the level of their junction; and that, when the specific gravity of the air is diminished, the mercury will fall, and will again rise when it regains its former wards with density.

Prop. II.

89. If any body is immersed in a fluid, or floats on its surface, it is pressed upwards with a force equal to the weight of the quantity of fluid displaced.

Let m H be the section of a body immersed in the vessel AB filled with a fluid. Any portion mn of its upper surface is pressed downwards by the column immersed fluid CmnD (59.), but in the fluid, the similar portion EF of its lower surface is pressed upwards with a column of fluid equal to CEFD, therefore the part EF is pressed upwards with the difference of these forces, that is, with a force equivalent to the column of fluid mEFn, for CEFD — CmnD = mEFn. In the same way it may be shewn, that the remaining part FH is pressed upwards with a force equal to the weight of a column nFHo; and therefore it follows, that the rectangle mEHo is pressed upwards with a force equivalent to a column m EH o, that is, to the quantity of fluid displaced.

90. If the body floats in the fluid, like CH in the vessel AB, the same consequence will follow; for the body CH is evidently pressed upwards with a force equivalent to the column m EH o, that is, to the part immersed or the quantity of fluid displaced. Now, as the same may be demonstrated of every other section of a solid parallelopiped, we may conclude, that the proportion is true with respect to every solid whose section is rectangular.

91. When the solid has any other form, as CD, however irregular, we may conceive its section to be divided into any other a number of very small rectangles n o; then (57.) the small portion of the solid at n is pressed downwards by a column of particles m n, and the small portion at o is pressed upwards by a column of particles equal to n o; therefore the difference of these forces, viz. the column n o, is the force with which the portion o is pressed upwards. In the same manner it can be shewn, that every other similar portion of the lower surface of the solid CD is pressed upwards with a force equal to a column of particles whose height is equal to the vertical breadth of the solid; but all these columns of particles must occupy the same space as the solid itself; therefore any solid body immersed in a fluid, or floating on its surface, is pressed upwards with a force equal to the weight of the quantity of fluid displaced.

92. Cor. 1. When a body floats in a fluid, the weight of the quantity of fluid displaced is equal to the weight of the floating solid. For since the solid is in equilibrium with the fluid, the force which causes it to descend must be equal to the force which presses it upwards; but the force which keeps a part of the solid immersed in the fluid is the weight of the solid, and the force which presses the fluid solid upwards, and prevents it from sinking, is equivalent placed to the weight of the quantity of fluid displaced (89.); there- Of specific force these forces, and the weights to which they are equivalent, must be equal.

93. Cor. 2. A solid weighed in a fluid loses as much of its weight as is equal to the weight of the quantity of fluid displaced; for since the body is pressed upwards with a force equal to the weight of the fluid displaced (89.), this pressure acts in direct opposition to the natural gravity or absolute weight of the solid, and therefore diminishes its absolute weight by a quantity equal to the weight of the fluid displaced. The part of the weight thus lost is not destroyed: It is only sustained by a force acting in a contrary direction.

94. Cor. 3. A solid immersed in a fluid will sink, if its specific gravity exceed that of the fluid: It will float on the surface, partly immersed, if its specific gravity be less than that of the fluid; and it will remain wholly immersed wherever it is placed, if the specific gravities of the solid and fluid are equal. In the first case, the force with which the solid is pressed downwardly exceeds the upward pressure, and therefore it must sink. In the second case, the upward pressure exceeds the pressure downwards, and therefore the body must float; and, in the third case, the upward and downward pressures being equal, the solid will remain wherever it is placed.

95. Cor. 4. The specific gravities of two or more fluids are to one another as the losses of weight sustained by the same solid body, and specifically heavier than the fluids, when weighed in each fluid respectively. The solid in this case displaces equal quantities of each fluid; but the losses of weight are respectively as the absolute weights of the quantities displaced (Cor. 2.), therefore the specific gravities, which are as the absolute weights of equal quantities of any body (86.), must be as the losses of weight sustained by the immersed solid.

96. Cor. 5. The specific gravity of a solid is to that of a fluid as the absolute weight of the solid is to the loss of weight which it sustains when weighed in the fluid. For since the loss of weight sustained by the solid is equal to the absolute weight of the quantity of fluid displaced, or of a quantity of fluid of the same bulk as the solid, the specific gravities, which (86.) are in the ratio of the absolute weights of equal volumes, must be as the absolute weight of the solid to the loss of weight which it sustains.

97. Cor. 6. The specific gravity of a solid floating in a fluid, is to the specific gravity of the fluid itself, as the bulk of the part immersed is to the total bulk of the solid.

98. Cor. 7. Bodies which sustain equal losses of weight are of the same bulk. For, since the losses of weight are as the weights of the quantities of fluid displaced, and as the quantities displaced are as the bulks of the solids which displace them, the bulks must be equal when the losses of weight are equal.

99. The preceding corollaries may be expressed algebraically, and may be deduced from a general equation in the following manner. Let B be the total bulk of a floating body, and C the part of it which is immersed; let S be the specific gravity of the solid, and s that of the fluid. Then it is obvious, that the absolute weight of the solid will be expressed by $B \times S$, and the absolute weight of the fluid displaced by $C \times s$; for the fluid displaced has the same bulk as the part of the solid which is immersed. In order that an equilibrium may obtain between the solid and fluid, we must have $B \times S = C \times s$. Now, when $S > S$, we have $B > C$, so that the solid will float, which is the second case of Cor. 3.—When $S = S$ we have $B = C$, which is the third case of Cor. 3.—When $S < S$ we have $C > B$, that is, the body will sink below the surface; and it will descend to the bottom, for it cannot be suspended in the fluid without some power to support it; and if such a power were necessary, we should have $B \times S > C \times s$, which is contrary to the equation of equilibrium.

100. From the equation $B \times S = C \times s$ we have (Euclid VI. 16.) $S : s = B : C$, which is Cor. 6.—When the body is completely immersed we have $B = C$, in which case the equation becomes $B \times S = B \times s$; and when the solid is specifically heavier than the fluid, it will require a counterweight to keep the solid suspended in the fluid. Let W be the counterweight necessary for keeping the solid suspended in the fluid, then in the case of an equilibrium the equation will be $B \times s + W = B \times S$, or $B \times S - W = B \times s$, or $S \times B \times S - W = S \times B \times s$, whence (Euclid VI. 16.) $S : s = B \times S : B \times S - W$, which is Cor. 5.

101. If the same solid body is plunged in a second fluid of a different specific gravity from the first, let $\sigma$ be the specific gravity of the second fluid, and w the counterweight necessary to keep the solid suspended in it.—The equation for the first fluid was $B \times s + W = B \times S$ (100.), and the equation for the second fluid will be $B \times \sigma + w = B \times S$; therefore we shall have, by the first equation, $S \times B - W = S \times B$, and by the second $S \times B - w = S \times B$, consequently $S \times B : \sigma \times B = S \times B : W : S \times B - w$, or (Euclid V. 16.) $S : \sigma = S \times B : W : S \times B - w$, which is Cor. 4.; for the losses of weight in each fluid are evidently represented by $S \times B - W$ and $S \times B - w$.

102. If B and b express the bulks of two solids, S and s their specific gravities, $\sigma$ the specific gravity of the fluid, and W, w the counterweights which keep them in equilibrium with the fluid. Then with the solid S the equation will be $S \times B - W = \sigma \times B$ (101.); and with the solid s the equation will be $s \times b - w = \sigma \times b$. Therefore, if the two solids sustain equal losses of weight, we shall have $S \times B - W = s \times b - w$, since each side of the equation represents the loss of weight sustained by each solid respectively. Consequently, $\sigma \times B = \sigma \times b$, and dividing by $\sigma$, we have $B = b$, which is Corollary 7.

103. From the preceding proposition and its corollaries, Method of we may deduce a method of detecting adulteration in the detecting precious metals, and of resolving the problem proposed to adulterate Archimedes, by Hiero, king of Syracuse. Take a real guinea, and a counterfeit one made of copper and gold. If the latter be lighter than the former, when weighed in a pair of scales, the imposition is instantly detected; but should their weight be the same, let the two coins be weighed in water, and let the loss of weight sustained by each be carefully observed, it will then be found that the counterfeit will lose more of its weight than the unadulterated coin. For, since the specific gravity of copper exceeds that of gold, and since the absolute weights of the coins were equal, the counterfeit guinea must be greater in bulk than the real one, and will therefore displace a greater quantity of water, that is (93.), it will lose a greater part of its weight.

194. Hiero, king of Syracuse, having employed a goldsmith to make him a crown of gold, suspected that the proposed metal had been adulterated, and inquired at Archimedes if his suspicions could be verified or disproved without injuring the crown. The particular method by which Archimedes detected the fraud of the goldsmith is not certainly known; but it is probable that he did it in the following manner. A quantity of gold, of the same absolute weight as the crown, would evidently have the same bulk also, if the crown were pure gold, and would have a greater bulk if the crown were made of adulterated gold. By weighing, therefore, the quantity of gold and the crown in water, and observing their respective losses of weight, Archimedes found that the crown lost more of its weight than the quantity of gold; and therefore concluded, that as the crown must have displaced a greater portion of water than the piece of gold, its bulk must likewise have been greater, and the metal adulterated of which it was composed. Prop. III.

105. If two immiscible fluids, of different specific gravities, and a solid of an intermediate specific gravity, be put into a vessel, the part of the solid in the lighter fluid will be to the whole solid, as the difference between the specific gravities of the solid and the heavier fluid, is to the difference between the specific gravities of the two fluids.

Let AB (fig. 27.) be the vessel which contains the two fluids, suppose mercury and water, and the solid CD. The mercury being heavier than water, will sink to the bottom, and have mn for its surface, and the water will occupy the space AB mn. The solid having a greater specific gravity than water, will sink in the water (94.), but having a less specific gravity than mercury, it will float in the mercury. It will therefore be suspended in the fluids, having one portion C in the water, and the other portion D in the mercury. Now let S be the specific gravity of the mercury, s the specific gravity of the water, ε that of the solid, C the part of the solid in the water, and D the part in the mercury. Then the bulk of the solid is C + D, and its weight ε × C + D. The quantity of water displaced by the part C, or the loss of weight sustained by the part C, will be C × s; and the quantity of mercury displaced, or the loss of weight sustained by part D, will be D × S. But as the solid is suspended in the fluids, and therefore in equilibrium with them, the whole of its weight is lost. Consequently, the part of its weight which is lost in the water, added to the part lost in the mercury, must be equal to its whole weight; that is, C × s + D × S = ε × C + D, or sC + SD = εC + εD. Transposing sC and SD, we have sC - εC = SD - εD, or C × s - ε = D × S - ε, and (Euclid VI. 16.) C : D = s - ε : S - ε. Then, by inversion and composition (Euclid V. Propositions B and 18.) C : C + D = S - ε : S - ε. Q. E. D.

106. Cor. 1. From the analogy C : D = s - ε : S - ε, we learn that the part of the solid in the heavier fluid, is to the part in the lighter fluid, as the difference between the specific gravities of the solid and the lighter fluid, is to the difference between the specific gravities of the solid and the heavier fluid.

107. Cor. 2. When s is very small compared with S, we may use the analogy C : C + D = s - ε : ε, though in cases where great accuracy is necessary this ought not to be done. When the specific gravity of a body, lighter than water, is determined by comparing the part immersed with the whole body, there is evidently a small error in the result; for the body is suspended partly in water and partly in air. It is, in fact, a solid of an intermediate specific gravity floating in two immiscible fluids, and therefore its specific gravity should be ascertained by the present proposition.

Prop. IV.

108. If two bodies, whether solid or fluid, be mixed together so as to form a compound substance, the bulk of the heavier is to the bulk of the lighter ingredient, as the difference between the specific gravities of the compound, and the lighter ingredient, is to the difference between the specific gravities of the compound and the heavier ingredient.

Let S and s be the specific gravities of the two ingredients, ε the specific gravity of the compound, and B, b the bulks of the ingredients; then the bulk of the compound will be B + b, and its weight ε × B + b. The weight of the ingredient B will be B × S, and that of the other ingredient b × s; and as the weight of the compound must that of the equal to the weight of its ingredients, we have the following equation ε × B + b × s = BS + bs, and by transposing b and BS, we shall have B × s - BS = b × s - b × ε, or given:

\[ B \times s - S = b \times s - ε \]

therefore (Euclid VI. 16.)

\[ B : b = s - ε : S - ε \]

Q. E. D.

109. In the preceding proposition, it has been taken for granted that the magnitude of the compound is exactly equal to the sum of the magnitudes of the two ingredients. This, however, does not obtain universally either in fluids or solids; for an increase or diminution of bulk often attends the combination of two different ingredients. A greater cubical inch of alcohol, for example, combined with a cubical inch of water, will form a compound which will measure less than two cubical inches; and a cubical inch of tin when incorporated in a fluid state with a cubical inch of lead, will form a compound, whose bulk will exceed two cubical inches. The preceding proposition, however, is even in these cases, of great use in ascertaining the increase or decrease of bulk sustained by the compound, by comparing the computed with the observed bulk.

Prop. V. Problem.

110. How to determine the specific gravities of bodies, whether solid or fluid.

The simplest and most natural way of finding the specific gravities of bodies would be to take the absolute weights of a cubic inch, or any other determinate quantity, of each mine the substance; and the number thus found would be their specific gravities. But as it is difficult to form two bodies of the very same size, and often impossible, as in the case of precious stones, to give a determinate form to the substance under examination, we are obliged to weigh them in a fluid, and deduce their specific gravities from the losses of weight which they severally sustain. Water is a fluid which is always employed for this purpose, not only because it can be had without difficulty, but because it can be procured of the same temperature, and of the same density, in every part of the world. The specific gravity of water is always called 1,000, and with this, as a standard, the specific gravity of every other substance is compared. Thus, if a certain quantity of water weighed four pounds, and a similar quantity of mercury five pounds, the specific gravity of the mercury would be called 134, because as 4 : 5 = 1 : 134. In order, therefore, to determine the densities of bodies, we have occasion for no other instrument than a common balance with a hook fixed beneath one of its scales. When fitted up in this way, it has been called the hydrostatic balance, which has already been described under the article Hydrostatic Balance.

111. When the substance is heavier than its bulk of water.—Suspend the solid by means of a fine silver wire to find the hook beneath the scale, and find its weight in air. Fill a jar with pure distilled water, of the temperature of gravity of 62° of Fahrenheit's thermometer, and find the weight of a solid when immersed in this fluid. The difference of these weights is the loss of weight sustained by the solid. Then (96.), as the loss of weight is to the weight of the solid in air, so is 1,000 the specific gravity of water to a Of specific fourth proportional, which will be the specific gravity of Gravities, the solid. But as the third term of the preceding analogy is always 1,000, the fourth proportional, or density of the solid, will always be had by dividing the weight of the solid in air by its loss of weight in water. If the solid substance consists of grains of platinum or metallic filings, place it in a small glass bucket. Find the weight of the bucket in air, when empty, and also its weight when it contains the substance. The difference of these weights will be the weight of the substance in air. Do the very same in water, and its weight in water will be had. Its specific gravity will then be found as formerly. If the body is soluble in water, or so porous as to absorb it, it should be covered with varnish or some unctuous substance. When it is weighed in water, it should never touch the sides of the glass jar, and it must be carefully freed from any bubbles of air that happen to adhere to it.

112. When the substance is lighter than its bulk of water.

To find the specific gravity of a body, and also of the compound body, both in air and in water; and by subtracting their weight in water from their weight in air, find how much weight they have severally lost. Then say as the difference between their losses of weight is to the weight of the light body in air, so is 1,000 to the specific gravity of the body.

113. When the substance is a powder which absorbs water, or is soluble in it.

To find the specific gravity of the powder with the powder to be examined; and having rammed it as close as possible to the very top, find the weight of the powder. Remove the powder from the phial, and fill it with distilled water, and find its weight. The weight of the powder, divided by the weight of the water, will be the specific gravity of the former. See art. 124.

114. When the substance is a fluid, its specific gravity may be determined very accurately by the method in the preceding article, or by the following method, deduced from article 95.—Take any solid specifically heavier than water, and the given fluid. Find the loss of weight which it sustains in water, and also in the given fluid. Then, since the specific gravities are as the losses of weight sustained by the same solid, the specific gravity of the fluid required will be found by dividing the loss of weight sustained by the solid in the given fluid, by the loss of weight which it sustains in water.

Sect. II. On the Hydrometer.

Hydrometer invented by Hypatia.

115. In order to determine, with expedition, the strength of spirituous liquors, which are inversely proportional to their specific gravities, an instrument more simple, though less accurate, than the hydrostatic balance, has been generally employed. This instrument is called a hydrometer, sometimes an arcometer or gravimeter, and very erroneously a hygrometer by some foreign authors. It seems to have been invented by Hypatia, the daughter of Theon Alexandrinus, who flourished about the end of the fourth century; though there is some foundation for the opinion that the invention is due to Archimedes.

116. The hydrometer of Fahrenheit, which is one of the simplest that has been constructed, is represented in fig. 28, and may be formed either of glass or metal. A B is a cylindrical stem, and C, D two hollow balls appended to it. Into the lower ball D is introduced a quantity of mercury, sufficient to make the ball C sink to F, a little below the surface of distilled water. If this apparatus be plunged into a fluid lighter than water, the ball C will sink farther below the surface; and, if it be immersed in a heavier fluid, it will rise nearer the surface. In this way we can tell whether one fluid is more or less dense than another. Gravities. But in order to determine the real specific gravities of the fluids, the hydrometer must either be loaded with different weights, or have a scale A B engraved on its stem. The former of these methods was employed by Fahrenheit. Having placed some small weights on the top A, he marked any point E, to which the instrument sunk in distilled water. By weighing the instrument thus loaded, he found the weight of a quantity of water equal to the part immersed (92). When the hydrometer was placed in a fluid denser than water, he loaded it with additional weights till it sunk to the same point E. The weight of the hydrometer being again found, gave him the weight of a quantity of the denser fluid equal to the part immersed; weights, but as the part immersed was the same in both cases, the weights of the hydrometer were equal to the absolute weights of equal quantities of the two fluids; and, consequently, the specific gravities of the water and the other fluid were in the ratio of these weights. When the fluid, whose density is required, has less specific gravity than water, some of the weights are to be removed from the top A, till the instrument sinks to E; and the density of the fluid to be determined as before.—Instead of making the weight of the hydrometer variable, it is more simple, though with less accuracy, to have a scale of equal parts upon the stem engraved AB. In order to graduate this scale, immerse the hydrometer in distilled water, at the temperature of 60° Fahrenheit, so that it may sink to B near the bottom of the stem, which may be easily effected, by diminishing or increasing the quantity of mercury in the ball D. At B place the number 1,000, which shews that every fluid, in which the hydrometer sinks to B, has its specific gravity 1,000, or that of distilled water. The hydrometer is then to be plunged in another fluid less dense than water, suppose oil, whose specific gravity may be .900, and the point A marked, to which it sinks. Every fluid, therefore, in which the hydrometer sinks to A, has its specific gravity .900; and if the scale AB be divided into equal parts, every intermediate degree of specific gravity between .900 and 1,000 will be marked. If the scale AB be divided into four parts in the points E, F, G, the fluid in which the hydrometer sinks to G will have .975 for its specific gravity; the specific gravity of that in which it sinks to F will be .950, and so on with the other points of division. If it is required to extend the range of the instrument, and to make it indicate the densities of fluids specifically lighter than water, we have only to load it in such a manner as to make it sink to the middle of the scale F in distilled water; and by taking two fluids, between whose densities the specific gravity of every other fluid is contained, excepting mercury and metals in a fluid state, to determine, as before, the extremities of the scale.

117. When the weight of the hydrometer is variable, let E be the point to which it sinks in two different fluids, and W be the absolute weight necessary to make it sink to E in the denser fluid, and W ± p the weight necessary to make it sink to the same point in the lighter fluid. Let S, s be the specific gravities of the two fluids, and V the volume of the part of the hydrometer that is constantly immersed. Then \( W = \frac{S}{s} \times V \), \( W \pm p = s \times V \).

From the first equation we have, \( V = \frac{W}{S} \), and from the second, \( V = \frac{W \pm p}{s} \). second equation \( V = \frac{W + p}{s} \), consequently \( \frac{W + p}{s} = W \),

and by reduction \( s = \frac{s \times W + p}{W} \). Thus, by knowing \( W \)

and the weight \( p \), and also \( S \) the specific gravity of one of

the fluids, which will be 1.000 if that fluid be water, we

can find \( s \) the specific gravity of the other fluid.

118. When the weight of the hydrometer is constant,

and the density of the fluid indicated by the depth to which

it descends, let \( F, E \) be the points to which it sinks in two

different fluids, whose specific gravities are \( S, s \), \( W \) the

absolute weight of the hydrometer, \( V \) the volume of the

part immersed when the hydrometer has sunk to \( E \), and

\( v \) its volume when sunk to \( F \). Then (99.), we have

\( W = S \times V \), and \( W = s \times v \), consequently \( s \times v = S \times V \),

and \( s = \frac{S \times V}{v} \). If the absolute weight \( W \), therefore, of

the hydrometer be known, and also the volumes \( V, v \), and

the specific gravity \( S \) of one of the fluids, which may be

water, the specific gravity of the other fluid may be deter-

mined by the preceding formula. When the figure of the

hydrometer is regular, the volumes \( V, v \), may be deter-

mined geometrically; but as the instrument is generally of

an irregular form, the following methods should be em-

ployed:

Jones's Hydrometer.

Jones's hy-

drometer, Holborn, is a simple and accurate instrument, and requires

only three weights to discover the strengths of spirituous

liquors from alcohol to water. Like other instruments of the

same kind, it is adjusted to the temperature of 60° of Fahrenheit;

but as every change of temperature produces a change

in the specific gravity of the spirits, Mr Jones found it ne-

cessary to attach a thermometer to the instrument, and

thus make a proper allowance for every variation of tem-

perature. Almost all bodies expand with heat and con-

tract with cold; and as their volume becomes different at

different temperatures, their specific gravities must also

be variable, and will diminish with an increase of

temperature. M. Homberg, and M. Eisenclim found

that the absolute weight of a cubic inch of brandy was four

drams 42 grains in winter, and only four drams 32 grains

in summer, and that the difference in spirits of nitre was

still greater. It has been found, indeed, upon an average,

that 32 gallons of spirits in winter will expand to 33 gal-

lons in summer. As the strength of spirituous liquors is

inversely as their specific gravities, they will appear much

stronger in summer than in winter. This change in their

strength had been formerly estimated in a rough way; but

by the application of the thermometer, and by adjusting its

divisions experimentally, Mr Jones has reduced it to pretty

accurate computation. It has already been stated (109.)

that where two substances are combined, the magnitude of

the compound body is sometimes greater and sometimes

less than the sum of the magnitudes of the two ingredients,

and that this mutual penetration particularly happened in

the mixture of alcohol and water. In strong spirits, this

concentration is sometimes so great, as to produce a diminu-

tion of four gallons in the 100; for if to 100 gallons of

spirit of wine found by the hydrometer to be 66 gallons in

the 100 over proof, you add 66 gallons of water to reduce

it to proof, the mixture will consist only of 162 gallons in-

stead of 166 of proof spirits. This mutual penetration of

the particles of alcohol and water has also been considered

in Mr Jones's hydrometer, which we shall now describe

with greater minuteness.

120. In fig. 29, the whole instrument is represented with

the thermometer attached to it. Its length Gravities.

AB is about 9\(\frac{1}{2}\) inches; the ball C is made of

hard brass, and nearly oval, having its conju-

gate diameter about 1\(\frac{1}{2}\) inches. The stem

AD is a parallelopiped, on the four sides of

which the different strengths of spirits are

engraved; the three sides which do not ap-

pear in fig. 29, are represented in fig. 30, with

the three weights numbered 1, 2, 3, cor-

responding with the sides similarly marked at the

top. If the instrument,

when placed in the spirits,

sinks to the divisions on

the stem without a weight,

their strength will be shewn

on the side AD marked 0

at the top, and any degree

of strength from 7\(\frac{1}{2}\) gallons

in the 100 to 47 in the

100 above proof, will thus

be indicated. If the hy-

drometer does not sink to

the divisions without a

weight, it must be loaded

with any of the weights 1,

2, 3, till the ball C is com-

pletely immersed. If the weight No. 1, is necessary, the side

marked 1 will shew the strength of the spirits, from 46 to 13

gallons in the 100 above proof. If the weight No. 2 is em-

ployed, the corresponding side will indicate the remainder of

overproof to proof, marked P in the instrument, and likewise

every gallon in 100 under proof, down to 29. When the

weight No. 3, is used, the side similarly marked will shew

any strength from 30 gallons in the 100 under proof, down

to water, which is marked W in the scale. The small

figures as 4 at 65, 3) at 61, 2) at 48 (fig. 29.) indicate the

diminution of bulk which takes place when water is mixed

with spirits of wine in order to reduce it to proof; thus, if

the spirit be 61 gallons in the 100 over proof, and if 61

gallons of water are added in order to render it proof, the

magnitude of the mixture will be 3\(\frac{1}{2}\) gallons less than the

sum of the magnitudes of the ingredients, that is, instead

of being 161 it will be only 157\(\frac{1}{2}\) gallons. The thermo-

meter F connected with the hydrometer, has four columns

engraved upon it, two on one side as seen in the figure,

and two on the other side. When any of the scales upon

the hydrometer, marked 0, 1, 2, 3, are employed, the

column of the thermometer similarly marked must be used,

and the number at which the mercury stands carefully ob-

served. The divisions commence at the middle of each

column which is marked 0, and is equivalent to a tempera-

ture of 60° of Fahrenheit; then, whatever number of di-

visions the mercury stands above the zero of the scale, the

same number of gallons in the 100 must the spirit be reck-

oned weaker than the hydrometer indicates, and whatever

number of divisions the mercury stands below the zero,

so many gallons in the 100 must the spirit be reckoned

stronger.

Nicholson's Hydrometer.

121. A considerable improvement on the hydrometer

Nicholson's

it capable of ascertaining the specific gravities both of solider-

and fluids. F is a hollow ball of copper attached to the

dish AA by a stem B, made of hardened steel. To the

lower extremity of the ball is affixed a kind of iron stirrup

FF; carrying another dish G of such a weight as to keep

the stem vertical when the instrument is afloat. The parts of the hydrometer are so adjusted, that when the lower dish G is empty, and the upper dish AA contains 1000 grains, it will sink in distilled water at the temperature of 60° Fahrenheit, so that the surface of the fluid may cut the stem DB at the point D. In order to measure the specific gravities of fluids, let the weight of the instrument, when loaded, be accurately ascertained. Then, this weight is equal to that of a quantity of distilled water at the temperature of 60°, having the same volume as that part of the instrument which is below the point D of the stem. If the hydrometer, therefore, is immersed to the point D in any other fluid of the same temperature, which may be done by increasing or diminishing the weights in the dish AA, the difference between this last weight and 1000 grains will express the difference between equal bulks of water and the other fluid. Now as the weight of the mass of water is equal to the weight of the instrument, which may be called W, the above-mentioned difference or D must be either added to or subtracted from W, (according as the weight in the dish AA was increased or diminished) in order to have the weight of an equal bulk of the fluid; then \( W \pm D \) will be to W as the specific gravity of the given fluid is to that of water. This ratio will be expressed with considerable accuracy, as the cylindrical stem of the instrument being no more than \( \frac{1}{30} \)th of an inch in diameter, will be elevated or depressed nearly an inch by the subtraction or addition of \( \frac{1}{30} \) of a grain, and will, therefore, easily point out any changes of weight, not less than \( \frac{1}{30} \) of a grain, or \( \frac{1}{30} \) of the whole, which will give the specific gravities to five places of figures. The solid bodies whose specific gravities are to be determined by this hydrometer, must not exceed 1000 grains in weight. For this purpose, immerse the instrument in distilled water, and load the upper dish AA till the surface of the water is on a level with the point D of the stem. Then, if the weights required to produce this equilibrium be exactly 1000 grains, the temperature of the water will be 60° of Fahrenheit; but if they be greater or less than 1000 grains, the water will be colder or warmer. After noting down the weight necessary for producing an equilibrium, unload the upper dish, and place on it the body whose specific gravity is required. Increase the weight in the upper dish, till the instrument sinks to the point D, and the difference between this new weight and the weight formerly noted down, will be the weight of the body in air. Place the body in the lower dish G, and add weights in the upper dish till the hydrometer again sinks to D. This weight will be the difference between 1000 grains and the weight of the body in water; and since the weight of the body in air, and its weight in water, are ascertained, its loss of weight will be known, and consequently its specific gravity (96).

Wilson's Beads.

122. In order to determine the strength of spirits with the greatest expedition, Professor Wilson of Glasgow employed a very simple method. His hydrometer consists of a number of glass beads, the specific gravities of each of which vary in a known ratio. When the strength of any spirit is to be tried, the glass beads, which are all numbered, are to be thrown into it. Some of those whose specific gravity exceeds that of the spirit will sink to the bottom, while others will swim on the top, or remain suspended in the fluid. That which neither sinks to the bottom nor swims on the surface, will indicate by its number Of Specific the specific gravity of the spirits (78). These beads have Gravities been greatly improved by Mrs Lovi, and are numbered in a variety of ways, either to show specific gravities or the strength of spirits.

Barometrical Areometer.

123. This name may be appropriately given to an instrument which is more useful for the purposes of illustration than of measurement. If two immiscible liquids are poured into a two-branched tube ABC, the one into the branch BC, and the other into the branch AC, till they balance each other, their specific gravities will be to one another inversely as the heights of each column. Thus, if we pour in mercury at A, and water at B, so that when the surface of the mercury is at D, that of the water is at E, we shall find that if the column of mercury DF is two inches, that of the water EG will be 27 inches, and their specific gravities will be as 27 to 2, or as 13\(\frac{1}{2}\) to 1. If we pour in at B linseed oil in place of water, the height EF will be 29 inches, and the specific gravity of the oil 0.931; because \( \frac{27}{2} = 13\frac{1}{2} : 0.931 \). By thus using mercury as the balancing column, the specific gravities of all fluids that do not mix with it, or act upon it, may be readily ascertained. The results thus obtained are not affected by the admission of the air at the open ends A and B, because the same weight of air presses upon the two balancing columns. But if we pour in mercury at A till the bent tube ACB contains above thirty inches of it, and close up the end A, and remove the air from above the mercury in AC, the column of mercury being no longer pressed down by the air in AC, will be pressed up to near the top of the tube AA by the pressure of the column of air in BC, and the instrument becomes a barometer, a column of air balancing a column of mercury. In this case, the tube BI becomes unnecessary, and the mercury may be enclosed in a glass ball at I, with an opening to admit the air.

Say's Stereometer.

124. Captain M. K. Say, of the French Engineers, invented a very ingenious instrument for measuring the specific gravity of liquid bodies, soft bodies, porous bodies, and powders, as well as that of solids; and he published a detailed description of it, illustrated with various figures, in the Annales de Chimie for 1797.

Many years afterwards, Professor Leslie brought forward the same instrument, under the name of a Conimeter, as a new invention, but without any variation of the principle. The following general description will give our readers an idea of the very beautiful principle on which this instrument is founded: Let AE be a glass tube, about three feet long, and open at both ends, the upper part AB being about 4-10ths, and the lower part BE about 2-10ths, of an inch in diameter. The upper edge of the tube is ground smooth, so that it can be shut air-tight by a piece of ground plate-glass, and the upper tube AB communicates with the lower one by a very narrow slit at B, which allows air to pass through it, but prevents sand or powder from passing.

The powder, sand for example, is put into the tube AB, and the lower tube BE is plunged into the open vessel F, containing mercury, till the mercury rises exactly within

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1 See also Nicholson's Journal, 4to, vol. i. p. 325.