**HYDROMECHANICS** (ὑδρομηχανικά), the science of the
mechanics of water and fluids in general, including *hydrostatics*
or the mathematical theory of fluids in equilibrium, and *hydromechanics*,
the theory of fluids in motion. The practical application
of hydromechanics forms the province of hydraulics (*q.v.*).

*Historical.*—The fundamental principles of hydrostatics were first
given by Archimedes in his work Περὶ τῶν ὀχουμένων, or *De iis quae*
*vehuntur in humido*, about 250 B.C., and were afterwards applied
to experiments by Marino Ghetaldi (1566–1627) in his *Promotus*
*Archimedes* (1603). 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.

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, and about 120 B.C. the fountain
of compression, the siphon, and the forcing-pump were invented by
Ctesibius and Hero. The siphon is a simple instrument; but the
forcing-pump is a complicated invention, which could scarcely
have been expected in the infancy of hydraulics. It was probably
suggested to Ctesibius by the *Egyptian Wheel* or *Noria*, which was
common at that time, and which was a kind of chain pump, consisting
of a number of earthen pots carried round by a wheel. In
some of these machines the pots have a valve in the bottom which
enables them to descend without much resistance, and diminishes
greatly the load upon the wheel; and, if we suppose that this valve
was introduced so early as the time of Ctesibius, it is not difficult
to perceive how such a machine might have led to the invention of
the forcing-pump.

Notwithstanding these inventions of the Alexandrian school, its
attention does not seem to have been directed to the motion of
fluids; and 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. In his work *De aquaeductibus*
*urbis Romae commentarius*, he considers the methods
which were at that time employed for ascertaining the quantity of
water discharged from ajutages, and the mode of distributing the
waters of an aqueduct or a fountain. He remarked that the flow of
water from an orifice depends 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 law of the velocities of running water as
depending upon the depth of the orifice, the want of precision which
appears in his results is not surprising.

Benedetto Castelli (1577–1644), and Evangelista Torricelli (1608–1647),
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 satisfactorily
explained 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 where the water rushed through a small ajutage 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 the proposition that the velocities of liquids are as the
square root of the head, apart 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 projectorum*, and it was confirmed
by the experiments of Raffaello Magiotti on the quantities
of water discharged from different ajutages under different pressures
(1648).

In the hands of Blaise Pascal (1623–1662) hydrostatics assumed
the dignity of a science, and in a treatise on the equilibrium of
liquids (*Sur l’équilibre des liqueurs*), found among his manuscripts
after his death and published in 1663, the laws of the equilibrium
of liquids were demonstrated in the most simple manner, and amply
confirmed by experiments.

The theorem of Torricelli was employed by many succeeding
writers, but particularly by Edmé Mariotte (1620–1684), whose
*Traité du mouvement des eaux*, 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 committed considerable mistakes.
Others he treated very superficially, and in none of his experiments
apparently did he attend to the diminution of efflux arising from the
contraction of the liquid 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 through friction. His contemporary
Domenico Guglielmini (1655–1710), who was inspector of
the rivers and canals at Bologna, 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 supposed 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.

The effects of friction and viscosity in diminishing the velocity of
running water were noticed in the *Principia* of Sir Isaac Newton,
who threw much light upon several branches of hydromechanics.
At a time when the Cartesian system of vortices universally prevailed,
he found it necessary to investigate that hypothesis, and in
the course of his investigations he showed that the velocity of any
stratum of the vortex is an arithmetical mean between the velocities
of the strata which enclose it; and from this it evidently follows
that the velocity of a filament of water moving in a pipe is an arithmetical
mean between the velocities of the filaments which surround
it. Taking advantage of these results, Henri Pitot (1695–1771)
afterwards showed 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 called the “cataract,”
being an hyperboloid generated by the revolution of an hyperbola
of the fifth degree around the axis of the cylinder which should pass
through the orifice, and the second the remainder of the water in
the cylindrical vessel. He considered the horizontal strata of this
hyperboloid as always in motion, while the remainder of the water
was in a state of rest, and imagined that there was a kind of cataract
in the middle of the fluid. When the results of this theory were
compared with the quantity of water actually discharged, Newton
concluded that the velocity with which the water issued from the
orifice was equal to that which a falling body would receive by
descending through half the height of water in the reservoir. This
conclusion, however, is absolutely irreconcilable with the known
fact that jets of water rise nearly to the same height as their reservoirs,
and Newton seems to have been aware of this objection. Accordingly,
in the second edition of his *Principia*, which appeared in 1713,
he reconsidered his theory. He had discovered a contraction in the
vein of fluid (*vena contracta*) which issued from the orifice, and found
that, at the distance of about a diameter of the aperture, the section
of the vein was contracted in the subduplicate ratio of two to one.
He regarded, 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, though still open to
serious objections. Newton was also the first to investigate the
difficult subject of the motion of waves (*q.v.*).

In 1738 Daniel Bernoulli (1700–1782) published his *Hydrodynamica*
*seu de viribus et motibus fluidorum commentarii*. His theory of
the motion of fluids, the germ of which was first published in his
memoir entitled *Theoria nova de motu aquarum per canales quocunque*
*fluentes*, communicated to the Academy of St Petersburg as
early as 1726, was founded on two suppositions, which appeared to
him conformable to experience. He supposed that the surface of
the fluid, contained in a vessel which is emptying itself by an orifice,
remains always horizontal; and, if the fluid mass is conceived to be
divided into an infinite number of horizontal strata of the same
bulk, that these strata remain contiguous to each other, and that
all their points descend vertically, with velocities inversely proportional
to their breadth, or to the horizontal sections of the
reservoir. In order to determine the motion of each stratum, he
employed the principle of the *conservatio virium vivarum*, and
obtained very elegant solutions. But in the absence of a general
demonstration of that principle, his results did not command the
confidence which they would otherwise have deserved, and it
became desirable to have a theory more certain, and depending solely
on the fundamental laws of mechanics. Colin Maclaurin (1698–1746)
and John Bernoulli (1667–1748), who were of this opinion,
resolved the problem by more direct methods, the one in his *Fluxions*,
published in 1742, and the other in his *Hydraulica nunc primum*
*detecta, et demonstrata directe ex fundamentis pure mechanicis*, which
forms the fourth volume of his works. The method employed by
Maclaurin has been thought not sufficiently rigorous; and that of
John Bernoulli is, in the opinion of Lagrange, defective in clearness
and precision. The theory of Daniel Bernoulli was opposed also by
Jean le Rond d’Alembert. When generalizing the theory of pendulums
of Jacob Bernoulli (1654–1705) 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