When a vessel is filled with a homogeneous fluid, and it is in equilibrium, all the particles of the fluid are pressed equally in all directions. This law was known to Archimedes, and its knowledge enabled him to detect the fraud committed by the gold smith upon Hiero, King of Syracuse. The first regular work upon Hisdrodynamics was written by Sextus Julius Frentinus, inspector of the public fountains at Rome under the Emperors Nerva and Trajan. He laid down the law, that water which flows in a given time, from a given orifice, does not depend _merely_ upon the magnitude of the orifice, but upon the _head_ or height of the fluid in the vessel. From that period until the 17th century none of the principles upon which this cause depends, were much studied, nor the doctrine of fluids much known. At length Gallileo the astronomer, by his discovery of the uniform acceleration of gravity, paved the way for a rapid improvement in hydrodynamics. Gallileo was acquainted with the fact that water could not be made to rise more than a certain height in a common pump; but he was entirely unacquainted with the reason. His pupil, Torricelli, and his friend, Viviani, discovered that it was owing to the pressure of the external air, and thus the problem was solved. Mariotte, who introduced experimental philosophy into France, was the first who announced that fluids suffer a retardation from the friction of their particles against the sides of tubes; and he shewed that this was the case even though the tubes were made of the _smoothest glass_. From his works, which were published after his death, in 1684, it appears that though he was thus acquainted with the principle upon which it is explained, he was unacquainted with the _vena contracta_. About that time this subject began to be much more studied in Italy. Dominic Guglielmini, a celebrated engineer, in 1697, published a very learned work upon the friction and resistance of fluids; and from that period to this the learned of all nations have admitted, that this resistance and retardation of fluids, owing to their friction, did take place in a moving fluid. This work, as connected with the motion of rivers and water in open canals, is one of deep interest in natural philosophy; and it is one, which in this age of improvements, should not be neglected in this country. Sir Isaac Newton, whose capacious mind grasped at every kind of knowledge, struggled hard to detect the reason of this resistance. In his 2nd book of his "Principia," propositions 51, 52 and 53, he lays down certain hypotheses, from which it results, that the filaments (as he calls them,) of a fluid, in a pipe, will be kept back by their adhesion to the sides of the tube, and that the next filaments will be kept back, though in a less degree, by their adhesion to the first filaments, and so on, until the velocity of the fluid will be greatest at the centre. Now if we apply this principle to the discharge of a fluid through a plain aperture,