408 D'ALEMBERT. 



of the author's preference of this over the common demon- 

 stration, is not at all satisfactory. His proof consists 

 in supposing the body to move on a plane sliding in two 

 grooves parallel to one side of the parallelogram, and at 

 the same time carried along in the direction of the other 

 side. This is not one whit more strict and rigorous than 

 the ordinary supposition of the body moving along a 

 ruler parallel to one side, while the ruler at the same 

 time moves along a line parallel to the other side. Indeed 

 I should rather prefer this demonstration to D'Alembert's. 

 The 'Traite de Dynarnique' appeared in 1743, and in 

 the following year its fundamental principle was applied 

 by the author to the important and difficult subject of 

 the equilibrium, and motion of fluids, the portion of the 

 ' Principia 7 which its illustrious author had left in its least 

 perfect state. Pressed by the difficulty, of the inquiry 

 which is one of the most important in Hydrodynamics, the 

 motion of a fluid through an orifice in a given vessel, and 

 despairing of the data affording the means of a strict 

 and direct solution, Newton had recourse to assump- 

 tions marked by the most refined ingenuity, but 

 admitted to be gratuitous and to be unauthorized by the 

 facts. The celebrated Cataract is of this description. 

 He supposes ('Principia/ lib. ii. prop. 36,) that a body of 

 ice shaped like the vessel, comes in contact with the 

 upper surface of the liquid and melts immediately on 

 touching it, so as to keep the level of the fluid always the 

 same, and that a cataract is thus formed, of which the 

 upper surface is that of the fluid, and the lower that of 

 the orifice. His first investigation assumed the issuing 

 column to be cylindrical, but he afterwards found that 

 the lateral pressure and motion gave it the form of a 



