MECHANICAL PRINCIPLES OF FLUIDS. 69 



When a fluid spouts from an orifice in a vessel, 

 Castelli saw that the velocity of efflux depends on 

 the depth of the orifice below the surface : but he 

 erroneously judged the velocity to be exactly pro- 

 portional to the depth. Torricelli found that the 

 fluid, under the inevitable causes of defect which 

 occur in the experiment, would spout nearly to the 

 height of the surface : he therefore inferred, that 

 the full velocity is that which a body would acquire 

 in falling through the depth ; and that it is conse- 

 quently proportional to the square root of the 

 depth. This, however, he stated only as a result 

 of experience, or law of phenomena, at the end of 

 of his treatise, De Motu Naturaliter Accelerate, 

 printed in 1643. 



Newton treated the subject theoretically in the 

 Principia (1687) ; but we must allow, as Lagrange 

 says, that this is the least satisfactory passage of 

 that great work. Newton, having made his experi- 

 ments in another manner than Torricelli, namely, 

 by measuring the quantity of the efflux instead of 

 its velocity, found a result inconsistent with that 

 of Torricelli. The velocity inferred from the quan- 

 tity discharged, was only that due to half the depth 

 of the fluid. 



In the first edition of the Principia*, Newton 

 gave a train of reasoning by which he theoretically 

 demonstrated his own result, going upon the prin- 

 ciple, that the momentum of the issuing fluid is 



8 B. ii. Prop, xxxvii. 



