MECHANICAL PRINCIPLES OF FLUIDS. 349 



surface : but lie erroneously judged the velocity to be exactly propor- 

 tional to the depth. Torricelli found that the fluid, under the inevit- 

 able 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 consequently 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 his treatise, De Motu Naturaliter 

 Accelerate, printed in 1643. 



Newton treated the subject theoretically in the Pnncipia (1687); 

 but we must allow, as Lag-range says, that this is the least satisfactory 

 passage of that great work. Newton, having made his experiments 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 quantity discharged, was 

 only that due to halfihQ depth of the fluid. 



In the first edition of the Principia* Newton gave a train of reason- 

 ing by which he theoretically demonstrated his own result, going 

 upon the principle, that the momentum of the issuing fluid is equal 

 to the momentum which the column vertically over the orifice would 

 generate by its gravity. But Torricelli's experiments, which had 

 given the velocity due to the whole depth, were confirmed on repeti- 

 tion : how was this discrepancy to be explained ? 



Newton explained the discrepancy by observing the contraction 

 which the jet, or vein of water, undergoes, just after it leaves the 

 orifice, and which he called the vena contracta. At the orifice, the 

 velocity is that due to half the height ; at the vena contracta it is 

 that due to the whole height. The former velocity regulates the 

 quantity of the discharge; the latter, the path of the jet. 



This explanation was an important step in the subject ; but it made 

 Newton's original proof appear very defective, to say the least. In 

 the second edition of the Principia (1714), Newton attacked the 

 problem in a manner altogether different from his former investigation. 

 He there assumed, that when a round vessel, containing fluid, has a 

 hole in its bottom, the descending fluid may be conceived to be a 

 conoidal mass, which has its base at the surface of the fluid, and its 

 narrow end at the orifice. This portion of the fluid he calls the cat- 

 aract ; and supposes that while this part descends, the surrounding 



3 B. ii. Prop, xxxvii. 



