of Natural Philosophy. 137 



forces are not equal. But the demonstration is such as 

 would prove them no less equal, in all cases whatever. — 

 We must confess ourselves at a loss to assign any consistent 

 meaning to the term " centrifugal force," ia relation to or- 

 bits not circular. Is this force measured by the distance by 

 which a revolving body would be more remote from the 

 centre, in a given small time, if the centripetal force were 

 suspended, than it actually is while the centripetal force 

 acts .'' If so, the centripetal and centrifugal forces are al- 

 ways equal, for the same point of an orbit, whatever be its 

 figure. Or is it measured by the absolute increase of dis- 

 tance from the centre, which would take place in a given 

 small time, if the body were abandoned to its projectile 

 force ? If so, in passing from the higher to the lower ap- 

 sis of an excentric orbit, the centrifugal force is a negative 

 quantity. 



Lemma 4. The inference concerning the equality of the 

 arc to the chord and tangent in their vanishing state, is in- 

 conclusive when the tangent is less than the arc, as it will be 

 in certain positions of the subtense. The demonstration 

 may be rendered complete in the following manner : — AB ; 

 AD4-DB : : AC : AB + BC. But AB + BC is ultimately 

 equal to AC ; hence AD + RB is ultimately equal to ABj 

 and much more is the arc AB equal to the chord AB, 



The first Part of Book III. which treats of Hydrostatics^ 

 presents us with several instances of explicit or implied er- 

 ror ; particularly prop. 2, prop. 13, Schol. and propositions 

 24, 26, 31, and S6. But we have no room to dwell upon 

 them ; and shall therefore pass directly to the second Part, 

 which is devoted to Pneumatics. 



Prop. 51. The force with which wind strikes a sail of 

 given dimensions, was stated in former editions as varying 

 in the duplicate ratio of the cosine of the angle of incidence. 

 In the last edition, the term sine is substituted for cosine. 

 The phrase " angle of incidence" was before used in the same 

 sense as in optics : it is now employed in the ordinary sense 

 of mechanics. But this correction goes but little way to- 

 wards freeing the proposition from exception. If the sail 

 be supposed confined to move in the same direction witli 

 the wind, which the demonstration seems to imply, a reso- 

 lution of the force on a third independent account was ne- 

 cessary, which would have reduced that part of the force 



Vol. Ill No. 1. 18 



