Wakelin. — Fallacies in the Theory of Circular Motion. 137 



a blow is of the nature of an infinite force, while it might be said that the 

 centripetal force is infinitesimal, it being the force of a pressure. In such 

 statements as these, however, the phrase " of the nature of an infinite 

 force," is itself a vague and indefinite expression. 



The clearest objection that can be raised against the common measure 

 of centripetal force, is that which can be urged against the assertion that a 

 mass revolving in a circle with uniform velocity, is every instant trying to 

 fly off at a tangent to its orbit. The words "instant" and "tangent" 

 here have their mathematical meanings. If the string holding the revolving 

 mass in its circular orbit is inextensible in a mathematical sense, then it is 

 difficult to conceive how the stone can possibly be doing otherwise than 

 trying to go off at a tangent to its orbit. But this will only make it less 

 necessary for the centripetal force to be great in amount, even when the 

 velocity is considerable. If the stone is every " instant'' trying to fly off at 

 a "tangent" in a strictly mathematical sense, then every "instant" the 

 stone is going in a direction at right-angles to the string ; how then can it 

 possibly exert a pull alorig the string ? Of course practically no string is 

 inextensible, but theoretically, the more the string is made inextensible, the 

 less should be the force necessary to retain the stone in its orbit. And yet 

 for all that, some force is necessary, for how could a stone be deflected from 

 a straight line unless a force acted upon it. Let a ball strike a smooth 

 surface very obliquely, in a direction almost parallel with the surface, and 

 it will be deflected from its straight course ; but how small relatively to that 

 of the striking ball would be the force that deflected the ball in the least 

 degree only out of its course. Is not this effect similar to the effect of a 

 centripetal force ? If so, can the force possibly be so great as shown in a 

 previous paragraph it would be if the formula is correct ? 



Perhaps the foregoing reasons may be considered a sufficient cause for 

 a reconsideration of the formula giving the measure of centripetal force. 

 Let us examine with great care, and step by step, the process by which 

 this formula has been obtained. A large number of treatises on astronomy 

 and mechanics, including the best and most commonly used, deduce the 

 formula from one of the two following propositions : — 



Theoeem. — If two straight hues cut one another within a circle, the 

 rectangle contained by the segments of one of them, shall be equal to the 

 rectangle contained by the segments of the other. {Euclid, Third Book, 

 Prop. 35.) The case is taken where the diameter bisects a chord. 



Theorem. — If from any point without a circle two straight lines be 

 drawn, one of which cuts the circle, and the other touches it, the rectangle 

 contained by the whole Ime which cuts the circle and the part of it without 

 the circle, shall be equal to the square on the line which touches it. 



11 



