Waxeuw.— Fallacies in the T. heory 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 tnextensible 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 along 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 
@ 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 
® 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 :— 
Tarorem.—If two straight lines cut one another within a circle, the 
Tectangle 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. 
Tatorem.—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 line which cuts the circle and the part of it without 
the anh shall be equal to the square on the line which touches it. 
