138 Transactions.— Miscellaneous. 
(Euclid, Third Book, Prop. 36.) The particular case taken is where the 
point outside the circle is the extremity of the tangent, and a diameter pro- 
duced to meet the point. Mr. Todhunter deduces the formula apparently 
from the parallelogram of velocities. 
The method of deriving the formula for the measure of centripetal force 
is very clearly and precisely given by Mr. (I think now Professor) Goodeve 
in his ‘* Principles of Mechanics.” He takes the thirty-sixth Proposition of 
the Third Book of Euclid, where it is proved that the square of the tangent 
is equal to the rectangle contained by the diameter produced to meet the 
tangent and the part produced. Accordingly, in the annexed figure, the 
square on the line AP will be equal to the rectangle DP, PB. In the 
work referred to, the explanation given is nearly as 
follows :—The rectangle DP, PB is equal to DB, 
PB together with the square on PB. When the 
W angle is made very small, the square on PB may be 
I neglected, and then the square on the tangent AP 
is equal to the rectangle DB, BP. Now DB is the 
diameter of the circle; then (AP)*—2r ° PB. (1.) 
In the limit this is mathematically exact. Let the 
body revolving with uniform motion be supposed passing through the point 
B by the end of the time ¢. If no force had deflected the body it would have 
pursued a straight course along the tangent, and would have reached the 
point P at the end of the time ¢, or to speak more exactly it would very 
nearly have reached that point, because AP the tangent is greater than 
AB the arc. When the arc, however, is extremely small, the difference | 
between the arc of the angle and its tangent is inappreciable—in the limit 
they coincide. The body was deflected from its course the length PB. It 
is pulled through the distance PB, that is it falls through that distance. 
From this geometrical construction we can now derive an algebraical equa- 
tion. The line AP is equal to tv, and the distance fallen through, namely 
PB, is equal to 4 ft. Here v of course is the velocity of the body, and f 
stands as usual for the acceleration. The time that would have been taken 
by the body to move from A to P is, of course, the same that it took to fall 
from P to B, that is, if the angle represented by the are, or tangent, be very 
small. In the figure the angle is very much exaggerated for the sake of 
clearness, but the arc taken should not be greater than a degree when the 
error will be very small. Bearing these considerations in mind, we o 
proceed to evolve from the equation we have obtained the measure of centr!- 
petal force. The equation before given may be conveniently put distinctly. 
hus:— 
AP xiv (2) 
PB=4 ft? (8) 
