under Gravity, suggested by a proposition of Galileo's. 229 



In this the two variables 6 </>, or a ft, are separated and 

 the discussion now turns on the values of the function 



/(0 7 ) = cosec6M2A0-(l + cos6O}, 

 or, reinstating «, 



/(a y) = cosec a {sin y(2 cos a— lj— (sin 2 y— sin 2 a)i}. 



When /[* y) +/(/3 y) is positive the two-chord path is the 

 quicker. 



Now it is easjr to verify that if y<45° the function y' is 

 positive for all values of the other angle. So the two chords 

 are quicker than the one provided the level to which the 

 velocity is due is below the centre of the circle, and this is 

 precisely the limitation imposed by Galileo. On the other 

 hand if y>60° then, for all values of a. and /3 which are less 

 than y,/is negative and the single-chord path is the quicker. 



What happens for values of y between 45° and 60° can be 

 seen by examining the graphs of the function/, drawn on 

 fig. 2 for y = 45°, 50°, 55°, 60° and a<y. The curve for 

 y = 45° touches the axis at the origin. For higher values of 

 y it sinks below the axis and then rises, crossing at a point 



Fijr. 2. 



given by sec a = 4 sin 2 y — 1. This zero of the function 

 moves to the right as y increases and coincides with the end- 

 point a = y when y = 60°. For still larger values of y, as has 

 already been said, the part of the curve which concerns us 

 lies wholly below the origin. 



