under Gravity, suggested by a proposition of Galileo's. 239 



through the distance between the points, The ordinates of 

 the curves show the amounts by which the corresponding 

 times exceed this. 



Fur. 11. 



The curve marked "line" gives the motion along the 

 straight line of given length at different slopes, in accordance 

 with the value of cosec^ 0. That marked " circle " gives the 

 times along what Galileo supposed to be the brachistochrone, 

 viz. a circular arc with horizontal tangent at the lowest point. 



The " two-line " curve corresponds to the quickest path 

 along two straight lines, as discussed in § 2. 



The curve marked " cycloid '' shows the true minimum 

 time along a cycloid with vertical cusp at the upper point. 

 This is calculated by the formula i|r sini/3cosec^, where yjr is 

 connected with the slope /3 by the relation 



tan ft = sin 2 \^/(-^r — sin ^r cos ^r). 



Of course it is easiest to plot the graph by beginning with 

 assumed values for yjr. 



The graph for the "circular arc of quickest descent''' is 

 got by picking out the minima of the different curves on 

 tig. 10. When this is done it is found that the resultant 

 curve lies so close to the " cycloid " curve as to make it 

 impossible to separate them on the scale of our figure. This 

 is an unexpected result. For ft = the circle time is about 

 1*78, the cycloid time is s/ 'it = 1 , 11, and the curves at other 

 points are closer than this. 



It will be noticed that Galileo's curve approaches very 

 closely to the true minimum at /3 = 37^ c or thereabouts, the 

 difference in the times amounts to about | per cent. 



