77 



with the same vertex, and with a smaller parameter, b, is 

 drawn nearer the straight line, its sectors are - (x + 3C). 



Now the times in the first parabola, or the areas, at 

 any two points referred to the abscissae x and z, being 



(x -f 3 C), and ^~ (z + 3 C), the times or areas 



in the second parabola will be ~ (x -f 3 C), and ^ 



(z + 3 C), respectively ; and therefore it is evident that 

 the areas at the distances x and z, in the one curve are 

 in the same proportion to one another with the areas in 

 the other curve at those distances. If the parameter be 

 continually diminished of the second curve, until that curve 

 coincides with the axis, the same proportion holds ; and 

 the times, therefore, in falling through the axis, will be as 

 the areas of the first curve, corresponding to the points of 

 that axis. And so it may be shown in the ellipse and 

 hyperbola. 



Hence it follows, that in the case of the parabola, the 

 velocity of the falling body in any given point is equal to 

 that with which the body would, moving uniformly, de 

 scribe a circle having for its centre, the centre to which 

 the body is falling, and for its diameter the distance of 

 the given point from that centre. In the circle, the ve 

 locity at the given point is to the velocity in the circle 

 described from the centre, with the distance of the given 

 point for the radius, as the square root of the distance fallen 

 through to that of the whole distance of the point where the 

 fall begins. Thus let d be the distance of the given point 

 to which the body has fallen, D the distance of the point 

 at which it began to fall ; the velocity in the case of a para 

 bola is equal to that of the body moving in a circle, whose 



