OF DEFLECTIVE FORCES. 149 



to the verse sine of an arc which increases 

 uniformly. 



For since the accelerating force, in the 

 direction of the curve, is always to the 

 force of gravity as AB to BC, or as BC f_ 

 to the constant quantity BD, it varies as 

 BC, or as its double, CE, the arc to be c|J 

 described, and CE being called s, the force 



dz ^ 



g -— must vary as s (208). If therefore any two arcs be 

 as 



supposed to be equally divided into an equal number of 

 evanescent spaces, the force will be every where as the 

 space to be described ; and it may be considered, for each 

 space, as equable, and the increments of the times, and 

 consequently the whole times, will be equal. Supposing 

 the generating circle to move uniformly, the velocity of 

 the describing point C will always be as CD (209), or 

 since AD : CD:: CD : BD, and CD = V(AD.BD) as 

 v^AD,- but the velocity of a body falling in DA, or 

 descending in FC, varies in the same ratio (232, 230, 

 286) ; therefore if the velocity at E be equal to that which 

 a body acquires by falling through GE, the describing 

 point C will always coincide with the place of a heavy 

 body descending in FCE ; and the velocity of the point 

 of contact D is half that of C at E (209), it would there- 

 fore describe a space equal to GE in the time that a body 

 would fall through GE, and will describe FG in a time 

 which is to that time as FG to GE, or as half the circum- 

 ference of a circle to its diameter, and this will be the 

 time of descent in a cycloidal arc. And since FC=i2DB 

 — ^2BC, FC is equal to the verse sine of the angle CBD, 

 to the radius 2BD : but the angle CAD increasing uni- 



