20 



NEWTON S PRINCIPIA. 



generated by the uniform progression or motion of other 

 quantities, and examining the relations which the smallest 

 conceivable spaces thus generated by this motion bear to 

 one another, and to the spaces generated at the moment of 

 their inception, or when they are nascent, which is termed 

 their prime ratio, and at the moment of their vanishing, 

 or when they are evanescent, which is termed their ultimate 

 ratio. Thus a point moving along in a straightforward 

 direction generates a straight line ; a line moving parallel 

 to itself, or two. lines moving at right angles to one another, 

 generate a rectangle : one line moving, while a point in it 

 moves along it so that its progress on the moving line 

 always bears a given ratio to the progress the line has 

 made (m . A P = P M), describes a triangle ; the same 

 motion, if the progress of the point bears a variable 



relation to that of the line (x. AP = PM ; x.xAP 

 being some function of A P), describes a curve line and 

 curvilinear area ; and so of solids, which are generated by 

 the motion of planes. 



It follows from this mode of generation that if the length 

 of any curve line be divided into an infinite number of 

 lines, the sum of these will not differ from the curve 

 line by any assignable quantity, nor will each differ from 

 a straight line ; and if its area be divided into an infinite 

 number of smaller areas by lines drawn parallel to the 



