28 NEWTON S PRINCIPIA. 



dx, which being put = gives us # = &amp;gt;; or, at the 

 extremity of the line b, the curve approaches the nearest ; 

 and that whatever be its parameter; for a has vanished 

 from the equation. 



Again, we have seen that the ultimate ratio of the 

 sum of all the rectangles M P x P Q, contained by the 

 ordinates and the increments of the abscissa to the curve s 

 area A P M is that of equality ; or, in other words, that 

 the differential of a curvilinear area being the rectangle con 

 tained by the ordinate and the differential of the abscissa, 

 or y d x, the integral of this, or the sum of all those small 

 rectangles, is equal to the area. In this expression, then, 

 let y be inserted in terms of x, and the integral gives the 

 area. Thus in the parabola y= Va~x; therefore d x Va x 

 is the differential of the area, and its integral, or which is 



P 2y*dy . 2 y 3 2 y 2 

 the same thing, the integral of - -, is - x , or. x 



CL O CL o CL 



2 



xy, that is, - xy, or two- thirds of the rectangle of the 

 o 



co ordinates ; as we also know from conic sections. 



Next, we have seen that the ratio of the infinitely small 

 rectilinear sides into which a curve line may be divided 

 (each of those small lines being the hypothenuse of a 

 right-angled triangle, the sides of which are the differentials 

 N T, M N of the co-ordinates), to the infinitely small 

 portions of the curve itself is that of equality ; therefore 

 the differential of the curve is equal to the square root of 

 the sum of the squares of the differentials of the ordinate 

 and abscissa, and that differential is equal to 



Hence in the circle, an arc whose cosine is x and radius 



r is equal to the integral of , ----- - . And an arc whose 



V r 2 x 2 



cosine is rx, is equal to the integral of - 



