NEWTON S PBINCIPIA. 



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line whose progressive motion generated the curvilinear 

 area, the sum of these infinitely narrow areas will differ 

 from the area of the curve by a difference less than any 

 assignable quantity, nor will each differ from a rectangle ; 

 in other words, the ratio of the nascent curve line and 

 nascent curvilinear area will be that of equality with the 

 small lines and small rectangles, and the ultimate ratio of 

 the sums of the lines and rectangles to the whole curve 

 line and curvilinear area, respectively, will be that of 

 equality : Or to put it otherwise, if the axis of the curve 

 be divided into parts P P, &c., and the area into spaces 

 PMEP, &c., by ordinates PM, PK, &c., and the num 

 ber of these spaces be increased, and their breadth PP 

 be diminished indefinitely, which is the operation of the 



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generative motion of PM, the size of each of the small 

 spaces MNRO (by which the curvilinear areas differ from 

 the rectangles) diminishes indefinitely, and the ultimate 

 ratio of all the curve areas PMEP, and all the rectangles 

 PNRP, becomes that of equality, and therefore the sum 

 of evanescent differences N M O R, NROR, &c., whereby 

 the whole curvilinear area differs from the whole amount 

 of the rectangles P NRP, becomes less than any assignable 

 quantity, or the curvilinear area coincides with the sum 

 of the rectangles. And so of the sum of all the diagonals 

 MR, RR, &c., which becomes the curve line MR A. 

 Hence we infer that the amount of these small spaces 



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