On prime and ultimate Ratios, ISO 



The first method which the ancients made use of for this 

 purpose was the method of exhaustions : an example of the 

 use of this method is given by Euclid in the second proposi- 

 tion of his twelfth book, where he compares the circle and 

 square, and proves that any two circles are to each other 

 as the squares on their diameters : the same method was 

 also made use of by Archimedes in determining the quadra- 

 ture of the parabola. The argument here made use of is 

 called reducth ad absurdum, which, though strictly logical, 

 is often tedious, because every proposition must be divided 

 into two cases, in one of which it must be shown that the 

 former of the two quantities to be compared is not greater 

 than the latter ; in the other, that it is not less, it was 

 with a view of shortening this mode of reasoning, that Ca- 

 valeriiis invented the method of indivisibles : in this method, 

 every line is supposed to be made up of a number of other 

 lines whose lengths are indefinitely short, and every curvi- 

 linear figure is considered as a polygon of an indefinite 

 number of sides : — these principles were in many cases 

 extremely easy and convenient, and produced true results; 

 they, however, often led their followers into perplexities, 

 and sometimes into error. 



It was to avoid the tediousness of the method of ex- 

 haustions, and the errors in the method of indivisibles, that 

 Newton invented his method of prime and ultimate ratios ; 

 the principles of which he laid down in the first lemma of 

 the Principia, as observed above. Several eminent mathema- 

 ticians have endeavoured to demonstrate Newton's lemma: 

 it however certainly admits of no direct proof; it is itself a 

 definition, and requires only to be illustrated, or explained. 

 By introducing the doctrine of motion into geometry, 

 much has been effected. Newton employed his method 

 of prime and ultimate ratios to the quadrature of all kinds 

 of spaces, by supposing one or more of the sides of the 

 figure to be in motion, and to generate those figures by the 

 motion of their extreme points. The application to right- 

 lined figures was natural and easy; and to apply it to a 

 square, we will suppose that the square is generated by the 

 motion of two right lines perpendicular to each other, and 

 that move parallel to two other right lines placed at right 

 angles. 



Let x denote the length of each side at any given posi- 

 tion of those lines, and let i be the increase, in the length 

 of each side, caused by the motion of the two moveable 

 sides; then, x+x will be the length of each side so in- 

 creased, 



