A Theory of Fluxions. 



341 



Fig. 8. 



Fig. 7. 



AT. X' If C 



C"# 



2. From the before mentioned parts, select the pair NBCT, 

 MECX, the part NBCT being removed to prevent the figure 

 from being too much crowded. Draw the line CM' from C, 

 to the middle of the arc ME, and divide NBCT and MECX 

 by drawing the ordinates NT, M'X', from the points of in- 

 tersection of CM' with the curves. Let the division N'BCT 

 be represented by a', and M'ECX' by ri. Proceeding in the 

 same manner, subdivide N'BCT' bv the line N"T", and 

 M'ECX' by the line M"X",and let N"BCT"=a", and M'ECX" 

 =n", and so on. By the foregoing lemma, these divisions 

 form a series of proportional quantities, that is, 



a \nl\d iriy.a" : n"\\a!" : n"'::,&c. 



3. Let the parallelograms NnCT, N'n'CT, NV'CT", &c, 

 be represented by A, A', A", &c, and the parallelograms 

 MroCX, MWCX, M'W'CX;', &c, by N, N', N", &c., and they 

 will form the following series of proportional quantities, 



A : N: :A' : N': :A" : N": :A"' : N'": :, &c. 



4. As we proceed in these two series, the differences NBrc, 

 N'Bn', N"Bn", &c, and MEw, M'Em', M"Em", &c, between 

 the curvilineal spaces and the rectilineal ones continually 

 lessen, and the curvilineal expressions of the antecedents 

 approach towards the rectangular expressions of them, and 

 both converge towards the ordinate BC, as their limit ; and 

 in like manner the consequents converge towards the ordi- 

 nate EC, as their limit. Here, although we cannot arrive at 

 the nascent and evanescent terms themselves, yet we can 

 ascertain the relation between the prime and ultimate ratios ; 

 for suppose the divisions are continued, until these differen- 



