64 LIMITS AND FLUXIONS 



fluxion of j|/^, let x =y^. Take u~z^^ the u—x=z —y 

 y.zz-\-2y-\-yy^ or z^y : u—x=i : zz-\-zy-\-yy. li 

 now j/ and z approach continually until they coincide 

 with an intermediate ordinate, then z=y and the 

 chord through the extremities of the ordinatesi and 

 z will likewise coincide with the tangent. Therefore, 

 the ordinate is to the subtangent as i is to 3j/^. 

 Hence the proportion i : 3XF=>' : i', or i'=3>x^, the 

 fluxion required. The same argument is applied to 

 y^". In these demonstrations appeal is made to a 

 geometrie figure, and no attention is directed to the 

 ratio z—y \u—x for the difficult case when j = 5. 

 The author remarks that " though we commonly 

 say that . . . mjy'*''^ is the Fluxion ofy ; yet that 

 expression is not sufficiently accurate : Therefore, 

 the sense in which we desire to be understood is, 

 that I : 7ny"^~^ : : y : 7nyy'"'^, that is, unity is to 

 uiy"'''^, or j> is to myy'"~^^ as the fluxion or velocity 

 with which y is generated, is to the fluxion, or 

 contemporary velocity with which j/"' is generated, 

 and so for the rest " (p. 79). Thus, the emphasis. 

 is placed upon the ratios of velocities. 



Anonymous Trans lation ^ of Neiuton's 

 ' ' Method of Fluxio ns,'' 1737 



i^^a. Colson's translation from the Latin ol' 

 Newton's Method of Fluxions, published in 1736,, 

 was foUowed in 1737 by a second translation, which 



^ A Treatise of the Method of Fluxions and Infinite Series^ With its 

 Application to the Geonictry of Curve Lines. By Sir Isaac Newton, Kt, 

 Translatcdfrom the Latin Orioinal not yet published. Designed by the^ 

 Author for the Use of Lear neri. London, MDCCXXXVII. 



