VOL. XXIX.] PHILOSOPHICAL TRANSACTIONS. IQl 



by; or, to speak more accurately, they are considered with the quantities of 

 the motions by which they are supposed to be generated; for the fluxions are 

 proportional to the velocities, only when the moving quantities, which produce 

 the flowing quantities considered, are equal. These quantities of motion, or 

 velocities, when the moving quantities are equal, are what Sir Isaac Newton 

 calls fluxions. As in the method of increments there are second, third, and 

 other increments; so in the method of fluxions there are second, third, and 

 other fluxions; the fluxions themselves being considered as quantities that are 

 formed by motion, the quantity of which motion is their fluxions. As the 

 method of increments consists of two parts: one being concerned in finding 

 the increments from the integrals given, and the other in finding the integrals, 

 having the increments given; so the method of fluxions consists of two parts: 

 the one showing how to find the fluxions, having the fluents given; and the 

 other showing how to find the fluents freed from fluxions, having given the 

 relations of the fluxions, whether compounded with their fluents or otherwise. 

 The principles of this method may all be drawn directly as a corollary from the 

 principles of the method of increments. For Sir Isaac Newton having demon- 

 strated, Phil. Nat. Princ. Math. sect. 1, and in the beginning of his treatise 

 De Quadratura Curvarum, that the fluxions of quantities are proportional to 

 their nascent or evanascent increments, if in any proposition relating to incre- 

 ments you make the increments to vanish, and to become equal to nothing, and 

 for their proportion put the fluxions, you will have a proposition that will be 

 true in the method of fluxions. This is but a corollary to Sir Isaac Newton's 

 demonstration of the fluxions being proportional to the nascent increments. 

 For this reason, to make the method of fluxions to be understood more 

 thoroughly, I thought it proper to treat of these two methods together, and I 

 have handled them promiscuously, as if they were but one method. Some 

 people, because that the fluxions are proportional to the nascent increments of 

 quantities, have thought that by the method of fluxions Sir Isaac Newton has 

 introduced into mathematics the consideration of infinitely small quantities; as 

 if there were any such thing as a real quantity infinitely small. But in this they 

 are mistaken, for Sir Isaac only considers the first or last ratios of quantities, 

 when they begin to be, or when they vanish, not after they have become 

 something, or just before they vanish; but in the very moment when they do 

 so. In this case, quantities are not considered as infinitely little; but they are 

 really nothing at the time that Sir Isaac takes the proportions of their fluxions; 

 and the truth of this method is demonstrated from the principles of the method 

 of increments, in the same manner as the ancients demonstrated their conclu- 

 sions in the method of exhaustions, by a deductio ad absurdum. 



