igO PHILOSOPHICAL TRANSACTIONS. [aNNO 1715. 



have endeavoured to explain in this treatise. For it being the foundation of 

 the method of fluxions, that the fluxions of quantities are proportional to the 

 nascent increments of those quantities, in order to understand that method 

 thoroughly, I found it necessary to consider well the properties of increments 

 in general. And from those properties I saw it would be easy to draw a perfect 

 knowledge of the method of fluxions; for if in any case the increments are 

 supposed to vanish, and to become equal to nothing, their porportions become 

 immediately the same with the proportions of the fluxions. In this method I 

 consider quantities, as formed by a continual addition of parts of a finite mag- 

 nitude, and those parts I call the increments of the quantities they belong to, 

 because that by the addition of them the quantities are increased. These parts 

 being considered as formed in the same manner by a continual addition of other 

 parts, thence follows the consideration of second increments, and so on to 

 third, fourth, and other increments of a higher kind. For example, if ar stands 

 for any number in the series O, 1, 4, 10, 20, 35, &c. in which the numbers are 

 formed by a continual addition of the numbers in the series J, 3, 6, lo, 1 5, &c. 

 then the numbers in the latter series are called the increments of the numbers 

 in the foregoing series; thus, for example, if to the third number (4) in the 

 first series, I add the corresponding third number (6) in the second series, I 

 shall produce the next, that is the fourth number (10) in the first series; and 

 so of the rest. Any number in the first series being called .r, the corresponding 

 number, which is its increment, in the second series, I express by .r. And 

 these numbers .r being formed in the same manner by the numbers in the series 

 f , 2, 3, 4, 5, &c. I call these last numbers .r, they being the first increments of 

 the numbers a:, and the second increments of the numbers x; and so on. 

 Hence having given any series of numbers that are called by a general character 

 37, their increments are found by taking their difi^erences; thus in the present 

 example, the first increments f in the series 1, 3, 6, 10, 15, &c. are found by 

 taking the diflferences of the numbers x in the series 1, 4, 10, 20, 35, &c. and 

 the second increments x in the series 1, 2, 3, 4, 5, &c. are found in the like 

 manner, by taking the diff^erences of the numbers a.; and so of the third and 

 other increments. This method consists of two parts: one is concerned in 

 showing how to find the relations of the increments of several variable quan- 

 tities, having given the relation of the quantities themselves; and the other is 

 concerned in finding the relations of the integral quantities themselves freed 

 from the consideration of their increments, having given the relations of the 

 Increments; either simply, or being any how compounded with their integral 

 quantities. In the method of fluxions, quantities are not considered with their 

 parts, but with the velocities of the motions they are supposed to be formed 



