22 NEWTON S PKINCIPIA. 



or quantities N M O R, formed by multiplying together 

 two evanescent quantities, is as nothing in comparison with 

 the rectangles P M O P formed by only one evanescent 

 quantity multiplied into a finite quantity, and may be 

 neglected in any equation that expresses the relations of 

 those rectangles with each other. But if some other 

 quantities be found which are, in comparison with these 

 small ones, themselves infinitely small, the areas formed by 

 multiplying this second set of small quantities may be 

 rejected in any equation expressing the relations of those 

 first small quantities. 



Thus we have the origin and constitution of quantities 

 which in the Newtonian scheme are called fluxions of 

 different orders, because conceived to express the manner 

 of the generation of quantities by the motion of others, 

 and in Leibnitz s language are called infinitesimals or 

 differences, because conceived to express the constant addi 

 tion of one indefinitely small quantity to another. Ob 

 taining the fluxions, or the differences, from the quantity 

 generated by the motion or by the addition, is called the 

 direct method; obtaining the quantity generated from the 

 fluxions, or finding the sum of all the differences, is called 

 the indirect method. The one theory calls the direct 

 method that of finding fluxions, the indirect that of finding 

 fluents ; the other theory calls the former differentiation, 

 or finding differentials, the latter integration, or finding 

 integrals. The two systems, therefore, in no one respect 

 whatever differ except in their origin and language; 

 their rules, principles, applications, and results, are the 

 same. 



A different symbol has been used in the two systems ; 

 Newton expressing a fluxion by a point or dot, and the 

 fluxion of that fluxion, or a second fluxion, by two dots, 

 and so on. Leibnitz prefixes the letter d, and its powers 



