66 KANSAS Academy of science. 



But without discussing the merits or the principles of either of these methods, it is 

 proposed in this memoir to furnish a method that is rigorously exact, and does noi 

 depend upon any suppositions or theories, fictitious or otherwise. Sir Isaac Newton 

 based his Calculus, or system of fluxions, as he called it, upon the doctrine of limits j 

 but by reference to his Principia, Book II, Section II, Lemma II, it may be seen 

 that there was in his mind a better and a rigorous method by which to obtain th& 

 desired results. 



In order to illustrate the superiority of the Newtonian method over all others, 

 take for example the proposition that the differential of the product of two 

 variables is the differential of the first into the second, plus the differential of the 

 second into the first. 



First, by the infinitesimal method: 



M G E Let AB--=.r, and BF ^y. Let-— BC — dx, and FG 



or DE^ dy. Then the area of ABFK will be repre- 

 sented by xy. Now, we shall represent the original 

 value of ABFK by the equation ^t^^xy. 



By taking on increments the consecutive value of 

 the function becomes %i + du = xy -J- ydx + xdy -(- dxdy^ 

 Subtracting the first equation from the second, the re- 

 sult becomes du = ydx -\- xdy -\- dxdy. It is easily seen 

 that ydx and xdy are infinitesimals of the first order, but dxdy, being the product of 

 two infinitesimals, each of the first order, constitutes an infinitesimal of the second 

 order, and therefore must be dropped. But dxdy is the area of FDEG, and how- 

 ever small dx and dy may be, their product does represent some value. The method 

 is very pretty, and reaches conclusions with less labor than by the doctrine of 

 limits. Let us now examine Newton's Lemma II. It is as follows: "The moment 

 of any genitum is equal to the moments of each of the generating sides drawn into 

 the indices of the powers of those sides, and into their coetficients continually." 

 Newton then goes on to explain his Lemma thus: "I call any quantity a genitum 

 which is not made by addition or subtraction of divers parts, but is generated or 

 produced in arithmetic by the multiplication, division, or extraction of the root of 

 any terms whatsoever; in geometry, by the invention of contents and sides, or of 

 the extremes and means of proportionals. Quantities of this kind are products, 

 quotients, roots, rectangles, squares, cubes, square and cube sides, and the like. 

 These quantities I here consider as variable and indetermined, and increasing or 

 decreasing, as it were, by a perpetual motion or flux, and I understand their mo- 

 mentaneous increments or decrements by the name of moments; so that the incre- 

 ments may be esteemed as added or affirmative moments, and the decrements a& 

 subducted or negative ones. But take care not to look upon finite particles as such. 

 Finite particles are not moments, but the very quantities generated by the moments. 

 We are to conceive them as the just nascent principles of finite magnitudes. Nor 

 do we in this Lemma regard the magnitude of the moments, but their first propor- 

 tion, as nascent. It will be the same thing if, instead of moments, we use either 

 the velocities of the increments and decrements ( which may also be called the 

 motions, mutations, and fluxions of quantities ), or any finite quantities pro- 

 portional to those velocities. The coefficient of any generating side is the 

 quantity which arises by applying the genitum to that side." Newton then 

 says that "the sense of the Lemma is, that if the moments of any quantities, 

 A, B, C, etc., increasing or decreasing by a perpetual flux, or the velocities 

 of the mutations which are proportional to them, be called a, b, c, etc., 

 the moment or mutation of the generated rectangle AB will be aB-\- 1>A; the moment 



