156 



In this manner Sir Ifaac Newton inveftigated tlie limiting ratio of the arc to the 

 chord, tang. &c. In this way alfo the limiting ratio of the increments of the alge- 



n — I 



braical quantities x and x is had. Let o be the increment of x, and n.v o-f"" . 



«— 3 n 



n — I .V »:+ &c. is the increment of .V ; the general ratio then of thefe increments is 



2 



n— I n~2 



equal to the ratio t : nx + n.n—i x o -\- &c. In this feries it may readily 



be Ihewn that o may be taken fo fmall that the fum of the terms after the firft may be lefs 



g— I 

 than any quantity that can be affigned ; fo that, according to the definition i : tix 

 is the limiting ratio required. 



Here is no fliifting of the quertion, as it has been termed, by firfl talcing o a real mag- 

 nitude and then no magnitude, nor any introdudion of infinitesimal quantities. The quef- 

 tion is concerning the limit of the ratio of the increments not concerning the ratio itfelf ; • the 

 general ratio of the increments is equivalent to a ratio, one term of which is fixed, and the 

 other compofed of a fixed, and variable quantity ; this variable quantity being fufceptible of 

 a lefs value than any that can be affigned, the ratio of the fixed quantities is by the definition 

 the limiting ratio. 



2. For the ready folution of the problem, the following propofition is often of the greateft 

 importance, and is one of the principal fources of compendium derived from this doflrine. 



Prop. A. In deducing a hmit of a variable quantity, depending upon other variable quan- 

 tities, the limits of thefe variable quantities may be ufed inftead of the quantities themfelves in 

 any part of the procefs. 



Let L be the limit of any variable quantity M involved in the procefs ; then in 

 any of the Heps L may be ufed for M, and the conclufion will be true. For let f = M 

 — L exprefs the general relation between L and M, then, without conCdering the limit, in 

 the conclufion will be found inftead of L, L + e. 



But to obtain the limit required, the limits of the variable quantities mull be ufed, and 

 therefore now L fubftituted for 'L, -\- e. 



This propofition relates both to geometrical magnitudes and analytical quantities. 



Any fteps, which we can demonftrate will lead us to a true conclufion, mud be confidered 

 with a reference to that conclufion, as logical. Such are the fteps in which we fubftitute 

 limits for the quantities themfelves, although the quantities are never accurately equal to 

 their limits. By fubjoining the word ultimo to fuch fteps we refer to the general 

 conclufion, and intend thefe fteps are only true with a reference to that. From be- 

 ing thus enabled to (horten the fteps of a demcnftration arifes much of the value of 

 the method, whether it be applied to geometrical demcnftration or analytical procefles. 



This 



