1 7S Mel hod of Progre<isions. 



But it will be shown iti my next letter that m may be taken 

 of such a magnitude that the sum of the series 1 + 2" + 3" +• 



4"+ ., .. yw' will not sensibly differ from -^,-— ; and at 



the same time the distance — will be so inconceivably small that 



III ■' 



ax' "*" ^ . 



~- will express the true area of the curve. 



If the ordinate y be expressed by ax' + /', then 



n + \ 



~ + Ix = the area. 



H+ 1 — 



It would be easy to multiply examples, which would afford a 

 happy illustration of many of the methods of obtaining fluents ; 

 as it is easy to perceive that the i of the method of fluxions is 



equivalent to — in this method. It will also be observed that 



if the series does not begin from zero a correction will be neces- 

 sary, and may be made in the same manner as in fluxions. 



in the preceding illustration the ordinates are supposed to be 

 perpendicular to the abscissas ; should they be inclined at any 



?! + 1 



, , , , , , , Sin. c X ff.r 



ancle c, tlie area would be expressed by ; . 



What has been shown respecting the quadrature of curves is 

 equally applicable to other objects of calculation, such as the 

 contents of solids, mechanical effects, and inquiries of a like na- 

 ture. In this letter I shall only offer another example, which is 

 to find the length of a curve. 



Let the abscissa be divided into vi parts, and to each point of 

 division let an ordinate be drawn perpendicular to the abscissa. 

 Then, if the square of the difference between two ordinates, at 



the distance -- aijart, be added to the square of — the sum will 



be equal to the square of the small portion of the curve inter- 

 cepted between those ordinates. 



Hence, if?; = .x", the length z of the curve may be exhibited 

 as follows; 



1)1 m m 



0:-^=.lx /i+4_iy+(i+i^_-__)~+&c. • 1 



The progression will generally become more complicated when • 



the 



