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XXVI. So77ie Account of a Method which viay le applied to the 

 same Purposes as Sir Isaac Newton's Method of Fluxions. 

 By Mr. Thomas Tredgold. 



Letter I. 

 To Mr. Tilloch. 



Sir, — 1 ou are well aware that there are not many mathema- 

 ticians who are perfectly satisfied with the manner of establishing 

 the first principles of the fluxional calculus ; and the logical basis 

 of the differential calculus is still more objectionable. Perhaps 

 the method, which I intend partly to explain in this letter, will 

 be thought less objectionable in its principles, and that these 

 principles being more obvious it may be applied with greater cer- 

 tainty, and be more capable of improvement. I submit it how- 

 ever, with diffidence, to the opinions of those who have more ex- 

 tended knowledge, and a greater share of experience. 



My method will be best explained by showing examples of its 

 application, which with your permission I will lay before your 

 readers, successively at such times as may be most convenient to 

 myself; applying it in the first instance to the quadrature of curves. 



Let X be the abscissa, and y the corresponding ordinate of a 

 curve, and let the relation of x to yhe expressed by the equation 

 xy = ax^. Also, suppose the abscissa to be divided into m equal 

 parts, and an ordinate to be drawn at each point of division. It 

 is obvious that the distance between the points of division will be 



equal to — , and the abscissas, and their corresponding ordinates, 

 may be expressed as under, for each point of division. 



• r . X '?.!• .'I.r 'i.l- 1"r 



Abscissas — , — , — , — , .. .. • 



m 111 ni m '"■ 



Ordmates --^^, a(--),a{-), o(-), .. < - ). 



Now, if the arithmetical mean between two adjoining ordi- 

 nates be multiplied by the distance l)etween them, and this di- 

 stance be very small, the product will not sensibly differ from 

 the true area of the curvilinear space, of which the two sides are 

 bounded l)y the two ordinates. Therefore, the area of the curvi- 

 linear space, which is bounded bv the ordinate y, abscissa .r, and 

 I the curve, may be expressed by the series ; 



'<'''+>..r")x (x"+o".."). x ('i\r'' + '.i"x").x U^\"x' +m ".l').' 



I em ' 2;;j ' 2/u ">n ' 



Or, ax X ^-^ ±1- -^'^ -f- ■• .. .. 0!i-z}L±tL, 



Vol.57. No. '27:i..l/c/-t7i 1 Sin. . Z nut 



